Some Basic Results On The Mathematic Structures Of H_∞ Control Theory

In this dissertation, the following basic problems of H∞ control theory have been studied:the constructive proof of L2 space’s decomposition; the relationships of transfer function’s H∞-norm about linear system [A,B,C,0] and linear system [A,B,C,D], the simplification proof of some theorem; and the algorithms about H∞-norm; the component items and algorithms of quadratic self-adjoint matrix polynomial’s eigenvalues. The detail interpretations and three examples are given to explain these studied results how to be used in the H∞ control theory and its applications.First, the function space L2’s decomposition is studied. To any function in L2, we present a method in detail that we construct two functions in H2 and H1/2 correspondence with the function in L2. Where H2 is defined as usual. We present that which function in L2 is correspondence with the given pair of functions in H2 and H1/2 respectively.We also prove the uniqueness of a class of functions.We also strictly prove that the function space H2 ’s orthogonal complement H1/2 is really as usual definition.The H1/2’s usual definition is a set of complex function x(s).The function x(s) is analytic on the open right-half-plan(Res>0) and uniformly square integrable(Lebesgue).Second, we studied the condition that H∞-norm is less than 1 in transfer function of linear stable system [A,B,C,0] and [A,B,C,D]-We presented a equivalent judgement theory that system [A,B,C,D]’s H∞-norm is less than 1,if and only if system [A+BR-1DTC,BR-1/2,(I+DR-1DT)1/2C,O]’s H∞-norm is less than 1; We offered an equation judgement theory that system [A,B,C,D]’s H∞-norm is less than 1.The equation judgement theory was deduced from equivalent judgement theory. We established on one-to-one correspondence relationship between the linear stable system [A,B,C,0]’s judgement theory and stable system [A,B,C,D]’s.The proof of judgement theories about system [A,B,C,D]’s H∞-norm was simplified. We presented two deductions by which we can conclude that system [A.B,C,D]’S H∞-norm is whether or not less than y by judging system [A+BR-1DTC,BR-2,(I+DR-1DT)2C,O]’s H∞-norm is whether or not less than y.We established a relation between the linear system [A,B,C,D]’s H∞-norm being whether or not less than y and the system [A,B,C,DJ’s Hamilton matrix having or having not zero point on the axis of imaginaries.The characteristics of the Hamilton matrix’s eigenvalue and its eigenpolynomial were studied.We offered an equivalence theorem between judging the system [A,B,C,D]’s Hamilton matrix having or having not a zero point on the axis of imaginaries and its judging polynomial having or having not a zero point on the axis of reals. We presented an algorithm in detail that estimates Hamilton matrix having or having not a zero point on the axis of imaginaries.We designed an algorithm in detail that calculate the system [A,B,C,D]’s H∞-norm.We also realized the algorithm, offered a example and analysis its result. We studied the method presenred by Madhu N.Belur and C.Praagman) for computing the system [A,B,C,D]’s H∞norm by directly computing the isolated common zero of two special bivariate polynomials .A new judgement method about isolated common zero of two bivariate polynomials is presented.In chapter 4, the quadratic self-adjoint matrix polynomial’s properties were studied in detail. We presented the relation between the quadratic self-adjoint matrix polynomial’s eigenvalue and its corresponding quadratic polynomial’s root for any non-zero vector. The relation is that the quadratic self-adjoint matrix polynomial’s eigenvalue is greater or smaller than its corresponding quadratic polynomial’s root for a given non-zero vector. The interval’s bounds of parameter were given accurately.If the parameter is a number in the interval, then ensure that the quadratic self-adjoint matrix polynomial is negative. The linearly independent of eigenvectors corresponding to the positive(or negative) sign characteristic were demonstrated with simpler ways than before.We designed a parameter bound algorithm about the quadratic self-adjoint matrix polynomial that can judge the quadratic self-adjoint matrix polynomial to be positive,negative or not either. The algorithm is realized.We established a relation between the algorithm and the robust control research about linear system with uncertainty parameter. We offered the theories for establishing the relation and demonstrated them. We offered two examples for explaining the bounds of the quadratic self-adjoint matrix polynomial how to be used for optimizing performance to the closed loop system. One of the examples is state-feedback,another is output-feedback. We explained why there are many state-feedback controllers to satisfy the design demand. Because we described accurately the bounds of parameter about quadratic self-adjoint matrix polynomial be negative,we conclude that the objective of the closed-loop performance can be improved if there existed controller for the linear system with uncertainty parameter.