Sliding Mode Applications to Optimal Filtering and Control

Abstract

In practical applications, a control system operates under uncertainty conditions that may be generated by parameter variations or external disturbances. Consider a real trajectory of the disturbed control system $$ \dot{x}(t)=f(x(t))+B(t)u+g_1(x(t),t)+g_2(x(t-\tau ),t). ~~ (5.1) $$ Here, x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, f(x(t)) is a known function determining the proper state dynamics, the rank of matrix B(t) is complete and equal to m for any t > t0, and the pseudoinverse matrix of B is uniformly bounded: $$ \|B^{+}(t)\| \leq b^{+},\,b^{+}=const>0,\ B^{+}(t) := [B^{T}(t)B(t)]^{-1}B^{T}(t), $$ and B + (t)B(t) = I, where I is the m-dimensional identity matrix. Uncertain inputs g1 and g2 represent smooth disturbances corresponding to perturbations and nonlinearities in the system. For g1,g2, the standard matching conditions are assumed to be held: \(g_1,g_2\in \mbox{span} B\), or, in other words, there exist smooth functions γ1,γ2 such that $$g_1(x(t),t)=B(t)\gamma_1(x(t),t),~~ (5.2)$$ $$g_2(x(t-\tau ),t)=B(t)\gamma_2(x(t- \tau),t),$$ $$||\gamma_1(x(t),t)||\le q _1||x(t)|| +p_1,\ q_1,p_1>0,$$ $$||\gamma_2(x(t-\tau ),t)||\le q _2||x(t-\tau )||+p_2,\ q_2,p_2>0.$$ The last two conditions provide reasonable restrictions on the growth of the uncertainties.Let us also consider the nominal control system $$ \dot{x}_0(t)=f(x_0(t))+B(t)u_0(x_0(t-\tau ),t), ~~ (5.3) $$ where a certain delay-dependent control law u0(x(t − τ),t) is realized. The problem is to reproduce the nominal state motion determined by (5.3) in the disturbed control system (5.1).The following initial conditions are assumed for the system (5.3) $$ x(s)=\phi(s), ~~ (5.4) $$ where φ(s) is a piecewise continuous function given in the interval [t0 − τ,t0].Thus, the control problem now consists in robustification of control design in the nominal system (5.3) with respect to uncertainties g1,g2: to find such a control law u = u0(x(t − τ),t) + u1(t) that the disturbed trajectories (5.1) with initial conditions (5.4) coincide with the nominal trajectories (5.3) with the same initial conditions (5.4).KeywordsOptimal Control ProblemSliding ModeOptimal RegulatorOptimal FilterQuadratic Cost FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.