Soft Constrained Model Predictive Control with Robust Stability Guarantees

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X Soft Contrained Model Predictive Control with Robut Stability Guarantee Melanie N. Zeilinger, Member, IEEE, Manfred Morari, Fellow, IEEE, and Colin N. Jone, Member, IEEE Abtract Soft contrained MPC i frequently applied in practice in order to enure feaibility of the optimization during online operation. Standard technique offer global feaibility by relaxing tate or output contraint, but cannot enure cloedloop tability. Thi paper preent a new oft contrained MPC approach for tracking that provide tability guarantee even for untable ytem. Two type of oft contraint and lack variable are propoed to enlarge the terminal contraint and relax the tate contraint. The approach enure feaibility of the MPC problem in a large region of the tate pace, depending on the impoed hard contraint, and tability i guaranteed by deign. The optimal performance of the MPC control law i preerved whenever all tate contraint can be enforced. Aymptotic tability of all feaible reference teadytate under the propoed control law i hown, a well a inputto-tate tability for the ytem under additive diturbance. The oft contrained method can be combined with a robut MPC approach, in order to exploit the benefit of both technique. The propertie of the propoed method are illutrated by numerical example. Index Term Soft contraint, Model Predictive Control I. INTRODUCTION In control ytem, there are generally two type of contraint: thoe originating from phyical limitation of the actuator or the ytem itelf, including critical bound related to, e.g., afe operation of the plant, or contraint derived from deired ytem pecification. While input contraint can therefore generally not be exceeded, tate or output contraint can either be hard if they fall under the firt category, or they can be conidered oft. Violation may then in practice be tolerated for hort time period, e.g. becaue of unexpected diturbance. Model predictive control (MPC) i a ucceful paradigm for the control of contrained ytem and offer guaranteed contraint atifaction a well a tability in cloedloop, when all contraint are enforced in the MPC problem [1]. Impoing hard tate or output contraint can, however, be overly conervative or render the optimization problem infeaible in cloed-loop operation. One poible remedy i to imply remove the contraint for ome portion of the prediction horizon until the problem become feaible. However, thi may lead to large contraint violation in cloed-loop, when M.N. Zeilinger i with the Department of Electrical Engineering and Computer Science, UC Berkeley, CA 94720, USA, and the Department of Empirical Inference, Max Planck Intitute for Intelligent Sytem, Tübingen, Germany, mzeilinger@berkeley.edu. C.N. Jone i with the Laboratoire d Automatique, École Polytechnique Fédérale de Lauanne (EPFL), CH 1015 Lauanne, Switzerland, colin.jone@epfl.ch. M. Morari i with the Automatic Control Laboratory, ETH Zurich, CH 8092 Zurich, Switzerland, morari@control.ee.ethz.ch. Manucript received May 21, 2013; revied December 2, implementing the firt control input of the horizon, without any poibility to tune the amount of violation. A popular approach i a o called oft contrained technique, where tate or output contraint are relaxed and the ize of the violation i penalized in the cot. While thi recover feaibility of the MPC problem and offer the ability to tune the performance, tandard oft contrained MPC cheme generally do not provide tability guarantee. In thi paper we propoe a oft contrained linear MPC approach for tracking that guarantee tability even for openloop untable ytem. Although oft contraint are widely ued in practical implementation of MPC, thi topic ha received comparably little attention in the literature. In [2], a condition on the quadratic penalty on the output contraint violation i derived to guarantee tability for ingle-input ingle-output ytem. In [3], MPC with hard input and oft output contraint i conidered and tability i proven for open-loop table ytem by howing that the tability proof in MPC extend to thi cae. Thi reult alo hold for marginally table ytem, if the horizon length i ufficiently long, which i however difficult to chooe in practice. The performance of oft contrained MPC for relaxing output contraint wa invetigated in [4]. The ue of exact penalty function in order to enforce hard contraint when poible i dicued in [5], [6], in which cae the tability propertie are preerved in the feaible et of the correponding hard contrained problem. In [7], a Youla parametrization i employed and robut tability of the hard or oft contrained problem i enforced by adding an LMI to the MPC problem. The ue of barrier function to replace contraint preented in [8] i a related idea, however the key difference i that the barrier impoe a penalty inide the contraint et, wherea in oft contrained cheme a penalty i only impoed on the contraint violation. A a reult, thi approach only provide tability in the feaible et of the correponding hard contrained problem. The method propoed in thi paper ha the advantage that it i conceptually imilar to a tandard oft contrained technique uually applied in practice, but it alo maintain the deirable propertie of MPC. Feaibility of the MPC problem i enured in a large region of the tate-pace, which depend on the impoed hard contraint. Stability i guaranteed by deign, while allowing to tune the ytem performance and the amount of contraint violation. The method i baed on the MPC approach for tracking introduced in [9] and ue a finite horizon with a terminal weight a well a a terminal contraint. All input contraint are hard contraint, while tate contraint are oftened in two way. Since a complete relaxation of the terminal contraint lead to a lo of the

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X tability propertie, we retrict the amount of relaxation by uing an enlarged terminal et. All other tate contraint are oftened by the introduction of two type of lack variable, which i a crucial item for proving tability. The approach allow for any poitive definite, convex penalty function on the contraint violation. Here, we include a quadratic and an l 1 - or l -norm penalty in order to allow for better tuning and for exact penalty function, which preerve the optimal MPC behavior whenever the tate contraint can be enforced [4], [5]. The propoed problem etup reult in a convex econdorder cone program (SOCP), which can be olved efficiently uing, e.g., interior-point method [10] [13]. We how that aymptotic tability of all feaible reference teady-tate in the abence of diturbance i guaranteed within the feaible et of the oft contrained MPC problem. In addition, input-to-tate tability of the propoed cheme under additive diturbance i proven. The robut invariant et, in which input-to-tate tability can be guaranteed, depend on the maximum diturbance ize. Uing the preented oft contrained method, tability can be provided in a potentially much larger et than with a hard contrained method and unexpected diturbance can be tolerated by relaxing tate contraint. The oft contrained cheme can alo be combined with a robut MPC framework. The advantage of both technique can be exploited in order to account for a certain diturbance ize with a robut deign, while dealing with exceeding diturbance by mean of oft contraint. We how that the tability reult extend to the combined robut and oft contrained approach. A numerical example demontrate that the propoed cheme provide feaibility and tability for a large region of the tate pace and that ignificant diturbance can be tolerated. Application to a large-cale example how that the oftcontrained MPC problem can be olved with computation time in the milliecond range even for ignificant problem dimenion. Thi paper extend the initial work in [14] to a oft contrained method for tracking and propoe a new combination of the oft contrained cheme with robut MPC, including new theoretical and numerical reult. The outline of the paper i a follow: After reviewing ome preliminary reult in Section II, Section III introduce the problem and the propoed oft contrained MPC formulation for tracking. Aymptotic tability of the nominal ytem under the reulting control law i proven in Section IV. Section V how input-to-tate tability of the uncertain ytem under the oft contrained control law a well a the combined robut and oft contrained approach. The propertie of the preented approach are illutrated in