Robust and Optimal Control, 2014, 03rd.
Glory of LQG Control
LQG (Linear Quadratic Gaussian) Control
Special Issue on Linear-Quadratic-Gaussian Problem
IEEE TAC Special Issue,16 - 6, 1971 (About 340 pages)
3. Robustness and Uncertainty
3.1 Why Robustness?
[SP05, Sec. 4.1.1, 7.1, 9.2]
3.2 Representing Uncertainty
[SP05, Sec. 7.2～7.4]
3.3 Uncertain Systems
[SP05, Sec. 8.1～8.3]
3.4 Systems with Structured Uncertainty
[SP05, Sec. 8.2]
M.Athans
Linear System
Cost Function
Reference:
[SP05] S. Skogestad and I. Postlethwaite,
Multivariable Feedback Control; Analysis and Design,
Second Edition, Wiley, 2005.
2
Glory and Drawback of LQG Control
Optimal
Control
Theory
LQG
Gain [dB]
L.S.Pontryagin
H.H.Rosenbrock (UMIST), IEEE TAC Special Issue, 16 - 6, 1971
R.E.Kalman
Stability
Theory
A.M.Lyapunov
“Gap between Theory and Practice”
(1890)
R.Bellman
Feedback
Theory
Phase [°]
Linear System
Theory
Drawback of LQG Control
Stability Margin in Multivariable Systems
Good, Bad, or Optimal?
Frequency [rad/s]
essential requirement … that changes of loop gains …
in all combinations, should leave the system with an
adequate stability margin.
H.W.Bode
H.Nyquist
3
Catastrophe of LQG
Applications of LQG Control
4
Blind Spot of LQG Control
Stability Margin of LQ Control
1964 Circle Criterion Inverse Problem
A.E.Bryson. Jr., IEEE TAC, 22 - 5, 1977
In the frequency domain, the vector locus of the open loop transfer
function
never enters the circle centered at
with radius 1
F-8C Crusader Aircraft
(i) Gain Margin：
(ii) Phase Margin： More than or equal 60°
(iii) Allowable Range of Gain Decrease：Until 50% (1/2)
Trident Submarine (1975)
Stability Margin in
Multivariable Systems
When is a Linear Control
System Optimal?
Multivariable LQ
Discussions
… very limited success …
… not very practical …
5
R.E. Kalman, ASME,
86 - D, 1964
Nyquist Plot of
M.Safonov
6
1
Robust and Optimal Control, 2014, 03rd.
Blind Spot of LQG Control
Stability Margin of LQG Control (Robustness)
Blind Spot of LQG Control
Nyquist Plot of
(i) 状態フィードバックという現実的で
はない制御則が金科玉条であり，そ
れを補う観測器も次数の点で実用性
に乏しい．
(ii) 定常特性がほとんど無視されてい
た．たとえば，最適レギュレータはイン
パルス上の外乱しか処理できない．
J.Doyle, G.Stein, IEEE TAC, 24 - 4, 1979
LQG Regulator
Phase Margin： 15°Oops…
木村, “多変数制御系の理論と応用-I,”
システムと制御, Vol. 22, No. 5, pp. 293-301, 1978
Nyquist plot for the resulting
observer-based controller is
shown in Fig. 2. Oops… less
than 15°phase margin.
7
Representing Uncertainty
Weaker as controller
in order to weaken
high freq.
8
伊藤, 木村, 細江, “線形制御系の設計理論,”
計測自動制御学会編, コロナ社, 1978
Uncertainty Regions
[SP05, Ex., p. 265]
Idealization
Simplification
Integrator
(-20dB/dec)
Representing Uncertainty in SISO Systems
System and Model
Real Physical
System
Phase Delay -90°
(high frequencies)
フィードバック制御系では高周波雑音
を抑制するため，開ループ伝達関数
の高周波特性は減衰の大きい方がよ
く, 実際の制御系では，必ずしも円条
件を満足させないのが普通である．と
はいっても，最適レギュレータの重要
性は，少しも減ぜられていない．
Observation
[SP05, p. 265]
First Order Plant Model
Case 1: Uncertain Gain
Nominal Value
Uncertainty
[Ex.]
