[Lecture Notes in Control and Information Sciences] Advanced Topics in Control and Estimation of...

8

Non Linear Systems – Measurement

Output-Feedback Control

8.1 Introduction and Problem Formulation

We consider the system that is described by (1.11) and (1.12). We also considerthe following controlled output:

zt =

[h(xt, t)ut

], t ∈ [0,∞), (8.1)

where h : Rn × [0,∞) → Rr is a Borel measurable function, and letYt = {ys : s ≤ t}. In Chapters 6 and 7 we have considered the correspondingproblems of H∞state-feedback and H∞state estimation and obtained the suf-ficient conditions that guarantee the dissipation of the corresponding treatedsystems and assure a given performance level. In the present chapter we applythe results that have been obtained in the last two sections to the problemof H∞output-feedback control of the above mentioned stochastic nonlinearsystem.

Let α(·, ·) be positive Borel function on Rn × [0,∞) (where Rn × [0,∞)is endowed with the Borel σ-algebra). In what follows it will be assumedthat E{α(x, t)} < ∞ for all t ∈ [0,∞) and for all F -measurable, Rn-valuedrandom variables which satisfy E{||x||2} <∞. The control objective is to findan output-feedback controller ut = u(Yt, t) such that, for a given γ > 0, thefollowing H∞ criterion is satisfied.

E{∫ t2

t1

‖zt‖2 dt} ≤ γ2E{α(xt1 , t1) +∫ t2

t1

γ2 ‖vt‖2 dt} (8.2)

for all 0 ≤ t1 < t2, for all F0-measurable x0 with E{||x0||2} < ∞, and forall disturbances vt in Au (provided Au is nonempty). Whenever the system(1.11) satisfies the above inequality, it is said to possess an L2-gain that isless than or equal to γ. Note that for the infinite time-horizon t2 = ∞, and itis required that v satisfies:

∫∞0

||vt||2dt <∞.

E. Gershon & U. Shaked: Advanced Topics in Control & Estimation, LNCIS 439, pp. 139–154.DOI: 10.1007/978-1-4471-5070-1_8 c© Springer-Verlag London 2013

140 8 Non Linear Systems – Measurement Output-Feedback Control

8.2 Stochastic H∞ OF Control

We consider now the system (1.11) together with the observations (1.12) andthe controlled output (8.1). As the state xt of the plant is not available, wefollow the common practice (the certainty equivalence approach) of replacingthe state that is to be processed by the controller with an estimator outputxt. A natural choice of an estimator (see, e.g. [73] for the deterministic case)is:

dxt=f(xt, t)dt+g(xt, t)u∗t (xt)dt+ g1(xt, t)v

∗t (xt)+K(xt, t)(dyt−h2(xt, t)dt

−g3(xt, t)v∗t (xt)dt),(8.3)

where K(xt, t) is the estimator gain, an n× r matrix,

u∗t (x)=−1

2[I+

1

2gT(x, t)Vxx(x, t)g(x, t)]

−1gT(x, t)V Tx (x, t)

and

v∗t (xt)=1

2[γ2I− 1

2U(xt, t)]

−1gT1(xt, t)VTx (xt, t).

Using yt of (1.12) in (8.3), we arrive at the following augmented system.

dxet = fe(xet ,K, t)dt+ ge1(x

et ,K, t)[vt − v∗t (xt)]dt+ ge2(xt, t)[vt − v∗t (xt)]dW 2

t

+Ge(xet , t)dW1t ,

(8.4)where K = K(x, t), while

xet = col{xt, xt}, W 1t = col{Wt, W

1t , W

2t , W

3t }, (8.5)

fe(xet ,K, t)

=

[f(xt, t) + g1(xt, t)v

∗t (xt) + g(xt, t)u

∗t (xt)

f(xt, t)+g(xt, t)u∗t (xt)+g1(xt, t)v

∗t (xt)+K(xt, t)(h2(xt, t)−h2(xt, t))

],

(8.6)

ge1(xet ,K, t) = col{g1(xt, t), K(xt, t)g3(xt, t)}, ge2(xt, t) = col{g2(xt, t), 0},

(8.7)

Ge(xet , vt) =

[g(x, t)u∗(x) G(xt, t) g2(xt, t)v∗t (xt) 0

0 0 0 K(xt, t)G2(xt, t)

], (8.8)

and

h2(x, t) = h2(x, t)− g3(x, t)v∗t (x) h2(x, t) = h2(x, t)− g3(x, t)v∗t (x). (8.9)

Note that occasionally (whenever found convenient) we will use the abbrevi-ation fe(xet , t) for f

e(xet ,K, t). We now have the following theorem.

8.2 Stochastic H∞ OF Control 141

Theorem 8.1. Consider the stochastic system (1.11) together with the aug-mented system (1.12) and the controlled output (8.1). Assume there is a pos-itive function V : Rn × [0,∞) → R+, with V ∈ C2,1 so that it satisfies theHJI (6.8) of Theorem 6.2.1. Assume also that there are: a positive functionW : R2n × [0,∞) → R+ and a matrix K(x, t) which satisfy the following HJIfor some γ > 0.

