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Modern control design, specifically pole-placement and observer-based state feedback. It covers the design of a state feedback control to place closed-loop poles, the use of state estimators for systems with unavailable states, and the design of a full-state observer. The document also includes examples and MATLAB code.
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CHAPTER
MODERN
CONTROL
DESIGN
The classical design techniques of Chapters 6 and 7 are based on the root-locus and
frequency response that utilize only the plant output for feedback with a dynamic
controller. In this final chapter on design, we employ modern control designs that
require the use of all state variables to form a linear static controller. Modern control
design is especially useful in multivariable systems; however, in this chapter the ideas
of state-space design are illustrated using single-input, single-output systems.
One approach in modern control systems accomplished by the use of state feed-
back is known as pole-placement design. The pole-placement design allows all roots
of the system characteristic equation to be placed in desired locations. This results in
a regulator with constant gain vector K.
The state-variable feedback concept requires that all states be accessible in a
physical system, but for most systems this requirement is not met; i.e., some of the
states are inaccessible. For systems in which all states are not available for feedback,
a state estimator (observer) may be designed to implement the pole-placement design.
The other approach to the design of regulator systems is the optimal control
problem where a specified mathematical performance criterion is minimized.
8.1. Pole-Placement Design 171
8.1 Pole-Placement Design
The control is achieved by feeding back the state variables through a regulator with
constant gains. Consider the control system presented in the state-variable form
x _(t) = Ax(t) + Bu(t) (8.1)
y (t) = Cx(t)
Consider the block diagram of the system shown in Figure 8.1 with the following state
feedback control
u(t) = Kx(t) (8.2)
where K is a 1 n vector of constant feedback gains. The control system input r (t)
is assumed to be zero. The purpose of this system is to return all state variables to
values of zero when the states have been perturbed.
....................................................................................................................... ..... .
.........
......
.
................................................................................................................................................................................................................................................................................................................................................ ..... .
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.
Plant
................................................................................... ..... .
.........
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.
. .. ... .. ... .. ... .. ... .. ... .. ... .. ... ..............................................................................................................................................................................................
......
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........... .....
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n
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. .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... ..........................................................................................................................................................
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........... .....
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. .. ... ... .. ... ..
r (t) = 0 u(t) y (t)
x
n
(t)
x
(t) x
(t)
FIGURE 8.
Control system design via pole placement.
Substituting (8.2) into (8.1), the closed-loop system state-variable representation
is
x _(t) = (A BK )x(t) = A
f
x(t) (8.3)
The closed-loop system characteristic equation is
jsI A + BKj = 0 (8.4)
Assume the system is represented in the phase variable canonical form as follows
2 6 6 6 6 6 6 4
x _
x _
.
.
.
x _
n 1
x _
n
3 7 7 7 7 7 7 5
=
2 6 6 6 6 6 6 4
0 1 0 : : : 0
0 0 1 : : : 0
.
.
.
0 0 0 : : : 1
a
0
a
a
: : : a
n 1
3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4
x
x
.
.
.
x
n 1
x
n
3 7 7 7 7 7 7 5
+
2 6 6 6 6 6 6 4
0
0
.
.
.
0
3 7 7 7 7 7 7 5
u(t)
(8.5)
172 8. Modern Control Design
Substituting for A and B into (8.4), the closed-loop characteristic equation for the
control system is found
jsI A + BKj = s
n
+ (a
n 1
+ k
n
)s
n 1
- + (a
+ k
)s + (a
0
+ k
) = 0 (8.6)
For the specified closed-loop pole locations
; : : : ;
n
, the desired charac-
teristic equation is
c
(s) = (s +
) (s +
n
) = s
n
+
n 1
s
n 1
- +
s +
0
= 0 (8.7)
The design objective is to find the gain matrix K such that the characteristic
equation for the controlled system is identical to the desired characteristic equa-
tion. Thus, the gain vector K is obtained by equating coefficients of equations (8.6)
and (8.7).
k
i
=
i 1
a
i 1
(8.8)
If the state model is not in the phase-variable canonical form, we can use the trans-
formation technique of Chapter 3 Section 3.5 to transform the given state model to
the phase-variable canonical form. The gain factor is obtained for this model and then
transformed back to confirm with the original model. This procedure results in the
following formula, known as Ackermann’s formula.
K =
0 0 0 1
S
1
c
(A) (8.9)
where the matrix S is given by
S =
B AB A
B : : : A
n 1
B
(8.10)
and the notation
c
(A) is given by
c
(A) = A
n
+
n 1
A
n 1
- +
A +
0
I (8.11)
The function [K, A
f
] = placepol(A, B, C, p) is developed for the pole-placement
design. A, B, C are system matrices and p is a row vector containing the desired
closed-loop poles. This function returns the gain vector K and the closed-loop system
matrix A
f
. Also, the MATLAB Control System Toolbox contains two functions for
pole-placement design. Function K = acker(A, B, p) is for single input systems, and
function K = place(A, B, p) , which uses a more reliable algorithm, is for multi-input
systems.
The condition that must exist to place the closed-loop poles at the desired lo-
cation is to be able to transform the given state model into phase-variable canonical
form. That is, the controllability matrix S, given in (8.10), must have a nonzero deter-
minant. This characteristic is known as controllability.
Example 8.
For the plant
2
4
x_
x _
x _
3
5
=
2
4
1 0 0
1 2 0
3
5
2
4
x
x
x
3
5
+
2
4
0
0
3
5
u
8.1. Pole-Placement Design 173
y =
x
design a state feedback control to place the closed-loop pole at 3 j 4 and 8.
Obtain the initial condition response, given x
(0) = 1 , x
(0) = 1 and x
(0) = 1.
The following commands
A=[-1 0 0; -1 -2 0; 1 0 0];
B=[1; 0; 0];
C=[1 1 0 ]; D=0;
j=sqrt(-1);
P=[-3+j4 -3-j4 -8]; % desired closed-loop poles
[K,Af]=placepol(A,B,C,P); % returns gain K and closed-loop
% system matrix
% initial condition response
t=0:.02:2;
r=zeros(1,length(t)); % generates a row of zero input
x0=[1 1 -1]; % initial state
[y,x]=lsim(Af, B, C, D, r, t, x0); % initial state response
subplot(2,2,1), plot(t, x(:,1)),title('x_1(t)'), grid
subplot(2,2,2), plot(t, x(:,2)),title('x_2(t)'), grid
subplot(2,2,3), plot(t, x(:,3)),title('x_3(t)'), grid
subplot(2,2,4), plot(t, y),title('y(t)'), grid
subplot(111)
result in
Feedback gain vector K
Uncompensated Plant
Transfer function:
s^2 + s
-----------------
s^3 + 3 s^2 + 2 s
Compensated system closed-loop
Transfer function:
s^2 + s
-------------------------
s^3 + 14 s^2 + 73 s + 200
Compensated system matrix A - B*K
-12 -51 -
-1 -2 0
The results are given in Figure 8.2.
174 8. Modern Control Design
0 0.5 1 1.5 2
− 2
0
x
(t)
0 0.5 1 1.5 2
−0.
0
0.
x
(t)
0 0.5 1 1.5 2
− 1
−0.
0
0.
x
(t)
0 0.5 1 1.5 2
− 2
0
y(t)
FIGURE 8.
Initial condition response for Example 8.1.
Example 8.
A single-input single-output plant has the transfer function
G(s) =
s(s + 1)(s + 4)
Obtain a state model for the plant and design a state feedback that will place the
closed-loop pole at 2 j 2 and 5. Also, obtain the closed-loop transfer function
for the controlled system.
The following commands
num=4; den=[1 5 4 0];
[A, B, C, D]=tf2ss(num, den) % converts transfer function
% to state model
j=sqrt(-1);
P=[-2+j2 -2-j2 -5]; % desired closed-loop poles
[K,Af]=placepol(A,B,C,P); % returns gain K & closed-loop
% system matrix
result in
A =
-5 -4 0
8.2. Controllability 175
B =
0
0
C =
D =
0
Feedback gain vector K
Uncompensated Plant
Transfer function:
-----------------
s^3 + 5 s^2 + 4 s
Compensated system closed-loop
Transfer function:
-----------------------
s^3 + 9 s^2 + 28 s + 40
Compensated system matrix A - B*K
-9 -28 -
From the above results the closed-loop transfer function for the controlled plant is
T (s) =
C (s)
R (s)
=
s
+ 9 s
+ 28 s + 40
8.2 Controllability
A system is said to be controllable when the plant input u can be used to transfer
the plant from any initial state to any arbitrary state in a finite time. The plant de-
scribed by (8.1) with the system matrix having dimension n n is completely state
controllable if and only if the controllability matrix S in (8.10) has a rank of n. The
function S=cntrable(A, B) is developed which returns the controllability matrix S
and determines whether or not the system is state controllable.
