arXiv:1503.03314v1 [math.OC] 11 Mar 2015 rosre ros h neligoe-opotmlcon- optimal open-loop underlying the outliers, errors, sensor observer disturbances, or of presence Fur- result the behavior. may in closed-loop intothermore, which unstable even taken design, or MPC not suboptimal in the usually in are explicitly optimization account required aspects several the algorithmic and example, of For implementation, MPC theoretical actual remain. disadvantages between their gap and a concepts is often as there well However, as more [29] and and therein. [22] more references see is processes, a industrial that as to concept considered applied be control can accepted linear MPC widely both and for systems, system nonlinear closed-loop and the sta- of the account properties concerning into bility results theoretical taken various exist be problem. There can optimization corresponding input the a on within and explicitly pre- both constraints states the controlled, potential system on be and the to Based objective system cost the fashion. user-defined time of the next horizon dynamics the to receding dicted at applied a repeated is in is input step optimization control the and optimal first plant the the Only of state. is element system evolving that the problem by control parametrized suitable optimal a fashion, open-loop horizon finite-horizon receding a in instant, sampling T ∗ hswr a upre yteDush Forschungsgemein- Deutsche the by supported was work This uain B4521 n lse fEclec nSimulati in Excellence of 310/2). Cluster EXC and Technology, 425/2-1, Com- EB and Control putation, in Ways Novel (Emmy-Noether-Grant, schaft banasaiiigfebc a ysliga each at solving by law feedback stabilizing a obtain h n togycne ucin h eut r lutae b illustrated are results The t function. as convex sequence strongly input control and based necessarily stabilizing the not characterize are to schemes asympto MPC constrain guarantee stabilizing and proposed performance to like allow properties further that discuss results present We control. ai dao oe rdciecnrl(P)i to is (MPC) control predictive model of idea basic e nti ae,w netgt h s frlxdlogarithmic relaxed of use the investigate we paper, this In eae oaihi are ucinBased Function Barrier Logarithmic Relaxed oe rdcieCnrlo ierSystems Linear of Control Predictive Model .Introduction I. nvriyo ttgr,Paewlrn ,750Sutat Ge Stuttgart, 70550 9, Pfaffenwaldring Stuttgart, of University hita elradCrsinEbenbauer Christian and Feller Christian nttt o ytm hoyadAtmtcControl Automatic and Theory Systems for Institute e-mail: { christian.feller,ce Abstract on emnmzro lblydfie,cniuul differentiab continuously defined, globally a of minimizer he aifcini eedneo h neligrelaxation. underlying the of dependence in satisfaction t en fanmrclexample. numerical a of means y 1 } i tblt ftecrepnigcoe-opsse,and system, closed-loop corresponding the of stability tic in e 8 0 2 7.Hwvr w andisadvantages main two However, 27]. 12, optimiza- to 10, on-line [8, allow iterative see thus, tion, any without and, MPC reformulation implement solution unconstrained optimal the the track of asymptotically that al- continuous-time gorithms for open-loop accessible the problem makes control particular, approach optimal based In function tai- barrier techniques. on the optimization based which implementation numerical design, efficient lored reformu- MPC an based the for allows into function then integrated barrier already a is Thus, lation optimiza- unconstrained problem. even to tion or reduc- problem constrained time control equality same optimal an the open-loop at underlying while the ing system closed-loop stability the asymptotic based guarantee of function to barrier allow references, approaches above [24]. MPC the algorithms is in it optimization shown like interior-point As terms in function barrier done suitable also of func- means cost the by into schemes, op- tion incorporated MPC open-loop are the problem based control in recently function timal occurring barrier extended constraints which In been inequality MPC, the 13]. has 11, based and [10, [35] function in given in barrier is hand introduced of other was concept the on on schemes approach the algorithms MPC particular by MPC between One and gap 36]. one the 28, the reduce 9, to 5, allows [2, that oper- e.g. closed-loop desired see in the ation, running of when some preserve properties still theoretical possible, if implemen- de- that, MPC on tations algorithmic spent reliable been and recent efficient has com- In signing effort a algorithm. research control to considerable