Wt(xe, t) + Wxe(xe, t)fe(xe, t) + 1

4Wxe(xe, t)ge1(xe, t)[γ2I − 1

2 U(xe, t)]−1

ge1(xe, t)T WT

xe(xe, t) + 12Tr

{(Ge)T Wxexe(xe, t)Ge(xe, t)

}

+he(xe, t)The(xe, t) ≤ 0 ∀xe ∈ R2n

(8.10)where

U(xe, t) = [ge2(xe, t)]T Wxexe(xe, t)ge2(x

e, t), he(xe, t) = u∗(x) − u∗(x) (8.11)

such that

γ2I − 1

2[ge2(x

e, t)]T Wxexe(xe, t)ge2(xe, t) ≥ αI

for some positive number α, and for all xe ∈ R2n. Then, the closed-loop systemwith the control

u∗t (x) = −1

2[I +

1

2gT (x, t)Vxx(x, t)g(x, t)]

−1gT (x, t)Vxx(x, t)

is dissipative with respect to the supply rate γ2||v||2 − ||z||2, it possesses astorage function

S(xe, t) = V (x, t) + W (xe, t),

and it has an L2-gain ≤ γ.

Proof: Application of Theorem 6.2.1 yields

Vt(x, t)+Vx[f(x, t)) + g1(x, t)v+g(x, t)u∗(x)]+‖z‖2−γ2 ‖v‖2

+ 12Tr{GT (x, t)Vxx(x, t)G(x, t)}

+ 12u

∗T (x)gT (x, t)Vxx(x, t)g(x, t)u∗(x)}

≤∥∥ 12 [I +

12 g

T (x, t)Vxx(x, t)g(x, t)]−1gT (x, t)V T

x (x, t)+ u∗(x)∥∥2

−∥∥∥v − 1

2γ2 gT1 (x, t)V

Tx (x, t)

∥∥∥2

= ||u∗(x) − u∗(x)||2 − γ2||v − v∗(x)||2 = ||he(xe)||2 − γ2||r||2

(8.12)

where r = r(x) = v − v∗(x).Define

S(xe, t) = V (x, t) + W (xe, t).

Thus, S is positive definite and satisfies:


S(0, t) = 0 ∀t ≥ 0.

Obviously, the infinitesimal generator of the process xe satisfies:

L{S(xe, t)} = L{V (x, t)}+ L{W(xe, t)}

where:

L {V (x, t)} = Vt(x, t) + Vx(x, t) {f(x, t) + g1(x, t)v + g(x, t)u∗(x)}+ 1

2Tr{GT (x, t)Vxx(x, t)G(x, t)

}+ 1

2u∗T (x)gT (x, t)Vxx(x, t)g(x, t)u∗(x)

L{W (xe, t)

}=Wt(x

e, t)+Wxe[fe(xe, t)+ge1(xe, t)r]

+12Tr

{(G(xe, t))T WxexeGe(xe, t)

}.

In view of (8.12) we have:

L {V (x, t)} + ||z||2 − γ2||v||2 ≤ ||he(xe)||2 − γ2||r||2.

By the HJI (8.10) it follows that

L{W (xe, t)

}+ ||he(xe)||2 − γ2||r||2 ≤ 0.

Therefore,

L {S(xe, t)}+||z||2−γ2||v||2 = L {V (xe, t)}+||z||2−γ2||v||2+L{W (xe, t)

}≤ 0.

This implies that S(xe, t) is a storage function for the closed-loop system withthe supply rate ||z||2− γ2||v||2, which implies that the closed-loop system hasan L2-gain property (i.e., the inequality (8.2) is satisfied)

Remark 8.1. As in the deterministic case it is not easy to find a matrixK(x, t)depending on x and t alone, such that HJI (8.10) is satisfied (see [73] forthe deterministic case where a local solution is offered). In what follows weoffer two ways for approaching this problem which yield, in certain cases, asatisfactory solution for K(x, t).

Consider first the case for which G2 = 0. Then, the part of the latter inequalitythat contains K is given by:

Γ (K) = Wx(xe, t)K(x, t)(h2(x, t) − h2(x, t)) + 1

4 (Wx(xe, t)K(x, t)g3(x, t)

+Wx(xe, t)g1(x, t))(γ

2I− 12 U(x

e, t))−1(Wx(xe, t)K(x, t)g3(x, t)

+Wx(xe, t)g1(x, t))

T

= 14γ2 [Wx(x

e, t)K(x, t)− Ψ(xe, t)]Ra(x, t)[KT (x, t)WT

x (xe, t)− ΨT (xe, t)]

− 14γ2 [Ψ(x

e, t)Ra(x, t)ΨT (xe, t)− Wx(x

e, t)g1(x, t)(I − 12γ2 U(x

e, t))−1

gT1 (x, t)WTx (xe, t)],

(8.13)


where

Ψ(xe,t)=−[2γ2(h2(x,t)−h2(x,t))T+Wx(xe,t)g1(x, t)R(x, t)]R

−1a (x,t)

R(x, t) = (I− 12γ2 g

T2 (x, t)Wxx(x

e, t)g2(x, t))−1gT3 (x, t)

and Ra(x, t) = g3(x, t)R(x, t).

The gain matrix K(x, t) that minimizes Γ (K), and thus leads to a minimumleft hand side in (8.10), is clearly one that satisfies Wx(x

e, t)K(x, t) = Ψ(xe, t).Unfortunately, the latter equation may not possess a solution for K whichdepends only on x and t. One way to circumvent this difficulty is to chooseK(x, t) s.t.

