376 069 Multivariable feedback control V2 · 376_069 Multivariable feedback control V2 1 of 47...

Observability

376_069 Multivariable feedback control V2 1 of 47

Observability

Background • Formalized by R.E. Kalman in 1960 • Key concept in dynamic systems and

control and estimation theory • If a system is observable, we can

reconstruct its state based on input-output records

• Fundamental importance for - observer design - Kalman filter design - systems identification • Will present "modern" and classical

observability tests

• Used with controllability to understand MIMO input-output properties

MIMO pole-zero cancellations minimum realizations

Observability


Observability definition • Given the nonlinear system dx(t)/dt = f(x(t),u(t)) ;x(0) = ξ y(t) = g(x(t),u(t)) • The system is observable iff one can

calculate the fixed (but arbitrary) initial state ξ based upon finite-time measurements of

- The control input u(t), 0 ≤ t ≤ T - The output y(t), 0 ≤ t ≤ T • Otherwise the system is called

unobservable

Observability


State reconstruction

ξ = initial state u(t) dynamic y(t) system • System observable ⇒ can determine

numerical value of ξ • Knowledge of - initial state ξ - control u(t), 0 ≤ t ≤ T - dynamics dx(t)/dt = f(x(t),u(t)) Will yield state trajectory x(t), 0 ≤ t ≤ T Via numerical integration x(t) = ξ +

0( ( ), ( ))

tf x u dτ τ τ∫

Observability


Remarks • No easy test for general nonlinear systems

• Easy test exist for FDLTI dynamic systems. Test is the mathematical dual of controllability test, but concepts are fundamentally different

• Two tests "modern" - modal approach "classical" - via Caley-Hamilton theorem • Warning A dynamic model which is mathematically observable, might have poor state reconstruction accuracy from a practical point of view

Observability


Observability in LTI systems • Complete state-space model dx(t)/dt = Ax(t) + Bu(t) ;x(0) = ξ y(t) = Cx(t) + Du(t) • For observability studies we need only

analyze dx(t)/dt = Ax(t) ;x(0) = ξ y(t) = Cx(t) • Because x(t) = eAtξ + ( )

0( )

tA te Bu dτ τ τ−∫

y(t) = Cx(t) + Du(t) = C eAtξ + ( )

0( )

tA tCe Bu dτ τ τ−∫ + Du(t)

= C eAtξ + known time function • Known time functions can be subtracted

out

Observability


Modal approach • Dynamic model: x(t) ∈ Rn ; y(t) ∈ Rp dx(t)/dt = Ax(t) ;x(0) = ξ y(t) = Cx(t) • Individual outputs: k = 1, 2, ..., p

C =

c'1c'2!

c'p

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

thus: yk(t) = c'kx(t) ; c'k ∈ Rn • Recall '

1( ) ( )i

n ti i

ix t v e wλ ξ

==∑

thus: ' '

1( ) ( ) ( )i

n ti ik k

iy t c v e wλ ξ

==∑

• Insight

The size of the scalar (c'kvi) determines the

contribution of the ith mode to the kth output

Observability


Modal unobservability • Output structure: k = 1, 2, ..., p '

1( ) ( ) ( )i

n ti ik k

iy t c v e wλ ξ

=′=∑

• ith mode is unobservable in kth output iff: (c'kvi) = 0 (orthogonal) • If (c'kvi) = 0 for all k = 1, 2, ..., p

or Cvi = 0 , then the ith mode is unobservable (from all outputs) • Implication - pick initial condition ξ collinear to vi, ξ = αvi - we cannot detect ξ in output vector y(t) • A system is unobservable iff one or more

of its modes are unobservable

Observability


Modal observability • ith mode is observable (in one or more

outputs) iff: Cvi ≠ 0 vi : ith right eigenvector of A, i.e. Avi = λivi • The system dx(t)/dt = Ax(t) + Bu(t) ;x(0) = ξ y(t) = Cx(t) + Du(t) Is observable iff all of its modes are observable Test: Cvi ≠ 0 for all i = 1, 2, ..., n • Notation Refer to observability of matrix pair [A, C] A = n x n matrix C = p x n matrix

