Computation, Approximation and Stability of Explicit FeedbackMin-MaxNonlinearModelPredictiveControl

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1 Computation, Approximation and Stability of Explicit FeedbackMin-MaxNonlinearModelPredictiveControl Alexandra Grancharova a, Tor A. Johansen b a Institute of Control and System Research, Bulgarian Academy of Sciences, Sofia, Bulgaria b Department of Engineering Cybernetics, Norwegian University of Science and Technology, 749 Trondheim, Norway Abstract This paper presents an approximate multi-parametric Nonlinear Programming (mp-nlp) approach to explicit solution of feedback min-max NMPC problems for constrained nonlinear systems in the presence of bounded disturbances and/or parameter uncertainties. It is based on an orthogonal search tree structure of the state space partition and consists in constructing a piecewise nonlinear (PWNL) approximation to the optimal sequence of feedback control policies. Conditions guaranteeing the robust stability of the closed-loop system in terms of a finite l -gain are derived. Key words: Min-max model predictive control; Multi-parametric programming; l -stability. Introduction Nonlinear Model Predictive Control (NMPC) involves the solution at each sampling instant of a finite horizon optimal control problem subject to nonlinear system dynamics and state and input constraints (Keerthi & Gilbert, 988; Mayne & Michalska, 99). A novel approach for NMPC design for constrained input-affine nonlinear systems is suggested in (Bacic, Cannon, & Kouvaritakis, 3), which deploys state space partitioning and graph theory to retain the on-line computational efficiency. In (Mhaskar, El-Farra, & Christofides, 6), a Lyapunov-based NMPC design for input-affine nonlinear systems is proposed that guarantees stabilization and state and input constraints satisfaction from an explicitly characterized set of initial conditions. The benefits of an explicit solution, in addition to the efficient on-line computations, include also verifiability of the implementation, which is an essential issue in safety-critical applications. In (Johansen, ; Johansen, 4; Grancharova, Johansen, & Tøndel, 7), approximate approaches to explicit piecewise linear (PWL) NMPC solutions for general constrained nonlinear systems, based on multi-parametric Nonlin- This paper was not presented at any IFAC meeting. Corresponding author T. A. Johansen. Tel Fax addresses: alexandra.grancharova@abv.bg, Tor.Arne.Johansen@itk.ntnu.no. ear Programming (mp-nlp) ideas (Fiacco, 983), have been developed. Notice that the off-line computational complexity of the explicit approaches tends to increase rapidly with the number of states. Approximate explicit NMPC solutions can also be computed using dynamic programming approximations (Bertsekas and Tsitsiklis, 998; Lincoln and Rantzer, 6). Since models are only an approximation of the real process, it is important for NMPC to be robust with respect to model uncertainties and disturbances. One approach to robust NMPC design is to optimize the nominal performance while guaranteeing robust feasibility and robust stability of the closed-loop system. Thus in (Mhaskar, 6), a Lyapunov-based robust NMPC design for input-affine nonlinear systems subject to uncertainty and input constraints is developed, which allows for an explicit characterization of the closedloop stability region. Another robust NMPC strategy consists of solving a min-max problem to optimize the robust performance while enforcing the state and input constraints for all possible uncertainties. The min-max robust MPC was first proposed in (Campo & Morari, 987). There are two formulations of min-max NMPC: the open-loop and the closed-loop formulation (see (Magni & Scattolini, 7) for review of the min-max NMPC approaches). The open-loop min-max NMPC (Michalska & Mayne, 993; Limon, Alamo, & Camacho, ; Magni & Scattolini, 7) guarantees the robust stability and the robust feasibility of the system, but it may be very conservative since the control sequence Preprint submitted to Automatica 6 December 8

2 has to ensure constraints fulfilment for all possible uncertainty scenarios without considering the fact that future measurements of the state contain information about past uncertainty values. As a result, the open-loop min-max NMPC controllers may have a small feasible set and a poor performance. An approximate mp-nlp approach to explicit solution of open-loop min-max NMPC problems has been suggested in (Grancharova & Johansen, 5). The conservativeness of the open-loop approaches is overcome by the closed-loop min-max NMPC (Magni, De Nicolao, Scattolini, & Allgöwer, 3; Magni & Scattolini, 7; Limon, Alamo, Salas & Camacho, 6), where the optimization is performed over a sequence of feedback control policies. With the closed-loop approach, the min-max NMPC problem represents a differential game where the controller is the minimizing player and the disturbance is the input of the maximizing player ( the nature ) (Magni et al., 3). The controller chooses the control input as a function of the current state so as to ensure that the effect of the disturbance on the system output is sufficiently small for any choice made by the nature. In this way, the closed-loop min-max NMPC would guarantee a larger feasible set and a higher level of performance compared to the open-loop min-max NMPC (Magni et al., 3). Recently, several approaches have been developed for explicit solution of min-max MPC problems for special classes of uncertain nonlinear systems. Thus, for constrained linear systems with polytopic uncertainty, approaches for explicit solution of the open-loop and the closed-loop min-max MPC problems have been developed, respectively in (Cychowski & O Mahony, 5) and in (Wan & Kothare, 3; Björnberg & Diehl, 6; Muñoz de la Peña, Bemporad, & Filippi, 6). The method in (Bemporad, Borrelli, & Morari, 3) applies to linear systems with polyhedral parametric uncertainty and additive bounded disturbances and both the openloop and the closed-loop min-max control problems are solved explicitly. Approaches