1 ECE-320: Linear Control Systems
Homework 4
Due: Tuesday January 10 at the beginning of class 1) Consider the plant
0 1
3 ( )
0.5
p
G s
s s
α α
= =
+ +
where 3 is the nominal value of α0 and 0.5 is the nominal value ofα1. In this problem we will investigate the sensitivity of closed loop systems with various types of controllers to these two parameters. We will assume we want the settling time of our system to be 0.5 seconds and the steady state error for a unit step input to be less than 0.1.
a) (ITAE Model Matching) Since this is a first order system, we will use the first order ITAE model,
( ) o
o
o
G s s
ω ω
= +
i) For what value of ωowill we meet the settling time requirements and the steady state error requirements?
ii) Determine the corresponding controllerG sc( ).
iii) Show that the closed loop transfer function (using the parameterized form of G sp( ) and the controller from part ii) is
0
1 0
8
( 0.5) 3
( )
8
( ) ( 0.5)
3
o
s G s
s s s
α
α α
+ =
+ + +
iv) Show that the sensitivity of G so( )to variations in α0 is given by 0
0 8
G s
S s
α =
+
v) Show that the sensitivity of G so( )to variations in α1 is given by
1 2
0.5 8.5 4
o
G s
S
s s
α
− =
+ +
b) (Proportional Control) Consider a proportional controller, with kp =2.5.
i) Show that the closed loop transfer function is
0
1 0
2.5 ( )
2.5
o
G s s
α
α α
=
2 ii) Show that the sensitivity of G so( )to variations in α0 is given by
0 0
0.5 8
G s
S s α
+ =
+
iii) Show that the sensitivity of G so( )to variations in α1 is given by
1
0.5 8
o
G
S s α
− =
+
c) (Proportional+Integral Control) Consider a PI controller with kp =4 and ki =40.
i) Show that the closed loop transfer function is
0
1 0
4 ( 10) ( )
( ) 4 ( 10)
o
s G s
s s s
α
α α
+ =
+ + +
ii) Show that the sensitivity of G so( )to variations in α0 is given by
0
0 2
( 0.5) 12.5 120
G s s
S
s s
α = +
+ +
iii) Show that the sensitivity of G so( )to variations in α1 is given by
1 2
0.5 12.5 120
o
G s
S
s s
α
− =
+ +
d) Using Matlab, simulate the unit step response of each type of controller. Plot all responses on one graph. Use different line types and a legend. Turn in your plot and code.
g) Using Matlab and subplot, plot the sensitivity to α0 for each type of controller on one graph at the top of the page, and the sensitivity to α1 on one graph on the bottom of the page. Be sure to use different line types and a legend. Turn in your plot and code. Only plot up to about 8 Hz (50 rad/sec) using a semilog scale with the sensitivity in dB (see below). Do not make separate graphs for each system!
In particular, these results should show you that the model matching method, which essentially tries and cancel the plant, are generally more sensitive to getting the plant parameters correct than the PI controller for low frequencies. However, for higher frequencies the methods are all about the same.
Hint: If ( ) 2 2 2 10
s T s
s s
=
+ + , plot the magnitude of the frequency response using:
T = tf([2 0],[1 2 10]); w = logspace(-1,1.7,1000); [M,P]= bode(T,w);
3
semilogx(w,Mdb); grid; xlabel('Frequency (rad/sec)'); ylabel('dB');
2) For the following two circuits,
show that the state variable descriptions are given by
[
]
1 1
( ) ( ) ( )
( ) ( ) 0 [0] ( )
1
1 1
( ) ( ) ( )
b
L L L
in B in
c c c
a a
R
t L L t L t
v t y t R v t
t t
i i
t R C
i d
v v v
d
C R
t
C
− −
= + = +
−
1
( ) ( ) ( ) ( )
( ) ( ) ( ) 0 ( )
( ) 1 ( ) ( )
0 0
a b
a
L a b L a b L a
a b in in
c c a b c a b
R R
t L R R L t R R t R
L R R v t y t v t
t t R R t R
R
i i i
v R
t
C d
v v
d
− −
= + + + = − +
+ +
4 3) For the plant
2 2 ( )
1 2
1
p
n n
K G s
s ζ s
ω ω
=
+ +
a) If the plant input isu t( ) and the output isx t( ), show that we can represent this system with the differential equation
2 2
( ) 2 n ( ) n ( ) n ( )
x t + ζω x t +ω x t =Kω u t
b) Assuming we use states q t1( )=x t( )and q t2( )=x t( ), and the output is x t( ), show that we can write the state variable description of the system as
[
]
[ ]
1 1
2 2
2 2
1 2
0 1 0
( ) ( )
( ) 2
( ) ( )
( )
( ) 1 0 0 ( )
( )
n n n
q t q t
d
u t K
q t q t
dt
q t
y t u t
q t
ω ζω ω
= +
− −
= +
or
( ) ( ) ( ) ( ) ( ) ( )
q t = Aq t +Bu t y t =Cq t +Du t
Determine the A, B, C and D matrices.
c) Assume we use state variable feedback of the form u t( )=G r tpf ( )−kq t( ), where r t( )is the new input to the system, Gpf is a prefilter (for controlling the steady state error), and kis the state variable feedback gain vector. Show that the state variable model for the closed loop system is
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
pf
pf
q t A Bk q t BG r t
y t C Dk q t DG r t
= − +
= − +
or
( ) ( ) ( )
( ) ( ) ( )
q t Aq t Br t
y t Cq t Dr t
= +
= +
d) Show that the transfer function (matrix) for the closed loop system between input and output is given by
1 ( )
( ) ( )( ( ))
( ) pf pf
Y s
G s C Dk sI A Bk BG DG
R s
−
= = − − − +
and ifD is zero this simplifies to
1 ( )
( ) ( ( ))
( ) pf
Y s
G s C sI A Bk BG
R s
−
= = − −
e) Assume r t( )=u t( ) and D=0. Show that, in order for lim ( ) 1
t→∞ y t = , we must have
1 1
( )
pf G
C A Bk − B
− =
−
5
If matrix Pis given as
a b
P
c d
=
then
1 1 d b
P
c a
ad bc
− = −
−
−
and the determinant of Pis given by ad−bc. This determinant will also give us the characteristic polynomial of the system.
4) For each of the systems below:
• determine the transfer function when there is state variable feedback
• determine if k1and k2exist (k=
[
k1 k2]
) to allow us to place the closed loop poles anywhere. That is, can we make the denominator look like 21 0
s +a s+a for any a1 and any a0. If this is true, the system is said to be controllable.
a) Show that for
1 0 0
1 1 1
q= q+ u
[
0 1]
[0]y= q+ u
the closed loop transfer function with state variable feedback is
2 ( 1) ( )
( 1)( 1 )
pf
s G
G s
s s k
− =
− − +
b) Show that for
0 1 0
0 1 1
q= q+ u
[
0 1]
[0]y= q+ u
the closed loop transfer function with state variable feedback is 2
2 1
( )
( 1)
pf sG G s
s k s k
=
+ − +
c) Show that for
0 1 0
1 1 1
q= q+ u
[
1 0]
[0]y= q+ u
the closed loop transfer function with state variable feedback is 2
2 1
( )
( 1) ( 1)
pf G G s
s k s k
=