Observability


Detectability • If mode vieλit is unobservable but stable,

i.e. Re{λi} < 0, then mode “i” is called detectable

• If all unobservable modes are detectable,

then [A, C] is called detectable • Notes - if [A, C] is observable, then it is

detectable - if every unstable mode is observable, then

[A, C] is detectable

Observability


Classical observability test • Form the n x (p x n) observability matrix Mo = [C' A'C' (A')2C' ... (A')n-1C'] • If out of the p x n columns of Mo there are

n that are linearly independent, i.e. Rank (Mo) = n Then [A, C] is observable • If Rank (Mo) < n Then [A, C] is unobservable. It may be detectable • Dual to controllability/stabilizability result • No modal information

• No detectability information

Multivariable Transmission Zeros


Multivariable transmission zeros

What will we discuss? • Review of SISO zeros • Definition of MIMO transmission zeros • Calculation of MIMO zeros

- General eigenvalue problem • Implications

- time-domain - frequency-domain



SISO zeros • Numerical example y(s) = g(s) u(s) g(s) = 1

(s 2)ss

++

- zero location: s = -1 - pole locations: s = 0 , s = -2 • Input-output property g(-1) = 0 • Plant absorbs input “frequency” u(s) = 11s+ ⇒ u(t) = e-t y(s) = 1 1

1( 2)s

ss s+

++ = 1( 2)s s+

y(t) = 0.5(1 - e-2t) - y(t) does not contain input “frequency” e-t



SISO pole-zero cancellations • Transfer function y(s) = g(s)u(s) g(s) =

1 2(s p )(s p )s z−

− −

- in general z ≠ p1, z ≠ p2

• Impulse response (u(s) = 1) has solution y(t) = k1ep1t + k2ep2t

• If z = p

1, impulse response is

y(t) = c1ep2t

and if z = p2 y(t) = c2ep1t

• Pole-zero cancellation makes a pole

frequency (natural frequency) invisible in the output impulse response

- loss of observability?



General SISO case (I) • Transfer function

g(s) = 1 1 1 0

1 1 1 0

......

m m m mn n n

s s ss s s

β β β βα α α

− −− −

+ + + ++ + + +

= 1

1

( )( )( )( )

mm kkn

ii

s zB sA ss p

β=

=

−=

−

∏

∏

• Zeros are roots of numerator polynomial

B(s) • Poles are roots of denominator polynomial

A(s) • SISO state-space model dx(t)/dt = Ax(t) + bu(t) ;x(0) = ξ y(t) = c'x(t) • Transfer function in terms of state-space

description g(s) = c'(sΙ - A)-1b = ( )

( )B sA s



General SISO case (II) • Define (sΙ - A)-1 = 1

det(sI A)− Q(s)

⇒ c'(sΙ - A)-1b = ' ( )

det(sI A)c Q s b

− = ( )( )B sA s

• Numerator polynomial is of degree at most

n - 1 (strictly proper system assumed - no feed through term)

• Zeros are roots of B(s) = c'Q(s)b

- zeros depend on A, b, c - both numerical values and directions of b

and c must be known to evaluate zeros • Denominator polynomial: A(s) = det(sΙ - A)

- Poles depend only on A matrix



Impact of zeros on transient response

• SISO plant

g(s) = 2

1 s b1

bs s

⎛ ⎞⎜ ⎟⎝ ⎠+

+ +

Unity DC Gain Poles at s = 1 3

2 2j− ± Zero at s = -b Show unit step response as b = 2 → b = 0.1 (minimum phase system) b = -0.1 → b = -2 (non-minimum phase system)



Minimum phase step response

b=2

b=0.2

b=0.1

b=0.4



Non-minimum phase step response

b=-0.1

b=-0.2

b=2



Definition of MIMO transmission zeros

• Assume "square" plant model (m=p) dx(t)/dt = Ax(t) + Bu(t) ;x(t)∈Rn; u(t)∈Rm y(t) = Cx(t) + Du(t) ;y(t)∈Rm G(s) = C(sΙ - A)-1B+D ;m x m TFM • Definition: plant has a zero at frequency zk

if there exists ξk ∈ Rn ; uk ∈ Rm , (not both zero) so that the solution of

dx(t)/dt = Ax(t) + Bukezkt ;x(0) = ξk y(t) = Cx(t) + Du(t) has the property that y(t) ≡ 0 for all t ≥ 0 • Generalization of "input absorbing

frequency" concept for SISO systems



Definition applied to SISO example

• Plant: g(s) = 1

( 1)ss s

++

• Controllable state-space model

dx(t)/dt = 0 10 1⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

x(t) + 01⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

u(t)

y(t) = [1 1] x(t) • Use u(t) = ukezkt=2e-t (uk = 2, zk = -1)

u(s) = 21s+

x(0) = ξ = 11

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦−

• Solution y(s) = C(sΙ - A)-1ξ + g(s)u(s)

