Predictive control for markov jump systems with markov switching autoregressive multiplicative noise

Assume that the plant to be controlled can be described by the following model ^xk+1 ^{- A[0}k+1^]xk + ^B[0k +1^, Jk+1^]uk > ⁽¹⁾ Jk+1 ^{- a[0}k+1^]Jk +^P[0k+1^] + ^^[0k+1^]wk+1- ⁽²⁾ V (3) i=1 V V V a[0_t] = xe,,«m] = SQaP''⁰!⁰*] = хе„ст';а',ст' e R^?X?,P' e R^?, (4) /=1 /=1 /=1 m₊i,y_k+i\ = {в'[а'у_к]+в'т +B\&w_k+l]),£',F„B' £ 1Г"^ХИ", (5) where x_k e M"¹ is the vector of states, u_k e M"" is the vector of control inputs, y_k el' is a sequence of stochastic vectors, w_k e R' is a zero mean independent random sequence; 9i,k+1 (i = 1,v ) are the components of the vector 9k+1, 9k = [S(Tk, 1), ..., S(Tk, v)]^T, S(Tk,j) is a Kronecker function; {Tk; k = 0, 1, 2, .} is a finite-state discrete-time homogeneous Markov chain taking values in {1, 2, v} with transition probability matrix P = [P/]. All of the elements of matrix B are assumed to be linear functions of vector y. We impose the following inequality constraints on the control inputs (element-wise inequality): ^Uk ^^Sk^Uk^^Uk = (6) where S_k e , «Г*.«Г" ^e · For control of system (1)-(5) we synthesize the strategies with a predictive control model. At each step k we minimize the quadratic criterion with a receding horizon ^Jк +m\ к ^- E{2 Xk+i^Rk+i^Xk+i + ^uk+i-1\k^Rk+i-1^uk+i-1\k \ ^xk,⁰k,Л^}, i-1 on trajectories of system (1) over the sequence of predictive controls Uk\k, ..., Uk+m- 1\k dependent on information up to time k, under constraints (6), where R^- > 0,R_{k +i}_₁ > 0 are given symmetric weight matrices of corresponding 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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