Section VI by numerical example. II. PRELIMINARIES A polyhedron i the interection of a finite number of halfpace P = {x Ax b} and a polytope i a bounded polyhedron. If A R m n, then A i R n i the vector formed by the i th row of A. If b R m i a vector, then b i i the i th element of b. Given a equence u [u 0,, u N 1 ], u j denote the j th element of u. If a equence depend on a parameter denoted by u(x), u j (x) denote it j th element. If x R n i a vector and Q i a poitive emi-definite matrix, then x 2 Q = xt Qx and [x] + = max{0, x} taken elementwie. A function γ : R 0 R 0 i of cla K if it i continuou, trictly increaing and γ(0) = 0 [15]. If in addition γ() a, then it i of cla K. A function β : R 0 R 0 R 0 i of cla KL if for each fixed t 0, β(, t) i of cla K, for each fixed 0, β(, ) i non-increaing and β(, t) 0 a t [15]. Conider the dicrete-time uncertain linear ytem x(k + 1) = Ax(k) + Bu(k) + w(k), k N (1) that i ubject to the following contraint: x(k) X R n, u(k) U R m, (2) where x(k) i the tate, u(k) i the control input and w(k) W R n i a bounded diturbance at the k th ample time. X {x G x x f x } and U {u G u u f u }, where G x R px n, f x R px and G u R pu m, f u R pu, are polytopic contraint on the tate and input that each contain the origin in their interior. W i a convex and compact diturbance et that contain the origin. When it i convenient, we make ue of the lighter notation x + = Ax+Bu+w, where x + denote the ucceor tate at the next ampling time. The nominal model of ytem (1) decribe the ytem conidering no diturbance, given by x(k + 1) = A x(k) + Bū(k). (3) The olution of the uncertain ytem controlled by the control law u(k) = κ(x(k)) at ampling time k for the initial tate x(0) and for a equence of diturbance w i denoted a φ κ (k, x(0), w). A teady-tate and input pair z (x, u ) of the nominal ytem (3) i characterized by the condition (I A)x = Bu. The contraint limit the et of feaible teady-tate to S {(x, u ) (x, u ) X U, (A I)x + Bu = 0} and in the oft contrained cae to S {(x, u ) u U, (A I)x + Bu = 0}. The et of admiible teady-tate for tracking i given by S tr {(x, u ) (1 + ξ)g x x f x, (1 + ξ)g u u f u, (A I)x + Bu = 0} S, where 0 < ξ 1 i a mall poitive contant, retricting the reference to the interior of the contraint. While the ytem under conideration may be untable, it i aumed to atify the following tanding aumption: Aumption II.1. The pair (A,B) i tabilizable. The following tandard definition can be found in [16]: Definition II.2 ((Robut) poitively invariant et). A et S R n i a robut poitively invariant (RPI) et of ytem x + = f(x) + w, if f(x) + w S for all x S, w W. S i called a poitively invariant (PI) et of ytem x + = f(x), if f(x) S for all x S. Stability of an uncertain ytem will be analyzed uing the framework of input-to-tate tability (ISS): Definition II.3 (Regional ISS [17], [18]). Given an RPI et Γ R n containing the origin in it interior, ytem x(k + 1) = f(x(k)) + w(k) i Input-to-State Stable (ISS) in Γ with

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X repect to w(k) W, if there exit a KL-function β and a K-function γ uch that for all initial tate x(0) Γ and for all diturbance equence w [w j ] j 0 with w j W: φ κ (k, x(0), w) β( x(0), k) + γ( w [0,k 1] ) k 0, where w [0,k 1] max{ w j, j [0, k 1]}. Note that the condition for input-to-tate tability reduce to that for aymptotic tability, if w j = 0 for all j 0. Theorem II.4 (Regional ISS [19], [20]) Let Γ be an RPI et for ytem x(k + 1) = f(x(k)) + w(k) and S Γ be a compact et, both including the origin a an interior point. If there exit a function V : R n R +, uitable K -cla function α 1, α 2, α 3 and a K-cla function γ uch that: V (x) α 1 ( x ) x Γ, V (x) α 2 ( x ) x S, V (f(x) + w) V (x) α 3 ( x ) + γ( w ) x Γ, w W, (4a) (4b) (4c) (4d) V ( ) i called an ISS Lyapunov function in Γ and the ytem x(k+1) = f(x(k))+w(k) i ISS in Γ with repect to w W. A. MPC for Tracking Piecewie Contant Reference We conider the problem of tracking a given reference teady-tate z r (x r, u r ) S tr tarting from a given initial tate x. Thi work employ the tracking formulation introduced in [9] a the bai for the propoed oft contrained cheme, offering recurive feaibility and an enlarged region of attraction compared to the tandard approach of applying a change of variable [20], [21]. An artificial teady-tate i introduced, which may deviate from the deired reference if the latter i not a feaible target from the current tate, and can be een a an artificial et point that i imultaneouly teered to the reference. The cot i then deigned for tracking the artificial teady-tate, where a penalty term on the deviation between the artificial and the real teady-tate enure convergence to the deired reference. The reulting nominal MPC problem for tracking P N (x, z r ) i given by: N 1 V N (x, u, z, z r ) l(x i x, u i u ) + V f (x N x ) i=0 V N(x, z r ) min x,u,z V N (x, u, z, z r ) + V o (x x r, u u r ) (5a) (5b).t. x 0 = x, (5c) x i+1 = Ax i + Bu i, (5d) (x i, u i ) X U, (5e) x N X f (x, u ), (5f) (x, u ) S, (5g) for i = 0,..., N 1, where x = [x 0, x 1,, x N ] and u = [u 0,, u N 1 ] denote the tate and input equence, the tage cot i defined a l(x, u) x 2 Q + u 2 R, V f (x) x 2 P i a terminal penalty function and Q, R and P are ymmetric poitive definite matrice. In thi tracking formulation z = (x, u ) denote the artificial teady-tate and input pair, X f (x, u ) i a compact terminal et for tracking that i parameterized by the teady-tate, and V o (, ) : R n R m R + i a poitive definite cot on the tracking offet. We refer to [9], [22] for more detail on thi approach. Problem P N (x, z r ) implicitly define the et of feaible control equence U N (x, z ) {u x.t. (5c) (5f) hold} and feaible initial tate X N {x z S.t. U N (x, z ) }. Note that the feaible et i independent of the given reference. The reulting MPC control law for tracking i given in a receding horizon fahion by κ(x, z r ) = u 0(x, z r ), (6) where u (x, z r ) i the optimal olution to Problem P N (x, z r ). Aumption II.5. It i aumed that for any given (x, u ) S, V f (x x ) i a Lyapunov function in X f (x, u ) and that X f (x, u ) i a PI et for the nominal ytem (3) under the local control law for tracking κ f (x) = K(x x )+u, which can be tated a the following condition: A1: V f ((A + BK)(x x )) V f (x x ) l(x x, K(x x )) x X f (x, u ). A2: X f (x, u ) X, Ax + Bκ f (x) X f (x, u ), κ f (x) U x X f (x, u ). Theorem II.6 (Nominal tability under κ(x, z r ) [9]) Let (x r, u r ) S tr be a given reference teady-tate. x r i aymptotically table for the cloed-loop ytem x + = Ax + Bκ(x, z r ) with region of attraction X N. The extenion of thi tracking cheme to a robut tubebaed MPC framework for ytem with bounded additive diturbance of the form (1) wa conidered in [23]. III. SOFT CONSTRAINED MPC - PROBLEM SETUP Thi paper develop a oft contrained MPC method baed on the tracking formulation P N (x, z r ), which provide tability guarantee in the preence of oft contraint. We firt briefly dicu in the following why the tability propertie are lot uing a tandard oft contrained technique, and then preent the new formulation. A. Problem Statement A commonly applied approach i to relax all tate contraint by the introduction of lack variable ɛ i and to minimize the amount of contraint violation by including penalty function on the lack variable in the MPC cot, i.e. to replace x i X in (5e) with G x x i f x + ɛ i and to add N 1 i=0 l ɛ(ɛ i ) to the cot function in (5b), where ɛ i 0 and l ɛ i a poitive definite function. Stability would be preerved in thi cae when impoing a terminal et, in which all tate and input contraint are atified. Note that the method in [33] can be conidered a an approach with a hard terminal contraint, but ince the terminal et i R n for table ytem, global tability can be hown. For marginally table or untable ytem a conidered in thi paper, a hard terminal contraint repreent a ignificant limitation. In order to enure feaibility of the MPC problem in a large region of the tate pace, an extremely long prediction horizon would have to be choen or it would