for explicit solution of robust finite horizon optimal control problems for constrained piecewise affine systems with bounded disturbances have been proposed, based on an open-loop formulation in (Mukai, Azuma, Kojima, & Fujita, 3), and on a closed-loop formulation in (Kerrigan & Mayne, ; Raković, Kerrigan, & Mayne, 4). Methods for explicit solution of min-max MPC or H problems for constrained linear systems with additive bounded uncertainties are suggested in (Muñoz de la Peña, Alamo, Ramirez & Camacho, 7) for the open-loop formulation, and in (Kerrigan & Maciejowski, 4; Mayne, Raković, Vinter & Kerrigan, 6) for the closed-loop formulation. However, the problem of obtaining an explicit solution of closed-loop min-max NMPC problems for general nonlinear systems with state and input constraints has not been considered until now. This paper considers the feedback min-max NMPC problem for general constrained nonlinear discrete-time systems in the presence of bounded disturbances and/or parameter uncertainties as formulated in (Magni & Scattolini, 7). The present work suggests an approximate mp-nlp approach to explicit solution of the feedback min-max NMPC problem, which consists in constructing a piecewise nonlinear (PWNL) approximation to the optimal sequence of feedback control policies. Conditions guaranteeing the l -stability of the closed-loop system are derived. Mp-NLP problem Consider the discrete-time nonlinear system: x(t + ) = f(x(t),u(t),w(t)) y(t) = h(x(t),u(t),w(t)), () where x(t) R n, u(t) R m, y(t) R s and w(t) R q are the state, input, output and disturbance variable. The following constraints are imposed: u min u(t) u max, y min y(t) y max. () It is assumed that (Magni et al., 3): A. f and h are C functions with f(,,) =, h(,,) =. A. y min < < y max and u min < < u max. A3. Let X be a non-empty set, containing the origin as an interior point and t be a positive integer. The system () is zero-state detectable in X, i.e. x() X and u( ) such that constraints () are satisfied t and x(t) X, t t, we have y(t) w= =, t t lim t x(t) =. A4. Given a positive constant γ, the disturbance w is such that: w(t) γ y(t), t t. (3) Let x(t) = x and u(t) = u. Then, the space of the admissible disturbances is denoted by W A (u,x) R q. As mentioned in (Magni et al., 3), inequality (3) can also represent a wide class of modeling errors. As in (Magni et al., 3), first a H control problem is defined: Definition (H control problem). Design a statefeedback control law: u = k(x) (4) guaranteeing that the closed-loop system () (4) with input w W A (u,x) and output y has a finite l -gain γ in a bounded positively invariant set Ω, that is, x(t) Ω: i. x(t + i) Ω, i >. ii. u min k(x(t + i)) u max and y min h(x(t + i),k(x(t + i)),w(t + i)) y max, i. iii. There exists a positive definite function β(x(t)), such that T : T T y(t + i) γ w(t + i) + β(x(t)) (5) i= i=

3 for any non-zero w W A (u,x). The following assumption is made (Magni et al., 3): A5. Suppose that there exists an auxiliary control law u = k a (x) that solves the H control problem, with a domain of attraction Ω a, whose boundary is assumed to be a level curve of a positive function V ka (x) such that: V ka (f(x,k a (x),w)) V ka (x) ( y γ w ), x Ω a, w W A (u,x) (6) and V ka () =. Definition (Admissible disturbance realization). Let K = {k,k,...,k N } {k (x t t ),k (x t+ t ),..., k N (x t+n t )} be a vector of feedback control policies and N be a finite horizon. Consider the closed-loop system for i =,,,...,N : x t+i+ t = f(x t+i t,k i (x t+i t ),w t+i ) y t+i t = h(x t+i t,k i (x t+i t ),w t+i ) (7) with initial state x t t = x. Then, the disturbance realization W = {w t,...,w t+n } R qn is admissible for the given K and x if the following holds: w t+i γ y t+i t, i =,,,...,N. (8) The space of the admissible disturbance realizations over horizon N and corresponding to the given K and x is denoted by W B (K,x) R qn. It is supposed that a full measurement x of the state is available at the current time t. We consider the feedback min-max NMPC problem (Magni & Scattolini, 7): Definition 3 (Constrained feedback min-max NMPC problem). Suppose that assumptions A A5 hold. For the current x, the feedback min-max NMPC solves the following optimization problem: V o max(x) = min K max W W B (K,x) J(K,x,W) (9) subject to x t t = x and: y min y t+i t y max, i =,...,N () u min u t+i u max, i =,,...,N () x t+n t Ω a () u t+i = k i (x t+i t ), i =,,...,N (3) x t+i+ t = f(x t+i t,u t+i,w t+i ), w t+i W A (u t+i,x t+i t ), i N (4) y t+i t = h(x t+i t,u t+i,w t+i ), w t+i W A (u t+i,x t+i t ), i N (5) and the cost function given by: J(K,x,W) = N [ yt+i t γ w t+i ] i= +V ka (x t+n t ). (6) Here, N is the finite horizon and γ is the l -gain which is interpreted as the disturbance attenuation level. Note that in (9) (6) w t+i denotes a single disturbance at time instant t + i, while W is an admissible disturbance realization as specified in Definition. An auxiliary control law k a (x) is typically obtained by solving the H control problem for the linearized system (Michalska & Mayne, 993). Thus, a practical way to compute a nonlinear control k a (x) satisfying assumption A5 is suggested in (Magni et al., 3). An optimal solution to the feedback min-max NMPC problem (9) (6) is denoted K o = {k o,k o,...,k o N } {k o (x t t ),k o (x t+ t ),...,k o N (x t+n t)} and the control input is chosen according to the receding horizon policy u(x t t ) = k o (x t t ). It is assumed that: A6. Each feedback control policy k i (x t+i t ), i =,,...,N, has the form: k i (x t+i t ) = α i k a (x t+i t ) + r i (ξ i,x t+i t ) = g i (p i,x t+i t ), (7) where p i = [αi T ξi T]T R ni are the parameters that need to be optimized, k a (x t+i t ) is an auxiliary control law that satisfies assumption A5, and r i (ξ i,x t+i t ) is a continuous function with r i (ξ i,) =. In general, the parameterization of the form (7) would lead to an approximate solution to the feedback minmax NMPC problem (9) (6). Denote with P the whole set of parameters that need to be determined, i.e. P = [p T p T... p T N ]T R np, where n p = N n i. Then, the i= worst-case value of cost function (6) with respect to the disturbances is denoted by: V max (P,x) = max J(P,x,W). (8) W W B (P,x) Note that the argument K is now substituted with the argument P. The optimization problem (9) (6) can be formulated in a compact form as follows: Problem P V o max(x) = min P max W W B (P,x) J(P,x,W) (9) subject to G(P,x,W), W W B (P,x). () Problem P defines an mp-nlp, since it is NLP in P parameterized by x. We remark that the constraints function G(P,x,W) in () is implicitly defined by () (5). Define the set of N-step robustly feasible initial states: X f = {x R n G(P,x,W), W W B (P,x) for some P R np }. () If the problem (9) (6) is feasible, then X f is a nonempty set. Then, due to assumption A, the origin is an interior point in X f. 3

