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## 7.3.1 gain

We consider a linear time-invariant system

 (7.17)

and the cost functional

 (7.18)

with . In this cost, the squared norms of the input and the output inside the integral are multiplied by scalar weights of the opposite signs (and we assume the two weights to have been normalized so that their product equals ). Consequently, it is clear that the optimal cost will now be nonpositive (just set ). Nevertheless, as we will see, the form of the optimal solution is very similar to the one we saw in Section 6.2 and can be established using similar calculations. Suppose that there exists a matrix with the following three properties:

1. .
2. is a solution of the ARE

 (7.19)

3. The matrix is Hurwitz.7.2

Then, we claim that the optimal cost is

 (7.20)

and the optimal control is the linear state feedback

 (7.21)

(Notice the wrong" signs in the formulas (7.19)-(7.21) compared to Section 6.2; this sign difference could be reconciled by working with instead of here.)

To prove this claim, let us define the function . Its derivative along solutions of the system (7.17) is , which is easily checked to be equivalent to

 (7.22)

We now introduce the auxiliary finite-horizon cost

 (7.23)

Using the formula (7.22) and noting that the first term on its right-hand side vanishes by (7.19), we can rewrite this cost as

which makes it clear that (7.20) and (7.21) are the optimal cost and optimal control for the cost functional (7.23). We want to show that they are also optimal for the original cost functional (7.18). To this end, we first note that since , the following bound holds for all :

On the other hand, we can pass to the limit as in the already established relation

where . In view of the fact that is a Hurwitz matrix, the closed-loop system is exponentially stable. We thus obtain , and the desired result is proved.

While the formulas appearing here and in Section 6.2 are similar, the meanings of the two problems are very different. The cost (7.18) no longer reflects the objective of keeping both and small. Instead, this cost is small when is large relative to . We can regard here not as a control that regulates the output but as a disturbance that tries to make the output large, with the optimal input being in some sense the worst-case disturbance. Let us try to formulate this idea in more precise terms. The fact that (7.20) is the optimal cost for the functional (7.18) implies that the inequality

 (7.24)

holds for all , with the equality achieved by the optimal control . From now on we focus on the case when . Specializing (7.24) to this case, after simple manipulations we reach

 (7.25)

The fraction on the left-hand side of (7.25) is the ratio of the norms7.3 of the input and the output, and the supremum is being taken over all nonzero inputs with finite norms. If we view the system (7.17), with the zero initial condition, as an input/output operator from to , then (7.25) says that the induced norm of this operator does not exceed . This induced norm is called the gain of the system.

We see that if, for a given value of , we can find a matrix with the three properties listed at the beginning of this subsection, then the system's gain is less than or equal to . (We do not know whether is actually achieved by some control; note that the optimal control (7.21) is excluded in (7.25) because it is identically 0 when .) A converse result also holds: if the gain is less than , then a matrix with the indicated properties exists. If we sidestep the original optimal control problem and only seek sufficient conditions for the gain to be less than or equal to , then it is not hard to see from our earlier derivation that the conditions on the matrix can be relaxed. Namely, it is enough to look for a symmetric positive semidefinite solution of the algebraic Riccati inequality

 (7.26)

The formula (7.22) then yields

Integrating both sides from to an arbitrary time , rearranging terms, and using the definition of and the fact that , we have

and in the limit as we again arrive at (7.25).

In the frequency domain, the system (7.17) is characterized by the transfer matrix . Using Parseval's theorem, it can be shown that the gain equals the largest singular value of supremized over all frequencies ; for systems with scalar inputs and outputs, this is just where is the transfer function. In view of this fact, the gain is also called the norm.

Next: 7.3.2 control problem Up: 7.3 Riccati equations and Previous: 7.3 Riccati equations and   Contents   Index
Daniel 2010-12-20