4 Melanie N. Zeilinger, Member, IEEE, Manfred Morari, Fellow, IEEE, and Colin N. Jone, M IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X have to be adapted online; both approache are undeirable for implementation. Similarly, tability would be guaranteed Abtract R + i a mall poitive contant and c R px vector with c i T 1 2 G T x,i 2 i = 1,..., p x. i a contant by uing an infinite horizon, which i however intractable in The modification introduced IndexinTerm Problem P the preence of additional hard contraint. N (x, z r) are explained in the following ection and are illutrated in Figure 1. If the terminal contraint i relaxed by a lack Softvariable contraint, thatmodel i Predictive Control minimized in the cot, the tability guarantee i lot even in the nominal cae. The tability proof employing the optimal MPC x cot a a Lyapunov function fail for two poible reaon. x 2 1 x 3 X If the terminal tate i outide the region where a control x law tabilizing the uncontrained ytem i feaible, no input 0 = x Ef (x N,x ) equence i available for proving a decreae in the cot. If the local control law atifie the input contraint, but the tate ɛ 2 ET (x,u ) ɛ contraint are violated, a decreae in the cot can no longer 1 ɛ 3 =0 Cf be guaranteed due to the addition of the lack penaltie to the ɛ 0 ɛ cot function. B. Soft Contrained MPC Problem Formulation A tability guarantee by mean of the tandard tability proof in MPC ha to be acrificed in exchange for a complete relaxation of the terminal contraint. In thi work, we therefore propoe to ue a retricted relaxation by mean of an enlarged terminal et that enforce only the input contraint. In addition, two different type of lack variable are employed, which will be key in proving (input-to-tate) tability in a large feaible et. The propoed oft contrained MPC problem P N (x, z r) i given by: Problem P N (x, z r) (Soft contrained MPC problem) N 1 VN (x, u, z, ɛ, z r ) V N (x, u, z, z r ) + l ɛ (ɛ ) + VN (x, z r ) = min V N (x, u, z, ɛ, z r ) x,u,z,ɛ i=0 l ɛ (ɛ i + ɛ ) (7a).t. x 0 = x, (7b) x i+1 = Ax i + Bu i, (7c) G u u i f u, (7d) G x x i f x + ɛ + ɛ i, (7e) x N E f (x, u ), (7f) (x, u ) S, (7g) (1 + ξ)g x x f x + ɛ, (7h) c x N x T f x + ɛ G x x ɛ i 0, ɛ 0, (7i) for i = [0,..., N 1], where ɛ = colin.jone@epfl.ch. [ɛ 0,..., ɛ N 1, ɛ ] are the lack variable correponding to the tate equence x. The offet cot i defined a V o (x x r, u u r ) = ρ x x x r p + ρ u u x r p and the penalty function on the lack variable i taken a l ɛ (ɛ) = ɛ 2 S + ρ ɛ ɛ p, where S i a ymmetric poitive emi-definite matrix, p {1, }, and ρ x, ρ u, ρ ɛ R + are poitive contant weight. We ue an invariant ellipoidal terminal et, given by { } Ef (x, u ) x x x 2 T 1 r(x, u ), (8) where T R n n i a ymmetric poitive definite matrix and r : S [0, 1) i a quadratic poitive definite function. ξ Fig. 1. Illutration of the optimal lack variable ɛ, ɛ i, i = 0, 1, 2, 3, the terminal et Ef (x, u ), the caled terminal et E T (x N, x ) and enlarged terminal et Cf for an initial tate x outide X. Problem P N (x, z r) implicitly define the et of feaible control equence U N (x, z ) = {u x.t. (7b) (7d), (7f) hold} and feaible initial tate X N {x z S.t. U N (x, z ) }. For a given tate x X N and reference z r S tr, Problem P N (x, z r) reult in a convex econd order cone program (SOCP) and it olution yield the optimal control equence u (x, z r ). Note that SOCP can be efficiently olved uing, e.g., interior-point method [10] [13]. The implicit optimal oft contrained MPC control law i then given in a receding horizon fahion by κ (x, z r ) u 0 (x, z r ). (9) 1) Relaxation of the Terminal Contraint: The terminal et for tracking in (7f) i relaxed by two effect: i) by allowing the artificial et point to move to any teady-tate (x, u ) S atifying only the input contraint, ii) by allowing the tate contraint to be violated in the terminal et Ef (x, u ). Thi reult in an enlarged terminal et Cf {x (x, u ) S.t. x Ef (x, u )} for the MPC problem, given by the et of all tate x, for which there exit a teady-tate uch that the terminal contraint i atified. Aumption III.1. For any given (x, u ) S, E f (x, u ) i M.N. Zeilinger i with the Department a PIofet Electrical underengineering the localand control Computer lawscience, κ UC Berkeley, CA 94702, USA, and the Inference, Max Planck Intitute (7j) for Intelligent Sytem, Tübingen, Germany, f (x) mzeilinger@berkeley.edu. = K(x x ) + u C.N. Jone i with the Laboratoire atifying d Automatique, the following École Polytechnique condition: Fédérale de Lauanne (EPFL), CH 1015 Lauan A3: κ f (x) U x E f (x, u ) A4: Ax + Bκ f (x) x 2 T = (A + BK)(x x ) 2 T x x 2 T. It i further aumed that the et Cf i compact. M. Morari i with the Automatic Control Laboratory, ETH Zurich, CH 8092 Zurich, Switzerland, morari@control.ee.ethz Manucript received May 21, 2013; revied December 2, Note that compared to condition A2 in Aumption II.5, condition A3 only enforce the input contraint. Condition A4 i lightly tronger than et invariance and i required for proving tability in Section IV. If K i taken a the infinite horizon LQR control law, which i a common choice in MPC, a matrix T atifying condition A4 i, e.g., given by the olution to the dicrete-time algebraic Riccati equation. Compactne of Cf i required to enure boundedne of the

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X feaible et and in turn uniform continuity of the optimal value function VN (x, z r) (Lemma V.1). It can be eaily atified by impoing a large upper bound on the teady-tate x. Lemma III.2 A matrix T and function r defining the invariant ellipoidal target et E f (x, u ) in (8) uch that Aumption III.1 i atified can be computed by olving a convex linear matrix inequality (LMI). Proof: See appendix. Remark III.3. The amount, by which the terminal et can be enlarged depend on S. An increae can only be achieved if the reference teady-tate i not the only teady-tate, i.e x can differ from x r, and the impoed hard contraint are not limiting the ize of the terminal et to be alway contained in the tate contraint. Remark III.4. The combination of moving the artificial teady-tate and neglecting tate contraint offer a ignificant increae of the terminal et, which i the reaon for chooing the tracking formulation [9]. The propoed cheme could alo be applied to a tandard MPC formulation by allowing tate contraint to be violated in the terminal et, which would, however, reult in a maller terminal and hence feaible et. 2) Slack Variable: We now explain the crucial item in the propoed oft contrained cheme, the lack variable ɛ and ɛ i that are ued to often all tate contraint: ɛ repreent the amount of contraint relaxation that i neceary in order to include the ellipoid ET (x N, x ) for a particular value of (x, u ) into the relaxed tate contraint, where E T (x N, x ) { x } x x 2 T x N x 2 T (10) i a caling of the ellipoidal terminal et Ef (x, u ) containing x N on it boundary. Thi can be expreed a { } max G x,i x x x 2 x T x N x 2 T f x,i + ɛ,i i = 1,..., p x, reulting in the following condition [24]: T 1 2 G T x,i 2 T 1 2 (xn x ) 2 f x,i + ɛ,i G x,i x i = 1,..., p x, which correpond to (7i) with c i = T 1 2 G T x,i 2 i = 1,..., p x and i a collection of p x convex econd order cone contraint. ɛ i in (7e) repreent the additional contraint violation of each tate x i for i = 0,..., N 1 with repect to the tate contraint relaxed by ɛ. The ue of the lack variable ɛ defined by (7i) enure that the terminal tate, which i contained in ET (x N, x ), will lie inide the tate contraint relaxed by the amount ɛ and will not require a further relaxation of the tate contraint, i.e. ɛ N = 0, where ɛ N i the lack variable of the terminal tate defined by G x x N f x + ɛ + ɛ N. Thi provide feaibility of the hifted equence uing the hifted lack variable with the lat lack variable being zero. Contraint (7h) additionally enforce that the teady-tate x alway ha to lie in the interior of the contraint relaxed by ɛ by an amount ξ, which i a uerpecified mall, poitive parameter. For a tate that i cloe to the artificial teady-tate, thi enure that the teady-tate can alway be hifted toward the reference without increaing the lack variable. A will be hown in Section IV, thee item provide that the optimal cot function i till a Lyapunov function and are hence crucial for proving tability of the propoed oft contrained MPC cheme. Remark III.5. By Aumption III.1, the et E T (x N, x ) i a poitively invariant et under the local control law κ f (x). Remark III.6. Defining ɛ by the incluion of E f (x, u ) into the relaxed tate contraint would alo provide ɛ N = 0 and would allow for proving aymptotic tability. We have choen E T (x N, x ) here, ince it reult in a lack cloer to the actual contraint violation of the terminal tate. Remark III.7. While a trictly poitive value of ξ in contraint (7h) i required to prove tability of the cloed-loop ytem (Lemma IV.2), the particular choice i not crucial and for ξ 1, it will have a negligible or no effect on the ytem behavior. Remark III.8 (Hard tate contraint). For eae of notation, we aume the relaxation of all tate contraint except the terminal contraint in P N (x, z r). However, the reult directly extend to the cae where ome of the tate contraint are conidered a hard contraint with only minor notational change. 