15.An introduction to H~ control theory

(iii)CR ◦ CR = CR;

(iv)OR ◦ OR = OR;

(v)OR is invertible.

Proof (i) Since −A is Hurwitz, by part (i) of Theorem 5.32 there is a unique symmetric matrix P so that (−At, P, bbt) is a Liapunov triple. What’s more, the proof of Theorem 5.32 gives P explicitly as

Z ∞

P = e−Atbbte−Att dt.

0

Now one sees trivially that (A, −P, −b, bt) is also a Liapunov triple. This part of the proposition now follows because CR = −P.

(ii) The proof here is exactly as for part (i).

(iii) This follows from the characterisations of CR and CR given in Proposition 15.24. (iv) This follows from the characterisations of OR and OR given in Proposition 15.24. (v) Since OR is square, injectivity is equivalent to invertibility. Suppose that OR is not

invertible. Then, since OR is positive-semidefinite, there exists x Rn so that xtORx = 0, or so that Z ∞

xte−Attccte−Atxdt.

0

injective, and from part (v) of Proposition 15.25, OR is injective. Thus u = 0 if and only if

Thus we have a characterisation of all nonzero singular values of the Hankel operator as eigenvalues of an n × n matrix. This is something of a coup. We shall suppose that the nonzero singular values are arranged in descending order σ1 ≥ σ2 ≥ · · · ≥ σk, so that σ1 denotes the largest of the singular values. We shall call σ1, . . . , σk the Hankel singular values for the Hankel operator R.

Now we wish to talk about the “size” of a Hankel operator R. Since R is a linear map between two inner product spaces—from RH+2 to RH−2 —we may simply define its norm in the same manner in which we defined the induced signal norms in Definition 5.19. Thus we

This is called the Hankel norm of the Hankel operator R. The following result follows easily from Theorem 15.26 if one knows just a little more operator theory than is really within the confines of this course. However, it is an essential result for us.

Krein [1968a, 1968b, 1971]. Our interest in this section is in a special version of this rather general work, as we are only interested in rational functions, whereas Nehari was interested in general H∞ functions.

15.28 Theorem Let R0 RH−∞, let σ1 > 0 be the largest Hankel singular value for R0, and let (R1, R2) RH+2 × RH−2 be a σ1-Schmidt pair. Then

= k Rk .

Now let (R1, R2) be a σ1-Schmidt pair and write Q = (R0 − R)R1 for R RH+∞. Since R1 RH+2 , R0 (R1) RH+2 . Since RR1 RH+2 , Π−(Q) = Π−(R0R1) = R0 (R1). Therefore,

we compute

0≤ kQ − R0 (R1)k22

=kQk22 + h R0 (R1), R0 (R1)i2 − 2 hQ, R0 (R1)i2

=kQk22 + h R0 (R1), R0 (R1)i2 − 2 Π−(Q), R0 (R1) 2

=kQk22 − h R0 (R1), R0 (R1)i2

=kQk22 − R1, R0 R0 (R1) 2

=kQk22 − σ12 hR1, R1i2

=kQk22 − σ12 kR1k22

≤ kR0 − Rk2∞ kR1k22 − σ12 kR1k22 = kR0 − Rk2∞ − σ12 kR1k22

≥ 0.

This shows that Q = R0 (R1), or, equivalently,

(R0 − R)R1 = R0 (R1) = σ1R2,

significant, and it is perhaps not transparent how Hankel operators and Nehari’s Theorem relate in any way to the model matching problem. The relationship is, in fact, quite simple, and in this section we give a simple algorithm for obtaining a solution to the model matching problem using the tools of this section. However, as with Nevanlinna-Pick theory, there is a drawback in that on occasion a hack will have to be employed. Nonetheless, the process is systematic enough.

Let us come right out and state the algorithm.

15.29Model matching by Hankel norm approximation Given T1, T2 RH+∞.

15.4A robust performance example

15.5 Other problems involving H∞ methods

It turns out that the robust performance problem is only one of a number of problems falling under the umbrella of H∞ control. In this section we briefly indicate some other problems whose solution can be reduced to a model matching problem, and thus whose solution can be obtained by the methods in this chapter.