Topic #15
16.30/31 Feedback Control Systems
State-Space Systems
• Closed-loop control using estimators and regulators.
• Dynamics output feedback
• “Back to reality”
• Reading: FPE 7.6
Fall 2010
16.30/31 15–2
Combined Estimators and Regulators
• Now evaluate stability and/or performance of a controller when K
designed assuming
u(t) = −Kx(t)
but implemented as
u(t) = −Kx̂(t)
• Assume we have designed a closed-loop estimator with gain L
x̂˙(t) = (A − LC)x̂(t) + Bu(t) + Ly(t)
ŷ(t) = Cx̂(t)
• The closed-loop system dynamics are given by:
ẋ(t)
x̂˙(t)
y(t)
ŷ(t)
u(t)
=
=
=
=
=
Ax(t) + Bu(t)
(A − LC)x̂(t) + Bu(t) + Ly(t)
Cx(t)
Cx̂(t)
−Kx̂(t)
• Which can be compactly written as:
��
�
� �
�
A
−BK
x˙ (t)
x(t)
=
⇒ ẋcl (t) = Acl xcl (t)
˙
x̂(t)
x̂(t)
LC A − BK − LC
• This does not look too good at this point – not even obvious that
the closed-system is stable.
λi(Acl ) =??
November 5, 2010
Fall 2010
16.30/31 15–3
• Can fix this problem by introducing a new variable x̃ = x − x̂ and
then converting the closed-loop system dynamics using the
similarity transformation T
��
�
� �
�
I 0
x(t)
x(t)
x̃cl (t) �
=
= T xcl (t)
x̂(t)
x̃(t)
I −I
• Note that T = T −1
• Now rewrite the system dynamics in terms of the state x̃cl (t)
Acl ⇒ T Acl T −1 � Ācl
• Since similarity transformations preserve the eigenvalues we are
guaranteed that
λi(Acl ) ≡ λi(Ācl )
• Work through the math:
�
��
��
�
I 0
A
−BK
I 0
Ācl =
I −I
LC A − BK − LC
I −I
�
��
�
A
−BK
I 0
=
A − LC −A + LC
I −I
�
�
A − BK BK
=
0
A − LC
• Because A¯cl is block upper triangular, we know that the closed-loop
poles of the system are given by
det(sI − Ācl) � det(sI − (A − BK)) · det(sI − (A − LC)) = 0
November 5, 2010
Fall 2010
16.30/31 15–4
• Observation: Closed-loop poles for this system consist of
the union of the regulator poles and estimator poles.
• So we can just design the estimator/regulator separately and com­
bine them at the end.
• Called the Separation Principle.
• Keep in mind that the pole locations you are picking for these two
sub-problems will also be the closed-loop pole locations.
• Note: Separation principle means that there will be no ambiguity or
uncertainty about the stability and/or performance of the closed-loop
system.
• The closed-loop poles will be exactly where you put them!!
• And we have not even said what compensator does this amazing
accomplishment!!!
November 5, 2010
Fall 2010
16.30/31 15–5
The Compensator
• Dynamic Output Feedback Compensator is the combination of
the regulator and estimator using u(t) = −Kx̂(t)
x̂˙(t) = (A − LC)x̂(t) + Bu(t) + Ly(t)
⇒ x̂˙(t) = (A − BK − LC)x̂(t) + Ly(t)
u(t) = −Kx̂(t)
• Rewrite with new state xc(t) ≡ x̂(t)
ẋc(t) = Acxc(t) + Bcy(t)
u(t) = −Ccxc(t)
where the compensator dynamics are given by:
Ac � A − BK − LC , Bc � L , Cc � K
• Note that the compensator maps sensor measurements to actuator
commands, as expected.
• Closed-loop system stable if regulator/estimator poles placed in the
LHP, but compensator dynamics do not need to be stable.
λi(A − BK − LC) =??
November 5, 2010
Fall 2010
16.30/31 15–6
• For consistency with the implementation of classical controllers, define
the compensator transfer function as
Gc(s) = Cc(sI − Ac)−1Bc
so that
u(t) = −Gc(s)y(t)
• Note that it is often very easy to provide classical interpretations (such
as lead/lag) for the compensator Gc(s).
• Can implement this compensator with a reference command r(t) using
feedback on e(t) = r(t) − y(t) rather than just −y(t)
r
e
−
Gc(s)
u
Gp(s)
y
⇒ u(t) = Gc(s)e(t) = Gc(s)(r(t) − y(t))
• Still have u(t) = −Gc(s)y(t) if r(t) = 0.
• So now write state space model of the compensator as
ẋc(t) = Acxc(t) + Bce(t)
u(t) = Ccxc(t)
• Intuitively appealing setup because it is the same approach used
for the classical control, but it turns out not to be the best approach
⇒ More on this later.
