Control Theory of Descriptor Systems Lecture Notes...Control laws based on linearization work...

Control Theory of Descriptor SystemsLecture Notes

Lena ScholzTU Berlin (WS 2014/15)

February 6, 2015

CONTENTS

1 Introduction 5

2 Solvability 9

2.1 Linear DAEs with constant coefficients . . . . . . . . . . . . . . . . . . . . . 11

2.2 Linear DAEs with variable coefficients . . . . . . . . . . . . . . . . . . . . . 16

2.3 Nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Feedback Regularization 31

3.1 Linear Descriptor Systems with constant coefficients . . . . . . . . . . . . . 32

3.2 Linear Descriptor Systems with variable coefficients . . . . . . . . . . . . . 35

3.3 Nonlinear Descriptor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Control theoretical concepts 43

4.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3

CONTENTS

4.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Staircase forms and system properties 73

6 Optimal Control Problems 91

4

CHAPTER

1

INTRODUCTION

A general control system can be written in the form

0 = F (t, x, x, u), x(t0) = x0, (1.1a)

y = G(t, x, u), (1.1b)

where F : I × Dx × Dx × Du → Rl and G : I × Dx × Du → Rp are continuous functions,Dx,Dx ⊆ Rn and Du ⊆ Rm are open, x0 ∈ Rn and I = [t0, tf ] ⊂ R. Equation (1.1a) is calledstate equation and (1.1b) is called ouput equation. The continuous differentiable functionx : I→ Rn is called the state of the system, u : I→ Rm the input or control and y : I→ Rpis the output of the system.

x

F

u y

Figure 1.1: Representation of a general control system

Notation 1.1. Utilize the convenient notation

x(t) =d

dtx(t), x(t) =

d2

dt2x(t), . . .

for derivatives and

F,x :=∂

∂xF (t, x, x, u), F,x :=

∂

∂xF (t, x, x, u)

for partial derivatives.

5

1 Introduction

u m1

m2

`

x1

x3x2

g

Figure 1.2: Cart pendulum

If F,x is regular, the state equation (1.1a) can be reformulated as ordinary differentialequation (ODE)

x = φ(t, x, u)

by use of the implicit function theorem. Here, we will also allow that F,x is rank deficient.In this case, (1.1a) contains differential and algebraic equations, i.e., (1.1a) is a differential-algebraic equation (DAE). In the control community, the system (1.1) is called a descriptorsystem. Systems of the form (1.1) arise for example in mechanical, electrical and chemicalengineering.

Example 1.2 (Cart Pendulum). Consider a rigid pendulum of length ` with point massm2 attached to a cart with mass m1 that only moves in horizontal direction. The situationis depicted in Figure 1.2.

We have the following notation.

m1 mass of the cartm2 mass of the pendulum` length of the pendulumg gravityx1 horizontal position of the cart(x2, x3) position of the mass m2

u external force acting on the car.

The motion of the system can be described by the Euler-Lagrange equations (ELE). TheLagrange function is given by

L(x, x, λ) = T (x, x)− U(x)−nc∑k=1

λkgk(x),

6

where T (x, x) denotes the kinetic energy, U(x) denotes the potential energy and g1(x) =0, . . . , gnc(x) = 0 denote the (ideal) constraints that restrict the motion of the system. The

vector λ =[λ1 . . . λnc

]>consists of the Lagrange multipliers. Introduce w =

[x λ

]>.

Then the Euler-Lagrange equations are given by

d

dt

(∂

∂wL(w, w)

)− ∂

∂wL(w, w) = Fex, (1.2)

where Fex denotes some external actions (forces). In the case of the cart pendulum, we havethe kinetic energy T = 1

2m1x21 + 1

2m2

(x2

2 + x23

), the potential energy U = m2gx3 and the

constraint g(x) = (x2 − x1)2 + x23 − `2 = 0. This yields the Lagrange function

L =1

2m1x

21 +

1

2m2

(x2

2 + x23

)−mgx3 − λ

((x2 − x1)2 + x2

3 − `2).

Introducing x4 = x1, x5 = x2 and x6 = x3, the ELE (1.2) is given by

x1 = x4

x2 = x2

x3 = x3

m1x4 = 2λ(x2 − x1) + u

m2x5 = −2λ(x2 − x1)

m2x6 = −2λx3 −m2g

0 = (x2 − x1)2 + x23 − `2.

(1.3)

Since we are only interested in the position of the pendulum, the output equation has theform

y =

[0 1 0 0 0 0 00 0 1 0 0 0 0

]x =

[x2

x3

],

where x =[x1 x2 · · · x6 λ

]>. Accordingly, we have n = 7 = l, m = 1 and p = 2.

Linearization of (1.1) along a reference trajectory leads to a linear descriptor system withvariable coefficients of the form

E(t)x(t) = A(t)x(t) +B(t)u(t) + f(t), x(t0) = x0,

y(t) = C(t)x(t) +D(t)u(t) + g(t),(1.4)

with continuous matrix functions E,A : I → Rl×n, B : I → Rl×m, C : I → Rp×n and D :I→ Rp×m and continuous inhomogeneities f : I→ Rl, g : I→ Rp. Similarly, linearization of(1.1) along a constant reference trajectory leads to a linear descriptor system with constantcoefficients

Ex(t) = Ax(t) +Bu(t) + f(t), x(t0) = x0,

y(t) = Cx(t) +Du(t) + g(t),(1.5)

with E,A ∈ Rl×n, B ∈ Rl×m, C ∈ Rp×n and D ∈ Rp×m. The leading function E(t) or thematrix E, respectively, are allowed to be (pointwise) singular.

7

1 Introduction

Remark 1.3. In standard state-space systems (i.e. LTV or LTI) one has E(t) = In = Eand l = n. Hence, they are special cases of (1.4) and (1.5).

Remark 1.4 (Linearization principle). Control laws based on linearization work locally forthe original nonlinear system.

Topics of this course

• Solvability (consistency, regularity for descriptor systems)• Regularization (index reduction) for example by feedback regularization• Control theoretical aspects: C-controllability, R-controllability, I-controllability• Stability and Stabilization• Optimal control problems (also for nonlinear)• Numerics: staircase form (GUPTRI), derivative array approach

Example 1.5 (Linearization of the cart pendulum). We linearize the equations of motion(1.3) of Example 1.2 along the equilibrium solution x =

[x1 . . . x6, λ

]=[0 0 −` 0 0 0, m2g

2`

].

Using the decomposition xi = xi + xi for i = 1, . . . , 6 and λ = λ+ λ yields

˙x1 + ˙x1 = x4 + x4

˙x2 + ˙x2 = x5 + x5

˙x3 + ˙x3 = x6 + x6

m1

(˙x4 + ˙x4

)= 2 (x2 + x2 − x1 − x1)

(λ+ λ

)+ u

m2

(˙x5 + ˙x5

)= −2 (x2 + x2 − x1 − x1)

(λ+ λ

)m2

(˙x6 + ˙x6

)= −2 (x3 + x3)

(λ+ λ

)−m2g

0 = (x2 + x2 − x1 − x1)2 + (x3 + x3)2 − `2.

(1.6)

Neglecting the quadratic terms xixj and xiλ results in

11

1m1

m2

m2

0

︸︷︷︸

E

x =

0 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 0−m2g

`m2g` 0 0 0 0 0

m2g` −m2g

` 0 0 0 0 00 0 −m2g

` 0 0 0 2`0 0 −2` 0 0 0 0

︸︷︷︸

A

x+

0001000

︸︷︷︸B

u+

00000

−m2g0

︸︷︷︸

f

,

y =

[0 1 0 0 0 0 00 0 1 0 0 0 0

]︸︷︷︸

C

x

8

CHAPTER

2

SOLVABILITY

A first question in the analysis of descriptor systems is the existence and uniqueness ofsolutions of (1.1), given by

0 = F (t, x, x, u), x(t0) = x0, (2.1)

0 = y −G(t, x, u). (2.2)

For ODEs we can employ the implicit function theorem to transform (1.1a) to

x = f(t, x, u). (2.3)

If f is a smooth function (or Lipschitz continuous with respect to the second argument) theODE theory (for example Picard-Lindel"off) ensures a unique solution x(t) for everyinitial condition x0 = x(t0) and any given continuous input function u. In the general caseof descriptor system this is no longer true as the following examples illustrate.

Example 2.1. Consider the system[0 10 0

] [x1

x2

]=

[1 00 0

] [x1

x2

]+

[01

]u,

[x1(0)x2(0)

]=

[x1,0

x2,0

],

which also reads as

x2 = x1, 0 = u.

We have an algebraic condition for the input u, which implies that the system is only solvableif u ≡ 0. On the other side, x1 is not uniquely determined (it can be arbitrarily chosen) andcan be seen as a control steering the component x2.

9

2 Solvability

Example 2.2. The system1 0 00 1 00 0 0

x1

x2

x3

=

0 1 00 0 10 1 0

x1

x2

x3

+

0 01 00 −1

[u1

u2

](2.4)

consists of two differential equations for x1 and x2 and one algebraic relation for x2 andu2. The third component is not explicitly determined. Differentiating the last equation of(2.4) and substituting into the first equation of (2.4) yields

0 = x2 − u2 =⇒ 0 = x2 − u2 = x3 + u1 − u2 =⇒ x3 = −u1 + u2.

The component x3 is implicitly defined and u2 must be differentiable. Accordingly, system(2.4) reads as

x1 = u2 (can be stirred by u2),

x2 = u2

x3 = −u1 + u2.


] [x1

x2

]=

[−1 10 0

] [x1

x2

]+

[1 00 1

] [u1

u2

].

Differentiation of the first equation gives

x1 = x2 + u1.

Substitute this into the second equation to end up with

x2 =1

2(u2 − u1) . (2.5)

Equation (2.5) gives a unique solution for every initial value x2(t0) whenever u2 and u1 areintegrable functions. The first component x1 and its initial value are uniquely defined by thefirst equation.

Definition 2.4.

1. A function x : I → Rn is called a (classical) solution of (1.1a) if x ∈ C1(I,Rn) and xsatisfies (1.1a) pointwise for some given input function u.

2. A function x : I→ Rn is called a solution of the initial value problem (IVP) consistingof (1.1a) and x(t0) = x0 ∈ Rn, if x is a solution of (1.1a) and satisfies x(t0) = x0.

3. An initial value x0 ∈ Rn is called consistent , if the corresponding IVP has at leastone solution.

10

2.1 Linear DAEs with constant coefficients

Remark 2.5. There also exist weaker solvability concepts (e.g. weak solutions or im-pulsive smooth solutions) that can be used to handle inconsistencies ow less smoothnessrequirements. [later]

In some applications (e.g. robust control) it is important to know whether the system issolvable for every input function u and every initial value x0 that is consistent with thisinput.

Definition 2.6. A control problem (1.1a) is called consistent , if there exists an inputfunction u for which (1.1a) has a solution. It is called regular if it has a unique solution forevery initial value that is consistent for the system with input u.

For given input u the system (1.1a) represents a differential-algebraic equation (DAE).Therefore the solvability theory for descriptor systems is strongly related to the theory forDAEs.


Consider the linear DAE

Ex = Ax+ f(t), (2.6)

with E,A ∈ Rl×n and f : I → Rl, x : I → Rn. Note that descriptor systems are a specialcase of (2.6) by setting f(t) = Bu(t) for given input u. The solution behavior of the systemdepends on the properties of the matrix pair (E,A) or equivalently the matrix pencil λE−Afor some λ ∈ C.

Definition 2.7. A matrix pencil λE−A or a pair (E,A) with E,A ∈ Rl×n is called regularif l = n and det(λE −A) 6= 0 for some λ ∈ C. Otherwise it is called singular .

Example 2.8 (Regular matrix pencil). The matrix pencil

(E,A) =

0 1 00 0 00 0 0

,1 0 0

0 1 00 0 1

is regular, since det(λE −A) = −1 6= 0 for all λ ∈ C.

Example 2.9 (Singular matrix pencil). The matrix pencil

(E,A) =

0 1 00 0 10 0 0

,1 0 0

0 0 00 0 1

is singular, since det(λE −A) = 0 for all λ ∈ C.

11

2 Solvability

Definition 2.10. Two pairs of matrices (E,A) and (E, A) are called (strongly) equivalentif there exist nonsingular matrices W ∈ Rl×l and T ∈ Rn×n such that

E = WET and A = WAT.

In this case, we write (E,A) ∼ (E, A).

Lemma 2.11. The matrix pair (E,A) is regular if and only if every strongly equivalentpair (E, A) is regular.

Proof. Let W,T ∈ Rn×n such that E = WET and A = WAT . Then, we have

det(λE − A) = det (W (λE −A)T ) = det(W ) det(T )︸︷︷︸6=0

det(λE −A),

which completes the proof.

Theorem 2.12 (Weierstraß canonical form). Let λE − A be regular. Then there existnonsingular matrices W,T ∈ Rn×n such that

λWET −WAT = λ

[Inf 0

0 N

]−[J 00 In∞

](WCF)

is in Weierstraß canonical form with J,N in Jordan canonical form, N nilpotent with indexof nilpotency ν, i.e. Nν = 0, Nν−1 6= 0. The number ν is called the index of λE −A or theindex of the DAE (2.6) and is denoted with ν = ind(E,A).

Proof. Since (E,A) is regular, there exists λ0 ∈ C with det(λ0E−A) 6= 0 and hence λ0E−Ais nonsingular.

(E,A) = (E,A− λ0E + λ0E)

∼(−(λ0E −A)−1E,−(λ0E −A)−1(A− λ0E + λ0E)

)=

((A− λ0E)−1E, I + λ0(A− λ0E)−1E

).

Furthermore there exists a nonsingular matrix S ∈ Rn×n such that S(A− λ0E)−1S−1 is inJordan canonical form, i.e.

S(A− λ0E)−1S−1 =

[J 0

0 N

],

where J is nonsingular (part belonging to the nonzero eigenvalues) and N is nilpotentstrictly upper triangular. Then the matrix I +λ0N is nonsingular upper triangular and wehave

(E,A) ∼([J 0

0 N

],

[I + λ0J 0

0 I + λ0N

])∼

([I 0

0 (I − λ0N)−1N

],

[J−1 + λ0I 0

0 I

]),

12


with (I + λ0N)−1N is strictly upper triangular and nilpotent. Transferring J−1 + λ0I and(I + λ0N)−1N to Jordan canonical form yields the desired form (WCF).

For regular matrix pairs (E,A) the linear DAE (2.6) can be transformed into Weierstraßcanonical form by

WETT−1x = WATT−1x+Wf(t)

⇐⇒[Inf 0

0 N

] [˙x1˙x2

]=

[J 00 In∞

] [x1

x2

]+

[f1

f2

]

by using the variable transformation x =

[x1

x2

]= T−1x and f =

[f1

f2

]= Wf. The resulting

system is decoupled into the ordinary differential equation

˙x1 = Jx1 + f1, (2.7)

which is called the differential part (also known as dynamic part or slow part) of the system(2.6) and the algebraic equation

N ˙x2 = x2 + f2, (2.8)

also known as the algebraic part or slow subsystem of (2.6). To see, that (2.8) is indeed analgebraic equation, we consider (2.6) in its matrix form

0 ∗. . .

. . .

. . . ∗0

˙x2,1......

˙x2,n∞

=

x2,1

...

...x2,n∞

+

f2,1

...

...

f2,n∞

,

where ∗ ∈ {0, 1}. The last equation uniquely determines x2,n∞ = −f2,n∞ . Using thisrelation, one can substitute in the second last equation and solve the algebraic equation.Continuing with this procedure uniquely determines all components of x2. In particular,the following result holds.

Lemma 2.13. The solution x2 of the algebraic equation (2.8) is given by

x2 = −ν−1∑i=0

N if(i)2 (t), (2.9)

where ν denotes the index of nilpotency of N .

Proof. Let D = ddt denote the differentiation operator. Since N is constant matrix, D and

N commute and also (ND)ν = NνDν = 0. Moreover,

(I −ND)

ν−1∑i=0

(ND)i = I −NνDν = I.

13

2 Solvability

Finally, (2.8) can be rewritten as (I −ND)x2 = −f2, which yields

x2 = −(I −ND)−1f2 = −ν−1∑i=0

N iDif2 = −ν−1∑i=0

N if(i)2 .

Thus, x2 is uniquely determined by the algebraic equation (2.9). The transformed initialvalue x2,0 has to satisfy the algebraic equation (2.9), i.e. it has to be consistent sinceotherwise the system is not solvable. Moreover, the inhomogeneity f2 (and hence also f)has to be (ν − 1) times continuously differentiable. On the other side, system (2.7) has aunique solution x1 for any initial value x1,0 and any inhomogeneity f1 given by

x1(t) = eJtx1,0 +

∫ t

0eJ(t−s)f1(s)ds.

The backtransformation finally gives the solution x = T x.

Remark 2.14. In the descriptor setting the inhomogeneity f(t) is given by f(t) = Bu(t).

Thus

[f1

f2

]= WBu(t) =

[B1

B2

]u(t) and B2u(t), i.e. u(t) has to be (ν−1) times continuously

differentiable. For the existence of a classical solution x we need u ∈ Cν(I,R). Thuspiecewise continuous control functions (or bang-bang control) might not work. Consistencyof initial conditions may depend on the derivatives of the input function u(t).

We summarize the previous results in the following theorem.

Theorem 2.15 (Existence and Uniqueness of solutions). Consider a linear DAE with con-stant coefficients (2.6) with regular matrix pair (E,A) and inhomogeneity f ∈ Cν(I,Rn)where ν = ind(E,A). Then it holds that:

1. The DAE (2.6) is solvable.

2. An initial value x0 ∈ Rn is consistent if and only if

x2,0 = −ν−1∑i=1

N if(i)2 (0),

where T−1x0 =

[x1,0

x2,0

]and Wf =

[f1

f2

]with T,W ∈ Rn×n that transform (E,A) to

Weierstraß canonical form (WCF).

3. Every initial problem (2.6) with consistent initial value x0 is uniquely solvable.

14


Definition 2.16. The set of consistent initial values is defines as

X 0c :=

{x0 = T

[x1,0

x2,0

] ∣∣ x1,0 ∈ Rnf , x2,0 = −ν−1∑i=0

N if (i)(0)

}.

We conclude that if (E,A) is regular, x0 ∈ X 0c and u(t) is ν times continuously differentiable,

then

Ex = Ax+Bu, x(0) = x0

has a unique (classical) solution. To clarify the dependency of the solution x on the initialvalue and the input, we write x(t;x0, u).

Theorem 2.17. If the pair of matrices (E,A) is regular, then the control problem (1.5) isconsistent and regular.

Proof. Taking u(t) ≡ 0, we have Ex = Ax + f with regular matrix pair (E,A). Thus,for consistent initial value x(0) = x0 there exists a (unique) solution. Hence, the controlproblem is consistent. Regularity follows from Theorem 2.15.

Theorem 2.18. If (E,A) with E,A ∈ Rl×n is a singular matrix pair, then the controlproblem (1.5) is not regular.

