Efficient implementation of Constrained Min-Max Model Predictive Control with Bounded Uncertainties

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1 Efficient implementation of Constrained Min-Max Model Predictive Control with Bounded Uncertainties D.R. Ramírez 1, T. Álamo and E.F. Camacho2 Departamento de Ingeniería de Sistemas y Automática, Universidad de Sevilla Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n 4192 Sevilla, SPAIN Phone: , Fax: , {danirr,alamo,eduardo}@cartuja.us.es Abstract Min-Max Model Predictive Control MMMPC is one of the strategies used to control plants subject to bounded additive uncertainties. The implementation of MMMPC suffers a large computational burden, especially when hard constraints are taken into account, due to the complex numerical optimization problem that has to be solved at every sampling time. This paper shows how to overcome this by transforming the original problem into a reduced min-max problem whose solution is much simpler. In this way, the range of processes to which MMMPC can be applied is considerably broadened. Proofs based on the properties of the cost function and a simulation example are given in the paper. 1 Introduction Mathematical models, especially control models which have to be kept simple, can only describe the dynamics of a process in an approximate way. There are different approaches for modelling uncertainties mainly depending on the type of technique used for designing the controllers. The approach considered here is that of global uncertainties Camacho and Bordóns, In this way, uncertainties will be considered to affect the 1-step ahead prediction equation, i.e. the uncertainties will affect the prediction capability of the model. Min-max control techniques have a great computational burden in common Lee and Yu, 1997; Scokaert and Mayne, 1998 which limits the range of processes to which they can be applied. Furthermore, the computational burden is even greater when hard constraints are taken into account. For fast dynamics the min-max problem cannot be solved numerically, and approximate solutions have to be used Ramírez et al., 21. However, these techniques impose great rigidity in the 1 Corresponding author. 2 The authors acknowledge MCYT-Spain for funding this work contract QUI C2-1. controller parameters, as well as a certain degree of approximation error. Recently, the MMMPC control law, traditionally regarded as highly nonlinear, has proven to be piecewise affine when a quadratic Ramírez and Camacho, 21 or 1-norm based criterion Bemporad et al., 21 is used as the cost function. With these results, together with those obtained when multiparametric mathematical programming is applied Bemporad et al., 21, explicit forms of the control law can be built. However, the number of regions in which the state space has to be partitioned grows with the prediction horizon as in a combinatorial explosion. Thus, storage requirements and searching time for the appropiate region can be very high for practical values of the prediction and control horizons. Furthermore, if the process model changes, the computation of the regions has to be done again. This paper shows a way of implementing Constrained MMMPC that requires only a fraction of the time required by the usual min-max solvers. The method is based on transforming the original min-max problem into a reduced min-max problem whose solution is much simpler. Thus, for many processes in which time constants are measured in seconds or minutes, the reduced min-max problem can be solved on line using standard numerical algorithms. In this paper it is presented a way to obtain a reduced min-max problem using a candidate solution e.g. the solution of the min-max problem solved in the previous sampling time. This reduced problem can be solved in a fraction of the time required by the original problem. The paper is organized as follows: section 2 presents the basic Constrained Min-Max MPC with bounded global uncertainties algorithm. Some useful properties and definitions are introduced in section 3. The strategy to obtain the reduced min-max problem is presented in section 4 and illustrated in section 5. Finally section 6 presents conclusions and questions to be addressed in future works.

