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)
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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
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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
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)
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