for
for
Uncertainty
any
Average
Case 2: Uncertain Gain/Time Constant
Mathematical
Model
Analysis
Model vs. System
Multiplicative Uncertainty
9
10
Obtaining Weight
[SP05, p. 267]
[SP05, p. 268]
: Perturbed Plant Model
: Nominal Plant Model
Step 1. Select a nominal model
Step 2. At each frequency, find the smallest radius
: Uncertainty Weight
any
includes the possible plants
which
:
A Set of Plant Models
ボード線図
ボード線図
20
20
10
10
0
-10
振幅(dB)
(dB)
振幅
to cover the set:
Magnitude [dB]
Step 3. Choose a weight
Disc Uncertainty
Center:
Radius:
-20
-20
-30
-30
-40
-40
-50
-50
-60
-60
-70
-70
-80
-2
-80
10 -2
10
11
-1
10 -1
10
0
10 0
10
1
10 1
10
周波数 (rad/sec)
周波数 (rad/sec)
2
10 2
10
Frequency [rad/s]
3
10 3
10
4
10 4
10
12
2
Robust and Optimal Control, 2014, 03rd.
Uncertainty Weight
[SP05, Ex. 7.6] Time-delay Variations
[SP05, p. 273]
： (Approximately) the frequency at which
the relative uncertainty reaches 100%.
： Magnitude of
at high frequency
： Relative uncertainty at steady-state
(p. 269)
Step 1. Nominal Model:
Step 2.
20
Step 3.
10
Magnitude [dB]
Frequency at which
the relative uncertainty
exceeds 100%
？
×
0
-10
-20
-30
-40
-50
-60 -3
10
Phase Information: Lost
-2
10
10
-1
0
10
10
1
10
2
Frequency [rad/s]
13
14
Representing Uncertainty in MIMO Systems
Report
Multiplicative (Output) Uncertainty
MATLAB: Robust Control Toolbox ver. 5.0
Robust Control Toolbox
- Analysis and Synthesis Toolbox
Uncertainty Weight
LMI Control Toolbox
Computer Access:
You can use MATLAB 2013a at GSIC. If you want to know
how to use, ask Learning Assistants (Wasa, Sugimoto, Funada).
Place: S5-303A e-mail: ta@fl.ctrl.titech.ac.jp
15
［Ex.］ Spinning Satellite: Uncertainty Weight [SP05, p. 295]
Uncertain Plant Model (Real System)
16
［Ex.］ Spinning Satellite: Uncertainty Weight [SP05, p. 295]
Step 2.
MATLAB Command
k1 = ureal('k1',1,'Per',[-40 40]);
k2 = ureal('k2',1,'Per',[-40 40]);
L1 = ureal('L1',0,'Range',[0 0.04]);
L2 = ureal('L2',0,'Range',[0 0.04]);
f1 = k1*tf([-L1/2 1],[L1/2 1]);
f2 = k2*tf([-L2/2 1],[L2/2 1]);
f = [f1 0;0 f2];
farray = usample(f,100);
Uncertain Gain:
Time Delay Variation:
100 randomly generated parameters
Multiplicative (Output) Uncertainty
Parray=farray*Pnom;
Parray=frd(Parray,logspace(0,4,100));
Eo=(Parray-Pnom)*inv(Pnom);
figure
sigma(Eo,'b-');
hold on; grid on;
Step 1. Nominal Model:
17
18
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Robust and Optimal Control, 2014, 03rd.
［Ex.］ Spinning Satellite: Uncertainty Weight [SP05, p. 295]
［Ex.］ Spinning Satellite: Time Response for Uncertain Plant
Step 3.
