Min-Max Model Predictive Control Implementation Strategies

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1 y y Outline Why Min-Max Model Predictive Control? Min-Max Model Predictive Control Implementation Strategies Min-Max Model Predictive Control using openloop predictions. Min-Max Model Predictive Control using closedloop predictions. Daniel R. Ramirez & Eduardo F. Camacho Why Min-Max Model Predictive Control? Model Predictive control is a success both the industrial community and academia: Applications (Qin y Badgwell, ). % Refining and petrochemical industry. Hundreds of paper published. Why Min-Max Model Predictive Control? Pros: Multivariable from the beginning. Deadtime compensation. Optimal design. Guaranteed stability (academic version). Able to consider constraints. Cons: Computational burden. Need of a model. Robust Modelling errors (uncertainties, disturbances) Model can degrade the closed loop performance. Predictive Control Why Min-Max MPC? Why Min-Max Model Predictive Control? RobustMPC: Robust stability guaranteed: Keep the state within a region. Robust constraint satisfaction. Better performance against uncertainties More Robustness MPC MMMPC Improve performance in presence of uncertainties and disturbances: Improving robustness. Take into account uncertainties in the design. Min-Max formulation: worst case optimized design Min-Max MPC (MMMPC) Campo y Morari, segundos segundos

2 Why Min-Max Model Predictive Control? Why Min-Max Model Predictive Control?. This increase in robustness also appear in real world applications Control Engineering Practice, MPC MMMPC More robustness can also help avoding feasibility problems: Constraint violation due to modelling errors. Unfeasible optimization problem. More robustness can also help.... muestras. Δ u k -. - muestras muestras -. muestras Why Min-Max Model Predictive Control? Why Min-Max Model Predictive Control? Even when there are advantages the number of reported applications is very low? Computational Burden issues Tiempo (seg.) - Berenguel et al. Kim et al. Álvarez et al. horizonte de predicción Desiderable properties of an Min-Max MPC: Far lower computational burden, complexity growing with prediction horizon at non exponential rate. Use efficient numerical algorithms (such as in QP) to compute control signal. Flexible implementation, able to change parameters in real-time. Able to use constraints. Low error if an approximation of the optimizer is used. Robust stability guaranteed. Types of Min-Max MPC Types of Min-Max MPC A broad division of Min-Max MPC can be established between those which use open-loop predictions and those with closed-loop predictions (note that despite its type all of them are feedback controllers). Open-loop prediction based controllers are the most mature algorithms dating back to (Campo and Morari). They suffer from conservatism in the predicted evolution of the process that can lead to poor performance or feasibility problems. However, properly tuned they work well as proved later. Although its computational burden grows exponentially with certain parameters, there exists techniques to implement it in a wide class of systems. On the other hand, Closed-loop prediction based controllers have lower conservatism and better feasibility properties, because they take into account the fact that the control law is applied in a feedback manner. The price to be paid is an even greater computational burden and fewer implementation options (in fact there is no any single reported application to a real process). These controllers date from (Lee and Yu) and (Scokaert et al) and usually rely in recursive min-max problems or the optimization of a set control policies instead of a sequence of control signal values. Finally there is a comprimise between these two types of controllers called the semi-feedback approach, that can be used with open-loop prediction controllers to add a certain degree of closed-loop behavior in the predictions.

3 Open loop vs close loop prediction Open loop vs close loop prediction Open loop vs close loop prediction Min-Max Model Predictive Control based on Open-Loop Predictions MMMPC using Open-Loop Predictions Predictive Control past future Robust Model Predictive Control Explicit implementation Implementation using a nonlinear bound Implementation using a QP based bound Applications and Examples Output predictions y(t+k) Manipulated variables t t+ u(t+k) t+n At each sampling time t: Get new measures y(t) Estimate x(t) Solve a finite time optimal control problem to obtain u * (t) Apply the first component of u * (t)

