自适应模糊控制作为一种研究非线性控制的有效方法引起了人们的广泛关注.其成功应用在于自适应模糊逻辑系统具有一致逼近的特性，能够在任意精度上逼近一个定义在致密集上的连续非线性函数^[1]。在自适应模糊控制中，采用“if-then”规则来构建自适应模糊逻辑系统，用来逼近系统中未知不确定部分.以此为基础设计控制器，当对模型施加恰当的控制，可使系统有期望的输出。文献[2-4]给出了几种单输入单输出和多输入多输出非线性系统的自适应模糊控制方案，所设计的控制器可使非线性系统稳定并使系统输出跟踪期望信号。

在实际工程中存在着许多非线性系统，都不可避免地存在着多种时延因素，比如机械臂系统，由于惯性效应导致连杆之间的时延行为发生，还有电网和核反应堆系统等，时延的存在使得控制器设计不同于对传统非线性控制器设计，从而非线性时延系统的稳定性问题备受关注。文献[5-16]提出了多种行之有效的非线性时延系统的模糊控制方案。这些控制手段不但具有很好的理论突破，而且还成功的应用到工程设计中。文献[5-7]提出了采用时延模糊T-S模型的建模方法。但是该方法忽略了对非线性时延函数的建模误差，可是建模误差的忽略会影响系统的稳定性。文献[8-10]假定非线性时延函数满足匹配条件。尽管匹配条件实现了对误差的建模，然而匹配条件是强假设条件，不易寻求。文献[11-14]考虑建模误差和非线性时延函数有上界。虽然上界比匹配条件降低了保守性，可是上界同样会给控制器的设计增加约束。文献[15-16]提出的模糊自适应方法与反推技术相结合的控制方法。然而反推技术算法复杂，这给控制器的设计增加了难度。

研究工作拓展了文献[2]中的基于T-S模型的自适应模糊系统，推广到时延情形，构建了基于模糊T-S模型的自适应时变时延模糊逻辑系统。用该模糊逻辑系统来逼近未知非线性时变时延函数，从而实现了对一类多输入多输出非线性时变时延系统的建模，以此为基础，提出了一种自适应模糊跟踪控制方案。与文献[5-7]只考虑线性化之后的线性系统相比，笔者没有忽略建模误差，从非线性系统出发设计控制器，从而降低了保守性。与文献[8-14]相比，采用自适应时变时延模糊逻辑系统逼近非线性时延函数，从而克服了对时延函数做匹配条件假设和上界条件假设的不足，同时也降低了不等式的阶数，减少了求解不等式的难度。

具体设计思路：针对一类多输入多输出非线性时变时延系统，构建了基于模糊T-S模型的自适应时变时延模糊逻辑系统用来逼近未知非线性时变时延函数，通过反复调整模糊系统的权值、中心和幅度实现对未知非线性时变时延函数的近似。在自适应算法中，采用跟踪误差来调整自适应时变时延模糊逻辑系统中的参数。应用H_∞补偿器来抵消模糊逼近误差和外部扰动。根据Lyapunov稳定性理论，证明了闭环系统的稳定性并满足期望的H_∞跟踪性能。机械臂的仿真结果表明了该方案的可行性。

1 问题描述

考虑如下多输入多输出非线性时变时延系统

$ \begin{array}{*{20}{c}} {{{\dot x}_1} = {x_2},}\\ \cdots \\ {{{\dot x}_{\left( {{\beta _1} - 1} \right)}} = {x_{{\beta _1}}},}\\ {{{\dot x}_{{\beta _1}}} = {f_1}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}\left( {t - {\tau _1}\left( t \right), \cdots ,\mathit{\boldsymbol{x}}\left( {t - {\tau _r}\left( t \right)} \right)} \right)} \right. + }\\ {\sum\limits_{i = 1}^m {{g_{1i}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}\left( {t - {\tau _1}\left( t \right)} \right), \cdots ,\mathit{\boldsymbol{x}}\left( {t - {\tau _r}\left( t \right)} \right)} \right){u_i} + {d_1}} ,} \end{array} $

(1)

$ \begin{array}{*{20}{c}} {{{\dot x}_{\left( {{\beta _1} + 1} \right)}} = {\mathit{\boldsymbol{x}}_{\left( {{\beta _1} + 2} \right)}},}\\ \cdots \\ {{{\dot x}_n} = {f_m}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}\left( {t - {\tau _1}\left( t \right)} \right), \cdots ,\mathit{\boldsymbol{x}}\left( {t - {\tau _r}\left( t \right)} \right)} \right)}\\ {\sum\limits_{i = 1}^m {{g_{mi}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x}}\left( {t - {\tau _1}\left( t \right)} \right), \cdots ,\mathit{\boldsymbol{x}}\left( {t - {\tau _r}\left( t \right)} \right)} \right){u_i} + {d_m}} ,} \end{array} $

$ \begin{array}{*{20}{c}} {{y_1} = {x_1},}\\ \cdots \\ {{y_m} = {x_{\left( {n - {\beta _m} + 1} \right)}},}\\ {x\left( t \right) = \mathit{\Xi }\left( t \right),t \in \left[ { - \zeta ,0} \right],} \end{array} $

其中x=[x₁, …, x₁^(β₁-1), …, x_(n-βm+1), …, x_(n-βm+1)^(β_m-1)]^T∈Rⁿ、u=[u₁, …, u_m]^T和y=[y₁, …, y_m]^T分别是系统的状态、输入和输出向量，状态是可量测的，β₁+β₂+…+β_m=n，f_i、g_ij(i, j=1, …, m)为充分光滑连续函数，d_i(i=1, …, m)是外部扰动，Ξ(t)连续，表示系统的初始状态，τ_i(t)(i=1, 2, …, r)表示时变时延，ζ=max{τ_i(t)|1≤i≤r}。

引进时延算子σ_i(t):σ_i(t)x(t)=x(t-τ_i(t))(i=0, 1, …, r), 其中τ₀(t)=0，τ_i(t) > 0(i=1, …, r)。令σ(t)=[σ₀(t) σ₁(t) … σ_r(t)]，于是，含有时变时延的非线性向量函数和含有时变时延非线性矩阵函数可表示为

