Controller Design* [9#]

Trade-offs in MIMO feedback

design [9.1]

c c

c

q- - - - ? -

�6

6r +

-K G +

+

d

y

++

n

u

Figure 9.1: One degree-of-freedom feedback

y(s) = T (s)r(s) + S(s)d(s) − T (s)n(s) (9.1)

u(s) = K(s)S(s) [r(s) − n(s) − d(s)] (9.2)

1/17

Based on lecture notes from "Regelsysteme II", a course presented at ETH Zurich in 2005*# Refers to relevant section in:S. Skogestad, I. Postlehwaite, Multivariable Feedback Control,Second Edition, Wiley, Chichester, 2005. ISBN-10 0-470-01168-8

TU Wien ACIN course 376.069 Multivariable feedback control - Winter Semester 2015

Closed-loop objectives:

1. For disturbance rejection make σ̄(S) small.

2. For noise attenuation make σ̄(T ) small.

3. For reference tracking make σ̄(T ) ≈ σ(T ) ≈ 1.

4. For control energy reduction make σ̄(KS) small.

5. For robust stability in the presence of an additive

perturbation make σ̄(KS) small.

6. For robust stability in the presence of a

multiplicative output perturbation make σ̄(T )

small.

The closed-loop requirements 1 to 6 cannot all be

satisfied simultaneously. Feedback design is therefore

a trade-off over frequency of conflicting objectives.

2/17

Over specified frequency ranges, we can approximate

the closed-loop requirements by the following

open-loop objectives:

1. For disturbance rejection make σ(GK) large;

valid for frequencies at which σ(GK) ≫ 1.

2. For noise attenuation make σ̄(GK) small; valid

for frequencies at which σ̄(GK) ≪ 1.

3. For reference tracking make σ(GK) large; valid

for frequencies at which σ(GK) ≫ 1.

4. For control energy reduction make σ̄(K) small;

valid for frequencies at which σ̄(GK) ≪ 1.

5. For robust stability to an additive perturbation

make σ̄(K) small; valid for frequencies at which

σ̄(GK) ≪ 1.

6. For robust stability to a multiplicative output

perturbation make σ̄(GK) small; valid for

frequencies at which σ̄(GK) ≪ 1.

3/17

log magnitude

Performanceboundary

Robust stability, noise attenuation,control energy reduction boundary

��(GK)

�(GK)

log(w)

wl

wh

Figure 9.2: Design trade-offs for the multivariable loop transfer function

4/17

General control problem

formulation [9.3.1]

- -

�

-w z

vu

P

K

Figure 9.8: General control configuration

z

v

= P (s)

w

u

=

P11(s) P12(s)

P21(s) P22(s)

w

u

(9.24)

u = K(s)v (9.25)

5/17

The state-space realization of the generalized plant

P is given by

Ps=

A B1 B2

C1 D11 D12

C2 D21 D22

(9.26)

z = Fl(P,K)w (9.27)

where

Fl(P, K) = P11 + P12K(I − P22K)−1P21 (9.28)

H∞ control involve the minimization of the

H∞ norms of Fl(P, K)

6/17

The term H∞

The H∞ norm of a stable scalar transfer function

f(s) is simply the peak value of |f(jω)| as a function

of frequency, that is,

‖f(s)‖∞ ∆= max

ω|f(jω)| (2.101)

The symbol ∞ comes from:

maxω

|f(jω)| = limp→∞

(∫ ∞

−∞|f(jω)|pdω

)1/p

The symbol H stands for “Hardy space”, and H∞ is

the set of transfer functions with bounded ∞-norm,

which is simply the set of stable and proper transfer

functions.

7/17

H∞ optimal control [9.3.4]

With reference to the general control configuration ofFigure 9.8, the standard H∞ optimal control problem isto find all stabilizing controllers K which minimize

‖Fl(P,K)‖∞ = maxω

σ̄(Fl(P,K)(jω)) (9.42)

This has a time domain interpretation as the induced

(worst-case) 2-norm. Let z = Fl(P,K)w, then

‖Fl(P,K)‖∞ = maxw(t) 6=0

‖z(t)‖2

‖w(t)‖2(9.43)

where ‖z(t)‖2 =√∫∞

0

∑i |zi(t)|2dt is the 2-norm of

the vector signal.