2 2 Min-Max MPC with bounded global uncertainties The objective of MPC control is to compute the future control sequence ut, ut + 1,..., ut + N 1 in such a way that the optimal j-step ahead predictions yt + j t are driven close to the set point sequence wt, wt + 1,..., wt + N 1 for the prediction horizon. The way in which the system will approach the desired trajectories will be indicated by a function J which depends on present and future control signals and uncertainties. When bounded uncertainties are considered explicitly, it would seem that more robust control is obtained if the controller has tried to minimize the objective function for the worst situation. Furthermore, if hard constraints have to be taken into account the control sequence will be computed solving the following min-max problem: ux = arg min max u U θ Θ Jθ, u, x 1 where θ represents the sequence of future uncertainties, x the process state and U usually is a convex set which can be described as { u : Au b} and Θ = { θ R N2 /θ θ θ}. Note that a polytopic terminal constraint devised to provide robust stability can also be included within the constraints Au b see Kerrigan 2 and references therein. Furthermore, as it is shown in section 5, the effect of the uncertainties can be easily taken into account in the output constraints. Thus, it is ensured that for all uncertainties values within the bounds, the constraints would be satisfied. The cost function of the resulting min-max problem can be written as: with: J s x = min max u U θ Θ Jθ, u, x = min u U J u, x J u, x = max Jθ, u, x 2 θ Θ The function to be minimized J u, x is the maximum of the quadratic objective function Jθ, u, x which measures how well the process output follows the reference trajectories. N 2 N u Jθ, u, x = y k+j k w k+j 2 + λ u k+j 1 2 j=n 1 j=1 where: = 1 z 1, N 1 and N 2 define the beginning and end of the cost horizon, N u is the control horizon and y k+j k is the output prediction. When a global uncertainties approach is used, the way to model the uncertainties is to assume that all modelling errors are globalized in a vector of parameters, so that the plant can be described by the following family of models: ŷt+1 = ˆfyt,..., yt n a, ut,..., ut n b +θt In this work the prediction model used is a CARIMA model with integrated uncertainties: Az 1 yt = z d Bz 1 ut 1 + θt with θt Θ. Notice that the only assumption made here is that θt is bounded. It can be a function of past inputs and outputs. 3 Properties and definitions Several properties as well as some definitions will be presented in this section. When the prediction model is linear, the cost function takes the form of a quadratic criterion such as 2 and λ >, then the objective function J is convex on θ and u. This implies that the maximum of J will be attained at one of the 2 N2 vertexes of the polytope Θ Bazaraa and Shetty, In the following, J i u, x denotes Jθ i, u, x, where θ i is the uncertainty extreme realization related to vertex i. Also, when U is convex, any local minimizer of J will be the global minimizer Bazaraa and Shetty, Consider the set of j-ahead optimal predictions for j = 1,..., N 2, which can be written in condensed form as Camacho and Bordóns, 1999: y = G u u + G θ θ + F x x where y = [y k+1 y k+n2 ] T, θ = [θ k+1 θ k+n2 ] T, u = [ u k u k+nu 1] T and x represents the process state in terms of present and past outputs and past inputs. Without loss of generality, consider a constant set point of w k+j = for j = 1,..., N 2. Then, function J i u, x becomes: J i u, x = G u u + G θ θ i + F x x G u u + G θ θ i + F x x which yields: +λu T u J i u, x = u T Mu + 2Nx + n i u + x T Cx + 2d T i x + h i with M = G T u G u + λi, C = F T x F x, N = G T u F x, n i = G T u G θ θ i, d i = F T x G θ θ i, h i = θ T i GT θ G θθ i. Note that for positives values of λ, M = M T >. The solution of the min-max problem can occur on the constrained minima of one of the quadratic curves or on the constrained minimizer of the intersection of several curves. This situation is illustrated in figure 1. Consider the set of active vertexes, defined as follows:

3 J * J * J 1 J 2 J 1 J 2 min max a J * b J * a b Figure 2: Two min-max problems with the same solution a Full min-max with all curves b Reduced min-max with only curves related to active vertex intersection J 1 J 2 min max c J 1 J 2 d Figure 1: Possible locations of the solution of constrained min-max problem for two quadratic functions: a curve minima b as in a but attained on an active constraint c curves intersection d as in c but attained on an active constraint. Definition 1 Ix, the set of active vertexes is given by: Ix = {i : J i ux, x = J s x} 3 It is easy to see that the solution of 1 is also the solution of a reduced min-max problem given by: ux = arg min u U max i Ix J iu, x 4 This situation is illustrated in figure 2. As the number of active vertexes is generally much lower than the total number of vertexes, problem 4 can be solved with little computational burden. Property 1 For a given process state x and u c + u c, it can easily be seen that for any quadratic curve J i it is true that: J i u c + u c, x = J i u c, x + u T c M u c + 2b T i