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{F}}_{\sigma \left( t \right)}}\left( x \right) \buildrel \Delta \over = F\left( {x,x\left( {t - {\tau _1}\left( t \right)} \right), \cdots ,x\left( {t - {\tau _r}\left( t \right)} \right)} \right) = }\\ {\left[ {{f_i}\left( {x,x\left( {t - {\tau _1}\left( t \right)} \right), \cdots ,x\left( {t - {\tau _r}\left( t \right)} \right)} \right)} \right],\left( {i = 1,2, \cdots ,m} \right),}\\ {{\mathit{\boldsymbol{G}}_{\sigma \left( t \right)}}\left( x \right) \buildrel \Delta \over = G\left( {x,x\left( {t - {\tau _1}\left( t \right)} \right), \cdots ,x\left( {t - {\tau _r}\left( t \right)} \right)} \right) = }\\ {\left[ {\left( {{g_{ij}}\left( {x,x\left( {t - {\tau _1}\left( t \right)} \right), \cdots ,x\left( {t - {\tau _r}\left( t \right)} \right)} \right)} \right.} \right],}\\ {\left( {i,j = 1,2, \cdots ,m} \right),} \end{array} $

其中F_σ(t)(x)是m维的列向量，G_σ(t)(x)是m阶的方矩阵。从而，非线性系统(1)可改写为

$ \dot x = \mathit{\boldsymbol{A}}x + \mathit{\boldsymbol{B}}\left[ {{\mathit{\boldsymbol{F}}_{\sigma \left( t \right)}}\left( x \right) + {\mathit{\boldsymbol{G}}_{\sigma \left( t \right)}}\left( x \right)u + d} \right], $

(2)

$ \begin{array}{*{20}{c}} {y = \mathit{\boldsymbol{C}}x,}\\ {x = \mathit{\Xi }\left( t \right),t \in \left[ { - \zeta ,0} \right],} \end{array} $

式中A=diag[A₁, …, A_m], B=diag[B₁, …, B_m], C=diag[C₁, …, C_m]，${A_i} = \left[ {\begin{array}{*{20}{c}} 0&1&0& \cdots &0\\ \vdots &0&1& \ddots&\vdots \\ \vdots&\vdots&\ddots&\ddots &0\\ 0&0& \cdots&\ddots &1\\ 0&0& \cdots&\cdots &0 \end{array}} \right] \in {R^{{\beta _i} \times {\beta _i}}}$, B_i=[0, …, 0, 1]^T∈R^β_i, C_i=[1, 0, …, 0]∈R^β_i, d=[d₁, …, d_m]^T, F_σ(t)(x)和G_σ(t)(x)是未知部分。

对给定的参考信号y_r1, …, y_rm, 定义跟踪误差为e₁=y_r1-y₁, …, e_m=y_rm-y_m。令y_r=[y_r1, …, y_rm]^T, y_r^(β)=[y_r1^(β₁), …, y_rm^(β_m)]^T, Y_m=[y_1r, …, y_1r^(β₁-1), …, y_mr, …, y_mr^(β_m-1)]^T，e=[e₁, …, e₁^(β₁-1), …, e_m, …, e_m^(β_m-1)]^T。

控制任务：求一个反馈控制u=u_σ(t)(x|Θ₁, Θ₂, α, δ)和调整参数Θ₁、Θ₂、α和δ的自适应律，使闭环系统满足期望的H_∞跟踪性能并使系统输出快速跟踪参考信号。

假定1 ∀x∈U, U是致密集，G_σ(t)(x)是非奇异的。

假定2 τ_i(t)≤τ_iM，$\dot \tau $_i(t)≤τ_iD(i=1, 2, …, r)，τ_iM和τ_iD为正常数。

2 自适应时变时延模糊逻辑系统

自适应模糊逻辑系统(adaptive time-varying delay fuzzy logic systems)具有一致逼近特性，能够在任意精度上逼近一个定义在致密集上的连续非线性函数。构建基于模糊T-S模型的自适应时变时延模糊逻辑系统来逼近m维非线性时变时延向量函数F_σ(t)(x)和m阶非线性时变时延矩阵函数G_σ(t)(x)，自适应参数为权值Θ₁和Θ₂、中心α和幅度δ。对向量函数的第k个分量的逼近形式如下

R^l：若x₁是A₁^l, …, x_n是A_n^l，则$\hat f$_kσ(t)(x)=a₀^l+$\sum\limits_{i = 1}^n {\left( {a_i^l{x_i} + \sum\limits_{j = 1}^r {a_{ij}^l{x_i}\left( {t - {\tau _j}\left( t \right)} \right)} } \right)} $, (l=1, 2, …p),

于是, 对向量函数的第k个分量的逼近如下

$ \begin{array}{*{20}{c}} {{{\hat f}_{k\sigma \left( t \right)}}\left( {x\left| {{\theta _k},\alpha ',\delta '} \right.} \right) = }\\ {\frac{{\sum\limits_{l = 1}^p {\left( {a_0^l + \sum\limits_{i = 1}^n {\left( {a_i^l{x_i} + \sum\limits_{j = 1}^r {a_{ij}^l{x_i}\left( {t - {\tau _j}\left( t \right)} \right)} } \right)} } \right)} \prod\limits_{i = 1}^n {{\mu _{F_i^l}}\left( {{x_i},{{\alpha '}_l},{{\delta '}_l}} \right)} }}{{\sum\limits_{l = 1}^p {\prod\limits_{i = 1}^n {{\mu _{F_i^l}}\left( {{x_i},{{\alpha '}_l},{{\delta '}_l}} \right)} } }} = }\\ {\mathit{\boldsymbol{\xi }}_k^{\rm{T}}\left( {x,\alpha ',\delta '} \right){\theta _k}X,} \end{array} $

(3)

式中模糊基函数ξ_k^T(x, α′, δ′)=(ξ_k¹, …, ξ_k^p)∈R^p, $\xi _k^l = \frac{{\prod\limits_{i = 1}^n {{\mu _{F_i^l}}\left( {{x_i},\alpha _l',\delta _l'} \right)} }}{{\sum\limits_{l = 1}^p {\prod\limits_{i = 1}^n {{\mu _{F_i^l}}\left( {{x_i},\alpha _l',\delta _l'} \right)} } }}$，μ_Fi^l为隶属度函数，模糊基函数中心α′=[α′₁, …, α′_p]∈R^p，模糊基函数幅度δ′=[δ′₁, …, δ′_p]∈R^p，权值θ_k=(θ_k¹, …, θ_k^p)∈R^{p×(n(r+1)+1)}, θ_k^l=(a₀^l a₁^l a₁₁^l … a_1r^l … a_n^l a_n1^l … a_nr^l)∈R^(n(r+1)+1)(l=1, 2, …p)；X=(1 x₁ x₁(t-τ₁(t)) … x₁(t-τ_r(t)) … x_n x_n(t-τ₁(t)) … x_n(t-τ_r(t)))^T∈R^(n(r+1)+1)。同理，对矩阵函数分量的逼近有类似形式。于是得到对m维非线性时变时延向量函数F_σ(t)(x)和m阶非线性时变时延矩阵函数G_σ(t)(x)的逼近形式