8.3 Observer Design
In the pole-placement approach to the design of control systems, it was assumed that
all state variables are available for feedback. However, in practice it is impractical to
install all the transducers which would be necessary to measure all of the states. If the
state variables are not available because of system configuration or cost, an observer
176 8. Modern Control Design
....................................................................................................... ........ ...
............
....
.......................................................................................................................................................................................................................................................................................... ........ ...
............
....
Plant
....................................................................................................... ........ ...
............
.... . .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ......
..
...
...
..
...
...
..
...
...
..
...
. .. ... .. ... .. ... .. ... .. .. ... ... .. ... .. .. ... ... .. ... .. .. ... ... .. ... .. .. ... .. ... ... .. .. ... .. ... ... ................................................................................................................................................................
..... ....... ...... ...
Observer
. ... .. .. ... ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... ..............................................................
..... ........ ...... ..
Controller
....................................................................
...
..
...
..
...
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...
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...
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...
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...
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...
..
...
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...
..
...
..
...
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...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
.. .. ... .. ... ... ..
. .. ... .. ... ... ..
r (t) = 0 u(t) y (t)
Estimated
State x_
FIGURE 8.
State feedback design with an observer.
or estimator may be necessary. The observer is an estimator algorithm based on the
mathematical model of the system. The observer creates an estimate x^ (t) of the states
from the measurements of the output y (t). The estimated states, rather than the actual
states, are then fed to the controller. One scheme is shown in Figure 8.3.
Consider a system represented by the state and output equations
x _(t) = Ax(t) + Bu(t) (8.12)
y (t) = Cx(t) (8.13)
Assume that the state x(t) is to be approximated by the state ^x(t) of the dynamic
model
_
^x(t) = A ^x(t) + Bu(t) + G(y (t) y^ (t)) (8.14)
y ^ (t) = C ^x(t) (8.15)
Subtracting (8.14) from (8.12), and (8.15) from (8.13), we have
( x_(t)
_
^x(t)) = A(x(t) ^x (t)) G(y (t) y^ (t)) (8.16)
(y (t) y^ (t)) = C(x(t) ^x(t)) (8.17)
where x(t) ^x(t) is the error between the actual state vector and the estimated vec-
tor, and y (t) y^ (t) is the error between the actual output and the estimated output.
Substituting the output equation into the state equation, we obtain the equation for the
error between the estimated state vector and the actual state vector.
e _(t) = (A GC)e(t) = A
e
e(t) (8.18)
where
e(t) = x(t) ^x(t) (8.19)
If G is chosen such that eigenvalues of matrix A GC all have negative real parts,
then the steady-state value of the estimated state vector error e(t) for any initial con-
dition will tend to zero. That is, ^x(t) will converge to x(t). The design of the observer
is similar to the design of the controller. However, the eigenvalues of A GC must
8.3. Observer Design 177
be selected to the left of the eigenvalues of A. This ensures that the observer dynamic
is faster than the controller dynamic for providing a rapid updated estimate of the state
vector.
The estimator characteristic equation is given by
jsI A + GCj = 0 (8.20)
For a specified speed of response, the desired characteristic equation for the
estimator is
e
(s) = s
n
+
n 1
s
n 1
- +
s +
0
= 0 (8.21)
Thus, the estimator gain G is obtained by equating coefficients of (8.20) and
(8.21). This is identical to the pole-placement technique, and G is found by the appli-
cation of Ackermann’s formula
G =
e
(A)
2
6
6
6
4
C
C A
.
.
.
C A
n 1
3
7
7
7
5
1
2
6
6
6
4
0
0
.
.
.
