Sliding Mode Control and Observation 学习 Chapter 1: Terminal Sliding Mode 终端滑模

The advantage of terminal sliding mode (TSM) control lies in that it solves the infinite time convergence problem of linear sliding mode (LSM) control. Based on the power sliding hyper plane, the finite time convergence can be achieved, and the convergence performance and robustness of the system are improved…
Considering a simple second-order system：
$$\begin{gathered} {\dot{x}}_1=x_2 \\ {\dot{x}}_2=f+u \end{gathered}$$
and assuming $f=1$.

1. LSM
   Designing the LSM surface and controller as
   $$\begin{gathered} s_1=x_2+c{x}_1 \\ {u}_1=-\rho sign(s_1)-c{x}_2 \end{gathered}$$
   Constructing a Lyapunov function as: $V_1=\frac{1}{2}{s}^2$, we have
   $$\begin{gathered} {\dot{V}}_1=s_1{\dot{s}}_1 \\ =s_1(-\rho sign(s_1)+f) \\ \le -(\rho -f)|s_1| \\ =-\eta \sqrt{2}\sqrt{V_1(0)} \end{gathered}$$
   Solving the above equation, we have
   
   Thus, the system states $x_1,x_2$ will reach the sliding surface in finite time $t_r$.
   When $s_1=0$, we have
   
   That is the infinite time convergence propety of LSM.

Choosing $c=1,\rho =2$, the system convergence process can be obtained in the following:



It can be calculated that $t_r=2$ s, as shown in Figure 1. And after that, the system enters the sliding mode, which is shown in Figure 2-3;

2. TSM
   Design the TSM surface and controller as
   $$\begin{gathered} s_2=x_2+c{x}_1^{p/q} \\ {u}_2=-\rho sign(s_2)-c\frac{p}{q}{x}_1^{p/q-1}{x}_2 \end{gathered}$$
   Constructing a Lyapunov function as: $V_2=\frac{1}{2}{s}_2^2$, we have the same reaching time as $V_1$, thus, here is omitted;
   When $s_2=0$, we have
   
   where $x_1(0)$ is the value of $x_1$ when $s=0$.
   The finite time convergence propety of TSM is obtained.
   Choosing $c=1,\rho =2,p/q=5/7$, the system convergence process can be obtained in the following figures:
   
   
   
   It it obviously shown in Figure 1 that $t_r=2$ which is equal to that in LSM, due to the same reaching law.
   However, when $s=0$ and $x_1=0.7591$, the reaching phase ends and sliding phase begins. The finite time of sliding phase can be calculated as
   $$t_s=\frac{x_1(0{)}^{1-p/q}}{c(1-p/q)}=\frac{0.7591^{1-5/7}}{1-5/7}=3.23s$$
   which is shown in Figure 2.
   Here the proof is completed.
   Reference:
   Man, Z., Paplinski, A.P., & Wu, H.R. (1994). A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators. IEEE Trans. Autom. Control., 39, 2464-2469.

clc, clear, close all

ts = 0; h  = 0.001; tf = 8;

c = 1;
rho = 2;
p = 5/7;

x1 = 1; x2 = 1;

x1out = []; x2out = []; sout = []; uout = [];
x1aout = []; x2aout = []; saout = []; uaout = [];

sig = @(x,p) abs(x)^p*sign(x);
for t = ts:h:tf
    % LSM
%     s = x2 + c * x1;
%     u = -c*x2 - rho * sign(s);
    
    %TSM
    s = x2 + c*sig(x1,p);
    u = -c*p*sig(x1,p-1)*x2 - rho * sign(s);
    
    f = 1;
    x2dot = f + u;
    x2 = x2 + x2dot * h;
    x1dot = x2;
    x1 = x1 + x1dot * h;
    
    x1out = [x1out;x1];
    x2out = [x2out;x2];
    uout = [uout;u];
    sout = [sout;s];
    
end

t = ts:h:tf;

figure
plot(t,sout);
legend('s')
grid on;

figure
plot(t,x1out,t,x2out);
legend('x1','x2')
grid on;

figure
plot(x1out, x2out);
grid on;
xlabel('x1')
ylabel('x2')

figure
plot(t,uout);
grid on;
legend('u');