4 In parametric programming problems one seeks the solution P o (x) as an explicit function of the parameters x in some set X X f R n (Fiacco, 983). However, in the general case, an exact explicit solution of problem P with the associated shape of the state space partition can not be found. Therefore, it would be necessary to develop methods for approximate explicit solution by preliminary specifying the structure of the partition. In this paper, we suggest practical computational methods for constructing an explicit approximate solution of feedback min-max NMPC problems for general constrained nonlinear systems based on an orthogonal structure of the state space partition. Since the regions in the partition do not overlap (except at the boundary), the approximation corresponds to orthogonal basis-functions that form a complete basis on the space of continuous functions. This ensures an arbitrarily good approximation if the optimal solution is a continuous function. Note that this type of partition does not impose any restrictions on the class of problems that can be solved. 3 Approximate mp-nlp approach to explicit feedback min-max model predictive control The mp-nlp approaches in (Johansen, ; Johansen, 4; Grancharova et al., 7) build explicit suboptimal NMPC solutions assuming that the process model is known exactly. In (Grancharova & Johansen, 5), an mp-nlp approach to explicit approximate solution of open-loop min-max NMPC problems has been suggested. In this paper, an approximate mp-nlp approach to explicit solution of the feedback min-max NMPC problem (Definition 3) is proposed. In contrast to the method in (Grancharova & Johansen, 5) where a sequence of control actions is optimized, here the optimization is performed over a sequence of feedback control policies. Another difference from all previously developed approximate mp-nlp approaches, where a piecewise linear solution is obtained, is that the presented method constructs an explicit approximate solution, which represents a piecewise nonlinear function. 3. Non-convexity and close-to-global solution From a physical insight on the considered system (), it is supposed that the disturbance w can vary in the range: w min w(t) w max, () with known w min, w max. The procedure used to generate a discrete set of admissible disturbance realizations is: Procedure (Generation of discrete set of admissible disturbance realizations). Consider system (), where w(t) [w min ; w max ]. Let N be a finite horizon and K = {k,k,...,k N } be a vector of feedback control policies where each feedback function k i (x), i =,...,N, has the form (7). Suppose that the initial state of the system () is x t t = x and let j max be a positive integer. Then, for a given vector P = [p T p T... p T N ]T of parameters of K, a finite set W (P,x) = {W,W,...,W NW } of admissible disturbance realizations is generated where each realization W s = {w s t,w s t+,...,w s t+i,...,ws t+n }, s =,,...,N W is determined by applying Algorithm. Algorithm. Generation of an admissible disturbance realization. Input: N, P = [p T p T... p T N ]T, x, j max. Output: W s = {wt,w s t+, s...,wt+i s,...,ws t+n }.. Let i =.. while i N do 3. Let flag =, j =. 4. while flag = do 5. Generate value wt+i s [w min; w max ] by using random generator with uniform distribution. 6. j = j if wt+i s γ h(x t+i t,k i (x t+i t ),wt+i s ) then 8. Compute x t+i+ t = f(x t+i t,k i (x t+i t ),wt+i s ). 9. flag =.. else. if j > j max, terminate (an admissible disturbance realization is not found).. end if 3. end while 4. i = i end while In Algorithm, the parameter j max denotes the maximal allowed number of unsuccessful iterations and it is typically chosen to be j max = q, where q is the dimension of w. A special case is the case when the disturbance is of the form w(t) = d T y(t), where d R s is a vector of uncertain parameters with d min d d max. Then, the set of the admissible disturbance realizations can be generated by simulating the closed-loop system response for different values d s [d min ; d max ], s =,,...,N W of d. The procedure used to approximate problem P is: Procedure (Approximation of problem P). Suppose that assumptions A A6 hold. Let P be a given vector of parameters of the sequence K of feedback control policies. Suppose that a finite set W (P,x) = {W,W,...,W NW } of admissible disturbance realizations has been determined by applying Procedure. An estimate Ṽmax(P,x) of V max (P,x) is computed as follows: Ṽ max (P,x) = max J(P,x,W i). (3) W i W (P,x) Denote with G(P,x) the set of constraints functions: G(P,x) = {G(P,x,W i ), W i W (P,x)}. (4) Then problem P is approximated with the following mp- NLP problem: 4

