Execution time certification for gradient-based optimization in model predictive control

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1 Exection time certification for gradient-based optimization in model predictive control Giselsson, Ponts Pblished in: [Host pblication title missing] 2012 Link to pblication Citation for pblished version APA: Giselsson, P Exection time certification for gradient-based optimization in model predictive control. In [Host pblication title missing] pp IEEE--Institte of Electrical and Electronics Engineers Inc.. General rights Copyright and moral rights for the pblications made accessible in the pblic portal are retained by the athors and/or other copyright owners and it is a condition of accessing pblications that sers recognise and abide by the legal reqirements associated with these rights. Users may download and print one copy of any pblication from the pblic portal for the prpose of private stdy or research. Yo may not frther distribte the material or se it for any profit-making activity or commercial gain Yo may freely distribte the URL identifying the pblication in the pblic portal Take down policy If yo believe that this docment breaches copyright please contact s providing details, and we will remove access to the work immediately and investigate yor claim. L UNDUNI VERS I TY PO Box L nd Download date: 16. Apr. 2019

2 Exection time certification for gradient-based optimization in model predictive control Ponts Giselsson Department of Atomatic Control LTH Lnd University Box 118, SE Lnd, Sweden Abstract We consider model predictive control MPC problems with linear dynamics, polytopic constraints, and qadratic objective. The reslting optimization problem is solved by applying an accelerated gradient method to the dal problem. The focs of this paper is to provide bonds on the nmber of iterations needed in the algorithm to garantee a prespecified accracy of the dal fnction vale and the primal variables as well as garanteeing a prespecified maximal constraint violation. The provided nmerical example shows that the iteration bonds are tight enogh to be sefl in an inverted pendlm application. I. INTRODUCTION Model predictive control MPC is an optimization based control methodology that can handle state and control constraints see [9], [10] for thorogh descriptions of MPC. In the optimization problem a cost fnction is imized based on predicted ftre state and control trajectories and sbject to constraints. Optimal control and state trajectories are obtained and the first element in the inpt trajectory is applied to the system. This procedre is repeated every sampling instant which sets reqirements on the exection time of the optimization problem. The topic of this paper is to provide certificates for the exection time of the optimization algorithm sch that for every feasible initial condition the optimization algorithm provides a soltion within the sampling time. We consider linear time-invariant systems with polytopic constraints and qadratic cost and a dal accelerated gradient method [6] is sed to solve the reslting optimization problem. For accelerated gradient methods there are convergence rate reslts [13], [2], [17], [6] that depend explicitly on the norm of the difference between the optimal soltion and the initial iterate. If this norm can be bonded, a bond on the nmber of iterations to achieve a prespecified accracy of the fnction vale can be compted. This was done in [15] where inpt constrained MPC was considered. The condensed problem, i.e., the problem with all state variables eliated, was solved sing a fast gradient method. An iteration bond was obtained since the norm of the difference between the optimal soltion and the initial iterate is bonded by the size of the inpt constraint set. Accelerated gradient methods can also be applied to the dal problem [16], [6]. To compte a