Proof. Case 1 rank(λE −A) < n for all λ ∈ C.We choose u ≡ 0 and f(t) ≡ 0 and consider the homogeneous DAE Ex = Ax togetherwith x(0) = x0. Let λ1, . . . , λn+1 ∈ C be pairwise different. Then for every λi thereexists vi ∈ Cn\{0} with

(λiE −A)vi = 0

and the vi are linearly dependent. Hence, there exists αi ∈ C (i = 1, . . . , n + 1) notall of them being zero such that

∑n+1i=1 αivi = 0. Define x(t) =

∑n+1i=1 αivie

λit. Thenx(0) = 0 and

Ex(t) = E

n+1∑i=1

αiλieλit = A

n+1∑i=1

αivieλit = Ax(t).

Thus, x(t) is a solution of the homogeneous system with x(0) = 0. Since x(t) ≡ 0 isalso a solution, the solution is not unique, i.e. the control problem is not regular.

Case 2 rank(λE −A) = n for some λ ∈ C.Since (E,A) is singular, this implies l > n. With the variable transformation x(t) =eλtx(t) we get

E(eλt ˙x(t) + λeλtx(t)

)= Aeλtx(t) +Bu(t) + f(t).

15

2 Solvability

In particular, E ˙x(t) = (A − λE)x(t) + e−λtBu(t) + e−λtf(t). Since A − λE has fullcolumn rank n, there exists a nonsingular matrix T ∈ Rl×l such thtat T (A − λE) =[In0

], or more precisely

[E1

E2

]˙x =

[In0

]x+

[B1

B2

]u+

[f1

f2

]

with

[E1

E2

]= TE,

[B1

B2

]= TB, u(t) = e−λtu(t) and

[f1

f2

]= Tf The part (E1, In) is

regular since rank(λE1 − In) = n for λ = 0 and

E1˙x1 = x+B1u+ f1(t)

has a unique solution for every sufficiently smooth u(t) and every sufficiently smoothinhomogeneity f1(t) according to theorem 2.17. From E2

˙x = B2u + f2(t) we get aconsistency condition for B2u that must hold for the existence of a solution. Threeexists an arbitrary smooth inhomogeneity f2 for which E2

˙x 6= B2u+ f2(t) and hencethe system is not regular.

Remark 2.19. Note that for linear descriptor systems with constant coefficient the regu-larity of (E,A) is advantageous but not necessary (the system can still be consistent). Forsingular pairs (E,A) we can construct the Kronecker canonical form (KCF) instead of theWeierstraß canonical form (not in this lecture).

2.2 Linear DAEs with variable coefficients

Now we consider descriptor systems of the form

E(t)x(t) = A(t)x(t) +B(t)u(t) + f(t), x(t0) = x0

y(t) = C(t)x(t) +D(t)u(t) + g(t).(2.10)

In order to analyze the properties of the system we perform a behavior approach (first

suggested by Jan Willems ∼ 1990). We introduce the behavior vector z =

[xu

]and write

the state equation as [E(t) 0

]︸︷︷︸:=E

z =[A(t) B(t)

]︸︷︷︸:=A

z + f(t),

or equivalently asE(t)z(t) = A(t)z(t) + f(t) (2.11)

with E ,A : I→ Rl×(n+m).

16


Remark 2.20. The derivative of the input u occurs only formally in (2.11). Moreover, we

could also include the output equation by introducing z =[x> y> z>

]>and considering[

E(t) 0 00 0 0

]z =

[A(t) B(t) 0C(t) D(t) −Ip

]z +

[f(t)g(t)

].

However, since the output equation explicitly determines y, the output equation will notcontribute to the analysis and has not to be considered.

For system (2.11) we can apply the theory for nonsquare linear DAEs with variable co-efficients based on the strangeness-index concept (see also the DAE lecture). At first weconstruct the inflated system (or derivative array) obtained by the original DAE (2.11) andall derivatives(

d

dt

)i(E(t)z(t)) =

(d

dt

)i(A(t)z(t)) +

(d

dt

)if(t) i = 1, . . . , k

up to some order k of the form

Mk(t)vk(t) = Nk(t)vk(t) + hk(t),

where Mk,Nk : I→ R(k+1)l×(n+m) are given by

(Mk)i,j =

(ij

)E(i−j) −

(i

j + 1

)A(i−j−1), i, j = 0, . . . , k,

(Nk)i,j =

{A(i) , i = 0, . . . , k, j = 0,

0 otherwise,

(vk)j = z(j), j = 0, . . . , k,

(hk)j = f (j), j = 0, . . . , k.

Example 2.21. For k = 2, we add the first derivative

E z + E z = Az +Az + f

and the second derivative

Ez(3) + 2E z + E z = Az + 2Az +Az + f

of (2.11) to the DAE (2.11) to obtain E 0 0

E − A E 0

E − 2A 2E − A E

zz

z(3)

=

A 0 0

A 0 0

A 0 0

zzz

+

fff

.Hypothesis 1. There exists integers µ, a, d and v such that the inflated pair (Mµ,Nµ)associated with the pair of matrix-valued functions (E(t),A(t)) has the following properties:

17

2 Solvability

1. For all t ∈ I we haverank (Mµ(t)) = (µ+ 1) l − a− v

such that there exists a smooth matrix-valued function Z of size (µ+ 1)l× (a+ v) andpointwise maximal rank satisfying

Z>Mµ = 0.

2. For all t ∈ I we have

rank

Z>NµIn+m

0...0

= a.

This implies that without loss of generality Z can be partitioned as Z =[Z2 Z3

]with

Z2 of size (µ+ 1)l × a and Z3 of size (µ+ 1)l × v such that

A2 := Z>2 Nµ

In+m

0...0

has full row rank a and Z>3 Nµ

In+m

0...0

= 0. Furthermore, there exists a smooth

matrix-valued function T2 of size (n+m)× (n+m− a) and pointwise maximal ranksatisfying

A2T2 = 0.

Note that n+m− a = d+u, where u denotes the number of undetermined components.

3. For all t ∈ I we haverank(E(t)T2(t)) = d

such that there exists a smooth matrix-valued function Z1 of size l × d, where

d = l − a− vµ, with

vµ = l − rank([Mµ Nµ

])+ rank

([Mµ−1 Nµ−1

])with the convention rank

([M−1 N−1

])= 0. Moreover Z1 has pointwise maximal

rank satisfying

rank(Z>1 (t)E(t)

)= d.

Definition 2.22. The smallest possible µ in Hypothesis 1 is called the strangeness-indexor s-index of the behavior system (2.11). A behavior system (2.11) with µ = 0 is calledstrangeness-free

18


If Hypothesis 1 is satisfied for µ, a, d and v (we say that the s-index is well-defined), we canformulate the reduced systemE1(t)

00

z(t) =

A1(t)

A2(t)0

z(t) +

f1(t)

f2(t)

f3(t)

(d)

(a)

(v)

(2.12)

with E1 = Z>1 E , A1 = Z>1 A1, f1 = Z>1 f, A2 = Z>2 Nµ

In+m

0...0

, f2 = Z>2 hµ, f3 = Z>3 hµ.

Remark 2.23.

1. In principle, the state vector z can be partitioned in[z>1 z>2 z>3

]>with z1 ∈ Rd the

differential components, z2 ∈ Ra the algebraic components and z3 ∈ Ru the undeter-mined components (u = n+m− d− a, but this would mix up states x and controlsu.

2. For a linear DAE without input, i.e.

E(t)x(t) = A(t)x(t) + f(t)

the system is called regular if it satisfies Hypothesis 1 for l = n (and m = 0) andµ, a, d, v such that n = d + a (i.e. v = 0 and u = n − a − d = 0). It is called regularand strangeness-free if it satisfies Hypothesis 1 with µ = 0 and v = u = 0. In thefollowing we will say: The descriptor system (2.10) is regular and strangeness-free asa free system if it satisfies Hypothesis 1 for u(t) ≡ 0 and µ = 0,m = 0, l = n = d+ a.

3. For pairs of constant matrices (E,A) the s-index is always well-defined. The conditionfor regularity of the pencil λE−A can be replaced by the condition v = u = 0 implyingthat l = n+m. We have the relation that ν = ind(E,A) = µ+ 1.

Example 2.24. Consider the system[0 01 −t

] [z1

z2

]=

[−1 t0 0

] [z1

z2

]+

[f1

f2

]with n = l = 2. Hypothesis 1 is not satisfied for µ = 0. For µ = 1 we consider

(M1,N1) =

0 0 0 01 −t 0 01 −t 0 00 −1 1 −t

,−1 t 0 00 0 0 00 1 0 00 0 0 0

,

which has the following properties

19

2 Solvability

1. For all t ∈ I we have rank(M1) = 2. This implies 2l − a − v = 2 =⇒ a + v = 2.

Choosing Z> =

[1 0 0 00 1 −1 0

]we get Z>M1 = 0.

2. We compute

rank

(Z>N1

[I2

0

])= rank

[1 0 0 00 1 −1 0

]−1 t0 00 10 0

= rank

([−1 t0 −1

])= 2 = a

for all t ∈ I. Henceforth, we have v = 0, d = 0 and set Z2 = Z and A2 =

[−1 t0 −1

].

Choosing [·] = T2 ∈ R2×0 (the empty matrix) we have A2T2 = [·].

3. Finally, rank(ET2) = rank([·]) = 0 = d. Analogously we can choose Z1 = [·] ∈ R2×0

and Hypothesis 1 is satisfied for µ = 1, a = 2, d = v = 0. The corresponding reducedsystem is given by [

0 00 0

] [z1

z2

]=

[−1 t0 −1

] [z1

z2

]+

[f1

f2

],

with

[f1

f2

]= Z>

[f1 f2 f1 f2

]>=

[f1

f2 − f1

]. Simple computations show that the

result is given by [z1

z2

]=

[tf2 − tf1 + f1

f2 − f1.

]Remark 2.25.

1. In the reduced system (2.12) the third block row has v equations. Note that v is ingeneral larger than vµ, where

l = d+ a+ vµ.

2. The reduced system (2.12) is strangeness-free, i.e., it satisfies Hypothesis 1 for µ = 0.

Theorem 2.26 (Existence and Uniqueness). Let the strangeness-index of (E ,A) as in (2.11)be well-defined (i.e. (E ,A) satisfies Hypothesis 1 with constant values µ, a, d, v) and letf ∈ Cµ+1(I,Rl). Then we have:

1. The system (2.11) is solvable if and only if f3(t) ≡ 0 in (2.12).

2. An initial condition z(t0) = z0 is consistent with the system if and only if A2(t0)z0 +f2(t0) = 0.

3. The corresponding IVP is uniquely solvable if and only if in addition u = 0.

20


Remark 2.27. Note that the behavior system (2.11) has the same solution set as thereduced (strangeness-free) system (2.12) (since the variable z stays the same).

In the original control setting, the reduced formulation (2.12) takes the form

E1(t)x(t) = A1(t)x(t) +B1(t)u(t) +f1(t) (d)

0 = A2(t)x(t) +B2(t)u(t) +f2(t) (a)

0 = f3(t) (v)

y = C(t)x(t) +D(t)u(t) +g(t) (p)

(2.13)

with E1(t) = E1

[In0

], Ai = Ai

[In0

], Bi = Ai

[0Im

]for i = 1, 2.

Remark 2.28.

1. The submatrix A2 has been obtained from the block matrixA B

A B...

...

A(µ) B(µ)

by transformations from the left only. Therefore, we only need the derivatives ofthe coefficient matrices, but no derivatives of the input function u (the derivatives ofu occur only formally in the inflated pair, but are not used for the construction of(2.13).) Thus, we need no further smoothness requirements for the input u.

2. Since only transformations from the left are used the part stemming from the originalstates x and the part from the original inputs u is not mixed up.

3. Also initial conditions stay the same.

Lemma 2.29. A DAE of the formE1(t)00

z(t) =

A1(t)A2(t)

0

z(t) =

f1

f2

f3

(d)

(a)

(v)

is strangeness-free if and only if the matrix[E1(t)A2(t)

]has pointwise full row rank a+ d for all t ∈ I.

Proof. The proof is left as exercise.

21

2 Solvability


] [xx2

]=

[0 01 0

] [x1

x2

]+

[01

]u, x(t0) = x0.

In behavior form, the system is given by

(E ,A) =

([1 0 0

0 0 0

],

[0 0 0

1 0 1

])=

([E1

0

],

[A1

A2

]).

We have rank

([E1

A2

])= rank

([1 0 01 0 1

])= 2 and hence the system is strangeness-free

in the behavior form (i.e. µ = 0). However, for a given input u(t) (e.g. we consider thefree system with u(t) ≡ 0) we have to consider the DAE with

(E,A) =

([1 00 0

],

[0 01 0

])=

([E1

0

],

[A1

A2

]).

Here we have rank

([E1

A2

])= 1 < 2 and the system is not strangeness-free as (free) DAE.

In particular, x2 is not uniquely determined and hence the system is not regular.

Definition 2.31. Two pairs of matrix valued functions (E(t), A(t)) and (E(t), A(t)) withE,A, E, A : I → Rm×n are called globally equivalent if there exist pointwise nonsingularmatrix functions P ∈ C(I,Rm×m) and Q ∈ C1(I,Rn×n) such that

(E,A) ∼ (E, A) = (PEQ,PAQ− PEQ).

Remark 2.32. The additional term for the matrix A is based on the following fact. Sub-stitute x = Qx in Ex = Ax+ f to end up with

EQx+ EQx = AQx+ f ⇐⇒ EQx = (AQ− EQ)x+ f.

Theorem 2.33. Under some constant rank assumptions (see (A1), (A2) below), the reducedformulation (2.13) of the linear descriptor system (2.10) is globally equivalent to a controlsystem of the form

x1 = A13(t)x3+ A14(t)x4 + B12(t)u2+ f1(t) (d)

0 = x2 + B22(t)u2+ f2(t) (a− φ)

0 = A31(t)x1 + u1 + f3(t) (φ)

0 = + f4(t) (v)

y1 = x3 D12(t)u2+ g1(t) (ω)

y2 = C21(t)x1 +C22(t)x2 D22(t)u2+ g2(t) (p− ω).(2.14)

Remark 2.34. In the construction of (2.14) we want to avoid transformations that mix xand u. Thus, we are restricted in the choice of possible equivalence transformations.

22


Proof. We have to consider the matrix pair of the DAE in (2.13) in[x> u> y>

]>

(E , A) =

E1 0 00 0 00 0 00 0 0

,A1 B1 0A2 B2 00 0 0C D −Ip

d

a

v

p

,

where E1 has pointwise full row rank d (due to the construction). Then there exists pointwise

orthogonal functions U ∈ C(I,Rm×m) and V ∈ C1(I,Rd×d) such that U>E1V =[Σ1 0

]with Σ1 ∈ C(I,Rd×d) pointwise nonsingular. After renaming of matrices (which we dowithout further emphasis) we arrive at

(E , A) ∼

Σ1 0 0 00 0 0 00 0 0 00 0 0 0

,A11 A12 B1 0A21 A22 B2 00 0 0 0C1 C2 D −Ip

∼

Id 0 0 00 0 0 00 0 0 00 0 0 0

,A11 A12 B1 0A21 A22 B2 00 0 0 0C1 C2 D −Ip

.

To proceed, we need the following rank assumption.

Assume that A22 : I→ Ra×(n−d)) has pointwise constant rank a− φ. (A1)

Then we can do a column compression similar as above, i.e. there exist global equivalencetransformations such that

(E , A) ∼

Id 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

,A11 A12 A13 B1 0A21 Ia−φ 0 B2 0A31 0 0 B3 00 0 0 0 0C1 C2 C3 D −Ip

d

a− φφ

v

p

.

Since (E , A) is strangeness-free by construction, the block matrix [A2, B2] of the originalpair has pointwise full row rank a. Therefore, B3 of size φ×m has pointwise full row rankφ and hence

(E , A) ∼

Id 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

,A11 A12 A13 B11 B12 0A21 Ia−φ 0 B21 B22 0A31 0 0 Iφ 0 00 0 0 0 0C1 C2 C3 D1 D2 −Ip

d

a− φφ

v

p

.

23

2 Solvability

To proceed, we need a second rank assumption.

Assume that C3 of size p× (n− d− a+ φ) has pointwise constant rank ω. (A2)

Thus, we obtain

(E , A) ∼

Id 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

,

A11 A12 A13 A14 B11 B12 0 0A21 Ia−φ 0 0 B21 B22 0 0A31 0 0 0 Iφ 0 0 00 0 0 0 0 0 0 0C11 C12 Iω 0 D11 D12 −Iω 0C21 C22 0 0 D21 D22 0 −Ip−ω

.

Finally, the identity blocks can be used for block row and column eliminations. Hereby werestrict the column eliminations to those acting only on columns that belong to the samevariable x, y and u.

(E , A) ∼

Id 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

,

A11 0 A13 A14 0 B12 0 00 Ia−φ 0 0 0 B22 0 0A31 0 0 0 Iφ 0 0 00 0 0 0 0 0 0 00 0 Iω 0 0 D12 −Iω 0C21 C22 0 0 0 D22 0 −Ip−ω

.

In the last step we perform a global equivalence transformation with P = I and Q =diag(Q1, I, . . . , I), such that

(E , A) ∼

Q1 0 · · ·

0 0 · · ·... 0

,A11Q1 − Q1 0 A13 · · ·

0 · · ·A31Q1 · · ·

...

.

If we choose Q1 as the solution of the IVP

Q1 = A11Q1, Q1(t0) = Id on I,

the unique solvability of the IVP ensures that Q1 is pointwise nonsingular and (E , A) isglobally equivalent to the pair in (2.14).

Now we are able to characterize consistency and regularity of the descriptor system (2.10).

Corollary 2.35. Let the strangeness-index µ be well-defined for the system (2.11) in be-havior form. Furthermore, let the quantities φ and ω (as defined in the proof of Theorem2.33) be constant on I. Then, we have the following:

1. The linear descriptor system (2.10) is consistent if and only if either v = 0 or f4 ≡ 0.

24


2. If the system is consistent and if φ = 0, then for a given input u an initial conditionis consistent if and only if

x2(t0) = −B22(t0)u2(t0)− f2(t0)

(for the system in form (2.14)) holds.

3. The system (2.10) is regular and strangeness-free (as a free system with u(t) ≡ 0) ifand only if v = φ = 0 and d + a = n. (In this case there exists a unique solutionfor every sufficiently smooth u and f and consistent initial conditions, i.e. the system(2.10) is regular!)

Remark 2.36.

1. If the system (2.10) is consistent and v > 0 the equations corresponding to the fourthblock row of (2.14) describe the redundancies in the system that can simply be omit-ted.

2. Even for a consistent system (2.10) with consistent initial conditions the solution of thecorresponding IVP will in general not be unique (since x3 and x4 are not determined).

Proof. The global equivalence transformations leading to (2.14) do not change the solu-tion behavior, since we do not mix up the variables x, u and y (there is a one-to-onecorrespondence between the solution sets of (2.10) and (2.14) via pointwise nonsingulartransformations applied separately to x, u and y). Thus, it is sufficient to consider (2.14).

1. If v = 0 and f4 6= 0 there clearly exists no solution (independent of the choice of u).Conversely, if v = 0 or f4 ≡ 0 we can determine u as follows. Setting u2 = 0 andchoosing x3 = x4 = 0 we get x2 from x2(t) = −f2(t) and x1 as solution of

x1 = f1(t).

With this we can set u1 = −A31(t)x1 − f3(t) and have found a solution for x1, x2, x3

and x4.