4 Robust Predictive Control Based on models that take into account uncertainties: Robust Model Predictive Control System model: UNCERTAIN PREDICTIONS Bounded additive uncertainties past future y(t+k) Twostrategiestoconsideru(t): Open-loop predictions: Output Predictions Manipulated Variables u(t+k) t t+ t+n Semi-feedback predictions: Computed by the controller Robust MPC Min-max MPC ( - norm) w min-max v u MPC + - Plant K x past C y future The inner loop pre-stabilizes the nominal system Campo and showed that by using an - norm the min-max problem reduces to a linear programming problem. Although the algorithm was developed for the truncated impulse response, it can easily be extended to the other models used. The objective function is now described as y(t+k) Manipulated Variables u(t+k) Output predictions t t+ t+n Min-max MPC ( - norm) Min-max MPC ( - norm)

5 Min-max MPC ( - norm) Min-max MPC ( - norm) A LP (with many constraints: the vertices of the uncertainty polytope) Min-max MPC (-norm) Min-max MPC (-norm) Min-max MPC (quadratic cost) Min-max MPC (quadratic cost) Cost Function: The Min-Max Strategy: V * (x,u) is convex Robust constraint Satisfaction Constraints: The max function: Dependence on uncertainty in constraints can be removed offline (Lofberg,Kerrigan ) Convex function The maximum is attained at least at one of the vertices of Θ. (Bazaraa and Shetty )

6 Thus, the max function can be computed as: Number of vertices: Min-max MPC (quadratic cost) NP hard problem Exponential complexity As a result, the min-max problem is of the NP hard kind and in general it cannot be solved in real time except for slow processes and/or small horizons. Real time implementation of Min-Max MPC How to get an Min-Max MPC that can be computed in real time?? Some suggestions: Use an explicit implementation, in which the minmax problem is precomputed for feasible states. Solve an equivalent reduced min-max problem. Substitute the worst case cost for a low computational burden close approximation. MMMPC using Open-Loop Predictions Explicit implementation of Min-Max MPC Robust Model Predictive Control Explicit implementation Implementation using a nonlinear bound Implementation using a QP based bound Applications and Examples MMMPC not solvable on line An alternative: Multi parametric programming Explicit form control laws: Multi-parametric Programming Geometrical Approach Successfully applied to MPC: Quadratic norms LP based norms Nominal YES YES Min Max Open Loop predictions Yes YES Min Max Closed Loop predictions NO YES (Ramirez and Camacho Automatica ); (Muñoz de la Peña et.al. CEP, SCL ) Based on properties of cost function and the concept of active vertices (i.e., the vertices in which the worst case is attained at the solution). Itallows: To prove that the control law is PWA on the states. To find an explicit solution using a constructive algorithm. (Bemporad et.al. AUT, Bemporad et.al. TAC ) (Sakizlis et.al. AUT, Wan et.al. TAC ) Approximate closed loop explicit solutions Works well, but difficult to understand, complex specially with constraints, heuristic if need to be efficient.

7 Min-max as a Quadratic Program The min-max problem can be rewritten as: Min-max as a Quadratic Program Min max problem: Added and subtracted V(x,v,) Taking into account the cost definition: Epigraph approach: Quadratic dependence on w Affine dependence on x and v Maximization Min-max as a Quadratic Program Min max problem: Example Consider the double integrator with bounded additive uncertainties: linear constraints The uncertainty satisfies: The state and control input are constrained: Quadratic Programming problem Exponential number of constraints Semi-definite positive function MULTIPARAMETRIC THEORY (Tondel et.al. CDC ) The control performance objectives are described by A linear feedback law is considered: Example Example N = N = N= N= x x I= I= I= A= I= A= A= A= I= I= A=[ ] A=[ ] I= I= A=[ ] A=[ ] I= I= I= A= I= A= A= A= x x N= x x x x N= x x N= N= N= x x