bond on the nmber of iterations to achieve a prespecified accracy, a bond on the norm of the difference between the optimal dal variables and initial dal iterate is needed. This is more involved in the dal space than in a constrained primal space since dal variables are not chosen from a compact set. This is addressed in [16] where the eqality constraints are dalized and a bond on the norm of the optimal dal variables is obtained sing a recent reslt in [5]. The obtained bonds trn ot to be qite conservative. Another method to provide comptation time certificates in MPC is to bond the search time in the lookp table in explicit MPC [3], [1]. Practically this method is limited to small or medim-sized problems. For interior point methods, iteration bonds are available [11], these are, however, reported to be qite conservative [16], [11]. In this paper we consider the dal to the condensed problem, i.e., the dal to the problem where the state variables are eliated. The reslting optimization problem has only ineqality constraints and we apply the accelerated gradient method to the dal problem. To compte an iteration bond, we need a bond on the norm of the optimal dal variables. Using a reslt in [12] a bond to this norm can be compted if a Slater vector to the optimization problem is known. Comptation of the norm bond reqires that the distance from eqality in the ineqality constraints for the Slater vector is known, as well as the primal cost for the Slater vector. We will see that sch a Slater vector can be constrcted for almost all feasible initial conditions in the MPC case. The provided nmerical example shows that the presented bonds are tight enogh to give sefl bonds in an inverted pendlm application. II. PROBLEM SETUP AND PRELIMINARIES We consider the problem of controlling a linear dynamical system to the origin sbject to polytopic constraints. To achieve this we se MPC in which the following finite horizon optimization problem is solved at the crrent state x R n : V N x := x, N 1 1 x T t Qx t + T t R t xt NQ N x N t=0 s.t. x t, t X U, t = 0,...,N 1 x t+1 = Ax t +B t, t = 0,...,N 1 x N X f, x 0 = x 1

3 where x t R n, t R m and x = [x T 1,...,x T N ]T and = [ T 0,...,T N 1 ]T. We se the standard assmptions that Q 0, Q N 0 and R 0. The constraint sets are assmed to be polytopes X = {x R n C x x d x }, X f = {x R n C f x d f } U = { R m C d }. Throghot this paper we assme that the sets X, X f, and U are non-empty and compact and that 0 int X, 0 int X f, and 0 int U which implies that d x,d f,d > 0. By introdcing the following matrices B 0 0 A = A. A N, B =. AB A N 1 B AB B the predicted ftre state variables can be described in the crrent state x and control variables as We frther define x = A x+b. Q := blkdiagq,...,q,q N, R := blkdiagr,...,r, C x := blkdiagc x,...,c x,c f, d x := [d T x,...,d T x,d T f] T, C := blkdiagc,...,c, d := [d T,...,d T ] T. The optimization problem 1 can, sing these matrices, eqivalently be written as V N x = J N x, := 1 2 T H+ x T G+ 2 xt 1 F x s.t. g x, 0 2 where H = B T QB+R, G = A T QB, F = A T QA+Q, g x, = C d x and C = C C x B, d x = d d x C x A x. To solve 2 we introdce dal variables µ R p 0 for the ineqality constraints. The first p p dal variables in the dal variable vector µ correspond to the inpt constraints and the last p p dal variables correspond to the state constraints. If Slater s condition holds, we get the following dal problem cf. [4] V N x = max 1 µ 0 2 T H+ x T G+µ T C d x. As shown in [6] the dal problem becomes max 1 µ 0 2 CT µ+g T x T H 1 C T µ+g T x µ T d x. 3 We define the dal fnction D N x,µ = 1 2 CT µ+g T x T H 1 C T µ+g T x µ T d x which satisfies the following properties cf. [6]. Proposition 1: The dal fnction has Lipschitz continos gradient with Lipschitz constant L = CH 1 C T and the gradient is given by D N x,µ = CH 1 C T µ+g T x d x. This implies that the dal fnction