Model predictive control for stochastic systems with Markovian jumps and serially correlated parameters under constraints

Assume that the plant to be controlled can be described by the following model: x(k +1) = A[a(k +1), k + 1]x(k) + B[a(k +1), \(k +1), k + 1]u (k), (1) where x(k) e M"^x is the vector of state, u(k) e M^ is the vector of control inputs; A[a(k), k] e M"^xx"^x, B[a(k),n(k),k] e Кⁿx^xnu are system and input matrices, respectively; n(k) e К^q is assumed to be a stochastic time series and all of the elements of B[n(k),k] are assumed to be linear functions of n(k); {a(k); k = 0,1,2,...} is a finite-state discrete-time Markov chain taking values in {1,2,...,v} with known transition probability matrix and initial distribution. We assume that a(k) and n(k) are mutually independent and at the instant of decision making, the current state of the market is known, i.e., the Markov state {a(k)} is observable. Let F =( Fk )_k>₁ be the complete filtration with a-field Fk generated by the {r|(s): s = 0, 1, 2,.,k} that models the flow of information to time k. We allow the time series n(k) to be serially correlated. Let us assume that we know the first- and second-order conditional moments for the stochastic vector n(k) about F_k: E{(k + i)/ Fk} = n(k + i), E{(k + i)n^T (k + j)/ Fk} = ©_t] (k), (k = 0,1,2,...), (i, j = 1,2,...,l). We impose the following constraints on the decision variables: u_mm (k) < S(k)u (k) < u_max (k), S(k) e M^pxn, umin (k), Um_ax (k) e M^P. (2) To control system (1) subject to constraints (2), at each step k we minimize the quadratic criterion with a receding horizon m J(k + m /k) = E{^x^T (k + i)R₁(k + i)x(k + i) -R₂ (k + i)x(k + i) + u^T(k + i -1/k)R(k + i - 1)u(k + i -1/ k) / x(k), a(k), F_k}, i=1 where m is the prediction horizon; u(k/k),...,u(k+m-1/k) is the sequence of predictive controls under; R₁(k+i)>0, R₂(k+i)>0, and R(k+i)>0 are the weight matrices of corresponding dimensions. The model predictive control methodology was used to solve the problem. The optimal control strategies were synthesized under hard constraints imposed on the control variables.

Keywords

Authors

References