Page 773 - Mechanical Engineers' Handbook (Volume 2)

764 Control System Design Using State-Space Methods

˙
T
1
T
P(t) R (t) P(t)B(t)R (t)B (t)P(t) P(t)A(t) A (t)P(t) (26)
2
1
and the terminal condition
P(t ) P (27)
1 ƒ
Numerical solution of the matrix Riccati equation is a subject of great importance and of
considerable research. Some useful techniques have been briefly described by Kwakernaak
and Sivan. 3
Solution of the LQR problem simplifies as the terminal time t approaches infinity. It
1
can be shown then that the solution of the matrix Riccati equation approaches a steady-state
solution P (t) that is independent of P . The resulting steady-state control law
ƒ
s
1
u(t) RB (t)P (t)x(t) (28)
T
2
s
results in an exponentially stable closed-loop system if:
1. The linear system of Eq. (4) in Chapter 17 is uniformly completely state controllable.
T
2. The pair A(t), H (t) is uniformly completely reconstructible where H (t) is any matrix
r
r
T
such that H (t)H (t) equals R (t).
r
1
r
The matrix Riccati equation for the steady-state LQR problem simplifies to an algebraic
equation and P is a constant if the system matrices and the weighting matrices in the index
s
of performance are constant. The resulting algebraic Riccati equation is
T
1
T
R PBR B P AP PA 0 (29)
s
s
s
s
2
1
where P is a unique positive-definite solution of Eq. (29) and the resulting time-invariant
s
closed-loop system is asymptotically stable if:
1. The linear system of Eq. (6) in Chapter 17 is completely state controllable.
2. The pair A, H T r is completely observable (reconstructible), where H is any matrix
r
such that H H T r equals R .
1
r
Another version of the LQR problem involves minimization of the quadratic index of
performance for an LTI system over a finite time interval. If the weighting matrices are also
time invariant, in many cases the optimal feedback gains are constant over most of the time
interval of interest and vary with time only near the terminal time. Since constant feedback
gains are easier to implement in practice, implementation of constant gains over the entire
time interval would represent a nearly optimal solution that is practically more convenient. 3
3.2 The Discrete-Time LQR Problem
The results of the LQR problem for discrete-time systems parallel those for continuous-time
systems already stated. They are summarized here and described in greater length by Kwak-
3
ernaak and Sivan. The time-varying, discrete-time system is described by Eq. (10) in Chapter
17 and the initial condition
x(k ) x 0 (30)
0
The index of performance to be minimized by controller design, for the finite-time LQR
problem, is