3) Penalty Function: A penalty function on the lack variable i included in the cot in (7a), in order to minimize the contraint violation and to enure atifaction of the tate contraint whenever poible. The penalty can be choen a in tandard oft contrained cheme. In the propoed formulation, we include a quadratic penalty, which i often preferable for tuning the contraint violation [4], and an l 1 or l -norm penalty, in order to allow for exact penalty function. It i well-known that, when the weight on the l 1 or l -norm are ufficiently large and there exit a feaible olution to the hard contrained problem P N (x, z r ), then the optimal olution to the oft contrained problem P N (x, z r) correpond to that of the hard contrained problem [5], [6], [25]. Note that an l 1 or l -norm i alo ued in the offet cot for penalizing the deviation of the artificial from the reference teady-tate, in order to enforce the reference a the target point if it i feaible [22]. Remark III.9. The reult preented in thi paper hold for any poitive definite, convex penalty function on the lack variable, i.e. the quadratic penalty could alo be omitted. Note that the linear penaltie can be implified for implementation. The l 1 -norm can directly be replaced with the um of the lack variable, and the l -norm can be formulated by uing a ingle lack variable for the contraint relaxation at each tage that i then penalized in the cot. C. Soft Contrained MPC Propertie The oft contrained formulation P N (x, z r) enlarge the feaible et compared to the hard contrained problem ince

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X X N XN. By electing the prediction horizon accordingly, it can be choen to cover any polytopic region of interet up to the maximum tabilizable et for the input-contrained ytem, i.e. all initial tate for which there exit a feaible input at all time uch that the tate converge to the origin without conidering the tate contraint. In the following ection, we demontrate how the introduction of the previouly decribed component allow u to how that: 1) The optimal cot function VN (x, z r) i a Lyapunov function and all reference teady-tate z r S tr are aymptotically table for the controlled nominal ytem with an enlarged region of attraction compared to a tandard nominal MPC method (Section IV). 2) The reference teady-tate i ISS for the controlled ytem under additive diturbance uing the propoed oft contrained method a well a a combined robut and oft contrained approach (Section V). The region of attraction i enlarged compared to a pure robut MPC approach conidering the ame diturbance ize, which i demontrated by numerical example (Section VI). IV. NOMINAL STABILITY In the following, we prove that the reulting optimal oft contrained control law κ (x, z r ) in (9) aymptotically tabilize the nominal ytem in (3) in the enlarged PI et XN. For thi, we how in three tep that the optimal cot function of the oft-contrained MPC problem VN (x, z r) i a Lyapunov function. Lemma IV.1 Let u (x, z r ), x (x, z r ), x (x, z r ), u (x, z r ), ɛ (x, z r ) be the optimizer of P N (x, z r) for ome x XN and reference teady-tate z r S tr and let x + = Ax + Bκ (x, z r ). The hifted control equence u hift = [u 1 (x, z r ),..., u N 1(x, z r ), ũ(x, z r )], (11) with ũ(x, z r ) = K(x N (x) x (x, z r ))+u (x, z r ) i feaible for P N (x+, z r ) with teady-tate z hift = z (x, z r ) and lack variable ɛ hift = [ɛ 1 (x, z r ),..., ɛ N 1 (x, z r), 0, ɛ (x, z r )] and VN (x +, z r ) VN (x, z r ) l(x x (x, z r ), u 0 (x, z r ) u (x, z r )). (12) Proof: For brevity, we drop the dependence on (x, z r ). Feaibility of u hift for P N (x+, z r ) with z hift and ɛ hift follow from feaibility of u, x, u, ɛ at x and poitive invariance of ET (x N, x ). ɛ N = 0 reult from the fact that x N E T (x N, x ) and the definition of the lack variable in (7). (12) then follow from A1 in Aumption II.5 and tandard argument in MPC. Lemma IV.1 implie that the cloed-loop ytem converge to x. In order to achieve aymptotic convergence to the reference x r, we have to how that x imultaneouly converge to x r. We firt tate a lemma howing that if the tate i cloer to the artificial teady-tate x than ome fraction of the ditance between the artificial and the target teady-tate x r, then we can move the artificial teady-tate toward x r, while providing a decreae in the cot uing the auxiliary control law. Thi reult will then allow u to prove one of the main reult of thi paper in Theorem IV.3 and how aymptotic tability of x r for the cloed-loop ytem under the propoed oft contrained MPC control law. Lemma IV.2 Let (x a, u a ) be a teady-tate, u a, x a the input and tate equence generated by applying the auxiliary control law κ f (x) = ua + K(x x a ) tarting from x a 0 and let ɛ a be the aociated minimal lack. Denote x a,α = αx a + (1 α)x r, u a,α = αu a + (1 α)u r. There exit contant δ > 0 and α (0, 1) uch that x a 0 x a P (1 α) x a x r P δ implie that 1) the lack ɛ a α = ɛ a are feaible for x a and x a,α, 2) V N (x a, u a, z a,α, z r ) V N (x a, u a, z a, z r ) (1 α) 2 x a x r 2 P, and therefore V N (x a, u a, z a,α, ɛ a α, z r ) V N(x a, u a, z a, ɛ a, z r ) (1 α) 2 x a x r 2 P. (13) Proof: The reult i hown by proving 1) and 2) eparately. For 1), we will prove that there exit a δ > 0 uch that ɛ a,α = ɛ a i a feaible choice for any α 1 ( x a x r P δ)/ x a x r P α < 1. Feaibility of ɛ a i,α = ɛa i then follow from the ue of the ame tate equence. For 2), it i hown that the condition hold for α 2 α < 1, i.e. both 1) and 2) hold for max{α 1, α 2 } α < 1. Note that 2) i independent of the oft contrained formulation and can be hown a in the hard contrained cae in [26]. The proof i included in the appendix for completene. Proof of 1): Let A K A + BK. Uing the contraint on ɛ a, we can derive the following two condition on ɛ a,α. From (7h), (1 + ξ)g x x a f x + ɛ a and recalling that (1 + ξ)g x x r f x : (1 + ξ)g x x a,α α(f x + ɛ a ) + (1 α)f x f x + αɛ a f x + ɛ a,α. (14) By Aumption III.1 and uing T γ 2 P, for ome γ 1, we have that x a N xa 2 T xa 0 x a 2 T γ2 x a 0 x a 2 P γ2 δ 2. From (7i) we then obtain: c x a N x a,α T + G x x a,α (15a) = c x a N x a + (1 α)(x a x r ) T + αg x x a + (1 α)g x x r (15b) c x a N x a T + (1 α)c (x a x r ) T + (1/(1 + ξ))(f x + αɛ a ) (15c) 2γδc + (1/(1 + ξ))(f x + αɛ a ) f x + ɛ a,α. (15d) Condition (14) and (15) are atified for ɛ a,α max{αɛ a α, 1+ξ ɛa } and (1 α) x a x r P δ ξ 2γc f i(1+ξ) x,i i = 1,..., p x, howing that ɛ a,α = ɛ a i a feaible choice for any δ and α atifying the latter condition. Theorem IV.3 (Aymptotic Stability under κ (x, z r )) Let (x r, u r ) S tr be a reference teady-tate. x r i aymptotically table for the cloed loop ytem

7 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X x(k + 1) = Ax(k) + Bκ (x(k), z r ) with region of attraction XN. Proof: Uing Lemma IV.1 and IV.2, it can be hown that VN (x, z r) i a Lyapunov function by following the ame argument a in the hard contrained cae preented in [26]. The proof i included in the appendix. V. ROBUST STABILITY In practice, model uncertaintie or external diturbance caue a deviation from the nominal ytem dynamic in (3). There are two general approache to deal with diturbance. In the cae of linear ytem and under certain aumption on the MPC problem etup, the nominal control law offer inherent robut tability propertie [19], [27] and robut tability can be guaranteed in an RPI et that depend on the conidered diturbance ize. Robut MPC cheme, on the other hand, take a wort-cae diturbance ize explicitly into account by changing the problem formulation and/or tightening the contraint, e.g. uing a min-max or a tube-baed MPC approach (ee e.g. [1], [19], [20], [28] and the reference therein). In a hard contrained etup, both technique have potential limitation. Uing a nominal MPC cheme, the RPI et, in which tability can be guaranteed, may be prohibitively mall for the conidered diturbance ize. Uing a robut MPC cheme, the choice of the diturbance bound employed in the controller deign i often conervative, ince feaibility of the MPC problem may be lot if the diturbance exceed the predefined bound. The propoed oft contrained cheme can be ued to improve the propertie of both technique. The