Wx(xe, t)K(x, t) = Ψ(xe, t) + Φ(xe, t), (8.14)

where Φ(xe, t) is a function that allows a solution K∗ for (8.14) which isindependent of x. For this choice of K∗(x, t) the above Γ (K) becomes thefollowing.

Γ (K∗)= 14γ2 [Φ(x

e, t)Ra(x, t)ΦT (xe, t)−Ψ(xe, t)Ra(x, t)Ψ

T (xe, t)

+Wx(xe, t)g1(x, t)(I − 1

2γ2 U(xe, t))−1gT1 (x, t)W

Tx (xe, t)].

Assuming an existence of a solution K(x, t) to (8.14) for some functionΦ(xe, t), we have established the following theorem.

Theorem 8.2. Consider the stochastic system (1.11) together with the aug-mented system (8.4) and the controlled output (8.1). Assume there is a pos-itive function V : Rn × [0, T ] → R+, with V ∈ C2,1 so that it satisfies theHJI (6.8) of Theorem 6.2.1. Assume also that there are: a positive functionW : R2n × [0, T ] → R+ in C2,1 and a matrix K(x, t), which satisfy (8.14). Inaddition, let W satisfy the following HJI.

Wt(xe, t) + Wx(x

e, t)[f(x, t) + g1(x, t)v∗t (x) + g(x, t)u

∗t (x)]

+Wx(xe, t)[f(x, t) + g1(x, t)v

∗t (x) + g(x, t)u

∗(x)]

+ 12Tr

{G(x, t)T Wxx(x

e, t)G(x, t)}+ 1

2u∗T (x)gT (x, t)Wxx(x

e, t)g(x, t)u∗(x)

+ 12v

∗T (x)gT2 (x, t)Wxx(xe, t)g2(x, t)v

∗(x) + he(xe, t)The(xe, t)

+ 14γ2 [Φ(x

e, t)Ra(x, t)ΦT (xe, t)−Ψ(xe, t)Ra(x, t)Ψ

T (xe, t)

+Wx(xe, t)g1(x, t)(I − 1

2γ2 gT2 (x, t)Wxx(x

e, t)g2(x, t))−1gT1 (x, t)W

Tx (xe, t)] ≤ 0

∀xe ∈ R2n

W (xe, 0) = γ2||x− x||2.(8.15)

Then, the closed-loop system is dissipative with respect to the supply rateγ2||v||2 − ||z||2, with the storage function defined as


S(xe, t) = V (x, t) + W (xe, t),

and therefore has an L2-gain ≤ γ.

Remark 8.2. In Theorem 8.1 above, it was assumed that K(x, t) exists so that(8.10) is satisfied. Finding such K is a difficult task, in general. The methodthat is offered above allows a solution (that may be conservative) for K whichis independent of the system’s state x. It is noted that the inequality (8.15)is only sufficient, but by no means necessary.

8.2.1 Example

In order to demonstrate the applicability of the above approach we considerthe following special case of (1.11) and (1.12):

dxt = Axtdt+B1vtdt+ GxtdWt

dyt = C2xtdt+ ρ · sin(2(x2 − x1))dt+ κvtdt,(8.16)

where {xt}t≥0 ∈ R2 and yt ∈ R1. It is assumed that (6.8) is satisfied with

V (x) = xTPx. Choosing W (xe, t) = xeTQxe, eq.(8.13) now reads:

Ψ = −2γ2κ−1 [C2(x− x) + 2ρsin(x2 − x2 − x1 + x1)cos(x2 + x2 − x1 − x1)]

+2κ−1(x− x)−1PB1 − κ−1xeTQIB1

(8.17)

where I = [I2 0]T. Denoting also I = [I2 − I2]T and Υ = col{−1, 1, 1,−1}, we

find that: Ψ = xeT Ψ where

Ψ =[−2γ2κ−2ICT

2 − 4γ2κ−2ρΥsinc(x2−x2−x1+x1)cos(x2+x2−x1−x1)

+2κ−1(IPB1 −QIB1)].

(8.18)

For the above choice of W we find that Wx = xeTQI and Wx = xeTQ˜I where˜I = [0 I2]

T.

Denoting Φ = xeT Φ, (8.15) reduces now to

xeT[2Q

[I(A+ γ−2B1B

T1 P )− IB2B

T2 P + ˜I(A+ γ2B1B

T1 P −B2B

T2 P )

]

+GT ITQIG+ IPB2BT2 P I+4QIB1B

T1 I

TQ]xe+ κ2

4γ2xeT

[ΦΦT − Ψ Ψ

]xe≤ 0

(8.19)

Eq. (14) becomes: ˜IK(x) = Q−1[Ψ + Φ

]and then

K(x) = 2Q−12 κ−1(γ2κ−1CT

2 − PB1).

Replacing ΦΦT in (8.19) with its upper bound 16γ4κ−4ρ2[

1 −1−1 1

]and elim-

inating xe from both sides of (8.19), the resulting inequality in Q1 and Q2


is independent of x and x. The requirement in regard to Q2 then becomes asimple Lyaponov inequality, while the result for Q1 can be found using linearmatrix inequality solver.

