Research On Predictive Control And Model Free Control

Model predictive control(MPC) or receding horizon control enjoys a lotof industrial interest mainly due to its intrinsic ability to deal with hardconstraints on the inputs , states and outputs explicitly. The nominal closedloopstability of nominal constraint model predictive control has been solvedsystematically but such result for the robust model predictive control foruncertain linear systems is far from complete and therefore received muchattention in recent years. Most of these methods of robust model predictivecontrol for uncertain systems is based on those of the nominal system.However, when those methods which are available for nominal systems areextended to uncertain systems, the results, such as feasibility, stability andfeasible region, maybe not hold. The first part of this paper mainly concernsthe robust model predictive control problems for two uncertain systems. Thefeasibility and stability of these problems are considered based on the essentialdifference of the nominal systems and the uncertain systems. Moreover,some stable improved algorithms also are proposed.Generally speaking, conversional process control requires that a goodmathematical model of the controlled process be derived in order to designa adequate controller, at least the structure of the model is obtained in advance.During the design of model free control one do not have to establishthe mathematical model, the modeling of model free control is obtained byfeedback control. The existing method only use the measurement of input and output and its form is simple and less flexible. The second part of this paper mainly consider some improvements on the existing model free control.I. Robust model predictive control1. The ploytopic uncertain systemsConsider the following polytopic uncertain system:with state and input constraintswhereΩ= Co{[A₁|B₁], [A₂|B₂],…, [A_L|B_L]}, and at time k achieving thefollowing robust cost indexFirstly, the sufficient and necessary feasible condition of the algorithm withoutconstraints in [47] is considered. It is found that this algorithm guaranteesglobally asymptotically closed-loop stability once the feasible condition asfollowing is satisfied.Theorem 1 For any weighting matrices (?) > 0, R > 0 and for any initialstate value x(0), the optimization without constraints in [47] is feasible if andonly if there exists F and P > 0 satisfyingSecondly, the feasibility and stability of the robust predictive control algorithmwith N free control moves proposed in [36] is considered. It is foundthat this algorithm may not be feasible and stable if it is only initially feasible.Two sufficient conditions for the feasibility and stability of this algorithm ispresented. Theorem 2 If the robust predictive control algorithm with N free controlmoves in [36] is initial feasible, then it will be feasible for k > 0 if(H1). N=1.(H2). (?)k≥0, A(k) is stable.â–¡â–¡Moreover, an improved algorithm-variable horizon robust predictive controlalgorithm is proposed which guarantees the robust stability of the closed-loopsystem once the problem is initially feasible.Variable horizon robust predictive control algorithm:S1: Initialize horizon N≥1, state initial value x(0) and let k := 0;S2: Solve the optimization(2-11), apply the first control move u*(k|k)to thesystem and achieve the measured state x(k +1|k + 1);S3: Let k := k + 1, N := N - 1. If N > 0, go to Step2; otherwise go toStep4;S4: Solve the optimization (2-15) with the state measurement x(k|k), applythe optimum u*(k|k) = YG^-1x(k|k) to the system and achieve themeasured state x(k + 1|k + 1);S5: Let k := k + 1, goto Step 4.The stability of variable horizon robust predictive control algorithm is asfollows:Theorem 3 Given the uncertain system with input and state constraints,state and input weighting matrices, an initial control horizon N, if the variablehorizon robust predictive control algorithm is feasible at time k = 0, then itis feasible for all times k > 0 and the closed-loop system is robust stable.â–¡Though the variable horizon robust predictive control algorithm is superiorto the algorithm proposed in [36] it still has some disadvantages. Finally,a synthesis approach robust model predictive control algorithm is presented base on these results in [36] and [68]. This proposal uses time-varying sequencesof models in a polytope to forecast the model uncertainties and optimizesthe terminal constrained set, the local controller and the terminalcost on-line. Using standard techniques, the problem is reduced to a convexoptimization involving linear matrix inequalities(LMIs). The following resultis the closed-loop stability of the proposal.Theorem 4 The receding-horizon implementation of the synthesis approachof constrained robust model predictive control algorithm guarantees exponentialclosed-loop stability for a stabilizable plant, once a feasible solution ofthe optimization problem is found.