5 Problem P Ṽmax(x) o = min Ṽ max (P,x) subject to G(P,x). (5) P Thus, we can approximate the infinite number of constraints () with a finite amount of constraints which are function only of P and x. For a given min-max NMPC problem it would be necessary to analyze how the size of the set of admissible disturbance realizations generated with Procedure would effect the worst-case cost function value and the satisfaction of constraints in problem P. It should be expected that with the increase of the number of the generated disturbance sequences, the probability of satisfaction of the constraints in problem P would be higher. On the other hand, this will lead to an increase of the computational efforts related to the design of the explicit MPC controller. Therefore, for every specific min-max NMPC problem, a tradeoff should be made and a reasonable number of admissible disturbance realizations should be determined. Hereafter, let X R n be a hyper-rectangle where we seek an explicit approximate solution of problem P. Problem P can be non-convex with multiple local minima. Therefore, it would be necessary to apply an efficient initialization of problem P so to find a closeto-global solution. One possible way to obtain this is to find a close-to-global solution at a point v X by comparing the local minima corresponding to several initial guesses and then to use this solution as an initial guess at the neighbouring points v i X, i =,,...,N, i.e. to propagate the solution. The following procedure is used to generate a set of points V = {v,v,v,...,v N }. Procedure 3 (Generation of set of points). Consider any hyper-rectangle X X with vertices Λ = {λ,λ,...,λ N λ } and center point v. Consider also the hyper-rectangles X j X, j =,,...,N with vertices respectively Λ j = {λ j,λj,...,λj N λ }, j =,,...,N. Suppose X X... X N. For each of the hyper-rectangles X and X j X, j =,,...,N, denote the set of its facets centers with Φ j = {φ j,φj,...,φj N φ }, j =,,,...,N. Define the set { of all } points { V = {v,v,v,...,v N }, where N v i Λ j N Φ }, j i =,,...,N. j= j= For a hyper-rectangle in the n-dimensional state space, the number of its vertices is N λ = n and the number of its facets centers is N φ = n. The following procedure is applied to find a close-toglobal solution at the points v i V, i =,,,...,N : Procedure 4 (Close-to-global solution of problem P). Consider any hyper-rectangle X X with a set of points V = {v,v,v,...,v N } determined by applying Procedure 3. Then: a). If there exist local minima of problem P at the center point v of X, determine a close-to-global solution of P at v through the following minimization: P (v ) = arg P local i min {P local,...,p local } N P Ṽ max (P local i,v ).(6) Here, Pi local, i =,,...,N P correspond to local minima of the cost function Ṽmax(P,v ) obtained for a number of initial guesses Pi, i =,,...,N P. b). If a solution in step a) has been obtained, determine a close-to-global solution of problem P at the points v i V, i =,,...,N by applying Algorithm. Algorithm. Determination of a close-to-global solution of problem P at the points v i V, i =,,...,N. Input: Hyper-rectangle X with a set of points V = {v,v,v,...,v N }, data to problem P, close-to-global solution P (v ) at the center point v. Output: Close-to-global solution of problem P at the points v i V, i =,,...,N.. Let i =.. while i N do 3. Let V s = {v,v,v,...,v N } V be the subset of points at which a feasible solution of P has been already determined. 4. Find the point ṽ V s that is most close to the point v i, i.e. ṽ = arg min v V s v v i, and for which a solution P (ṽ) has been already obtained. 5. Solve P at the point v i with initial guess for the optimization variables set to P (ṽ). 6. If a solution of P at the point v i has been found, mark v i as feasible and add it to the set V s. Otherwise, mark v i as infeasible. 7. i = i end while 3. Computation of explicit approximate solution We restrict our attention to a hyper-rectangle X R n where we seek to approximate the close-to-global sequence of control policies K = {k,k,...,kn } {k(x t t ),k(x t+ t ),...,kn (x t+n t)}. We require that the state space partition is orthogonal and can be represented as a k d tree (Bentley, 975). The main idea of the approximate mp- NLP approach is to construct a piecewise nonlinear (PWNL) approximation K = { k, k,..., k N } { k (x t t ), k (x t+ t ),..., k N (x t+n t )} to the closeto-global feedback K = {k,k,...,kn } on X. The constituent sequences of nonlinear control policies are denoted with K Xi = { k,xi,..., k N,Xi } { k,xi (x t t ),..., k N,Xi (x t+n t )} and are defined on hyper-rectangles X i covering X. This means that a sequence K Xi is applied for x t t X i. Let 5

6 K X = { k,x,..., kn,x } be an approximation to the close-to-global solution K = {k,...,kn }, valid in X. Denote with P X = [p T,X... p T N,X ] T the parameters of KX. According to assumption A6, ki,x (x t+i t ) = g i (p i,x,x t+i t ), i =,,...,N. Let Ṽ max (P X,x) be the cost function value due to initial state x = x t t and sequence K X of control policies, i.e. Ṽ max (P X,x) = max J(P X W i W,x,W i ). Then, the (P X,x) approximate sequence K X = { k,x,..., k N,X } {g (p,x,x t t ),...,g N (p N,X,x t+n t )}, valid for x t t X, is computed with the following procedure: Procedure 5 (Computation of explicit approximate solution). Suppose that assumptions A A6 hold. Consider any hyper-rectangle X X with a set of points V = {v,v,v,...,v N } determined with Procedure 3. Suppose that a close-to-global solution of problem P at the points v i V, i =,,,...,N has been obtained by applying Procedure 4 and let Ṽ max(v i ), i =,,,...,N be the close-to-global cost function values. Compute the parameters P X = [p T,X... p T N,X ] T of the sequence K X = { k,x,..., k N,X } by solving the NLP: min P X N i= ( Ṽ max (P X,v i ) Ṽ max(v i ) +µ g (p,x,v i ) k (v i ) ) (7) subject to G(P X,v i ), v i V. (8) In (7), the parameter µ > is a weighting coefficient. Note that the sequence K X = { k,x,..., k N,X }, computed with Procedure 5, satisfies the constraints in problem P only for the discrete set of points V X. 3.3 Estimation of error bounds Suppose that the parameters P X of the sequence K X, valid in X, has been computed with Procedure 5. Then, for the cost function approximation error in X we have: ε(x) = Ṽ max (P X,x) Ṽ max(x) ε, x X. (9) The following procedure can be used to obtain an estimate ε of the maximal approximation error ε in X. Procedure 6 (Computation of error bound approximation). Consider any hyper-rectangle X X with a set of points V = {v,v,v,...,v N } determined by applying Procedure 3. Compute an estimate ε of the error bound ε through the following maximization: ε = max ( Ṽ max (P X,v i ) Ṽ max(v i )). (3) i {,,,...,N } The estimate ε represents a probable degree of suboptimality, since it depends on the finite set of admissible disturbance realizations generated with Procedure. 