November 5, 2010
Fall 2010
16.30/31 15–7
Mechanics
• Basics:
e(t) = r(t) − y(t),
Gc(s) :
u(t) = Gce(t),
y(t) = Gu(t)
ẋc(t) = Acxc(t) + Bce(t) ,
G(s) :
ẋ(t) = Ax(t) + Bu(t)
• Loop dynamics L(s) = G(s)Gc(s)
⇒
u(t) = Ccxc(t)
,
y(t) = Cx(t)
y(t) = L(s)e(t)
ẋ(t) = Ax(t) +BCc xc(t)
ẋc(t) =
+Ac xc(t) +Bce(t)
�
L(s) :⇒
ẋ(t)
ẋc(t)
�
�
A BCc
��
x(t)
xc(t)
0 Ac
�
�
�
� x(t)
y(t) = C 0
xc(t)
=
�
�
+
0
Bc
�
e(t)
• To “close the loop”, note that e(t) = r(t) − y(t), then
�
��
�
�
� �
��
�
��
�
� x(t)
A BCc
ẋ(t)
x(t)
0
=
+
r(t) − C 0
ẋc(t)
xc(t)
Bc
xc(t)
0 Ac
�
��
� �
�
A
BCc
x(t)
0
=
+
r(t)
xc(t)
Bc
−BcC Ac
�
�
�
� x(t)
y(t) = C 0
xc(t)
• Acl is not exactly the same as on page 15-2 because we have re­
arranged where the negative sign enters into the problem. Same
result though.
November 5, 2010
Fall 2010
16.30/31 15–8
Scaling DOFB Compensator
• DOFB = Dynamic Output Feedback
• Assume that the closed-loop system has been written for the system
on 15-6, then it is still possible that the DC gain of the response from
r to y is not unity
• Scalar input/output assumed for now
• Need to scale the closed-loop to fix this using r ⇒ N r
• Closed-loop dynamics are
ẋcl (t) = Acl xcl (t) + Bcl N r(t)
y(t) = Ccl xcl (t)
• So to have the TF of y/u have unity gain at DC, must have
�
�−1
N = − Ccl (Acl )−1Bcl
November 5, 2010
Fall 2010
16.30/31 15–9
Simple Compensator Example
• Let G(s) = 1/(s2 + s + 1) with state-space model given by:
�
�
� �
�
�
0 1
0
A=
, B=
, C= 1 0 , D=0
1
−1 −1
• Design the regulator to place the poles at s = −4 ± 4j
�
�
λi(A − BK) = −4 ± 4j ⇒ K = 31 7
• Regulator pole time constant τc = 1/ζωn ≈ 1/4 = 0.25 sec
• Put estimator poles with faster time constant τe ≈ 1/10
• Use real poles, so Φe(s) = (s + 10)2
�
�−1 �
�
C
0
L = Φe(A)
CA
1
⎛�
�
�2
� �
�⎞ �
�−1 � �
0 1
0 1
100 0
1 0
0
⎠
= ⎝
+ 20
+
1
−1 −1
−1 −1
0 100
0 1
�
�� � � �
99 19
0
19
=
=
1
80
−19 80
November 5, 2010
Fall 2010
16.30/31 15–10
• Compensator:
Ac = �
A − BK −� LC
� �
� �
�
�
0 1
0 �
19 �
−
=
31 7 −
1 0
1
80
−1 −1
�
�
−19 1
=
−112 −8
�
Bc = L =
19
80
�
�
�
Cc = K = 31 7
1
2
3
4
5
6
7
a=[0 1;−1 −1];b=[0 1]';c=[1 0];d=0;
k=acker(a,b,[−4+4*j;−4−4*j]);
l=acker(a',c',[−10 −10]')';
%
% For state space for G c(s)
%
ac=a−b*k−l*c;bc=l;cc=k;dc=0;
• Compensator transfer function:
Gc(s) = Cc(sI − Ac)−1Bc �
= 1149
U
E
s + 2.553
s2 + 27s + 264
• Note that the compensator has a low frequency real zero and two
higher frequency poles.
• Thus it looks like a “lead” compensator.
November 5, 2010
Fall 2010
16.30/31 15–11
Fig. 1: Simple plant – compensator ∼ lead 2–10 rads/sec.
Fig. 2: Loop transfer function L(s) shows the slope change near ωc = 5
rad/sec. Note that we have a large PM and GM.
November 5, 2010
Fall 2010
16.30/31 15–12
Fig. 3: Freeze compensator poles and zeros; look at root locus of closed-
loop poles versus an additional loop gain (nominally α = 1.)
Fig. 4: Closed-loop transfer function.
