Управление дискретными динамическими системами со случайными зависимыми параметрами при ограничениях.pdf Lately there has been a steadily growing need and interest in systems with stochasticparameters and/or multiplicative noise. The same systems have been gaining greater acceptancein many engineering and finance applications.Optimization techniques to various control and estimation problems for such systemshave been intensively studied in the literature.In particular, a linear quadratic control for systems with random independent parametersis studied in [1-3]. In [4-8] the authors consider stochastic optimal controlproblem of systems with dependent parameters which switch according to a Markovchain.In above-mentioned papers there are no constraints on the state and control variables.However, constraints arise naturally in many real world applications.In recent years considerable interest has been focused on model predictive control(MPC), also known as receding horizon control (RHC), as an appropriate and effectivetechnique to solve the dynamic control problems having input and state/output constraints.The basic concept of MPC is to solve an open-loop constrained optimizationproblem at each time instant and implement only the first control move of the solution.This procedure is repeated at the next time instant. Some of the recent works on thissubject can be found in literature [9-20].MPC for constrained discrete-time linear systems with random independent parametersis considered in [15, 16]. In [17, 18] MPC of linear with random dependent parameterssystems under constraints is examined, where parameters evolution is describedby multidimensional stochastic difference equations. In [19] the MPC problemof discrete-time Markov jump with multiplicative noise linear systems subject to constraintson the control variables is solved.In this paper we consider MPC for constrained discrete-time with stochastic nonlinearparameters systems. The main novelty of this paper is that no assumption is madeabout the form of the function describing parameters evolution. The first and secondconditional distribution moments, the conditional autocorrelations and the mutual crosscorrelationsare known only. The performance criterion is composed by a linear combinationof a quadratic and liner parts. The open-loop feedback control strategy is derivedsubject to hard constraints on the input variables. Predictive strategies computation includesthe decision of the sequence of quadratic programming tasks.The approach is advantageous because the rich arsenal of methods of non-linear estimationor if no model can be found that describes the underlying non-linear structureadequately, then the results of nonparametric estimation may be used directly for describingcharacteristics of non-linear time series.1. Problem formulationWe consider the following discrete-time with non-linear stochastic parameters systemon the probabilistic space (ƒ, F , P):( 1) ( ) [ƒ( 1), 1] xk+ =Axk+B k+ k+u(k), (1)where x(k) is the nx-dimensional vector of state, u(k) is the nu-dimensional vector ofcontrol; ƒ(k) (k=0,1,2…) denotes a sequence of depended q-dimensional random vectors.A, B[ƒ(k),k] are the matrices with appropriate dimensions, where B[ƒ(k),k] linearlydepend on ƒ(k).Let F =( Fk )k≥1 is the flow of sigma algebras defined on (ƒ, F , P), where Fk denotesthe sigma algebra generated by the {(x(s), ƒ(s)): s=0, 1, 2,…,k} up to time instantk and it means the information (measurements) before the time k.We assume that we know conditional moments for the process ƒ(k) about Fk :M{ƒ(k+i)/Fk}=ƒ(k+i), (2){ƒ( ) ƒ( ) ƒ( ) ƒ( ) } (),( 0,1,2,...), ( , 1, 2,...)./ TM k i k i k j k j k ijkk ij⎡⎣ + − + ⎤⎦⎡⎣ + − + ⎤⎦ = ƒ= =F(3)In that follows, we use notation: for any matrix ƒ[ƒ(k),k], dependent on ƒ(k),ƒ(k) =E{ƒ[ƒ(k),k]/Fk} without indicating the explicit dependence of matrices onƒ(k). Also we use the standard notation, for square matrix A, A≥0 (A>0, respectively) todenote that the matrix A is positive semidefinite (positive definite), and tr(A) to representthe trace of A.We impose the following inequality constraints on the control:umin(k)≤S(k)u(k)≤umax(k), (4)where S(k) is the matrix with appropriate dimension.The cost function of the RHC is defined as a function, composed by a linear combinationof a quadratic part and a linear part, which is to be minimized at every time k{ } 1 31( / ) ( ) ( , ) ( ) ( , ) ( )/ mTkiJ k m k M x k i R k i x k i R k i x k i=+ =ƒ + + − + F +{ } 12 40( / ) (,)( / ) (,)( 1/ )/ , mTkiM u k i k R k iu k i k R k iu k i k−=+ƒ + + − + − F (5)on trajectories of system (1) over the sequence of predictive controls u(k/k), …,u(k+m−1/k) dependent on the state x(k), under constraints (4); R1(k,i)≥0, R2(k,i)>0,R3(k,i) ≥0, R4(k,i) ≥0 are given symmetric weight matrices of corresponding dimensions;m is the prediction horizon.Only the first control vector u(k/k) is actually used for control. Thereby we obtaincontrol u(k) as a function of state x(k), i.e. the feedback control. This optimization processis solved again at the next time instant k+1 to obtain control u(k+1). The synthesis ofpredictive control strategies leads to the sequence of quadratic programming problems.2. Model predictive control strategies designConsider the problem of minimizing the objective (5) with respect to the predictivecontrol variables u(k+i/k), subject to constraints (4).Theorem. The set of predictive controls U(k)=[uT(k/k), …,uT(k+m-1/k)]T, such that itminimizes the objective (5) subject to (4), for each instant k is defined from the solvingof quadratic programming problem with criterion( / ) 2 ( ) ( ) ( ) ( ) Y k+m k =⎡⎣ xTk G k −F k⎤⎦U k +UT(k)H(k)U(k) (6)under constraintsUmin(k)≤S(k)U(k)≤Umax(k), (7)where S(k)=diag(S(k),...,S(k+m−1)),Umin (k)=[umTin (k),...,umTin (k+m−1)]T ,Umax (k)=[umTax (k),...,umTax (k+m−1)]T ,H(k), G(k), F(k) - are the block matrices11 12 121 22 21 2( ) ( ) ( )( ) ( ) ( ) ( ) ,( ) ( ) ( )mmm m mmH k H k H kH k H k H k H kH k H k H k⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎢⎣ ⎥⎦

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