4.4 Evaluating Quadratic Indices

In this section we apply the Lyapunov theory to evaluate the quadratic indices

subject to

where Q = C ^T C such that the pair ( A, C) is observable, and the matrix A is asymptotically stable. We can reformulate the above problem as follows. Evaluate the quadratic performance indices

subject to

where the pair ( A, C) is observable, and the matrix A is asymptotically stable.

To evaluate

we note that

where P and Q = C ^T C satisfy the Lyapunov matrix equation, A ^T P + PA = ? Q. Hence,

Integrating both sides of (4.10) with respect to t yields

because, by assumption, A is asymptotically stable and therefore lim _{t ??} x( t) = 0 for all x(0). Furthermore, P = P ^T > 0, and thus J ₀ > 0 for all x(0) ? 0.

Repeating the above arguments leads to a similar expression for J ₁. First observe that

where P satisfies the Lyapunov matrix equation, A ^T P + PA = ? Q. Hence,

Because P = P ^T > 0, we can solve

for P ₁ = P ^T ₁ > 0. The above equation can also be represented as

Substituting (4.13) into (4.12) yields

Using similar arguments, we can evaluate J _r, r