2. Let the system be consistent with φ = 0. Then the system reduces to

x1 = A13(t)x3 +A14(t)x4 +B12(t)u2 + f1

0 = x2 +B22(t)u2 + f2.

For every fixed input u2 this is a strangeness-free DAE (due to Lemma 2.29). Thesecond equation represents the algebraic part. Since x3 and x4 are undetermined thesolution will not be unique (in general).

3. Assume that v = φ = 0 and d+ a = n. Then (2.14) reduces to

x1 = B12(t)u2 + f1

0 = x2 +B22(t)u2 + f2.

25

2 Solvability

This system is uniquely solvable for every input u2 and every inhomogeneity andconsistent initial conditions. Moreover, it is strangeness-free for u2 = 0. Converselylet the system be regular and strangeness-free for u = 0. We have

x1 = A13x3 +A14 + f1 (d)

0 = x2 + f2 (a− φ)

0 = A31x1 + f3 (φ)

0 = f4 (v).

The last equation restricts the inhomogeneity, hence v = 0. If φ > 0 we have eithers-index bigger zero (for A31 6= 0) or a consistency condition for f3 (A31 = 0) andhence φ = 0. If d + a 6= n there are free solution components, which contradicts theassumption. Hence d+ a = n.

Example 2.37. Consider the control problem[0 01 ηt

] [x1

x2

]=

[−1 −ηt0 −(1 + η)

] [x1

x2

]+

[10

]u, η ∈ R.

The reduced formulation of the behavior system (2.13) takes the form (see Ex. I.3)[1 ηt0 0

] [x1

x2

]=

[0 −(1 + η)−1 −ηt

] [x1

x2

]+

[01

]u.

The transformation into the global equivalent form (2.14) looks as follows

(E , A

)≡([

1 0 00 0 0

],

[0 −1 0−1 0 1

])by using Q =

1 −ηt 00 1 00 0 1

with A22 = 0 of size a× (n− d) = 1× 1 of rank 0 = d− φ. This implies φ = 1. The systemin form (2.14) is given by([

1 0 00 0 0

],

[0 −1 0−1 0 1

])=

([Id 0 00 0 0

],

[0 A14 0A31 0 Iφ

])with d = 1, a = 1, φ = 1. The system is consistent since v = 0, but not regular andstrangeness-free as a free system since φ 6= 0 (cp. Ex. I.3).

2.3 Nonlinear systems

Recall the nonlinear system (1.1) of the form

F (t, x, x, u) = 0,

y −G(t, x, u) = 0.

26


As in Section 2.2 we use a behavior approach, i.e. we set z =

[xu

]and consider

F (t, z, z) = 0 (2.15)

instead of (1.1). Since the output equation does not influence the consistency and regularity,we do not consider it in our analysis. Similar as before, we introduce a nonlinear derivativearray

Fk(t, z, z, . . . , z(k+1)

)=

F (t, z, z)ddtF (t, z, z)

...(ddt

)kF (t, z, z)

= 0.

Also in the nonlinear case we can formulate a Hypothesis.

Hypothesis 2. Consider a system of nonlinear DAEs (2.15) in behavior form. Then thereexist integers µ, r, a, d and v such that the set

Lµ ={zµ ∈ I× Rn+m × . . .× Rn+m

∣∣Fµ(zµ) = 0}

is nonempty and such that for every point z0µ =

(t0, z0, z0, . . . , z

(µ+1)0

)∈ Lµ, where z

(j)0

denotes an algebraic variable, there exists a neighborhood of z0µ in which the following prop-

erties hold:

1. The set Lµ ⊆ R(µ+2)(n+m)+1 forms a manifold of dimension (µ+ 2)(n+m) + 1− r.

2. We have rank(Fµ,[z,z,...,z(µ+1)]

)= r on Lµ.

3. We have

corank(Fµ,[z,z,...,z(µ+1)]

)− corank

(Fµ−1,[z,z,...,z(µ)]

)= v

on Lµ, where corank(F−1,z) = 0 by convention.

4. We have rank(Fµ,[z,...,z(µ+1)]

)= r− a on Lµ and there exist smooth full rank matrix-

valued functions Z2 of size (µ+ 1)l × a and T2 of size (n+m)× (n+m− a) definedon Lµ respectively, that satisfy

Z>2 Fµ,[z,...,z(µ+1)] = 0,

rank(Z>2 Fµ,z

)= a,

Z>2 Fµ,zT2 = 0.

5. We have rank(F,zT2) = d = l−a− v on Lµ and there exists a smooth full rank matrixvalued function Z1 defined on Lµ such that Z>1 F,zT2 has full rank.

27

2 Solvability

Remark 2.38.

1. Compared to the linear setting we have Mµ∼= Fµ,[z,z,...,z(µ+1)] and Nµ ∼= Fµ,z.

2. For a matrix A ∈ Rm×n the corank(A) := m − rank(A) is the codimension of therange of A.

3. A nonempty set Lµ ⊆ Rn that is locally diffeomorphic to an open set V in Rr, i.e.the set can locally be parametrized by r scalars, is called a manifold of dimension r.

This means that for each z0 ∈ Lµ, z0 partitioned into[x>0 y>0

]>, x0 ∈ Rr, y0 ∈ Rn−r,

there exists a neighborhood V ⊆ Rr of x0 and a neighborhood U of z0 ∈ Lµ such that

U := Lµ ∩ U ={g(x)

∣∣x ∈ V } ,where g : V → U is a diffeomorphism.

Definition 2.39. The smallest possible µ in Hypothesis 2 is called the strangeness-indexof the DAE (2.15).

If F is sufficiently smooth and satisfies Hypothesis 2 with µ, r, a, d, v, then (locally) we canderive a reduced system for the form

F1(t, z, z) = 0 (d differential equations)

F2(t, z) = 0 (a algebraic equations)(2.16)

with F1 : I × Dz × Dz → Rd, F2 : I × Dz → Ra, where F1 = Z>1 F (t, z, z) and F2 =Z>2 Fµ(t, z,H(ω0)).

Remark 2.40.

1. The reduced system (2.16) can be constructed locally using the implicit functiontheorem.

2. The reduced system (2.16) is strangeness-free, and every solution of the original sys-tem (2.15) also solves (2.16). However, (2.16) may still contain underdeterminedcomponents since m+ n ≥ d+ a.

3. Let z0µ = (t0, z0, z0, . . . , z

(µ+1)0 ) ∈ Lµ be fixed. By Hypothesis 2, Lµ is a manifold

of dimension n = (µ + 2)(n + m) + 1 − r that can locally be parametrized by nparameters. Choosing n parameters ω out of (t, z, z, . . . , z(µ+1)), then there exists aneighborhood V ⊆ Rn of ω0, as part of z0

µ corresponding to ω, and a neighborhood

U ⊆ R(µ+2)(n+m)+1 of z0µ such that

U := Lµ ∩ U ={g(ω)

∣∣ω ∈ V } ,28


where g : V → U is a diffeomorphism. Thus Fµ(zµ) = 0 holds locally if and only ifzµ = g(ω) for some ω ∈ U . In particular, there exists a function H : V → R(µ+1)(n+m)

such that (z, . . . , z(µ+1)

)= H(ω) for all ω ∈ V.

This implies Fµ(t, z,H(ω)) = 0 and in particular Fµ(t, z,H(ω0)) = 0. Thus,

F2(t, z) = Z>2 Fµ(t, z,H(ω0)),

F1(t, z, z) = Z>1 F (t, z, z)

are defined locally in a neighborhood of z0µ ∈ Lµ. For more details see the DAE lecture

or [6].

We could split z into[z1 z2 z3

]with z1(t) ∈ Rd differential components, z2(t) ∈ Ra

algebraic components and z3(t) ∈ Ru undetermined components (u = m+ n− a− d). But,this would mean we mix input and state variables (u and x) as components of x. Thus, weproceed as follows. From Hypothesis 2 we get

F2,zT2 = Z>2 Fµ,zT2 = 0

rank(T2) = n+m− arank(F1,zT2) = rank(Z>1 F,zT2) = d.

Choosing T ′2 such that[T ′2 T2

]is nonsingular, we get

rank

[F1,z

F2,z

]= rank

[F1,zT

′2 F1,zT2

F2,zT′2 0

]= rank(F1,zT2) + rank(F2,zT

′2) = d+ a.

Thus,[F>1,z F>2,z

]>has pointwise full row rank and hence[

F1,x 0

F2,x F2,u

]of size (d+ a)× (n+m) has full row rank. Note that fixing a control u will in general not

give a strangeness-free regular reduced problem, since[F>1,x F>2,x

]>may be singular.

Question Can we choose a control u such that the resulting reduced problem is regularand strangeness-free? Answer: yes (see TODO 3.3).

As a consequence, we get the following result.

Theorem 2.41. Let F in (1.1) be sufficiently smooth and satisfies Hypothesis 2 withµ, a, d, v. If v = 0 and n = a+ d and the reduced problem (2.16) satisfies the rank condition

rank

[F1,x

F2,x

]= a+ d,

then the control problem (1.1) is regular.

29

2 Solvability

Example 2.42. We consider the descriptor system

F (t, x, x, u) =

[x2

log(x2) + sin(u)

]= 0,

with n = 2,m = 1. The corresponding behavior system with z =[z>1 z>2 z>3

]>=[

x>1 x>2 x>3]>

takes the form

F (t, z, z) =

[z2

log(z2) + sin(z3)

]= 0.

We check Hypothesis 2 for µ = 0.

L0 ={

(t, z1, z2, z3, z1, z2, z3)∣∣ z2 = 0, z2 = exp(− sin(z3))

}⊆ R7

and L0 is a manifold of dimension 5 = (µ+ 2)(n+m) + 1− r = 7− r and hence r = 2 (canbe parametrized by (t, z1, z3, z1, z3)).

F0,z =

[0 0 00 1

z2cos(z3)

]=

[0 0 00 exp(sin(z3)) cos(z3)

]on L0

F0,z =

[0 1 00 0 0

].

Thus, rank(F0,[z,z]) = 2 = r, corank(F0,[z,z]) = 0 = (2− 2) = v and rank(F0,z) = 1 = r − a.

This implies a = 1. With Z>2 =[0 1

]we obtain Z>2 F0,z = 0 and

rank(Z>2 F0,z) = rank([

0 exp(sin(z3)) cos(z3)])

= 1 = a.

Choosing

T2 =

1 00 − cos(z3)0 exp(sin(z3))

we get Z>2 F0,zT2 = 0 and finally rank(F,zT2) = rank

[0 − cos(z3)0 0

]= 1 = d if e.g. z3 ∈

[−1, 1] (z3 corresponds to the control). We conclude that Hypothesis 2 is satisfied for z3 ∈[−1, 1] with µ = 0, a = 1, d = 1, v = 0 and with Z>1 =

[1 0

]we obtain Z>1 F,zT2 =[

0 − cos(z3)]

of rank 1 = d. With these choices for Z1, Z2, the reduced problem is thesame as the original control problem,

F (t, x, x, u) =

[x2

log(x2) + sin(u)

]= 0, u(t) ∈ [−1, 1].

For the free system with u(t) = 0, we get F =

[x2

log(x2)

]= 0 and this system is not

strangeness-free, since [F1,x

F2,x

]=

[0 10 1

x2

]is singular. The free system satisfies Hypothesis 2 for µ = 1, a = 1, d = 0, v = 1, r = 3. Inparticular, x1 is undetermined.

30

CHAPTER

3

FEEDBACK REGULARIZATION

In the control setting, system properties can be modified using feedback control. E.g. onecan use

u = K(t, x) with K : I× Dx → Rm (state feedback),

u = F (t)x+ w(t) with F ∈ C(R,Rm,n), w : I→ Rm (proportional state feedback),

u = Fx+ w(t) with F ∈ Rm,n, w : I→ Rm (proportional state feedback),

u = K(t, y) with K : I× Rp → Rm (output feedback),

u = F (t)y + w(t) with F ∈ C(R,Rm,p), w : I→ Rm (proportional output feedback),

u = Fy + w(t) with F ∈ Rm,p, w : I→ Rm (proportional output feedback),

u = K(t, x) with K : I× Dx → Rm (state derivative feedback),

u = K(t, y) with K : I× Rp → Rm (output derivative feedback)

and combinations of the above are possible. The situation of such closed-loop systems isdepicted in Figure 3.1. In particular, it is possible to achieve index reduction and regular-

X

F

u y

x

X

F

u y

Figure 3.1: Closed-loop systems

ization by using feedback control.

31

3 Feedback Regularization

3.1 Linear Descriptor Systems with constant coefficients

Consider system (1.5), given by

Ex = Ax+Bu+ f(t), x(0) = x0,

y = Cx+Du+ g(t).

In the following, we can assume without loss of generality that D = 0 in (1.5) (i.e. there isno direct feed-through of the input in the output equation). If D 6= 0 we can consider anextended descriptor system of the form[

E 00 0

]ξ =

[A 00 I

]ξ +

[BD2

]u,

y(t) =[C −D1

]ξ,

(3.1)

where D = D1D2 is a factorization of D (e.g. D1 = I,D2 = D). The original system (1.5)is equivalent to (3.1) in the sense, that if x(t) is a solution of (1.5) for a given input u(t),

then ξ(t) =[x(t)> −(D2u(t))>

]>solves (3.1). Note that the part −D2u in ξ occurs only

formally in (3.1). Now, applying a proportional state feedback u = Fx + w to (1.5), weobtain the closed-loop system

Ex = Ax+B(Fx+ w) + f = (A+BF )x+Bw + f.

Analogously, if we apply proportional output feedback u = Fy+w, we obtain the closed-loopsystem

Ex = Ax+B(Fy+w) + f = Ax+BF (Cx+ g) +Bw+ f = (A+BFC)x+Bw+BFg+ f.

Theorem 3.1 (Feedback Regularization). Consider a control system given by (E,A,B,C).

1. There exists a matrix F ∈ Rm,n such that the matrix pair (E,A+BF ) is regular andof index ν = ind(E,A+BF ) ≤ 1 if and only if E,A ∈ Rn,n and

rank[E AS∞ B

]= n,

where S∞ is a matrix with range(S∞) = ker(E) (i.e. the columns of S∞ span thekernel of E).

2. There exists a matrix F ∈ Rm,p such that the matrix pair (E,A + BFC) is regularand of index ν = ind(E,A+BFC) ≤ 1 if and only E,A ∈ Rn,n and

rank[E AS∞ B

]= n and rank

ET>∞AC

= n,

where S∞ as above and range(T∞) = ker(E>).

32

3.1 Linear Descriptor Systems with constant coefficients


] [x1

x2

]=

[1 00 0

] [x1

x2

]+

[01

]u

y =[0 1

] [x1

x2

].

For the free system without inputs (i.e. u = 0) we have to consider

(E,A) =

([0 10 0

],

[1 00 0

]),

which is singular. Choose S∞ =[1 0

]>we have rank

[E AS∞B

]= rank

[0 1 1 00 0 0 1

]=

2 = n. Thus, choosing F =[1 0

], then

(E,A+BF ) =

([0 10 0

],

[1 01 0

])is regular with ν = 1. With T∞ =

[0 1

]>we have rank

[E> A>T∞ C>

]>= 1 6= n.

Thus, there exists no regularizing proportional output feedback.

Proof of Theorem 3.1. The matrices E,A have to be square since otherwise the closed-loopsystems cannot be regular. There exist nonsingular matrices P,Q ∈ Rn,n such that

PEQ =

[Ir 00 0

], Q =

[Q1 Q2

], P =

[P1

P2

]and the columns of Q2 span ker(E) and the columns of P>2 span ker(E>). Setting

PAQ =

[A11 A12

A21 A22

], PB =

[B1

B2

], CQ =

[C1 C2

],

we have

rank[E AQ2 B

]= rank

[PE PAQ2 B

] Q II

= rank

[PEQ PAQ2 B

]= rank

[Ir 0 A11 B1

0 0 A22 B2

]= r + rank

[A22 B2

] != n.

Thus, the matrix[A22 B2

]must satisfy

rank[A22 B2

]= n− r. (3.2)

33


Analogously, we deduce from

rank

EP2AC

= rank

PEWP2AQCQ

= rank

Ir 00 0A21 A22

C1 C2

= r + rank

[A22

C2

]!

= n,

that the matrix[A>22 C>2

]>must satisfy

rank

[A22

C2

]= n− r. (3.3)

We only need to prove 2 (since 1 follows from C = I). The matrix pair (E,A + BFC) isregular and of index ≤ 1 if and only if the matrices are square and A22 + B2FC2 is eithernot present or nonsingular (see Ex. 1.2). There exist nonsingular matrices U, V ∈ Rn−r,n−rsuch that

U−1A22V =

[I 00 0

].

We set A22 := V

[I 00 0

]U−1. Then it holds that

A22A22A22 =

(U

[I 00 0

]V −1

)(V

[I 00 0

]U−1

)(U

[I 00 0

]V −1

)= U

[I 00 0

]V −1 = A22,

and

A22 +B2FC2 =[A22 B2

] [A22 00 F

] [A22

C2

]. (3.4)

Assume that F ∈ Rn,p exists such that A22 +B2FC2 is nonsingular, then from (3.4) follows(3.2) and (3.3). Conversely, assume that (3.2) and (3.3) hold. Then

rank([A22 B2FC2

])= rank

(U−1

[A22 B2FC2

]V)

= rank

([I 00 0

]+

[B1

B2

]F[C1 C2

])

= rank

[I 0 B1

0 0 B2

]I 0 00 0 00 0 F

I 00 0

C1 C2

with U−1B2 =

[B>1 B>2

]>and C2V =

[C1 C2

]. Note that due to (3.2) and (3.3) the left

and right matrix in the last equality have full rank and so have B2 and C2. Hence, thereexist nonsingular matrices U1, U2, V1, V2 such that

U−11 B2V1 =

[I 0

]and U−1

2 C2V2 =

[I0

].

Then

rank (A22 +B2FC2) = rank

[I 0 B11 B12

0 0 I 0

]I 0 0 00 0 0 00 0 F11 F12

0 0 F21 F22

I 00 0C11 IC12 0

.

34

3.2 Linear Descriptor Systems with variable coefficients

Choosing F11 = I, F12 = 0, F21 = 0 and F22 = 0 gives

rank (A22 +B2FC2) = rank

([I +B11C11 B11

C11 I

])= n− r.

Remark 3.3. The properties of a control system (1.5) can also be altered by use of deriva-tive feedback control of the form u = −Gy+v(t) with G ∈ Rm,p or a combined proportionaland derivative feedback u = Fy −Gy + v(t) leading to the closed-loop system

(E +BGC)x = (A+BFC)x+Bv + f .

For C = I we have the special case of state feedback. The regularization by feedback problemthen takes the form: For given E,A,B,C find F,G such that (E+BGC,A+BFC) is regularand of index ν ≤ 1. For this problem, similar conditions as in the previous theorem can bederived (see e.g. Bunse-Gerstner, Mehrmann, Nichols, 1992).


Consider the descriptor system (1.4) given by

E(t)x = A(t)x+B(t)u+ f(t), x(t0) = x0

y = C(t)x+D(t)u+ g(t).

Again, we assume that D(t) ≡ 0. In behavior form (2.11) we have

E z = Az + f(t)

with z =[x> u>

]>, E =

[E 0

],A =

[A B

]. If the strangeness index µ of the system in

behavior form is well-defined we can formulate the reduced system (2.12) (using Hypothesis1 E1(t)

00

z =

A1(t)

A2(t)0

z +

f1

f2

f3

dav

with characteristic values d, a, v. These characteristic values are invariant under propor-tional feedback.

Theorem 3.4. Consider a linear descriptor system of form (1.4) and assume that thestrangeness index of the system in behavior form (2.11) is well-defined. Then, the charac-teristic values d, a and v are invariant under proportional state feedback u = F (t)x+w andproportional output feedback F (t)y + w.

35


Proof. In the behavior setting, proportional state feedback takes the form[xu

]=

[In 0F (t) Im

]︸︷︷︸

=:Q

[xu

],

i.e., is is equivalent to a change of basis of z. Note that this works only, since E =[E 0

].

Similarly, proportional output feedback is an equivalence transformation in the more generalbehavior approach, where we also include yxu

y

=

In 0 00 Im F (y)0 0 Ip

and premultiply by P =

[I B(t)F (t)0 I

]. Since (global) equivalence transformations of

(E ,A) do not change the characteristic quantities, µ, a, d and v are invariant for both typesof feedback.

Corollary 3.5. Let the strangeness index µ be well-defined for the system (2.11) in behaviorform and let the quantities φ and ω (as in Theorem 2.33) be constant on I. Then, thereexists a state feedback u = F (t)x+ w such that the closed-loop system

E(t)x = (A(t) +B(t)F (t))x(t) +B(t)w + f(t) (3.5)

is regular (as a free system, i.e. w ≡ 0) if and only if v = 0 and d+ a = n.

Proof. Since proportional state feedback can be written as (global) equivalence transfor-mation of the pair (E ,A) in behavior form, it holds that first applying the feedback to theoriginal system (1.4) and then computing the reduced system formulation is the same asfirst computing the reduced system and then applying the feedback. Thus, the closed-loopsystem (3.5) is regular as a free system if and only if the reduced formulation (2.12) withinserted feedback is regular and strangeness-free as a free system. Under the assumptionsof Theorem 2.33, it is sufficient to consider the system in the form

x1 = A13x3 +A14x4 +B12u2 + f1, (d)

0 = x2 +B22u2 + f2, (a− φ)

0 = A31x1 + u1 + f3, (φ)

0 = f4. (v)

The output equation is not involved in our considerations, thus we can formally set ω = 0,such that x3 does not appear. First, we assume that v = 0 and d + a = n, such that wehave to consider

x1 = A14x4 +B12u2 + f1, (d)

0 = x2 +B22u2 + f2, (a− φ)

0 = A31x1 + u1 + f3, (φ)

x =

x1

x2

x4

da− φφ

.

36


Thus, x4 and u1 are both of size φ. Applying the proportional state feedback[u1

u2

]=

[x4 −A31x1 + w1

w + 2

],

we get the closed-loop system

x1 = A14x4 +B12w2 + f1, (d)

0 = x2 +B22w2 + f2, (a− φ)

0 = x4 + w1 + f3, (φ)

and for w =[w>1 w>2

]>= 0 we have

x1 = A14x4 + f1, (d)

0 = x2 + f2, (a− φ)

0 = x4 + f3, (φ)

which is regular and strangeness-free. Conversely, let (3.5) be regular as a free system withw = 0. Then, necessarily, we need v = 0 and l = n = a+ d.

Corollary 3.6. Let the strangeness index µ be well-defined for the system (2.11) in behaviorform and let the quantities φ and ω (as in Theorem 2.33) be constant on I. There exists anoutput feedback u = F (t)y + w such that the closed-loop system

E(t)x = (A(t) +B(t)F (t)C(t))x+B(t)w + f(t) +B(t)F (t)g(t)

is regular (as a free system) if and only if v = 0 and d+ a = n and φ = ω.

Proof. As in the proof of Corollary 3.5 we can consider the reduced formulation (2.14) forD = 0 instead of (1.4).

⇐=: Assume that v = 0, d + a = n and φ = ω in (2.14). Thus, the unknown x4 does notappear in (2.14) and u1 and y1 are of the same size. Using the feedback

u1 = y1 + w1,

u2 = w2,

yields the closed-loop system

x1 = A13x3 +B12w2 + f1, (d)

0 = x2 +B22w2 + f2, (a− φ)

0 = A31x1 + y1 + w1 + f3 = A31x1 + x3 + w1 + f3 + g1. (φ).

For w =[w>1 w>2

]>= 0 this system is regular and strangeness-free, since

rank

([E1

A2

])= rank

Id 0 00 Ia−φ 0A31 0 Iφ

= d+ a = n.

37


=⇒: Let the closed-loop system be regular as a free system. Then, necessarily, v = 0 anda+ d = n. For a feedback matrix

F =

[F11 F12

F21 F22

]φ

m− φ,

we obtain A+BFC as

A+BFC =

0 0 A13 A14

0 Ia−φ 0 0A31 0 0 0

+

0 B12

0 B22

Iφ 0Iφ 0

[F11 F12

F21 F22

] [0 0 Iω 0C21 C22 0 0

]

=

B21F22C21 B12F22C22 A13 +B12F12 A14

B22F22C21 I +B22F22C22 B22F21 0A31 + F12C21 F12C22 F11 0

(d)(a− φ)

(φ)

of size (d+ a)× (d+ a). Thus, the closed-loop system is regular and strangeness-freefor w = 0 if and only if Id 0 0 0

B22F22C21 I +B22F22C22 B22F21 0A31 + F12C21 F12C22 F11 0

has pointwise full row rank (is piecewise nonsingular). This implies ω − φ = 0 ⇐⇒φ = ω.

3.3 Nonlinear Descriptor Systems

Consider system (1.1) given by

F (t, x, x, u) = 0, x(t0) = x0

y −G(t, x) = 0

and the corresponding reduced system formulation (2.16) given by

F1(t, x, x, u) = 0 x(t0) = x0

F2(t, x, u) = 0

y −G(t, x) = 0

obtained by applying Hypothesis 2. Now the question is: Can we find a feedback controlu such that the reduced problem is regular and strangeness-free? Necessarily, we needn = d+ a. Applying a state feedback u = K(t, x) leads to the closed-loop system

F1(t, x, x,K(t, x)) = 0

F2(t, x,K(t, x)) = 0.(3.6)

38


This closed-loop system is regular and strangeness-free if and only if[F1,x

F2,x + F2,uK,x

]is pointwise nonsingular. (3.7)

Since the reduced system (2.16) is defined only locally, it is sufficient to satisfy condition(3.7) only locally. Thus, we can restrict to linear state feedback Kx(t) + w(t) such thatK = K,x. In Section 2.3 we have seen that[

F1,x 0

F2,x F2,u

]has full row rank d + a. Thus using E1(t) := F1,x, A2(t) := F2,x, B2(t) := F2,u similar asin (2.13), the existence of a suitable feedback matrix K = K,x follows from Corollary 3.5.The control function w(t) can be used to satisfy initial conditions of the form

u(`)(t0) = Kx(`)0 + w(`)(t0) = u

(`)0 for ` = 0, . . . , µ+ 1.

Altogether, we have proved the following theorem.

Theorem 3.7. Assume that the control problem (1.1) in behavior form satisfies Hypothesis2 with characteristic values values µ, a, d, v and let d+ a = n. Furthermore, let

zµ,0 = (t0, x0, u0, . . . , x(µ+1)0 , u

(µ+1)0 ) ∈ Lµ.

Then there (locally) exists a state feedback u = K(t, x) satisfying

u0 = K(t0, x0) and

u0 = K,t(t0, x0) +K,x(t0, x0)x0,

such that the closed-loop reduced problem (3.6) is regular and strangeness-free.

Example 3.8. Consider the descriptor system from Example 2.42 given by

F (t, x, x, u) =

[x2

log(x2) + sin(u)

]= 0

already in reduced form. We have already see in Example 2.42 that the free system withu(t) ≡ 0 is not strangeness-free. To obtain a regular and strangeness-free closed-loop systemwe need to find K such that [

F1,x

F2,x + F2,uK,x

]is nonsingular. We have [

F1,x 0

F2,x F2,u

]=

[0 1 00 1

x2cos(u)

].

39


At z0,0 = (t0, x1,0, x2,0, u0, x1,0, x2,0, u0)> ∈ L0 given by z0,0 = (0, 0, 1, 0, 0, 0, 0) we have[F1,x 0

F2,x F2,u

]=

[0 1 00 1 1

]and choosing K =

[1 0

]gives [

F1,x

F2,x + F2,uK,x

]=

[0 11 1

]nonsingular at z0,0, i.e. u(t) = Kx+ w(t) = x1 + w(t). Observe that

0 = u0 = u(t0) = x1(t0) + w(t0) = x1,0 + w(t0) = w(t0)

0 = u0 = u(t0) = x1(t0) + w(t0) = x1,0 + w(t0) = w(t0).

Thus, we can choose w(t) ≡ 0. The corresponding closed-loop system is given by

x2 = 0,

0 = log(x2) + sin(x2),

which is regular and strangeness-free in a neighborhood of z0,0. For x1(0) = x1,0 = 0, x2(0) =x2,0 = 1 we get the unique solution

x1(t) ≡ 0

x2(t) ≡ 1.

Now, we want to consider also output feedback of the form u = K(t, y). Inserting thiscontrol in the reduced problem (2.16) yields the closed-loop reduced problem

F1(t, x, x,K(t, G(t, x))) = 0

F2(t, x,K(t, G(t, x))) = 0.

Again, the condition[F1,x

F2,x + F2,uK,yG,x

]=

[F1,x 0

F2,x F2,u

] [I

K,yG,x

]is nonsingular (3.8)

has to be satisfied. For y = x we are again in the state feedback case. Again, we canproceed as in Section 3.2 for linear systems. Setting E1 := F1,x, A2 := F2,x, B2 := F2,u andC := G,x. We can determine the quantities φ and ω locally at a point (t0, z0, z0) givenzµ,0 ∈ Lµ as in Theorem 2.33. In this way we get a nonlinear version of Corollary 3.6.

Corollary 3.9. Suppose that the control problem (1.1) in generalized behavior form using

z =[x> u> y>

]>satisfies Hypothesis 2 with characteristic values µ, a, d, v and d + a = n. Furthermore, letφ = ω locally for the system at (t0, z0, z0) given by zµ,0 ∈ Lµ. Then there exists an outputfeedback u = K(t, y) satisfying u0 = K(t0, y0) and u0 = K,t(t0, y0) +K,y(t0, y0)y0 such thatthe closed-loop reduced problem is regular and strangeness-free.

40


Proof. Under the given assumptions we can apply the same theory as for the linear problemto obtain a suitable matrix K,y sucht that (3.8) holds. Then, we can use the linear outputfeedback u(t) = Ky(t) + w(t), where the control function w(t) is used to satisfy the giveninitial conditions.

41

CHAPTER

4

CONTROL THEORETICALCONCEPTS

4.1 Controllability

In contrast to standard state-space systems there are several different notions of controllabil-ity for descriptor systems. Unfortunately, there is no uniform terminology in the literature.We consider system (1.1) of the form

F (t, x, x, u) = 0 x(t0) = x0,

y −G(t, x) = 0.

Definition 4.1.

1. The descriptor system (1.1) is called completely controllable (C-controllable) if for anygiven initial state x0 ∈ Rn and final state xf ∈ Rn there exists a control input u thattransforms the system from x0 to xf in finite time tf ≥ t0 (i.e. ∃u, tf <∞ such thatx(tf ;u, x0) = xf ).

2. For (1.1) a set Rx0 is called reachable from x0 ∈ Rn if for all xf ∈ Rx0 there exists anadmissible control input u that transfers the system from x0 to xf in finite time (i.e.∃u, tf <∞ such that x(tf ;u, x0) = xf ∈ Rx0).

Rx0 :={xf ∈ Rn

∣∣ ∃ u, tf <∞ : x(tf : u, x0) = xf}⊆ Rn.

43

4 Control theoretical concepts

Let R :=⋃x0∈X

t0cRx0 denote the reachable set (where X t0c ⊆ Rn is the set of all

consistent initial values x0 at time t0). The system (1.1) is called controllable withinthe reachable set (R-controllable), if any state in R can be reached from any consistentinitial state x0 in finite time (i.e. for any x0 ∈ R, xf ∈ R ∃ u, tf < ∞ such thatx(tf ;u, x0) = tf ).

Remark 4.2.

1. R-controllability is sometimes also called finite dynamics controllability.

2. In general, descriptor systems will not be C-controllable, since algebraic constraintsfix the solution onto a certain solution manifold. In the case E = In, R-controllabilitycoincides with C-controllability.

Example 4.3. Consider the descriptor system[0 01 0

] [x1

x2

]=

[0 11 0

] [x1

x2

]+

[01

]u.

The reachable set is given by R = {(x1, x2) ∈ R2∣∣ x2 = 0} and the system is R-controllable.

For descriptor systems another phenomenon arises if input functions are used that are onlypiecewise continuous. Since the solution may depend on derivatives of the input it mayhappen that no classical solution exists. Thus, a descriptor system can adopt a generalizedsolution (i.e. a distribution as solution).

Example 4.4. Consider the descriptor system[0 10 0

] [x1

x2

]=

[1 00 1

] [x1

x2

]+

[−1−1

]u.

Let the input u be given by

u(t) =

{0, 0 ≤ t ≤ 1,

1, 1 ≤ t ≤ tf ,

i.e. u is only piecewise continuous. The solution is given by

x1(t) = u+ u

x2(t) = u.

Thus, for the given u, no solution in the classical sense exists. Nevertheless, the stateresponse of the system can be depicted as in Figure 4.1. and (x1, x2) is a solution in thedistributional sense.

Definition 4.5 (first rough version). A descriptor system (1.1) is called impulse controllable(I-controllable) if for any given initial state x0 ∈ Rn there exists an admissible control inputu that transforms the system to some impulsive state in finite time.

44

4.1 Controllability

t

x2

1

1t

x1

1

1

Figure 4.1: Impulse behavior of solutions. The impuse in x1 is due to the jump behavior inthe input.

It can be shown that I-controllability is equivalent to the ability to cancel all impulsivestates by choosing a suitable u. This can be done by state feedback control (i.e. for everyinitial state x0 there exists a state feedback control such that the closed-loop system has noimpulsive solutions).

For regular time-invariant (LTI) descriptor systems of the form

Ex = Ax+Bu, x(0) = x0

y = Cx(4.1)

with E,A ∈ Rn,n, B ∈ Rn,m, C ∈ Rp,n, there exists purely algebraic characterizations of thedifferent controllability concepts. Without loss of generality, we assume that r = rank(E) <n. Then there exist nonsingular matrices T,W ∈ Rn,n such that the system is (strongly)equivalent to

x1 = Jx1 +B1u, x1(0) = x1,0 (4.2a)

Nx2 = x2 +B2u, x2(0) = x2,0 (4.2b)

y = C1x1 + C2x2. (4.2c)

with

WET =

[Inf 0

0 N

], WAT =

[J 00 In∞

],

CT =[C1 C2

], WB =

[B1

B2

], T−1x =

[x1

x2

]and let ν = ind(E,A). We call (4.2a) the slow subsystem of dimension nf and (4.2b) thefast subsystem of dimension n∞. Then, we know that the state response of (4.2) is

x1(t) = eJtx1(0) +

∫ t

oeJ(t−s)B1u(s)ds (t > 0)

x2(t) = −ν−1∑i=0

N iB2u(i).

Thus, an admissible control function (for a classical solution) has to satisfy u ∈ Cν−1p (I,Rm)

(i.e. (ν − 1)- times piecewise continuously differentiable). For any t > 0 the state response

45


x(t) = T[x>1 x>2

]>is uniquely determined by the initial condition x1(0), the control input

u(s), 0 ≤ s ≤ t, and the time point t. In particular, the initial condition x2(0) has to beconsistent (i.e. x2(0) is uniquely determined) and only x1(0) can be choosen arbitrarily. Inthe following, we denote by R0 the reachable set of (4.2) from the zero initial conditionx1(0) = 0 (and consistent x2(0)).

Lemma 4.6. For any polynomial p(t) 6≡ 0 consider the matrix

W (p, t) =

∫ t

0p(s)eA1sB1B

>1 e

A>1 sp(s)ds,

with A1 ∈ Rn,n, B1 ∈ Rn,m. Then, it holds that

Im(W (p, t)) = Im[B1 A1B1 . . . An−1

1 B1

]for any t > 0.

Proof. The statement of the lemma is equivalent to the statement that

ker(W (p, t)) =

n−1⋂i=0

ker

(B>1

(A>1

)i).

Let x ∈ ker(W (p, t)), then

x>W (p, t)x =

∫ t

0x>p(s)eA1sB1B

>1 e

A>1 sp(s)x ds =

∫ t

0‖B>1 eA

>1 sp(s)x‖22︸︷︷︸≥0

ds = 0

and hence B>1 eA>1 sp(s)x = 0 for 0 ≤ s ≤ t. The polynomial p(s) has a finite number of

roots in [0, t], thus we have

B>1 eA>1 sx = 0, 0 ≤ s ≤ t.

Since s is arbitrary, we have x ∈⋂n−1i=0 ker

(B>1

(A>1)i)

(by Cayley-Hamilton) and thus

ker(W (p, t)) ⊆ ker(B>1

(A>1)i)

. For x ∈ ker(B>1

(A>1)i)

the reverse of this process yields

x ∈ ker(W (p, t)), which implies

ker

(B>1

(A>1

)i)⊆ ker(W (p, t)).

Lemma 4.7. Let xi ∈ Rn, i = 0, 1, . . . , ν − 1 and t1 > 0. Then there exists a polynomialp(t) ∈ Rn of order ν − 1 such that p(i)(t1) = x1 for i = 0, 1, . . . , ν − 1.

46

4.1 Controllability

Proof. Straightforward by setting

p(t) = x0 + x1(t− t1) + . . .+1

(ν − 1)!xν−1(t− t1)ν−1.

Theorem 4.8. Let R0 be the reachable set of (4.2) from the zero initial condition x1(0) = 0.Then

R0 = Im[B1 JB1 . . . Jnf−1B1

]⊕ Im

[B2 NB2 . . . Nn∞−1B2

].

Remark 4.9. In Theorem 4.8, ⊕ is meant as the cartesian product, i.e. ⊕ = ×, as this isthe common notation in the literature.

Proof. For x1(0) = 0 and t > 0 the state response of (4.2) is given by

x1(t) =

∫ t

0eJ(t−s)B1u(s)ds, x2(t) = −

ν−1∑i=0

N iB2u(i)(t).

Obviously, x2(t) ∈ Im[B2 NB2 . . . Nn∞−1B2

](since ν ≤ n∞). Furthermore, there

exists (exercise) analytic functions βi(t) ∈ R, i = 0, . . . , nf − 1 such that

eJt = β0(t)I + β1(t)J + . . .+ βnf−1(t)Jnf−1

(see Exercise). Thus,

x1(t) =

∫ t

0eJ(t−s)B1u(s)ds =

nf−1∑i=0

J iB1

∫ t

0βi(t− s)u(s)ds︸︷︷︸

:=w(t)

∈ Im[B1 JB1 . . . Jnf−1B1

]

for all t > 0. Hence

x(t) =

[x1(t)x2(t)

]∈ Im

[B1 JB1 . . . Jnf−1B1

]⊕ Im

[B2 NB2 . . . Nn∞−1B2

].