November 5, 2010
Fall 2010
Code: Dynamic Output Feedback Example
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
% Combined estimator/regulator design for a simple system
% G= 1/(sˆ2+s+1)
% Jonathan How
% Fall 2010
%
%close all;clear all
set(0,'DefaultLineLineWidth',2)
set(0,'DefaultFigureColor','None')
set(0,'DefaultlineMarkerSize',8);set(0,'DefaultlineMarkerFace','b')
set(0, 'DefaultAxesFontSize', 12);set(0, 'DefaultTextFontSize', 12);
%
a=[0 1;−1 −1];b=[0 1]';c=[1 0];d=0;
k=acker(a,b,[−4+4*j;−4−4*j]);
l=acker(a',c',[−10 −11]')';
%
% For state space for G c(s)
%
ac=a−b*k−l*c;bc=l;cc=k;dc=0;
19
20
21
G=ss(a,b,c,d);
Gc=ss(ac,bc,cc,dc);
22
23
24
25
f=logspace(−1,2,400);
g=freqresp(G,f*j);g=squeeze(g);
gc=freqresp(Gc,f*j);gc=squeeze(gc);
26
27
28
29
30
31
32
33
34
35
36
figure(1);clf;orient tall
subplot(211)
loglog(f,abs(g),f,abs(gc),'−−');axis([.1 1e2 .02 1e2])
xlabel('Freq (rad/sec)');ylabel('Mag')
legend('Plant G','Compensator Gc','Location','East');
grid
subplot(212)
semilogx(f,180/pi*angle(g),f,180/pi*angle(gc),'−−');
axis([.1 1e2 −200 50])
xlabel('Freq (rad/sec)');ylabel('Phase (deg)');grid
37
38
L=g.*gc;
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
figure(2);clf;orient tall
subplot(211)
loglog(f,abs(L),[.1 1e2],[1 1]);axis([.1 1e2 .2 1e2])
xlabel('Freq (rad/sec)');ylabel('Mag')
legend('Loop L');
grid
subplot(212)
semilogx(f,180/pi*phase(L.'),[.1 1e2],−180*[1 1]);
axis([.1 1e2 −290 0])
xlabel('Freq (rad/sec)');ylabel('Phase (deg)');grid
%
% loop dynamics L = G Gc
%
al=[a b*cc;zeros(2) ac];
bl=[zeros(2,1);bc];
cl=[c zeros(1,2)];
dl=0;
figure(3)
rlocus(al,bl,cl,dl)
%
% closed−loop dynamics
% unity gain wrapped around loop L
%
acl=al−bl*cl;bcl=bl;ccl=cl;dcl=d;
%
% scale closed−loop to get zero SS error
N=inv(ccl*inv(−acl)*bcl)
67
68
69
70
71
72
73
74
75
hold on;plot(eig(acl),'d','MarkerFaceColor','b');hold off
grid on
axis([−25 5 −15 15])
%
% closed−loop freq response
%
Gcl=ss(acl,bcl*N,ccl,dcl*N);
gcl=freqresp(Gcl,f*j);gcl=squeeze(gcl);
76
November 5, 2010
16.30/31 15–13
Fall 2010
77
78
79
80
81
82
16.30/31 15–14
figure(4);clf
loglog(f,abs(g),f,abs(gcl),'−−');
axis([.1 1e2 .01 1e2])
xlabel('Freq (rad/sec)');ylabel('Mag')
legend('Plant G','closed−loop G {cl}');grid
title(['Factor of N=',num2str(round(1000*N)/1000),' applied to Closed loop'])
83
84
85
figure(5);clf
margin(al,bl,cl,dl)
86
87
88
89
90
91
figure(1);export
figure(2);export
figure(3);export
figure(4);export
figure(5);export
fig
fig
fig
fig
fig
reg
reg
reg
reg
reg
est1
est2
est3
est4
est5
−pdf;
−pdf;
−pdf;
−pdf;
−pdf;
92
93
94
95
96
97
98
99
100
101
102
103
figure(6);clf
% new form on 16−7
bcl=[−b;−b];
N=inv(ccl*inv(−acl)*bcl)
Gnew=ss(acl,bcl*N,ccl,dcl*N);
[yorig,t]=step(Gcl,2.5);
[ynew]=step(Gnew,t);
plot(t,yorig,t,ynew);setlines(2);
legend('Original','New');grid on
ylabel('Y');xlabel('Time (s)')
figure(6);export fig reg est6 −pdf;
November 5, 2010
Fall 2010
16.30/31 15–15
Mag
8·14·20
Fig. 5: Example #1: G(s) = (s+8)(s+14)(s+20)
0
10
G
Gc
−1
0
10
10
1
10
Freq (rad/sec)
2
10
3
10
Phase (deg)
50
0
−50
−100
G
Gc
−150
−200 −1
10
0
10
1
10
Freq (rad/sec)
2
10
3
10
Loop L
Mag
0
10
−2
10
−1
10
0
10
1
10
Freq (rad/sec)
2
10
3
10
Phase (deg)
0
−50
−100
−150
−200
−250
−1
10
November 5, 2010
0
10
1
10
Freq (rad/sec)
2
10
3
10
Fall 2010
16.30/31 15–16
8·14·20
Fig. 6: Example #1: G(s) = (s+8)(s+14)(s+20)
Bode Diagram
Gm = 11 dB (at 40.4 rad/sec) , Pm = 69.7 deg (at 15.1 rad/sec)
Magnitude (dB)
50
0
−50
−100
Phase (deg)
−150
0
−90
−180
−270
−360
−1
10
0
10
1
10
Frequency (rad/sec)
2
3
10
10
2
10
Plant G
Gcl unscaled
Gcl scaled
1
Mag
10
0
10
−1
10
−2
10
−1
10
November 5, 2010
0
10
1
10
Freq (rad/sec)
2
10
3
10
Fall 2010
16.30/31 15–17
8·14·20
Fig. 7: Example #1: G(s) = (s+8)(s+14)(s+20)
Root Locus
150
0.7
0.56
0.42
0.28
0.14
140
120
100
100 0.82
80
Imaginary Axis
0.91
60
50
40
0.975
20
0
20
0.975
40
−50
0.91
60
80
−100 0.82
100
120
0.7
0.56
−150
−150
0.42
−100
0.28
0.14
−50
140
0
50
100
Real Axis
Root Locus
25
0.86
0.76
0.64
0.5
0.34
0.16
20
0.94
15
Imaginary Axis
10
0.985
5
50
0
40
30
20
10
−5
−10
0.985
−15
0.94
−20
0.86
−25
−50
−45
−40
0.76
−35
−30
0.64
−25
−20
Real Axis
0.5
−15
0.34
−10
0.16
−5
0
� – closed-loop poles, � -- Plant poles, � – Plant zeros, � – Compensator poles, � – Compensator zeros
• Two compensator zeros at -21.54±6.63j draw the two lower frequency plant poles further into the LHP.
