Adaptive Control - Simulation Experiment 1 Designing Adaptive Laws Using Lyapunov's Stability Theory

一、问题描述

Let the state equation of the control object be
$${\dot{\mathbf{x}}}_p={\mathbf{A}}_p(t)x_p+{\mathbf{b}}_p(t)u\text{\hspace{1em}(1)}$$
式中
$${\mathbf{A}}_p=\left[\begin{array}{cc}0& 1\\ -6& -7\end{array}\right],{\mathbf{b}}_p=\left[\begin{array}{l}2\\ 4\end{array}\right]\text{\hspace{1em}(2)}$$
The equation of state of the reference model is
$${\dot{\mathbf{x}}}_m={\mathbf{A}}_m{x}_m+{\mathbf{b}}_{m}r\text{\hspace{1em}(3)}$$
式中
$${\mathbf{A}}_m=\left[\begin{array}{cc}0& 1\\ -10& -5\end{array}\right],{\mathbf{b}}_m=\left[\begin{array}{l}1\\ 2\end{array}\right]\text{\hspace{1em}(4)}$$
The adaptive law is designed using Lyapunov stability theory.

二、问题建模

Because of the parameters of the control object（状态矩阵 ${\mathbf{A}}_p$ 和控制矩阵 ${\mathbf{b}}_p$ ）一般是未知的,and cannot be directly adjusted.So in order to change the dynamic characteristics of the control object,Feedforward control plus feedback control is required.

控制信号 $u$ by the feedforward signal $Kr$ 和反馈信号 $F{x}_p$ 组成,即
$$u=Kr+F{\mathbf{x}}_p\text{\hspace{1em}(5)}$$
式中,$r$ 为 $m$ 维输入向量,${\mathbf{x}}_p$ 为 $n$ 维状态向量,$K$ 为 $m\times m$ Feedforward gain matrix,$F$ 为 $m\times n$ 反馈增益矩阵;Specifically in this simulation experiment,输入向量维度 $m=1$,State vector dimension $n=2$.

将(5)Substitute the formula into the state equation of the control object,可得
$${\dot{\mathbf{x}}}_p=\left[{\mathbf{A}}_p(t)+{\mathbf{b}}_p(t)F\right]{\mathbf{x}}_p+{\mathbf{b}}_p(t)Kr\text{\hspace{1em}(6)}$$
Let the generalized state error vector of the system be
$$\mathbf{e}={\mathbf{x}}_m-{\mathbf{x}}_p\text{\hspace{1em}(7)}$$
by the equation of state of the reference model,结合(6)式及(7)式,可得：
$$\dot{\mathbf{e}}={\mathbf{A}}_m\mathbf{e}+\left({\mathbf{A}}_m-{\mathbf{A}}_p-{\mathbf{b}}_{p}F\right){\mathbf{x}}_p+\left({\mathbf{b}}_m-{\mathbf{b}}_{p}K\right)r\text{\hspace{1em}(8)}$$
在理想情况,即 $e\to 0$ 的情况下,(8)The last two terms of the right-hand side of the equation should be equal0.Set up the feedforward gain matrix $K$ and the feedback gain matrix $F$ 的理想值分别为 $\bar{K}$ 和 $\bar{F}$.

can finally be(8)written as
$$\dot{\mathbf{e}}={\mathbf{A}}_m\mathbf{e}+{\mathbf{b}}_m{\bar{K}}^{-1}\Phi {\mathbf{x}}_p+{\mathbf{b}}_m{\bar{K}}^{-1}\Psi r\text{\hspace{1em}(9)}$$
式中,$\Phi =\bar{F}-F$ 为 $m\times n$ 矩阵,$\Psi =\bar{K}-K$ 为 $m\times m$ 矩阵.