8 Example Feedback PT- x x I= A= I= A= I= A=[ I= ] A=[ ] N = N = I= A= I= A= I= A=[ I= ] A=[ ] Set of active vertices is very small Scaled laboratory process nd order system Fast dynamics Ts =. s. (Explicit solution) I= A= I= A= I= A= I= A= x x Heat In Heat Stored Heat Out Results of the MMMPC Comparisons MMMPC with linear feedback u k N=, λ = Different controllers MPC MPC Different positions of the inlet throttle from o to o N=, λ = MMMPC MMMPC N=, λ = N=, λ = Applied with success MMMPC with linear feedback MMMPC with linear feedback (Muñoz de la Peña et.al CEP ) Explicit implementation: Pros and Cons MMMPC using Open-Loop Predictions Possibility to implement explicit min-max MPC control laws. using standard multi-parametric techniques. Well suited for fast systems and embedded applications. But The exponential number of constraints limits the value of the prediction horizon. The implementation is not flexible, the user cannot change parameters without recomputing the controller. Robust Model Predictive Control Explicit implementation Implementation using a nonlinear bound Implementation using a QP based bound Applications and Examples

9 Using an upper bound of the worst case cost A computationally cheaper upper bound is minimized instead of the worst case cost : Strategy based on a nonlinear bound Main contribution: Previous results: LMI methods Kothare et.al, Lofberg BQP Algorithms Alamo el.al Strategy based in a nonlinear bound, Ramirez et al., JPC Strategy based in QP problems, Alamo et al., Automatica Lower computational burden New upper bound of the maximization problem not based on LMIs or complex algorithms. Main Properties: Low computational burden. Based on matrix computations. Close to the worst case cost. Computing the nonlinear upper bound Computing the nonlinear upper bound The quadratic maximization problem can be rewritten as: The idea to obtain the bound is: If T is a diagonal positive definite matrix such that: Asume that ε =, then: Binary optimization problem then: Note that matrix H depends on (x,u) Computation of T: LMI or Proposed method based on simple matrix computations Computing the nonlinear upper bound Computing the nonlinear upper bound T will be obtained adding to H a series of matrices of the form: Vectors v i are computed so that matrix H is partially diagonalized: positive definite matrix The v i will be computed in such a way that allow to obtain a close bound with low complexity. If Free parameter This idea is applied recursively

10 Computing the nonlinear upper bound Nonlinear upper bound algorithm The error introduced at each diagonalization step must be minimum: cc Let For k= to n- Let H sub = [T ij ] for i,j= k,...,n Compute α for H sub Form Form End For When the procedure is over, the bound is computed as σ u (H) = trace(t) Computing the control signal Accuracy of the nonlinear upper bound At each sampling time, solve: It will be compared against the LMI bound: Deviation for a set of random matrices of the form H=H 'H. and apply the first component of u * (x) using a receding horizon strategy. If at any step of the diagonalization procedure, all the elements of H sub are the best bound will be No need to continue the computation loop Nesterov et al, Typical matrices in MPC problems are outside this zone, because the mean of its elements are different from zero Desviación de la cota LMI (%) Dimension matriz media de los elementos de H Computational burden of the bound MMMPC using Open-Loop Predictions Typical matrices in MPC problems are outside this zone, because the mean of its elements are different from zero Aceleración Dimensión matriz - - media de los elementos de H Robust Model Predictive Control Explicit implementation Implementation using a nonlinear bound Implementation using a QP based bound Applications and Examples Relative speed up from the LMI bound Solver LMI by F. Rendl

11 QP based strategy It has two desiderable properties: Robust stability guaranteed. Implementation based on an optimization problem similar to that of conventional MPC Quadratic Programming (QP). Outline of the strategy:. Obtain an initial estimation of the solution of the min-max problem.. Obtain a quadratic function that bounds the worst case cost.. Compute some parameters denoted α k for the initial estimation.. Use a diagonalization procedure to obtain the quadratic function.. Compute the control signal as the minimizer of the quadratic function. QP based strategy Robust stability guaranteed, but requires semi-feedback for non open-loop stable sytems: The control law of the original MMMPC original is computed by solving at each step t: Includes a terminal cost Terminal region Stability conditions imposed on Terminal Cost and Region Stabilizing design Mayne, Conditions on the terminal region Stabilizing design Admissible robust invariant set QP problems based strategy Step : Initial estimation of the solution of the min-max problem The maximum cost can be computed as: Maximal robust invariant set computed for the systema controlled by: An easily computable (although not very good) upper bound is : If there exists a solution for x(k) Then exists solution for x(k+) Guarantees robust feasibility Condition on the terminal cost Convergence to the origin (Optimal cost Lyapunov function) Initial estimation QP equivalent Slack variables Step : Computing the quadratic function Step : Computing the quadratic function Step.: Compute the parameters α k for the initial estimation Taking into account that if it is assumed ε=: Step.: Compute a matrix denoted by Use the diagonalization algorithm of the nonlinear bound over to obtain the α k : The resulting diagonal matrix, denoted by σ u (v) verifies:

