Model predictive control for discrete systems with state andinput delays.pdf One of the modern formalized approaches to the system control synthesis based onmathematical methods of optimization is control methods using predictive models -Model Predictive Control (MPC).his approach began to develop in the early 1960s. It was developed for equipmentand process control in petrochemical and energy industries for which the application oftraditional synthesis methods was extremely complicated according to mathematicalmodel's complication. During the last years application was considerably extended coveringtechnologic fields [1], economic system control [2], inventory control [3] and investmentportfolio control [4].The results of this paper extend the results of the paper [5].1. Problem StatementSuppose the object is described by the following state-space system of lineardifferenceequations11rt t i t i t h tix Ax A + x− Bu − w== +ƒ + + , xk= xk, (k= −r,0 ), ui=ui, (i= −h,−1), (1)ƒt= Hxt+vt, (2)yt= Gxt. (3)Here nxt  R ( xt= xt, t= −r,...,−1,0 , xt is considered to be given) is the object state,mut R is the control input (ut=ut, t = -h,-h+1,…,-1. ut is given), pyt  R is theoutput (which is to be controlled), lƒt  R is the observation (measured output), r, hare the state and input delay values respectively. Further, the state noise wt and measurement noise vt are assumed to be Gaussian distributed with zero mean and covariancesW and V respectively, i.e.M{wtwkƒ}=Wƒt,k, M{vtvkƒ}=Vƒt,k ,where ƒt,k is the Kronecker delta. This model is used to make predictions about plantbehavior over the prediction horizon, denoted by N, using information (measurementsof inputs and outputs) up to and including the current time t.It is supposed the plant operates under the constrained conditions:a1 ≤ S1xt ≤ a2, (4)ƒ1(xt−h) ≤ S2ut−h ≤ ƒ2(xt−h) (5)Here S1 and S2 are structural matrices that are composed of zeros and units, identifyingconstrained components of vectors xt and ut ; a1, a2, ƒ1(xt), ƒ2(xt) are given constantvectors and vector-functions.The problem is to determine an acting strategy on the base of the observation ƒt accordingto which the output vector of the system yt will be close to the reference takinginto account constraints on the state and input.2. PredictionWith the Gaussian assumptions on state and measurement noise it is possible tomake optimal (in the minimum variance sense) predictions of state and output using aKalman filter, see e.g. [6].Let xˆi| j and yˆi| j be estimates of the state and output at time i given information upto and including time j where j ≤ i. Then1| |1 | 1 | 11ˆ ˆ ˆ ( ˆ )rt t tt i t it i t h t t ttix+ Ax− Ax− − − Bu− K Hx−== +ƒ + + ƒ − , xˆk|k−1 = xk , k= −r,0 ,yˆt+1|t = Gxˆt+1|t ,1 ( ) t t t K APH HPH V− = ƒ ƒ+ ,1Pt1 W APtA APtH (HPtH V) HPt A ƒ ƒ ƒ − ƒ+ = + − + , P0 =Px0 , (6)where Px0 is the given initial value of the variance matrix. Equation (6) for Pt is knownas the discrete-time Riccati-equation.MPC usually requires estimates of the state and/or output over the entire predictionhorizon N from time t + 1 until time t + N, and can only make these predictions basedon information up to and including the current time t. Equations (6) can be used to obtainxˆt+1|t , yˆt+1|t . Optimal state/output estimates from time t + 2 to t + N can be obtainedas follows1| | | 1 |1ˆ ˆ ˆrt i t t it j t i j t j t h itjx+ + Ax+ Ax+ − − − Bu− +== +ƒ + , i= 1,N, (7)yˆt+i|t = G xˆt+i|t , i= 1,N. (8)In the above the notation ut-h+i|t is used to distinguish the actual input at time t+i,namely ut−h+i, from that used for prediction purposes, namely ut-h+i|t.Equation (7) can be expanded in terms of the initial itial state xˆt+1|t and future controlactions ut−h+i|t as follows1 11 1 1| 1| | 1 |1 1 1ˆ ˆ ˆi r ii ik i kt it t t j t k j t j t h ktk j kx A x A Ax А Bu− −− − − − −+ + + − − − − += = == +ƒ ƒ +ƒ , i= 1,N. (9)Now in terms of predicting the output, equation (8) can be expanded in terms of theabove expression for xˆt+1|t , which results in a series of equations that provide optimaloutput predictions. The key point to note is that each output prediction is a function ofthe initial state xˆt+1|t and future inputs ut−h+i|t only:1 11 1 1| 1| | 1 |1 1 1ˆ ˆ ˆi r ii ik i kt it t t j t k j t j t h ktk j ky GA x G A Ax G A Bu− −− −− − −+ + + − − − − += = == + ƒ ƒ + ƒ , i= 1,N. (10)This series of prediction equations can be stated in an equivalent manner using matrixvector notation. Denote1||ˆˆˆt ttt NtxXx++⎡ ⎤= ⎢⎢ ⎥⎥⎢⎣ ⎥⎦

Скачать электронную версию публикации