？
MATLAB Command
time = 0:0.01:3;
step_ref = ones(1,length(time));
Filter = tf(1,[0.1 1]);
step_ref_filt = lsim(Filter,step_ref,time);
ref = [step_ref_filt'; zeros(1,length(time))];
〇
Manual Fitting
MATLAB Command
Auto Fitting
MATLAB Command
r0 = 0.4; rinf = 2.5; tau = 0.04;
wM = tf([tau r0], [tau/rinf 1]);
WM = eye(2)*wM;
sigma(WM,'r');
[Usys,uInfo] = ucover(Parray,Pnom,1,’InputMult');
wM = uInfo.W1;
WM = eye(2)*wM;
sigma(WM,'r');
Order of
[yhi1,t] = lsim(Pnom,ref,time);
plot(t,yhi1,'r-');
plot(time,ref,'g-.');
For nominal model
19
Uncertain Systems
Unstructured Uncertainty [SP05, p. 293]
Unstructured Uncertainty
figure
hold on; grid on;
Parray=farray*Pnom;
for i = 1 : 100
[yhi,t] = lsim(Parray(:,:,i),ref,time);
plot(t,yhi(:,1),'b-');
plot(t,yhi(:,2),’g-');
end
20
[SP05, pp. 113, 543]
Perturbed Model Set
Multiplicative
（Output）
Multiplicative
（Input）
Inverse Multiplicative
（Output）
Inverse Multiplicative
（Input）
Additive
Upper Linear Fractional Transformation (LFT)
Inverse Additive
21
Systems with Structured Uncertainty
Input Multiplicative/Diagonal Uncertainty
[SP05, p. 296]
[Ex.]
Additive, Input and Output Multiplicative Uncertainty
[Ex.]
noise
Input
external disturbances
Actuators
System
22
Aileron
Canard
Flap
noise
Sensors
Rudder
Output
Elevator
Process
Elevon
NASA HIMAT
X-29 Aircraft
×
Block Diagonal
Stability Margin in Multivariable Systems
Block Diagonal
23
A.E.Bryson. Jr., IEEE TAC, 22 - 5, 1977
24
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Robust and Optimal Control, 2014, 03rd.
Structured Uncertainty
Big Picture
[SP05, p. 296]
[SP05, pp. 12, 289]
Structured Uncertainty
Unstructured
Block Diagonal
LQG
: Generalized Plant
: Controller
25
3. Robustness and Uncertainty
26
Next Class
4. Robust Stability and Loop Shaping
3.1 Why Robustness?
[SP05, Sec. 4.1.1, 7.1, 9.2]
3.2 Representing Uncertainty
[SP05, Sec. 7.2～7.4]
4.1 Robust Stability
[SP05, Sec. 7.5, 8.4, 8.5]
3.3 Uncertain Systems
[SP05, Sec. 8.1～8.3]
4.2 Robust Stabilization
[SP05, Sec. 7.5, 8.4, 8.5]
4.3 Mixed Sensitivity and Loop Shaping
3.4 Systems with Structured Uncertainty
[SP05, Sec. 2.6, 2.8, 9.1]
[SP05, Sec. 8.2]
1st Report
Reference:
Reference:
[SP05] S. Skogestad and I. Postlethwaite,
Multivariable Feedback Control; Analysis and Design,
Second Edition, Wiley, 2005.
[SP05] S. Skogestad and I. Postlethwaite,
Multivariable Feedback Control; Analysis and Design,
Second Edition, Wiley, 2005.
Coprime Factor Uncertainty [SP05, p. 304]
Parametric Uncertainty: State Space
[Ex.]
[SP05, p. 292]
[Ex.]
Loop Shaping
29
30
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Robust and Optimal Control, 2014, 03rd.
Parametric Uncertainty: State Space (Cont.)
Diagonal Uncertainty
[SP05, p. 292]
[SP05, pp. 289, 296, 300]
Allowed Structure
Parametric Uncertainties
Nonparametric Uncertainties
Allowed Perturbations
cf. Linear parameter varying (LPV) system
Polytopic-type system
Affine parameter-dependent system
Gain Scheduled
Problem
31
32
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