$ {{\mathit{\boldsymbol{\hat F}}}_{\sigma \left( t \right)}}\left( x \right) = \mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( {x,\alpha ,\delta } \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X,{{\mathit{\boldsymbol{\hat G}}}_{\sigma \left( t \right)}}\left( x \right) = \mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( {x,\alpha ,\delta } \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar X, $

(4)

式中模糊基函数矩阵Ψ(x, α, δ)=diag[ξ₁^T(x, α′, δ′), …, ξ_m^T(x, α′, δ′)]∈R^m×pm, 权值Θ₁=[θ₁^T, θ₂^T, …, θ_m^T]^T∈R^{pm×(n(r+1)+1)}，权值Θ₂=(θ_ij)_m×m∈R^{pm×m(n(r+1)+1)}，中心$\alpha = {\rm{diag}}\left[ {\underbrace {\alpha ', \cdots ,\alpha '}_{m个}} \right] \in {R^{m \times pm}}$，幅度$\delta = {\rm{diag}}\left[ {\underbrace {\delta ', \cdots ,\delta '}_{m个}} \right] \in {R^{m \times pm}}$，X同上，$\bar X = {\rm{diag}}\left[ {\underbrace {X, \cdots ,X}_{m个}} \right]$。

定义参数误差$\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}$₁=Θ₁-Θ₁^*，$\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}$₂=Θ₂-Θ₂^*，α=α-α^*, δ=δ-δ^*，其中Θ₁^*，Θ₂^*，α^*和δ^*分别为权值，中心和幅度的最优值。则自适应时变时延模糊逻辑系统对非线性时变时延向量函数F_σ(t)(x)和非线性时变时延矩阵函数G_σ(t)(x)的逼近误差有如下形式

$ \begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat F}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{F}}_{\sigma \left( t \right)}}\left( x \right) = \left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - } \right.}\\ {\left. {\delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_1}X + \left( {\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) + \tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X + {w_1},} \end{array} $

(5)

$ \begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat G}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{G}}_{\sigma \left( t \right)}}\left( x \right) = \left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - } \right.}\\ {\left. {\delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\tilde \Theta }}_2}\bar X + \left( {\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) + \tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar X + {w_2},} \end{array} $

(6)

其中Ψ(x)=Ψ(x, α, δ)，Ψ_α(x)=diag[ξ_1α′^T(x, α′, δ′), …, ξ_mα′^T(x, α′, δ′)]，Ψ_δ(x)=diag[ξ_1δ′^T(x, α′, δ′), …, ξ_mδ′^T(x, α′, δ′)]，ξ_iα′^T(x, α′, δ′)和ξ_iδ′^T(x, α′, δ′)分别表示ξ_i^T关于α′和δ′的偏导数，w₁和w₂是残差项。

注1：为了证明所构建的自适应时变时延模糊逻辑系统的一致逼近特性，设Y为由式(3)的模糊逻辑系统组成的集合。可以证明1)(Y，d_∞)是一个代数，2)(Y，d_∞)能离析U上各点，3)(Y，d_∞)在U上任意点上均不为零。根据Stone-Weierstrass定理和1)，2)，3)可以推出自适应时变时延模糊逻辑系统的泛逼近性。由式(3)可以看出，模糊系统的输出为模糊基函数加权平均与权值的乘积，模糊基函数的中心和幅度也在线调整，从而该模糊系统不但了实现了模糊模型的自动更新，而且能不断修正各隶属度函数，使得模糊逼近更准确，提高了逼近精度。模糊规则通常可选择5-7条，也可适当增加模糊规则数，隶属度函数可选择不对称高斯函数、梯形函数以及钟形函数等。从而自适应模糊逻辑系统的模糊基函数具有更大的可变性和延展性，提高自适应模糊逻辑系统的逼近能力。当逼近误差的范数小于一个数量级10^-5时，逼近精度是可接受的，逼近误差可作为程序结束的条件。

3 控制器设计

采用模糊控制律

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{u}} = {{\hat G}_{\sigma \left( t \right)}}{{\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2},\alpha ,\delta } \right.} \right)}^{ - 1}}\left[ { - {{\hat F}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1},\alpha ,\delta } \right.} \right) + } \right.}\\ {\left. {y_r^{\left( \beta \right)} + {\mathit{\boldsymbol{K}}^{\rm{T}}}e - {u_{com}}} \right],} \end{array} $

(7)

式中$\mathit{\boldsymbol{\hat F}}$_σ(t)(x|Θ₁, α, δ)和$\mathit{\boldsymbol{\hat G}}$_σ(t)(x|Θ₂, α, δ)分别是对未知函数F_σ(t)(x)和G_σ(t)(x)的近似，由式(4)给出，K^T是反馈增益阵，使得A-BK^T的特征多项式是Hurwitz的，u_com是H_∞补偿器，用来补偿外部扰动和逼近误差。

因为$\dot Y$_r=AY_r+By_r^(β)，把式(7)代入式(2)得误差动态方程为

$ \begin{array}{*{20}{c}} {\dot e = \left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right)e + \mathit{\boldsymbol{B}}\left[ {\left( {{{\mathit{\boldsymbol{\hat F}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{F}}_{\sigma \left( t \right)}}\left( x \right)} \right) + } \right.}\\ {\left. {\left( {{{\mathit{\boldsymbol{\hat G}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{G}}_{\sigma \left( t \right)}}\left( x \right)} \right)u - d} \right] + \mathit{\boldsymbol{B}}{u_{{\rm{com}}}}。} \end{array} $

(8)

定义最优参数Θ₁^*、Θ₂^*、α^*和δ^*

$ \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1^ * \buildrel \Delta \over = \arg \mathop {\min }\limits_{{\mathit{\Theta }_1} \in {\mathit{\Omega }_1}} \left[ {\mathop {\sup }\limits_{x \in U} \left\| {{{\mathit{\boldsymbol{\hat F}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{F}}_{\sigma \left( t \right)}}\left( x \right)} \right\|} \right], $

$ \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2^ * \buildrel \Delta \over = \arg \mathop {\min }\limits_{{\mathit{\Theta }_2} \in {\mathit{\Omega }_2}} \left[ {\mathop {\sup }\limits_{x \in U} \left\| {{{\mathit{\boldsymbol{\hat G}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{G}}_{\sigma \left( t \right)}}\left( x \right)} \right\|} \right], $