nominal oft contrained method offer inherent robut tability propertie and can thereby provide tability in a potentially much larger RPI et, ince tate contraint can be relaxed. It i important to note that, while tability i formally only guaranteed within the RPI et, the control law i defined everywhere in the enlarged feaible et. If contraint atifaction hould be guaranteed for a certain expected ize of the diturbance, the oft contrained cheme can be combined with a robut MPC approach. While the robut method deign the problem for a certain diturbance bound, the ue of the propoed oft contrained formulation enure feaibility and tability of the MPC problem if the diturbance exceed thi bound. Conervatim in the choice of the diturbance bound for robut MPC can thereby be avoided and the ytem performance improved. Robut tability under both the nominal oft contrained MPC cheme a well a the combination with a robut MPC approach i proven in the following uing the framework of input-to-tate tability. A. ISS of Nominal Soft Contrained MPC Aume that the ytem i ubject to an additive uncertainty a given in (1). Becaue of the diturbance, the hifted equence u hift in (11) may no longer be feaible for P N (x+, z r ). For all x + XN there doe, however, exit a feaible olution to P N (x+, z r ) and input-to-tate tability can be hown in an RPI et XW X N. It i given by the maximum robut poitively invariant et for the controlled uncertain ytem x + = Ax+Bκ (x, z r )+w under the optimal oft contrained MPC control law in (9). We make ue of the following reult in order to how that the uncertain ytem in (1) under the nominal control law κ (x, z r ) i input-to-tate table with repect to the (unpecified) diturbance et W in Theorem V.2. Lemma V.1 (Continuity of VN (x)) Conider problem P N (x, z r). The optimal value function VN (x, z r) uniformly continuou in x on XN. Proof: Continuity follow directly from continuity and convexity of the cot function and the contraint in (7) a well a compactne of the contraint et for all x XN (Theorem in [29]), where the latter i provided by the fact that the terminal et i compact (Aumption III.1). Uniform continuity then follow, ince XN i compact (e.g. Propoition 5 in [20]). Theorem V.2 (ISS under κ (x, z r )) Let (x r, u r ) S tr be a given reference teady-tate. x r i ISS for the cloed loop ytem x(k+1) = Ax(k)+Bκ (x(k), z r )+w(k) with repect to w(k) W with region of attraction X W. Proof: From the proof of Theorem IV.3 and Lemma V.1 it follow that VN (x, z r) i a uniformly continuou Lyapunov function and hence there exit a K-cla function γ( ), uch that V N (y, z r) V N (x, z r) γ( y x ) (ee, e.g., [20], A.11) a well a a K -cla function α 3 ( ) uch that V N (Ax + Bκ (x, z r ), z r ) V N (x, z r) α 3 ( x x r ). It follow from thee fact that VN (x +, z r ) VN (x, z r ) = VN (Ax + Bκ (x, z r ) + w, z r ) VN (x, z r ) + VN (Ax + Bκ (x, z r ), z r ) VN (Ax + Bκ (x, z r ), z r ) α 3 ( x x r ) + γ( w ), i.e. V N (x, z r) i an ISS-Lyapunov function with repect to w W and x r i ISS for the cloed-loop ytem. The uncertain ytem controlled by the oft contrained control law κ (x, z r ) i hence input-to-tate table againt ufficiently mall diturbance. Since the RPI et XW depend on W, the ize of the diturbance and the correponding region, for which tability can be formally guaranteed, depend on the particular ytem of interet. In the following ection we will how that the previouly preented reult can be directly extended to the combination of a robut and oft contrained MPC framework, in order to take advantage of both propertie. B. Combination of Robut and Soft Contrained MPC We aume in the following that the ytem i affected by two uncertaintie: x(k + 1) = Ax(k) + Bu(k) + w 1 (k) + w 2 (k), (16) where w 1 W 1, w 2 W 2 and W 1, W 2 are convex and compact et that each contain the origin. The diturbance w 1 i explicitly taken into account by uing a robut MPC i

8 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X technique, which provide contraint atifaction and tability in the preence of w 1. Feaibility and tability in the preence of w 2 i guaranteed by mean of the propoed oft contrained cheme. In thi work, we apply the tube-baed robut MPC approach for linear ytem [28]. The method i baed on the ue of a feedback policy of the form u = ū + K(x x) that bound the effect of the diturbance w 1 and keep the tate x of the uncertain ytem under w 1 cloe to the tate of the nominal ytem in (3). The ue of tightened tate and input contraint enure feaibility of the uncertain ytem in (16) depite the diturbance w 1 : X = X Z { x Gx x f x }, Ū = U KZ { u G u u f u }, with fx,i = f x,i h Z (G T x,i ), i = 1,..., p x, f u,i = f u,i h Z (K T G T u,i ), i = 1,..., p u, where Z i an RPI et for the controlled ytem x + = (A+BK)x+w 1 and h Z (a) = up x Z a T x i the upport function of Z evaluated at a. Note that the tightened tate and input contraint again reult in compact polytope. See [28] for a detailed decription of the method. The tracking approach decribed in Section II wa extended to a robut tube-baed MPC method for tracking in [23], which will be combined in the following with the oft contrained cheme propoed in Section III. Problem P r N (x, z r) (Robut oft contrained MPC problem) V r N (x, z r ) = min x,ū, z, ɛ,z r V N ( x, ū, z, ɛ, z r ) + V f (x x 0 ).t. x x 0 Z, (7c) (7j), where in the contraint (7c)-(7j), f x, f u and E f (x, u ) are replaced with f x, f u and Ē f ( x, ū ), repectively. The condition on the robut terminal et for tracking Ē f ( x, ū ) are obtained by replacing the input contraint U in Aumption III.1 with Ū. Compared to [28], we propoe to augment the cot with the term V f (x x 0 ), which offer the advantage of directly providing an ISS Lyapunov function (Theorem V.4, ee alo [26], [30] for more detail). Remark V.3. The ue of tightened tate contraint in the oft contrained formulation, i.e. replacing f x with f x, ha the advantage that the behavior of the robut MPC controller i recovered if the contraint can be enforced. Stability would, however, alo be provided by uing f x. The et of feaible initial tate of the robut oft contrained problem i denoted by XN r. The robut formulation doe not change the problem tructure and Problem P r N (x, z r) again reult in a convex SOCP. For a given tate x XN r, the olution of P r N (x, z r) yield the optimal control equence ū r (x, z r ) and the optimal firt tube center x r 0 (x, z r ). The robut oft contrained control law i then given in a receding horizon fahion by κ r (x, z r ) ū r 0 (x, z r ) + K(x x r 0 (x, z r )). (17) Input-to-tate tability will in the following be hown for the robut invariant et XW r X N r, given by the maximum robut poitively invariant et for the controlled uncertain ytem x + = Ax + Bκ r (x, z r ) + w 1 + w 2 with w 1 W 1, w 2 W 2. Let S tr {(x, u ) S tr (1+ξ)G x x f x, (1+ξ)G u u f u }. tr S Theorem V.4 (ISS under κ r (x, z r )) Let (x r, u r ) be a given reference teady-tate. x r i ISS for the cloed loop ytem x(k + 1) = Ax(k) + Bκ r (x(k), z r ) + w 1 (k) + w 2 (k) with repect to w 1 (k) W 1 and w 2 (k) W 2 with region of attraction XW r. Proof: The firt part of the proof aume w 2 = 0 and follow imilar tep a in Section IV to how that VN r (x, z r) i an ISS Lyapunov function with repect to w 1. A more detailed verion of thi part of the proof can alo be found in [26]. The econd part of the proof then how that VN r (x, z r) i alo an ISS Lyapunov function with repect to w 2. In the following, we omit the dependence of the optimal olution on (x, z r ). Let w 2 = 0, i.e. x + = Ax+Bκ r (x, z r )+w 1. Lemma IV.1 extend to the robut problem etup howing feaibility of the hifted equence with ɛ hift = [ ɛ r 1,..., ɛ r N 1, 0, ɛr ]. Uing uniform continuity of V f ( ), x + = x r 1 +(A+BK)(x x r 0 )+w 1 and Aumption II.5, it can be hown that there exit a K-cla function γ 1 ( ) uch that V f (x + x r 1 ) V f (x x r 0 ) x x r 0 2 Q + γ 1( w 1 ). Uing tandard argument and convexity of 2 Q, it then follow that VN r (Ax + Bκ r (x, z r ) + w 1, z r ) VN r (x, z r ) VN( x hift, ū hift, z r, ɛ hift, z r ) + V f (x + x r VN( x r, ū r, z r, ɛ r, z r ) V f (x x r x r 0 x r 2 Q x x r 0 2 Q + γ 1 ( w 1 ) 1 ) 0 ) 1 x xr 2 Q + γ 1 ( w 1 ). (18) 2 Following imilar argument a in the proof of Theorem IV.3 (x, z r) it can be hown that there exit K -cla function α( ), α( ) uch that VN r (x, z r) α( x x r ) x XN r r and VN (x, z r) α( x x r ) x Ēf (x r, u r ) Z. The condition that remain to be hown i that (18) alo provide a trict decreae that i a function of x x r 2 Q. Lemma IV.2 directly extend to the robut problem formulation by replacing the contraint with the tightened form (Part 1)) and ince the additional cot term V f (x x 0 ) doe not contain the teady-tate x and i irrelevant for the argument (Part 2)). Conidering the ame cae and argument a in the proof of Theorem IV.3, it can then be hown that there exit a K -cla function α 3 ( ) uch that and uing optimality of V r N VN r (Ax + Bκ r (x, z r ) + w 1, z r ) VN r (x, z r ) α 3 ( x x r ) + γ 1 ( w 1 ). For the econd part of the proof, uniform continuity of the optimal value function VN r (x, z r) follow a in the non-robut cae from the proof of Lemma V.1. Therefore, there exit a K-cla function γ 2 ( ), uch that VN r r (y) VN (x) γ 2 ( y x ) and we obtain VN r (Ax + Bκ r (x, z r ) + w 1 + w 2, z r ) VN r (x, z r ) α 3 ( x x r ) + γ 1 ( w 1 ) + γ 2 ( w 2 ), proving the reult. Theorem V.4 prove ISS of the uncertain ytem in (16) controlled by κ r (x, z r ) in (17) with repect to the diturbance

