ABSTRACT
In the modern control design literature, besides eigenstructure assignment, there is a main result called “linear quadratic optimal control” (LQ). The two designs are quite opposite in direction. The eigenstructure assignment, especially the analytical eigenvector assignment, is designed mainly from the bottom up, based on the given plant system’s structure. On the contrary, the LQ control is designed from top down, based on a given and abstract optimal criterion as
(9.1)
where Q and R are symmetrical, positive semi-definite and symmetrical, positive definite matrices, respectively. The LQ design is aimed at minimizing J of (9.1) under the constraint (1.1a)
Inspection of (9.1) shows that to minimize or to have a finite value of J, x(t→∞) must be 0. Hence the control system must be stable (see Definition 2.1). In addition, among the two terms of J, the first term reflects the smoothness and quickness of x(t) before it converges to 0, while the second term reflects the control energy, which is closely related to control gain and system robustness (see Example 8.7). Hence the LQ design can consider both performance and robustness.