8.2.2 The Case of Nonzero G2

In the case where G2(x, t) is not identically zero, Ge(xe, vt) of (8.4e) shouldbe replaces by:Ge(xe, vt) = diag{G(x, t), K(x, t)G2(x, t)}and the term 1

2Tr{G2(x, t)TK(x, t)T Wxx(x

e, t)K(x, t)G2(x, t)}should thus be added to Γ (K) of (8.13). The latter term does not allowcompletion of the terms in Γ (K) to squares and thus solutions for K thatachieve the prescribed L2-gain bound can be obtained by one of the followingmethods.

Solution no. 1: One may solve for K in (8.14). In this case, due to theadditional term in Γ (K), the positive term12Tr

{G2(xt,t)

T [ΨT (xe, t)+ΦT (xe, t)]W−1xx (xe, t)[Ψ(xe, t)+Φ(xe, t)]G2(xt, t)

}is added to the left side of (8.15).

Solution no. 2: Since the above solution entails an over design one mayconsider, similarly to the method used in linear gain scheduling with uncer-tainty in the input or the output matrices (see, e.g., [41], p. 7-4), the followingmodified system which contains a simple linear lowpass component of largebandwidth between the measured output and the point where the noise signalvt is applied. The effect of this component, which will be a part of the con-troller, on the solution is negligible whenever its bandwidth is very large incomparison with the system ‘bandwidth’. We may now formulate the abovein the following way. Define a new state ζ as

dζt = −ρζtdt+ ρh2(xt, t)dt+ ρG2(xt, t)dW3t

and we measure now the filter output corrupted with the deterministic noiseg2(xt, t)vtdt that is,

dyt = ζtdt+ g3(xt, t)vtdt.

Replacing xt in xet of (8.4b) with col{xt, ζt} and defining xt to be the estimator

of the latter augmented vector we obtain that dxt satisfies the following:

dxtΔ=

[dxtdζt

]=

[f(xt, t)

ρh2(xt, t)− ρζt

]dt+

[g(xt, t)

0

]u∗t (xt)dt

+

[g1(xt, t)

0

]v∗t (xt)dt+

[K(xt, ζt, t)

ρK1(xt, ζt, t)

](dyt − ζtdt− g3(xt, t)v∗t (xt)).

Denoting then:


fe(xet , t)= col{f(xt, t) + g1(xt, t)v∗t (xt)+g(xt, t)u∗t (xt), −ρ(ζt−h2(xt, t)),

f(xt, t)+g(xt, t)u∗t (xt)+g1(xt, t)v

∗t (xt)+K(xt, t)[ζt− ζt+g3(xt, t)v∗t (xt)

−g3(xt, t)v∗t (xt)],

ρ(h2(xt, t)− ζt)+ρK1(xt, ζt, t)[ζt− ζt+g3(xt, t)v∗t (xt)−g3(xt, t)v∗t (xt)]}

g1(xet ,t)=col{g1(xt, t), 0, K(xt, ζt, t)g3(xt, t), ρK1(xt, ζt, t)g3(xt, t)},

ge2(xet ,t)=col{g2(xt, t), 0, 0, 0},

Ge(xet , t)=

⎡⎢⎢⎢⎢⎣

g(x, t)u∗(x) G(xt, t) g2(xt, t)v∗t (xt) 0

0 0 0 ρG2(xt, t)

0 0 0 0

0 0 0 0

⎤⎥⎥⎥⎥⎦

and defining W et = col{Wt,W

1t ,W

2t ,W

3t } we obtain:

dxet = fe(xet , t)dt+ g1(xet , t)[vt(xt)− v∗t (xt)] + Ge(xet , t)dW

et

andxet = col{xt, ζt, xt}.

(8.20)

Applying the result of Theorem 8.2 to the augmented system (8.20) we obtainthat the minimizing gain matrix col{K, ρK1} satisfies

Wx(xe, t)

[K(xt, t)ρK1(xt, t)

]= Ψ(xe, t) + Φ(xe, t)

and

Ψ(xe, t) = −[2γ2(ζt− ζt+g3(xt, t)v∗t (xt)−g3(xt, t)v∗t (xt))T

+Wx(xe, t)g1(xt, t)(I − 1

2γ2 gT2 (xt, t)Wxx(x

e, t)g2(xt, t))−1gT3 (xt, t)]R

−1a ,

(8.21)

where Φ(xe, t) is a function that allows a solution col{K,K1} for (8.21) thatis independent of xt, and where W (xe, t) is a positive function W : R2(n+p) ×[0,∞) → R+ that satisfies the following inequality.

Wt(xe, t)+Wx(x

e, t)[f(xt, t)+g1(xt, t)v∗t (xt)+g(xt, t)u

∗t (xt)]

−ρWζ [ζt−h2(xt, t)]+ρWζt[h2(xt, t)− ζt]+ 1

2Tr{G(x, t)T Wxx(x

e, t)G(x, t)}

+Wx(xe, t)[f(xt, t)+g1(xt, t)v

∗t (xt)+g(xt, t)u

∗(xt)]

+ ρ2

2 Tr{G2(x, t)

T Wζζ(xe, t)G2(x, t)

}+ he(xe, t)The(xe, t)

+Wx(xe, t)g1(xt, t)(I − 1

2γ2 gT2 (xt, t)Wxx(x

e, t)g2(xt, t))−1gT1 (xt, t)W

Tx (xe, t)]

+ 14γ2 [Φ(x

e, t)Ra(xt, t)ΦT (xe, t)−Ψ(xe, t)Ra(xt, t)Ψ

T (xe, t) ≤ 0∀xe ∈ R2n.

