| 引用本文: | 李铁钧.一类非线性系统的可逆性[J].控制理论与应用,1984,1(1):87~97.[点击复制] |
| Li Tiejun.INVERTIBILITY OF A CLASS OF NONLINEAR SYSTEMS[J].Control Theory and Technology,1984,1(1):87~97.[点击复制] |
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| 一类非线性系统的可逆性 |
| INVERTIBILITY OF A CLASS OF NONLINEAR SYSTEMS |
| 摘要点击 1194 全文点击 512 投稿时间：1982-12-11 修订日期：1983-05-04 |
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| DOI编号 |
| 1984,1(1):87-97 |
| 中文关键词 |
| 英文关键词 |
| 基金项目 |
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| 中文摘要 |
| 本文讨论定义在实解析流形上的如下形状的非线性系统的可逆性{dx(t)/dt=A（x（t））+m∑i=1ui（t）Bi（x（t）），y（t）=C（x（t））
本文主要结果是分别得到了如上系统具有可逆性的必要条件和充分条件。充分条件的证明是构造性的，因而可用来具体构造由系统的输出信息重现输入信息的逆系统。 |
| 英文摘要 |
| In this paper we discuss the invertibility for nonlinear systems of the form {dx(t)/dt=A（x（t））+m∑i=1ui（t）Bi（x（t）），y（t）=C（x（t））(1) where x∈M, Mis a connected real analytic manifold; A, B1, B2,…,Bm are real analytic vector fields on M; ui(t), i=1,2,…,m are real, analytic functions from [0,+∞]into , we denote u(t)=(u1(t))，(u2(t))，…(um(t)); C is a real analytic mapping from M into R, i.e.,c(x)=(c1(x), c2(x),… ,cp(x)), ci, i=1,2,…,p are real analytic funtions
For any initial state x0∈M and control function u(t), we denote the output of system (1) by y(t,u,x0).
The nonlinear system(1) is invertible at x0∈M if for any two distinct control functions u(t).v(t),y(t,u,x0)≠y(t,V,x0).(1) is invertible if there exists an open and dense submanifold M0 of M such that (1) is invertible for all x0∈M.
For the system (1) we define aif={∞ if adkABi(ci)=0, k=0,1,2,… l if adk kABi(ci)=0, k=0,1,2,…, l-1 and adl ABi(ci) adfABi(ci) ≠0
i=1,2,…,m, j=1,2,…,p
and let[a]={aij}
The main results of this paper are;
Theorem 2. For system (1), suppose that p≥m and the following conditions are satisfied;
(i) In every row of [a] there exists elements different from ∞, locating in different columns. Without loss of generality we may assume that they are a11,a22,…,amm,
(ii) aij |
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