$ \begin{array}{l} {\alpha ^ * } \buildrel \Delta \over = \arg \mathop {\min }\limits_{\alpha \in {\mathit{\Omega }_3}} \left[ {\mathop {\sup }\limits_{x \in U} \left( {\left\| {{{\mathit{\boldsymbol{\hat F}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{F}}_{\sigma \left( t \right)}}\left( x \right)} \right\| + } \right.} \right.\\ \left. {\left. {\left\| {{{\mathit{\boldsymbol{\hat G}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{G}}_{\sigma \left( t \right)}}\left( x \right)} \right\|} \right)} \right], \end{array} $

$ \begin{array}{l} {\delta ^ * } \buildrel \Delta \over = \arg \mathop {\min }\limits_{\delta \in {\mathit{\Omega }_4}} \left[ {\mathop {\sup }\limits_{x \in U} \left( {\left\| {{{\mathit{\boldsymbol{\hat F}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{F}}_{\sigma \left( t \right)}}\left( x \right)} \right\| + } \right.} \right.\\ \left. {\left. {\left\| {{{\mathit{\boldsymbol{\hat G}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2},\alpha ,\delta } \right.} \right) - {\mathit{\boldsymbol{G}}_{\sigma \left( t \right)}}\left( x \right)} \right\|} \right)} \right], \end{array} $

其中U={x∈Rⁿ}，Ω₁={Θ₁=(θ_i)_m×1∈R^{pm×(n(r+1)+1)}}，Ω₂={Θ₂=(θ_ij)_m×m∈R^{pm×m(n(r+1)+1)}}，Ω₃={α=diag[α₁, …, α_m]∈R^m×pm}，Ω₄={δ=diag[δ₁, …δ_m]∈R^m×pm}。

于是，定义最优逼近误差为w

$ \begin{array}{*{20}{c}} {w = \left( {{{\mathit{\boldsymbol{\hat F}}}_{\sigma \left( t \right)}}\left( {x\left| {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1^ * ,{\alpha ^ * },{\delta ^ * }} \right.} \right) - {\mathit{\boldsymbol{F}}_{\sigma \left( t \right)}}\left( x \right)} \right) + }\\ {\left( {{{\mathit{\boldsymbol{\hat G}}}_{\sigma \left( t \right)}}\left( {x\left| {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1^ * ,{\alpha ^ * },{\delta ^ * }} \right.} \right) - {\mathit{\boldsymbol{G}}_{\sigma \left( t \right)}}\left( x \right)} \right)u。} \end{array} $

(9)

由式(5)和式(6)，改写式(9)，从而w=w₁+w₂u。于是，式(8)可改写为

$ \begin{array}{*{20}{c}} {\dot e = \left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right)e + \mathit{\boldsymbol{B}}\left[ {\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - } \right.} \right.}\\ {\left. {\left. {\delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_1}X + \left( {\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) + \tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X} \right] + }\\ {\mathit{\boldsymbol{B}}\left[ {\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_2}\bar X + \left( {\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) + } \right.} \right.}\\ {\left. {\tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar X} \right]u + \mathit{\boldsymbol{B}}\bar w + \mathit{\boldsymbol{B}}{u_{com}}。} \end{array} $

(10)

其中w=w-d。

根据跟踪误差e，选择模糊逻辑系统的参数自适应律

$ {{\mathit{\boldsymbol{ \boldsymbol{\dot \varTheta} }}}_1} = - {\eta _1}{\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right)^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}P\left( {e + 2\mu \dot e} \right){\mathit{\boldsymbol{X}}^{\rm{T}}}, $

(11)

$ {{\mathit{\boldsymbol{ \boldsymbol{\dot \varTheta} }}}_2} = - {\eta _2}{\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right)^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}P\left( {e + 2\mu \dot e} \right){\left( {\bar Xu} \right)^{\rm{T}}}, $

(12)

$ \dot \alpha = - {\eta _3}{\mathit{\boldsymbol{B}}^{\rm{T}}}P\left( {e + 2\mu \dot e} \right){\left( {{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right)\left( {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X + {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar Xu} \right)} \right)^{\rm{T}}}, $

(13)

$ \dot \delta = - {\eta _4}{\mathit{\boldsymbol{B}}^{\rm{T}}}P\left( {e + 2\mu \dot e} \right){\left( {{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)\left( {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X + {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar Xu} \right)} \right)^{\rm{T}}}, $

(14)

其中η₁，η₂，η₃，η₄和μ是正常数，Ψ(x), Ψ_α(ξ), Ψ_δ(x)分别为模糊基函数矩阵，模糊基函数矩阵关于中心的偏导数，模糊基函数矩阵关于幅度的偏导数。

采用H_∞补偿器u_com来补偿外部扰动和逼近误差，H_∞补偿器如下

$ {u_{com}} = - \left( {\frac{1}{\alpha }} \right){\mathit{\boldsymbol{B}}^{\rm{T}}}Pe, $

(15)

其中外部扰动有界，逼近误差的范数小于数量级10^-5，对称正定矩阵P由下面不等式给出

$ \mathit{\boldsymbol{S}} = \left[ {\begin{array}{*{20}{c}} {{s_{11}}}&{{R_{1M}}}& \cdots &{{R_{rM}}}&{\mu \left( {{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right)}^{\rm{T}}} + \frac{1}{\gamma }\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P}}} \right)}\\ * &{ - \left( {1 - {\tau _{1D}}} \right){R_1} - {R_{1M}}}& \cdots &0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots \\ *&*&\cdots &{ - \left( {1 - {\tau _{rD}}} \right){R_r} - {R_{rM}}}&0\\ *&*&\cdots&* &{ - 2\mu \mathit{\boldsymbol{P}} + \sum\limits_{i = 1}^r {\tau _{iM}^2{\mathit{\boldsymbol{R}}_{iM}}} + \frac{{4{\mu ^2}}}{{{\rho ^2}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P}}} \end{array}} \right] < 0, $

(16)

式中s₁₁=(A-BK^T)P+P(A-BK^T)-$\frac{2}{\gamma }$PBB^TP+$\sum\limits_{i = 1}^r {{R_i}} - \sum\limits_{i = 1}^r {{R_{iM}}} $+$\frac{4}{{{\rho ^2}}}$PBB^TP，R_i > 0，R_iM > 0(i=1, 2, …, r)，ρ²≥2γ > 0，由假设2知τ_iM和τ_iD为正常数，*表示对称位置元素。控制系统框图如图 1所示。

4 稳定性分析

定理1 对多输入多输出非线性时变时延系统(1)，选择模糊控制律(7)，自适应时变时延模糊逻辑系统(4)，参数自适应律(11-14)，H_∞补偿器(15)，则闭环系统(8)是有界稳定的并满足H_∞跟踪性能