3.4 Approximate mp-nlp algorithm Assume the tolerance ε > of the cost function approximation error is given. Denote with S X the volume of a given hyper-rectangular region X X R n, i.e. S X = n x i, where x i is the size of X along the i= state variable x i. Let S min be the minimal allowed volume of the regions in the partition of X. The following algorithm is proposed to compute the explicit approximate feedback min-max NMPC controller on X: Algorithm 3. Explicit feedback min-max NMPC. Input: Data to problem P, the number N of internal regions (used in Procedure 3), the parameter µ (used in Procedure 5), the approximation tolerance ε. Output: Partition Π = {X,X,...,X NX } and associated PWNL control function K Π = { K X, K X,..., K XNX }.. Initialize the partition to the whole hyper-rectangle, i.e. Π = {X}. Mark the hyper-rectangle X as unexplored, flag :=.. while flag = do 3. while unexplored hyper-rectangles in Π do 4. Select any unexplored hyper-rectangle X Π. 5. Compute a solution to problem P at the center point v of X by applying Procedure 4a. 6. if P has a feasible solution at v then 7. Define a set of points V = {v,v,v,...,v N } by applying Procedure Compute a solution to problem P for x fixed to each of the points v i, i =,,...,N by applying Procedure 4b. 9. if P has a feasible solution at all points v i, i =,,...,N then. if X then. Let K X = k a (x).. If X Ω a, mark X as explored and feasible. Otherwise, mark X to be split. 3. else 4. Compute a sequence K X = { k,x,..., kn,x } of control policies using Procedure 5, as an approximation to be used in X. 5. if a sequence of control policies was found then 6. Compute an estimate ε of the error bound ε in X by applying Procedure If ε > ε, mark the hyper-rectangle X to be split. Otherwise, mark X as explored and feasible. 8. else 6

7 9. Compute the volume S X of the hyperrectangle X. If S X < S min, mark X infeasible and explored. Otherwise, mark X to be split.. end if. end if. else 3. Compute the volume S X of the hyperrectangle X. If S X < S min, mark X infeasible and explored. Otherwise, mark X to be split. 4. end if 5. else 6. Compute the volume S X of the hyperrectangle X. If S X < S min, mark X infeasible and explored. Otherwise, mark X to be split. 7. end if 8. end while 9. flag := 3. if hyper-rectangles in Π that are marked to be split then 3. flag := 3. while hyper-rectangles in Π that are marked to be split do 33. Select any hyper-rectangle X Π marked to be split. 34. Split X into hyper-rectangles X,..., X Ns by applying heuristic splitting rules. Mark X,..., X Ns unexplored, remove X from Π, and add X,..., X Ns to Π. 35. end while 36. end if 37. end while In step 34, the heuristic splitting rules from (Grancharova et al., 7) are applied to partition a given hyper-rectangle X. Thus, if a sequence of control policies valid in X is computed, but the required accuracy is not achieved, then X is split by a hyperplane through its center and orthogonal to that axis where a maximal reduction of the approximation error can be achieved. If there is no feasible solution of problem P at the center point v of X, or the NLP problem (7) (8) is infeasible, then X is split by a hyperplane through its center and orthogonal to an arbitrary axis. If some of the points associated to X are feasible and others are not, then X is split into hyper-rectangles such that some of them will include only feasible points. 4 Stability 4. Computation of approximate region of attraction Let X Π = NX X i, X i Π be the set associated to the i= partition Π obtained with Algorithm 3. Consider the suboptimal closed-loop system: x(t + ) = f(x(t), k (x(t)),w(t)) (3) y(t) = h(x(t), k (x(t)),w(t)), (3) where k (x(t)) = [I... ] K is the approximate PWNL feedback law determined with Algorithm 3 and is defined on the set X Π. The fact that the explicit MPC controller is specified for an initial condition x(t) X Π does not imply that x(t) is within the region of attraction for the system (3) (3). Therefore, the set X Π may not be a domain of attraction for this system. In fact, although a feasible control law exists at state x(t) X Π, the successor state x(t + ) may go out of the set X Π. Moreover, the set X Π may not be convex (see the simulation example in section 5). Therefore, first it would be necessary to find a set Ω X Π, which is an inner convex approximation of the set X Π. Then, a convex set Ω Ω should be determined such that Ω Ω a and for every initial state that belongs to the set Ω, the state trajectory of the system (3) (3) will lie in the set Ω. This is specified in the following definition. Definition 4 (Approximate region of attraction for the suboptimal closed-loop system). Let Π = {X,X,...,X NX }, X Π = NX X i, X i Π and i= K = { k, k,..., k N } be respectively the state space partition, the associated set in the state space and the approximate PWNL sequence of feedback control policies, determined with Algorithm 3. Let P = [ p T p T... p T N ]T be the parameters of K. Assume that XΠ is a nonempty set. Suppose that there exist polyhedral sets Ω and Ω, such that Ω a Ω Ω X Π. Let E Ω = {x j x j Ω, j =,...,N p } denote a finite set of randomly generated points. Let the state of the system (3) (3) at time t is x t t = x j E Ω. Consider a finite set W ( P,x j ) = {W,W...,W NW } of admissible disturbance realizations W s = {wt,w s t+, s...,wt+n s }, s =,...,N W, generated by applying Procedure. Let X s,j = {x s,j t+ t,xs,j t+ t,...,xs,j t+n t } denote the state trajectory of system (3) (3) obtained with K and corresponding to initial state x j E Ω and disturbance realization W s W ( P,x j ), i.e.: x s,j t+i+ t = f(xs,j t+i t, k (x s,j t+i t ),ws t+i) i =,,,...,N, (33) where k (x s,j t+i t ) = [I... ] K. Then, if: X s,j Ω, x j E Ω and W s W ( P,x j ), (34) the set Ω is referred to as an approximate region of attraction for the suboptimal closed-loop system (3) (3). Let S Ω and S Ω denote the volumes of the polyhedral sets Ω and Ω defined as their Lebesgue measures, i.e. S Ω = Ω dx and S Ω = Ω dx. The volume of the set 7