• Compensator poles are at much higher frequency.
• Looks like a lead compensator.
November 5, 2010
Fall 2010
16.30/31 15–18
Mag
0.94
Fig. 8: Example #2: G(s) = s2 −0.0297
0
10
G
Gc
−1
0
10
10
1
10
Freq (rad/sec)
2
10
3
10
Phase (deg)
50
0
−50
−100
G
Gc
−150
−200 −1
10
0
10
1
10
Freq (rad/sec)
2
10
3
10
Loop L
Mag
0
10
−2
10
−1
10
0
10
1
10
Freq (rad/sec)
2
10
3
10
Phase (deg)
0
−50
−100
−150
−200
−250
−1
10
November 5, 2010
0
10
1
10
Freq (rad/sec)
2
10
3
10
Fall 2010
16.30/31 15–19
0.94
Fig. 9: Example #2: G(s) = s2 −0.0297
Bode Diagram
Gm = 11.8 dB (at 7.41 rad/sec) , Pm = 36.6 deg (at 2.76 rad/sec)
Magnitude (dB)
50
0
−50
−100
Phase (deg)
−150
−135
−180
−225
−270
−2
10
−1
0
10
1
2
10
10
Frequency (rad/sec)
3
10
10
2
10
Plant G
Gcl unscaled
Gcl scaled
1
Mag
10
0
10
−1
10
−2
10
−1
10
November 5, 2010
0
10
1
10
Freq (rad/sec)
2
10
3
10
Fall 2010
16.30/31 15–20
0.94
Fig. 10: Example #2: G(s) = s2 −0.0297
Root Locus
15
0.7
0.82
Imaginary Axis
10
0.56
0.42
0.28
0.14
0.91
5 0.975
20
0
17.5
15
12.5
10
7.5
5
2.5
−5 0.975
−10
0.91
0.82
−15
−20
0.7
0.56
−15
0.42
−10
0.28
0.14
−5
0
5
Real Axis
2
Root Locus
2
0.64
0.76
0.5
0.34
0.161.75
1.5
1.5
1.25
0.86
1
1
0.75
Imaginary Axis
0.94
0.5
0.5
0.985
0.25
0.985
0.25
0
−0.5
0.5
0.94
−1
0.75
1
0.86
1.25
−1.5
0.76
0.64
−2
−2
−1.5
1.5
0.5
−1
0.34
0.161.75
−0.5
20
Real Axis
0.5
1
1.5
2
� – closed-loop poles, � -- Plant poles, � – Plant zeros, � – Compensator poles, � – Compensator zeros
• Compensator zero at -1.21 draws the two lower frequency plant poles further into the LHP.
• Compensator poles are at much higher frequency.
• Looks like a lead compensator.
