Section 10–8 / Quadratic Optimal Regulator Systems 795
whereTis a nonsingular matrix. Then Equation (10–115) can be written as
which can be rewritten as
The minimization of Jwith respect to Krequires the minimization of
with respect to K. (See Problem A–10–16.) Since this last expression is nonnegative, the
minimum occurs when it is zero, or when
Hence,
(10–117)
Equation (10–117) gives the optimal matrix K. Thus, the optimal control law to the quad-
ratic optimal control problem when the performance index is given by Equation (10–114)
is linear and is given by
The matrix Pin Equation (10–117) must satisfy Equation (10–115) or the following
reduced equation:
(10–118)
Equation (10–118) is called the reduced-matrix Riccati equation. The design steps may
be stated as follows:
1.Solve Equation (10–118), the reduced-matrix Riccati equation, for the matrix P.
[If a positive-definite matrix Pexists (certain systems may not have a positive-
definite matrix P), the system is stable, or matrix A-BKis stable.]
2.Substitute this matrix Pinto Equation (10–117). The resulting matrix Kis the
optimal matrix.
A design example based on this approach is given in Example 10–9. Note that if the
matrixA-BKis stable, the present method always gives the correct result.
Finally, note that if the performance index is given in terms of the output vector
rather than the state vector, that is,
then the index can be modified by using the output equation
to
(10–119)
and the design steps presented in this section can be applied to obtain the optimal
matrixK.
J=
3
q0(x C QCx+u* Ru)dt
y=Cx
J=
3
q0