This document contains 13 examples of exercises related to control systems. The examples involve tasks such as deriving state space models, bringing feedback control systems into generalized standard form, designing controllers, computing norms, and parametrizing stabilizing controllers. The examples cover topics including disturbance decoupling, observer design, state feedback, and coprime factorizations.

1. AKS
ROBUST CONTROL
Exercise Problems
Erstellt von
Prof. Dr.-Ing. Steven X. Ding
M. Sc. Christopher Reimann
Automatisierungstechnik und komplexe Systeme (AKS)

2. Example 1
Please derive the state space model of the system shown in the following figure
a
2s
1s
F
1v2v
2d 1d
c
1m2m
w1Fw2F
Figure 1 Structure diagram of a coupled car
with constant parameters denoting masses, friction coefficients (proportional to the
respective velocity), spring stiffness. The variables denote the velocity of car 1 and 2
respectively, the position, the distance between two cars, driving force, unknown
disturbance forces. Assume that the spring force is expressed by
1m ,m2
1)
1 2d ,d
c 1 2v ,v
1 2s ,s a F w1 w2F ,F
s 2F c(s s= − . It is known that is
control input, and are measured outputs. (The control objective is to keep the velocities
F
1s a 1 2v v=
and the velocity follows a given reference signal ).1v refv
Example 2
For the system in Example 1, assume that the control objective is to keep the velocities v1 =
v2 and to keep the velocity v1 following a given reference signal . A dynamic output
feedback controller controller in the form of
refv
1s
F K(s)
a
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
is used in the feedback control loop. Please bring the feedback control system into the
generalized standard form.
Example 3
Given a system described by
1

3. d
d
x Ax Bu E w
y Cx Du F w
= + +
= + +
&
It is controlled by an observer based feedback controller
x r
ˆ ˆ ˆx Ax Bu L(y y)
ˆ ˆy Cx Du
ˆu F x F r
= + + −
= +
= +
&
Control goal is to make the output y to follow given reference signal r.
Please bring the feedback control system into the generalized standard form.
Example 4
Given a system as follows
rv u y
W(s) 1F (s) P(s)
−
2F (s)
It is desired that
• the plant output y should track a reference signal r, and
• the control effort is as small as possible.
Please bring the feedback control system into the generalized standard form.
Example 5
Suppose that areN(s),M(s),U(s),V(s) RH∞ transfer function matrices, both
and are right coprime factorizations. Please show that
1
N(s)M (s)−
1
U(s)V (s)−
1
M(s) O M(s) U(s)
O V(s) N(s) V(s)
−
⎡ ⎤ ⎡
⎢ ⎥ ⎢
⎣ ⎦ ⎣
⎤
⎥
⎦
is also a right coprime factorization.
2

4. Example 6
Given a linear time-invariant plant 22y G (s)u= with transfer function
22 2
1
G (s)
s
=
Please write down a parametrisation of all proper real-rational controllers u K that
stabilizes .
(s)y=
22G (s)
Example 7
Given a linear time-invariant plant y G(s)u= with transfer function
s 1
G(s)
s(s 2)
−
=
−
Please prove that every controller that stabilizes G(s) is itself unstable.
Example 8
Suppose that a PI controller
2
1
k
K(s) k
s
= +
stabilizes a linear time-invariant plant y G(s)u= with transfer function
c
G(s) , b 0, c 0
s b
= > <
−
Please verify that the above PI controller can be written in the form of the parametrization of
stabilizing controller with some Q(s) RH∞∈ .
(Hint: Find a such thatQ(s) RH∞∈ 1ˆ ˆK (Y MQ)(X NQ)−
= − + − )
Example 9
(1) Given a system as shown below
3

5. 1 2
1 11 12
2 21 22
A B B
z w
C D D
y u
C D D
⎡ ⎤
⎡ ⎤ ⎡ ⎤⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦
where
[ ]
1 2
1 2
11 12 21 22
0 3 0 3
A , B , B
0 2 1 1
1 0
C 1 0 , C
0 1
D 0, D 0,D 0,D 0
−⎡ ⎤ ⎡ ⎤ ⎡
= = =⎢ ⎥ ⎢ ⎥ ⎢−⎣ ⎦ ⎣ ⎦ ⎣
⎡ ⎤
= = ⎢ ⎥
⎣ ⎦
= = = =
⎤
⎥
⎦
Please design a disturbance decoupling controller such that the closed-loop transfer function
from w to z is zero.
(2) Given a system
2
1 s 4s 2
z ws 1 (s 1)(s 2)
y u1 1
s 1 s 1
⎡ ⎤+ +
⎢ ⎥⎡ ⎤ ⎡ ⎤+ + +⎢ ⎥=⎢ ⎥ ⎢ ⎥
⎢ ⎥⎣ ⎦ ⎣ ⎦
−⎢ ⎥
+ +⎣ ⎦
Please design a disturbance decoupling controller such that the closed-loop transfer function
from w to z is zero.
Example 10
(1) Given a system:
[ ]
1 1 1
x x
0 2 1
y 1 0 x
−
u
⎡ ⎤ ⎡ ⎤
= +⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦
=
&
Please compute the norm of G ( .2H yu s)
(2) Given
1
G(s)
s 2
=
+
Please compute the H∞ norm of .G(s)
4

6. Example 11
Given a plant model
d
d
x Ax Bu E d
y Cx Du F d
= + +
= + +
&
where is the state vector, is the control input vector, d is the unknown disturbance
vector, is the measured output vector, are known matrices of compatible
dimensions. In order to estimate the state vector , a state observer of the following form
x u
y dA,B,E ,C,D,Fd
x
ˆ ˆ ˆx Ax Bu L(y Cx Du)= + + − −&
is used, where is the observer gain matrix to be determined.L
(1) Please derive the dynamics of the state estimation error e , where ˆe x x= − .
(2) Please write down the transfer function matrix from the unknown disturbance d to
the estimation error .
edG (s)
e
(3) Given a constant . Please show how to find the observer gain matrix L so that the
error dynamics is stable and the H
0α >
∞ -norm of is smaller thated α,G (s)
(4) Given constants and . Please show how to find the observer gain matrix so
that the following conditions are satisfied:
0α > 0δ > L
- the error dynamics is stable and the H∞ -norm of is smaller that ,edG (s) α
- the real part of all the eigenvalues of A LC− is smaller than −δ.
(5) Given a constant . Please show how to find the observer gain matrix so that the
error dynamics is stable and the generalized -norm of is smaller that .
0γ > L
2H edG (s) γ
Example 12
Given a system
x 2x u, x(0) 3
z x du
= + =
= +
&
where x is the state vector, u is the control input, z is the controlled output vector, d is a
constant number. A state feedback controller u=Kx, where the feedback gain matrix K is
constant, is used.
(1) Please find K that minimizes 2
z .
(2) Please also compute the minimal value of 2
z .
5

7. Example 13
Given a system
[ ]
[ ]
x 2x u 1 0 d
y x 0 1 d
1 0
z x u
0 1
= + +
= +
⎡ ⎤ ⎡ ⎤
= +⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
&
where is the state, is the control input, is the unknown disturbance vector, is the
measured output, is the controlled output vector. In the feedback control loop, an output
feedback control law of the following form is used
x u d y
z
u K(s)y=
(1) Please find so that the -norm of , which denotes the closed-loop transfer
function matrix from to , is minimized.
K(s) 2H zdG (s)
d z
(2) Please compute the minimum value of zd 2
G (s) .
6

8. 1
I. EXTRA TASK ON 08.11.2017
A. Example α
Given the SISO System
G(s) =
1
s
find a coprime factorization with its inverse in RH∞.
II. EXTRA TASKS ON 29.11.2017
A. Example β
Given a linear time-invariant plant y = G(s)u with transfer function
G(s) =
s − 3
(s − 1)(s − 2)
please write down an observer/residual generator-based realization of controller parametrization.
B. Example γ
Given the following controller
˙xk =


1 −1
0 1

 xk +


−2
1

 y
u = 1 2 xk
please write down a parametrization form of all the plants that can be stabilized by the above
controller.
November 6, 2017 DRAFT

9. 2
III. EXTRA TASKS ON 31.01.2018
A. Example δ
Consider the standard feedback control loop with
y(s) = Gyu(s)u(s) = ( ˆM(s) + ˆ∆M )−1
( ˆN(s) + ˆ∆N )
where ( ˆM(s), ˆN(s)) is the left-coprime pair of the nominal system
G0 : ˙x = Ax + Bu, y = Cx.
Assume that [ ˆ∆M
ˆ∆N ] ∞ ≤ 0.1 and
K(s) = −(ˆY (s) + M(s)Q(s))( ˆX(s) − N(s)Q(s))−1
ˆY (s) = F(sI − A − BF)−1
L
ˆX(s) = C(sI − A − BF)−1
L + I
M(s) = F(sI − A − BF)−1
B + I
N(s) = (C + DF)(sI − A − BF)−1
B + D
where F, L are given and Q(s) ∈ RH∞.
(1) Please give the upper bound of


ˆX(s) − N(s)Q(s)
−(ˆY (s) + M(s)Q(s))


∞
such that the closed-loop
system is stable.
(2) Please propose a design scheme for F, L (LMI condition) such that the closed-loop is
stable when Q(s) = 0.
November 6, 2017 DRAFT

10. 3
B. Example ǫ
Given the following system:
plant:
˙x = Ax + Bu + Edd
y = Cx + Du + Fdd
observer:
˙ˆx = Aˆx + Bu + L(y − Cˆx + Du)
e = x − ˆx
˙e = ˙x − ˙ˆx
˙x − ˙ˆx = (A − LC)e + (Ed − LFd)d
Ged(s) = (sI − A + LC)−1
(Ed − LFd)
Solve the following tasks:
1) Find observer gain L s.t. Ged(s) ∞ ≤ ∞.
2) Find observer gain L s.t.
a) error dynamic is stable and Ged(s) ∞ ≤ ∞.
b) the real-part of all eigenvalues of (A-LC) is smaller then −δ.
3) Find observer gain L s.t. error dynamic is stable and generalized H2-norm of Ged(s) is
smaller than γ.
November 6, 2017 DRAFT