On the other hand, let

x =

[x1

x2

]∈ Im

[B1 JB1 . . . Jnf−1B1

]⊕ Im

[B2 NB2 . . . Nn∞−1B2

],

with x1 ∈ Im[B1 JB1 . . . Jnf−1B1

]and x2 ∈ Im

[B2 NB2 . . . Nn∞−1B2

]. Thus,

there exists wi ∈ Rn∞ , i = 0, . . . , ν − 1 such that

x2 = −ν−1∑i=0

N iB2wi.

47


From Lemma 4.7 for any fixed t > 0 there exists a polynomial p(s) of order ν − 1 such thatp(i)(t) = wi. Thus, using the input function u(t) = u1(t) + p(t) we get the system response

x1(t) =

∫ t

0eJ(t−s)B1u1(s)ds+

∫ t

0eJ(t−s)B1p(s)ds

and

x1 := x1 −∫ t

0eJ(t−s)B1p(s)ds ∈ Im

[B1 JB1 . . . Jnf−1B1

]for some fixed t > 0. For any fixed t > 0 let q(s) = sν(s − t)ν 6= 0. From Lemma 4.6 we

deduce the existence of z ∈ Rnf such that W (q, t)z = x1. Setting u1(s) = q(s)2B>1 eJ>(t−s)z

for 0 ≤ s ≤ t we get the system response

x1(t) =

∫ t

0eJ(t−s)B1q(s)

2B>1 eJ>(t−s)zds+

∫ t

0eJ(t−s)B1p(s)ds

=

∫ t

0q(s)eJ(t−s)B1B

>1 e

J>(t−s)q(s)dsz +

∫ t

0eJ(t−s)B1p(s)ds

= W (q, t)z + x1 − x1 = x1

and

x2(t) = −ν−1∑i=0

N iB2u(i)(t) = −

ν−1∑i=0

N iB2(u1(t)︸︷︷︸=0

+ p(i)(t)︸︷︷︸=wi

)

= −ν−1∑i=0

N iB2wi = x2.

Hence, x ∈ R0 and the result follows.

Example 4.10.

1. Consider the system

x1 =

[1 10 1

]x1 +

[01

]u, x1(0) = x0

1,

0 = x2 +[−1 0

]u,

with n = 4, nf = 2, n∞ = 2, which is already in Weierstraß canonical form (WCF).The reachable set from x1(0) = 0 is given by

R0 = Im[B1 JB1

]⊕[B2 NB2

]= R2 ⊕ (R⊕ {0}) = R3 ⊕ {0}.

2. Consider the system [0 10 0

]x2 = x2 +

[−1−1

]u.

48

4.1 Controllability

The reachable set is given by R0 = R2 and the state response of this system is givenby

x2(t) = −ν−1∑i=0

N iB2u(i)(t) =

[u+ uu.

]Thus, for any w =

[w1 w2

]> ∈ R2 and t1 > 0 we can choose u(t) such that u(t1) =w2, u(t1) = w1 − w2 and we get

x2(t1) =

[w1

w2

].

Note that such a control strategy may require high input energy, since for large ‖w1−w2‖ u will also be large.

For the next results, we need the following reminder from control theory.

Lemma 4.11 (Hautus-Popov-Lemma). The following are equivalent:

1. The system x = Ax+Bu is C-controllable.

2. rank(K) = rank[B AB . . . An−1B

]= n.

3. If z is eigenvector of A>, then z>B 6= 0.

4. rank[λI −A B

]= n for all λ ∈ C.

Theorem 4.12.

1. The slow subsystem (4.2a) is C-controllable if and only if

rank[λE −A B

]= n for all λ ∈ C, λ finite.

2. The following statements are equivalent:

a) The fast subsystem (4.2b) is C-controllable.

b) rank[B2 NB2 · · · Nν−1B2

]= n∞.

c) rank[N B2

]= n∞.

d) rank[E B

]= n.

e) For any nonsingular matrices Q1 and P1 satisfying

E = Q1

[I 00 0

]P1, let QB =

[B1

B2

].

49


Then B2 is of full row rank, rank(B2) = n− rank(E).

3. The following statements are equivalent:

a) The system (4.2) is C-controllable.

b) The slow and fast subsystems (4.2a) and (4.2b) are both C-controllable.

c) rank[B1 JB1 · · · Jnf−1B1

]= nf and rank

[B2 NB2 · · · Nν−1B2

]= n∞.

d) rank[λE −A B

]= n for all finite λ ∈ C and rank

[E B

]= n.

e) rank[αE − βA B

]= n for all (α, β) ∈ C2\{(0, 0)}.

Proof.

1. The slow subsystem is an ODE, thus the controllability conditions for standard LTIsystems apply and (4.2a) is C-controllable if and only if rank

[λI − J B1

]= nf for

all finite λ ∈ C. Furthermore, we have

rank[λE −A B

] (WCF)= rank

[λWET −WAT WB

]= rank

[λI − J 0 B1

0 λN − I B2

].

The matrix λN − I is nonsingular for any finite λ ∈ C and hence

rank[λE −A B

]= n∞ + rank

[λI − J B1

]= n.

2. 2a ⇐⇒ 2b By definition the fast subsystem (4.2b) is C-controllable if the reachableset is

Im[B2 NB2 . . . Nν−1B2

]= Rn∞ ⇐⇒ rank

[B2 NB2 . . . Nν−1B2

]= n∞.

2b ⇐⇒ 2c The system (N,B2) is C-controllable (as a standard LTI system) if andonly if rank

[λI −N,B2

]= n∞ for all λ ∈ C. Thus, this condition holds for all

λ ∈ σ(N) = {0} since N nilpotent and thus

rank[λI −N B2

]= n∞ for all λ ∈ C ⇐⇒ rank

[−N B2

]= rank

[N B2

]= n∞.

2c ⇐⇒ 2d We have

rank[E B

]= rank

[WET WB

]= rank

[Inf 0 B1

0 N B2

]= nf +rank

[N B2

].

Thus, rank[N B2

]= n∞ ⇐⇒ rank

[E B

]= n.

50

4.1 Controllability

2d ⇐⇒ 2e Similar as the previous statement.

3. 3a ⇐⇒ 3c Let the system be C-controllable and let x1(0) = 0. Then for any t1 > 0and w ∈ Rn, there exists an admissible control input u ∈ Cν−1

p such that x(t1) =w. Thus

R0 = Im[B1 JB1 · · · Jnf−1B1

]⊕ Im

[B2 NB2 · · ·Nν−1B2

]= Rn

⇐⇒ rank[B1 JB1 · · · Jnf−1B1

]= nf and rank

[B2 NB2 · · ·Nν−1B2

]= n∞.

On the other hand, let the rank conditions hold. Then, we know that

Rx1(0) = R0 +

{[x1

x2

] ∣∣ x1 = eJtx1(0) ∈ Rnf , x2 = 0 ∈ Rn∞}

= Rn

and thus the system (4.2) is C-controllable.

3b ⇐⇒ 3c Follows from 1 and 2.

3b ⇐⇒ 3d Clear from 1 and 2.

3d ⇐⇒ 3e rank[λE −A B

]= rank

[αβE −A B

]= rank

[αE − βA B

].

Example 4.13. Consider the system

x1 =

[1 10 1

]x1 +

[01

]u

0 = x2 +

[−10

]u.

We have rank[B1 JB1

]= rank

[0 11 1

]= 2 and rank

[B2 BN2

]= rank

[−1 00 0

]= 1 < 2.

Thus, the system is not C-controllable, while the slow subsystem is C-controllable.

Remark 4.14. For systems with large state dimension n, the criteria given in the previoustheorem are not suitable for numerical computations, since the system decomposition into(WCF) or the eigenvalues are needed. A better way for numerics is via staircase forms (seechapter 5).

A regular linear time-invariant descriptor system is R-controllable (i.e. controllable withinthe reachable set) if for any given tf > 0 and x1(0) ∈ R, w ∈ R, there exists an admissiblecontrol input u ∈ Cν−1

p such that x(tf ) = w.

Theorem 4.15. The following statements are equivalent:

1. The system (4.2) is R-controllable.

51


2. The slow subsystem is C-controllable.

3. rank[λE −A B

]= n for all finite λ ∈ C.

4. rank[B1 JB1 · · · Jnf−1B1

]= nf .

Proof.

1 ⇐⇒ 1 By definition the system is R-controllable if

R0 = Im[B1 JB1 · · · Jnf−1B1

]⊕ Im

[B2 NB2 · · ·Nν−1B2

]= Rnf ⊕ Im

[B2 NB2 · · · Nν−1B2

].

Thus, Im[B1 JB1 · · · Jnf−1B1

]= Rnf ⇐⇒ the slow subsystem (4.2a) is C-

controllable.

2 ⇐⇒ 3 follows directly from Theorem 4.12.

Thus, C-controllability implies R-controllability. The converse direction is not true in gen-eral.

Example 4.16.

1. Consider the system from Example 4.13 given by

x1 =

[1 10 1

]x1 +

[01

]u

0 = x2 +

[−10

]u.

The slow subsystem is C-controllable as we have already seen and hence the system isR-controllable.

2. Let N be nilpotent and consider the system Nx = x + Bu. The system consists onlyof the fast subsystem and this is always R-controllable.

Corollary 4.17. Consider system (4.1) with regular matrix pencil λE − A. Then, thesystem is C-controllable if and only if the system is R-controllable and rank

[E B

]= n.

Proof. Clear from previous results.

52

4.1 Controllability

Exkurs: Generalized Functions and Distributional Solutions

We want to allow for jumps/discontinuities in the solution x(t) at a number of distinctpoints in I in a distributional setting. In the following, let Dn = C∞0 (R,Rn) denote theset of infinitely differentiable functions with values in Rn and compact support in R. Theelements of Dn are called test functions.

Definition 4.18. A linear functional f : Dn → Rn with

f(α1φ1 + α2φ2) = α1f(φ1) + α2f(φ2) for all φ1, φ2 ∈ Dn, α1, α2 ∈ R

is called a generalized function or a distribution if it is continuous, that is f(φ)→ 0 in Rnfor all sequences (φi)i∈N with φi → 0 in Dn. [A sequence (φi(t))i∈N converges to zero if all

functions φi vanish outside a bounded interval and(φ

(q)i (t)

)i∈N

converges uniformly to zero

for all q ∈ N0.] We denote the space of all distributions acting on Dn by Cn.

In order to use distributions in the framework of descriptor systems we need the notionof derivatives and primitives of distributions. The q-th order derivative f (q), q ∈ N0 of adistribution f ∈ Cn is defined by

f (q)(φ) = (−1)qf(φ(q)) for all φ ∈ Dn.

The functional f (q) is linear and continuous, thus every distribution has derivatives ofarbitrary order in Cn. For a distribution f ∈ Cn any distribution X ∈ Cn that satisfies

X(φ) = f(φ) for all φ ∈ Dn

is called a primitive of f , i.e. X is a solution of X = f . For A ∈ C∞(R,Rm,n) and x ∈ Cn

multiplication by a matrix-valued function is defined by

Ax(φ) = x(A>φ) for all φ ∈ Dn.

Example 4.19. The Dirac delta distribution δα ∈ Cn is defined by δα(φ) = φ(α) for allφ ∈ Dn, α ∈ R. It can be loosely thought of as

δα(x) =

{+∞ x = α,

0 otherwise.

Remark 4.20. Since for a given φ ∈ Dn and t > 0 sufficiently large it holds that

φ(0) = −(φ(t)− φ(0)) = −φ(t)∣∣t0

= −∫ t

0φ(t)dt = −

∫ ∞0

φ(t)dt = −∫RH(t)φ(t)dt =: H(φ),

where H(t) =

{0 for t < 0,

1 for t ≥ 0is the Heaviside function. We find the relation δ0 = H. We

can also define the shifted version of H by Hα(t) := H(t− α) and δα = Hα.

53


Two distributions f1, f2 ∈ Cn are equal if f1(φ) = f2(φ) for all φ ∈ Dn. In the following,a function x : I → Rn, I ⊆ R is treated as a function defined on R by setting x(t) = 0for t 6∈ I. Futhermore, nonsmooth behavior of the solution is restricted to happen at acountable number of points τj ∈ T ⊆ R.

Definition 4.21. Suppose that the set T = {τj ∈ R∣∣ τj < τj+1, j ∈ Z} has no accumulation

points. A generalized function/distribution x ∈ Cn is called impulsive smooth if it can bewritten in the form

x = x+ ximp, x =∑j∈Z

xj , (4.3)

where xj ∈ C∞([τj , τj+1],Rn) for all j ∈ Z and the impulsive part ximp has the form

ximp,j =

qj∑i=0

cijδ(i)τj , cij ∈ Rn, qj ∈ N0. (4.4)

The set of all impulsive smooth distributions is denoted by Cnimp(T).

Lemma 4.22.1. A distribution x ∈ Cnimp(T) uniquely determines the decomposition (4.3).2. For x ∈ Cnimp(T) we can asign a function value x(t) for every t ∈ R\T by x(t) = xj(t)

for t ∈ (τj , τj+1) and limits x(τ−j ) = limt→τ−jxj−1(t) and x(τ+

j ) = limt→τ+jxj(t) for

every τj ∈ T.3. All derivatives and primitives of x ∈ Cnimp(T) are again in Cnimp(T).4. The set Cnimp(T) is a vector space and closed under multiplication with functions A ∈C∞(R,Rm,n).

Proof. See [6, Lemma 3.75].

We can also introduce a measure for the smoothness of impulsive smooth distributions asfollows.

Definition 4.23. The impulse order of x ∈ Cnimp(T) at τj ∈ T is defined as iord(x)∣∣τj

:=

−q−2 if x can be associated with a continuous function in [τj−1, τj+1] and q, with 0 ≤ q ≤ ∞is the largest integer such that

x∣∣[τj−1,τj+1]

∈ Cq([τj−1, τj+1],Rn).

It is defined as iord(x)∣∣τj

:= −1 if x can be associated with a function that is continuous in

[τj−1, τj+1] except at t = τj and it is defined as

iord(x)∣∣τj

:= max{i ∈ N0

∣∣ 0 ≤ i ≤ qj , cij 6= 0}

otherwise. The impulse order of x is defined as iord x := maxτj∈T iord(x)∣∣τj

.

54

4.1 Controllability

Lemma 4.24. Let x ∈ Cnimp(T) and A ∈ C∞(R,Rm,n). Then iord Ax ≤ iord x with equalityfor m = n and A(τj) invertible for each τj ∈ T.

Proof. See [6].

Example 4.25. Consider the model of a differentiator circuit (see Figure 4.2) described bythe following DAE

−

+

A

u(t)

C

R

C: linear capacitorR: linear resistorA: linear amplifierxi: node potential at node i

Figure 4.2: Model of a differentiator circuit

x1 − x4 = u(t),

C(x1 − x2) +1

R(x3 − x2) = 0,

x3 = A(x4 − x2),

x4 = 0,

with input voltage u(t) =

{0 for t < 0,

1 for t ≥ 0.

With x4 = 0, x1 = u(t), x1 = u and x3 = −Ax2 we get

x2 = − 1

CR(A+ 1)x2 + u.

Due to the jump in the input voltage, u is not differentiable. For u = H and A → ∞, themodel equations take the form

x1 − x4 = H,

C(x1 − x2) +1

R(x3 − x2) = 0,

x2 = 0,

x4 = 0,

with solution x1 = H, x2 = 0, x3 = −RCH = −RCδ0 and x4 = 0. The system is a DAEof index ν = 2 (or µ = 1) and iord f = −1. For a consistent initial value, e.g. x(−1) = 0we have a unique solution x with iord x = 0.

55


For the special case of regular time-invariant DAEs of the form Ex = Ax + f with f ∈Cnimp(T) of impulse order iordf = q ∈ Z ∪ {−∞} we can proceed as follows. First, we cantransform the matrix pair (E,A) into (WCF)

(E,A) ∼ (WET,WAT ) =

([Inf 0

0 N

],

[J 00 In∞

]).

Thus, we get

x1 = Jx1 + f1, (4.5a)

Nx2 = x2 + f2, (4.5b)

where

[x1

x2

]= T−1x. For the distributional ODE (4.5a) we can consider the fundamental

solution matrix X(t) that satisfies

X(t) = JX(t), X(t0) = I,

i.e. X(t) = eJ(t−t0) ∈ C∞(R,Rnf ,nf ). Thus, a distribution x ∈ Cnimp(T) solves (4.5a) if and

only if z = X−1x ∈ Cnimp(T) solves

z = g1 = X−1f1,

i.e. z is a primitive of g1. Since f ∈ Cnimp(T) we have f1 ∈ Cnfimp(T) and hence g1 ∈ C

nfimp(T)

with iord g1 = iord f1 by Lemma 4.24. Using the decomposition (4.3) given by g1 =g1 + g1,imp with

g1 =∑j∈Z

g1,j and g1,imp =∑j∈Z

qi∑i=0

cijδ(i)τj

with the convention that g1,imp,j =∑qj

i=0 cijδ(i)τj = 0 if qj < 0, a primitive of g1 has the

form

z = c+

∫ t

t0

∑j∈Z

g1,j(s)ds+

∫ t

t0

∑j∈Z

qi∑i=0

cijδ(i)τj

= c+

∫ t

t0

∑j∈Z

g1,j(s)ds+∑j∈Z

(∫ t

t0

c0jδτj +

qi∑i=1

cijδ(i)τj

)

= c+

∫ t

t0

∑j∈Z

g1,j(s)ds+∑j∈Z

c0jHτj +∑j∈Z

qi−1∑i=0

cijδ(i)τj

with some c ∈ Rnf . Hence, every primitive z of g1 has impulse order q − 1 and so everysolution of (4.5a). For the initial value it holds that

z(t0) = c+∑j∈Z

c0jHτj (t0) for t0 ∈ R\T

56

4.1 Controllability

and

z(τ−i ) = c+∑j∈Z

c0j limt→τ−i

Hτj (t) = c+∑

j∈Z,τj>τi

c0j ,

z(τ+i ) = c+

∑j∈Z

c0j limt→τ+i

Hτj (t) = c+∑

j∈Z,τj≥τi

c0j .

Thus, there exists a unique solution of (4.5a) in Cnimp(T) satisfying one of the initial condi-tions

x1(t0) = x1,0, x1(τ−i ) = x1,0, x1(τ+i ) = x1,0

with τi ∈ T. For the algebraic part (4.5b) we use

f2 = f2 + f2,imp ∈ Cn∞imp(T).

The unique solution of (4.5b) is given by

x2 = −ν−1∑i=0

N i(f

(i)2 + f

(i)2,imp

)∈ Cn∞imp(T)

with iord(x2) ≤ q + ν − 1. Similar as before, consistency of an initial value implies thatx2(t0) = x2,0 for t0 ∈ R\T.

Theorem 4.26. Let (E,A) be regular with ν = ind(E,A). Let xj0 ∈ C∞([τj−1, τj ],Rn) be

given and let f = f + fimp, f =∑

i∈Z fi, where fj = Exj0 − Axj0. Then, the followingstatements hold:

1. The DAE

Ex = Ax+ f with xj−1 = xj0 (4.6)

has a unique solution x ∈ Cnimp with iord x ≤ iord f + ν − 1.2. Let x = x+ ximp, x =

∑i∈Z xi be the unique solution of (4.6). Then x = x− xj−1 is

the unique solution of

E ˙x = Ax+ f + Exj,0δτj , xj−1 = 0,

where xj,0 = xj−1(τj) and f = f − fj.