$ \begin{array}{*{20}{c}} {\int_0^T {{{\mathit{\boldsymbol{\tilde e}}}^{\rm{T}}}\left( { - S} \right)\tilde e{\rm{d}}t} \le {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( 0 \right)Pe\left( 0 \right) + \sum\limits_{i = 1}^r {\int_{ - {\tau _i}\left( 0 \right)}^0 {{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( s \right){R_i}e\left( s \right){\rm{d}}s} } + }\\ {\sum\limits_{i = 1}^r {{\tau _{iM}}\int_{ - {\tau _{iM}}}^0 {\left( {s - {\tau _{iM}}} \right){{\mathit{\boldsymbol{\dot e}}}^{\rm{T}}}\left( s \right){R_{iM}}\dot e\left( s \right){\rm{d}}s} } + }\\ {\frac{1}{{{\eta _1}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_1^{\rm{T}}\left( 0 \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_1}\left( 0 \right)} \right) + \frac{1}{{{\eta _2}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_2^{\rm{T}}\left( 0 \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_2}\left( 0 \right)} \right) + }\\ {\frac{1}{{{\eta _3}}}tr\left( {{{\mathit{\boldsymbol{\tilde \alpha }}}^{\rm{T}}}\left( 0 \right)\tilde \alpha \left( 0 \right)} \right) + \frac{1}{{{\eta _4}}}tr\left( {{{\mathit{\boldsymbol{\tilde \delta }}}^{\rm{T}}}\left( 0 \right)\tilde \delta \left( 0 \right)} \right) + {\rho ^2}\int_0^T {\left( {{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}\mathit{\boldsymbol{\bar w}}} \right){\rm{d}}t} 。} \end{array} $

(14)

其中e=[e^T(t) e^T(t-τ₁(t)) … e^T(t-τ_r(t)) $\dot e$(t)]^T，S < 0；正定矩阵R_i > 0，R_iM > 0(i=1, 2, …, r)，R_i和R_iM满足式(16)。

证 选取Lyapunov函数

$ \begin{array}{*{20}{c}} {V = \frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}Pe + \frac{1}{2}\sum\limits_{i = 1}^r {\int_{t - {\tau _i}\left( t \right)}^t {{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( s \right){\mathit{\boldsymbol{R}}_i}e\left( s \right){\rm{d}}s} } + }\\ {\frac{1}{2}\sum\limits_{i = 1}^r {{\tau _{iM}}\int_{t - {\tau _{iM}}}^t {\left( {s - \left( {t - {\tau _{iM}}} \right)} \right){{\mathit{\boldsymbol{\dot e}}}^{\rm{T}}}\left( s \right){\mathit{\boldsymbol{R}}_{iM}}\dot e\left( s \right){\rm{d}}s} } + \frac{1}{{2{\eta _1}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_1^{\rm{T}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_1}} \right)}\\ { + \frac{1}{{2{\eta _2}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_2^{\rm{T}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_2}} \right) + \frac{1}{{2{\eta _3}}}tr\left( {{{\tilde \alpha }^{\rm{T}}}\tilde \alpha } \right) + \frac{1}{{2{\eta _4}}}tr\left( {{{\mathit{\boldsymbol{\tilde \delta }}}^{\rm{T}}}\mathit{\boldsymbol{\tilde \delta }}} \right)。} \end{array} $

$ \begin{array}{*{20}{c}} {\dot V = \frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\left[ {{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right)}^{\rm{T}}}\mathit{\boldsymbol{P}} + \mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right) - \frac{2}{\alpha }\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P}}} \right]\mathit{\boldsymbol{e + }}}\\ {\frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{PB\bar w + }}\frac{1}{2}{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{Pe + }}\frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\sum\limits_{i = 1}^r {{\mathit{\boldsymbol{R}}_i}e} - }\\ {\frac{1}{2}\sum\limits_{i = 1}^r {\left( {1 - {\tau _{iD}}} \right){\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {t - {\tau _i}\left( t \right)} \right){\mathit{\boldsymbol{R}}_i}e\left( {t - {\tau _i}\left( t \right)} \right)} + }\\ {\frac{1}{2}{{\mathit{\boldsymbol{\dot e}}}^{\rm{T}}}\sum\limits_{i = 1}^r {\tau _{iM}^2{\mathit{\boldsymbol{R}}_{iM}}\mathit{\boldsymbol{\dot e}}} - \frac{1}{2}\sum\limits_{i = 1}^r {\left( {{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right) - {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {t - } \right.} \right.} }\\ {\left. {{\tau _i}\left( t \right)} \right){\mathit{\boldsymbol{R}}_{iM}}\left( {e\left( t \right) - e\left( {t - {\tau _i}\left( t \right)} \right) + \left[ {{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{PB}}\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - } \right.} \right.} \right.}\\ {\left. {\left. {\alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_1}X + \frac{1}{{{\eta _1}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_1^{\rm{T}}{{\mathit{\boldsymbol{ \boldsymbol{\dot {\tilde \varTheta}} }}}_1}} \right)} \right] + }\\ {\left[ {{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{PB}}\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_2}\bar Xu} \right. + }\\ {\left. {\frac{1}{{{\eta _2}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_2^{\rm{T}}{{\mathit{\boldsymbol{ \boldsymbol{\dot {\tilde \varTheta}} }}}_2}} \right)} \right] + \left[ {{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{PB}}\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right)\left( {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X + {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar Xu} \right)} \right. + }\\ {\left. {\frac{1}{{{\eta _3}}}tr\left( {{{\mathit{\boldsymbol{\tilde \alpha }}}^{\rm{T}}}\dot {\tilde \alpha} } \right)} \right] + \left[ {{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{PB}}\tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)\left( {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X + {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar Xu} \right) + \frac{1}{{{\eta _4}}}tr\left( {{{\mathit{\boldsymbol{\tilde \delta }}}^{\rm{T}}}\dot {\tilde \delta} } \right)} \right]。} \end{array} $

(18)

