摘要
本文主要研究了鲁棒自适应控制的两个理论问题，分为以下两部分：
一．相对阶n~*=3的具有未建模动态系统的鲁棒直接型模型参考自适应控制
考虑下面的单输入单输出系统其中u_p，y_p∈R~1分别是系统的输入和输出，参数a_i和b_j(i=0，…，n-1，j=0，…，m-1)是未知常数，Δ_k(s)(k=1，2)是关于输入和输出的未建模动态，μ_k≥0(k=1，2)是参数。
控制目标是设计具有未规范化自适应律的直接型模型参考自适应控制律u_p，使得闭环系统的所有信号有界，同时使跟踪误差尽可能地小，这里其中y_m，r∈R~1，r(t)是参考输入，它是分段连续和一致有界的。
对该系统，作如下假设：
(P_1) Z_p(s)是Hurwitz多项式；
(P_2) G_p(s)的相对阶n~*=n-m=3；
(P_3) 高频增益K_p的符号已知，且存在常数，使得。
对参考信号y_m(t)作如下假设：
(M_1)：Z_m(s)和R_m(s)是阶次分别为q_m和P_m的首一Hurwitz多项式，且P_m≤n；
(M_2)：参考模型的相对阶p_m-q_m=3．；
(M_3)：参考输入r。
该部分对于相对阶记=3的具有未建模动态的一类系统，给出了具有未
规范化自适应律的普棒直接型模型参考自适应控制器的设计方法，分析了闭
环系统的稳定性和性能
二.具有时变参数的分散系统的自适应反推控制
考虑如下有N个关联子系统组成的复杂系统
活沉(亡)==A。‘(忿)x区(亡)+b沉(t)。‘(t)+
从(t)=呱(1+户“△‘*(s))x。‘(t)+
扬(t，脚)，
肠△*j(，)脚，
“半
N︻，﹂笋Nl，1比
其中xo‘(t)任R叭，夕*(t)〔R，。‘(t)任R分别是第葱个子系统的状态，输出和输
入;A。‘(t)，b。‘(t)是时变矩阵，Co‘是常数矩阵;凡(t，场)。R“‘和△勺(s)场(‘笋J)
分别表示第j个子系统到第乞个子系统的静态干扰和动态干扰项;△“(习是
第葱个子系统的未建模动态，户勺>0是参数.
控制目标是对每一个子系统设计一个局部自适应控制器，使整个关联系
统稳定并且使所有的输出y，渐近地充分小.
对该系统，作如下假设:
假设1
(l)对每一个子系统，系统的阶n‘和相对阶八=n、一m*己知;
(2)系统的高频增益b黔(t)的符号已知，并且存在一个常数占>0，使
1乙:n’(t)}>占，V亡全。;
(3)多项式B“(s)=b尹:(‘)sm!+…+公(t)s+b牙(t)是一致Hurwitz的，即
B，，(s)的所有零点z，，(t)(J=1，…，。，)满足Re(z‘，(t))三一占，Vt全。，占>o是一
常数.
假设2
(l)时间函数a，(艺)#t，bj(t)是可微且有界的，且它们的导数是分段连续的
并且有界，即存在常数庆l>0，刀、:>0使得}d、(t)}三月‘1，!bj(t)}三风2‘·
(2)10‘(‘)l和I。‘(‘)l‘分上界分另，J是M0‘和姚‘，其中“、(‘)议砖而，氏(‘)-
(b少‘(t)，…，b分(t)，a犷‘一‘(‘)，…，a牙(t))T是未知的参数向量.
假设3
对于非线性关联项凡(t，外)，满足“凡(t，巧)lI三凡助}，其中凡>0.
假设4
△以s)“=l，…，N;j二1，…，川稳定且严格正则并且带有单位高频增
益.
该部分针对一类有N个具有任意阶的关联子系统组成的分散系统，继
续研究了鲁棒自适应控制问题，严格地给出了闭环系统的性能分析.
This paper mainly deals with two theoretical problems of robust adaptive control. It is composed of two parts.
一. Robust Direct Model Reference Adaptive Control for Relative Degree n* = 3 System with Unmodeled Dynamics
Consider the following SISO plant
where are the plant input and output, respectively, , parameters ai and bj(i = 0,...,n -1, j = 0, ...,m -1) are unknown constants, k(s)(k = 1,2) are the unmodeled dynamics of the system, (k = 1,2) are parameters.
The control objective is to design a robust direct model reference adaptive control law up with unnormalized adaptive law such that all signals in the closed-loop plant are bounded and the tracking error e1(t) yP(t)-ym(t) is small enough, where
where ym,r R1, r(t) is a piecewise continuous uniformly bounded signal.
We give the following assumptions for the plant:
(P1) Zp(s) is Hurwitz polynomial;
(P2) The relative degree of Gp(s) is n* = n - m = 3;
(P3) The sign of the plant high-frequency gain Kp is unknown and there exist a constant K > 0 such that |KP| > K.
For the reference model , we need assumptions:
(M1) Zm(s),Rm(s) are monic Hurwitz polynomials of degree qm,pm respectively ,where pm < n;
(M2) The relative degree of the reference model equals to n* = pm - qm= 3;
(M3) The reference input
In this part, for a kind of systems with unmodeled dynamics and the relative degree n* = 3, we give the design method of a robust direct model reference adaptive controller with unnormalized adaptive law, and analyze stability and performance of the closed-loop system.
二. Adaptive Backstepping Control of Decentralized Plants with Time-varying Parameters
Consider the following large-scale system consisting of N interconnected sub-systems,and the ith subsystem is modelled by
where are the local states,output and input, respectively; are time-varying matrices,Coi is constant matrice; and denote,respectively,the static interactions and the dynamic interactions from the jth subsystem to the ith subsystem; ii(s) is the unmodeled dynamics in the ith subsystem;and uij> 0 are parameters.
Control objective is to design a local adaptive controller for each subsystem such that the overall interconnected system is stable and all the outputs yi is sufficiently small asymptotically.
For the plant , we need the following assumptions:
Asumption 1
(1) The plant order ni and its relative degree are known;
(2) The sign of the plant high-frequency gain bmii(t) is knowm and there exist a constant > 0 such that
(3) The polynomial is uniformly Hur-witz, i.e. all zeros of Bit(s) satisfy the equality: Re(zij(t)) <
-6, Vt >0 for some constant 6 >0. Asumption 2
(1) Time functions ai(t) and bj(t) are difFerentiable and bounded, and there derivatives are both piecewise continuous and bounded, i.e. there exist constants such that
(2) Upper bound and known,where unknown parameter vectors.
Asumption 3
For the nonlinear interactions fij(t,yj), we have where
> 0.
Asumption 4
are stable and strictly proper with unity high- frequency gains.
In this part, for a large-scale system consisting of N interconnected subsystems with arbitrary degree, we continue to study the robust adaptive control problem and give rigorous performance analysis of the closed-loop system.
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