3
7
7
7
5
(8.22)
and the notation
e
(A) is given by
e
(A) = A
n
+
n 1
A
n 1
- +
A +
0
I (8.23)
The function [G, A
e
] = observer(A, B, C, p
e
) is developed for the estimator. p
e
is
the desired estimator eigenvalues. This function returns the gain vector G and the
closed-loop system matrix A
f
.
The necessary condition for the design of an observer is that all the states can
be observed from the measurements of the output. This characteristic is known as
observability.
Example 8.
For the plant
x_
x _
=
x
x
+
0
u
y =
x
design a full-state observer such that the observer is critically damped with eigenval-
ues at 8 and 8.
The following commands
A=[ 0 1; 16 0];
B=[0; 1];
C=[1 0]; D=0;
Pe=[-8 -8]; % desired observer eigenvalues
[K,Ae]= observer(A,B,C,Pe); % returns gain K and closed-loop
result in
178 8. Modern Control Design
Estimator gain vector G
Open-loop Plant
Transfer function:
--------
s^2 - 16
Error matrix A - G*C
8.4 Observability
A system is said to be observable if the initial vector x(t) can be found from the
measurement of u(t) and y (t). The plant described by (8.12) is completely state ob-
servable if the inverse matrix in (8.22) exists. The function V=obsvable(A, C) returns
the observability matrix V and determines whether or not the system is state observ-
able.
8.5 Combined Controller-Observer Design
Consider the system represented by the state and output equations (8.12) and (8.13)
with the state feedback control based on the observed state x^(t) given by
u(t) = K ^x(t) (8.24)
Substituting in (8.12), the state equation becomes
x _(t) = (A BK)x (t) + BKe(t) (8.25)
Combining the above equation with the error equation given by (8.19), we have
x_(t)
e_(t)
=
A BK BK
0 A GC
x(t)
e(t)
(8.26)
The function [K, G, A
c
] = placeobs(A, B, C, p, p
e
) is developed for the com-
bined controller-observer design. A is the system matrix, B is the input column vector,
and C is the output row vector. p is a row vector containing the desired closed-loop
poles and p
e
is the desired observer eigenvalues. The function displays the gain vec-
tors K and G , open-loop plant transfer function, and the controlled system closed-loop
transfer function. Also, the function returns the gain vector K , and the combined sys-
tem matrix in (8.26).
8.5. Combined Controller-Observer Design 179
Example 8.
For the system of Example 8.3, design a controller-observer system such that the
desired closed-loop poles for the system are at 1 j 2. Choose the eigenvalues of
the observer gain matrix to be p
e
= p
e
= 8.
The following commands
A=[ 0 1; 16 0];
B=[0; 1];
C=[1 0]; D=0;
j=sqrt(-1);
P=[-1+j2 -1-j2]; % desired regulator roots
Pe=[-8 -8]; % desired observer roots
[K,G,Af]= placeobs(A,B,C,P,Pe); % returns gain K,G & closed-loop
% system matrix
result in
Feedback gain vector K
Estimator gain vector G:
Open-loop Plant
Transfer function:
--------
s^2 - 16
Controller-estimator
Transfer function:
496 s + 2192
----------------
s^2 + 18 s + 117
Controlled system closed-loop
Transfer function:
496 s + 2192
------------------------------------
s^4 + 18 s^3 + 101 s^2 + 208 s + 320
Combined controller observer system matrix
-5 -2 21 2
0 0 -16 1
0 0 -64 0
From the above results the controller-observer transfer function is
G
ce
=
496 s + 2192
s
+ 18 s + 117
The closed-loop transfer function for the controlled plant is
T (s) =
C (s)
R (s)
=
496 s + 2192
s
+ 18 s
+ 101 s
+ 208 s + 320
180 8. Modern Control Design
8.6 Optimal Regulator Design
The object of the optimal regulator design is to determine the optimal control law
u
(x; t) which can transfer the system from its initial state to the final state (with zero
system input) such that a given performance index is minimized. The performance
index is selected to give the best trade-off between performance and cost of control.
The performance index that is widely used in optimal control design is known as the
quadratic performance index and is based on minimum-error and minimum-energy
criteria.