Proof. See [6].

Remark 4.27. Setting fj = Exj0 −Axj0 forces xj0 to be a solution of

Ex = Ax+ fj for t ∈ [τj , τj+1] for all j ∈ Z. (4.7)

57


Thus, the inhomogeneity fj is modified in such a way that a given initial condition x(τ−j ) =

xj,0(τj) is made consistent for (4.7). Since x = ˙x+ ximp +∑

(xi(τi)− xi−1(τi)) δτi we have

E(

˙x+ ximp +∑

(xi(τi)− xi−1(τi)) δτi

)= A (x+ ximp) + f + fimp

= A (x+ ximp) +∑

j∈Z,j 6=ifi + fimp + Exj0 −Ax

j0.

Setting x = x− xj−1 and f = f − fj this can be expressed as

E ˙x = Ax+ f + Exj,0δτj , xj−1 = 0,

where xj,0 = xj−1(τj).

In the descriptor setting we consider

Ex = Ax+Bu (4.8)

with input function u ∈ C∞p (R,Rn) (infinitely often piecewise continuously differentiable).Thus, u ∈ Cnimp(T) with iord(u) = q ≤ −1 and we can consider (4.8) in a distributionalframework.

Remark 4.28. If we consider piecewise continuous distributions, then u ∈ Cν−1p is sufficient

(see [4]).

The corresponding system in (WCF) has the form

x1 = Jx1 +B1u. (4.9a)

Nx2 = x2 +B2u. (4.9b)

Since iord(u) = q ≤ −1 we have that iord(x1) = q − 1 ≤ −1 and thus (at least) x1 ∈C0(R,Rn), i.e. there are no impulse terms in the system response x1. The system responseof the algebraic part (fast subsystem) takes the form

x2 = −ν−1∑i=0

N iB2u(i)

with iord(x2) ≤ q + ν − 1 ≤ ν − 2. From Theorem 4.26 we get that x2 = x2 − x2,j−1 is theunique solution of

N ˙x2 = x2 +B2u+Nx2j,0δτj , x2,j−1 = 0,

where x2j,0 = x2,j−1(τj) and u = u− uj . Thus,

x2 = −ν−1∑i=0

N iB2u(i) −

ν−1∑i=0

N i+1x2j,0δ(i)τj = −

ν−1∑i=0

N iB2u(i)

︸︷︷︸=P2

−ν−1∑i=1

N ix2j,0δ(i−1)τj︸︷︷︸

=P1

.

There exist impulsive terms in the state response x2 either due to the initial condition (P1)or due to possible jump behaviors in u and u(i) (P2).

58

4.1 Controllability

Theorem 4.29. Consider (4.8) with regular pair (E,A). Then there exists u ∈ C∞p such

that ∆τjx := x(τ+j ) − x(τ−j ) = w (the jump in x at τj) for some τj ∈ T if and only if

w ∈ 0⊕ Im[B2 NB2 · · · Nν−1B2

].

Proof.

”⇒” ∆τjx =

[0

∆τjx2

]since ∆τjx1 = 0 for u ∈ C∞p , τj ∈ T. Thus,

∆τjx2 = x2(τ+j )− x2(τ−j ) = −

ν−1∑i=0

N iB2

(u(i)(τ+

j )− u(i)(τ−j )).

Hence, ∆τjx2 ∈ Im[B2 NB2 · · ·Nν−1B2

].

”⇐” Choose w0, . . . , wν−1 such that w = −∑ν−1

i=0 NiB2wi. Thus, we can choose

u(t) =

{w0 + (t− τj)w1 + 1

2(t− τj)2w2 + . . .+ 1(ν−1)!(t− τj)

ν−1wν−1 if t ≥ τj ,0 if t < τj .

and ∆τjx2 = −∑ν−1

i=0 NiB2∆τju

(i) = −∑ν−1

i=0 NiB2wi = w.

We define the mapping Iτj : Rn∞ → Cnimp by Iτj (w) :=

[0

I2,τj (w)

]with

I2,τj (w) := −ν−1∑i=0

δ(i−1)τj N iw for τj ∈ T.

Note that Iτ=0(x2(0)) represents the impulse behavior in x(t) at the initial time point t0 = 0caused by the initial condition x2(0), and Iτj (w) includes all possible impulse terms in x(t)at τj .

Theorem 4.30. Consider (4.8) with regular pair (E,A). For any w ∈ Rn∞ there existsu ∈ C∞p such that the impulsive part of x at τj denoted by ximp,j is given by ximp,j = Iτj (w)if and only if

I2,τj (w) ∈[NB2 N2B2 · · ·Nν−1B2

],

i. e.(I2,τj (w)

)(φ) ∈

[NB2 N2B2 · · ·Nν−1B2

]for all φ ∈ Dn.

Proof. Since x1,imp = 0 we have ximp =

[0

x2,imp

]and ximp,j =

[0

x2,imp,j

]. Similar as before

using Theorem 4.26

x2 = −ν−1∑i=0

N iB2u(i) −

ν−1∑i=1

N ix2j,0δ(i−1)τj

59


and iord(x2) ≤ ν − 2 and hence

x2,imp,j = −ν−1∑i=1

N ix2j,0δ(i−1)τj .

Furthermore, we have I2,τj (w) =∑ν−1

i=1 δ(i−1)τj N iw. Thus, x2,imp,j = I2,τj (w) if and only if

−Nx2j,0 = Nw. Assuming that there exists an appropriate input u such that −Nx2j,0 =

Nw, we get by using x2,j,0 = x2,j−1(τj) and x2,j−1(τj) = −∑ν−1

i=0 NiB2u

(i)j−1(τj) that

w ∈ Im[B2 NB2 · · · Nν−1B2

]+ Ker(N).

We have the decomposition w = w+ w with w = Im[B2 NB2 · · ·Nν−1B2

], w ∈ Ker(N)

and w =∑ν−1

j=0 NjB2wj for some wj ∈ Rn∞ . Thus

I2,τj (w) =

ν−1∑i=1

δ(i−1)τj N i(w + w) =

ν−1∑i=1

δ(i−1)τj N i

ν−1∑j=0

N jB2wj

=ν−1∑i=1

ν−1∑j=0

δ(i−1)τj N i+jB2wj .

which implies I2,τj (w) ∈ Im[NB2 N2B2 · · · Nν−1B2

]. On the other hand, if I2,τj (w) ∈

Im[NB2 N2B2 · · · Nν−1B2

], then

N iw ∈ Im[NB2 N2B2 · · · Nν−1B2

]= N Im

[B2 NB2 · · · Nν−2B2

]for i = 1, . . . , ν − 1 and in particular

Nw ∈ N Im[B2 NB2 · · · Nν−2B2

]⇐⇒ w ∈ Im

[B2 NB2 · · · Nν−1B2

]+Ker(N).

We can find an appropriate u by using Theorem 4.29.

Definition 4.31. The system (4.8) is called impulse controllable ( I-controllable) if for anyinitial condition x(0), τj ∈ T and w ∈ Rn∞ there exists an admissible control functionu ∈ C∞p such that ximp,j = Iτj (w).

Theorem 4.32. The following statements are equivalent.

1. The system (4.8) is I-controllable.2. The fast subsystem (4.9b) is I-controllable.3. Ker(N) + Im

[B2 NB2 · · · Nν−1B

]= Rn∞.

4. Im(N) = Im[NB2 N2B2 · · · Nν−1B2

].

5. Im(N) + Im(B2) + Ker(N) = Rn∞.

6. rank

[E 0 0A E B

]= n+ rank(E).

Proof.

60

4.1 Controllability

1 ⇐⇒ 2 Clear since ximp =

[0

ximp,2

].

2 ⇐⇒ 3 Follows from Theorem 4.30.

3 ⇐⇒ 4 Follows since

Im[NB2 N2B2 · · · Nν−1B2

]= N Im

[B2 NB2 · · · Nν−1B2

]= N

(Im[B2 · · · Nν−1B2

]+ Ker(N)

).

4 =⇒ 5 Follows from

Im(N) + Im(B2) + Ker(N) = Im[NB2 · · · Nν−1B2

]+ Im(B2) + Ker(N)

= Im[B2 NB2 · · · Nν−1B2

]+ Ker(N) = Rn∞ ,

where the last equality follows from 3.

5 =⇒ 3 It holds that Im(N i+1) + Im(N iB2) = Im(N i) for i = 1, . . . , ν − 1. This implies

Rn∞ = Ker(N) + Im(B2) + Im(NB2) + . . .+ Im(Nν−1B2) + Im(Nν)

= Ker(N) + Im[B2 NB2 · · · Nν−1B2

].

4 ⇐⇒ 6 Consider the Kalman decomposition [5] of the matrix pair (N,B2), i. e. thereexists a nonsingular matrix V such that

(N,B2) ∼ (V −1NV, V −1B2) =

([N11 N12

0 N22

],

[B21

0

])n1

n2

with (N11, B21) is C-controllable, i. e. Im[B21 N11B21 · · · Nν−1

11 B21

]= Rn1 .

Thus,

Im(N) = Im

([N11 N12

0 N22

])(4)= Im

[NB2 N2B2 · · ·Nν−1B2

]= Im

[N11

0

]⇐⇒ n2 = 0 or N22 = 0.

61


Moreover, it holds that

rank

([E 0 0A E B

])= rank

([WET 0 0WAT WET WB

])

= rank

Inf 0 0 0 0

0 N 0 0 0J 0 Inf 0 B1

0 In∞ 0 N B2

= 2nf + rank

([N 0 0In∞ N B2

])

= 2nf + rank

N11 N12 0 0 0

0 N22 0 0 0In1 0 N11 N12 B21

0 In2 0 N22 0

.

We know that rank[N11 B21

]= n1. Hence, we have

rank

([E 0 0A E B

])= 2nf + n1 + n2 + rank

([N11 −N12N22

0 −N222

])= n+ nf + rank

([N11 N12N22

0 N222

]).

Thus rank

([E 0 0A E B

])= n+ rank(E) = n+nf + rank

([N11 N12

0 N22

])if and only

if

rank

([N11 N12

0 N22

])= rank

([N11 N12N22

0 N222

]).

Since N22 is nilpotent, this consequently holds if and only if n2 = 0 or N22 = 0.

Example 4.33. Consider again the descriptor system from Example 4.13, given by

x1 =

[1 10 1

]x1 +

[01

]u

0 = x2 +

[−10

]u.

We have already seen that the system is R-controllable (Example 4.16) but not C-controllable(Example 4.13). We have

Im(N) + Ker(N) + Im(B2) = Im

[0 00 0

]+ Ker

[0 00 0

]+ Im

[−10

]= {0}+ R2 + Im

[−10

]= R2.

62

4.1 Controllability

This implies that the system is I-controllable. Alternatively, we have

rank

([E 0 0A E B

])= rank

I2 0 0 0 00 0 0 0 0∗ 0 I2 0 e2

0 I2 0 0 −e1

= 6 = n+ rank(E).

Theorem 4.34. The following statements are equivalent.

1. The system (4.8) is I-controllable.2. rank

([E AS∞ B

])= n where S∞ is a matrix with Im(S∞) = Ker(N).

3. There exists F ∈ Rm,n such that (E,A+BF ) is regular and ν = ind(E,A+BF ) ≤ 1.4. rank

([N K∞ B2

])= n∞ where Im(K∞) = Ker(N).

5. rank

([N 0 0In∞ N B2

])= n∞ + rank(N).

Proof.

1 ⇐⇒ 2 As in the proof of the Theorem on feedback regularization (Theorem 3.1) thereexist nonsingular matrices P and Q such that

PEQ =

[Ir 00 0

], PAQ =

[A11 A12

A21 A22

], PB =

[B1

B2

]and rank(E) = r, Q =

[Q1 Q2

]. In particular, Q2 = S∞. Thus, we have

rank([E AS∞ B

])= rank

([Ir 0 A12 B1

0 0 A22 B2

])= r + rank

([A22 B2

]).

Analogously,

rank

([E 0 0A E B

])= rank

Ir 0 0 0 00 0 0 0 0

A11 A12 Ir 0 B1

A21 A22 0 0 B2

= 2r + rank

([A22 B2

]).

Thus, we have rank([E AS∞ B

])= n if and only if rank

([A22 B2

])= n − r if

and only if rank

([E 0 0A E B

])= n+ r.

2 ⇐⇒ 3 Clear (see Section 3.1).

2 ⇐⇒ 4 Let S∞ such that ES∞ = 0, then WETT−1S∞ =

[Inf 0

0 N

]T−1S∞ = 0 with

T−1S∞ =

[S1

S2

]. In particular, we have S1 = 0 and NS2 = 0, which implies that

S2 = K∞. The rest follows immediately as before.

63


1 ⇐⇒ 5 Follows from the previous theorem.

Definition 4.35. A system (4.8) is called strongly controllable (S-controllable) if it is R-controllable and I-controllable, i. e. if rank

[λE −A B

]= n for all finite λ ∈ C and

rank[E AS∞ B

]= n, where Im(S∞) = ker(E).

The relationship between the various controllability concepts can be presented as

=⇒ R-controllabilityC-controllability =⇒ S-controllability =⇒ R- and I-controllability

=⇒ I-controllability

Remark 4.36.1. The ability to cancel all impulses in the system response by choosing a suitable state

feedback control such that the resulting closed-loop system is regular and of indexν ≤ 1 is often used as definition for I-controllability. With this regard, linear time-varying and nonlinear descriptor systems can be handled as in chapter 3.

2. C-controllability and R-controllability for linear time-varying descriptor systems canbe treated via appropriate staircase forms (see chapter 5).

3. Nonlinear descriptor systems are usually handled by local linearization.

4.2 Observability

Definition 4.37. The descriptor system (1.1) is called completely observable (C-observable)if the initial condition x(0) can be uniquely determined by u(t) and y(t) for 0 ≤ t <∞.

This means that for a C-controllable system the state x(t) can be uniquely determined fromu and y by observing the initial condition and constructing the system response at any timet ≥ 0.

Definition 4.38 (Alternative definition). The system (1.1) is C-observable if the zerooutput y(t) ≡ 0 with u(t) ≡ 0 implies that the system has only the trivial solution x(t) ≡ 0.

Definition 4.39. The system (1.1) is called observable within the reachable set (R-observable)if any state in the reachable set can be uniquely determined by y(t) and u(t) for t ≥ 0.

Thus, we need an appropriate projection to variables that are associated with the dynamicalpart of the system. For general nonlinear systems this is difficult to obtain.

64

4.2 Observability

Remark 4.40.1. R-observability is sometimes also called finite dynamics observability.2. While C-observability reflects the reconstruction ability of the whole state x(t) from

measured output together with the control input, R-observability characterized theability to reconstruct only the reachable states. Thus, C-observability implies R-observability.

Definition 4.41. A descriptor system (1.1) is called impulse observable (I-observable) if theimpulse behavior in the state response x(t) can be uniquely determined from the impulsebehavior of the output and jump behavior in the input.

Theorem 4.42. Consider a regular linear descriptor system of the form (4.1).

1. Let u(t) ≡ 0. Then y(t) ≡ 0 for all t ≥ 0 if and only if

x0 ∈ Ker

C1

C1J...

C1Jnf−1

⊕Ker

C2

C2N...

C2Nν−1

, x = T−1x =

[x1

x2

]as in (4.2).

2. The slow subsystem (4.2a) is C-observable if and only if rank

[λE −AC

]= n for all

finite λ ∈ C.3. The following statements are equivalent:

a) The fast subsystem (4.2b) is C-observable.

b) rank

C2

C2N...

C2Nν−1

= n∞.

c) Ker

[NC2

]= {0}.

d) rank

[NC2

]= n∞.

e) rank

[EC

]= n.

f) For any two nonsingular matrices P1, Q1 satisfying

P1EQ1 =

[I 00 0

], let CP1 =

[C1 C2

].

Then C2 is of full column rank, rank(C2) = n− rank(E).4. The following statements are equivalent:

a) The system (4.1) is C-observable.b) The slow and fast subsystem (4.2a) and (4.2b) are both C-observable.

c) rank

[λE −AC

]= n for all λ ∈ C and rank

[EC

]= n.

65


d) rank

[αE − βA

C

]= n for all (α, β) ∈ C2\{(0, 0)}.

Proof.

1. For u(t) ≡ 0 the state response of (4.2) is given by x1(t) = eJtx1(0) and x2(t) =

−∑ν−1

i=1 Nix2(0)δ

(i−1)0 and y(t) = y1(t) + y2(t) = C1x1(t) + C2x2(t). Thus

y(t) ≡ 0 ⇐⇒ C1x1(t) + C2x2(t) = 0 for all t ≥ 0

⇐⇒ C1x1(t) = 0 and C2x2(t) = 0 for all t ≥ 0,

where the last iff follows from the decomposition in the smooth and impulsive part.

If y1(t) = C1x1(t) = C1eJtx1(0) ≡ 0 for all t ≥ 0, then also y

(i)1 (t) ≡ 0 for all

i = 0, . . . , nf − 1 and all t ≥ 0. In particular, we have y(i)1 (0) = 0 and hence

C1

C1J...

C1Jnf−1

x1(0) = 0 =⇒ x1(0) ∈ Ker

C1

C1J...

C1Jnf−1

.Moreover,

y2(t) = C2x2(t) = −ν−1∑i=1

C2Nix2(0)δ

(i−1)0 ≡ 0

⇐⇒ C2Nix2(0) = 0 for i = 0, . . . , ν − 1

⇐⇒ x2(0) ∈ Ker

C2

C2N...

C2Nν−1

.

Thus, x(0) =

[x1(0)x2(0)

]∈ Ker

C1

C1J...

C1Jnf−1

⊕Ker

C2

C2N...

C2Nν−1

.

2. The slow subsystem is a standard LTI system. Thus, (4.2a) is C-observable if and only

if (J,C1) is C-observable if and only if rank

[λI − JC1

]= nf for all λ ∈ C. Furthermore

rank

[λE −AC

]= rank

[λWET −WAT

CT

]= rank

λI − J 00 λN − IC1 C2

= n∞ + rank

[λI − JC1

].

66

4.2 Observability

3. By definition C-observability of the fast subsystem means that for y2(t) ≡ 0 for t ≥ 0

with u(t) ≡ 0 it follows that x2(0) = 0. By 1 this is equivalent to Ker

C2

C2N...

C2Nν−1

= {0}

and hence 3a ⇐⇒ 3b. Furthermore,

rank

C2

C2N...

C2Nν−1

= n∞ ⇐⇒ rank[CT2 NTCT2 · · · (NT )ν−1C2

]= n∞

⇐⇒ NT ξ2 = ξ2 + CTu is C-controllable (Theorem 4.12).

The above system is called the dual system of (4.2b). By Theorem 4.12 we have that3b-3f are equivalent.

4. Follows from 1 and similar as in 3 by Theorem 4.12.

Example 4.43. Consider again the descriptor system from Example 4.13 with additionaloutput equation, given by

x1 =

[1 10 1

]x1 +

[01

]u

0 = x2 +

[−10

]u

y =[1 0

]x1.