The Lyapunov function is chosen as ：
$$V=\frac{1}{2}\left[{\mathbf{e}}^T\mathbf{P}\mathbf{e}+\mathrm{tr}\left({\Phi }^T{\Gamma }_1^{-1}\Phi +{\Psi }^T{\Gamma }_2^{-1}\Psi \right)\right]\text{\hspace{1em}(10)}$$
式中,$\mathbf{P}$ 为 $n\times n$ A dimensionally positive definite symmetric matrix,${\Gamma }_1$ 和 ${\Gamma }_2$ 均为 $m\times m$ A dimensionally positive definite symmetric matrix;符号 $\mathrm{tr}$ 表示矩阵的迹.

求(10)The derivative of the formula with respect to time,得
$$\dot{V}=\frac{1}{2}\left[\dot{\mathbf{e}}\mathbf{P}\mathbf{e}+{\mathbf{e}}^T\mathbf{P}\dot{\mathbf{e}}+\mathrm{tr}\left({\dot{\Phi }}^T{\Gamma }_1^{-1}\Phi +{\Phi }^T{\Gamma }_1^{-1}\dot{\Phi }+{\dot{\Psi }}^T{\Gamma }_2^{-1}\Psi +{\Psi }^T{\Gamma }_2^{-1}\dot{\Psi }\right)\right]\text{\hspace{1em}(11)}$$
将(9)式代入(11)式,Then according to the properties of the matrix trace,于是有
$$\begin{array}{rl}\dot{V}=& \frac{1}{2}{\mathbf{e}}^T\left(\mathbf{P}{\mathbf{A}}_m+{\mathbf{A}}_m^{\mathbf{T}}\mathbf{P}\right)\mathbf{e}+\mathrm{tr}\left({\dot{\Phi }}^T{\Gamma }_1^{-1}\Phi +{\mathbf{x}}_p{\mathbf{e}}^T\mathbf{P}{\mathbf{b}}_m{\bar{K}}^{-1}\Phi \right)\\ & +\mathrm{tr}\left({\dot{\Psi }}^T{\Gamma }_2^{-1}\Psi +r{\mathbf{e}}^T\mathbf{P}{\mathbf{b}}_m{\bar{K}}^{-1}\Psi \right)\end{array}\text{\hspace{1em}(12)}$$
To satisfy Lyapunov's second method,需保证(12)The formula is negative definite,对应的情况为(12)The first term of the formula is negative definite,The latter two are both zero.

因为 ${\mathbf{A}}_m$ is a stable matrix,A positive definite symmetric matrix can be selected $Q$,使 $\mathbf{P}{\mathbf{A}}_m+{\mathbf{A}}_m^{\mathbf{T}}\mathbf{P}=-\mathbf{Q}$ 成立.At the same time according to the above corresponding situation,$\Phi$ 和 $\Psi$ 的选择如下：
$$\begin{array}{rl}\dot{\Phi }& =-{\Gamma }_1{\left({\mathbf{b}}_m{\bar{K}}^{-1}\right)}^T\mathbf{P}\mathbf{e}{\mathbf{x}}_p^T\\ \dot{\Psi }& =-{\Gamma }_2{\left({\mathbf{b}}_m{\bar{K}}^{-1}\right)}^T\mathbf{P}\mathbf{e}{r}^T\end{array}\text{\hspace{1em}(13)}$$
当 ${\mathbf{A}}_p$ 和 ${\mathbf{b}}_p$ constant or slowly changing,Adaptive regulation can be obtained：
$$\begin{array}{rl}F(t)& ={\int }_0^t{\Gamma }_1{\left({\mathbf{b}}_m{\bar{K}}^{-1}\right)}^T\mathbf{Pe}{\mathbf{x}}_p^{T}d\tau +F(0)\\ K(t)& ={\int }_0^t{\Gamma }_2{\left({\mathbf{b}}_m{\bar{K}}^{-1}\right)}^T\mathbf{Pe}rd\tau +K(0)\end{array}\text{\hspace{1em}(14)}$$
An additional point to note is that,According to the above steps, the requirements of the adaptive adjustment law are obtained ${\mathbf{x}}_p$ 与 $r$ 线性独立.The two conditions are independent $r(t)$ is a square wave signal with a certain frequency or is $q$ A segmented continuous signal composed of sinusoidal signals of different frequencies,其中 $q>n/2$ 或 $q>(n-1)/2$.

