Model predictive control with quadratic criterion for jump Markov discrete linear systems under constraints

Let the control object be described by the equation x(k +1) = A[a(k + 1)]x(k) + B[a(k + 1)]u(k), where x(k) e R is the vector of state, u(k) e R is the vector of control, a(k) (k= 0,1,2...) denotes a time-invariant Markov chain taking values in a finite set of observable states {1,2,.. ,,v} with the known transition probability matrix P = |P j J ((, je{1,2,...,v}) , P_ji = P{a(k + 1) =a_j|a(k) =a,J, £ P_jj = 1, and the initial distributionp_t = P{a(0) =i} (i =1,2,...,v), £ p_t = 1. j=1 i=1 The state a of the Markov chain a(k) selects the system matrices A[a(k)] and B[a(k)] from the sets A={ A' e R" : i = 1,v } и B={ B' e R"" : i = 1,v } respectively. It is assumed that the state of the Markov chain is observable at the time instant k. On control variables the following constraints are imposed: Um_m(k) < 5(k)u(k) < Um_ax(k), (2) where 5(k) e R -, Um_m (k), u^ (k) e R . To control system (1), we synthesize model predictive control strategies. At each step k we minimize the quadratic criterion with a receding horizon J(k+m/k) = E jl, x(k +i)R₁x(k+i) + u(k +i-1/k)Ru(k +i-\/k)jx(k),a(k) = a_y} (3) on trajectories of system (1) over the sequence of predictive controls u(k/k),...,u(k + m- 1/k), which depend on system's state and on the state of Markov chain at the moment k under constraints (2), where m is a prediction horizon, R₁ > 0, R > 0 are weight matrices of corresponding dimensions, k is a current moment. The synthesis of predictive control strategies is reduced to the sequence of quadratic programming tasks.

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