对任意的正常数μ > 0，有下面等式(19)和不等式(20)成立

$ \begin{array}{*{20}{c}} {0 = \frac{1}{2}\left( {2\mu {{\mathit{\boldsymbol{\dot e}}}^{\rm{T}}}P\mathit{\boldsymbol{\dot e}} + \mu {{\mathit{\boldsymbol{\dot e}}}^{\rm{T}}}P\left\{ {\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right)e + } \right.} \right.}\\ {\mathit{\boldsymbol{B}}\left[ {\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_1}X + \left( {\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) + } \right.} \right.}\\ {\left. {\left. {\tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X} \right] + \mathit{\boldsymbol{B}}\left[ {\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - } \right.} \right.}\\ {\left. {\left. {\delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_2}\bar X + \left( {\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar X} \right]u + }\\ {\left. {\mathit{\boldsymbol{B\bar w}} + \mathit{\boldsymbol{B}}{u_{com}}} \right\} + \mu \left\{ {\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right)e + \mathit{\boldsymbol{B}}\left[ {\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - } \right.} \right.} \right.}\\ {\left. {\left. {\alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_1}X + \left( {\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X} \right] + }\\ {\mathit{\boldsymbol{B}}\left[ {\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_2}\bar X + \left( {\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) + } \right.} \right.}\\ {\left. {{{\left. {\left. {\left. {\tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right){\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar X} \right]u + \mathit{\boldsymbol{B\bar w}} + \mathit{\boldsymbol{B}}{u_{com}}} \right\}}^{\rm{T}}}P\dot e} \right)。} \end{array} $

(19)

$ \begin{array}{*{20}{c}} {\frac{1}{2}\mu \left( {{{\mathit{\boldsymbol{\dot e}}}^{\rm{T}}}\mathit{\boldsymbol{PB\bar w + }}{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}{B^{\rm{T}}}P\mathit{\boldsymbol{\dot e}}} \right) \le }\\ {\frac{1}{2}\left( {\frac{{4{\mu ^2}}}{{{\rho ^2}}}{{\mathit{\boldsymbol{\dot e}}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}P\dot e + \frac{{{\rho ^2}}}{4}{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}\mathit{\boldsymbol{\bar w}}} \right)。} \end{array} $

(20)

又因为

$ \frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{PB\bar w}} + \frac{1}{2}{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}Pe \le \frac{1}{2}\left( {\frac{4}{{{\rho ^2}}}{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}Pe + \frac{{{\rho ^2}}}{4}{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}\mathit{\boldsymbol{\bar w}}} \right)。$

(21)

于是，对式(18)左右两端分别加式(19)的左右两端，再由不等式(20)(21)，得到

$ \begin{array}{*{20}{c}} {\dot V \le \frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\left[ {{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right)}^{\rm{T}}}\mathit{\boldsymbol{P}} + \mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right) - } \right.}\\ {\left. {\frac{2}{\alpha }\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P + }}\frac{4}{{{\rho ^2}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P}}} \right]\mathit{\boldsymbol{e + }}\frac{1}{2}\left( {\frac{4}{{{\rho ^2}}}{e^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}Pe + } \right.}\\ {\left. {\frac{{{\rho ^2}}}{4}{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}\mathit{\boldsymbol{\bar w}}} \right) + \frac{1}{2}{\mathit{\boldsymbol{e}}^{\rm{T}}}\sum\limits_{i = 1}^r {{R_i}e} - \frac{1}{2}\sum\limits_{i = 1}^r {\left( {1 - {\tau _{iD}}} \right){\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {t - } \right.} }\\ {\left. {{\tau _i}\left( t \right)} \right){\mathit{\boldsymbol{R}}_i}e\left( {t - {\tau _i}\left( t \right)} \right) + \frac{1}{2}{{\dot e}^{\rm{T}}}\sum\limits_{i = 1}^r {\tau _{iM}^2{\mathit{\boldsymbol{R}}_{iM}}\dot e - } }\\ {\frac{1}{2}\sum\limits_{i = 1}^r {\left( {{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( t \right) - {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {t - {\tau _i}\left( t \right)} \right){\mathit{\boldsymbol{R}}_{iM}}\left( {e\left( t \right) - e\left( {t - {\tau _i}\left( t \right)} \right)} \right.} \right.} + }\\ {\frac{1}{2}\left( {\frac{{4{\mu ^2}}}{{{\rho ^2}}}{{\dot e}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}P\dot e + \frac{{{\rho ^2}}}{4}{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}\mathit{\boldsymbol{\bar w}}} \right) + \frac{1}{2}\left[ {2\mu {{\dot e}^{\rm{T}}}P\dot e + } \right.}\\ {\mu {{\mathit{\boldsymbol{\dot e}}}^{\rm{T}}}P\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right)e + \mu {\mathit{\boldsymbol{e}}^{\rm{T}}}{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{K}}^{\rm{T}}}} \right)}^{\rm{T}}}P\dot e - }\\ {\left. {\left( {\frac{\mu }{\alpha }} \right){{\dot e}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}Pe - \left( {\frac{\mu }{\alpha }} \right){\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{B}}^{\rm{T}}}P\dot e} \right] + }\\ {\left[ {{{\left( {e + 2\mu \dot e} \right)}^{\rm{T}}}\mathit{\boldsymbol{PB}}\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - } \right.} \right.}\\ {\left. {\left. {\delta {\mathit{\Psi }_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_1}X + \frac{1}{{{\eta _1}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_1^{\rm{T}}{{\mathit{\boldsymbol{ \boldsymbol{\dot {\tilde \varTheta}} }}}_1}} \right)} \right] + }\\ {\left[ {{{\left( {e + 2\mu \dot e} \right)}^{\rm{T}}}\mathit{\boldsymbol{PB}}\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( x \right) - \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right) - } \right.} \right.}\\ {\left. {\left. {\delta {\mathit{\Psi }_\delta }\left( x \right)} \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_2}X + \frac{1}{{{\eta _2}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_2^{\rm{T}}{{\mathit{\boldsymbol{ \boldsymbol{\dot {\tilde \varTheta}} }}}_2}} \right)} \right] + }\\ {\left[ {{{\left( {e + 2\mu \dot e} \right)}^{\rm{T}}}\mathit{\boldsymbol{PB}}\tilde \alpha {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\alpha }\left( x \right)\left( {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X + {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar Xu} \right) + } \right.}\\ {\left. {\frac{1}{{{\eta _3}}}tr\left( {{{\mathit{\boldsymbol{\tilde \alpha }}}^{\rm{T}}}\dot {\tilde \alpha} } \right)} \right] + \left[ {{{\left( {e + 2\mu \dot e} \right)}^{\rm{T}}}\mathit{\boldsymbol{PB}}\tilde \delta {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_\delta }\left( x \right)} \right.}\\ {\left. {\left( {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1}X + {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2}\bar Xu} \right) + \frac{1}{{{\eta _4}}}tr\left( {{{\mathit{\boldsymbol{\tilde \delta }}}^{\rm{T}}}\dot {\tilde \delta} } \right)} \right]。} \end{array} $

(22)