November 5, 2010
Fall 2010
16.30/31 15–21
Mag
8·14·20
Fig. 11: Example #3: G(s) = (s−8)(s−14)(s−20)
0
10
G
Gc
−1
0
10
10
1
10
Freq (rad/sec)
2
10
3
10
Phase (deg)
50
0
−50
−100
G
Gc
−150
−200 −1
10
0
10
1
10
Freq (rad/sec)
2
10
3
10
Loop L
Mag
0
10
−2
10
−1
10
0
10
1
10
Freq (rad/sec)
2
10
3
10
Phase (deg)
0
−50
−100
−150
−200
−250
−1
10
November 5, 2010
0
10
1
10
Freq (rad/sec)
2
10
3
10
Fall 2010
16.30/31 15–22
8·14·20
Fig. 12: Example #3: G(s) = (s−8)(s−14)(s−20)
Bode Diagram
Gm = −0.9 dB (at 24.2 rad/sec) , Pm = 6.67 deg (at 35.8 rad/sec)
Magnitude (dB)
50
0
−50
−100
−150
−200
−135
Phase (deg)
−180
−225
−270
−315
−360
−1
10
0
1
10
2
3
10
10
Frequency (rad/sec)
4
10
10
2
10
Plant G
Gcl unscaled
Gcl scaled
1
Mag
10
0
10
−1
10
−2
10
−1
10
November 5, 2010
0
10
1
10
Freq (rad/sec)
2
10
3
10
Fall 2010
16.30/31 15–23
8·14·20
Fig. 13: Example #3: G(s) = (s−8)(s−14)(s−20)
Root Locus
200
0.76
0.64
0.5
0.34
0.16
150
100
0.64
0.5
0.34
−100
−50
Real Axis
150 0.86
Imaginary Axis
100
0.94
50 0.985
250
0
200
50
−50 0.985
−100
0.94
−150 0.86
0.76
−200
−250
−200
−150
0.64
0.76
20
0.5
0.34
50
100
150
0.16
20
15
15 0.86
Imaginary Axis
0
Root Locus
25
25
10
0.16
10
0.94
5
5 0.985
0
−5 0.985
−10
5
0.94
10
−15 0.86
−20 0.76
0.64
−25
−25
−20
15
20
0.5
−15
0.34
−10
0.16
−5
250
Real Axis
5
10
15
20
25
� – closed-loop poles, � -- Plant poles, � – Plant zeros, � – Compensator poles, � – Compensator zeros
• Compensator zeros at 3.72±8.03j draw the two higher frequency plant poles further into the LHP. Lowest
frequency one heads into the LHP on its own.
• Compensator poles are at much higher frequency.
• Not sure what this looks like.
November 5, 2010
Fall 2010
16.30/31 15–24
(s−1)
Mag
Fig. 14: Example #4: G(s) = (s+1)(s−3)
0
10
G
Gc
−1
0
10
10
1
10
Freq (rad/sec)
2
10
3
10
Phase (deg)
50
0
−50
−100
G
Gc
−150
−200 −1
10
0
10
1
10
Freq (rad/sec)
2
10
3
10
Loop L
Mag
0
10
−2
10
−1
10
0
10
1
10
Freq (rad/sec)
2
10
3
10
Phase (deg)
0
−50
−100
−150
−200
−250
−1
10
November 5, 2010
0
10
1
10
Freq (rad/sec)
2
10
3
10
Fall 2010
16.30/31 15–25
(s−1)
Fig. 15: Example #4: G(s) = (s+1)(s−3)
Bode Diagram
Gm = −3.4 dB (at 4.57 rad/sec) , Pm = −22.4 deg (at 1.41 rad/sec)
Magnitude (dB)
20
0
−20
−40
−60
Phase (deg)
−80
−120
−150
−180
−210
−2
10
−1
0
10
1
2
10
10
Frequency (rad/sec)
3
10
10
2
10
Plant G
Gcl unscaled
Gcl scaled
1
Mag
10
0
10
−1
10
−2
10
−1
10
November 5, 2010
0
10
1
10
Freq (rad/sec)
2
10
3
10
Fall 2010
16.30/31 15–26
(s−1)
Fig. 16: Example #4: G(s) = (s+1)(s−3)
Root Locus
50
50
0.64
0.5
0.38
0.24
0.12
40
40 0.76
30
30
0.88
20
Imaginary Axis
20
10
0.97
10
0.97
10
0
−10
−20
−30
20
0.88
30
−40 0.76
40
0.64
−50
−50
0.5
−40
−30
0.38
0.64
0.76
8
500
−20
−10
Real Axis
0.5
0.34
10
20
0.16
8
6
6 0.86
Imaginary Axis
0.12
Root Locus
10
10
4
0.24
4
0.94
2
2 0.985
0
−2 0.985
−4
2
0.94
4
−6 0.86
−8 0.76
0.64
−10
−10
−8
6
8
0.5
−6
0.34
−4
0.16
−2
100
Real Axis
2
4
6
8
10
� – closed-loop poles, � -- Plant poles, � – Plant zeros, � – Compensator poles, � – Compensator zeros
• Compensator zero at -1 cancels the plant pole. Note the very unstable compensator pole at s = 9!!
• Needed to get the RHP plant pole to branch off the real line and head into the LHP.
• Other compensator pole is at much higher frequency.
• Note sure what this looks like.
November 5, 2010
Fall 2010
16.30/31 15–27
B747 DOFB
• The previous B747 controller assumed full state feedback - would like
to extend that now to consider output feedback using the output of
the washout filtered yaw rate – e
• To place the estimator poles, take the approach of just increasing the
frequency of the LQR closed-loop poles by a factor of 2:
λd(Aest) = 2 × λ(Alqr )
• Then use pole placement to determine the estimator gains place.m
1
2
3
4
5
6
Gp=WashFilt*sys*H;
set(Gp, 'statename', {'xwo' 'beta' 'yaw rate' 'roll' 'phi' 'xa'});
set(Gp, 'inputname', {'rudder inputs' 'aileron'},...