8.3 Norm-Bounded Uncertainty 147

8.3 Norm-Bounded Uncertainty

The above results were based on the assumption that the system’s parametersare completely known. In the present section we consider the case where theseparameters are uncertain. We consider the following system:

dxt=(f(xt, t) +Δf)dt+(g1(xt, t) +Δg)vtdt+g2(xt, t)vtdW2t +G(xt, t)dW

1t

+ g(xt, t)utdt+g(xt)utdWt

(8.22)

z(t)=h(xt, t) +Δh (8.23)

where ut ∈ Rm is the control input which is taken to be identically zero, where{xt}t≥0 ∈ Rn is a solution to (8.22) with the initial condition x0, {vt}t≥0 ∈Rm1 is an exogenous disturbance and {W 1

t }t≥0 ∈ R, {W 2t }t≥0 ∈ R1 are

Wiener processes. Also, zt ∈ Rr is the output vector to be regulated. Thematrices Δf, Δg and Δh are assumed to be continuous functions of xt and tthat possess the following structure:

[Δf(x, t) Δg(x, t)

]= H1(x, t)F (x, t)

[E1(x, t) E2(x, t)

],

Δh(x, t) = H2(x, t)F (x, t)E1(x, t),

where H1(x, t) : Rn × [0,∞) → Rn×n1 , H2(x, t) : Rn × [0,∞) → Rr×n1 ,E1(x, t) : R

n × [0,∞) → Rn2 , E2(x, t) : Rn × [0,∞) → Rn2×m1 and F (x, t) :

Rn × [0,∞) → Rn1×n2 are continuous functions and where

F (x, t)FT (x, t) ≤ In1 , ∀x ∈ Rn and t ∈ [0, ∞). (8.24)

Applying the BRL obtained in Theorem 6.2.1 by replacing f , g1 and h byf +Δf , g1 +Δg and h+Δh, respectively, the corresponding HJI becomes:

Vt(x, t) + Vx(x, t)[f(x, t) +H1(x, t)F (x, t)E1(x, t)]+14Vx(x, t)(g1(x, t)

+H1(x, t)F (x, t)E2(x, t))[γ2I− 1

2U(x, t)]−1(gT1 (x, t)+E

T2 (x, t)F

T (x, t)HT1 (x, t))

Vx(x, t)+12Tr{GT (x, t)Vxx(x, t)G(x, t)}

+(hT (x, t) + ET1 (x, t)F

T (x, t)Ht2(x, t))(h(x, t) +H2(x, t)F (x, t)E2(x, t))

≤ 0 ∀x ∈ Rn, ∀t ≥ 0,

where U(x, t) is defined in (6.7). Applying Schur complement to the latter,and denoting:

Γ (x, t)

=

⎡⎢⎢⎣

Vt(x, t)+Vx(x, t)f(x, t)+1

2Tr{GT (x, t)Vxx(x, t)G(x, t)}Vx(x, t)g1(x, t) hT (x, t)

∗ −[4γ2I−2U(x, t)] 0∗ ∗ −Ir

⎤⎥⎥⎦

(8.25)


the following inequality is obtained;

Γ (x, t)+

⎡⎣ 0

0H2(x, t)

⎤⎦F (x, t)[E1(x, t) 0 0

]+

⎡⎣E

T1 (x, t)00

⎤⎦FT (x, t)

[0 0 HT

2 (x, t)]

+

⎡⎣Vx(x, t)H1(x, t)

00

⎤⎦F (x, t)[ 1

2E1(x, t) E2(x, t) 0]

+

⎡⎢⎣12E

T1 (x, t)

ET2 (x, t)0

⎤⎥⎦FT (x, t)

[HT

1 (x, t)VTx (x, t) 0 0

]≤ 0 ∀x ∈ Rn, ∀t ≥ 0.

Using the fact that for any two matrices α and β of compatible dimensionsand for any positive scalar ε the following holds:

αβT + βαT ≤ εααT + ε−1ββT ,

together with the bound of ( 8.24), we obtain the following:

Theorem 8.3. Consider the system described by (8.22)–(8.24), where ut ≡ 0,

with the controlled output of (8.23), and the supply rate S(v, z) = γ2 ‖v‖2 −‖z‖2. Suppose there is a positive function V (x, t) ∈ C2,1. Let V (x, t) satisfyγ2I − 1

2U(x, t) ≥ αI for some α > 0, and for all x, t, where U(x, t) is definedin (6.7) and assume that the following HJI is satisfied for some positive scalarsε1 and ε2:⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Γ (x, t)

⎡⎣Vx(x, t)H1(x, t)

00

⎤⎦ ε1

⎡⎣12E

T1 (x, t)

ET2 (x, t)0

⎤⎦⎡⎣ 0

0H2(x, t)

⎤⎦ ε2

⎡⎣E

T1 (x, t)00

⎤⎦

∗ −ε1In1 0 0 0∗ ∗ −ε1In2 0 0∗ ∗ ∗ −ε2In1 0∗ ∗ ∗ ∗ −ε2In2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦≤0 (8.26)

∀x ∈ Rn and ∀t≥0 where Γ (x, t) is defined in (8.25). Then, the system (8.22)is dissipative with respect to the supply rate S(v, z).

