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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
376_069 Multivariable feedback control V2 2 of 47
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
376_069 Multivariable feedback control V2 3 of 47
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
376_069 Multivariable feedback control V2 4 of 47
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
376_069 Multivariable feedback control V2 5 of 47
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
376_069 Multivariable feedback control V2 6 of 47
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
376_069 Multivariable feedback control V2 7 of 47
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
376_069 Multivariable feedback control V2 8 of 47
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
376_069 Multivariable feedback control V2 9 of 47
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
376_069 Multivariable feedback control V2 10 of 47
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
376_069 Multivariable feedback control V2 11 of 47
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
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 12 of 47
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
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 13 of 47
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?
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 14 of 47
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
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 15 of 47
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
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 16 of 47
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)
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 17 of 47
Minimum phase step response
b=2
b=0.2
b=0.1
b=0.4
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 18 of 47
Non-minimum phase step response
b=-0.1
b=-0.2
b=2
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 19 of 47
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
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 20 of 47
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
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 21 of 47
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
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 22 of 47
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
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 23 of 47
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
Multivariable Transmission Zeros
376_069 Multivariable feedback control V2 24 of 47
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
376_069 Multivariable feedback control V2 25 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 26 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 27 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 28 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 29 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 30 of 47
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⎡ ⎤
⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 31 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 32 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 33 of 47
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]
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 34 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 35 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 36 of 47
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
Multivariable Pole-Zero Cancellations
376_069 Multivariable feedback control V2 37 of 47
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
376_069 Multivariable feedback control V2 38 of 47
Introduction to feedback systems
Issues to be discussed • Definition of feedback problem
• Mathematics of SISO and MIMO feedback loops
Introduction to feedback systems
376_069 Multivariable feedback control V2 39 of 47
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?
Introduction to feedback systems
376_069 Multivariable feedback control V2 40 of 47
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
Introduction to feedback systems
376_069 Multivariable feedback control V2 41 of 47
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
Introduction to feedback systems
376_069 Multivariable feedback control V2 42 of 47
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
Introduction to feedback systems
376_069 Multivariable feedback control V2 43 of 47
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
Introduction to feedback systems
376_069 Multivariable feedback control V2 44 of 47
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
Introduction to feedback systems
376_069 Multivariable feedback control V2 45 of 47
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
Introduction to feedback systems
376_069 Multivariable feedback control V2 46 of 47
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
Introduction to feedback systems
376_069 Multivariable feedback control V2 47 of 47
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