â–¡Let∑_N denote the feasible region of the algorithm with horizon length N.The monotonicity of∑_N respect to horizon length N is analyzed and it isfound the size of∑_N is non-decreasing (with respect to set inclusion) withhorizon length N:Theorem5∑₀(?)∑₁(?)…(?)∑_N(?)….â–¡2. The systems with bounded additive uncertaintiesConsider the following state-space model with bounded additive uncertaintieswith state and input constraintsand a quadratic performance indexThe open-loop min-max model predictive control is based on finding the controlcorrection sequence u that minimizes J(θ,u, x) for the worst case of the predicted future evolution of the process state or output signal. This isaccomplished through the solution of a min-max problem:the feasibility and stability of open-loop min-max model predictive control(OL-MMMPC) for systems with additive bounded uncertainties are considered.It is found that the OL-MMMPC may not be feasible and stable if itis only initially feasible. A sufficient condition for the feasibility and stabilityof OL-MMMPC is presented.Theorem 6 Provided that the system matrix A is stable, i.e.ρ(A) < 1,the constraint setΩ(?) X satisfiesand the weighting matrices Q = P = 0. If the OL-MMMPC optimizationproblem is feasible at time k = 0, then it is feasible for all times k > 0 andthe set E :=⊕_i=1^∞A^i-1D(?) is robustly asymptotically stable for the closed-loopsystemwith attraction domain F*.â–¡Then an improved OL-MMMPC algorithm is proposed which guarantees therobust stability of the closed-loop system once the problem is initially feasible.Improved OL-MMMPC algorithm.S1: Initialize horizon N, x(0)∈F* and let k := 0;S2: Solve the optimization (2.3) and apply the first control move u*(k|k) tothe system and achieve the measured state x(k + 1|k + l); S3: Let k:= k+1, N := N -1. If N > 0, go to Step2; otherwise go toStep4;S4: Apply u*(k|k) = -Kx(k|k) to the system and achieve the measured statex(k +1|k+1);S5: Let k := k + 1, goto Step 4.Theorem 7 Given the uncertain system with input and state constraints,state and input weighting matrices Q, P and R, an initial control horizon N,and some pre-chosen K. Assume that the system matrices and the terminalconstraint set satisfy (A1)-(A3). If the improved OL-MMMPC algorithmis feasible at time k = 0, then it is feasible for all times k > 0 and the setΣ':=⊕_i=1^∞(A-BK)^i-1D(?) is robustly asymptotically stable for the closed-loop systemwith attraction domain F*.â–¡II. Some improvements on model free controlConsider a industrial process S that can be described, under some mildconditions, asA universal model as follows of the system S is founded firstlyBased on the measurement of the system and some estimate method, theestimate value of (?)(k) denoted by (?)(k) can be achieved and the control lawis also obtained:Firstly, a simple and reasonable convergence condition of the existingmodel free control is given. LetTheorem 8 If (?)k and i = 1,…, n, there exists 0 <α≤βsatisfying:then there exists a properλ, such thatλ_k=λandIt is easier to be satisfied and is more reasonable comparing to the existingresults.Then, some improvements on model free control are considered.1. The model free controller with dead zoneIn order to avoid frequent movement of control action and eliminate thevibration of the system in some conditions in computer control system, illuminatedby PID controller with dead zone, one can use the model free controllerwith dead zone as followswhere2. The model free controller with static gainNo other information about the process is used in the existing model freecontroller except the output and input measurements. If we can obtain easilysome information about the process and apply it in the existing methods,some improved result may be achieved. Here combined the static gain for aclass of system we present the model free controller with static gain where3. The model free controller with constraintsThere are many constraints in industrial processes and these constraintsshould be considered when a control law be solved. Here, the following con-straints are considered:where the constraints on input is hard while on output is soft. This problem isreduced to a multi-objective programming (MOP) and a multilevel orderingmethod is used to accomplish the programming with respect to the sense ofthe control problem.The 1st level: Feasibilityachieved the optimal solutions s_o⁺,s_o^-≥0. If the problem is feasible, thens_o⁺=s_o^-=0.The 2nd level: Performance of trackingother constraints.achieved the optimal solution e* =minJ₂. If the output at next time canreach the set-point value y₀ under the conditions of constraints then e* = 0. The 3rd level: Control movesother constraints.In the end of this paper, the relations between predictive control and PIDcontrol, between model free control and PID control are simply analyzed. Itis found that both predictive control and model free control have the formof classical PID control. It helps one to understand the new-style controlmethods according to the classical PID control and design the PID controlparameters with the new-style control methods.