12 Let: Step : Computing the signal control The control signal will be computed solving: Stability The proposed control law guarantees the closed loop robust stability. Outline of the proof: Let J(x(t)) be the optimal cost of the original min-max problem. It can be proved that using the proposed control law there exists γ > such that: Thisimpliesthat: The suboptimality is bounded: That is, from any x(t) the state evolves into: It can escape Stability MMMPC using Open-Loop Predictions ThestateisalwayssteeredintoΦ ε, but it can be escape from it. The set to which the state can evolve from Φ ε is: Robust Model Predictive Control Explicit implementation Implementation using a nonlinear bound Implementation using a QP based bound Applications and Examples A simulated example A simulated example Two tank process: Ogunnaike y Ray, Tanks sections: m and m. a = a =. m min -. % of the liquid is pumped back to tank. h o =a h F F Controller based on the nonlinear bound ε =. N=, Nu=, vertices niveles caudales R h o =a h F muestras

13 Computational burden Accuracy of the nonlinear bound Aceleración (media y máx-mín) horizonte de predicción Aceleración (media y máx-mín) horizonte de predicción Desviation from the exact optimal cost and from the optimal cost obtained using the LMI bound. Average, max and min speed-up over the exact MMMPC. Average, max and min speed-up over the minimization of the LMI bound. Accuracy of the nonlinear bound A simulated example coste óptimo coste óptimo Controller based on QP problems ε =. N=, Nu= niveles caudales... muestras Optimal cost with the nonlinear bound and the LMI bound for a simulation (N u =,N=). muestras Optimal cost with the nonlinear bound and the exact MMMPC for a simulation. desviación muestras A simulated example A simulated example costes óptimos muestras costes óptimos desviación (%) muestras Aceleración (media y máx-mín) Optimal cost of the exact MMMPC and the QP based controller (Nu=N=). Optimal cost of the exact MMMPC when it is used both the exact solution and that of the QP based controller, that is horizonte de predicción Average, max and min speed-up for the QP based controller over the exact MMMPC.

14 u Application to a pilot plant Application to a pilot plant Multilevel Step sequence Model by Santos et al., TT o C.. error V % -. Tank with a.kw heater. Tiempo (minutos) -. Tiempo (minutos) Recirculation through a heat exchanger. Industrial instrumentation. Connected to Simatic-IT. It emulates a CSTR using the heater to simulate the heat of an exothermic reaction. Intercambiador de Calor Identification of a first order model by Least Squares Sampling time: seg. Deadtime rounded up to. Smith Predictor-like correction. Normey y Camacho, Controller based on the nonlinear bound ε =. N=, Nu= Application to a pilot plant TT o C V % C A Tiempo (minutos)... Tiempo (minutos) C A TT o C V % Application to a pilot plant.... Tiempo (minutos) Disturbances in the input (%) C A TT o C V %..... Tiempo (minutos) Disturbances in the inlet flow (%) Application to a pilot plant Application to a pilot plant Controller based on QP problems ε =. N=, Nu= T V % C A tiempo... Tiempo (minutos) C A T v. Tiempo (minutos).... Tiempo (minutos) Disturbances in the inlet flow (%) T v MMMPC NL MPC MMMPC QP ref MMMPC NL MPC MMMPC QP tiempo Comparison for a set point tracking experiment using: MPC. MMMPC based on the nonlinear bound. MMMPC based on QP problems.