'outputname', {'filtered yaw rate' 'bank angle'});
[Ap,Bp,Cp,Dp]=ssdata(Gp);
[Klqr,S,Elqr]=lqr(Ap,Bp(:,1),Cp(1,:)'*Cp(1,:),0.1)
7
8
9
Eest=real(Elqr)*2+2*imag(Elqr)*sqrt(−1);
Kest=place(Ap',Cp(1,:)',Eest);Kest=Kest';
• Note the transposes on the A and C here in the place command
• Yields the gains
�
�
Lest = −31.500 5.2985 −7.854 −122.6895 121.7798 54.858
• Much larger that LQR gains because the poles are moving further
November 5, 2010
Fall 2010
16.30/31 15–28
• Now form DOFB compensator as before:
1
2
3
4
5
6
7
8
na=size(Ap,1);
% Form compensator and closed−loop
% see 15−5
ac=Ap−Bp(:,1)*Klqr−Kest*Cp(1,:);bc=Kest;cc=Klqr;dc=0;
Gc=ss(ac,bc,cc,dc);
% see 15−7
acl=[Ap Bp(:,1)*cc;−bc*Cp(1,:) ac];bcl=[zeros(na,1);bc];ccl=[Cpbeta,zeros(1,na)];dcl=0;
Gcl=ss(acl,bcl,ccl,dcl);
• Consider the DOFB compensator for this system – it is 6th order
because the augmented plant is (4+1+1) – Gc pole-zero map and
closed-loop poles shown
3
Gc zeros
Gc poles
2
1
0
−1
−2
−3
−5
−4
−3
−2
−1
0
Root Locus
3
2
Imaginary Axis
1
0
−1
−2
−3
−5
−4
−3
−2
Real Axis
November 5, 2010
−1
0
Fall 2010
16.30/31 15–29
• Now consider the time response again to an initial perturbation in β:
1
OL
LQR
Gain
DOFB
0.8
0.6
0.4
β
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
5
10
15
Time
20
25
30
• Time response looks good – perhaps even better than LQR, but
note that is not inconsistent as this DOFB might be using more
control effort, and LQR balances that effort with the state cost
1
LQR x2
2
DOFB x
LQR u2
DOFB u2
0.9
0.8
x2 and u2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
Time
November 5, 2010
8
10
Fall 2010
Code: B747 LQR, Estimator and DOFB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
% B747 lateral dynamics
% Jonathan How, Fall 2010
% working from Matlab Demo called jetdemo
clear all;%close all
%
A=[−.0558 −.9968 .0802 .0415;
.598 −.115 −.0318 0;
−3.05 .388 −.4650 0;
0 0.0805 1 0];
B=[ .00729 0;
−0.475
0.00775;
0.153
0.143;
0
0];
C=[0 1 0 0; 0 0 0 1];D=[0 0; 0 0];
sys = ss(A,B,C,D);
set(sys, 'inputname', {'rudder' 'aileron'},...
'outputname', {'yaw rate' 'bank angle'});
set(sys, 'statename', {'beta' 'yaw rate' 'roll rate' 'phi'});
[Yol,Tol]=initial(ss(A,B,[1 0 0 0],zeros(1,2)),[1 0 0 0]',[0:.1:30]);
[V,E]=eig(A)
%
% CONTROL − actuator dynamics are a lag at 10
actn=10;actd=[1 10]; % H(s) in notes
H=tf({actn 0;0 1},{actd 1;1 1});
%
tau=3;washn=[1 0];washd=[1 1/tau]; % washout filter on yaw rate
WashFilt=tf({washn 0;0 1},{washd 1;1 1});
%
Gp=WashFilt*sys*H;
set(Gp, 'statename', {'xwo' 'beta' 'yaw rate' 'roll' 'phi' 'xa'});
set(Gp, 'inputname', {'rudder inputs' 'aileron'},...
'outputname', {'filtered yaw rate' 'bank angle'});
[Ap,Bp,Cp,Dp]=ssdata(Gp);
[Klqr,S,Elqr]=lqr(Ap,Bp(:,1),Cp(1,:)'*Cp(1,:),0.1);rifd(Elqr)
%
% gain feedback on the filter yaw rate
figure(1);clf
rlocus(Ap,Bp(:,1),−Cp(1,:),0);
sgrid([.1 .2 .3 .4],[.7 .8 .9 1]);grid on;axis([−1.4 .1 −1.2 1.2])
Kgain=−2;
Egain=eig(Ap−Bp(:,1)*Kgain*Cp(1,:));rifd(Egain)
hold on;plot(Egain+eps*sqrt(−1),'bd');hold off
export fig b747 lqr1 −pdf
44
45
46
47
48
Cpbeta=[0 1 0 0 0 0]; % performance output variable
Ggain=ss(Ap−Bp(:,1)*Kgain*Cp(1,:),Bp,Cpbeta,0);
xp0=[0 1 0 0 0 0]';
[Ygain,Tgain]=initial(Ggain,xp0,Tol); %'
49
50
51
52
53
54
55
figure(2);clf
srl(ss(Ap,Bp(:,1),Cp(1,:),0));
sgrid([.1 .2 .3 .4],[.7 .8 .9 1])
axis([−1.25 1.25 −1.25 1.25]);grid on
hold on;plot(Elqr+eps*sqrt(−1),'bd');hold off
export fig b747 lqr2 −pdf
56
57
58
59
60
61
62
63
64
% close the LQR loop