It is often computationally (and theoretically)

simpler to design a sub-optimal one (i.e. one close to

the optimal controller in the sense of the H∞ norm).

Let γmin be the minimum value of ‖Fl(P,K)‖∞ over

all stabilizing controllers K. Then the H∞sub-optimal control problem is: given a γ > γmin,

find all stabilizing controllers K such that

‖Fl(P,K)‖∞ < γ

8/17

Matlab command:>> [K,N,gamma] = hinfsyn(P,ny,nu)

Mixed-sensitivity H∞ control [9.3.5]

To optimize performance, minimize ‖w1S‖∞,

to minimize control inputs, minimize ‖w2KS‖∞.

Compromise: ∥∥∥∥[w1S

w2KS

]∥∥∥∥∞

(9.52)

General setting: disturbance d as a single exogenous

input, error signal z =[zT1 zT

2

]T, where

z1 = W1y and z2 = −W2u, (Figure 9.10).

c cq q- -

-

-

?

-

-

6

�

-

w = dz1

z2

z

P

K

G

W1

−W2

y-

+

Setpoint r = 0 vu

+

+

Figure 9.10: S/KS mixed-sensitivity optimization in standard form (regulation)

9/17

Thus z1 = W1Sw and z2 = W2KSw and:

P11 =

W1

0

P12 =

W1G

−W2

P21 = −I P22 = −G(9.53)

where the partitioning is such that

z1

z2- - -v

=

P11 P12

P21 P22

w

u

(9.54)

and

Fl(P,K) =

W1S

W2KS

(9.55)

10/17

Another useful mixed sensitivity optimization

problem, is to find a stabilizing controller which

minimizes ∥∥∥∥∥∥

W1S

W2T

∥∥∥∥∥∥

(9.56)

∞

The S/T mixed-sensitivity minimization problem canbe put into the standard control configuration as

shown in Figure 9.12.

cq q-

-

-

?

-

-

�

-

w = rz1

z2

z

P

K

G

W1

W2

vu

+-

Figure 9.12: S/T mixed-sensitivity optimization in standard form

11/17

P11 =

W1

0

P12 =

−W1G

W2G

P21 = I P22 = −G(9.57)

12/17

Robust performance

Analysis:

RP ⇐⇒ µ∆̃(N(jω)) < 1, for all ω

∆̃ = diag(∆,∆p)

N(s)

∆

we

z v

Synthesis:

minK(s) stabilizing

γ

subject to:

µ∆̃(Fl (P (jω),K(jω))) < γ, for all ω P (s)

K(s)

∆

we

ym u

z v

13/17

Weighted sensitivity [2.8.2]

Typical specifications in terms of S:

1. Minimum bandwidth frequency ω∗B .

2. Maximum tracking error at selected frequencies.

3. System type, or alternatively the maximum

steady-state tracking error, A.

4. Shape of S over selected frequency ranges.

5. Maximum peak magnitude of S, ‖S(jω)‖∞ ≤M .

Specifications may be captured by an upper bound,

1/|wP (s)|, on ‖S‖.

14/17

10−2

10−1

100

101

10−2

10−1

100

101

Magnitude

Frequency [rad/s]

|1/wP ||S|

(a) Sensitivity S and performance weight wP

10−2

10−1

100

101

0

1

2

3

Magnitude

Frequency [rad/s]

‖wP S‖∞

(b) Weighted sensitivity wP S

Figure 2.28: Case where |S| exceeds its bound 1/|wP |, resulting in ‖wP S‖∞ > 1

15/17

|S(jω)| < 1/|wP (jω)|, ∀ω (2.103)

⇔ |wP S| < 1, ∀ω ⇔ ‖wP S‖∞ < 1 (2.104)

Typical performance weight:

wP (s) =s/M + ω∗

B

s+ ω∗BA

(2.105)

10−2

10−1

100

101

102

10−2

10−1

100

Magnitude

Frequency [rad/s]

M

A

ω∗B

Figure 2.29: Inverse of performance weight. Exact and

asymptotic plot of 1/|wP (jω)| in (2.105)

16/17

To get a steeper slope for L (and S) below the

bandwidth:

wP (s) =(s/M1/2 + ω∗

B)2

(s+ ω∗BA

1/2)2(2.106)

17/17