15 Feedback PT- System Identification Heat In Scaled laboratory process nd order system Fast dynam ics T s =. s. Heat Stored (Explicit solution) Heat Out CARIMA ARX. e k.. e k ΔA(z )y B(z k = z d B(z)u k d )Δu k d + θ k One step identification error for ARX model One step identification error for CARIMA model. Least squares identification method Ts =. Delay = Random binary input Identification set/ Validation set Best fitting Error bounding CARIMA model System Identification Delay Compensation Best model: Smit h Predict or St rat egy: A(z ) =.z.z B(z ) =.+.z +.z ŷ k+d = +d k +( k d ) Delay = error bound =. u k PT - z -- d,k-d + - CARIMA ARX ΔA(z )y B(z k = z d B(z)u k d )Δu k d + θ k G(z) + d,k + + ŷ k+d Reference Traking Explicit control law In order t o evalut at e reference t raking, t he st at e vector is augmented: (Bemporad et.al. ) h i T z k = ŷ k+d ŷ k+d Δu k Δu k Δu k r k ( r k ) = z T k Qz k Fixed reference over the predicition horizon Q uadratic crit erion IF L = h Lz k THEN Δu k = i T h i z k

16 Results of the MMMPC MMMPC with Linear Feedback N=, λ =, α=. N=, λ =., α=. u k MMMPC N=, MMMPC λ =, α =. N=, λ =., α=. N=, λ =, α=. u k Different controllers W ell tuned MM M PC Agresive cont rol law Open loop predictions Linear feedback: Δu k = Kz k + v k Linear controller Feedback on the predictions without complexity increase (Bemporad et,al, ) Output Worst case evolution of a system Nominal Linear feedback Time step k MMMPC with Linear Feedback Results of the MMMPC with linear feedback Linear feedback: ì Δu k = Kz k + v k MMMPC with MMMPC linear feedback, with linear N=, feedback λ =, uα = k N=, λ = Different controllers W ell tuned MM M PC N=, λ = v k MMMPC + Δυ k Kz PT - Pre-cont rolled system ( linear model) yk u k N=, λ = N=, λ = Softer control law Feedback in the predictions Comparisons Comparisons MPC MPC Different positions of the inlet throttle from o to o MMMPC MMMPC u k MMMPC MMMPC MMMPC with linear feedback MMMPC with linear feedback MMMPC with linear feedback u k MMMPC with linear feedback Control Input

17 MMMPC using Closed-Loop Predictions Min-Max Model Predictive Control using Closed-Loop Predictions Feedback Min-Max MPC Approximate Multi-parametric Programming Decomposition algorithm Example Open Loop Min-Max MPC Feedback Min Max MPC Feedback Min-Max MPC Feedback Min Max MPC Feedback Min-Max MPC subject to: Hard problem References: subject to: Notion of FEEDBACK (Receding horizon scheme) LMI approaches Multi-parametric approaches Large scale deterministic equivalents (Kothare et.al. Automatica ) (Bemporad et.al. TAC ) (Scockaert et.al. TAC ) How to implement it?? The Closed-Loop Min-Max MPC Clearly an open problem. Suggestions: Approximate Multi-parametric Programming Decomposition Algorithm Dynamic Programming can be used to state the closed-loop min-max MPC. The problem can be expressed as the recursive problem. J * t (x (t)) J t (x (t), u (t)) s.t. R x x (t) + R u u(t) r F (x (t), u (t), Ø (t)) є X (t + ) J t (x (t), u (t) L (x (t), u (t)) + J * t+ (x (t + )) WhereX (t+) is the region where function J * t+ (x (t+) is defined.

