Homogeneous Sliding Mode Control (HSMC)

$\textbf{Model of the control system:}$

$\dot x\!=\!{\color{blue}A}x+{\color{blue}B}(u+\gamma(t,x)), \;\;\; y\!=\!Cx, \;\;\; x\!\in\! \mathbb{ R } ^n, \;\;\; u\!\in\! \mathbb{ R } ^m, \;\;\; {\color{blue}A}\!\in\! \mathbb{ R } ^{n\times m}, \;\;\; {\color{blue}B}\!\in\! \mathbb{ R } ^{n\times m}, \;\;\; {\color{blue}C}\!\in\! \mathbb{ R } ^{p \times n} $ where $y\in \mathbb{ R } ^p$ is a controllable output and $\gamma:\mathbb{ R } \times \mathbb{ R } ^n \mapsto \mathbb{ R } ^m$ is an unknown uniformly bounded function.

$\textbf{Control law:}$

$u_{hsmc}={\color{magenta}{K_0}}x+ {\color{magenta}K}\mathbf{ d } (-\ln \|{\color{blue}C}x\|_{\mathbf{ d } }){\color{blue}C}x, \quad {\color{magenta}{K_0}}\in \mathbb{ R } ^{m\times n}, \quad {\color{magenta}K}\in \mathbb{ R } ^{m\times p}, \quad $

where $\mathbf{ d } (s)=e^{s{\color{magenta}{G_{\mathbf{ d } }}}}$ is a dilation in $\mathbb{ R } ^p$, ${\color{magenta}{G_{\mathbf{ d } }}}\in \mathbb{ R } ^{p\times p}$ and the homogeneous norm $\|x\|_{\mathbf{ d } }$ is induced by the weighted Euclidean norm $\|x\|=\sqrt{x^{\top} {\color{magenta}P} x}$ in $\mathbb{ R } ^p$, ${\color{magenta}P}\in \mathbb{ R } ^{p\times p}$.

$\textbf{Properties:}$

enforces sliding mode on the surface $Cx=0$ in a finite time

$Cx(t)=0, \quad \forall t\geq \|Cx(0)\|_{\mathbf{ d } }/ {\color{blue} \rho}; $
the output dynamics $\dot y=C\dot x$ is $\mathbf{ d } $-homogeneous of degree $-1$.
ejection of the matched perturbation $\gamma$ if

$ |\gamma(t,x)|\leq {\color{blue}\gamma_{\max}}.$

HSMC Design

The function ${\color{red} { \texttt{hsmc_design }} } $ computes parameters $ {\color{magenta}{K_0,K,G_d}}$ and ${\color{magenta} P}$ of HPC for given $ {\color{blue}A}, {\color{blue}B}, {\color{blue}C}, {\color{blue}\rho}>0$ and ${\color{blue}\gamma_{\max}}\geq 0$

$\textbf{Input parameters}: {\color{blue}A}, {\color{blue}B}, {\color{blue}C}, {\color{blue}\rho}$ (by default $\rho=1$) and ${\color{blue}{\gamma_{\max}}}$ (by default $\gamma_{\max}=0$)

$\textbf{Output parameters}: {\color{magenta}{K_0,K,G_d}} $ and $ {\color{magenta} P} $

HSMC Implementation

The function ${\color{red} { \texttt{e_smc }} } $ computes $\textit{explicit} $ discretization of $u_{hsmc}$

$\textbf{Input parameters}: x$, ${\color{blue}C}, {\color{magenta}{K_0, K}}, {\color{magenta}{G_{\mathbf{ d } }, P}} $

$\textbf{Output parameters}: u_{hsmc}$

The function ${\color{red} { \texttt{si_hpc }} } $ computes $ \textit{semi-implicit}$ discretization of $u_{hsmc}$

$ \textbf{Input parameters}: h\, $ (sampling period), $x, {\color{blue}{A,B}}, {\color{blue}C}, {\color{magenta}{K_0,K}}, {\color{magenta}{G_{\mathbf{ d } },P}} $

$ \textbf{Output parameters}: u_{hsmc}$

Use ${\color{red} { \texttt{demo_hsmc.m}} } $ from $ \texttt{HCS Toolbox} $ as a demo of HSMC design

