Modern Control Engineering

798 Chapter 10 / Control Systems Design in State Space

solves the continuous-time, linear, quadratic regulator problem and the associated Riccati

equation. This command calculates the optimal feedback gain matrix Ksuch that the

feedback control law

minimizes the performance index

subject to the constraint equation

Another command

[K,P,E] = lqr(A,B,Q,R)

returns the gain matrix K, eigenvalue vector E, and matrix P, the unique positive-definite

solution to the associated matrix Riccati equation:

If matrix A-BKis a stable matrix, such a positive-definite solution Palways exists. The

eigenvalue vector Egives the closed-loop poles of A-BK.

It is important to note that for certain systems matrix A-BKcannot be made a sta-

ble matrix, whatever Kis chosen. In such a case, there does not exist a positive-definite

matrixPfor the matrix Riccati equation. For such a case, the commands

K = lqr(A,B,Q,R)

[K,P,E] = lqr(A,B,Q,R)

do not give the solution. See MATLAB Program 10–18.

EXAMPLE 10–10 Consider the system defined by

Show that the system cannot be stabilized by the state-feedback control scheme

whatever matrix Kis chosen. (Notice that this system is not state controllable.)

Define

Then

= B

• 1 - k 1
  0

1 - k 2

2

R

A-BK= B

- 1

0

1

2

R - B

1

0

RCk 1 k 2 D

K=Ck 1 k 2 D

u=-Kx

B

x# 1

x# 2

R = B

- 1

0

1

2

RB

x 1

x 2

R+ B

1

0

Ru

PA+A P-PBR-^1 B P+Q= 0

x

=Ax+Bu

J=

3

q

0

(x Qx+u Ru)dt

u=-Kx

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