can be maximized sing an accelerated gradient method [13], [2], [17], [6]. The algorithm presented in [6] with a cold-starting strategy, i.e., µ 0 = 0 is presented below. Algorithm 1: Accelerated gradient algorithm Initialize µ 0 = µ 1 = 0 and 1 = H 1 G T x. For k 0 k = H 1 C T µ k +G T x ũ k = k + k 1 k +2 k k 1 { µ k+1 = max 0,µ k + k 1 k +2 µk µ k L Cũ k d x } Before we state the convergence rate properties of the algorithm, we introdce the set of optimal dal variables M x = {µ R p 0 D N x,µ V N x}. We also introdce X N which is the steerable set defined as X N = { x R n there exist s.t. C d x}. Remark 1: From [14], we know that the steerable set X N is convex and that 0 X N. We also denote by x the optimal soltion to 2 with initial condition x and σh the smallest eigenvale to H. Proposition 2: Sppose that x X N. For any µ M x Algorithm 1 has the following convergence rate properties: 1 The dal fnction converges as D N x,µ D N x,µ k 2L µ 2 2 The primal variable rate of convergence is, k 1. 4 k +1 2 k x 2 4L µ 2 σhk +1 2, k The constraint violation is bonded by g x, k g x, x 2 4L2 µ 2, k 1. k +1 2 Proof. Argment 1 is proven in [2], [17], [6] and argment 2 is proven in [6]. To prove the third argment we have g x, k g x, x 2 = = C k d x C x d x 2 = D N x,µ k D N x,µ 2 2L D N x,µ,µ k µ +D N x,µ D N x,µ k 2LD N x,µ D N x,µ k. The first ineqality comes from [13, Theorem 2.1.5] since D N is convex. The second ineqality is de to first order

4 optimality condition [13, Theorem 2.2.5] for the convex fnction D N. It is left to apply Argment 1 to prove the reslt. The objective of the paper is to, a priori, compte bonds on the nmber of iterations needed to achieve a prespecified dal fnction, primal variable, and constraint satisfaction tolerance when initializing the algorithm with µ 0 = 0. These bonds shold ideally hold for any initial state x X N. In this paper we will show how to compte bonds that hold for any x βx N where β 0,1 and βx N is defined as βx N := { x R n 1 β x X N}. From the definition and Remark 1 we conclde that βx N X N and that 0 βx N. Before we proceed with the presentation we introdce the following definition. Definition 1: We define κ 1 as the smallest scalar sch that for every x X N the following holds V N x κ J N x,. Remark 2: The optimal soltion to J N x, is c x = H 1 G T x. The corresponding cost becomes J N x, = 1 2 xt GH 1 G T x x T GH 1 G T x+ 1 2 xt F x = 1 2 xt F GH 1 G T x. By defining P := F GH 1 G T where P 0 we get V N x κ J N x, = κ 2 xt P x. Also, note that we have A. Notation V N x J N x, = 1 2 xt P x. We denote byrthe real nmbers and byr 0 non-negative real nmbers. The norm refers to the Eclidean norm or the indced Eclidean norm nless otherwise is specified and x,y = x T y. Frther σh denotes the largest singlar vale of H and σh denotes the smallest singlar vale of H. Frther [ ] i denotes the i:th element in the vector. Ths, if we can find a Slater vector for any initial condition x βx N we can bond the norm of the optimal Lagrange mltipliers, µ. In the following lemma we show how to constrct a Slater vector to 2 for any initial state x βx N. Before we present the lemma the following notation is introdced; d := [d T,d T x] T and d := j [d] j which implies that d > 0. Lemma 2: For every x βx N with β 0,1, ū x = β x/β is a Slater vector to the optimization problem 2. The Slater vector satisfies γ x,ū x 1 βd. Proof. Since x βx N we have by definition that x/β X N. The optimal control trajectory at x/β is x/β. Since x/β X N the optimal control trajectory is feasible, i.e., the following holds g x x β, β = C x/β d C x A x/β +B 0. x/β d x For any x βx N we have for the chosen Slater vector ū x = β x/β that g x,β x β = C β x/β d C x A x+bβ x/β d x βc = x/β d +β 1d βc x A x/β +B x/β d x +β 1d x C = β x/β d 1 βd C x A x/β +B x/β d x 1 βd x 1 βd. 1 βd x This gives γ x,ū x = 1 j p [ g x,β x/β] j This completes the proof. 