Since

rank

[C1

C1J

]= rank

[1 01 1

]= 2 = nf and rank

[C2

C2N

]= rank

[0 00 0

]= 0 < n∞,

we find that the slow subsystem is C-observable while the fast subsystem is not.

Theorem 4.44. Consider a regular linear system of the form (4.1). Then the system (4.1)is R-observable if and only if the slow subsystem (4.2a) is C-observable, i. e.

rank

[λE −AC

]= n for all finite λ ∈ C.

67


Proof. Any reachable state x(t) = T

[x1(t)x2(t)

]has the form

x1(t) = eJtx1(0) +

∫ t

0eJ(t−s)B1u(s)ds,

x2(t) = −ν−1∑i=0

N iB2u(i)(t),

y(t) = y1(t) + y2(t) = C1x1(t) + C2x2(t),

i. e. x2(t) is uniquely determined by u(t) and y1(t) = C1x1(t) = y(t)− C2x2(t) is uniquelydetermined by y(t) and u(t). Thus, a reachable state x(t) can be reconstructed from y(t)and u(t) if and only if x1(t) can be uniquely determined by y1(t) and u(t), i. e. the slowsubsystem (4.2a) is C-observable.

Corollary 4.45. The system (4.1) is C-observable if and only if the system is R-observable

and rank

[EC

]= n.

To prove the dual properties for I-observability we use the following lemma.

Lemma 4.46. Consider a regular system in WCF (4.2). Then, the impulsive part of theoutput at τj ∈ T is yimp,j ≡ 0 for all j ∈ Z for u(t) ≡ 0 if and only if

x2j,0 ∈ Ker

C2NC2N

2

...C2N

ν−1

, x2j,0 = x2,j−1(τj), τj ∈ T.

Proof. Since yimp,j = Cximp,j = C2x2,imp,j we have yimp,j = −∑ν−1

i=1 C2Nix2j,0δ

(i)τj for all

τj ∈ T. Thus, yimp,j ≡ 0 if and only if C2Nix2j,0 = 0 for i = 1, . . . , ν − 1 if and only if

x2j,0 ∈ Ker

C2NC2N

2

...C2N

ν−1

.

Theorem 4.47. Consider system (4.1). Then, the following statements are equivalent.

1. The system (4.1) is I-observable.2. The fast subsystem (4.2b) is I-observable.

3. Ker

C2

C2N...

C2Nν−1

∩ Im(N) = {0}.

68

4.2 Observability

4. Ker

C2N...

C2Nν − 1

= Ker(N).

5. Ker(N) ∩Ker(C2) ∩ Im(N) = {0}.6. Let ([

N>11 N>21

0 N>22

],

[C>21

0

])n1

n2

be the Kalman decomposition [5] of (N>, C>2 ), where (N11, C21) is C-observable. Then

either n2 = 0 or N22 = 0 and rank(N11) = rank

[N11

N21

].

7. rank

E A0 E0 C

= n+ rank(E).

Proof. Assume that u ∈ C∞p . From ximp =

[0

x2,imp

]and

yimp = C1x1,imp + C2x2,imp = C2x2,imp = y2,imp.

Hence, we know 1 ⇐⇒ 2.

2 ⇐⇒ 4 For u(t) ≡ 0 we have x2,imp = −∑ν−1

i=1 Nix2j,0δ

(i−1)τj ≡ 0 ⇐⇒ Nx2j,0 = 0. Thus

from Lemma 4.46 we get 2 ⇐⇒ Ker(N) = Ker

C2NC2N

2

...C2N

ν−1

⇐⇒ 4.

3 ⇐⇒ 4 Assume that Ker

C2N...

C2Nν−1

= Ker(N). Then for any

w ∈ Ker

C2...

C2Nν−1

∩ Im(N),

there exists β such that

w = Nβ ∈ Ker

C2...

C2Nν−1

=⇒ β ∈ Ker

C2N...

C2Nν−1

= Ker(N).

69


Conversely, assume that

Ker

C2

C2N...

C2Nν−1

∩ Im(N) = {0}.

Then, for any w ∈ Ker

C2N...

C2Nν−1

it holds that Nw ∈ Ker

C2...

C2Nν−1

⊆ Ker(N).

Since Ker(N) ⊆ Ker

C2N...

C2Nν−1

the equality follows.

3 ⇐⇒ 5 Similar as before.

Assume that Ker(N) = Ker

CN2...

C2Nν−1

. Since Ker(N) + Im(N>) = Rn∞ we have that

Im(N>) = Im[N>C>2 · · ·

(N>)ν−1

C>2

]. Then the equivalence of 4,5,6 and 7 follows

from Theorem 4.32 (I-controllability).

Definition 4.48. A descriptor system (4.1) is called strongly observable (S-observable) ifit is R-observable and I-observable.

Note that controllability and observability are dual concepts, i. e. the following resultshold:

Theorem 4.49 (Duality Principle). Consider a linear descriptor system of the form (4.1).The the following holds:

1. The system (4.1) is C-controllable if and only if the dual system given by

E>ξ = A>ξ + C>u

y = B>ξ(4.10)

is C-observable.2. The system (4.1) is R-controllable if and only if the dual system (4.10) is R-observable.3. The system (4.1) is I-controllable if and only if the dual system (4.10) is I-observable.

Proof. Follows from the previous discussion.

70

4.2 Observability

The relationship between the different observability concepts are as follows:

=⇒ R-observabilityC-observability =⇒ S-observability =⇒ R- and I-observability

=⇒ I-observability

Corollary 4.50. The following statements are equivalent:

1. The system (4.8) is I-observable.

2. rank

ET>∞AC

= n, where Im(T∞) = Ker(E>).

3. rank

NK∞C2

= n∞, where Im(K∞) = Ker(N>).

Remark 4.51. For the existence of a feedback F ∈ Rm,p such that (E,A+BFC) is regularand of index ν ≤ 1 we need I-observability and I-controllability.

71

CHAPTER

5

STAIRCASE FORMS AND SYSTEMPROPERTIES

The results presented in the previous chapters that are based on the Weierstraß canonicalform (WCF) are useful from a theoretical point of view. However, it is well known thatthe numerical computation of the (WCF) in finite precision is ill-conditioned and smallperturbations can radically change the kind and number of blocks in the (WCF).

A better numerical way are staircase algorithms that use a sequence of rank decisions andtransformations with orthogonal matrices to transform the system into a suitable condensedform.

We consider linear descriptor systems of the form

Ex = Ax+Bu

y = Cx(5.1)

with E,A ∈ Rl,n, B ∈ Rl,m and C ∈ Rp,n.

Lemma 5.1. There exist orthogonal matrices P ∈ Rl,l and Q ∈ Rn,n such that

PEQ =

E11 0 E13

0 0 00 0 0

rs

l − r − s, PAQ =

A11 A12 A13

A21 A22 A23

0 0 A33

, PB =

B1

B2

0

,CQ =

[C1 C2 C2

]with r = rank(E) and s = rank(B2).

73

5 Staircase forms and system properties

Proof. By row compression there exists an orthogonal matrix P ∈ Rl,l such that

P[E B A

]=

E11 E12 E13 B1 A11 A12 A23

0 0 0 B2 A21 A22 A23

0 0 0 0 A31 A32 A33

rs

l − r − s

with r = rank(E) and s = rank(B2) (e.g. by using SVD or QR decomposition). Then, weconsider the matrix [

A33 A31 A32

E13 E11 E12

].

There exists an orthogonal matrix Q ∈ Rn,n such that[A33 A31 A32

E13 E11 E12

]Q =

[A33 0 0E13 E11 0

] [q = l − s− r

r

](via column compression). Setting

Q =

0 I 00 0 II 0 0

Q0 0 II 0 00 I 0

it follows that PEQ, PAQ, PB are in the desired form.

Theorem 5.2. Consider (E,A,B,C) with E,A ∈ Rn,n, B ∈ Rn,m, C ∈ Rp,n (i. e. l = n).Then there exist orthogonal matrices P and Q such that

PEQ =

E11 0 E13

0 0 E23

0 0 E33

t1t2t3

, PAQ =

A11 A12 A13

A21 A22 A23

0 0 A33

, PB =

B1

B2

0

,CQ =

[C1 C2 C2

],

(5.2)

where t2 = n− t1 − t3 and

1. rank(E11) = t1,2. rank(B2) = t2,3. A33 is a block upper triangular matrix with square diagonal blocks and4. E33 is a block upper triangular matrix with zero diagonal blocks and the same block

decomposition as A33.

Proof. We inductively apply Lemma 5.1 starting with

P (1)EQ(1) =

E(1)11 0 E

(1)13

0 0 00 0 0

rsq, P (1)AQ(1) =

A(1)11 A

(1)12 A

(1)13

A(1)21 A

(1)22 A

(1)23

0 0 A(1)33

, P (1)B =

B(1)1

B(1)2

0

,CQ(1) =

[C

(1)1 C

(1)2 C

(1)2

].

74

If rank(E(1)11 ) = r we have the desired form. Otherwise, we apply Lemma 5.1 for

E =

[E

(1)11 00 0

], A =

[A

(1)11 A

(1)12

A(1)21 A

(1)22

], B =

[B

(1)1

B(1)2

], C =

[C

(1)1 C

(1)2

]and obtain P and Q. Setting

P (2) =

[P 00 I

]P (1) and Q(2) = Q(1)

[Q 00 I

]we get

P (2)EQ(2) =

E

(2)11 0 E

(2)13 ∗

0 0 0 ∗0 0 0 ∗0 0 0 ∗

, P (2)AQ(2) =

A

(2)11 A

(2)12 A

(2)13 A

(2)14

A(2)21 A

(2)22 A

(2)23 A

(2)24

0 0 A(2)33 A

(2)34

0 0 0 A(2)44

.

We can continue inductively until E(k)11 has full rank. In every step the size of the blocks

E(k)11 is decreased at least by 1, thus we obtain the desired form after a maximum of n

steps.

As a consequence of Theorem 5.2 we can separate the parts of the system (5.1) that are notI-controllable.

Corollary 5.3. Consider the system (5.1) with matrices E,A,B transformed as in (5.2)(Theorem 5.2). Then the subsystem consisting of[

E11 00 0

] [x1

x2

]=

[A11 A12

A21 A22

] [x1

x2

]+

[B1

B2

]u

is I-controllable.

Proof. With S∞ =[0 I

]>we have

rank([E AS∞ B

])= rank

([E11 0 A12 B1

0 0 A22 B2

])= t1 + t2

of full rank.

If we want to use output feedback control we also have to consider the matrix C.

75


Theorem 5.4. Consider (E,A,B,C) as in Theorem 5.2. Then there exist orthogonalmatrices P and Q such that

PEQ =

E11 0 0 E14

0 0 0 E24

E31 E23 E33 E34

0 0 0 E44

t1t2t3t4

, PAQ =

A11 A12 0 A14

A21 A22 0 A24

A31 A32 A33 A34

0 0 0 A44

PB =

B1

B2

B3

0

t1t2t3t4

, CQ =[C1 C2 0 C4

](5.3)

with

1. rank(E11) = t1,2. rank(C2) = t2,3. A33 and A>44 are block lower triangular matrices with square diagonal blocks,4. E33 and E>44 are block lower triangular matrices with zero diagonal blocks and the

same dimensions as the diagonal blocks of A33 and A>44,5. the subsystem consisting of[

E11 00 0

] [x1

x2

]=

[A11 A12

A21 A22

] [x1

x2

]+

[B1

B2

]u

y =[C1 C2

] [x1

x2

]is I-controllable and I-observable and

6. the subsystem consisting ofE11 0 00 0 0

E31 E32 E33

x1

x2

x3

=

A11 A12 0

A21 A22 0

A31 A32 A33

x1

x2

x3

+

B1

B2

B3

uy =

[C1 C2 0

] x1

x2

x3

is I-controllable.

Proof. At first we determine P1 and Q1 such that P1EQ1, P1AQ1, P1B and CQ1 are in theform (5.2) of Theorem 5.2. Then, we consider the subsystem consisting of

E =

[E11 00 0

]>, A =

[A11 A12

A21 A22

]>, B =

[C1 C2

]>

76

and apply Theorem 5.2 for (E, A, B), i.e. we determine P and Q such that P EQ, P AQand P B are in the form (5.2). Setting

P2 =

[Q> 00 It3

], Q2 =

[P> 00 It3

]and P = P2P1, Q = Q1Q2 we get the desired form (5.3). The properties 1-4 follow directlyfrom Theorem 5.2. Properties 5 and 6 follow from Corollary 5.3 and the duality principle(Theorem 4.49).

Based on the previous results we can decide if we can regularize the system by feedbackcontrol

Theorem 5.5. Consider the system (5.1) with matrices (E,A,B,C) transformed as in(5.2). Then there exists a matrix F ∈ Rm,n such that the pair (E,A+BF ) is regular if andonly if A33 is nonsingular.

Proof. Let F be partitioned as F =[F1 F2 F3

]. Then we have

det(λE − (A+BF )) = det

λE11 0 E13

0 0 E23

0 0 E33

−A11 A12 A13

A21 A22 A23

0 0 A33

−B1

B2

0

[F1 F2 F3

]= det

λE11 −A11 −B1F1 −A12 −B1F2 λE13 −A13 −B1F3

−A21 −B2F1 −A22 −B2F2 λE23 −A23 −B2F3

0 0 λE33 −A33

= det(λE33 −A33) det

(λ

[E11 00 0

]−[A11 +B1F1 A12 +B1F2

A21 +B2F1 A22 +B2F2

]).

If A33 is singular, we have

det(λE33 −A33) = det

λ

0 ∗ . . . ∗. . .

. . ....

. . . ∗0

−∗ . . . . . . ∗

. . ....

. . ....∗

= 0

for all λ ∈ C. Otherwise, if A33 is nonsingular, we have det(λE33 − A33) 6= 0 for allλ ∈ C. Since B2 has full rank there exists F2 such that A22 + B2F2 is nonsingular. LetF =

[0 F2 0

], then

det(λE − (A+BF )) = det(λE33 −A33) det

(λ

[E11 00 0

]−[A11 +B1F1 A12 +B1F2

A21 +B2F1 A22 +B2F2

])= det(λE33 −A33) det(λE11 − A11)︸︷︷︸

6≡0

det(A22 +B2F2)︸︷︷︸6=0

6≡ 0.

77


Theorem 5.6. Consider (5.1) with (E,A,B,C) given in the form (5.2) and A33 is non-singular. Then there exists F ∈ Rm,n such that (E,A+BF ) is regular and

ind(E,A+BF ) = ind

([0 E23

0 E33

]).

Proof. Let F =[F1 F2 F3

], then we have

(E,A+BF ) =

E11 0 E13

0 0 E23

0 0 E33

,A11 +B1F1 A12 +B1F2 A13 +B1F3

A21 +B2F1 A22 +B2F2 A23 +B2F3

0 0 A33

.

Since B2 has full rank we can choose F1 ∈ Rm,t1 and F2 ∈ Rm,t2 such that A21 +B2F1 = 0and A22 +B2F2 is nonsingular. With F3 = 0 we get

(E,A+BF ) =

E11 0 E13

0 0 E23

0 0 E33

,A11 +B1F1 A12 +B1F2 A13

0 A22 +B2F2 A23

0 0 A33

,

where (E11, A11 +B1F1) is regular and of index 0 since E11 is nonsingular. For the secondblock we have A22 +B2F2 and A33 nonsingular such that the index is given by

ind(E,A+BF ) = ind

([0 E23

0 E33

]).

Remark 5.7.

1. The index ind

([0 E23

0 E33

])is the minimal index we can reach by state feedback. If

we want to obtain a closed-loop system of index 1 we need that

[E23

E33

]= 0 (and A33

is nonsingular).

2. If[E>23 E>33

]> 6= 0 and A33 nonsingular, we have to consider a system of the formE11 0 E13

0 0 E23

0 0 E33

x1

x2

x3

=

A11 A12 A13

A21 A22 A23

0 0 A33

x1

x2

x3

+

B1

B2

0

u.From the last block equation we get A−1

33 E33x3 = x3 and A−133 E33 is block upper

triangular with zero blocks on the diagonal and therefore we have x3 ≡ 0. This meansthat the part of the system that is not I-controllable, i. e. x3, is not problematic sinceit vanishes identically (in particular is stable). The remaining system is I-controllable(see Corollary 5.3).

3. If A33 is not invertible the system cannot be regularized. In general this is assumedto be a modeling error.

78

Theorem 5.8. Consider (5.1) with (E,A,B,C) given in the form (5.2) and A33 nonsingu-

lar and[A>22 C>2

]>of full column rank. Then there exists F ∈ Rm,p such that (E,A+BFC)

is regular and

ind(E,A+BFC) = ind

([0 E23

0 E33

]).

Proof. Since B2 has full row rank and[A>22 C>2

]>has full column rank, there exists F

such that A22 + B2FC2 is nonsingular. Then the result follows in the same way as in theproof of Theorem 5.5 and Theorem 5.6.

Remark 5.9. In the case of output feedback we can consider the system in the form (5.3)of Theorem 5.4 given by

E11 0 0 E14

0 0 0 E24

E31 E23 E33 E34

0 0 0 E44

x1

x2

x3

x4

=

A11 A12 0 A14

A21 A22 0 A24

A31 A32 A33 A34

0 0 0 A44

x1

x2

x3

x4

+

B1

B2

B3

0

u

y =[C1 C2 0 C4

] x1

x2

x3

x4

.

Similar as before the system can be regularized by output feedback if and only if A33 andA44 are nonsingular. In this case we get x4 ≡ 0, i. e. the part that is not I-controllablevanishes identically. The part x3 belonging to the third block equation is dangerous sincethere can occur derivatives of the input u that cannot be observed if the index is not 0 (i. e.t3 > 0 and E33 6= 0).

Justified by the previous discussion it is often assumed that the system (5.1) is I-controllableand I-observable. If this is not the case then the parts that are not I-controllable and notI-observable can be removed using the transformation into the form (5.2) or (5.3). However,under the presence of rounding errors these components might still cause trouble.

Theorem 5.10 (Staircase form for linear descriptor systems, [2]). Let E,A ∈ Rn,n, B ∈Rn,m and C ∈ Rp,n. Then there exist orthogonal matrices U, V ∈ Rn,n, W ∈ Rm,m and

79


Y ∈ Rp,p such that

U>EV =

[ t1 n− t1t1 ΣE 0

n− t1 0 0

], U>AV =

t1 s2 t5 t4 t3 s6

t1 A11 A12 A13 A14 A15 A16

t2 A21 A22 A23 A24 0 0

t3 A31 A32 A33 A34 Σ35 0

t4 A41 A42 A43 Σ44 0 0

t5 A51 0 Σ53 0 0 0

t6 A61 0 0 0 0 0

,

U>BW =

k1 k2 k3

t1 B11 B12 0

t2 B21 0 0

t3 B31 0 0

t4 0 0

t5 0 0

t6 0 0

, Y >CV =

t1 s2 t5 t4 t3 s6

l1 C11 C12 C13 0 0 0

l2 C21 0 0 0 0 0

l3 0 0 0 0 0 0

(SCF)and the matrices ΣE ,Σ35,Σ44,Σ53 are nonsingular diagonal, B12 has full column rank, C21

has full row rank and the matrices[B21

B31

]∈ Rk1,k1 and

[C12 C13

]∈ Rl1,l1

with k1 = t2 + t3 and l1 = s2 + t5 are nonsingular.