respective years, leading the of infeasible, crash become plete may problem trol are ucin ntecneto iermdlpredictive model linear of context the in functions barrier na xlcttria e rsaecntan n allow and constraint state or set terminal explicit an on @ist.uni-stuttgart.de rmany ∗ The le, remain. On the one hand, the open-loop optimal con- paces. For any arbitrary set S, the expression S◦ will de- trol problem may still become infeasible since the barrier note the open interior and ∂S the boundary. Moreover, ⊤ functions are only defined within the interior of the corre- 1 := 1 1 of suitable dimension. sponding constraint sets. On the other hand, the barrier ···   function based approach inherently requires the use of a II. Problem Setup terminal set or terminal equality constraint, which is not desirable, and also typically not used, in practice. In this paper, we consider the control of linear time- In this paper, we show that both of these problems can invariant discrete-time systems of the form be eliminated by making use of so-called relaxed loga- x(k +1)= Ax(k)+ Bu(k) , (1) rithmic barrier functions, i.e., barrier functions that are smoothly extended by a suitable penalty term outside of where x(k) Rn refers to the vector of system states and the corresponding constraint set [3, 17, 23]. We discuss u(k) Rm refers∈ to the vector of system inputs, both at suitable relaxation procedures (Section A) and present re- time∈ instant k 0. Moreover, the matrices A Rn×n sults that allow to guarantee asymptotic stability of the and B Rn×m≥describe the corresponding system∈ dy- closed-loop system with and without making use of ter- namics,∈ where we assume (A, B) to be stabilizable. The minal sets (Sections B, C), to determine and control the control task is to regulate the system state to the origin maximal violation of input and state constraints in closed- while minimizing a given, user defined performance crite- loop operation (Section D), and to recover the optimal- rion and satisfying state and input constraints of the form ity properties of both nonrelaxed barrier function based x(k) and u(k) for all k 0. Here, Rn and and conventional linear MPC schemes (Section E). Fur- R∈m Xare typically∈ U considered≥ to be givenX⊂ convex sets thermore, based on the presented results, we provide in U⊂that contain the origin in their interior. In this paper, we Section V a step-by-step procedure for the constructive assume and to be polytopes defined as design of the overall MPC scheme. X U n The basic idea of relaxed barrier function based MPC is = x R : Cxx dx , (2a) X { ∈ m ≤ } closely related to approaches based on soft constraints = u R : Cuu du , (2b) or penalty functions, see e.g. [20, 37]. However, in these U { ∈ ≤ } approaches, closed-loop properties like stability or strict where C Rqx×n, C Rqu×m and d Rqx , d Rqu x ∈ u ∈ x ∈ ++ u ∈ ++ constraint satisfaction can usually only be guaranteed with qx, qu N . In linear MPC, this problem setup is ∈ + when making use of nonsmooth or even exact penalty usually handled by solving at each sampling instant an functions. The key feature of the relaxed barrier functions open-loop optimal control problem of the form discussed in this paper is that they result in a smooth for- N−1 mulation of the overall problem while still allowing for an ∗ J (x) = min ℓ(x ,u )+ F (x ) (3a) arbitrary close approximation of the corresponding non- N u k k N k=0 relaxed case. As a result, the stabilizing control input X s.t. x = Ax + Bu , x = x , (3b) can be characterized as the minimizer of a globally de- k+1 k k 0 fined, twice continuously differentiable, and strongly con- xk , k =0,...,N 1, xN f , (3c) ∈ X − ∈ X vex cost function, which makes it efficiently computable uk , k =0,...,N 1 . (3d) by standard nonlinear programming algorithms. Some ∈U − preliminary results on the concept of relaxed barrier func- for the current system state x = x(k) and a finite predic- N tion based MPC have been presented in [12]. In this pa- tion and control horizon N +. Here, the stage cost Rn Rm R ∈ Rn R per we significantly extend these results and provide an ℓ : + and the terminal cost F : + × → 2 2 → in-depth study of several theoretical and practical aspects are usually defined as ℓ(x, u)= x Q + u R and F (x)= 2 k k k k Sn of the presented linear MPC schemes. A particularly in- x P for appropriately chosen weight matrices Q +, k k m n ∈ teresting new result is the insight that the use of relaxed R S , P S . Furthermore, u = u0,...,uN−1 ∈ ++ ∈ ++ { } barrier functions allows to ensure global asymptotic sta- denotes the