(sΙ - A)-1 = 1 11( 1) 0

ss s s

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

−−

thus y(s) = 1 1 11 1 21 1 1( 1) ( 1)0 1

s sss s s ss

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦

⎣ ⎦ ⎣ ⎦

− ++ +− −−

= 2 2( 1) ( 1)s s s s− +− −

= 0 ;for all t



Issues to be addressed • Given state-space matrices, how do we

calculate - numerical values of MIMO zeros - directions associated with MIMO zeros

• Calculate above using the generalized

eigenvalue problem • Need insight in terms of plant input-output

properties - MIMO pole/zero cancellations - loss of observability/controllability



Generalized eigenvalue problem • Ordinary eigenvalue problem Avi = λivi w'iA = λiw'i ; A is n x n

λi[A] are the roots of det(λΙ - A) • Generalized eigenvalue problem

A: n x n matrix M: n x n matrix

Generalized eigenvalues: µi ; i= 1, 2, ..., p Generalized eigenvectors: αi (right) β'i (left) Aαi = µiMαi ⇒ (µiM - A)αi = 0 β'iA = µiβ'iM ⇒ β'i(µiM - A) = 0 Generalized eigenvalues µi are roots of: det(µM - A) • If M-1 exists, p = n If M is singular, then 0 ≤ p < n



Calculation of MIMO zeros • Square plant model dx(t)/dt = Ax(t) + Bu(t) ;x(t)∈Rn ;u(t)∈Rm y(t) = Cx(t) + Du(t) ;y(t)∈Rm G(s) = C(sΙ - A)-1B + D ;m x m • Transmission zeros location: s = z1, s = z2, ..., s = zp 0 ≤ p ≤ n • Generalized eigenvalue problem

zk I − A −B−C −D

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

ξkuk

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= 00

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

((n + m) x (n + m)) x ((n+m) x 1) = (n+m) x 1 • Solution of GEP provides locations and

directions of MIMO transmission zeros x(0) = ξk u(t) = ukezkt G(s) y(t) ≡ 0



MIMO zeros and the transfer function matrix

• Transfer function matrix G(s) = C(sΙ - A)-1B + D • Implication of GEP

0 = det kz I A BC D

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

− −− −

= det(zkΙ - A)det(C(zkΙ - A)-1B + D) = det(zkΙ - A)det(G(zk)) • No common poles or zeros (zk ≠ λi) ⇒ det(zkΙ - A) ≠ 0 Then for each zero at s = zk det(G(zk)) = 0 • If all poles and zeros are different, then

MIMO zeros are roots of det(G(s))

Multivariable Pole-Zero Cancellations


Multivariable pole-zero cancellations

Issues to be discussed • Impact of directional information in MIMO

poles and zeros

• Loss of controllability and observability in MIMO systems - numerical values of poles and zeros - directions of poles and zeros - pole-zero cancellations in SISO and

MIMO systems



SISO systems (I) • In SISO systems, described by SISO

transfer functions, pole-zero cancellations imply loss of controllability or observability

• Numerical example g(s) = ( 1)

( 1)( 2)( 3)s

s s s s+

+ + +

- Mode associated with pole s = -1 is

either uncontrollable or unobservable - More information needed



SISO systems (II) • System #1 u(t) 1

( 1)s s+ y1(t) ( 1)( 2)( 3)s

s s+

+ + y(t)

e-t mode unobservable in y(t) (observable in y1(t)) • System #2 u(t) ( 1)

( 2)( 3)s

s s+

+ + y1(t) 1

( 1)s s+ y(t)

e-t mode uncontrollable from u(t) (controllable from y1(t)) • Directional information important



A decoupled MIMO system u1(s) ( 1)

( 2)ss s

++

y1(s)

u2(s)

( 1)( 3)s

s s+ + y2(s)