Before we prove this Theorem, we draw several conclusions.

Corollary 5.11. Let E,A be in staircase form (SCF). Then the following statements areequivalent:

1. The pair (E,A) is regular and ν = ind(E,A) ≤ 1.2. s6 = t6 = 0 and A22 nonsingular.3. rank

[E AS∞

]= n, where range(S∞) = Ker(E).

4. rank

[E

T>∞A

]= n, where range(T∞) = Ker(E>).

Proof. We have

[E AS∞

]=

[ΣE 0 A12

0 0 A22

]and

[E

T>∞A

]=

ΣE 00 0

A21 A22

,

80

where

A12 =[A12 A13 A14 A15 A16

]

A22 =

A22 A23 A24 0 0A32 A33 A34 Σ35 0A42 A43 Σ44 0 00 Σ53 0 0 00 0 0 0 0

, A21 =

A21

A31

A41

A51

A61

.

Thus, the equivalences 2, 3 and 4 follows.

2 =⇒ 1 If s6 = t6 = 0 and A22 nonsingular, then A22 is nonsingular. Thus,

(E,A) ∼([

ΣE 00 0

],

[A11 − A12A22A21 0

0 I

]),

which is nonsingular and of index ν ≤ 1.1 =⇒ 3 If (E,A) is regular and of index ν ≤ 1 it holds

(E,A) ∼([I 00 0

],

[J 00 I

])and hence rank

([E AS∞

])= n.

Corollary 5.12. Let (E,A,B,C) be in staircase form (SCF). Then it holds:

1. The system (5.1) is I-controllable if and only if t6 = 0.2. The system is I-observable if and only if s6 = 0.3. rank

[E B

]= n if and only if t4 = t5 = t6 = 0.

4. rank

[EC

]= n if and only if t4 = t3 = s6 = 0.

5. The system (5.1) is C-controllable if and only if t4 = t5 = t6 = 0 and the system (5.1)is R-controllable (i. e. rank

[λE −A B

]= n for all λ ∈ C).

6. The system (5.1) is C-observable if and only if t4 = t3 = s6 = 0 and the system (5.1)is R-observable.

Proof. Follows directly from the staircase form (SCF).

Remark 5.13. We assume without loss of generality that k3 = l3 = 0 in the staircase form(SCF) since these parts have no influence on the system and can be omitted by defining anew input u or a new output y.

For the proof of Theorem 5.10 we can proceed using the Algorithm 1.

81


Algorithm 1 Staircase Form

Input: E,A ∈ Rn,n, B ∈ Rn,m, C ∈ Rp,nOutput: orthogonal matrices U, V ∈ Rn,n, W ∈ Rm,m, Y ∈ Rp,p such that U>EV ,U>AV , U>BW and Q>CV are in staircase form (SCF).

Set U := In, V := In,W := Im, Y := Ip.Step 1 Compute a SVD of E

E = UE

[ΣE 00 0

]V >E

with ΣE of size t1 × t1 nonsingular and diagonal. Update

E := U>EEVE =

[ t1 n− t1t1 ΣE 0

n− t1 0 0

], A := U>EAVE =

[ t1 n− t1t1 A

(1)11 A

(1)12

n− t1 A(1)21 A

(1)22

],

B := U>EB =

[ m

t1 B(1)1

n− t1 B(1)2

], C := CVE =

[ t1 n− t1p C

(1)1 C

(1)2

]

and set U := UUE , V := V VE .

Step 2 Compute SVDs of B(1)2 and C

(1)2 :

B(1)2 = UB

[ΣB 00 0

]V >B , C

(1)2 = UC

[ΣC 00 0

]V >C

with ΣB of size k1 × k1, ΣC of size l1 × l1 nonsingular and diagonal. Define

U1 =

[It1 00 U>B

]and V1 =

[It1 00 VC

]

82

and perform the updates

E := U1EV1 =

t1 l1 n− t1 − l1

t1 ΣE 0 0

k1 0 0 0

n− t1 − k1 0 0 0

,

A := U1AV1 =

t1 l1 n− t1 − l1

t1 A(2)11 A

(2)12 A

(2)13

k1 A(2)21 A

(2)22 A

(2)23

n− t1 − k1 A(2)31 A

(2)32 A

(2)33

,

B := U1BVB =

t1 m− t1

t1 B(2)11 B

(2)12

k1 ΣB 0

n− t1 − k1 0 0

,

C := U>CCV1 =

[ t1 l1 n− t1 − l1l1 C

(2)11 ΣC 0

p− l1 C(2)21 0 0

]

and U := UU1, V := V V1, Y := Y UC ,W := WVB.

Step 3 Compute SVDs of B(2)12 and C

(2)21 :

B(2)12 = U12

[Σ12 00 0

]V >12 , C

(2)21 = U21

[Σ21 00 0

]V >21

with Σ12 of size k2 × k2 and Σ21 of size l2 × l2. Update

B :=

[Ik1 00 V12

]=

k1 k2 k3

t1 B(3)11 B

(3)12 0

k1 ΣB 0 0

n− t1 − k1 0 0 0

,

C :=

[Il1 00 U>21

]C =

t1 l1 n− t1 − l1

l1 C(3)11 ΣC 0

l2 C(3)21 0 0

l3 0 0 0

Step 4 Compute SVD of A

(2)33 :

A(2)33 = UA

[Σ44 00 0

]V >A

with Σ44 of size t4 × t4 nonsingular an diagonal. Set n4,1 = n− t1 − k1 − t4,

83


n4,2 = n− t1 − l1 − t4 and perform the following updates:

E :=

It1 Ik1U>A

EIt1 Il1

VA

=

t1 l1 t4 n4,1

t1 ΣE 0 0 0

k1 0 0 0 0

t4 0 0 0 0

n4,2 0 0 0 0

A :=

It1 Ik1U>A

AIt1 Il1

VA

=

A

(33)11 A

(33)12 A

(33)13 A

(33)14

A(33)21 A

(33)22 A

(33)23 A

(33)24

A(33)31 A

(33)32 Σ44 0

A(33)41 A

(33)42 0 0

B :=

It1 Ik1U>A

B =

k1 k2 k3

t1 B(3)11 B

(3)12 0

k1 ΣB 0 0

t4 0 0 0

n4,2 0 0 0

C := C

It1 Il1VA

=

t1 l1 t4 n4,1

l1 C(3)11 ΣC 0 0

l2 C(3)21 0 0 0

l3 0 0 0 0

U := U

It1 Ik1UA

, V := V

It1 Il1VA

Step 5 Compute a permuted SVD of A(3)42 and A

(3)24 :

A(3)42 = U42

[0 Σ53

0 0

]V >42 , A

(3)24 = U24

[0 0

Σ35 0

]V >24

with Σ53 of size t5 × t5 and Σ35 of size t3 × t3 nonsingular and diagonal. We sett2 = k1 − t3, t6 = n−

∑5i=1 ti and

V5 =

It1

V42

It4V24

and U5 =

It1

U24

It4U42

84

perform the updates

E := U>5 EV5 =

ΣE 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

A := U>5 AV5 =

A11 A12 A13 A14 A15 A16

A21 A22 A23 A24 0 0A31 A32 A33 A34 Σ35 0A41 A42 A43 Σ44 0 0A51 0 Σ53 0 0 0A61 0 0 0 0 0

B := U>5 B =

B11 B12 0B21 0 0B31 0 00 0 00 0 00 0 0

where

[B21

B31

]is nonsinguar

C := CV5 =

C11 C12 C13 0 0 0C21 0 0 0 0 00 0 0 0 0 0

and V := V V5, U := UU5.

Remark 5.14.1. The condensed form (SCF) can be computed by 8 SVDs.2. Since only transformations with orthogonal matrices are used the algorithm is numer-

ically backward stable.3. Still, numerical rank decisions are critical since small perturbations can change the

numerical rank drastically.4. If one is only interested in state feedback control, then we can set C = I and neglect

the transformations that operate on C.

With the help of the condensed form (SCF) we can now construct feedback controls. In thefollowing, we consider proportional and derivative output feedbacks of the form

u = Fy −Gy + v

for system (5.1). Then the closed-loop system has the form

(E +BGC)x = (A+BFC)x+Bv

85


Theorem 5.15. Let (E,A,B,C) be given in the form (SCF).

1. If the system is I-controllable and I-observable, i. e.

rank[E AS∞ B

]= rank

ET>∞AC

= n,

then for all s ∈ N with 0 ≤ s ≤ t2 = s2 there exist matrices F,G ∈ Rm,p suchthat (E + BGC,A + BFC) is regular and ν = ind(E + BGC,A + BFC) ≤ 1 andrank(E +BGC) = t1 + s.

a) If s = t2, then this is achieved by derivative feedback alone with F = 0.b) If s = 0, then this is achieved by proportional feedback alone with G = 0.

2. If there exists F ∈ Rm,p such that (E,A+BFC) is regular and ν = ind(E,A+BFC) ≤1, then

rank[E AS∞ B

]= rank

ET>∞C

= n,

i. e. the system is I-controllable and I-observable.

Remark 5.16. For the case of proportional and derivative state feedback we can set C = I.

Proof.1. We can assume that (E,A,B,C) are in the form (SCF) with k3 = l3 = 0 and t6 =s6 = 0 (Corollary 5.12) as well as t2 = s2. Let

G =

[ l1 p− l1k1 G11 0m− k1 0 0

]∈ Rm,p, F =

[ l1 p− l1k1 F11 0m− k1 0 0

]∈ Rm,p

with

G11 =

[B21

B31

]−1 [Is 00 0

] [C12 C13

]−1 ∈ Rk1,l1

F11 =

[B21

B31

]−1

s t2 − s l1 − t2

s 0 0 0

t2 − s 0 φ 0

k1 − t2 0 0 0

[C12 C13

]−1 ∈ Rk1,l1

and φ = It2−s −[0 It2−s

]A22

[0

It2−s

]∈ Rt2−s,t2−s. Then we have

BGC =

t1 s n− t1 − s

t1 ∆11 ∆12 0

s ∆21 ∆22 0

n− t1 − s 0 0 0

86

with

∆11 = B11G11C11 ∈ Rt1,t1

∆12 = B11

[B21

B31

]−1 [Is0

]∈ Rt1,s

∆21 =[Is 0

] [C12 C13

]−1 ∈ Rs,t1

∆22 = Is

and

BFC =

t1 s t2 − s n− t1 − t2

t1 φ11 0 φ13 0s 0 0 0 0t2 − s φ31 0 φ33 0n− t1 − t2 0 0 0 0

∈ Rn,n

with

φ11 = B11F11C11 ∈ Rt1,t1

[0 C13 0

]= B11

[B21

B31

]−10 0 0

0 φ 00 0 0

∈ Rt1,n−t1

0C31

0

=

0 0 00 φ 00 0 0

[C12 C13

]−1C11 ∈ Rn−t1,t1

φ33 = φ.

It follows that

E −BGC =

ΣE + ∆11 ∆12 0∆21 Is 0

0 0 0

=

It1 ∆12 00 Is 00 0 I

ΣE + ∆11 −∆12∆21 0 0∆21 Is 0

0 0 0

and rank(E +BGC) = s+ rank(ΣE + ∆11 −∆12∆21). Furthermore, we have

∆11 −∆12∆21 = B11G11C11 −B11

[B21

B31

]−1 [Is0

] [Is 0

] [C12 C13

]−1

︸︷︷︸=G11

C11 = 0

87


and thus rank(E +BGC) = s+ rank(ΣE) = s+ t1. Due to the form of BFC we get

A+BFC =

A11 A12 A13 A14 A15

A21 A22 A23 A24 0A31 A32 A33 A34 Σ35

A41 A42 A43 Σ44 0A51 0 Σ53 0 0

+

φ11

[0 φ13

]0 0 0[

0φ31

] [0 00 φ33

]0 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

=

A11 A12 A13 A14 A15

A21 A22 A23 A24 0A31 A32 A33 A34 Σ35

A41 A42 A43 Σ44 0A51 0 Σ53 0 0

with

A22 = A22 +

[0 00 φ33

]=

[A22,1 A22,2

A22,3 A22,4

]+

[0 00 It2−s −A22,4

]=

[A22,1 A22,2

A22,3 It2−s.

]In particular,

t2 − s t5 t4 t3

t2 − s It2−s A23 A24 0t3 A32 A33 A34 Σ35

t4 A42 A43 Σ44 0t5 0 Σ53 0 0

as the lower right (n − t1 − s) × (n − t1 − s) principle of A + BFC is nonsingular.Thus, we have that (E + BGC,A + BFC) is regular and of index ν ≤ 1. Moreover,it is clear from the construction that F = 0 if s = t2 and G = 0 if s = 0.

2. If only proportional feedback is used then the matrices S∞ and T∞ are not changedby the feedback. Thus

rank[E AS∞ B

]= rank

[E (A+BFC)S∞ B

]= rank

ET>∞AC

= rank

ET>∞(A+BFC)

C

= n

(see Theorem 3.1).

Remark 5.17.1. If we use derivative feedback, then the existence of a feedback matrix G such that

(E + BGC,A) is regular and of index ≤ 1 is not sufficient for the system to be I-controllable and I-observable, since left and right nullspaces of E may change underderivative feedback.

88

2. To compute the regularizing feedback we need F11, G11 as defined in the previous

proof. Due to the construction of the form (SCF) the matrices

[B21

B31

]and

[C12 C13

]can be kept in factorized form as a product of an orthogonal matrix and a diagonalmatrix. Thus for the computation of F11, G11 or E +BGC, A+BFC we only haveto invert two diagonal matrices.

Corollary 5.18. Let (E,A,B,C) be given in the form (SCF). If

rank

[λE −AC

]= n for all λ ∈ C

and

rank[E AS∞ B

]= rank

ET>∞AC

= n

(i. e. the system is S-controllable and S-observable), then there exist F,G ∈ Rm,p and afeedback control u = Fy − Gy + v such that the closed-loop system (E + BGC,A + BFC)is S-controllable and S-observable with index ≤ 1 and rank(E +BGC) = t1 + s, where s isgiven such that 0 ≤ s ≤ t2.

Proof. From I-controllability and I-observability there follows the existence of F,G suchthat (E + BGC,A + BFC) is regular and of index ≤ 1 and still I-controllable and I-observable. The other conditions are invariant under feedback, thus the closed-loop systemis S-controllable and S-observable.

Corollary 5.19. Let (E,A,B,C) be given in the form (SCF).

1. There exists G such that E +BGC is nonsingular if and only if

rank[E B

]= rank

[EC

]= n.

2. There exists G ∈ Rm,p and a control u = −Gy + v such that the closed-loop system isC-controllable and C-observable with rank(E +BGC) if and only if

rank[αE − βA B

]= rank

[αE − βA

C

]= n for all (α, β) ∈ C2\{(0, 0}.

Proof.1. Assume first that there exists G such that E + BGC is nonsingular. Then from the

form (SCF) we get t3 = t4 = t+ 5 = t6 = s6 = 0 and hence

rank

[EC

]= rank

[E B

]= n.

The converse direction follows from Theorem 5.15 by choosing s = t2.

89


2. Since the rank conditions are invariant under feedback, the claim follows.

Remark 5.20.1. For linear systems with variable coefficients of the form

E(t)x(t) = A(t)x(t) +B(t)u(t), x(t0) = x0

condensed/staircase forms have been derived in [3] using special kinds of global an-alytic singular value decompositions (ASVDs) [1]. For the existence of an ASVD wehave to assume that E(t), A(t) and B(t) are analytic and rankE(t) = r for all t ∈ I,then

E(t) = U(t)

[Σ(t) 0

0 0

]V (t)>,

where U(t), V (t) are analytic and pointwise orthogonal. Regularization by propor-tional or derivative feedback can then be handled in a similar manner.

2. Nonlinear systems can be treated by local linearization.

Remark 5.21. In principle we now have two possibilities to check some of the systemproperties: either we use the derivative array approach as in chapter 3 or the staircaseform (SCF). The computation of the staircase form (SCF) is much more subtle numericallysince the consecutive rank decisions of transformed matrices have to be made in a properway. Also two-sided transformations are used that change the basis of the state space. Thestaircase form allows to check observability and controllability simultaneously but on thecost of changing the physical meaning of the state variables.

90

CHAPTER

6

OPTIMAL CONTROL PROBLEMS

We consider optimal control problems of the following form

Minimize J (x, u) =M(x(tf )) +

∫ tf

t0

K(t, x(t), u(t))dt (6.1a)

subject to F (t, x, x, u) = 0, x(t0) = x0 (6.1b)

with F ∈ C(I×Dx×Dx×Du,Rl) sufficiently smooth, I = [t0, tf ] ⊆ R, Dx,Dx ⊆ Rn, Du ⊆ Rmopen sets, x0 ∈ Dx, M : Dx → R and K : I× Dx × Du → R.

91

BIBLIOGRAPHY

[1] A. Bunse-Gerstner, R. Byers, V. Mehrmann, and N. K. Nichols. Numerical computationof an analytic singular value decomposition of a matrix valued function. Numer. Math.,60:1–40, 1991.

[2] A. Bunse-Gerstner, V. Mehrmann, and N. K. Nichols. Regularization of descriptorsystems by output feedback. IEEE Trans. Automat. Control, 39:1742–1748, 1994.

[3] R. Byers, P. Kunkel, and V. Mehrmann. Regularization of linear descriptor systemswith variable coefficients. SIAM J. Cont., 35:117–133, 1997.

[4] J. D. Cobb. On the solutions of linear differential equations with singular coefficients.J. Diff. Equations, 46:310–323, 1982.

[5] R.E. Kalman. Canonical structure of linear dynamical systems. Proc. Natl. Acad. Sci.U. S. A., 48(4):596–600, 1962.

[6] P. Kunkel and V. Mehrmann. Differential-Algebraic Equations. Analysis and NumericalSolution. European Mathematical Society, 2006.

93

INDEX

algebraic part, 13

consistentcontrol problem, 11, 15initial value, 10

control, 5control problem, 11

consistent, 11regular, 11

controllable∼ within the reachable set, 44C-∼, see completely controllablecompletely ∼, 43impulse ∼, 44, 60strongly sim, 64

DAE, see differential-algebraic equationderivative array, see inflated systemdescriptor system, 6differential part, 13differential-algebraic equation, 6Dirac delta distribution, 53distribution, 53dual system, 67dynamic part, 13

equivalentglobally, 22

strongly, 12Euler-Lagrange equation, 6

free system, 19

generalized function, 53

Heaviside function, 53

impulse controllable, 60impulsive smooth, 54index, 12inflated system, 17input, see control

jump, 59

Lagrangefunction, 6multiplier, 7

linearization, 7, 8principle, 8

matrixpair, 11pencil, 11

observable, 64∼ within the reachable set, 64

95

Index

completely sim, 64impulse sim, 65strongly sim, 70

output, 5equation, 5

reachable∼ set, 43, 44

regular, 19control problem, 11, 15matrix pencil, 11

s-index, see strangeness-indexsingular

matrix pencil, 11slow part, 13solution

classical, 10state, 5

equation, 5strangeness

-free, 18-index, 18, 28

Weierstraß canonical form, 12

96

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