The Model-Matching Error and Optimal Solution in Locally Convex Space

The model-matching error and the optimal solution in the Hardy space are extended to the locally convex space, and the model-matching error and the optimal solution in the locally convex space are achieved. Thereby the ordinary ∞ H -control theory is extended to with range in locally convex spaces through a form of a parameter vector. The algorithms of computing the infimal model-matching error and the infimal controller are presented.


INTRODUCTION
Assume that R is the real field and n R is the Cartesian product of n copies of R , here n is any positive integer, and that C is a complex plane.
To solve the problem for simplicity, we apply the ) (s G in the model matching problem to ) , ,where s in C , ξ in n R ,and ) , (locally convex space) for each fixed s in C and in ∞ H for each fixed ξ in n R .First, we extend several concepts.
Definition 1 The locally convex space ∞ VH consists of all complex-valued parameter functions ) , ( ξ s F of a complex variable s and a parameter ξ which are analytic and bounded about s in 0 Re > s ( for each fixed ξ in n R ).Similarly, we define the where Definition 2 The subset of ∞ VH consists of all real-rational functions of s and ξ ,will be denoted by (1) α will be called optimal,where α is a model-matching error.

When
) , ( ξ we have We conclude that ,will be viewed as p th power integrable functions about s and ξ .When We shall give in the form of parameter valued case the algorithms of computing the model-matching error α and the optimal controller Q .

THE MINIMAL REALIZATION
Definition 9 The linear time invarient system 1 S defined by Definition 10 The system 1 S described by ( 1) and ( 2) is completely observable if the observability matrix

Modern Applied Science
September, 2009 105 whose elements are rational functions of s ,we wish to find matrices where n I is the unit matrix of order n .
is termed a realization of and the inverse Laplace transform of we take the Laplace transform of ( 9) with zero initial conditions, we have Since from (10) the Laplace transform of the output is where the m r × matrix is an m p × strictly proper rationalfraction matrix of s (for any fixed ξ in n R ).
be the controllability and observability matrices in ( 5) and ( 6) respectively.We wish to show that if these both have rank n then ) , ( ξ s R has least order n .
Suppose that there exists a realization where 1 r and 1 m are positive integers, so that the rank of ix can not be greater than 1 n .That is, 1 n n < , so there can be no realization of ) , ( ξ s G having order less than n .

INFIMAL MODEL-MATCHING ERROR
The Lyapunov equations are Define the two controllability and observability gramians: are the unique solutions of ( 12) and ( 13) respectively.
Proof Using the definition we have are the unique solutions of ( 12).From the discussion above, the uniqueness is obvious.
are the unique solutions of (13) follows similarly. Q.E.D.
Definition 13 Suppose the linear operator , and strictly proper and analytic in 0 Re < s and ) , ( The first function on the left-hand side belong to 2 VH ; from ( 17 ; hence (21) dolds.
From Lemma 4, we can conceive Corollary 5 ) ( Proof from Theorem 3 there exists a function ) , Step Step ( 3 order n ,and is not, of course, unique.All such the above realizations infimum model-matching error α equals ) (ξ λ , the unique optimal X equals ) VL is the space of essentially bounded functions(for any fixed ξ in n R ).
p VRL space, p VRL ,will be viewed as a subset of p VL ,which consists of all real-rational functions of s and ξ .