sequence of control inputs over the predic- bility of the closed-loop system without making use of a tion horizon N, while f refers to a closed and con- X terminal set. vex terminal constraint set that may be used to guar- antee stability properties of the closed-loop system. Note Throughout the paper we will make use of the follow- that we make use of subindices to distinguish open-loop R R N ing notation. +, ++, and + denote the sets of non- predictions xk, uk from actual state and input trajecto- negative real, strictly positive real, and strictly positive ries x(k),u(k). The control law is obtained by solving (3) Sn Sn natural numbers. Furthermore, + and ++ refer to the at each sampling instant k 0 and applying u(k) = sets of positive semi-definite and positive definite matri- u∗(x(k)) in a receding horizon≥ fashion. Sufficient con- N 0 ces of dimension n +. For any given matrix M or ditions for the recursive feasibility of (3) as well as for the i i ∈ vector v, M and v refer to the i-th row or element and asymptotic stability of the closed-loop system are summa- ⊤ x M := √x Mx for any M S . A polytope is defined k k ∈ + rized in [22]. In the following, we will refer to this setup as the compact intersection of a finite number of halfs- as conventional linear MPC. Moreover, we will often use

2 + + x to denote the successor state and write (1) as x = Bu( ), Bx( ), and Bf ( ) are suitable convex barrier func- Ax + Bu. For given system state x = x(k) and input se- tions· with· domains · ◦, ◦, and ◦ with B (u) U X Xf u → ∞ quence u = u ,...,uN− , the resulting open-loop state for u ∂ , B (x) for x ∂ , and B (x) { 0 1} → U x → ∞ → X f → ∞ sequence is given by x(u, x) = x (u, x),...,xN (u, x) , for x ∂ f . Furthermore, the positive scalar ε > 0 { 0 } → X where the elements xk(u, x), k = 0,...,N are given ac- is the barrier function weighting parameter which deter- cording to (3b) with x0(u, x) = x. For a given optimal mines the influence of the barrier function values on the input sequence u∗(x), we write x∗(x) := x(u∗(x), x) as overall cost function. As outlined above, the goal is now ∗ ∗ well as x (x) := xk(u (x), x). Sometimes we also drop to choose the problem parameters P and f as well as k X the explicit dependence on the current system state for convex barrier functions Bu( ), Bx( ), and Bf ( ) in such ease of notation. a way that a linear MPC scheme· based· on (4)· asymp- totically stabilizes the origin for any arbitrary but fixed R III. Preliminary Results ε ++. If we want to employ standard MPC stabil- ity∈ concepts which are based on using the value function J˜∗ ( ) as a Lyapunov function for the closed-loop system, We introduce in this section some concepts and results on N · nonrelaxed logarithmic barrier function based MPC that this usually requires the overall cost function to be con- we will need in the remainder of the paper. In particular, tinuous as well as positive definite with respect to the we present a novel stability theorem which generalizes origin. In order to ensure the latter property for general different existing ideas and is applicable to a wide class barrier functions, the concept of gradient recentered bar- of barrier function based MPC approaches. rier functions has been introduced in [35]. In addition, a weighting based recentering approach for logarithmic bar- rier functions on polytopic sets has been proposed in [13]. A. Barrier Function Based Model Predictive Control The main idea in barrier function based MPC is to elimi- Definition 1 (Recentered log barrier function [13, 35]). Rr Rq×r Rq nate the inequality constraints from the above MPC open- Let = z : Cz d with C , d ++ P { ∈ ≤ } ∈ ∈ ◦ loop optimal control problem by making use of suitable be a polytopic set containing the origin and let B¯ : R ¯ q ¯ ¯ i Pi → barrier functions with a corresponding weighting factor. , B(z) = i Bi(z) with Bi(z) = ln( C z + d ) be =1 − − Based on this idea, it is possible to reformulate prob- the corresponding logarithmic barrier function. Then, the P ◦ R lem (3) as an equality constrained (or even unconstrained) function BG : + with P → strongly convex optimization problem, which can then be ⊤ BG(z)= B¯(z) B¯(0) [ B¯(0)] z (5) solved by means of tailored optimization procedures like, − − ∇ e.g., the Newton-method. In general, the exact solution defines the gradient recentered logarithmic barrier func- ◦ to the original problem, and hence also the stability prop- tion for the polytopic set . Furthermore, BW : R+ with P P → erties of the corresponding closed-loop system, are recov- q ered when the weighting factor of the barrier functions i BW (z)= (1 + w ) B¯i(z) B¯i(0) (6) approaches zero [6, ch. 11]. However, for an arbitrary − i=1 but fixed nonzero weighting, as it will necessarily occur X
