Fixed-time HPC

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

$\dot x={\color{blue}A}x+{\color{blue}B}u, \quad x\in \mathbb{ R } ^n, \quad u\in \mathbb{ R } ^m, \quad {\color{blue}A}\in \mathbb{ R } ^{n\times m}, \quad {\color{blue}B}\in \mathbb{ R } ^{n\times m} $ where the pair $\{A,B\}$ is controllable.

$\textbf{Control law:}$

\begin{equation} u_{fhpc}={\color{magenta}{K_0}}x+ \left\{ \begin{array}{ccc} \|x\|_{\mathbf{ d} _1}^{1+{\color{magenta}{\mu_1}}} {\color{magenta}K}\mathbf{ d}_1(-\ln \|x\|_{\mathbf{ d}_1})x & \text{ if } & x^{\top } {\color{magenta}P} x \leq 1\\ \|x\|_{\mathbf{ d}_2}^{1+{\color{magenta}{\mu_2}}} {\color{magenta}K}\mathbf{ d}_2(-\ln \|x\|_{\mathbf{ d}_2})x & \text{ if } & x^{\top } {\color{magenta}P} x >1 \end{array} \right. \end{equation} where ${\color{magenta}{\mu_10}}\!<\!0\!<\!{\color{magenta}{\mu_2}}$, $\mathbf{ d}_k(s)\!=\!e^{s(I_n+{\color{magenta}{\mu_k G_0}})}$ and $\|x\|_{\mathbf{ d}_k}$ is induced by $\|x\|\!=\!\sqrt{x^{\top}\!{\color{magenta}P} x}$, $k\!=\!1,2$

$\textbf{Properties:}$

fixed-time stabilization of linear plant: $ x(t)=\mathbf{0}, \quad \forall t\geq T_{\max}, \quad \forall x(0)\in \mathbb{ R } ^n, \quad T_{\max}=\frac{1}{-{\color{magenta}{\mu_1\rho}}}+\frac{1}{{\color{magenta}{\mu_2\rho}}}. $
local homogeneity of closed-loop system (of degree $\mu_1$ at $0$ and of degree $\mu_2$ at $\infty$) $\Rightarrow$ ISS with respect to measurement noises in $x$ and additive perturbations in the model;

Fixed-time HPC Design

The function ${\color{red} { \texttt{fhpc_design }} } $ computes parameters$ {\color{magenta}{K_0,K,G_0,P, \mu_1,\mu_2}}$ and ${\color{magenta} \rho}>0$ of Fixed-time HPC for given system matrix ${\color{blue}A} $ and control matrix ${\color{blue}B}.$

$\textbf{Input parameters}: {\color{blue}A} $ and $ {\color{blue}B}$

$ \textbf{Output parameters}: {\color{magenta}{K_0,K,G_0,P, \mu_1,\mu_2}} $ and $ {\color{magenta} \rho} $

Fixed-time HPC Implementation

The function ${\color{red} { \texttt{e_fhpc }} } $ computes $\textit{explicit} $ discretization of $u_{fhpc}$

$ \textbf{Input parameters}: x$, ${\color{magenta}{K_0, K}}, {\color{magenta}{\mu_1,\mu_2}}, {\color{magenta}{G_{0}, P}} $

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

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

$\textbf{Input parameters}: h \,$ (sampling period), $x, {\color{blue}A},{\color{blue}B}, {\color{magenta}{K_0, K}}, {\color{magenta}{\mu_1,\mu_2}}, {\color{magenta}{G_{0}, P}} $

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

Use ${\color{red} { \texttt{demo_fhpc.m}} } $ from $\texttt{HCS Toolbox} $as a demo of Fixed-time HPC design

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Example of Fixed-time Homogenenous Proportional Control (FHPC) design %% %% System: dx/dt=A*x+B*u %% %% 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) %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% %% Model of system %%%%%%%%%%%%%%%%%%%%%%% A= [28 26 15 93 28; 68 69 59 49 98; 90 13 7 65 4; 91 12 82 89 33; 75 19 72 54 97]/40; B= [10 33; 20 90; 30 50; 40 62; 50 58]/40; n=5;m=2; %%%%%%%%%%%%%%%%%% %% FHPC Design %%%%%%%%%%%%%%%%%% [K0 K G0 P mu1 mu2 rho]=fhpc_design(A,B); % design of FHPC Gd1=eye(n)+mu1*G0; Gd2=eye(n)+mu2*G0; %K0 - homogenization feedback gain %K - control gain %Gd1 - generator of dilation d1 %Gd2 - generator of dilation d2 %mu1 - negative homogeneity degree (close to 0) %mu2 - positive homogeneity degree (close to Inf) %P - shape matrix of the weighted Euclidean norm %settling time estimate T=1/(-mu1*rho)+1/(mu2*rho) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Numerical Simulation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t=0; Tmax=8; h=0.01; % sampling period x=[1;0;0;0;0]; tl=[t];xl=[x];ul=[]; noise=0; %magnitude of measurement noises disp('Run numerical simulation...'); [Ah, Bh]=ZOH(h,A,B); while t<Tmax xm=x+2*noise*(rand(n,1)-0.5); %u=(K0+K)*xm; %linear control (for comparison) %u=e_fhpc(xm,K0,K,G0,P,mu1,mu2,0.05); %explicit discretization of FHPC u=si_fhpc(h,xm,A,B,K0,K,G0,P,mu1,mu2); %semi-impicit discretization of FHPC x=Ah*x+Bh*u; % simulation of the system t=t+h; tl=[tl t]; xl=[xl x]; ul=[ul u]; end; ul=[ul u]; disp('Done!'); %%norm of the state at the time instant Tmax disp(['||x(Tmax)||=',num2str(norm(x))]) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Plot the simulation results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure; axes1 = subplot(1,2,1); %Plot the results 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({'FHPC,m=2'}); xlim(axes2,[0 Tmax]); ylim(axes2,[-10 10]); box(axes2,'on'); hold(axes2,'off'); set(axes2,'FontSize',30,'XGrid','on','YGrid','on'); legend2 = legend(axes2,'show'); set(legend2,'Interpreter','latex');