Abstract
In this chapter we introduce the concept of safety for control systems in both continuous and discrete time form. Given a system and a safe set, we say the system is safe if the system state remains inside the safe set for all initial conditions starting from the initial set. Control invariance can be employed to verify safety and design safe controllers. To this end, for general polynomial systems with semi-algebraic safe/initial sets, we show how Sum-of-Squares (SOS) programming can be used to construct invariant sets. For linear systems, evaluating invariance can be much more efficient by using ellipsoidal techniques and dealing with a series of SOS constraints. Following invariance analysis, safe control design and safety verification methods are proposed. We conclude this chapter by showing invariant set construction for both nonlinear and linear systems, and provide MATLAB code for reference.
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Appendix
Appendix
Program Code with SOSTOOLS
options.solver='sedumi'; % you need to add SeDuMi into work path % Constructing the vector field pvar x1 x2; x = [x1;x2]; f = [ x2; +x1+x1ˆ3/3+x2]; gx = [x1ˆ2+x2+1; x2ˆ2+x1+1]; limits = 1.5; % Define the safe set S >= 0 is safe bias = 0; S = -(x1-bias)ˆ2-(x2-bias)ˆ2+3; % Avoid a round region % Define the initial set I >= 0 I = -(x1-bias-0.4)ˆ2-(x2-bias-0.4)ˆ2+0.16; % Start from a round region % Initialize the SOSP prog = sosprogram(x); % The barrier function [prog,B] = sospolyvar(prog,monomials(x,0:4),'wscoeff'); % The multipliers [prog,sigma1] = sossosvar(prog,monomials(x,0:1)); [prog,sigma2] = sossosvar(prog,monomials(x,0:1)); [prog,sigma3] = sossosvar(prog,monomials(x,0:1)); % ============================= % Inequality constraints % Constraint 1: B(x) < 0 when S < 0 prog = sosineq(prog,-B+sigma1∗S-0.001); % 0.001 is used for strict inequality % Constraint 2: B(x) >= 0 when I >= 0 prog = sosineq(prog,B-sigma2∗I); % ============================= % Iterative procedure below, we interatively solve the problem, first find % a feasible control input % Initialize the control input SolU1 = 0; SolU2 = 0; % Initialize the barrier function g = [gx(1)∗SolU1; gx(2)∗SolU2]; prog_barrier = prog; expr = diff(B,x1)∗(f(1,1)+g(1,1))+diff(B,x2)∗(f(2,1)+g(2,1)); prog_barrier = sosineq(prog_barrier,expr); % Solve the problem prog_barrier = sossolve(prog_barrier,options); % Get the resulting invariant set SolB = sosgetsol(prog_barrier,B); clear prog_barrier % Initialize the polynomial multiplier SolLambda1 = 0; SolLambda2 = 0; SolLambda3 = 0; SolLambda4 = 0; SolLambda5 = 0; for i=1:20 % ============================ % First fix the multiplier, control input, solve for the invariant set % ============================ % Constraint 1: dB/dx∗f(x)+alpha B > 0 when B = 0 g = [gx(1)∗SolU1; gx(2)∗SolU2]; prog_barrier = prog; expr = diff(B,x1)∗(f(1,1)+g(1,1))+diff(B,x2)∗(f(2,1)+g(2,1)) +SolLambda1∗B; prog_barrier = sosineq(prog_barrier,expr); prog_barrier = sosineq(prog_barrier,-SolU1+limits-SolLambda 2∗B); prog_barrier = sosineq(prog_barrier,SolU1+limits-SolLambda 3∗B); prog_barrier = sosineq(prog_barrier,-SolU2+limits-SolLambda 4∗B); prog_barrier = sosineq(prog_barrier,SolU2+limits-SolLambda 5∗B); % Enlarge the area expr = B-sigma3∗SolB; prog_barrier = sosineq(prog_barrier,expr); % Solve the problem prog_barrier = sossolve(prog_barrier,options); % Get the resulting invariant set SolB = sosgetsol(prog_barrier,B); % Delete the temporary variable clear prog_barrier; % ============================ % Then fix the invariant set, solve for the control input prog_u = sosprogram(x); [prog_u,u1] = sospolyvar(prog_u,monomials(x,0:3)); [prog_u,u2] = sospolyvar(prog_u,monomials(x,0:3)); g = [gx(1)∗u1; gx(2)∗u2]; expr = diff(SolB,x1)∗(f(1,1)+g(1,1))+diff(SolB,x2)∗(f(2,1) +g(2,1))+SolLambda1∗SolB; prog_u = sosineq(prog_u,expr); % Control limits [prog_u,lambda2] = sospolyvar(prog_u,monomials(x,0:2)); [prog_u,lambda3] = sospolyvar(prog_u,monomials(x,0:2)); [prog_u,lambda4] = sospolyvar(prog_u,monomials(x,0:2)); [prog_u,lambda5] = sospolyvar(prog_u,monomials(x,0:2)); prog_u = sosineq(prog_u,-u1+limits-lambda2∗SolB); prog_u = sosineq(prog_u,u1+limits-lambda3∗SolB); prog_u = sosineq(prog_u,-u2+limits-lambda4∗SolB); prog_u = sosineq(prog_u,u2+limits-lambda5∗SolB); prog_u = sossolve(prog_u,options); % Get the control input SolU1 = sosgetsol(prog_u,u1); SolU2 = sosgetsol(prog_u,u2); % Get the multipliers SolLambda2 = sosgetsol(prog_u,lambda2); SolLambda3 = sosgetsol(prog_u,lambda3); SolLambda4 = sosgetsol(prog_u,lambda4); SolLambda5 = sosgetsol(prog_u,lambda5); % Delete the temporary variable clear prog_u; % ============================ % Then fix the invariant set, solve for the multipler lambda prog_lambda = sosprogram(x); [prog_lambda,lambda1] = sospolyvar(prog_lambda,monomials(x, 0:2)); g = [gx(1)∗SolU1; gx(2)∗SolU2]; expr = diff(SolB,x1)∗(f(1,1)+g(1,1))+diff(SolB,x2)∗(f(2,1)+ g(2,1))+lambda1∗SolB; prog_lambda = sosineq(prog_lambda,expr); % Solve the problem prog_lambda = sossolve(prog_lambda,options); % Get the resulting polynomial multipler SolLambda1 = sosgetsol(prog_lambda,lambda1); % Delete the temproray variable clear prog_lambda; end
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Wang, H., Margellos, K., Papachristodoulou, A. (2023). Assessing Safety for Control Systems Using Sum-of-Squares Programming. In: Kočvara, M., Mourrain, B., Riener, C. (eds) Polynomial Optimization, Moments, and Applications. Springer Optimization and Its Applications, vol 206. Springer, Cham. https://doi.org/10.1007/978-3-031-38659-6_7
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