defines the corresponding weight recentered logarithmic in all numerical implementations, stability of the origin is Rq barrier function, where the weighting vector w + by no means guaranteed. Several approaches towards the q ∈ is chosen in such a way that BW (0) = i=1(1 + stabilizing design of barrier function based MPC schemes i ∇ w ) B¯i(0) = 0. have been presented in [10, 11, 13, 35]. In the follow- ∇ P ing, we shortly summarize the main aspects of barrier In principle, the recentering can be seen as a modi- function based linear MPC and present a fairly general fication which preserves the main characteristics of the stability theorem for the considered problem setup with barrier function while ensuring that it is positive definite polytopic input and state constraints. Details on the dis- with respect to the origin. Both recentering approaches cussed MPC approaches can be found in the respective can in general be applied to our problem setup and will references. result in a continuously differentiable cost function that is positive definite and strongly convex in the optimization Let us consider in the following the barrier function based variable u. Moreover, the stability results presented in open-loop optimal control problem this work do hold independently of the underlying recen- N−1 tering method. J˜∗ (x) = min ℓ˜(x ,u )+ F˜(x ) (4a) Let us now turn towards the stability properties of the N u k k N k=0 closed-loop system when applying the barrier function X based MPC feedback u(k)=˜u(x(k)). s. t. xk+1 = Axk + Buk, x0 = x , (4b) Definition 2. For N N , let the feasible set N be ∈ + X where ℓ˜(x, u) := ℓ(x, u)+ εB (u)+ εB (x) and F˜(x) := defined as N := x : u = u ,...,uN− such u x X { ∈ X ∃ { 0 1} F (x)+ εBf (x) are the modified stage and terminal cost that uk , xk(u, x) for k = 0,...,N 1 and 2 2 2 ∈ U ∈ X − terms for ℓ(x, u) = x + u and F (x) = x , and xN (u, x) f . k kQ k kR k kP ∈ X }

3 Definition 3. In the following, the matrix AK := A+BK and [11]. We first show recursive feasibility of prob- ◦ describes the closed-loop dynamics for a given stabilizing lem (4). For any x0 N , there exists by definition ∈ Xu∗ ∗ ∗ local control law u = Kx and BK (x) := Bx(x)+ Bu(Kx) an optimal input sequence ˜ (x0)= u˜0,..., u˜N−1 that refers to the corresponding combined barrier function of guarantees strict satisfaction of all input,{ state, and} ter- input and state constraints for the set K := x : minal set constraints and results in a feasible open-loop Kx . X { ∈ X state sequence x∗(x )= x , x∗,...,x∗ with x∗ ◦. ∈ U} 0 { 0 1 N } N ∈ Xf The successor state is given as x+ = Ax + Bu˜∗ = x∗. As outlined above, different approaches towards the 0 0 0 1 Due to the linear dynamics and the properties of the stabilizing design of barrier function based MPC formu- + terminal set f , see A4 of Assumption 1, u˜ (x0) = lations, i.e., on how to choose the terminal cost matrix P , ∗ ∗ X ∗ u˜1,..., u˜N−1, KxN is a suboptimal but feasible input the terminal set f , and the corresponding barrier func- { } X + tion B ( ), have been presented in [10, 11, 35] and [13]. sequence for the initial state x0 that results in the in f x u+ + · the feasible open-loop state sequence (˜ (x0), x0 ) = In principle, all mentioned approaches are based on the + ∗ ∗ ∗ ∗ ◦ x0 , x2,...,xN , AK xN with AK xN f . This shows idea of choosing the terminal set f as a positively in- { ◦ } ∈ X X that for any x0 N , there exists a feasible input se- variant subset of the state space in which the function ∈ X + quence that ensures that the successor state x0 = Ax0 + BK ( ), i.e., the influence of the input and state constraint ∗ · Bu˜ lies again in the interior of the feasible set N , which barrier functions, can be upper bounded by a quadratic 0 X function. Then, based on this quadratic bound, the ter- guarantees recursive feasibility of the open-loop optimal minal cost matrix P is computed in such a way that it control problem (4). compensates for this influence and ensures that the bar- In the following, we show that the value function satisfies ˜ rier function based terminal cost F ( ) is a local control ∗ + ∗ ˜ ∗ ◦ · J˜ (x ) J˜ (x0) ℓ(x0, u˜ (x0)) x0 . (7) Lyapunov function for the auxiliary control law u = Kx. N 0 − N ≤− 0 ∀ ∈ XN While the approaches in [35] and [10] make use of ellip- + First, due to the suboptimality of u˜ (x0), it holds that soidal terminal