9 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X w 1 W 1 and w 2 W 2 and how that the reult preented for the oft contrained MPC method can directly be extended to the combination of a robut and oft contrained approach. VI. NUMERICAL EXAMPLES In thi ection, the propoed method for oft contrained and robut oft contrained MPC are illutrated for a mallcale example and computation time for a large-cale problem are provided. All et computation were carried out uing YALMIP [31] and the MPT toolbox [32]. x X 5 X W0.15 X W0.25 X 5 A. Illutrative Example Conider the following untable ytem: [ ] [ ] x(k + 1) = x(k) + u(k) + w(k). (19) The prediction horizon wa choen to be N = 5, the contraint on the tate and control input to x 5 and u 1, Q = I, R = 1 and S = 100I. The terminal cot function V f ( ) i taken a the uncontrained infinite horizon optimal value function for the nominal ytem with P = [ ] and κ f (x) = K(x x ) + u i the correponding optimal LQR controller. The exact penalty multiplier were choen a ρ ɛ = ρ x = ρ u = 100, which wa oberved to provide optimality in X N. For implicity, we take x r = 0, u r = 0 a the reference teady-tate for the following illutration. 1) Soft Contrained MPC: The feaible et X5 and the enlarged terminal et Cf for the oft contrained approach P N (x, z r) are illutrated and compared with the feaible et X 5 and terminal et C f for the hard contrained problem P N (x, z r ) in Figure 2, which demontrate that the oft contrained approach ignificantly enlarge the feaible et and thereby the region of attraction for the nominal cloed-loop ytem. Thi how that the propoed method provide the benefit of oft contraint and enure feaibility of the optimization problem in a large region while till guaranteeing tability of the cloed-loop ytem. x X 5 X 5 X C f x 1 Fig. 2. Feaible and terminal et for the oft contrained problem P N (x, zr) for N = 5 in comparion with the feaible and terminal et of the hard contrained problem P N (x, z r). C f x 1 Fig. 3. Feaible et and RPI et of P N (x, zr) for w {0.15, 0.25} together with a cloed-loop trajectory tarting at x(0) = [20, 1.25] T under a equence of extreme diturbance. Dot repreent the optimal teady-tate x (x(k), 0) at each ampling time. x X 5 X W0.25 X rh 5 X r 5 X r W x 1 Fig. 4. Feaible et and RPI et of P N (x, zr) and Pr N (x, zr) for w 1 = 0.1, w 2 = 0.15 together with cloed-loop trajectorie tarting at x(0) = [ ] T, where the dahed line olve P N (x, 0) and the olid line P r N (x, 0). X rh 5 i the feaible et for the robut hard contrained problem. We now analyze the robut tability propertie of the example ytem (19) under the oft contrained control law κ (x, z r ) in (9). Figure 3 how the ize of the RPI et XW w for two bound W w {w w w}, with w {0.15, 0.25} aeed by ampling. Note that for a robut tube-baed approach the feaible et i alway a ubet of X N. Thi demontrate the advantage of the oft contrained approach, where input-to-tate tability in the preence of a comparably large diturbance w W 0.25 can be guaranteed in the RPI et XW 0.25 X N. In addition, a cloed-loop trajectory tarting at x(0) = [ ] T under a equence of extreme diturbance with w(k) 2 = 0.25 k 0 i hown a well a the correponding optimal teady-tate at each ampling time, demontrating that the cloed-loop ytem i table and doe not leave the RPI et XW ) Robut Soft Contrained MPC: In the following, the propertie of the robut oft contrained MPC approach decribed in Section V-B are illutrated. Conider again ytem (19) with w(k) = w 1 (k) + w 2 (k) that i now ubject to two type of diturbance. Figure 4 how the comparion of the feaible et X5 r and the RPI et XW r 0.25 for w 1 X 5

10 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X 2014 Ww 1, w2 Ww 2 with w 1 = 0.1, w 2 = 0.15 in comparion with the feaible et X5 and the RPI et XW of the pure oft 0.25 contrained approach. The feaible et of the hard contrained robut MPC problem, i.e. Problem Pr N (x, zr ) with (x, u ) S, = 0, i = 0, w = w1 + w2 W0.25 i denoted by X5rh. Due to the tightening of the input contraint, the robut oft contrained approach ha a maller feaible et when compared to the pure oft contrained method. However, in comparion with the hard contrained robut MPC method, the feaible et for the combined approach i ignificantly larger, while till guaranteeing ISS with repect to w1 in X5r. The r RPI et XW i only lightly maller than X5r and input0.25 to-tate tability with repect to the combined diturbance w = w1 + w2 W0.25 i provided in a comparably large et. Cloed-loop trajectorie tarting from x(0) = [ ]T 136 hown for both9 Soft MPCcontrained with Robut Stability Guarantee are the Contrained robut oft and the pure oft contrained approach under a equence of extreme diturrobut Soft Contrained MPC bance kw1 (k)k2 = 0.1 and a diturbance kw2 (k)k2 W0.15 In the following the propertie of the robut oft contrained MPC approach decribed in k 0 that additionally affect the ytem at every third Section 9.7 aretime. illutrated. Conider ytem (9.17) with = w1 (k)+w 2 (k) that ampling Figure 4 again demontrate thatw(k) both approache i now ubject to two type of diturbance. Figure 9.5(b) how the comparion of the provide the cloed-loop ytem. The r feaible et input-to-tate X5r and the RPI ettability XW for wof 1 Ww, w2 Ww with w 1 = 0.1, w 2 = 0.1 in comparion the feaible etapproach XN and the RPI et XW the pure oft contrained robut oftwith contrained teer theof ytem earlier tor approach. The feaible et of the hard contrained robut Problem ward the origin and the trajectory remainmpc in Xproblem,, i.e. ince the W r rh 0.25 PN (x) with x = u = 0,! = 0,!i = 0, w = w1 + w2 W0.2 i denoted by X5. Due robut formulation i contraint, deignedthetorobut counteract the diturbance to the tightening of the input oft contrained approach ha a w contrained approach foroft a larger deviation ignificantly maller feaible et when comparedallow to the pure contrained method. 1. The oft robut MPC method, the feaible However, comparion the RPI hard contrained of the in tate withinwiththe et XW. Thi how that by 0.25 et for the combined approach i ignificantly larger while till guaranteeing ISS with uing a combination a robut and oftnominal contrained repect to w1 in XNr, which of i almot a large a the feaible etmethod, X5. The r r robutne againt a certain diturbance ize can be with provided RPI et XW i only lightly maller than X5 and input-to-tate tability repect to the combined diturbance w =ifwthe provided inthi a comparably 1 + wditurbance 2 W0.2 i till exceed while enuring tability bound. T large et. Cloed-loop trajectorie tarting from x(0) = [ 9.6 1] are hown for both approache under a equence of extreme diturbance w1 (k) = ±0.1 and a diturbance w2 (k) W0.1, k 0, of varying ize that additionally affect the ytem at every third B. Large-Scale Example ampling time. Figure 9.5(b) demontrate that both approache provide input-totate tability of the cloed-loop ytem,demontrate however the robut that oft contrained approach The following example the propoed provide a better performance, ince it i deigned for the diturbance w1 that contantly oft contrained MPC approach can be applied to problem affect the ytem. of ignificant ize. We conider two ytem of 6 and 12 ocillating mae, which are interconnected by pring and Large-Scale Example damper and are connected to wall on each ide. The actuator We now apply the oft contrained MPC approach to a large cale example and eexert tenion between different mae. The ytem with 12 timate the computational effort required to olve the correponding SOCP. Conider