• MIMO zeros: s = -1, s = 0 • MIMO poles: s = 0, s = -1, s = -2, s = -3 • MIMO system controllable and observable

even though there are MIMO poles and zeros in the same location

• Bottom line Directional information is important



Loss of observability • Model: x(t) ∈ Rn ; u(t) ∈ Rm ; y(t) ∈ Rm dx(t)/dt = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t)

G(s) = C(sΙ - A)-1B + D ;m x m • MIMO pole definition (λkΙ - A)vk = 0 • MIMO zero definition (zkΙ - A)ξk - Buk = 0 Cξk = 0 • If λk = zk and ξk = vk then Buk = 0 (uk = 0 in general) and Cvk = 0⇒kth mode is unobservable • If λk = zk but ξk ≠ vk then Cvk ≠ 0⇒kth mode is observable



Impact of unobservability • Hypothesis: kth mode is unobservable ⇒ Cvk = 0 Conclusion: This implies a MIMO pole-zero cancellation • Reasoning Let uk = 0 (λkΙ - A)vk = 0 ⇒ (λkΙ - A)vk - Buk = 0 Cvk = 0

⇒ λk is a MIMO zero with direction 0kv⎡ ⎤

⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦



Unobservability summary • Relation between unobservability and

MIMO pole-zero cancellation • Pole information (λkΙ - A)vk = 0 Location: s = λk Direction: vk • Zero information (zkΙ - A)ξk - Buk = 0 Cξk = 0 Location: s = zk

Direction: kku

ξ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

• Pole-zero cancellation if λk = zk and vk= ξk • Pole-zero cancellation implies that mode is

unobservable • Unobservability of mode implies pole-zero

cancellation



Loss of controllability • Model : x(t) ∈ Rn ; u(t) ∈ Rm ; y(t) ∈ Rm dx(t)/dt = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t)

G(s) = C(sΙ - A)-1B + D ;m x m • MIMO pole definition w'k(λkΙ - A) = 0 • MIMO zero definition η'k(zkΙ - A) - γ'kC = 0 η'kB = 0 • If λk = zk and η'k = w'k then γ'kC = 0 (γk = 0 in general) and w'kB = 0⇒kth mode is uncontrollable • If λk = zk but η'k ≠ w'k then w'kB ≠ 0⇒kth mode is controllable



Impact of uncontrollability • Hypothesis: kth mode is uncontrollable ⇒ w'kB = 0 Conclusion: This implies a MIMO pole-zero cancellation • Reasoning Let γk = 0 w'k(λkΙ - A) = 0 ⇒ w'k(λkΙ - A) - γ'kC = 0 w'kB = 0 ⇒ λk is MIMO transmission zero with direction [w'k 0]



Uncontrollability summary • Relation between uncontrollability and

MIMO pole-zero cancellation

• Pole information w'k(λkΙ - A) = 0 Location: s = λk Direction: w'k • Zero information η'k(zkΙ - A) - γ'kC = 0 η'kB = 0 Location: s = zk Direction: [η'k γ'k] • Pole-zero cancellation if λk = zk and w'k= η'k • Pole-zero cancellation implies that mode is

uncontrollable • Uncontrollability of mode implies pole-zero

cancellation



Impact on output response • System: x(0) = ξ dx(t)/dt = Ax(t) + Bu(t) y(t) = Cx(t) • Frequency domain output description

1

1

( ) (w' )

(w' B)u(s)

ni

ii

i ii

in

i

Cvy s sCvs

ξλ

λ

=

=

= −

+ −

∑

∑

(C vi) = 0 if ith mode is unobservable (w'iB) = 0 if ith mode is uncontrollable



Residue matrix • System: x(0) = ξ dx(t)/dt = Ax(t) + Bu(t) y(t) = Cx(t) • Transfer function matrix: ξ = 0 y(s) = G(s)u(s) G(s) = C(sΙ - A)-1B • Residue expansion G(s) =

1 1i i ii

n n

i i i

wBR Cvs sλ λ= =

=− −′∑ ∑

Ri = C viw'iB = residue matrix of ith pole • If ith MIMO pole at s = λi is cancelled by

MIMO zero, then Ri = 0 Generalization of SISO residue property



Design implications • Understand properties of mathematical

model of physical plant

• Plot poles and zeros in s-plane

• Calculate directions for each pole and zero - check for exact or near pole-zero

cancellations

• Understand controllability and observability properties of each mode

Introduction to feedback systems



Issues to be discussed • Definition of feedback problem

• Mathematics of SISO and MIMO feedback loops



Plant structure disturbances noise controls actuators plant sensors outputs dynamics u(t) y(t) • Homework example elevator δe(t) F - 8 θ pitch command aircraft angle flaperon δf(t) α angle of command attack • Issues

- How many controls? - Which outputs do we want to control? - Other sensor measurements?



