Mean-variance MPC for linear systems with Markovian jumps under constraints

We consider the following Markov jump linear system with multiplicative noise B [a(k +1), k +1] + £ Bj [a(k +1), k + 1]w (k +1) _ j=1 where x(k) is the n - dimensional vector of state, u(k) is the n - dimensional 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 transition probability matrix P = [Pj] , i, j e {1,2,..., v}, Pj= P {a(k +1) = aj |a(k) = a,}, }= 1, j=1 V and initial distribution p = P{a(0) = i}, i = 1,2,...,v; £p = 1; ro (k) are independent zero-mean i=1 random variables with unit variance and independent of the Markov chain a(k), k = 0,1,2.; A, B,[a(k),k], j = 1,.,n, - the matrixes of corresponding dimensions, Bj[a(k),k] e {B/'- }, j = 0,.,n, i = 1,.,v. Let y(k) = L(k)x(k) be the scalar output of the system (1), where L(k) is the vector of corresponding dimension. The following constraints are imposed on control actions u n(k) < S(k)u(k) < umax(k), (2) where S(k) is the matrix of corresponding dimension. For control of system (1) we synthesize the strategies with a predictive control model. At each step k we minimize the «mean-variance» criterion with a noving control horizon m J(k + m / k) = £p (k,i)M {(y(k + i) - M {y(k + i) / x(k), a(k)} ) / x(k), a(k)} -i=1 -p (k, i)M {y(k + i)/x(k), a(k)} + M {u (k + i -1/k) R(k, i - 1)u(k + i -1/k)/x(k), a(k)}, 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 at moment k, under constraints (2), R(k,i')>0 is the weigh matrix of corresponding dimension, p!(k,i)>0, p (k,i) >0 are weigh coefficients, m is the prediction horizon, k is the current moment. The synthesis of predictive control strategies is reduced to the solving of a sequence of quadratic programming tasks.

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