sets, the approaches presented in [11] and ˜∗ + ˜∗ ˜ u+ + ˜∗ JN (x0 ) JN (x0) JN (˜ (x0), x0 ) JN (x0), where [13] are based on polytopic terminal sets. In the follow- ˜ u+ − + ≤ − JN (˜ (x0), x0 ) denotes the value of the cost function ing, we present a set of sufficient stability conditions that + evaluated for the suboptimal input sequence u˜ (x0). is used in the remainder of this work and can be seen ˜ u+ + ˜∗ ˜ ∗ Moreover, JN (˜ (x0), x0 ) JN (x0) = F (AK xN ) as a generalization of the ideas presented in the above ˜ ∗ ˜ ∗ ∗ ˜ − ∗ ˜ ∗ − F (xN ) + ℓ(xN , KxN ) ℓ(x0, u˜0) ℓ(x0, u˜0) for any references. ◦ − ≤ − x0 since Sn Sm R ∈ XN Assumption 1. For Q +, R ++, ε ++ and ∈ ∈ ∈× a given stabilizing local control gain K Rm n let the F˜(A x∗ ) F˜(x∗ )+ ℓ˜(x∗ , Kx∗ ) (8a) ∈ K N N N N parameters of the barrier function based open-loop optimal ∗ − ∗ ∗ ∗ = A x 2 x 2 + x 2 + Kx 2 (8b) control problem (4) satisfy the following conditions. K N P N P N Q N R k k∗ −k k k∗ k k ∗ k + εBK (xN )+ εBf (AK xN ) εBf (xN ) (8c) A1: The barrier functions Bu( ) and Bx( ) are recentered ∗ ∗ − ∗ ◦ · · ε (Bf (AK x ) Bf (x )) 0 x . (8d) barrier functions for the sets and , respectively. ≤ N − N ≤ ∀ N ∈ Xf U X Sn ⊤ A2: There exists M +, s.t. BK (x) x Mx x , Here, the first inequality follows from the quadratic bound ∈ ◦ ≤ ∀ ∈ N ⊤ where K , 0 , is a convex, compact set. BK (x) γ x Mx x f and the suitable choice N ⊂ X ∈ N ≤ ∀ ∈ X ⊂ N Sn of the terminal cost matrix P , see A2 and A3 of Assump- A3: The matrix P ++ is a solution to the Lyapunov ∈⊤ ⊤ tion 1. The second inequality holds due to A5 of Assump- equation P = AK P AK + K RK + Q + εM. tion 1. A4: The terminal set f is convex and compact with 0 X + ∈ Finally, the considered problem setup and the design of f , f K , and x = AK x f x f . X X ⊂N ⊂X ∈ X ∀ ∈ X the barrier functions according to A1 of Assumption 1 ˜∗ ◦ R A5: The function Bf ( ) is a recentered barrier function ensure that JN : N + is a well-defined and positive · ◦ X → ∗ for the set f and Bf (AK x) Bf (x) 0 x . definite function with J˜ (x) whenever x ∂ . X − ≤ ∀ ∈ Xf N N Thus, in combination with the→ decrease ∞ property→ (7),X it Based on Assumption 1, we can state the following re- can be used as a Lyapunov function which proves asymp- sult on the stability of the closed-loop system. totic stability of the origin of system (1) under the feed- ∗ ◦ Theorem 1. Let Assumption 1 hold true and let the back u(k)=˜u (x(k)) for any x(0) . 0 ∈ XN feasible set N defined according to Definition 2 have X Remark 1. Note that the barrier function parameter nonempty interior. Then, the barrier function based R MPC feedback u(k)=˜u∗(x(k)) resulting from problem (4) ε ++ can in principle be decreased iteratively in each 0 sampling∈ step or between two consecutive sampling steps asymptotically stabilizes the origin of system (1) under strict satisfaction of all input and state constraints for without loosing feasibility or stability properties of the any initial condition x(0) ◦ . closed-loop system. This might be meaningful for numer- ∈ XN ical reasons or in order to enforce convergence to the op- Proof. The proof uses standard MPC stability arguments timal solution of the conventional MPC problem (3). We and comprises some of the main ideas presented in [10, 35] limit ourselves to fixed values of ε.

4 Different approaches on how to actually construct the tone and continuously differentiable function that satisfies neighborhood and the corresponding quadratic bound β(δ; δ)= ln(δ) as well as limz→−∞ β(z; δ)= . Then, N − ∞ for BK ( ) as well as suitable choices for the terminal set we call Bˆ : R R defined as · → f and the barrier function Bf ( ) have been proposed Xin [10, 11, 13, 35]. However, the fact· that the underlying ln(z) z>δ Bˆ(z)= − (9) barrier functions are only defined in the interior of the re- ( β(z; δ) z δ spective constraint sets may be problematic both from a ≤ practical and from a conceptual point of view. On the one the relaxed logarithmic barrier function for the set R+ and hand, violations of the state, input, and terminal set con- refer to the function β( ; δ) as the relaxing function. straints are not tolerated at all, which might cause severe · problems in the presence of uncertainties, disturbances, A graphical illustration of the basic idea is given in noise, observer errors, or sensor outliers. On the other Fig. 1. In general, it is advisable to choose the relax- ing function β( ; δ) as a strictly convex C2 function that hand, all existing stability concepts inherently require the · use of a suitable terminal set as they