Feedback topology

• Focus on one degree-of-freedom control

configuration

• Inputs - Reference inputs r (commands, setpoints) - Disturbances d (process noise, disturbance

variables) - Measurement noise n

• Key issues

- Stability of closed-loop process - Performance in the presence of model

uncertainty, disturbances and noise, i.e.: y ≈ r

- Other specs



SISO feedback systems

• Closed-loop system y(s) = t(s) r(s) t(s) = ( ) ( )

1 ( ) ( )g s k sg s k s+

• Stability (lumped systems) Closed-loop system is stable iff poles of t(s) are in the left-half of the s-plane • Need other tools for systems which are not

finite dimensional



MIMO feedback systems

• Closed-loop system y(s) = T(s) r(s) T(s) = [Ι + G(s)K(s)]-1G(s)K(s) = G(s)K(s)[Ι + G(s)K(s)]-1 • Proof: simple algebra y(s) = G(s)u(s) ;u(s) = K(s)e(s) ⇒ y(s) = G(s)K(s)e(s) e(s) = r(s) - y(s) ⇒ y(s) = G(s)K(s)[r(s) - y(s)] ⇒ [Ι + G(s)K(s)]y(s) = G(s)K(s)r(s) ⇒ y(s) = [Ι + G(s)K(s)]-1G(s)K(s)r(s) • Closed-loop system is stable iff poles of

T(s) in the left-half s-plane



Performance in SISO loops (I)

y(s) = ( ) ( ) ( )1 ( ) ( )

g s k s r sg s k s+ reference

+ 1 d( )1 ( ) ( ) sg s k s+ disturbance

- ( ) ( ) ( )1 ( ) ( )

g s k s n sg s k s+ sensor noise

• Note: y(s) = t(s)r(s) + s(s)d(s) - t(s)n(s) s(s) = 1

1 ( ) ( )g s k s+ sensitivity function



Performance in SISO loops (II) • True error e(s) = r(s) - y(s) • Error response: Want e(t) ≈ 0 e(s) = 1

1 ( ) ( )g s k s+ [r(s) - d(s)] + ( ) ( )1 ( ) ( )g s k sg s k s+ n(s)

• Note : e(s) = s(s)[r(s) - d(s)] + t(s)n(s) • Want s(s) = 0 and t(s) = 0 • Conflict s(s) + t(s) = 1 1

1 ( ) ( )g s k s+ +

g(s)k(s)1+ g(s)k(s) = 1



Performance in MIMO loops (I)

• Output response: want y(t) ≈ r(t) y(s) = [Ι + G(s)K(s)]-1G(s)K(s)r(s) + [Ι + G(s)K(s)]-1d(s) - [Ι + G(s)K(s)]-1G(s)K(s)n(s) • Note y(s) = T(s)r(s) + S(s)d(s) - T(s)n(s) S(s) = [Ι + G(s)K(s)]-1 :sensitivity TFM T(s) = [Ι + G(s)K(s)]-1G(s)K(s) :closed-loop TFM



Performance in MIMO loops (II) • True error e(s) = r(s) - y(s) • Error response: Want e(t) ≈ 0 e(s) = [Ι + G(s)K(s)]-1 [r(s) - d(s)] + [Ι + G(s)K(s)]-1G(s)K(s) n(s) • Note: e(s) = S(s)[r(s) - d(s)] + T(s)n(s) • Want S(s) = 0 and T(s) = 0 • Conflict S(s)+T(s)= Ι [Ι + G(s)K(s)]-1 + [Ι + G(s)K(s)]-1G(s)K(s) = Ι



Feedback design objectives • Stability

- nominal stability - stability-robustness

• Performance

- nominal performance - performance-robustness

• Performance attributes

- command following - disturbance rejection - insensitivity to sensor noise

• Must have performance specs • May not be able to meet specs

- unrealistic - inherent limitations

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376 069 Multivariable feedback control V2 · 376_069 Multivariable feedback control V2 1 of 47...

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