The above result assumed ut ≡ 0 in (8.22). If this is not the case and if

z(t)=col{h(xt, t) +Δh, ut} (8.27)

the following result replaces the one of Theorem 6.2.1.

Lemma 8.3.1 Consider the system described by (8.22)–(8.24) with the con-

trolled output of (8.27), and the supply rate S(v, z) = γ2 ‖v‖2−‖z‖2. Supposethere is a positive function V (x, t) ∈ C2,1. Let V (x, t) satisfy γ2I− 1

2U(x, t) ≥

8.3 Norm-Bounded Uncertainty 149

αI for some α > 0, and for all x, t, where U(x, t) is defined in (6.7) andassume that (8.26) is satisfied for some positive scalars ε1 and ε2 where to thefirst block on the diagonal of Γ (x, t) the term

−1

4Vx(x, t)g(x, t)[I +

1


−1gT (x, t)V Tx (x, t)

is added. Then, for

ut = −1

2[I +

1

2gT (x, t)Vxxg(x, t)]

−1gT (x, t)V Tx (x, t)

the system (8.22) is dissipative with respect to the supply rate S(v, z).

Considering next the case where the measurement is given by:

dyt = h2(xt, t)dt+ g3(xt, t)vtdt+G2(xt, t)dW3.

Once the BRL is obtained for systems with norm-bounded uncertainties theoutput-feedback problem can be solved as follows. Consider the following sys-tem.

dxt=(f(xt, t) +Δf)dt+g1(xt, t)vtdt+g2(xt, t)vtdW2t +G(xt, t)dW

1t

+g(xt, t)utdt+g(xt)utdWt

dyt = (h2(xt, t) +Δh2)dt+ g3(xt, t)vtdt+G2(xt, t)dW3

(8.28)

z(t)= col{h(xt, t) +Δh, ut},

where ut ∈ Rm is the control input, {xt}t≥0 ∈ Rn is a solution to (8.22) withthe initial condition x0, {vt}t≥0 ∈ Rm1 , {yt}t≥0 ∈ Rp is the measured outputand {W 1

t }t≥0 ∈ R, {W 2t }t≥0 ∈ R1 and {W 3

t }t≥0 ∈ Rm3 are Wiener processes.

Also, zt ∈ Rr is the output vector to be regulated. The matrices Δf, Δh2and Δh are assumed to be continuous functions of xt and t that possess thefollowing structure:

col{Δf(x, t), Δh(x, t), Δh2(x, t)} = col{H1(x, t), H2(x, t),

H3(x, t)}F (x, t)E1(x, t)(8.29)

where H1(x, t) : Rn × [0,∞) → Rn×n1 , H2(x, t) : Rn × [0,∞) → Rr×n1 ,H3(t) : Rn × [0,∞) → Rp×ny , E1(x, t) : Rn × [0,∞) → Rn2 , and F (x, t) :Rn × [0,∞) → Rn1×n2 are continuous functions and where F (x, t) satisfies(8.24).

Denoting:

Γ (x, t) =

⎡⎢⎢⎣Γ11(x, t) Vx(x, t)g1(x, t) hT (x, t)

∗ −[4γ2I−2U(x, t)] 0

∗ ∗ −Ir

⎤⎥⎥⎦


with

Γ11(x, t) = Vt(x, t)+Vx(x, t)f(x, t)+12Tr{GT (x, t)Vxx(x, t)G(x, t)}

− 14Vx(x, t)g(x, t)[I +

12 g

T (x, t)Vxx(x, t)g(x, t)]−1gT (x, t)V T

x (x, t)

we obtain the following inequality that corresponds to the HJI (8.26).

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Γ (x, t)

⎡⎣Vx(x, t)H1(x, t)

00

⎤⎦ ε1

⎡⎣12E

T1 (x, t)00

⎤⎦⎡⎣ 0

0H2(x, t)

⎤⎦ ε2

⎡⎣E

T1 (x, t)00

⎤⎦

∗ −ε1In1 0 0 0∗ ∗ −ε1In2 0 0∗ ∗ ∗ −ε2In1 0∗ ∗ ∗ ∗ −ε2In2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦≤0

∀x ∈ Rn, ∀t≥0.

Denoting then:

Δfe =

[H1(xt, t)

0

]F (xt, t)E1(xt, t) +

[0

K(xt, t)H3(t)

]F (xt, t)E1(xt, t)

the BRL of Theorem 8.3 is applied to the latter definition and to (8.4). De-noting

Γ e =

⎡⎢⎣Υ (xe, t) Wxe(xe, t)ge1(x

e, t) heT (xe, t)

∗ −[4γ2I − 2U(xe, t)] 0

∗ ∗ −I

⎤⎥⎦ ,

whereΥ (xe, t) = Wt(x

e, t)+Wxe(xe, t)fe(xe, t) + 12Tr{(Ge(xe, t))T Wxexe(xe, t)Ge(xe, t)}

and where U(xe, t) and he(xe, t) are defined in (8.11a,b) the following inequal-ity is obtained ∀xe ∈ R2n and ∀t≥0;

⎡⎢⎢⎢⎢⎣Γ e

⎡⎣ Wx(x

e, t)H1(x, t)+Wx(xe, t)K(xt, t)H3(x, t)

00

⎤⎦ ε

⎡⎣

12E

T1 (x, t)00

⎤⎦

∗ −εIn1 0∗ ∗ −εIn1

⎤⎥⎥⎥⎥⎦≤ 0,

(8.30)

where ε is a decision variable in R1. The solution to the output-feedbackproblem with the norm-bounded uncertainty described in (8.28)–(8.29) is de-scribed as follows.