are based on upper smoothly extends the natural logarithm at z = δ. In this case, Bˆ( ) is a strictly convex function that is twice bounding the barrier function BK ( ) by a quadratic func- · tion, which is of course only possible· locally in a region continuously differentiable and defined on z ( , ). ˆ ∈ −∞ ∞ around the origin. In the following section, we introduce Note that limδ→0 B(z) B(z) for any strictly feasible z R , which shows that→ the nonrelaxed formulation the concept of relaxed logarithmic barrier function based ∈ ++ MPC and show that it can be used to overcome all these can always be recovered by decreasing the relaxation pa- limitations, allowing for conceptually simpler and more rameter δ to zero. Note that we do not indicate the ex- reliable linear MPC schemes. plicit dependence of the relaxed barrier functions on the relaxation parameter δ for the sake of notational simplic- ˆ IV. Main Results ity. However, we will use B( ) to denote the relaxed ver- sion of a barrier function based· expression B( ). · The basic idea of relaxed logarithmic barrier functions is The first ideas on relaxed (or approximate) logarith- to smoothly extend a given logarithmic barrier function mic barrier functions with a quadratic relaxing function with a suitable, globally defined penalty function [3, 17, β( ; δ) seem to have been proposed in [3] and [23], re- · 23]. In the following, we provide an in-depth study of sev- spectively. In [17], the authors extended the concept to eral interesting theoretical and practical aspects of linear general polynomial penalty terms and applied it in the MPC approaches that are based on such relaxed logarith- context of continuous-time trajectory optimization. In mic barrier functions. In particular, we show that, on particular, the authors make use of the polynomial relax- the one hand, feasibility and stability properties of the ing function nonrelaxed formulation can always be recovered by ap- k proximating the original barrier functions close enough. k 1 z kδ βk(z; δ)= − − 1 ln(δ) , (10) On the other hand, we present novel linear MPC schemes k (k 1)δ − − "  # that, by exploiting the properties of relaxed logarithmic − barrier functions, do not require the use of a terminal set where k> 1 is an even integer. It is easy to verify that the and allow to guarantee global asymptotic stability of the function βk( ; δ) has all the desired properties mentioned · closed-loop system, see Section C. The benefits of global above. As reported in [17], already a quadratic extension stabilization and a simplified design procedure are bought based on k = 2 seems to work well in practice. with the loss of guaranteed input and state constraint In order to avoid large constraint violations, it may be satisfaction. However, as we will show in Section D, the beneficial if the relaxing function increases very rapidly relaxed barrier functions can always be designed in such outside the border of the feasible set. As an alternative a way that an a-priori defined tolerance for the maximal to the polynomial relaxation above, we proposed in [12] violation of state and input constraints is guaranteed for the following exponential relaxing function a certain set of initial conditions. We briefly discuss some z interesting performance and robustness properties of the βe(z; δ) = exp 1 1 ln(δ) , (11) − δ − − proposed MPC schemes in Section E.   which is an upper bound for the function βk( ; δ). In ′ ′ · A. Relaxed Logarithmic Barrier Function Based MPC fact, using the derivatives βk( ; δ), βe( ; δ) and the limit representation of the exponential· function,· it can be We begin our studies by introducing the concept of re- shown that βk(z; δ) βk (z; δ) βe(z; δ) and that laxed logarithmic barrier functions and discussing suit- ≤ +2 ≤ limk→∞ βk(z; δ) = βe(z; δ) z δ. Furthermore, for able realizations based on different relaxing functions. ∀ ≤ β( ; δ) = βk( ; δ) and β( ; δ)= βe( ; δ) it also holds that Definition 4 (Relaxed logarithmic barrier function). Bˆ(·z) B(z)· z R . We· assume· in the following that ≤ ∀ ∈ ++ Consider a scalar δ R++, called the relaxation param- one of the two functions βk( ; δ) and βe( ; δ) is used as eter, and let β( ; δ)∈ : ( ,δ] R be a strictly mono- relaxing function. · · · −∞ →

5 PSfrag replacements Based on these concepts, let us now consider the following 10 relaxed barrier function based MPC formulation 8

