Page 774 - Mechanical Engineers' Handbook (Volume 2)

3 The Standard Linear Quadratic Regulator Problem 765
J [(x )(k 1)R (k 1)x(k 1) u (k)R (k)u(k)] x (k )Px(k ) (31)
k 1
1
T
T
T 1
k k 0 1 2 1 ƒ 1
where the weighting matrices R (k), R (k), and P serve the same functions as R (t), R (t),
1
1
2
ƒ
2
and P for continuous-time systems and satisfy the same conditions. The values k , k are
ƒ
1
0
the initial and final time instants. A more general version of the J, including the term
T
x (k)R (k)u(k) within the summation sign, can be reduced to the form of Eq. (31) by ap-
12
propriate redefinition of the weighting matrices and control vector. 3
The solution of this control problem can be obtained using dynamic programming meth-
ods and is given by
u(k) K(k)x(k) (32)
where
T
K(k) {R (k) G (k)[R (k 1) P(k 1)]G(k)} 1
2 1
T
G (k)[R (k 1) P(k 1)]F(k) (33)
1
P(k)isa n n symmetric, positive-semidefinite matrix satisfying the matrix difference
equation
T
P(k) F (k)[R (k 1) P(k 1)][F(k) G(k)K(k)] k k , k 1 (34)
0
1
1
with the terminal condition
P(k ) P ƒ (35)
1
Unlike the matrix Riccati equation (26) for continuous-time systems, numerical solution of
the preceding matrix difference equations is straightforward for finite-time LQR problems.
The procedure involves solution of the difference equations backward in time:
1. Let k k 1. Then P(k 1) is equal to P and hence is known.
ƒ
1
2. Compute K(k) using Eq. (33) and the known value of P(k 1).
3. Compute P(k) using Eq. (34) and the known values of K(k) and P(k 1).
4. Reduce k by 1 and repeat 2 and 3 until k k .
0
The solution to the discrete-time LQR problem also simplifies as the terminal time k 1
approaches infinity. The solutions of the matrix difference equations (33) and (34) converge
to steady-state solutions K (k), P (k), which are independent of P . The resulting steady-state
ƒ
s
s
control law
u(k) K (k)x(k) (36)
s
results in an exponentially stable closed-loop system if:
1. The linear system of Eq. (10) in Chapter 17 is uniformly completely state control-
lable.
T
2. The pair F(k), H (k) is uniformly completely reconstructible where H (k)isany
r r
T
matrix such that H (k)H (k) equals R (k).
r r 1
Also, the matrices K and P are constants if the system matrices and the weighting
s
s
matrices in the index of performance of Eq. (31) are constants. They are given by solution
of the following algebraic equations: