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We consider a linear timeinvariant 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:

.

is a solution of the ARE

(7.19) 
 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 finitehorizon cost

(7.23) 
Using the formula (7.22) and noting that the first term on its righthand 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 closedloop 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 worstcase 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 lefthand side of (7.25) is the ratio of the
norms^{7.3} 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
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Daniel
20101220