HSMC with integral action
$\textbf{Model of the control system:}$
$\dot x\!=\!{\color{blue}A}x+{\color{blue}B}(u+\gamma+ p), \;\;\;y\!=\!Cx, \;\;\; x\!\in\! \mathbb{ R } ^n, \;\;\; u\!\in\! \mathbb{ R } ^m, \;\;\; {\color{blue}A}\!\in\! \mathbb{ R } ^{n\times m}, \;\;\; {\color{blue}B}\!\in\! \mathbb{ R } ^{n\times m}, \;\;\; {\color{blue}C}\!\in\! \mathbb{ R } ^{p\times n}$
where $y\in \mathbb{ R } ^p$ is a controlled output, $\gamma:\mathbb{ R } \times\mathbb{ R } ^{n}\to \mathbb{ R } ^m$ is an unknown (vanishing at $x=\mathbf{ 0 } $) function and $p\in \mathbb{ R } ^n$ is an unknown constant.
$\textbf{Control law:}$
$u_{hsmci}=u_{0}+\int^{t}_{0}u_{\mathrm{sint}} d\tau $
$u_{0}={\color{magenta}{K_0}}x+ \|{\color{blue}C}x\|_{\mathbf{ d } }^{0.5}{\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}, $
$u_{\mathrm{sint}}(Cx(\tau))={\color{magenta}{K_i}}\mathbf{ d } (-\ln \|{\color{blue}C}x\|_{\mathbf{ d } }){\color{blue}C}x,$
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
$\exists T=T(x(0)) : Cx(t)=0, \quad \forall t\geq T $
- rejection of the unknown constant perturbation $p$ and the (vanishing at $x=\mathbf{ 0 } $) matched perturbation $\gamma$ if
$|\gamma(t,x)|\leq {\color{blue}{\gamma_{\max}}}\|x\|_{\mathbf{ d } }^{1/2}.$
Design of HSMC with Integral action
The function ${\color{red} { \texttt{hsmc_design }} } $ computes parameters $ {\color{magenta}{K_0,K,K_i,G_d}}$ and ${\color{magenta} P}$ of HPC for given ${\color{blue}A}, {\color{blue}B}, {\color{blue}C},$ 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} $
Implementation of HSMC with integral action
- The function ${\color{red} { \texttt{e_hsmci }} } $ computes $\textit{explicit} $ discretization of $u_{hsmci}$
- $\textbf{Input parameters}: x, {\color{blue}C}, {\color{magenta} {K_0, K, K_i}}, {\color{magenta}{G_\mathbf{d}, P}} $
- $\textbf{Output parameters}: u_{0}$ and $u_{\mathrm{sint}}$
${\color{red} { \texttt{demo_fsmci.m}} } $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Example of disign of Homogenenous 2-Order Sliding Mode Control (H2SMC) % % System: dx/dt=A*x+B*(u+gamma(t,x)+p) % % 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,x) - unknown perturbation (vaninshing at x=0) % p - unknown constant (with unknown bound) % % Control aim: % % Given matrix C (k x n) and gamma_max the goal is to design SMC such that % % Cx(t)=0 for all t>=T, where T is a finite time % % for any p=const and |gamma(t,x)|<=gamma_max*sqrt(ncx) % where ncx - homogeous norm of C*x % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% %% Model of system %%%%%%%%%%%%%%%%%%%%%%% 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; %%%%%%%%%%%%%%%%%% %% H2SMC design %%%%%%%%%%%%%%%%%% rho=1; %convergence rate tuning parameter (learger rho faster the convergence) gamma_max=0.5; %magnitude of perturbation [K0 K Ki Gd P]=hsmci_design(A,B,C,rho,gamma_max); %design the parameteres of HSMC %K0 - homogenization gain %K - proportional gain %Ki - integral gain %Gd - generator of dilation %P - shape matrix of the weighted Euclidean norm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t=0; Tmax=5; h=0.001; % sampling period x=[1;0;2;0;1]; tl=[t];xl=[x];ul=[]; alpha=0; % no chattering attentuation %alpha=0.005; % in the noise-free case % alpha=0.1; % for the noise magnitude 0.001; p=1; %contstant perturbation with an unknown bound noise=0; % magnitude of measurement noises v=0; % for computation of integral term disp('Run numerical simulation...'); while t<Tmax xm=x+2*noise*(rand(5,1)-0.5); %(possibly) noised state measurement [u ui]=e_hsmci(xm,C,K0,K,Ki,Gd,P,alpha); % H2SMC %u=(K0+K*C)*xm; ui=Ki*C*xm; %linear PI control (for comparison) u=u+v; v=v+h*ui; %u=e_hsmc(xm,C,K0,K,Gd,P,alpha); %HSMC (for comparison reasons) gamma_t=abs(x(5))*gamma_max*[sin(2*t);cos(10*t)]; % matched perturbation x=x+h*A*x+h*B*(u+gamma_t+p); %esimulation 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,[-5 5]); 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({'H2SMC,m=2'}); xlim(axes2,[0 Tmax]); ylim(axes2,[-15 15]); box(axes2,'on'); hold(axes2,'off'); set(axes2,'FontSize',30,'XGrid','on','YGrid','on'); legend2 = legend(axes2,'show'); set(legend2, 'Interpreter','latex');