N−1 6 ˆ∗ ˆ ˆ JN (x; δ) = min ℓ(xk,uk)+ F (xN ) (12a) u − 4 k ln(z) X=0 β(z; δ) 2 s. t. xk+1 = Axk + Buk, x0 = x , (12b) β(0; δ)  0 where ℓˆ(x, u) := ℓ(x, u)+ εBˆ (u)+ εBˆ (x) for suitable re- δ z u x −1 0 1 2 laxed recentered logarithmic barrier functions Bˆ ( ) and u · z Bˆx( ) as discussed below. The term Fˆ( ) denotes a suit- able· relaxed terminal cost function which· we do not spec- Figure 1: Left: Principle of relaxed logarithmic barrier ify at the moment. As we will see in the Sections B and C, functions based on the relaxing function β(z; δ). Right: the choice of Fˆ( ) is crucial for ensuring stability proper- Weight recentered logarithmic barrier function (solid) and re- ties of the closed-loop· system. laxed weight recentered logarithmic barrier function for δ ∈ {0.01, 0.1, 0.5, 1} for the constraint −1 ≤ z ≤ 2 with z ∈ R. ˆ qu ˆ Assumption 2. The functions Bu(u) = i=1 Bu,i(u) qx we assume in the following that the terminal set can and Bˆ (x) = Bˆ (x) are relaxed recentered loga- f x i=1 x,i P be represented as X rithmic barrier functions for the polytopic sets and P R U X n and the relaxation parameter δ ++ is chosen such that f = x R : ϕ(x) 1 , (15) ∈ X { ∈ ≤ } Bˆu(0) = Bˆx(0) = 0 and Bˆu(0) = 0, Bˆx(0) = 0. ∇ ∇ where ϕ : Rn R is a continuously differen- → + Note that a δ satisfying Assumption 2 always exists. tiable, convex, and positive definite function that satis- In particular, if the relaxation parameter satisfies 0 < fies ϕ(AK x) ϕ(x) x f . When considering ellip- 1 qx 1 qu ≤ ∀ ∈ X δ min dx,...,dx , du, ...,du , suitable relaxed barrier soidal terminal sets as discussed in [35] and[10], ϕ( ) may ≤ { } ⊤ · functions can be easily constructed by simply relaxing the for example be chosen as ϕ(x) = x Pf x with a suitable Sn logarithmic terms of the recentered nonrelaxed formula- Pf ++. In [11] and [13], it is moreover shown how i i ∈ tion. Hence, for zi(x) := Cxx + dx, i =1,...,qx, we get smooth approximations of the Minkowski functional can for example − be used to define ϕ( ) for polytopic terminal sets of the n · form f = x R : Hf x 1 . i i Cxx X { ∈ ≤ } ln(zi(x)) + ln(d ) i zi(x) >δ ˆ − x − dx B1) Stabilization with strict constraint satisfaction Bx,i(x)= i (13)  i Cxx β(zi(x); δ) + ln(d ) i zi(x) δ In the following, we will show that asymptotic stability  x − dx ≤ of the closed-loop system as well as strict satisfaction of when using gradient recentered barrier functions and all input and state constraints can always be guaranteed for any feasible initial condition by making the relaxation i i (1 + wx) ln(zi(x)) + ln(dx) zi(x) >δ parameter δ R++ arbitrarily small. The key assumption Bˆx,i(x)= − ∈ i i underlying the following results is that both the state and ((1 + wx) β(zi(x); δ) + ln(dx)
zi(x) δ ≤ input constraint barrier functions and the terminal set (14)
constraint barrier function are relaxed. In particular, the when using weight recentered barrier functions, cf. Defi- barrier functions for the input and state constraints are nition 1. The barrier functions Bˆ ,i( ) for the input con- u chosen according to Assumption 2, while the terminal cost straints can be defined analogously. Note· that it is in prin- function is given by ciple possible to consider individual relaxation parameters ⊤ δi for the different constraints, but we restrict ourselves Fˆ(x)= x P x + εBˆf (x) , (16) to one overall δ for the sake of simplicity. While the relax- where Bˆ ( ) is a relaxed logarithmic barrier function for ation of the barrier functions directly implies that prob- f · the terminal set f . Assuming δ 1, the function Bˆ ( ) lem (12) is always recursively feasible, stability properties X ≤ f · of the resulting closed-loop system are by no means guar- can be defined as follows. anteed. This problem will be addressed in the following Assumption 3. The barrier function Bˆ ( ) for the ter- f · two sections. minal set f is a relaxed logarithmic barrier function of the form X B. Closed-Loop Stability: Terminal Set Based Ap- proaches ln(1 ϕ(x)) 1 ϕ(x) >δ Bˆf (x)= − − − (17) ( β(1 ϕ(x); δ) 1 ϕ(x) δ , In this section we present our main results on the closed- − − ≤ loop stability properties of relaxed logarithmic barrier where δ R++, δ 1, is the relaxation parameter and function based MPC schemes that make use of a suitable ϕ( ) is the∈ function≤ defining the terminal set according terminal set, cf. [12]. In accordance with [10, 11, 13, 35], to· (15).