三、问题求解

It can be seen from the above derivation,It is designed for adopting Lyapunov stability theoryMRACS,A feedforward gain matrix needs to be introduced $K$ and the feedback gain matrix $F$,The goal of the design is to determine $K$ 和 $F$ 的系数.

After introducing two gain matrices for adaptive control,The state equation of the tunable system becomes ：
$${\dot{\mathbf{x}}}_p=\left[{\mathbf{A}}_p(t)+{\mathbf{b}}_p(t)F\right]{\mathbf{x}}_p+{\mathbf{b}}_p(t)Kr\text{\hspace{1em}(15)}$$
It can be seen from the previous derivation,(14)式中的 ${\mathbf{b}}_m{\bar{K}}^{-1}$ 与 ${\mathbf{b}}_p$ 的关系如下：
$${\mathbf{b}}_m{\bar{K}}^{-1}={\mathbf{b}}_p=\left[\begin{array}{l}2\\ 4\end{array}\right]\text{\hspace{1em}(16)}$$
选取(14)Some of the adaptive parameters in the formula are as follows：
$$\mathbf{P}=\left[\begin{array}{ll}3& 1\\ 1& 1\end{array}\right],{\Gamma }_1={\Gamma }_2=1\text{\hspace{1em}(17)}$$
So the final adaptive law can be obtained：
$$\begin{array}{rl}F(t)& ={\int }_0^t\left[\begin{array}{ll}2& 4\end{array}\right]\left[\begin{array}{ll}3& 1\\ 1& 1\end{array}\right]\mathbf{e}{\mathbf{x}}_p^{T}d\tau +F(0)\\ K(t)& ={\int }_0^t\left[\begin{array}{ll}2& 4\end{array}\right]\left[\begin{array}{ll}3& 1\\ 1& 1\end{array}\right]\mathbf{erd}\tau +K(0)\end{array}\text{\hspace{1em}(18)}$$
Next, the above continuous adaptive law is discretized,For actual numerical simulation experiments.Let the numerical integration step size be $h$,The reference model state vector and the control object state vector at each moment are as follows：
$$\begin{array}{rl}{\mathbf{x}}_m(k+1)& ={\mathbf{x}}_m(k)+h\left[{\mathbf{A}}_m(k){\mathbf{x}}_m(k)+{\mathbf{B}}_m(k)r(k)\right]\\ {\mathbf{x}}_p(k+1)& ={\mathbf{x}}_p(k)+h\left[{\mathbf{A}}_p(k){\mathbf{x}}_p(k)+{\mathbf{B}}_p(k)u(k)\right]\end{array}\text{\hspace{1em}(19)}$$
Due to the requirements of the adaptive control law derived from the above ${\mathbf{x}}_p$ 与 $r$ 线性独立,即要求 $r(t)$ is a square wave signal with a certain frequency or is $q$ A segmented continuous signal composed of sinusoidal signals of different frequencies,其中$q>n/2$ 或 $q>(n-1)/2$.在本次实验中,$n=2$,Correspond to the requirements $q>1$,Therefore, in this experiment, we chose 3A segmented continuous signal composed of sinusoidal signals of different frequencies,The specific form of the input signal is as follows：
$$r(k)=\sin (0.01\pi k)+4\sin (0.2\pi k)+\sin (\pi k)\text{\hspace{1em}(20)}$$
The control signal introduced when we design the adaptive law $u$ The discretized form of is as follows：
$$u(k)=K(k)r(k)+F(k){\mathbf{x}}_p(k)\text{\hspace{1em}(21)}$$
最终,The adaptive law also needs to be discretized：
$$\begin{array}{rl}F(k)& =h\cdot \sum _{j=0}^k{\mathbf{b}}_p^T\mathbf{P}\mathbf{e}(k){\left({\mathbf{x}}_p(k)\right)}^T+F(0)\\ K(k)& =h\cdot \sum _{j=0}^k{\mathbf{b}}_p^T\mathbf{P}\mathbf{e}(k)r(k)+K(0)\end{array}\text{\hspace{1em}(22)}$$
After deriving all the adaptive laws and discretizing the corresponding laws,通过MATLABThe related simulation experiments were carried out.