Theorem 8.4. Consider the stochastic system (8.28)–(8.29) together with theaugmented system (8.4). Assume there exist a positive function V : Rn ×[0,∞) → R+, with V ∈ C2,1 and scalars ε1 and ε2 that satisfy the HJI

8.4 Infinite-Time Horizon Case: A Stabilizing H∞ Controller 151

(8.26) of Theorem 8.3. Assume also that there exist: a positive function W :R2n × [0,∞) → R+ a matrix K(x, t) and a scalar ε that satisfy (8.30) forsome γ ≥ 0, and that

γ2I − 1

2[ge2(x

e, t)]T Wxexe(xe, t)ge2(xe, t) ≥ αI

for some positive number α, and for all xe ∈ R2n . Then, the closed-loopsystem with the control

u∗t (x) = −1

2[I +

1


−1gT (x, t)Vxx(x, t)

and with the observer (8.3) is dissipative with respect to the supply rateγ2||v||2 − ||z||2, it possesses a storage function defined as S(xe, t) = V (x, t) +W (xe, t), and has an L2-gain ≤ γ.

Remark 8.3. Similar to the arguments of Remark 8.1 and Theorem 8.2, in thecase where G2 ≡ 0, the last theorem can be used to obtain a minimizing Kthat depends only on x and t by converting the matrix inequality (8.30) intoa scalar HJI using Schur complement. A completion to squares with respectto Wx(x

e, t)K(xt, t) can be obtained. Using then (8.14) a result similar to theone obtained in Theorem (8.3) can be achieved. The case where G2 is not zerocan be solved applying the method of Section 8.2.1.

8.4 Infinite-Time Horizon Case: A Stabilizing H∞Controller

In this section we consider the infinite-time horizon case. The problem ofsynthesizing an H∞ controller that renders a stable closed-loop system isinvestigated. Conditions under which the closed-loop system is asymptoticallystable, in both the probability sense and the mean-square sense are discussed.

We now state and prove two lemmas regarding the internal stability of theclosed-loop system.

Lemma 8.4.1 [Stability in probability]Assume there exists a positive function V (x, t) ∈ C2,1 such that inf

t>0V (x, t) →

∞ as ‖x‖ → ∞, satisfying the HJI (6.8) with hT (x, t)h(x, t) > 0 for all xand for all t ≥ 0. Assume also that there is a positive function W ∈ C2,1,W : R2n → R+, satisfying the HJI (8.10) with a strict inequality, so that

inft > 0, x ∈ Rn

W (xe, t) → ∞ as ||xe|| → ∞.

Then, the closed loop system is internally globally asymptotically stable inprobability.


Proof: Note that for v = 0 the HJI (6.8) reduces to

Vt(x, t) + Vx(x, t)f(x, t) − 14Vx(x, t)g(x, t)g

T (x, t)V Tx (x, t)

+ 12Tr{GT

1 (x, t)Vxx(x, t)G1(x, t)} + hT (x, t)h(x, t) ≤ 0 ∀x ∈ Rn, ∀t ≥ 0

which implies (LV )(x, t) ≤ −hT (x, t)h(x, t) < 0. On the other hand for v = 0the HJI (8.10) becomes

Wt(xe, t) + Wxe(xe, t)fe(xe, t)

+ 12Tr


}+ (he(xe, t)The(xe, t) < 0 ∀xe ∈ R2n.

This implies L0{W (xe, t)} < 0.Summarizing, one has now

L0{S(xe, t)} = L0{V (x, t)} + L0{W (xe, t)} < 0

which implies, by Theorem 1.3.1, that the closed loop system is internallyasymptotically stable in the probability sense.

The next lemma establishes the stability of the closed-loop in the meansquare sense.

Lemma 8.4.2 Assume there exists a positive function V (x, t) ∈ C2,1, withV (0, t) = 0 for all t ≥ 0, which satisfies the HJI of (6.8) for some γ > 0. Inaddition, let V satisfy

k1||x||2 ≤ V (x, t) ≤ k2||x||2 (8.31)

for all t ≥ 0 and for some positive numbers k1, k2. Furthermore, assume thatfor some k3, h

T2 (x, t)h2(x, t) ≥ k3||x||2 for all t ≥ 0 and for all x ∈ Rn.