Consider the plant described by
x _(t) = Ax(t) + Bu(t) (8.27)
The problem is to find the vector K (t) of the control law
u(t) = K(t)x(t) (8.28)
which minimizes the value of a quadratic performance index J of the form
J =
Z
t
f
t
0
(x
0
Qx + u
0
Ru)dt (8.29)
subject to the dynamic plant equation in (8.27). In (8.29) Q is a positive semidefinite
matrix and R is a real symmetric matrix. Q is positive semidefinite if all its principal
minors are nonnegative. The choice of the elements of Q and R allows the relative
weighting of individual state variables and individual control inputs.
To obtain a formal solution, we can use the method of Lagrange multipliers.
The constraint problem is solved by augmenting (8.27) into (8.29) using an n-vector
of Lagrange multipliers , . The problem reduces to the minimization of the following
unconstrained function
L(x; ; u; t) = [x
0
Qx + u
0
Ru] +
0
[Ax + Bu x_] (8.30)
The optimal values (denoted by the subscript ) are found by equating the partial
derivatives to zero.
@ L
@
= AX
+ Bu
x_
= 0 ) x_
= AX
+ B u
(8.31)
@ L
@ u
= 2 Ru
+
0
B = 0 ) u
=
R
1
0
B (8.32)
@ L
@ x
= 2 x
0
Q+
_
0
+
0
A = 0 )
_
= 2 Qx
A
0
(8.33)
Assume that there exists a symmetric, time varying positive definite matrix p (t) satis-
fying
= 2 p(t)x
(8.34)
Substituting (8.34) into (8.32) gives the optimal closed-loop control law
u
(t) = R
1
B
0
p(t)x
(8.35)
8.6. Optimal Regulator Design 181
Obtaining the derivative of (8.34), we have
_
= 2( p_ x
+ p x_
) (8.36)
Finally equating (8.33) with (8.36), we obtain
p _(t) = p(t)A A
0
p(t) Q + p(t)BR
1
B
0
p(t) (8.37)
The above equation is referred to as the matrix Riccati equation. The boundary
condition for (8.37) is p(t
f
) = 0. Therefore, (8.37) must be integrated backward in
time. Since a numerical solution is performed forward in time, a dummy time variable
= t
f
t is replaced for time t. Once the solution to (8.37) is obtained the solution
of the state equation (8.31) in conjunction with the optimum control equation (8.35)
is obtained.
The function [ , p, K, t, x]=riccati is developed for the time-domain solution of
the Riccati equation. The function returns the solution of the matrix Riccati equation,
p( ), the optimal feedback gain vector k( ), and the initial state response x(t). In or-
der to use this function, the user must declare the function [A; B; Q; R; t
0
; t
f
; x
0
] =system
(A; B; Q; R; t
0
; t
f
; x
0
) containing system matrices and the performance index ma-
trices in an M-file named system.m.
For linear time-invariant systems, since p_ = 0 , when the process is of infinite
duration, that is t
f
= 1 , (8.37) reduces to the algebraic Riccati equation
pA + A
0
p + Q pBR
1
B
0
p = 0 (8.38)
The MATLAB Control System Toolbox function [k, p]=lqr2(A, B, Q, R) can be
used for the solution of the algebraic Riccati equation.
Example 8.
Design a state feedback system for the plant described by
x_
x _
=
2
4
0 4 5
3
5
2
4
x
x
x
3
5
+
2
4
0
0
3
5
u
y =
x
Find the optimal control law to minimize the performance index
J =
Z
0
4 x
(t) + 3 x
+ 2 x
+
u
dt (8.39)
The admissible states and control values are unconstrained. The states are initially at
x
(0) = 2 , x
(0) = 0 and x
(0) = 2. For this system we have
A =
2
4
0 4 5
3
5
; B =
2
4
0
0
3
5
; Q =
2
4
3
5
; and R =
First an M-file named system.m is created and the function [A, B, Q, R, t
0
; t
f
; x
0
) =
system (A, B, Q, R, t
0
; t
f
; x
0
) is defined as follows.
182 8. Modern Control Design
− 2
0
Vector K(t) of the control law u(t)=−k(t)x(t)
− 2
− 1
0
Initial state response x
, x
, x
FIGURE 8.
Optimal feedback gain vector K (t) and initial condition response x (t).