8 X Π is S XΠ = NX X i dx, i.e. it represents the sum of the i= Lebesgue measures of all regions X i Π. Then, the following procedure is applied to compute an approximate region of attraction for the closed-loop system (3) (3): Procedure 7 (Computation of approximate region of attraction for the suboptimal closed-loop system). Let Π = {X,X,...,X NX }, X Π = NX X i, X i Π and i= K = { k, k,..., k N } be respectively the state space partition, the associated set in the state space and the approximate PWNL sequence of feedback control policies, determined with Algorithm 3. Assume the set X Π is non-empty. Suppose that there exist polyhedral sets Ω = {x X Π a i h i x b i, i =,,...,N Ω } and Ω = {x X Π a i h i x b i, i =,,...,N Ω }, such that Ω a Ω Ω X Π. Let E Ω = {x k x k Ω, k =,,...,N p } and E Ω = {x j x j Ω, j =,,...,N p } denote finite sets of randomly generated points. Then, for specified N Ω, N Ω, N p and N p, the approximate region of attraction for the closed-loop system (3) (3) is computed by implementing the following steps:. Determine the following polyhedron Ω = {x X Π a i h i, b h i x b i, i =,,...,N Ω }, where a i, i =,,...,N Ω are computed by solving the optimization problem: {a i,h i,b i,i =,,...,N Ω } = (35) arg min S Ω S XΠ subject to E Ω X Π. a i,h i,b i,i=,...,nω. Determine the approximate region of attraction as the following polyhedron Ω = {x X Π a i h i x b i, i =,,...,N Ω }, where a i, h i, b i, i =,,...,N Ω are computed by solving the optimization problem: {a i,h i,b i,i =,,...,N Ω } = arg min S Ω S Ω (36) a i,h i,b i,i=,...,nω subject to E Ω Ω, Ω a Ω, and condition (34). Problems (35) and (36) are nonlinear programming problems and nonlinear programming techniques (Bazaraa, Sherali & Shetty, 993) can be used to solve them. Further in the paper, the sets Ω and Ω determined with Procedure 7 will be denoted as Ω and Ω. After Procedure 7 is implemented, a partition Π RH = {R,R,...,R NR } is built such that Ω = NR R i. i= Each region R i Π RH represents either a hyperrectangular region, i.e. R i X j or a polyhedral region, i.e. R i = X j Ω, where X j Π. The PWNL function associated to the partition Π RH is defined as K Π RH = { K R, K R,..., K RNR }, where K Ri K Xj, i, K Xj K Π, given that R i X j or R i = X j Ω. As result, we obtain a partition Π RH and an approximate PWNL sequence of feedback control policies K RH = { k RH, k RH,..., k N RH } defined on the set Ω. 4. Stability result This section considers the stability of the closed-loop system: x(t + ) = f(x(t), k RH (x(t)),w(t)) (37) y(t) = h(x(t), k RH (x(t)),w(t)), (38) RH where k (x(t)) = [I... ] K RH is the approximate PWNL feedback law determined with Algorithm 3 and Procedure 7 and is defined on the approximate region of attraction Ω computed with Procedure 7. The following notation is introduced. Let N is the prediction horizon and x t t = x is the initial state of the system (37) (38). For any x Ω, let KN K RH = { k RH (x t t ),..., k N RH (x t+n t)} denote the approximate solution to the optimization problem P. Let X N = {x t+ t,...,x t+n t } and Y N = {y t t,...,y t+n t } denote the state and output trajectories of system (37) (38) obtained with K N and corresponding to a disturbance realization W N = {w t,...,w t+n } W B ( K N,x) (W B ( K N,x) R qn is the set of the admissible disturbance realizations over horizon N). Let V max (x,n) be the worst-case cost function value due to initial state x t t = x and sequence K N, i.e.: V max (x,n) = max J(x, K N,W N,N), (39) W N W B ( K N,x) where J(x, K N,W N,N) = [ y t+i t γ w t+i ]+ N i= V ka (x t+n t ). Consider the sequence K N+ = { k RH (x t t ),..., k N RH (x t+n t),k a (x t+n t )} for the problem P with horizon N +. Then, X N+ = {x t+ t,...,x t+n t, x t+n+ t } and Y N+ = {y t t,...,y t+n t,y t+n t } are the associated state and output trajectories of the system (37) (38) corresponding to initial state x t t = x and a disturbance realization W N+ = {w t,...,w t+n,w t+n } W C ( K N+,x) (W C ( K N+,x) R q(n+) is the set of the admissible disturbance realizations over horizon N + ). Let V max (x,n + ) be the worst-case cost function value due to initial state x t t = x and sequence K N+, i.e.: V max (x,n + ) = max J(x, K N+,W N+,N + ), (4) W N+ W C ( K N+,x) 8

9 where J(x, K N N+,W N+,N + ) = [ y t+i t i= γ w t+i ] + V ka (x t+n+ t ). Let P RH be the parameters of KRH. The following assumption is made on the solution K RH and the sets Ω and Ω resulting from Algorithm 3 and Procedure 7: A7. The constraints G( P RH,x,W) are satisfied for all x Ω and all W W B ( P RH,x). The sets Ω and Ω are such that Ω a Ω Ω X Π and x t+i+ t = f(x t+i t, k i RH (x t+i t ),w t+i ) Ω, x t t Ω, w t+i W A ( k i RH (x t+i t ),x t+i t ), i =,,,...,N. Here, the stability result is formulated: Theorem Given an auxiliary control law k a (x) and an associated invariant set Ω a, consider two positive constants γ and γ with γ γ <. Suppose that a nonempty region of attraction Ω and associated set Ω RH have been determined by applying Procedure 7. Let K with parameters P RH be the approximate PWNL feedback law determined with Algorithm 3 and Procedure 7. Consider the closed-loop system (37) (38), where krh (x(t)) = [I... ] K RH. Then, under assumptions A A7, the following holds for the closed-loop system (37) (38): i). In the absence of disturbance the origin is asymptotically stable for all x Ω. ii). In the presence of disturbance it has l -gain less than or equal to γ for all x Ω. Proof. i). From assumption A4 it follows that y t+i t w t+i, i. Then, the condition γ γ γ < leads to y t+i t > γ w t+i, i. Therefore, the stage cost L(y t+i t,w t+i ) = ( y t+i t γ w t+i ) is a positive definite function. Then, by taking into account that V ka (x) is a positive definite function too (cf. assumption A5), it follows: V max (x,n), x Ω. (4) In the absence of disturbance, the stage cost is L(y,) = L(h(x, k RH (x),),) and it is a positive definite function defined