mae i ofillutrated Fig.of5,12 where the ix which mae problem i the problem regulating a in ytem ocillating mae are interconnected 9.6. by pring-damper ytem and connected to ix wall mae on the ide, a hown in Fig.The obtained by conidering the firt from the left. The ix actuator exert tenion between different mae. The mae are 1, the pring u6 u3 F u1 u2 u4 u5 Figure 9.6: Sytem of ocillating mae. Fig. 5. Sytem of 12 ocillating mae. contant are 1, the damping contant are 0.1 and F = 1.05x1. The tate and input contraint are $u$ 1, $x$ 4, the horizon i choen a N = 5 and the weight mae have value 1 kg, the pring contant i k = 0.7 N/m, the damping contant i 0.1 N/m and an external force F = 0.5x1 i applied to the firt ma on the left. The control input are contrained to lie in ±1 N and the diplacement of the mae i contrained in ±4 m. The ytem i dicretized with ampling time t = 0.5. The parameter of the problem PN (x, zr ) are N = 10, Q = I and R = I, S = 100I, ρ = ρx = ρu = 100. In order to exemplify the computation time that can be achieved for the propoed oft contrained procedure, the reulting SOCP wa olved uing the olver MOSEK [12] and ECOS [13], which i particularly deigned to run on embedded 10 platform, on a MacBook Pro with Intel Core i7 CPU at 2.6 GHz. The problem dimenion, the ize of the reulting SOCP and the minimum, average and maximum computation time for the 6 and 12 mae problem uing 100 randomly ampled initial tate are given in Table I. The reult how that the SOCP can be olved in the milliecond range even for big problem dimenion, demontrating the practical applicability of the method. VII. C ONCLUSIONS In thi paper, a new oft contrained MPC method baed on a finite horizon MPC cheme for tracking wa introduced that provide cloed-loop tability even for untable ytem. The approach combine the benefit of a oft contrained cheme with the deirable propertie of MPC and enure feaibility of the optimization in a large region of the tate pace while guaranteeing cloed-loop tability. The propoed control law preerve the optimal tracking performance of the correponding hard contrained MPC approach whenever the tate contraint can be enforced. Aymptotic tability of all feaible reference teady-tate for the nominal ytem under the oft contrained control law wa hown, a well a inputto-tate tability in the preence of additive diturbance. The propoed oft contrained formulation wa combined with a robut approach, in order to provide contraint atifaction depite a certain diturbance ize, while enuring feaibility and tability for exceeding diturbance by mean of oft contraint. It wa hown that the reult on input-to-tate tability extend to thi cae. The preented numerical example illutrate the propertie of the new oft contrained cheme and demontrate that it can be applied to problem of ignificant ize in order to enure afety and feaibility during online operation. ACKNOWLEDGMENT The author thank A. Domahidi for hi valuable comment and upport with the implementation in ECOS. The reearch leading to thee reult wa upported by the European Union Seventh Framework Programme FP7/ under grant agreement number FP7-ICT A PPENDIX Detailed proof of Lemma III.2, reult 2) in Lemma IV.2 and Theorem IV.3 are provided in the following, where the lat two are imilar to the proof of Lemma 5.6 and 5.7 in [26], but are included to make the paper elf-contained. Proof of Lemma III.2: Conider a parametrization of the teady-tate by the parameter θ Rnθ [9]: x Mx = Mθ = θ, (20) u Mu where the column of M form a bai for the null pace of the matrix I A B, nθ i the dimenion of the null pace and Mx, Mu are appropriate partition of M. We define the augmented ytem v T = [xt xt θt ] with dynamic v + =

11 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X ECOS [13] MOSEK [12] M n m N nvar neq ineq (min / avg / max) (min / avg / max) / 20.2 / 25.0 m 10.8 / 13.0 / 18.9 m / / m 49.3 / 57.3 / 72.9 m TABLE I COMPUTATION TIMES FOR SOLVING THE SOFT CONSTRAINED PROBLEM P N (x, zr) FOR 100 RANDOMLY SAMPLED INITIAL STATES. NVAR, NEQ AND INEQ DENOTE THE NUMBER OF VARIABLES, EQUALITY CONSTRAINTS AND INEQUALITY CONSTRAINTS, RESPECTIVELY, IN THE RESULTING SOCP. A v,k v, where A v,k = [ ] A+BK 0 0 I. An ellipoidal invariant et of the form Ω f {v [ ] } R n v T Q 1 1 v 1 Q 1 2 can be computed by olving a convex LMI [10], where only the input contraint are conidered and the tate contraint are neglected: [ ] { [ ] [ ] Q1 Q 2 = argmin log det Q1 Q1 Q 1A T K Q 2 0, A Q K Q 1 Q 1 1,Q 2 Q K T G T u,j Q M T u G T u,j 2 2 f 2 u,j j = 1,..., p u }, (21) with A K = A + BK. Uing the relationhip θ = Mx T x + Mu T u thi can be directly tranformed into the ellipoidal terminal et Ef (x, u ) in (8) with T = Q 1 1 and r(x, u ) = (Mx T x +Mu T u ) T Q 1 2 (M x T x +Mu T u ). Note that if problem (21) i unbounded, boundedne can be impoed by etting a large upper bound on x, e.g. G x M x θ γf x for ome large contant γ > 0. Proof of reult 2) in Lemma IV.2: We denote A K A + BK, x a 0 x a 0 x a, x a x a x r, u a u a u r, ν = 1 α. By Aumption II.5, V f (x) i a Lyapunov function and P Q + K T RK, therefore A i K x Q A i K x P x P and A i K x K T RK A i K x P x P From the ue of the auxiliary control law, i.e. u a i = KAi K xa 0 +u a, and recalling that x a = Ax a +Bu a we obtain l(x a i x a,α, u a i u a,α) l(x a i x a, u a i u a ) = A i K x a 0 + ν x a 2 Q + KA i K x a 0 + ν u a 2 R A i K x a 0 2 Q KA i K x a 0 2 R, 2ν A i K x a 0 Q x a Q + ν 2 x a 2 Q + 2ν A i K x a 0 K T RK u a R + ν 2 u a 2 R 3ν 2 x a 2 P + 2ν 2 x a P u a R + ν 2 u a 2 R and imilarly V f (x a N xa,α) V f (x a N xa ) 3ν 2 x a 2 P. Uing V o (x a,α x r, u a,α u r ) = αv o ( x a, u a ), we get V N (x a, u a, z a,α, z r ) V N (x a, u a, z a, z r ) + ν 2 x a 2 P ν[3(n + 1) x a 2 P ν + 2N x a P u a R ν + N u a 2 Rν V o ( x a, u a )] 0, which i atified for 0 < ν hence α < 1, proving the reult. Proof of Theorem IV.3: V o( x a, ua ) 3(N+1) x a 2 P +2N xa P u a R+N u a 2 R and We will how that VN (x, z r) i a Lyapunov function. Let (x, u ) be the optimal teady-tate for tate x and reference z r. Feaibility of P N (x+, z r ) follow from feaibility of the hifted equence hown in Lemma IV.1. We have that VN (x, z r) x x 2 Q + V o(x x r ) α( x x r ) x XN. By optimality of VN (x, z r) we obtain VN (x, z r) V f (x x r ) α( x x r ) x Ef (x r, u r ) X, where α( ), α( ) are uitable K -cla function. The important condition to prove i that V = VN (x+, z r ) VN (x, z r) β x x r 2 Q with β > 0, which will be hown by conidering the following two cae. If the tate x + i ufficiently far away from the optimal artificial teady-tate at the previou time tep x, then the decreae obtained by the hifted equence when keeping the ame artificial teady-tate provide β > 0. If x + i cloe to x (and the decreae could potentially go to zero), then β > 0 i proven by howing that the teady-tate can be moved toward the target teady-tate uing the auxiliary control law. Let α (0, 1) and δ > 0 be contant atifying the condition in Lemma IV.2, a well a the additional condition {x x x P δ} Ef (x, u ) (x, u ) S. Cae 1: x + x P δ By the definition of x +, x P = P 1 2 x 2 and x a = Ax a + Bu a, we obtain: δ A(x x ) + B(u 0 u ) P A P x x 2 + B P u 0 u 2. Therefore either B P u 0 u 2 0.5δ A P x x 2 0.5δ x x 2 Q δ 2, or B P u 0 u 2 0.5δ u 0 u 2 R δ 2, where δ 0.5δ = c2 max( A P, B P ) and c 2 1 i uch that c 2 Q I, c 2 R I. From thi and (12) we then get V β x x r 2 P with β = min x XN (( x x 2 Q + u 0 u 2 R )/ x x r 2 Q ) and therefore β min x XN ( δ 2 / x x r 2 Q, 1) > 0. Cae 2: x + x P δ In thi cae, x + Ef (x, u ), and the optimal equence to regulate the ytem to the teady-tate (x, u ) i by applying the auxiliary control law κ tr f (x) = u + K(x x ). Cae 2a: (1 α) x x r P x + x P δ Following imilar argument a in Cae 1 it can be hown that (1 α) x x r P A P x x 2 + B P u 0 u 2 and by recalling that 2 Q 2 P there exit a contant δ (0, 1] uch that δ 2 (1 α) 2 x u 0 u 2 R x r 2 Q. From thi and (12) we get x x 2 Q + V 1 x x 2 Q δ 2 (1 α) 2 x x r 2 P 1 4 δ 2 (1 α) 2 x x r 2 Q

