Model predictive control of distributed stochastic hybrid systems with multiplicative noises under constraints

We consider the following complex Markov jump linear system composed of interconnected subsystems x^(q)(k +1) = A^(q)[a^(q) (k +1), k + 1]x^(q) (k) + B(^q)[a^(q) (k +1), k + 1]u ⁽q) (k) + ->-V gj^q)[a^(q)(k +1),k + 1]w(^q)(k + 1)u^(q)(k),q = 1S, ^ ^ j=1 where ^^q)(k) is the n(^q) - dimensional subsystem state vector, u^(q)(k) is the n(^q) - dimensional subsystem control vector; A^(q)[a^(q)(k),k], B/^q)[a^(q)(k),k], j = 0, ..., n are matrices of appropriate dimensions; w/^q)(k) is a sequence of white noises; a^(q)(k) denotes a time-invariant Markov chain taking values in a finite set of states {1, 2, ..., Vq}. These subsystems interact in the following way. The state of Markov chain a^(q)(k) of qth subsystem (q = 1, 2, ..., s) at the moment k depends on states of Markov chains a^(r)(k - 1) (r = 1, 2, ..., s) at the moment k - 1. Thus, the complex system dynamics depends on a discrete-time vector stochastic process a(k) = [a⁽¹⁾(k), a⁽²⁾(k), ..., a^(s)(k)]^T taking values in a finite set of states {q, jq}(q = 1, 2, ..., s; jq = 1, 2, ..., Vq). The process a(k) is a simple connected Markov chain with the transition probability matrix p,...,i,= ^p{«1^(k +¹⁾ = «u^,...^,«s(^k +¹ = «j^/a1^(k) = «И^,...^,as(^k) = «si,к £ p1,...,,^..j =¹ , J1^,-^,Js and the initial distribution Pj11 = P{«1(0) = Ji,-,«s(0) = Л},(л = irv^;..,js = TV,), £ P1 = 1. jj It is assumed that the Markov chain a(k) is observable at the moment k. The following constraints are imposed on each subsystem control effects umn (k) < s ^(q) (k)u^(q) (k) < u^ (k), q=1;,, (2) where S(k) is the matrix of corresponding dimension. For control of system (1) we synthesize the strategies with a predictive control model according to the following rule. At each step k we minimize the following quadratic objective with receding horizon J(k + m / k) = M{£ £(x^(q)(k + i))^TR(^q)(k + i)x^(q)(k + i) -R(^q)(k + i)x^(q)(k + i) + q=1i=1 +(u^(q)(k + i -1 / k))^TR^(q)(k + i - 1)u^(q)(k + i -1 / k)/x^(q)(k),a(k)}, on trajectories of system (1) over the sequence of predictive controls u^{( q)}(k +1 / k), l = 0, m -1, depending on the subsystem state at the current time kunder constraints (2); R^(q)(k + i) >0, R(^q)(k + i) >0, R^(q)(k + i) >0 are weigh matrices of appropriate dimensions; m is the prediction horizon; k is the current moment. The synthesis of predictive control strategies is reduced to the sequence of quadratic programming tasks.

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