1 β[d] j = 1 βd. By limiting the set of initial states, a Slater vector can be constrcted with a certain distance to eqality in the ineqality constraints. Using this reslt the following theorem provides a bond on the norm of the optimal dal variables. Theorem 1: For every x βx N we have that max µ κ 1 µ M x 21 βd x T P x. 6 III. LAGRANGE MULTIPLIER NORM BOUNDS All qantities in the bonds in Proposition 2 are known except for µ where µ M x. This section is devoted to bonding the norm of the optimal dal variables in 3 for any x βx N where β 0,1. The following reslt is sed to achieve this. Lemma 1: Assme that ū x is a Slater vector, i.e., that ū x satisfies Cū x < d x. Then max µ 1 µ M x γ x,ū x J N x,ū x V N x where γ x,ū x := 1 j p [ g x,ū x] j. Proof. A proof is provided in [12]. Proof. We will show that Lemma 1 gives 6 sing the Slater vector ū x = β x/β. We have J N x,β x/β = = 1 2 β x/β T Hβ x/β+ x T Gβ x/β+ 1 2 xt F x = β2 x 2 β T H x ] T [ x β +2 G x β β + ] T ] [ x [ x + F β β = β 2 V N x/β κ T ] [ x [ x 2 β] β2 P = κ β 2 xt P x

5 where the ineqality comes from Remark 2. From Lemma 1 and Lemma 2 we have max µ J N x,β x/β V N x µ M x γ x,β x/β 1 κ 1 βd 2 xt P x V N x κ 1 21 βd x T P x where the last ineqality is de to Remark 2. This completes the proof. Remark 3: If Definition 1 is changed sch that κ β1 is the smallest scalar sch that for all x β 1 X N and for someβ 1 0,1 we have an pper bond V N x κ β 1 2 xt P x. Then for every x β 2 X N where β 2 0,β 1 it is straightforward to verify that max µ µ M x κ β1 1 1 β 2/β 1 < κ 1 κ β β 2 /β 1 d x T P x. If 1 β 2 we get an improved bond on the norm of the dal variables compared to Theorem 1. The provided bond on the norm of optimal dal variables can, together with Proposition 2, be sed to bond the nmber of iterations to get a prespecified accracy in the fnction vale, primal variables and constraint violation. This is the topic of the following section. IV. ALGORITHM ITERATION BOUNDS In this section we provide bonds on the nmber of iterations within which a dal ǫ d -soltion, ǫ c constraint violation and ǫ p norm-distance to the primal optimal soltion are garanteed. The bonds are developed for the cold starting case, i.e., when the initial iterate is µ 0 = 0. A. Iteration bond to garantee dal ǫ-soltion The first bond is on the nmber of iterations within which a dal ǫ-soltion is garanteed. To avoid that scaling the Q and R-matrices give different bonds we se a relative tolerance. Theorem 2: Sppose that Algorithm 1 is initialized with µ 0 = 0. Then for every x βx N with β 0,1 we have if V N x D N x,µ k ǫ d V N x 7 k k d x := L x T P x ǫ d Proof. Ineqality 7 is eqivalent to κ βd D N x,µ D N x,µ k ǫ d D N x,µ for any µ M x. From Proposition 2 and Theorem 1 we have that D N x,µ D N x,µ k 2L µ 2 k Lκ β 2 d 2 k +12 xt P x 2. Since 1 2 xt P x V N x, we have that 2Lκ β 2 d 2 k +12 xt P x 2 ǫ d 1 implies 7. Rearranging the terms gives the reslt. 2 xt P x 8 Remark 4: By scaling the penalty-matrices Q a = aq, R a = ar we get H a = ah which implies L a = CHa 1 C T = 1 a CH 1 C T = 1 a L and P a = ap. Ths, sing a relative tolerance the same bond is obtained for every scaling factor a > 0. Remark 5: To get a bond that holds for all x βx N, k d x shold be maximized sbject to x βx N. An overestimator is to maximize k d x sbject to x βx which is readily available. The reslting maximization problem depends affinely on x T P x, hence the maximizing argment can be fond by maximizing x T P x on βx. This is a qadratic maximization problem that can be rewritten as a mixed integer linear program MILP as shown in [8]. MILP software prodce pper and lower bonds to the objective in each iteration and since an pper bond to the objective is enogh to compte an iteration bond, the optimization can be stopped when sfficient accracy is achieved. B. Iteration bond for constraint violation In this