Model Predictive Control for Nonlinear Stochastic Systems with Markovian Jumps under Constraints

Let the control object is described by the equation x(k +1) = Ax(k) + B[a(k +1),k + 1]u(k) + f (x(k), u(k), w(k), a(k +1)), (1) where x(k) is the n _x - dimensional vector of state, u(k) is the n _u - dimensional vector of control; w(k) is the n _w - dimensional vector of white noses with zero-mean and identity covariance matrix; 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 matrixP = [P j J (i, je{i,2,...,v}), P _} i = P {a(k+1)=a j |a(k)=a _i} , V V E P _ji = 1, and the initial distribution p _t = P {a(0) = i} (i = 1,2,..., v) , £ p _t = 1. j=1 i=1 It is assumed that the state of Markov chain is observable at time instant k, and w(k) is independent of the Markov chain a(k) (k = 0,1,2.). The function f is defined by its statistical properties as follows: M { f (x(k),u(k),w(k),a(k+1))/x(k),a(k)=a _j} = 0 , M { f (x(k ),u(k ),w(k ),a(k+1)) f (x( k ),u(k), w( k ),a( k+1) )/x(k ),a(k)=a _j } = T [a(k), k ] + +£ T (x (k)W'x(k)+u (k)M [a(k),k]u(k)) , for all x(k), where r=n(n+1)/2; T , W , and M = (C ) C (i = 1, r), T = (d ) D are positive semidefinite and symmetric matrices. The following constraints are imposed on the control variables umin(k) < S(k)u(k) < um _ax(k), (2) where S(k) is a 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 quadratic criterion with a receding horizon J (k+p/k) = M J E x (k+i) R ₁ (k ,i) x(k+i) +u (k+i-1/k )R ₂ (k,i-1)u(k+i-1 k)/x(k ),a(k)=a _j}, on trajectories of system (1) over the sequence of predictive controls u(k/k),.,u(k+p-1/k), which depend on system's state and on the state of Markov chain at moment k, under constraints (2), where R^k, i) > 0, R2(k, i) > 0are weigh matrices of corresponding dimensions; p is a prediction horizon, k is a current moment. The synthesis of predictive control strategies is reduced to the sequence of quadratic programming tasks.

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