Assume also that there is a positive function W ∈ C2,1. and W : R2n → R+

withk4||x||2 ≤ W (xe, t) ≤ k5||xe||2 xe ∈ R2n, t ≥ 0 (8.32)

which satisfies the following algebraic HJI:

Wxe(xe, t)fe(xe, t)+ 12Tr


}+ (he(xe, t)The(xe, t)

≤−Q(xe) ∀xe∈R2n

for some positive function Q(xe) with the property that

he(xe, t)The(xe, t) +Q(xe) ≥ k6||x||2 for all t ≥ 0, for all xe ∈ R2n,

and for some k6 > 0. Then, the closed-loop system (8.4) with v = 0 andu = − 1

2gT (x, t)V T

x (x, t) is exponentially stable in the mean square sense, andhas the property of L2 − gain ≤ γ, that is

8.4 Infinite-Time Horizon Case: A Stabilizing H∞ Controller 153

E{∫ ∞

0

‖zt‖2 dt} ≤ γ2E{‖xe0‖2+

∫ ∞

0

(‖vt‖2)dt} (8.33)

for all non-anticipative stochastic processes v that satisfy E{∫∞0

(‖vt‖2)dt} <∞, and whenever x0 satisfies E{W (xe0, 0)} ≤ γ2E{||xe0||2}.

Proof: The proof is a simple application of Theorem 1.3.2We consider now the time-invariant case with infinite-time horizon. De-

veloping an analog theory is straightforward, so we omit proofs and detaileddiscussions.

The time-invariant version of (1.11) and (1.12) is given by

dxt=f(xt)dt+g(xt)utdt+g1(xt)vtdt+G(xt)dW1t

dyt = h2(xt)dt+ g3(xt)vtdt+G2(xt)dW3

with the controlled output

z=col{h(x), u(x)}.

The time-invariant closed-loop system is now defined by:

dxet = fe(xet )dt+ ge1(x

et )[vt − v∗t (xt)]dt+Ge(xet )dW

et . (8.34)

We state now, without a proof the time-invariant analog of Theorems regard-ing stability.

Lemma 8.4.3 [Stability in probability: the invariant case]Assume there exists a positive function V (x) ∈ C2 such that V (x) → ∞ as‖x‖ → ∞, and assume V (x) satisfies the algebraic HJI:

Vx(x)f(x) − 14Vx(x)g(x)g

T (x)V Tx (x) + 1

4γ2Vx(x)g1(x)gT1 (x)Vx(x)

+ 12Tr{GT (x)Vxx(x)G(x)} + hT2 (x)h2(x) ≤ 0 ∀x ∈ Rn, ∀t ≥ 0

(8.35)

and hT2 (x, t)hT2 (x, t) > 0 for all x and for all t ≥ 0. Also assume that there is

a positive function W ∈ C2. and W : R2n → R+ with infx∈Rn

W (xe, t) → ∞ as

||x|| → ∞ which satisfies the HJI

Wxe(xe)fe(xe) + 14γ2 Wxe(xe)ge1(x

e)(ge1)T (xe)WT

xe(xe)

+ 12Tr

{(Ge(xe))T Wxexe(xe)Ge(xe)

}+ he(xe)The(xe) < 0 ∀xe ∈ R2n.

Then, the closed loop system is internally globally asymptotically stable inprobability, and has the property of L2-gain ≤ γ .

Lemma 8.4.4 Assume there exists a positive function V (x) ∈ C2, withV (0) = 0, which satisfies the HJI of (8.35) for some γ > 0. In addition,let V satisfy


k1||x||2 ≤ V (x) ≤ k2||x||2

for all t ≥ 0 and for some positive numbers k1, k2. Furthermore, assume thatfor some k3, h

T2 (x)h2(x) ≥ k3||x||2 for all x ∈ Rn. Assume also that there is

a positive function W ∈ C2 and W : R2n → R+ with

k4||x||2 ≤ W (xe) ≤ k5||xe||2 xe ∈ R2n,

which satisfies the following algebraic HJI:

Wxe(xe)fe(xe)+ 12Tr

{(Ge)T Wxexe(xe)Ge(xe)

}+he(xe)The(xe)

≤−Q(xe) ∀xe∈R2n

for some positive function Q(xe) with the property that he(xe)The(xe) +Q(xe) ≥ k6||x||2 for all xe ∈ R2n, and for some k6 > 0. Then, the closed-loopsystem (8.34), with v = 0 and u = − 1

2gT (x)V T

x (x), is exponentially stable inthe mean square sense, and has the property of L2 − gain ≤ γ, that is

E{∫ ∞

0

‖zt‖2 dt} ≤ γ2E{‖xe0‖2+

∫ ∞

0

‖vt‖2 dt} (8.36)

for all nonanticipative stochastic processes v that satisfy E{∫∞0

(‖vt‖2)dt} <∞, and whenever x0 satisfies E{W (xe0, 0)} ≤ γ2E{||xe0||2}.

8.5 Conclusions

In this chapter, we have formulated and solved an H∞ output-feedback con-trol problem for nonlinear stochastic systems. The theory which has beenintroduced in this work was facilitated by the concept of stochastic dissipa-tion. As we utilized a certainty equivalence principle, we have facilitated theestablishment of sufficient conditions, in terms of a pair of HJI, the solution ofwhich guarantees a controller that renders the underlying closed-loop systeman internal asymptotic stability in the mean square sense and L2-gain that isless than or equal to a prescribed attenuation level. We note that we bringin Chapter 13, Section 13.4, an example, which concerns the output-feedbackstabilization of an inverted pendulum.

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