令e=[e^T(t) e^T(t-τ₁(t)) … e^T(t-τ_r(t)) $\mathit{\boldsymbol{\dot e}}$(t)]^T，因为${{\mathit{\boldsymbol{\dot { \boldsymbol{\tilde \varTheta}} }}}_1} = {{\mathit{\boldsymbol{ \boldsymbol{\dot \varTheta} }}}_1},{{\mathit{\boldsymbol{\dot { \boldsymbol{\tilde \varTheta}} }}}_2} = {{\mathit{\boldsymbol{ \boldsymbol{\dot \varTheta} }}}_2},\dot {\tilde \alpha} = \dot \alpha ,\dot {\tilde \delta} = \dot \delta $由参数自适应律(11)和不等式(16)，整理式(22)可得

$ \begin{array}{*{20}{c}} {\dot V \le \frac{1}{2}{{\tilde e}^{\rm T}}S\tilde e + \frac{1}{2}{\rho ^2}{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}\mathit{\boldsymbol{\bar w}}}\\ { \le - \frac{1}{2}{\lambda _{\min }}\left( { - S} \right){{\left\| {\tilde e} \right\|}^2} + \frac{1}{2}{\rho ^2}{{\left\| {\bar w} \right\|}^2}。} \end{array} $

(23)

当‖e‖ > $\frac{\rho }{{{\lambda _{\min }}\left( { - S} \right)}}$‖w‖时，$\dot V$ < 0。从而，闭环系统(8)是有界稳定的。

对式(23)在[0, T]积分，得

$ \begin{array}{*{20}{c}} {\int_0^{\rm{T}} {{{\mathit{\boldsymbol{\tilde e}}}^{\rm{T}}}\left( { - S} \right)\tilde e{\rm{d}}t} \le {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( 0 \right)Pe\left( 0 \right) + }\\ {\sum\limits_{i = 1}^r {\int_{ - {\tau _i}\left( 0 \right)}^0 {{\mathit{\boldsymbol{e}}^{\rm{T}}}\left( s \right){R_i}e\left( s \right){\rm{d}}s} } + }\\ {\sum\limits_{i = 1}^r {{\tau _{iM}}\int_{ - {\tau _{iM}}}^0 {\left( {s - {\tau _{iM}}} \right){{\mathit{\boldsymbol{\dot e}}}^{\rm{T}}}\left( s \right){\mathit{\boldsymbol{R}}_{iM}}\mathit{\boldsymbol{\dot e}}\left( s \right){\rm{d}}s} } + }\\ {\frac{1}{{{\eta _1}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_1^{\rm{T}}\left( 0 \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_1}\left( 0 \right)} \right) + \frac{1}{{{\eta _2}}}tr\left( {\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}_2^{\rm{T}}\left( 0 \right){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varTheta} }}}_2}\left( 0 \right)} \right) + }\\ {\frac{1}{{{\eta _3}}}tr\left( {{{\mathit{\boldsymbol{\tilde \alpha }}}^{\rm{T}}}\left( 0 \right)\tilde \alpha \left( 0 \right)} \right) + \frac{1}{{{\eta _4}}}tr\left( {{{\mathit{\boldsymbol{\tilde \delta }}}^{\rm{T}}}\left( 0 \right)\tilde \delta \left( 0 \right)} \right) + {\rho ^2}\int_0^{\rm{T}} {\left( {{{\mathit{\boldsymbol{\bar w}}}^{\rm{T}}}\mathit{\boldsymbol{\bar w}}} \right){\rm{d}}t} 。} \end{array} $

证毕。

5 仿真算例

设多输入多输出非线性时变时延系统为2连杆机械臂系统^[19]

$ \begin{array}{*{20}{c}} {\ddot q\left( t \right) + C\left( {q,\dot q} \right)\dot q\left( t \right) + g\left( q \right) = }\\ {B\left( q \right)\mathit{\Gamma }\left( t \right) + \sum\limits_{i = 1}^r {{\xi _i}\left( t \right)q\left( {t - {\tau _i}\left( t \right)} \right) + d'} ,} \end{array} $

其中C(q, $\dot q$)=H^-1(q)C′(q, $\dot q$), g(q)=H^-1(q)g′(q)，B(q)_=H^-1(q)，d′=H^-1(q)d, q=[q₁, q₂]^T, ξ_i(t)是不确定性，τ_i(t)(i=1, 2, …, r)表示时变时延, d是外部扰动。具体数据参见文[20]。

令x₁=q₁, x₂=$\dot q$₁, x₃=q₂, x₄=$\dot q$₂，u₁=Γ₁, u₂=Γ₂，y₁=x₁, y₂=x₃，则非线性系统可改写为

$ \begin{array}{*{20}{c}} {\dot x = \mathit{\boldsymbol{A}}x + \mathit{\boldsymbol{B}}\left[ {{\mathit{\boldsymbol{F}}_{\sigma \left( t \right)}}\left( x \right) + {\mathit{\boldsymbol{G}}_{\sigma \left( t \right)}}\left( x \right)u + d'} \right],}\\ {y = Cx,}\\ {x = \mathit{\Xi }\left( t \right),t \in \left[ { - \zeta ,0} \right],} \end{array} $

A=diag(A₁, A₂), B=diag(B₁, B₂), C=diag(C₁, C₂), A₁=A₂=$\left[ {\begin{array}{*{20}{l}} 0&1\\ 0&0 \end{array}} \right]$，B₁=B₂=$\left[ {\begin{array}{*{20}{l}} 0&1 \end{array}} \right]$^T，C₁=C₂=$\left[ {\begin{array}{*{20}{l}} 1&0 \end{array}} \right]$，r=2, τ₁(t)=1+0.2cos(0.9t), τ₂(t)=1.1|sin(0.9t)|，ξ₁(t)=1+5sint, ξ₂(t)=1+5cost，d是有界噪声，上界为0.5，Ξ(t)=[0.4 0 -0.4 0]。

设计自适应模糊控制器，以跟踪信号y_r1与y_r2。y_r1与y_r2满足$\ddot y$_r1=-5y_r1-4$\dot y$_r1+r₁(t)，$\ddot y$_r2=-5y_r2-4$\dot y$_r2+r₂(t)，其中r₁(t)和r₂(t)均是幅值为1，周期为2π的方波信号。可得仿真结果如图 3-图 6所示。