u c with b i = Mu c + Nx + n i. to be solved at sampling time t could be replaced by an equivalent reduced min-max problem. The necessary steps to compute such estimation are given by: First, it is necessary to obtain a feasible candidate solution u c, which should be close to the optimal solution. In order to fulfill this requirement a number of procedures could be followed. Here it is suggested to use the optimal solution found in the previous sampling time, uxt 1. If uxt 1 is unfeasible at sampling time t the feasibility requirement can be fulfilled solving the following QP problem: ξ = arg min ξ 2 5 s.t. Aξ b Auxt 1 where A and b are the constraints matrix and vector of the min-max problem. Then, the candidate solution would be u c = uxt 1 + ξ. Alternatively, in 5 a shifted version of uxt 1 can be used, as it could be a better approximation for uxt because of the receding horizon strategy. Compute a tight ellipsoidal bounding for u c, where u c + u c is the optimal solution. A procedure to compute such ellipsoidal bounding is given in section 4.1. Reject as many vertexes as possible using the criterion given in section 4.2. Use the remaining vertexes to form I e x, which is a conservative estimation of Ix. 4 Online estimation of Ix This section shows how to build an estimation of the set of active vertexes for a given process state by means of a suboptimal candidate solution. With this estimation of Ix, which is denoted by I e x, the min-max problem 4.1 Ellipsoidal Bounding In the following, a procedure to find an ellipsoidal bounding of u c is given. First, for a given process state x = xt choose the following set: I = Ixt 1 {n vertexes with higher J i u c, x} 6

4 where n is a design parameter closely related to the degree of conservatism of the bounding of u c. Due to the optimality of u c + u c it is true that: J i u c + u c, x J u c + u c, x J u c, x i I Let λ i, λ i = 1, then it holds that: λ i J i u c + u c, x J u c, x 7 Applying property 1 on the left side of 7: u T c M u c + 2 λ i b T i u c + λ i J i u c, x J u c, x 8 On the other hand, the constraints can be written as: a T mu b r m m = 1,, n r where n r is the number of constraints in problem 1. The optimal solution is a feasible one, thus it is true that: a T mu c + u c b r m Let β m be scalars. It can easily be seen that: 2 n r β m a T m u c + u c b r m 9 Furthermore, adding inequalities 8 and 9, it is also true that: u T c M u c + 2 λ i b T i u c + λ i J i u c, x J u c, x + 2 n r which can be expressed as: u T c M u c + 2 λ i b i + + n r λ i J i u c, x J u c, x+2 β m a T m u c + u c b r m 1 n r u c β m a T mu c b r m This inequality describes an ellipsoid which can be rewritten as: where: A = u c a A 1 u c a 1 11 λ i b i + n r n r 2 n r M 1 λ i b i + + J u c, x λ i J i u c, x β m a T mu c b r m M 1 a = M 1 λ i b i + n r Ellipsoid 11 bounds u c, and its size is related to some extent to λ i, β m. Thus, it makes sense to find such λ i, β m which makes 11 as small as possible. This can be achieved solving the following QP problem: min s.a. λ i b i + + n r n r n r M 1 λ i b i 12 λ i J i u c, x 2 β m a T mu c b r m λ i i I β m m = 1,, n r λ i = 1 The optimal values of λ i, β m allow to define the least conservative ellipsoidal bounding of u c, which will be used in the exclusion criterion given in section 4.2. Finally it is noteworthy that the ellipsoid may not completely lay within the feasible space of problem 1. However, this is not a problem, it just leads to greater conservatism in the estimation of Ix. 4.2 Exclusion Criterion In this section an exclusion criterion devised to reject the majority of vertexes not in Ix is presented. Suppose that k Ix, then from the min-max concept: J i u c + u c, x J k u c + u c, x i 13 Let γ i be scalars satisfying γ i and γ i = 1, where i I. Then it is true that: γ i J i u c + u c, x J k u c + u c, x 14 Applying property 1: γ i J i u c, x J k u c, x 2 Furthermore, it is true that: b k γ i b i u c 15 γ i J i u c, x J k u c, x 2b T k u c + max 2γ i b T i u c 16 u c E where E is the ellipsoidal bounding of u c, computed as described in section 4.1. This bounding can be expressed as: E = { u c : u c a A 1 u c a 1} 17

5 The maximization problem in 16 can be solved analytically, thus it can be rewritten as: γ i J i u c, x J k u c, x 2b T k u c 18 i I +2 γ i b i A 1 γ i b i 2 γ i b T i a Furthermore, grouping terms it yields: Ji u c, x + 2b T i a 19 γ i 2 γ i b i A 1 γ i b i J k u c, x + 2b T k u c Inequality 19 describes a condition that must be fulfilled by all vertexes k Ix and any valid γ i, including those γ i that maximize the left side of 19. On the other hand, it can easily be seen that if a given vertex k does not satisfy 19 then k Ix. Thus, this fact can be exploited to reject many of the vertexes which are not active for x. These values of γ i that maximize the left side of 19 can be found solving: min 2 γ i b i A 1 γ i b i γ i J i u c, x γ i s.t. +2b T i a 2 γ i γ i = 1 i I The computed values of γ i together with those λ i and β i found when computing the ellipsoidal bounding can be used to reject non active vertexes. Taking into account 15 an online estimation of Ix can be obtained