on the set Ω which contains the origin in its interior (according to assumption A). Then, it follows from Lemma 4.3 from (Khalil, ) that there exist a K-function α ( x ) such that L(h(x, k RH (x),),) α ( x ), x Ω. Similarly, there exists a K-function α ( x ) such that V ka (x) α ( x ), x Ω a (the reader is referred to (Khalil, ) for the definition of K-functions). Assumption A5 holds also in the case of absence of disturbance and therefore the set Ω a is a positively invariant set for the nominal system (system (37) (38) with w(t) = ) in closed-loop with the auxiliary control law k a (x) and the inequality (6) takes the form: V ka (f(x,k a (x),)) V ka (x) +L(h(x,k a (x),),), x Ω a. (4) Therefore, according to Theorem with Assumption in (Lazar, Heemels, Bemporad & Weiland, 7) x = is asymptotically stable for all x Ω when w(t) =. ii). In a way similar to that in (Magni et al., 3), it can be proved that for the worst-case cost function values defined by (39) and (4) the following holds: V max (x,n + ) V max (x,n), x Ω. (43) Following similar arguments as in (Magni et al., 3) and by taking into account (43), for x Ω and for w t W A ( k RH (x),x), we have: V max (x,n) V max (f(x, k RH (x),w t ),N ) + RH { h(x, k (x),w t ) γ w t } V max (f(x, k RH (x),w t ),N) + RH { h(x, k (x),w t ) γ w t }. (44) Inequality (44) can be represented: V max (f(x, k RH (x),w t ),N) V max (x,n) RH { h(x, k (x),w t ) γ w t }. (45) Further, by considering that x t+ t = f(x, k RH (x),w t ) and y t t = h(x, k RH (x),w t ), the inequality (45) is written in the form: V max (x t+ t,n) V max (x,n) { y t t γ w t }. (46) In a similar way, it can be shown that: V max (x t+i+ t,n) V max (x t+i t,n) { y t+i t γ w t+i }, i =,,...,T. (47) After summing the inequalities (47) and by taking into account (4), we obtain: T i= y t+i t γ T i= w t+i + V max (x,n) (48) x Ω, T, W N W B ( K N,x). Therefore, the closed-loop system (37) (38) has l -gain less than or equal to γ in Ω. In the case when assumption A7 does not hold, no guarantee on the l -gain can be given and only an estimate of its upper bound can be computed. 9

10 5 Simulation example Consider a cart with a mass M moving on a plane (Magni et al., 3). The carriage is attached to the wall via a spring with elasticity ρ = ρ e x, where x is the displacement of the carriage from the equilibrium position associated with the external force u =. A damper with damping factor h d affects the system in a resistive way. The damping factor h d is an uncertain parameter and it is only known that h d = h d + h d, where h d =. and.5 h d.5. The system is described by the nonlinear discrete-time model (Magni et al., 3): x (t + ) = x (t) + T s x (t) (49) x (t + ) = x (t) T s ρ M e x(t) x (t) T s h d M x (t) +T s u(t) M + T sw(t), (5) ε(x ) = max( ε a, ε r min x X Ṽ max(x)), where ε a =.3 and ε r =. are the absolute and the relative tolerances. The state space partition of the feedback min-max NMPC controller (the set X Π ) and the associated sets Ω and Ω are shown in Fig.. It is noticed that in some part of the set X = [ 3, 5] [, ] a feasible solution does not exist. The number of the inequalities describing the sets Ω and Ω is specified to be 5 and Procedure 7 is applied to determine them. The set Ω is obtained graphically by minimizing the difference between its area and the area of the set X Π. The computations of the state trajectories of the suboptimal closed-loop system, performed according to equation (33), have shown that the set Ω can be determined simply by increasing the bound in one of the inequalities describing the set Ω..5 h d =.6 x space h d =. where x is the carriage velocity, w(t) = h d M x (t), T s =.4 is the sampling time, M = and ρ =.33. Like in (Magni et al., 3), we choose y = [x x u] T and it follows that w(t) = [ h d M ]y(t). Therefore, w(t) γ y(t) with γ =.5, according to assumption A4. The following input and state constraints are imposed on the system: x.5.5 h d =.6 X Π 4 u 4,.3 x.3. (5).5 Therefore, the disturbances vary in the range.3γ w.3γ. The horizon is N = 5 and the terminal constraint is: x t+n t Ω a, Ω a = {x R n x T Σx δ}, (5) where δ =. (Magni et al., 3) and Σ = [ ]. The mp-nlp approach described in section 3 is applied to design an explicit feedback min-max NMPC controller for the cart. The NMPC minimizes the worst-case of the cost function (6) subject to the system equations (49) (5) and the constraints (5) (5). In (6), it is chosen γ = and the terminal penalty is V ka = x T Σx (Magni et al., 3). Like in (Magni et al., 3), the feedback functions k i (x), i =,,...,N have the form: k i (p i,x) = α i k a (x) + ξ i, x + ξ i, x, (53) where p i = [α i ξ i, ξ i, ] T are the parameters that need to be optimized and k a (x) is the auxiliary control law. The determination of k a (x) is described in (Magni et al., 3). A set of three admissible disturbance realizations is generated which correspond to three values for the uncertain parameter h d ( h d =.5, h d =, h d =.5). One internal region X X is used in Procedure 3. In (7), it is chosen µ =. The approximation tolerance is chosen to be x Ω Ω Fig.. State space partition of the explicit feedback min-max NMPC (the set X Π), the associated sets Ω and Ω, and the state trajectories for h d =.6, h d =., h d =.6. In Fig., the optimal and the suboptimal feedback functions, respectively u (x,x ) = k (x,x ) and û(x,x ) = k (x,x ), are shown. The partition (the set Ω in Fig. ) has 537 regions and 4 levels of search. Totally, 3 arithmetic operations are needed in realtime to compute the control input (4 comparisons, multiplications, 6 additions and exponential). u * (x,x ) 4 4 x x 3 Fig.. The optimal (left) and the suboptimal (right) feedback functions (views rotated on 4 ). The performance of the suboptimal closed-loop system was simulated for initial state x() = [.6 ] T and for three values of h d. The response is depicted in the state u(x,x ) x 4 x