自适应模糊控制器的具体设计参数如下

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{u}} = {{\mathit{\boldsymbol{\hat G}}}_{\sigma \left( t \right)}}{{\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2},\alpha ,\delta } \right.} \right)}^{ - 1}}\left[ { - {{\mathit{\boldsymbol{\hat F}}}_{\sigma \left( t \right)}}\left( {x\left| {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1},\alpha ,\delta } \right.} \right) + } \right.}\\ {\left. {y_r^{\left( \beta \right)} + {\mathit{\boldsymbol{K}}^{\rm{T}}}e - {u_{com}}} \right],} \end{array} $

式中$\mathit{\boldsymbol{\hat F}}$_σ(t)(x)=Ψ(x, α, δ)Θ₁X，$\mathit{\boldsymbol{\hat G}}$_σ(t)(x)=Ψ(x, α, δ)Θ₂X，K^T=$\left[ {\begin{array}{*{20}{l}} 1&1&0&0\\ 0&0&2&2 \end{array}} \right]$，y_r^(β)=[$\ddot y$_r1, $\ddot y$_r2]^T，u_com=-$\left( {\frac{1}{\gamma }} \right)$B^TPe，γ=0.2 e=[e₁, $\dot e$₁, e₂, $\dot e$₂]^T是跟踪误差。

选择7条模糊规则, 隶属度函数选择如下

$ {R^{\left( j \right)}}:{\rm{if}}\;{x_1}\;{\rm{is}}\;F_1^j, \cdots ,{x_4}\;{\rm{is}}\;F_4^j,{\rm{then}}\;y\;{\rm{is}}\;{G^j}\left( {j = 1, \cdots ,7} \right)。$

$ {\mu _{F_i^1}}\left( {{x_i}} \right) = \frac{1}{{1 + \exp \left( {\frac{{{x_i} - {{\alpha '}_1}}}{{{{\delta '}_1}}}} \right)}}, $

$ {\mu _{F_i^2}}\left( {{x_i}} \right) = \exp \left[ { - {{\left( {\frac{{{x_i} - {{\alpha '}_2}}}{{{{\delta '}_2}}}} \right)}^2}} \right], $

$ {\mu _{F_i^3}}\left( {{x_i}} \right) = \exp \left[ { - {{\left( {\frac{{{x_i} - {{\alpha '}_3}}}{{{{\delta '}_3}}}} \right)}^2}} \right], $

$ {\mu _{F_i^4}}\left( {{x_i}} \right) = \exp \left[ { - {{\left( {\frac{{{x_i} - {{\alpha '}_4}}}{{{{\delta '}_4}}}} \right)}^2}} \right], $

$ {\mu _{F_i^5}}\left( {{x_i}} \right) = \exp \left[ { - {{\left( {\frac{{{x_i} - {{\alpha '}_5}}}{{{{\delta '}_5}}}} \right)}^2}} \right], $

$ {\mu _{F_i^6}}\left( {{x_i}} \right) = \exp \left[ { - {{\left( {\frac{{{x_i} - {{\alpha '}_6}}}{{{{\delta '}_6}}}} \right)}^2}} \right], $

$ {\mu _{F_i^7}}\left( {{x_i}} \right) = \frac{1}{{1 + \exp \left( {\frac{{{x_i} - {{\alpha '}_7}}}{{{{\delta '}_7}}}} \right)}}\left( {i = 1,2,3,4} \right)。$

令S₁=$\sum\limits_{j = 1}^7 {\prod\limits_{i = 1}^4 {{\mu _{F_i^j}}\left( {{x_i},\alpha _j',\delta _j'} \right)} } $, α′=(α′_j)_1×7, δ′=(δ′_j)_1×7，ξ^T(x, α′, δ′)=$\begin{array}{l} \left[ {\frac{{\prod\limits_{i = 1}^4 {{\mu _{F_i^j}}\left( {{x_i},\alpha _1'\delta _1'} \right)} }}{{\frac{{{S_1}, \cdots ,\prod\limits_{i = 1}^4 {{\mu _{F_i^j}}\left( {{x_i},\alpha _7'\delta _7'} \right)} }}{{{S_1}}}}}} \right] = \\ \;\;\;\;\;\;\;\;\;\;\left[ {{\xi _1}, \cdots ,{\xi _2}} \right] \end{array}$。

于是，关于隶属度函数的模糊基矩阵为

Ψ(x)=diag[ξ^T(x, α′, δ′), ξ^T(x, α′, δ′)]_2×14。

同理可得Ψ_α(x)=diag[ξ_α′^T(x, α′, δ′), …, ξ_α′^T(x, α′, δ′)]_14×14，Ψ_δ(x)=diag[ξ_δ′^T(x, α′, δ′), …, ξ_δ′^T(x, α′, δ′)]_14×14，ξ_α′^T(x, α′, δ′)和ξ_δ′^T(x, α′, δ′)分别表示ξ^T关于α′和δ′的偏导数, 自适应时变时延模糊逻辑系统中的参数Θ₁、Θ₂、α和δ的自适应律具有(11-13)的形式，其中η₁=1.5，η₂=0.8，η₃=0.02，η₄=0.01是可选择的正常数，Ψ(x)，Ψ_α(x)，Ψ_δ(x)是关于隶属度函数的矩阵, B=diag((0;1), (0;1)), X=diag[X, X]，X=(1 x₁ x₁(t-τ₁(t)) x₁(t-τ₂(t)) … x₄ x₄(t-τ₁(t)) x₄(t-τ₂(t)))^T∈R¹³，e=[e₁, $\dot e$₁, e₂, $\dot e$₂]^T，$\mathit{\boldsymbol{\dot e}}$=[$\dot e$₁，$\ddot e$₁，$\dot e$₂，$\ddot e$₂]^T。ρ=0.7, γ=0.2, μ=0.000 6，τ_1D=1.2，τ_2D=1.1，τ_1M=0.18，τ_2M=0.99，正定矩阵P由(16)求出

$ \mathit{\boldsymbol{P}} = \left[ {\begin{array}{*{20}{c}} {0.0209}&{0.0083}&0&0\\ {0.0083}&{0.0278}&0&0\\ 0&0&{0.0378}&{0.0234}\\ 0&0&{0.0234}&{0.0436} \end{array}} \right]。$

当r₁(t)和r₂(t)均是幅值为1，周期为π的方波信号。可得仿真结果如图 7-图 8所示。

仿真结果表明设计的自适应模糊控制器能快速的跟踪给定的参考信号。

6 结论

构建了基于模糊T-S模型的自适应时变时延模糊逻辑系统，通过在线自适应调整模糊系统的参数来逼近未知非线性时变时延函数，从而实现了对多输入多输出非线性时变时延系统的建模，以此为基础设计了模糊控制器，提出了一种自适应模糊跟踪控制方案。该方案从理论分析到仿真结果都表明了该方案的有效性。