可以得到2The case of the reference model value and the adjustable system value of the state vector of 1 dimension is as follows：

图1. Reference model value and tunable system value of the state vector

可以看到,Tunable systems do not track reference models very well,This is because there is no optimal match in this example.

附录：实现MATLAB代码

% Textbook exercises3.4-The adaptive law is designed using Lyapunov stability theory
clear, clc;
close all;

h=0.01;L=100/h;     % Numerical integration step size and simulation steps
% The coefficient matrix for the tunable system
Ap = [0 1;-6 -7];
Bp = [2; 4];
% The coefficient matrix of the reference model
Am = [0 1;-10 -5];
Bm = [1; 2];
% nis the row vector dimension、mis the dimension of the column vector,Bp是n*m的矩阵
n = size(Bp, 1);
m = size(Bp, 2);

P = [3 1;1 1];              % Calculated positive definite symmetric matrix for adaptive law

% Set the initial values ​​of all parameters
yr0 = zeros(m, 1);
xp0 = zeros(n, 1);
xm0 = zeros(n, 1);
u0 = zeros(m, 1);
e0 = zeros(n, 1);
F0 = zeros(m, n);           % Feedback gain matrix initial value
K0 = zeros(m, m);           % Feedforward gain matrix initial value

% Initially allocate parameter space
time = zeros(1, L);         % Used to record the moment of the simulation,Corresponds to the horizontal axis of the plot
yr = zeros(m, L);           % 输入信号(L个m维向量)
xp = zeros(n, L);           % The state vector of a tunable system(L个n维向量)
xm = zeros(n, L);           % The state vector of the reference model(L个n维向量)
u = zeros(m, L);            % 控制信号(L个m维向量)
e = zeros(n, L);            % Generalized state error vector for the system(L个n维向量)

for k = 1:L
    time(k) = k*h;
    % 输入信号
    yr(k) = 1*sin(0.01*pi*time(k))+4*sin(0.2*pi*time(k))+sin(1*pi*time(k));
    xp(:,k) = xp0+h*(Ap*xp0+Bp*u0);     % 计算xp
    xm(:,k) = xm0+h*(Am*xm0+Bm*yr0);    % 计算xm
    e(:,k) = xm(:,k)-xp(:,k);           % e=xm-xp
    
    % 代入F和Kadaptive control law
    F = F0+h*(Bp'*P*e0*xp0');
    K = K0+h*(Bp'*P*e0*yr0);

    % 控制信号u=K*r+F*xp（Kis the feedforward gain matrix,Fis the feedback gain matrix）
    u(:,k) = K*yr(k)+F*xp(:,k);
    
    % Assign the parameters obtained in this round to the initial values ​​of the parameters,It is convenient to use in the next iteration
    yr0 = yr(:,k);
    u0 = u(:,k);
    e0 = e(:,k);
    xp0 = xp(:,k);
    xm0 = xm(:,k);
    F0 = F;
    K0 = K;
end

subplot(2,1,1);
plot(time, xm(1,:), 'Color', 'b', 'LineWidth', 0.9);
hold on
plot(time, xp(1,:), 'Color', 'r', 'LineStyle', '--', 'LineWidth', 1.1);
xlabel('t');
ylabel('x_m_1(t)、x_p_1(t)');
legend('x_m_1(t)','x_p_1(t)');
hold off
subplot(2,1,2);
plot(time, xm(2,:), 'Color', 'b', 'LineWidth', 0.9)
hold on
plot(time, xp(2,:), 'Color', 'r', 'LineStyle', '--', 'LineWidth', 1.1)
xlabel('t');
ylabel('x_m_2(t)、x_p_2(t)');
legend('x_m_2(t)', 'x_p_2(t)');
hold off

参考书目

李言俊, 张科. Adaptive Control Theory and Applications[M]. 西北工业大学出版社, 2005.