11 space (Fig. ) and as trajectories in time (Fig. 3). It can be seen that the explicit feedback min-max NMPC controller brings the cart to the equilibrium despite of the presence of disturbance, and the constraints imposed on the system are satisfied. It can also be observed that the state trajectory does not leave the set Ω..5 u(t).5 u(t) related to a single region would increase approximately n +n n +n times. Further, by taking into account that the number of regions in the partition typically increases exponentially with the number of states, the total increase of the off-line computations would be approximately e (n n) ( n +n ) ( n +n ) times. Therefore, the development of a parallel computing algorithm to handle the off-line computational complexity is of a future interest Acknowledgements time x (t), x (t) x time x (t), x (t) x This work was sponsored by the Research Council of Norway, grants 649/V3, 5784/V3, 59556/I3. References.5.5 x time.5.5 x time Fig. 3. Control input and state trajectory for h d =.6 (left) and for h d =.6 (right). 6 Conclusions In this paper, an approximate mp-nlp approach to explicit solution of feedback min-max NMPC problems for constrained nonlinear systems in the presence of bounded disturbances and/or parameter uncertainties is suggested. It allows very efficient real-time implementation via a binary tree search, without real-time optimization. Under certain assumptions, conditions guaranteeing the stability of the suboptimal closed-loop system in terms of an upper bound of the l -gain are derived. However, in the general case, no guarantees on the stability can be given. Therefore, the future work may include the development of methods to estimate the upper bound of the l -gain when assumption A7 does not hold. Also, the elaboration of an approach to determine the minimal allowed number of admissible disturbance realizations generated with Procedure is a subject of a further research. In addition to the knowledge necessary to apply the approximate mp-nlp approaches, the developed method requires to specify the form of the feedback control policies (cf. assumption A6). Simple polynomial forms may be a reasonable choice, but it could be investigated if other functions are more appropriate. It should be noted that the offline computational complexity tends to increase rapidly with the number of states and this would restrict the application of the suggested approach only for systems with a few states. By considering the fact that for a hyper-rectangle in the n-dimensional state space, the number of its vertices is n and the number of its facets centers is n, then as the number of states increases from n to n the amount of the off-line computations Bacic, M., Cannon, M., & Kouvaritakis, B. (3). Constrained NMPC via state-space partitioning for input affine non-linear systems, International Journal of Control, 76, Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (993). Nonlinear programming: theory and algorithms. New York: Wiley. Bemporad, A., Borrelli, F., & Morari, M. (3). Min-max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48, Bentley, J. L. (975). Multidimensional binary search trees used for associative searching, Communications of the ACM, 8, Bertsekas, D. P., & Tsitsiklis, J. N. (998). Neuro-dynamic Programming. Athena Scientific, Belmont. Björnberg, J., & Diehl, M. (6). Approximate robust dynamic programming and robustly stable MPC. Automatica, 4, Campo, P. J., & Morari, M. (987). Robust model predictive control. Proceedings of the American Control Conference, 987, Minneapolis, Minn., vol., pp. 6. Cychowski, M., & O Mahony, T. (5). Efficient off-line solutions to robust model predictive control using orthogonal partitioning. Proceedings of the 6-th IFAC World Congress, July, 5, Prague, Czech Republic. Fiacco, A. V. (983). Introduction to sensitivity and stability analysis in nonlinear programming. Orlando, Fl: Academic Press. Grancharova, A., & Johansen, T. A. (5). Explicit min-max model predictive control of constrained nonlinear systems with model uncertainty. Proceedings of the 6-th IFAC World Congress, July, 5, Prague, Czech Republic. Grancharova, A., Johansen, T. A., & Tøndel, P. (7). Computational aspects of approximate explicit nonlinear model predictive control. In: Assessment and Future Directions of Nonlinear Model Predictive Control, Eds. R. Findeisen, F. Allgöwer and L. Biegler, Lecture Notes in Control and Information Sciences, vol.358, Springer-Verlag, pp.8 9. Johansen, T. A. (). On multi-parametric nonlinear programming and explicit nonlinear model predictive control. Proceedings of the IEEE Conference on Decision and Control,, Las Vegas, NV, vol. 3, pp Johansen, T. A. (4). Approximate explicit receding horizon control of constrained nonlinear systems. Automatica, 4, 93 3.

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Automatica, 39, Alexandra Grancharova received her M.Sc. and Ph.D. degrees in Automation of Industrial Processes from the University of Chemical Technology and Metallurgy, Sofia, Bulgaria. Since 994 she is with the Institute of Control and System Research at the Bulgarian Academy of Sciences (ICSR - BAS). From to 3 she held a postdoctoral position at the Department of Engineering Cybernetics at the Norwegian University of Science and Technology. Grancharova is since 4 an associate professor at ICSR - BAS. Her main research interests include explicit solution of constrained optimal control problems and modeling, simulation and optimal control of industrial processes. Tor A. Johansen received his Dr.Ing. (Ph.D) degree in Electrical and Computer Engineering from the Norwegian University of Science and Technology, Trondheim in 994. From 995 to 997 he was a research engineer with SINTEF Electronics and Cybernetics. Johansen is since 997 professor in Engineering Cybernetics at the Norwegian University of Science and Technology in Trondheim. He has been a research visitor at the University of Southern California, Technical University in Delft, and University of California in San Diego. He is a former associate editor of Automatica and IEEE Transactions on Fuzzy Systems. His research interests include optimization based control and industrial application of advanced control. He has published more than 6 journal papers.