section we bond the nmber of iterations within which a prespecified constraint violation is garanteed. We se the following relative tolerance g x, k ǫd. Theorem 3: Sppose that Algorithm 1 is initialized with µ 0 = 0. Then, g x, k ǫ c d holds for every x βx N if k k c x := Lκ 1 xt P x 1 βd 2 ǫ c 1. Proof. First note that if g x, k g x, ǫ c d then g x, k ǫ c d since g x, 0. From Proposition 2 and Theorem 1 we get g x, k g x, x 2L µ k +1 Lκ 1 xt P x 1 βd k +1. Setting this ǫ c d and rearranging the terms gives the iteration bond. Remark 6: This reslt can be sed in a constraint tightening approach to garantee a feasible soltion w.r.t. to the original constraint sets within k c x iterations. C. Primal variable iteration bond Using the same techniqes it is also possible to bond the nmber of iterations needed to garantee a primal soltion that is within a prespecified distance to the optimal soltion. Theorem 4: Sppose that Algorithm 1 is initialized with µ 0 = 0. Then, for every x βx N we have if k x ǫ p L κ 1 x T P x k k p x := 1. σh ǫ p 1 βd

6 Proof. From Proposition 2 and Theorem 1 we have k L 2 µ x σh k +1 L κ 1x T Px σh 1 βd k +1. Setting this ǫ p and rearranging gives the reslt. V. PRECONDITIONING There are two different ways of preconditioning the problem data to possibly achieve smaller iteration bonds. One is to do a variable change in the primal variables and another is to scale the matrices defining the ineqality constraints. We start by considering scaling the matrices defining the ineqality constraints. A. Scaling ineqality constraints All iteration bonds k d,k c,k p depend on L/d or L/d 2. By introdcing D = diagd and recalling the definition of L we get L/d 2 = CH 1 C T /λ 2 D. By scaling the ineqality constraints, this ratio can be imized to get less conservative bonds withot affecting the soltions of the optimization problem. We introdce the scaling matrix S = blkdiags,s x where S = diags 1,...,s p, S x = diags p+1,...,s p where p p and all elements s i > 0,i = 1,...,p. We get the following scaling SC Sd x. From the definition of C and d x we see that this is eqivalent to S C S d S x C x A x+b S x d x. The scaling of constraints will give as small bonds as possible if the scaling is chosen according to the following imization SCH 1 C T S S λ 2 SD. We introdce S = SD which is a diagonal matrix with strictly positive elements since it is a prodct of two diagonal matrices with strictly positive elements. This gives the eqivalent imization problem S SD 1 CH 1 C T D 1 S λ 2 S. 9 It was shown in [16, Lemma 1] that for invertible S, an optimal soltion is S = I. Since diagonal matrices with positive elements are a sbset of all invertible matrices, we get that S = I imizes 9. The optimal scaling becomes S = SD 1 = D 1. B. Preconditioning of primal variables When perforg a linear change of variables in primal variables, i.e., set q = T 1 where T is an invertible matrix, H, G and C mst be changed accordingly to not affect the primal optimal soltion. We get H q = T T HT, G q = GT and C q = CT. The Lipschitz constant does not change since L q = C q H 1 q C T q = CTT 1 H 1 T T T T C T = CH 1 C T = L. Straightforward verification of the algorithm when initialized with µ 0 = 0 gives that the µ k -seqence is identical whether sing the new variables q or the original variables. It is also straightforward to verify that the relation between the iterates in the new variable q k and the iterates in the original variable k is q k = T 1 k. Ths, we do not get better or worse convergence properties by preconditioning the primal variables. VI. NUMERICAL EXAMPLE We evalate the conservatism of the iteration bonds by applying them to a doble integrator system and a doble integrator with a pendlm attached. We consider the pendlm in [7] with pendlm length l = 0.4m. The cart has inner control loops that make it behave as a doble integrator. We