rejecting any vertex k that fulfills: γ i J i u c, x J k u c, x > max u c E 2 which can be expressed as: b k γ i b i u c γ i J i u c, x J k u c, x > 2 b k γ i b i a b k i I γ i b i A b 1 k i I γ i b i The set I e x, i.e. the conservative estimation of Ix, is then built from all vertexes not rejected with 21. Finally it is noteworthy that computing the ellipsoidal bounding as described in section 4.1 and the values of γ i obtained solving 2 requires solving a QP problem and a NLP problem. It may be argued that solving these problems can be computationally intensive, but it has to be done only once per sampling time. Furthermore, the percentage of rejected vertexes is so high that for the usual values of N 2 the whole procedure i.e. solving the QP and NLP problems, testing all vertexes and solving the reduced min-max problem takes only a fraction of the time required to solve the complete min-max problem. 5 An illustrative example The results obtained in section 4 will be illustrated with a simulation example. This example uses a first order prediction model, which is one of the more common choices for a wide class of industrial processes. The integrated uncertainties prediction model used is given by: y k+1 = 1.948y k.948y k u k +θ k+1 22 The controller parameters are: N u = 1, N 1 = 1, N 2 = 1, λ = 1 and the expected uncertainty values are.5 θt.5. Constraints on process output, control signal and control moves have been taken into account. Their values are:.5 y 1.5, u 8 and.75 u k.75. The constraints can easily expressed as Au b with: A = I N N I N N T T G u G u b = 1u 1u 1U 1ut 1 1U + 1ut 1 1y max i 1y + min i G θ θ i F x x G θ θ i + F x x The reference is set to 1, and the initial conditions are y k = y k 1 =. The min-max problem has been solved using standard optimization software. Figure 3 shows the process output, control signal and control moves. In the simulation a random disturbance has been added to the plant output to simulate more realistic conditions. On the other hand, at sampling time t = 61 an stationary step disturbance suddenly hits the process output, causing the great deviation from the set point seen in the output plot. This step disturbance disappears at t = 11. It can be seen in figure 3 that control signal and control moves become saturated when the output is far from the set point. The number of vertexes included in I e x at each sampling time is shown in figure 4. In this example, the parameter n in 6 is set to 5. It can be seen that most of the 124 vertexes are rejected every sampling time.

6 In fact, on the average per cent of the vertexes are rejected. In the set up used here which takes advantage of the matrix operations provided by Matlab the reduced min-max problem, together with the computations required to construct I e x, can be solved on the average 1 times faster than the original problem. It has to be expected that in another programming environment without built-in matrix functions this speedup of the computations would be much greater. vertexes in I e x samples y k.6 Figure 4: Number of vertexes included in I e x samples a However, many open questions remain to be addressed. It is worth investigating how to extend this approach to other formulations of Constrained Min-Max MPC such as closed loop strategies Scokaert and Mayne, 1998; Bemporad et al., 21. Also, techniques such as branch and bound could be applied to obtain nearly optimal solutions to the reduced min-max problem with lesser computational burden. u k u k samples b Figure 3: Simulation example: a Process output b control signal and control moves 6 Conclusions An efficient implementation of the Constrained MMMPC control law has been presented. The results presented in this paper broaden the range of processes to which, in practice, MMMPC can be applied. The strategy proposed in this work ensures much lower computational burden in most cases. This is accomplished by reducing the number of uncertainty extreme realizations to be considered, from 2 N2 to only a small fraction of them. References Bazaraa, M.S. and C.M. Shetty Nonlinear Programming. Theory and Algorithms. John Wiley & Sons. Bemporad, A., F. Borrelli and M. Morari 21. Robust Model Predictive Control: Piecewise Linear Explicit Solution. In: Proc. European Control Conference, ECC 1. Camacho, E.F. and C. Bordóns Model Predictive Control. Springer-Verlag. Kerrigan, E.C. 2. Robust Constraint Satisfaction: Invariant sets and Predictive Control. PhD thesis. University of Cambridge. Lee, J.H. and Zhenghong Yu Worst-case formulations of model predictive control for systems with bounded parameters. Automatica 335, Ramírez, D.R. and E.F. Camacho 21. On the piecewise linear nature of Min-Max Model Predictive Control with bounded uncertainties. In: Proc. 4th Conference on Decision and Control, CDC 21. Ramírez, D.R., M.R. Arahal and E.F. Camacho 21. Neural network based Min-Max MPC. Application to a heat exchanger. In: IFAC Workshop on Adaptation and Learning in Control and Signal Processing, ALCOSP 21. pp Scokaert, P.O.M. and D.Q. Mayne Min-max feedback model predictive control for constrained linear systems. IEEE Transactions on Automatic Control 438,