Acl=Ap−Bp(:,1)*Klqr;
Bcl=Bp(:,1);
% choose output to just state beta and control u
Ccl=[Cpbeta;Klqr];
Dcl=[0;0];
Glqr=ss(Acl,Bcl,Ccl,Dcl);
[Y,T]=initial(Glqr,xp0,Tol); %'
65
66
67
68
69
70
71
72
figure(3);clf
plot(Tol,Yol,T,Y(:,1),Tgain,Ygain);axis([0 30 −1 1]);
setlines(2)
legend('OL','LQR','Gain')
ylabel('\beta');xlabel('Time')
grid on
export fig b747 lqr3 −pdf
73
74
% estimator poles made faster by jfactor
November 5, 2010
16.30/31 15–30
Fall 2010
75
76
77
jfactor=2.05;
Eest=jfactor*Elqr;
Kest=place(Ap',Cp(1,:)',Eest);Kest=Kest';
78
79
80
81
82
83
84
85
86
87
88
89
90
na=size(Ap,1);
% Form compensator and closed−loop
% see 15−5
ac=Ap−Bp(:,1)*Klqr−Kest*Cp(1,:);bc=Kest;cc=Klqr;dc=0;
Gc=ss(ac,bc,cc,dc);
% see 15−7
acl=[Ap Bp(:,1)*cc;−bc*Cp(1,:) ac];
bcl=[zeros(na,1);bc];
% choose output to be state beta and control u
ccl=[Cpbeta,zeros(1,na);zeros(1,na) cc];
dcl=[0;0];
Gcl=ss(acl,bcl,ccl,dcl);
91
92
93
94
95
96
97
98
99
100
figure(5);clf
[p,z]=pzmap(Gc(1,1));
plot(z+sqrt(−1)*eps,'mo','MarkerFace','m')
hold on
plot(p+sqrt(−1)*eps,'bs','MarkerFace','b')
hold off
legend('Gc zeros','Gc poles','Location','NorthWest')
grid on;axis([−5.5 .5 −3 3])
export fig b747 lqr5 −pdf
101
102
103
104
105
106
107
108
109
110
111
112
113
114
figure(4);clf
rlocus(Gp(1,1)*Gc);axis([−5.5 .5 −3 3])
hold on
[p,z]=pzmap(Gp(1,1));
plot(p+sqrt(−1)*eps,'rx','MarkerFace','r')
plot(z+sqrt(−1)*eps,'ro','MarkerFace','r')
[p,z]=pzmap(Gc(1,1));
plot(p+sqrt(−1)*eps,'mx','MarkerFace','m')
plot(z+sqrt(−1)*eps,'mo','MarkerFace','m')
p=eig(acl);
plot(p+sqrt(−1)*eps,'bd')
hold off
export fig b747 lqr4 −pdf
115
116
117
% initial comp at zero
[Ycl,Tcl]=initial(Gcl,[xp0;xp0*0],Tol); %'
118
119
120
121
122
123
124
125
figure(3);clf
plot(Tol,Yol,T,Y(:,1),Tgain,Ygain,Tcl,Ycl(:,1));axis([0 30 −1 1]);
setlines(2)
legend('OL','LQR','Gain','DOFB')
ylabel('\beta');xlabel('Time')
grid on
export fig b747 lqr3a −pdf
126
127
128
129
130
131
132
133
134
135
figure(6);clf
plot(T,Y(:,1).ˆ2,Tcl,Ycl(:,1).ˆ2,T,Y(:,2).ˆ2,Tcl,Ycl(:,2).ˆ2);
axis([0 10 0 1]);
hold off
setlines(2)
legend('LQR xˆ2','DOFB xˆ2','LQR uˆ2','DOFB uˆ2')
ylabel('xˆ2 and uˆ2');xlabel('Time')
grid on
export fig b747 lqr3b −pdf
November 5, 2010
16.30/31 15–31
Fall 2010
16.30/31 15–32
Satellite Control Design
• Simple system model – goal is to point payload device (J2 = 0.1)
using torque inputs on the larger spacecraft (J1 = 1)
• Sensor is star tracker on payload (angle), and actuator is torque
applied to spacecraft
• Can form the state space model using the state x = [ θ2 θ̇2 θ1 θ̇1 ]T
• Plant: Gp(s) =
10bs + 10k
, k = 0.1 and b = 0.005
s2(s2 + 11bs + 11k)
• Rigid body part easily controlled using a lead, such as
3s + 1
Klead(s) = 1.5
s+5
2
2
10
10
Mag
Mag
Loop L
0
10
Plant G
Compensator Gc
−1
10
0
1
10
−2
10
2
10
0
10
10
−1
0
10
1
10
Freq (rad/sec)
10
2
10
Freq (rad/sec)
200
Phase (deg)
Phase (deg)
0
100
0
−100
−200 −1
10
0
1
10
2
10
−100
−200
−300
−1
10
0
10
1
10
Freq (rad/sec)
10
Freq (rad/sec)
Root Locus
3
3
0.5
0.38
0.28
0.17
0.08
2.5
0.64
2
2
1.5
0.8
1
1
0.94
Imaginary Axis
0.5
0
0.5
0.94
−1
1
0.8
1.5
−2
2
0.64
2.5
0.5
−3
−2
November 5, 2010
0.38
−1.5
0.28
−1
0.17
−0.5
Real Axis
0.08
30
0.5
1
2
10
Fall 2010
16.30/31 15–33
• So problems with this “non-collocated” control is apparent – the flex­
ible modes are driven unstable by the lead controller
• The problem with the lightly damped poles is that the loop phase
changes very rapidly – get wrong departure angle for the root locus.