12 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X and therefore β = 1 4 δ 2 (1 α) 2 > 0. Cae 2b: x + x P (1 α) x x r P δ In thi cae we can ue reult (13) in Lemma IV.2. Let u a, x a be the input and tate equence generated by applying the auxiliary control law κ tr f (x) = u + K(x x ) tarting from x + and let ɛ a be the aociated lack. We denote x,α = αx + (1 α)x r and imilarly for u,α. We how that α < 1, i.e. moving the teady-tate x toward x r, i feaible and provide the required cot decreae. Feaibility reduce to howing atifaction of the terminal contraint: x a N x,α P = x a N x + (1 α)(x x r ) P (A + BK) N (x + x ) P + (1 α) x x r P δ, which i atified for ome α < 1, ince (A + BK) N (x + x ) P < δ, proving that x a N E f (x,α, u,α ) by the definition of δ. In order to how the decreae, i.e. β > 0, we firt prove that the auxiliary control law provide a lower cot than the hifted equence. By optimality of the auxiliary control law for regulation to (x, u ), we have V N (x a, u a, z, z r ) V N (x hift, u hift, z, z r ). From x a i x P x + x P δ, the condition on δ in Lemma IV.2, i.e. 2γδc ξ 1+ξ f x, where T γ 2 P, and (1 + ξ)g x x f x + ɛ, we obtain c x a i x T 1 2 (f x G x x ξ ɛ ) i = 1,..., N. = 0 i feaible for x a, providing that ɛ a ɛ hift and hence VN (xa, u a, z a, ɛ a, z r ) VN (xhift, u hift, z, ɛ hift, z r ). It then follow from optimality of VN (x+, z r ), Lemma IV.1 and IV.2 that A a reult, ɛ a = ɛ, ɛ a i V VN(x a, u a, z,α, ɛ a α, z r ) VN (x, z r ) VN(x a, u a, z, ɛ a, z r ) VN (x, z r ) (1 α) 2 x x r 2 P VN(x hift, u hift, z, ɛ hift, z r ) VN (x, z r ) (1 α) 2 x x r 2 P x x 2 Q (1 α) 2 x x r 2 P 1 2 (1 α)2 x x 2 Q i.e. β = 1 2 (1 α)2 > 0. A decreae of the Lyapunov function with β > 0 i therefore guaranteed in all cae, which conclude the proof. REFERENCES [1] D. Q. Mayne, J. B. Rawling, C. V. Rao, and P. O. M. Scokaert, Contrained model predictive control: Stability and optimality, Automatica, vol. 36(6), pp , [2] E. Zafiriou and H.-W. Chiou, Output contraint oftening for SISO model predictive control, in Proc. of the American Control Conference, 1993, pp [3] A. Zheng and M. Morari, Stability of model predictive control with mixed contraint, IEEE Tranaction on Automatic Control, vol. 40, no. 10, pp , [4] P. O. M. Scokaert and J. B. Rawling, Feaibility iue in linear model predictive control, AIChE Journal, vol. 45, pp , [5] N. M. C. de Oliveira and L. T. Biegler, Contraint handling and tability propertie of model-predictive control, AIChE Journal, vol. 40, pp , July [6] E. C. Kerrigan and J. M. Maciejowki, Soft contraint and exact penalty function in model predictive control, in Proc. UKACC International Conference (Control), Cambridge, UK, Sept [7] S. C. Thomen, H. Niemann, and N. K. Poulen, Robut tability in predictive control with oft contraint, in Proc. of the American Control Conference, 2010, pp [8] A. G. Will and W. P. Heath, Barrier function baed model predictive control, Automatica, vol. 40, pp , [9] D. Limon, I. Alvarado, T. Alamo, and E. F. Camacho, MPC for tracking piecewie contant reference for contrained linear ytem, Automatica, vol. 44, pp , [10] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge Univerity Pre, [11] M. Anderen, J. Dahl, Z. Liu, and L. Vandenberghe, Interior-point method for large-cale cone programming, in Optimization for Machine Learning. MIT Pre, 2011, pp [12] E. D. Anderen, C. Roo, and T. Terlaky, On implementing a primaldual interior-point method for conic quadratic optimization, Mathematical Programming, vol. 95, pp , [13] A. Domahidi, E. Chu, and S. Boyd, ECOS: An SOCP olver for embedded ytem, in Proc. of the 2013 European Control Conference, July 2013, pp [14] M. N. Zeilinger, C. N. Jone, and M. Morari, Robut tability propertie of oft contrained MPC, in Proc. of the 49th IEEE Conf. on Deciion and Control, 2010, pp [15] M. Vidyaagar, Nonlinear Sytem Analyi, 2nd ed. Prentice Hall, [16] F. Blanchini, Set invariance in control, Automatica, vol. 35, pp , [17] Z.-P. Jiang and Y. Wang, Input-to-tate tability for dicrete-time nonlinear ytem, Automatica, vol. 37, pp , [18] E. D. Sontag and Y. Wang, New characterization of input-to-tate tability, IEEE Tranaction on Automatic Control, vol. 41, pp , [19] D. Limon, T. Alamo, D. M. Raimondo, D. Muñoz de la Peña, J. M. Bravo, A. Ferramoca, and E. F. Camacho, Input-to-tate tability: A unifying framework for robut model predictive control, Lecture Note in Control and Information Science, vol. 384, pp. 1 26, [20] J. B. Rawling and D. Q. Mayne, Model Predictive Control: Theory and Deign. Nob Hill Publihing, [21] J. Maciejowki, Predictive Control with Contraint. Prentice Hall, [22] A. Ferramoca, D. Limon, I. Alvarado, T. Alamo, and E. F. Camacho, MPC for tracking with optimal cloed-loop performance, Automatica, vol. 45, pp , Augut [23] D. Limon, I. Alvarado, T. Alamo, and E. F. Camacho, Robut tubebaed MPC for tracking of contrained linear ytem with additive diturbance, Journal of Proce Control, vol. 20, no. 3, pp , March [24] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrihnan, Linear Matrix Inequalitie in Sytem and Control Theory, er. Studie in Applied Mathematic. Philadelphia, PA: SIAM, Jun. 1994, vol. 15. [25] R. Fletcher, Practical Method of Optimization, 2nd ed. John Wiley and Son, New York, [26] M. N. Zeilinger, D. M. Raimondo, A. Domahidi, M. Morari, and C. N. Jone, On Real-time Robut Model Predictive Control, 2014, Automatica. To appear. [27] G. Grimm, M. J. Meina, S. E. Tuna, and A. R. Teel, Example when nonlinear model predictive control i nonrobut, Automatica, vol. 40, no. 10, pp , [28] D. Q. Mayne, M. M. Seron, and S. V. Rakovic, Robut model predictive control of contrained linear ytem with bounded diturbance, Automatica, vol. 41, no. 2, pp , [29] B. Bank, J. Guddat, D. Klatte, B. Kummer, and K. Tammer, Non-linear parametric optimization. Akademie-Verlag, Berlin, [30] M. N. Zeilinger, Real-time Model Predictive Control, Ph.D. diertation, ETH Zurich, [31] J. Löfberg, YALMIP : A Toolbox for Modeling and Optimization in MATLAB, in Proc. of the 2004 IEEE Int. Symp. on Computer Aided Control Sytem Deign, 2004, pp [32] M. Kvanica, Real-Time Model Predictive Control via Multi-Parametric Programming: Theory and Tool. VDM Verlag, 2009.

13 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. X, NO. X, X Melanie N. Zeilinger Melanie N. Zeilinger received the diploma in Engineering Cybernetic from the Univerity of Stuttgart, Germany, in She conducted her diploma thei reearch at the Department of Chemical Engineering, Univerity of California at Santa Barbara, USA, in In 2011 he received the Dr.c. degree with honor in Electrical Engineering from ETH Zurich, Switzerland. From he wa a potdoctoral fellow in the Automatic Control Laboratory at the École Polytechnique Fédérale de Lauanne (EPFL), Switzerland. She i currently a Marie Curie fellow at the Max-Planck Intitute for Intelligent Sytem, Tübingen, Germany, and a potdoctoral reearcher at the Univerity of California, Berkeley, USA. Her current reearch interet include real-time and ditributed control and optimization and the development of modeling and control technique for green energy-efficient technologie. Manfred Morari Manfred Morari wa head of the Department of Information Technology and Electrical Engineering at ETH Zurich from 2009 to Jan2012. He wa head of the Automatic Control Laboratory from 1994 to Before that he wa the McCollum-Corcoran Profeor of Chemical Engineering and Executive Officer for Control and Dynamical Sytem at the California Intitute of Technology. He obtained the diploma from ETH Zurich and the Ph.D. from the Univerity of Minneota, both in chemical engineering. Hi interet are in hybrid ytem and the control of biomedical ytem. In recognition of hi reearch contribution he received numerou award, among them the Donald P. Eckman Award, the John R. Ragazzini Award and the Richard E. Bellman Control Heritage Award of the American Automatic Control Council, the Allan P. Colburn Award and the Profeional Progre Award of the AIChE, the Curti W. McGraw Reearch Award of the ASEE, Doctor Honori Caua from Babe-Bolyai Univerity, Fellow of IEEE, IFAC and AIChE, the IEEE Control Sytem Technical Field Award, and wa elected to the National Academy of Engineering (U.S.). Manfred Morari ha held appointment with Exxon and ICI plc and erve on the technical adviory board of everal major corporation. Colin N. Jone Colin N. Jone i an Aitant Profeor in the Automatic Control Laboratory at the École Polytechnique Fédérale de Lauanne (EPFL) in Switzerland. He wa a Senior Reearcher at the Automatic Control Lab at ETH Zurich until 2010 and obtained a Ph.D. in 2005 from the Univerity of Cambridge for hi work on polyhedral computational method for contrained control. Prior to that, he wa at the Univerity of Britih Columbia in Canada, where he took a bachelor and mater in Electrical Engineering and Mathematic. Hi current reearch interet are in the area of high-peed predictive control and optimization, a well a green energy generation, ditribution and management.