18 The Closed-Loop Min-Max MPC () The Closed-Loop Min-Max MPC () x (t+) = A (Ø (t) x (t) + B (Ø (t)) u (t) + E (Ø (t)), with A (Ø (t)) = A o + A i Ø i (t), B (Ø (t)) = B + B i Ø i (t), E (Ø (t)) = E + E i Ø i (t). Where Ø i (t) denotes the ith component of the uncertainty vector Ø (t) є Θ. The stage cost of the objective function defined as: L (x (t+j), u (t+j) Q x (t+j) p + Ru (t+j) p With the terminal cost defined as J * t+n (x (t+n)) Px (t+n) p. The demostration is based on the following. If J (x (t), u) is a convex piecewise affine function (i.e., J (x (t), u) = max i=,.. s {L i u + H i x (t) + k i }), the problem: is equivalent to the following mp-lp problem: The solution is a piece affine function of the state. (Bemporad et. al. ) The Closed-Loop Min-Max MPC (). If J (u, x (t),θ) and g (u, x (t), θ) are convex functions in θ for all (u, x (t)) with θ єθ, where Θ is a polyhedron with vertices θ i (i=,, N Ø ) then the problem The Closed-Loop Min-Max MPC (). If J (u, x (t), θ) is convex and piecewise in x (t) u (i.e. J (u, x (t), θ) = L i (θ) u + H i (θ) x (t) + k i (θ)). and g (u, x (t), θ) is affine in x (t) and u (i.e. g (u, x (t), θ) = L g (θ) u + H g (θ) x (t) + k g (θ)). with L i, H i, k i, L g, H g, k g convex functions,. Then the min-max Problem is equivalent to the problem: is equivalent to the problem: The Closed-Loop Min-Max MPC () Let us now consider the first step of the dynamic programming problem (.) with L (x, u ) Qx (t +j) p + Ru (t+j) p The terminal cost The Closed-Loop Min-Max MPC () Appying the previous results, this is equivalent to the following mp-lp problem: and the linear system

19 The Closed-Loop Min-Max MPC () J * N- is the solution of the mp-lp problem which results in a piecewise affine function of x (N-). The corresponding control signal u * (N-) is also a continuours piecewise affine function of the state. The feasible set X (N-) is a convex polyhedron. The same recursively for j= N-, N-,, We can conclude that u * (t) is a piecewise affine function of state x (t). Notice that the argument does not hold when the objective funcion is quadratic as funtion (.) would not be piecewise affine convex with respect to the maximization variables θ (N-) Fast Implementation of MPC and Dead Time Considerations The number of regions grows exponentially with the horizon, dimension of state vector andnumberofverticesofuncertain polytope. The state vector dimension which can be very high for processes with long dead times. (as can be found frequently in industry) In the case of a dead time of d sampling instants, an augmented state vector x a (t) = [x (t) t u (t-) T u(t-d) T ]. Fast Implementation of MPC and Dead Time Considerations () MMMPC using Closed-Loop Predictions To overcome these problems. Use the predicted state to compute instead of Consider a process modelled by the reaction curve method with a dead time equal to its time constant. If the sampling time chosen is one-tenth of the time constant, then dim (x a (t)) = while dim (x (t))= Feedback Min-Max MPC Approximate Multi-parametric Programming Decomposition algorithm Example Multi-parametric Convex Programming Feedback MPC can be often solved by convex optimization: Multi-parametric Convex Programming Properties: piecewise affine (sub-)optimal solution Convex objective function Convex constraints z=optimization variables, x=parameters constraint satisfaction (sub-)optimality with guaranteed error bound Approximate multiparametric convex programming solver: A. Bemporad and C. Filippi, Approximate Multiparametric Convex Programming, CDC tree structure solution

20 Multi-parametric Convex Programming Application to: M.V.Kothare, V. Balakrishna, M.Morari Robust Constrained Model Predictive Control using Linear Matrix Inequalities, Automatica, Linear uncertain systems subject to constraints on (x,u) Multi-parametric Convex Programming Kothare s controller properties: The control law robustly regulates the system to the origin while assuring robust constraint satisfaction. Objective function Approximate control law: (Kothare et.al. Automatica ) The control law ultimately bounds the system in a region that contains the origin while assuring robust constraint satisfaction. Linear control law The gain matrix F is computed at each time step k Very efficient implementation (Muñoz de la Peña et.al.tac ) MMMPC using Closed-Loop Predictions Decomposition Algorithm Feedback Min-Max MPC Feedback Min-Max MPC Approximate Multi-parametric Programming Decomposition algorithm Example subject to: Linear systems with bounded uncertainties Additive uncertainties Polytopic uncertainties Decomposition Algorithm Decomposition Algorithm Cost functions based on a LP problem: Due to convexity of the prediction model and the cost function: Recursive application of mplp solvers Explicit form control laws: (Bemporad et.al. TAC ) Only extreme realizations have to be taken into account (finite dimensional problem) Time (Sckocaert et.al. TAC ) Root w= w=- w= w=- w= Cost functions based on a LP problem: Large scale linear problem (deterministic) (Kerrigan et.al. IJRNC ) Solve on-line large LP Complexity depends on state dimension w=- Multi-parametric approach Cost functions based on a QP problem: No similar result w(t)=(-,) Nodes of the tree grow exponentially with N Cost functions based on a QP problem: No similar result