${\color{red} { \texttt{demo_hsmc.m}} } $

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Example of the disign of a Homogenenous Sliding Mode Control (HSMC) for % % System: dx/dt=A*x+B*(u+gamma(t)) % % where % x - system state vector (n x 1) % u - control input (m x 1) % A - system matrix (n x n) % B - control matrix (m x m) % gamma(t) - unknown bounded perturbation % % Control aim: % % Given matrix C (k x n) the goal is to design a control law such that % independently of a bounded function gamma (with |gamma(t)|=T, where T is a finite time % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A=[0 3 0 0 0; -3 0 0 0 0; 0 0 1 -1 1; 0 1 0 2 1; 1 0 1 -1 3]; % system matrix B=[0 0; 0 0; 0 0; 1 0; 0 1]; % control matrix C=[0 0 1 0 0; 0 0 0 1 0; 0 0 0 0 1]; % sliding surface Cx=0 n=5;m=2; rho=1; %convergence rate tuning parameter (learger rho faster the convergence) gamma_max=0.5; %magnitude of perturbation [K0 K Gd P]=hsmc_design(A,B,C,rho,gamma_max); %design the parameteres of HSMC %K0 - homogenization gain %K - control gain %Gd - generator of dilation %P - shape matrix of the weighted Euclidean norm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t=0; Tmax=4; h=0.001; % sampling period x=[1;0;2;0;1]; tl=[t];xl=[x];ul=[]; disp(['Settling Time=',num2str(hnorm(C*x,Gd,P)/rho)]); alpha=0; % no chattering attentuation %alpha=0.005; % in the noise-free case % alpha=0.1; % for the noise magnitude 0.001; noise=0; % magnitude of measurement noises disp('Run numerical simulation...'); [Ah, Bh]=ZOH(h,A,B); while t<Tmax xm=x+2*noise*(rand(5,1)-0.5); %(possibly) noised state %u=(K0+K*C)*xm; % linear control (for comparison) u=e_hsmc(xm,C,K0,K,Gd,P,alpha); % explicit discretization of HSMC %u=si_hsmc(h,xm,A,B,C,K0,K,Gd,P,alpha); %semi-implicit discret. of HSMC gamma_t=gamma_max*[sin(2*t);cos(10*t)]; % matched perturbation x=Ah*x+Bh*(u+gamma_t); %simulation of the system t=t+h; tl=[tl t]; xl=[xl x]; ul=[ul u]; end; ul=[ul u]; disp('Done!'); %%%%%%%%%%%%%%%%%%%%%%%%% % Plot simulation results %%%%%%%%%%%%%%%%%%%%%%%%% figure; axes1 = subplot(1,2,1); hold(axes1,'on'); plot1 = plot(tl,xl,'LineWidth',2,'Parent',axes1); set(plot1(1),'DisplayName','$x_1$'); set(plot1(2),'DisplayName','$x_2$'); set(plot1(3),'DisplayName','$x_3$'); set(plot1(4),'DisplayName','$x_4$'); set(plot1(5),'DisplayName','$x_5$'); ylabel('$x$','Interpreter','latex'); xlabel('$t$','Interpreter','latex'); title({'n=5'}); xlim(axes1,[0 Tmax]); ylim(axes1,[-3 3]); box(axes1,'on'); hold(axes1,'off'); set(axes1,'FontSize',30,'XGrid','on','YGrid','on'); legend1 = legend(axes1,'show'); set(legend1,'Interpreter','latex'); axes2 = subplot(1,2,2); hold(axes2,'on'); plot2 = plot(tl,ul,'LineWidth',2); set(plot2(1),'DisplayName','$u_1$'); set(plot2(2),'DisplayName','$u_2$'); ylabel('$u$','Interpreter','latex'); xlabel('$t$','Interpreter','latex'); title({'HSMC,m=2'}); xlim(axes2,[0 Tmax]); ylim(axes2,[-12 12]); box(axes2,'on'); hold(axes2,'off'); set(axes2,'FontSize',30,'XGrid','on','YGrid','on'); legend2 = legend(axes2,'show'); set(legend2, 'Interpreter','latex');