choose sample time h = 0.02s as in [7]. We get the following discrete time dynamics for the pendlm system when the pendlm is in its inverted position cf. [7] x + = x The state variables are x = [p ṗ θ θ] T where p is cart position, ṗ is cart velocity, θ is pendlm angle and θ is pendlm anglar velocity. The doble integrator system is the system consisting of only the first two states, [p ṗ]. We have the following constraints 0.5 p ṗ θ θ 0.5. The objective is to imize N 1 t=0 x T t Qx t + T t R t +x T N Px N where Q = diag1,0.3,0.3,0.1, R = 0.1 and P is the infinite horizon cost for the nconstrained LQ-problem with weighting matrices Q and R. Frther we choose the teral set X f = X. In Figres 1 and 2 we compare the iteration bonds with the worst case actal nmber of iterations and the maximm nmber of iterations, k req, to garantee an exection time less than h = 0.02s on a machine with 1 Gflops/s compting power. If implemented wisely, the nmber of flops per iteration in Algorithm 1 is 2pN 2 +7pN and we get k req = 10 9 h 2pN 2 +7pN.

7 iterations k req k d d k c c k p p β Fig. 1. The figre shows the iteration bonds k d,k c,k p, the actal nmber of iterations kd act,kact c,kp act, and the maximal nmber of iterations needed to certify exection time within h=0.02s for the doble integrator system k req. iterations k req k d d k c c k p p β Fig. 2. The figre shows the iteration bonds k d,k c,k p, the actal nmber of iterations kd act,kact c,kp act, and the maximal nmber of iterations needed to certify exection time within h=0.02s for the pendlm system k req. In all examples, we se control horizon N = 10, accracy reqirements ǫ d = 0.01 and ǫ c = ǫ p = In Figre 1, the reslts for the doble integrator are presented. On the x-axis β in βx N is plotted and on the y-axis the iterations bonds and the actal nmber of iterations are plotted. We are able to certify that the optimization algorithm will terate with a close to optimal soltion for all x 0.925X N within the sampling time, h = 0.02s. We also see that for x 0.825X N we can garantee that the optimal soltion is fond in one iteration, i.e., that no constraints are active. In Figre 2, the reslts for the inverted pendlm system are presented. Also here we have β in βx N on the x-axis and the iteration bonds and the actal nmber of iterations on the y-axis. We are able to certify that for x 0.6X N that the reqired accracy is achieved within the sampling time, h = 0.02s. We can also certify that a dal ǫ d -soltion is fond within the sampling time for any x 0.9X N. We see that for large parts of the steerable set, X N, the iteration bonds give meaningfl reslts that can be sed to certify the MPC-controller with respect to exection time. VII. CONCLUSIONS AND FUTURE WORK We solve the optimization problems arising in MPC with linear dynamics, polytopic constraints, and a qadratic cost sing a dal accelerated gradient method [6]. By constrcting Slater vectors to the optimization problems, we are able to bond the norm of the optimal dal variables. This is sed to compte iteration bonds on the nmber of iterations within which a certain accracy of the dal fnction vale, constraint violation, and primal variables is garanteed. The provided nmerical example shows that the bonds are tight enogh to be sefl in a pendlm application. A ftre work direction is to search for tighter iteration bonds when sing warm-starting strategies. REFERENCES [1] Alessandro Alessio and Alberto Bemporad. A srvey on explicit model predictive control. In Nonlinear Model Predictive Control, pages [2] Amir Beck and Marc Tebolle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sciences, 21: , October [3] A. Bemporad, M. Morari, V. Da, and E.N. Pistikopolos. The explicit linear qadratic reglator for constrained systems. Atomatica, 381:3 20, Janary [4] Stephen Boyd and Lieven Vandenberghe. 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