21 Decomposition Algorithm Decomposition Algorithm Cost functions based on a LP problem: Equivalent problem. Multi-Stage Min Max Linear Problem. Main contribution It allows on-line implementation of feedback min-max MPC. Algorithm that solves (Muñoz de la Peña et.al. CDC ) (For a broader family of systems) Time z j w= Each node: Cost-to-go function V * j Set of variables z j Depends on uncertainty w= Root w=- w(t)=(-,) w=- w= w=- Based on the ideas introduced by Benders in MILP problems, LP problems, SP problems (Benders, NM ) Decomposition Algorithm Decomposition Algorithm Cost functions based on a QP problem: Equivalent problem. Multi-Stage Min Max Quadratic Problem. Time z j w= z i k = k = k = Allows implementation of feedback min max MPC with quadratic cost criterion. Root w= w=- w=- w= Function is PWA Algorithm converges to optimum Approximate solution w(t)=(-,) w=- Decomposition Algorithm MMMPC using Closed-Loop Predictions Approximation error z i k = k = k = Feedback Min-Max MPC Approximate Multi-parametric Programming Decomposition algorithm Example Function is PWQ Bound on the error is a parameter

22 Numerical Examples Numerical Examples Quadruple tank process Sampling time = s. Model of a real plant Computation times: It can be implemented up to N= More than nodes Only algorithm available in the literature. n x = Problem: Complexity still grows exponentially with N. Good performance... N= Proposed solution: RECOURSE HORIZON Still an open problem Finally, some papers D.R. Ramírez and E.F. Camacho, ON THE PIECEWISE NATURE OF MIN-MAX MODEL PREDICTIVE CONTROL WITH BOUNDED UNCERTAINTIES, IEEE CDC. D.R. Ramírez and E.F. Camacho, PIECEWISE AFFINITY OF MIN-MAX MPC WITH BOUNDED ADDITIVE UNCERTAINTIES AND A QUADRATIC CRITERION, Automatica,. D. Muñoz de la Peña, T. Álamo, D.R. Ramírez y E.F. Camacho, MIN-MAX MODEL PREDICTIVE CONTROL BASED ON A QUADRATIC PROGRAM, IET Proc. Control Theory and Applications,. T. Álamo, D.R. Ramírez y E.F. Camacho, EFFICIENT IMPLEMENTATION OF CONSTRAINED MIN-MAX MODEL PREDICTIVE CONTROL WITH BOUNDED UNCERTAINTIES: A VERTEX REJECTION APPROACH, Journal of Process Control,. D.R. Ramírez, T. Álamo, E.F. Camacho y D. Muñoz de la Peña, MIN-MAX MPC BASED ON A COMPUTATIONALLY EFFICIENT UPPER BOUND OF THE WORST CASE COST, Journal of Process Control,. T. Álamo, D.R. Ramírez, T. Álamo, D. Muñoz de la Peña y E.F. Camacho, MIN-MAX MPC USING A TRACTABLE QP PROBLEM, Automatica,. D. Muñoz de la Peña, A. Bemporad, and C. Filippi, ROBUST EXPLICIT MPC BASED ON APPROXIMATE MULTI- PARAMETRIC CONVEX PROGRAMMING, IEEE Transactions on Automatic Control,. David Muñoz de la Peña Sequedo, Teodoro Alamo Cantarero, A. Bemporad, Eduardo Fernández Camacho, A Decomposition Algorithm for Feedback Min-Max Model Predictive Control, IEEE Transactions on Automatic Control,.