• Can improve the design using notch filtering
• The notch has a lightly damped zero pair at a frequency just below the
resonant frequency of the system, resulting in a rapid phase change,
but in the opposite direction.
• Result is that the departure angle now to the left, not the right.
• Fixes instability problem - but damping of the resonant pole is
quite limited – need to find a better location for the notch zeros.
2
2
10
10
Mag
Mag
Loop L
0
10
0
10
Plant G
Compensator Gc
−1
10
0
1
10
−2
10
2
10
10
−1
0
10
1
10
Freq (rad/sec)
10
2
10
Freq (rad/sec)
Phase (deg)
Phase (deg)
0
200
100
0
−100
−200 −1
10
0
1
10
−100
−200
−300
2
10
−1
10
0
10
1
10
Freq (rad/sec)
10
Freq (rad/sec)
Root Locus
3
3
0.5
0.38
0.28
0.17
0.08
2.5
0.64
2
2
1.5
0.8
1
1
Imaginary Axis
0.94
0.5
0
0.5
0.94
−1
1
0.8
1.5
−2
2
0.64
2.5
0.5
−3
−2
November 5, 2010
0.38
−1.5
0.28
−1
0.17
−0.5
Real Axis
0.08
30
0.5
1
2
10
Fall 2010
16.30/31 15–34
• Lets try to do it using DOFB – nothing particularly complicated here
in the weights
1
2
3
4
5
6
7
8
9
kk=0.1;bb=0.05*sqrt(kk/10);J1=1;J2=0.1;
a=[0 1 0 0;−kk/J2 −bb/J2 kk/J2 bb/J2;0 0 0 1;kk/J1 bb/J1 −kk/J1 −bb/J1];na=size(a,1);
b=[0 0 0 1/J1]';c=[1 0 0 0];d=0;
k=lqr(a,b,c'*c,.1);
l=lqr(a',c',b*b',.01);l=l'
%
% For state space for G c(s)
%
ac=a−b*k−l*c;bc=l;cc=k;dc=0;
• Look at the resulting DOFB controller - × are the plant poles; • are
the compensator zeros; × are the compensator poles
Root Locus
3
0.78
0.64
0.5
0.34
0.18
2.5
0.87
Imaginary Axis
2
1.5 0.94
1
0.985
0.5
4
0
3.5
3
2.5
2
1.5
1
0.5
−0.5
0.985
−1
−4
−3.5
−3
−2.5
−2
−1.5
Real Axis
−1
−0.5
0
• So the DOFB controller puts zeros right near the plant poles – similar
to the notch filter – ensures that the departure angle of the poles is
to the left, not the right, so all the poles are stabilized.
• Appears to achieve much higher levels of damping in that pole
than the classical/notch filter design did.
November 5, 2010
Fall 2010
16.30/31 15–35
2
2
10
10
Mag
Mag
Loop L
0
10
Plant G
Compensator Gc
−1
0
10
1
10
−2
10
2
10
0
10
10
−1
0
10
1
10
Freq (rad/sec)
2
10
10
Freq (rad/sec)
200
Phase (deg)
Phase (deg)
0
100
0
−100
−200 −1
10
0
1
10
2
10
−100
−200
−300
−1
10
0
10
10
Bode Diagram
Gm = 1.87 dB (at 1.89 rad/sec) , Pm = 11.6 deg (at 1.49 rad/sec)
10
100
Magnitude (dB)
Plant G
closed−loop Gcl
1
10
Mag
2
10
Freq (rad/sec)
Factor of N=1 applied to Closed loop
2
1
10
Freq (rad/sec)
0
−100
−200
0
10
−300
0
Phase (deg)
−90
−1
10
−180
−270
−360
−2
10
−1
10
0
1
10
10
Freq (rad/sec)
November 5, 2010
2
10
−450
−2
10
−1
10
0
1
10
10
Frequency (rad/sec)
2
10
3
10
Fall 2010
16.30/31 15–36
DOFB Summary
• Separation principle gives a very powerful and simple way to develop
a dynamic output feedback controller
• Note that the designer now focuses on selecting the appropriate reg­
ulator and estimator pole locations.
• Once those and the implementation architecture (i.e., how refer­
ence input is applied, as on 15–5) are set, the closed-loop response
is specified.
• Can almost consider the compensator to be a by-product.
• The examples show that the design process is extremely simple.
November 5, 2010
MIT OpenCourseWare
http://ocw.mit.edu
16.30 / 16.31 Feedback Control Systems
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.