6 Note that the function Bˆf ( ) is continuously differen- Proof. The proof consists of three parts and is closely tiable, convex, and positive definite· by design. Consider related to that of Theorem 12 in [12]. First, we show now the following definition, which introduces a lower that the underlying input, state, and terminal set con- bound for the value of the relaxed barrier functions Bˆ ( ), straints are not violated for any x ˆN (δ); then we use x · 0 ∈ X Bˆu( ), and Bˆf ( ) evaluated at the borders of the respective standard MPC arguments to show that the value function constraint· sets.· Jˆ∗(x(k); δ) will decrease when applying the feedback u(k); finally, we use this result to conclude that the resulting Definition 5. Let the scalar β¯(δ) R be defined as ∈ ++ input and state sequences will also be strictly feasible at β¯(δ) = min β¯x(δ), β¯u(δ), β¯f (δ) , (18) all later time steps and that the origin of the closed-loop system is asymptotically stable. ¯ ∗ where βf (δ)= β(0; δ) and i) Let uˆ (x ) = uˆ∗,..., uˆ∗ , x∗(x ) = x ,...,x∗ 0 { 0 N−1} 0 { 0 N } ¯ ˆ i i denote the optimal open-loop input and state sequences βx(δ) = min Bx(x) Cxx = dx , i =1,...,qx , (19a) i,x { | } for a given x0 ˆN (δ). Since the cost function in (12) is ¯ ˆ j j a sum of positive∈ X definite terms, it holds that εBˆ (x∗) < βu(δ) = min Bu(u) Cuu = du , j =1,...,qu . (19b) x k u,j { | } ˆ∗ ˆ ∗ ˆ∗ ˆ ∗ JN (x0; δ), εBu(ˆuk) < JN (x0; δ), as well as εBf (xN ) < ¯ ¯ ¯ ˆ∗ ˆ ˆ ∗ ¯ Note that the values βx(δ), βu(δ), and hence also β(δ), JN (x0; δ) for all x0 N (δ) 0 . Hence, Bx(xk) < β(δ), ∗ ∈ X \{ } ∗ can be computed easily for a given δ R++ as the op- Bˆ (ˆu ) < β¯(δ), as well as Bˆ (x ) < β¯(δ). Due to ∈ u k f N timization problems in (19) are convex. Moreover, for Lemma 1 and the definition of β¯(δ), this implies that the case of gradient recentered relaxed logarithmic bar- x∗ ◦ andu ˆ∗ ◦ as well as x∗ ◦ for any k ∈ X k ∈ U N ∈ Xf rier functions, an explicit expression for a lower bound on x ˆ (δ) 0 . Of course, the case x = 0 is triv- ¯ 0 N 0 β(δ) has been given in [12]. The following Lemma shows ial.∈ Hence,X the\{ predicted} input and state sequences are that the sublevel sets of Bˆ ( ), Bˆ ( ), and Bˆ ( ) related to x · u · f · strictly feasible and the applied input results in a succes- β¯x(δ), β¯u(δ), and β¯f (δ) are always contained within the + ∗ ◦ sor state x0 = Ax0 + Buˆ0 N . sets , , and f , respectively. ii) Let us now consider x+∈ X ◦ . We can use basically X U X 0 ∈ XN Lemma 1. Let the Assumptions 2 and 3 hold and let the same arguments as in the proof of Theorem 1 to show the values β¯x(δ), β¯u(δ), and β¯f (δ) be defined according to that n ∗ ∗ ∗ Definition 5. Then it holds that ˆ = x R Bˆx(x) ˆ + ˆ ˆ ˆ SBx { ∈ | ≤ JN (x0 ; δ) JN (x0; δ) ℓ(x0, uˆ0(x0)) x0 N (δ). ¯ Rm ˆ ¯ − ≤− ∀ ∈ X βx(δ) , Bˆu = u Bu(u) βu(δ) , as (20) } ⊆ X S n{ ∈ | ≤ }⊆U u+ well as ˆ = x R Bˆf (x) β¯f (δ) f . In particular, we know that ˜ (x0) = SBf { ∈ | ≤ } ⊆ X ∗ ∗ ∗ x+ uˆ1,..., uˆN−1, KxN and (x0) = Proof. The result for the terminal set f follows directly { + ∗ ∗ }∗ ∗ X x0 , x2,...,xN , AK xN with AK xN f are sub- from the fact that Bˆ ( ) is convex and strictly monotone { } ∈ X f · optimal but feasible input and state sequences for the in the argument z = 1 ϕ(x). Considering the state + − initial state x0 N . Moreover, constraints, let us assume that there exists anx ¯ B ∈ X ∈ S x ˆ ∗ ˆ ∗ ˆ ∗ ∗ withx ¯ / , i.e., C x¯ > d . However, then there would F (AK xN ) F (xN )+ ℓ(xN , KxN ) (21a) ∈ X x x − exists a λ (0, 1) such that C λx¯ = λC x¯ d and ∗ 2 ∗ 2 ∗ 2 ∗ 2 x x x = AK xN P xN P + xN Q + KxN R (21b) Ci λx¯ = λC∈i x¯ = di for some i = 1,...,q . Now,≤ due to k k −k k k k k k x x x x ˆ ∗ ˆ ∗ ˆ ∗ ˆ + εBK (xN )+ εBf (AK xN ) εBf (xN ) (21c) the convexity and positive definiteness of Bx( ), it holds − ˆ ˆ ¯ · ˆ ∗ ˆ ∗ ∗ that Bx(λx¯) λBx(¯x) < βx(δ), which obviously is a ε Bf (AK xN ) Bf (xN ) 0 xN f . (21d) contradiction≤ to the definition of β¯ (δ). For the input ≤ − ≤ ∀ ∈ X x Here, the first inequality follows from the choice of constraints we can use similar arguments.
f and P according to Assumption 1 and the fact thatX ∗ ∗ ∗T ∗ ∗ Based on the above results, we now define a set of initial BˆK (x ) BK(x ) x Mx x f . The sec- N ≤ N ≤ N N ∀ N ∈ X conditions for which we can guarantee asymptotic stabil- ond inequality follows from the monotonicity of the re- ˆ ity of the origin as well as strict satisfaction of all input laxed logarithmic barrier function Bf ( ) and the assump- ∗ ∗ ∗ · and state constraints. tion that ϕ(AK xN ) ϕ(xN ) xN f . Based on ar- guments as in the proof≤ of Theorem∀ ∈ 1, X this result can be Definition 6. For a given δ R and a corresponding ∈ ++ used to prove the decrease property (20). β¯(δ) R++ according to Definition 5, let the set ˆN (δ) ∈ n ∗ X iii) The fact that the value function decreases shows that be defined as ˆN (δ) := x R Jˆ (x; δ) εβ¯(δ) . ∗ + ∗ + N Jˆ (x ; δ) Jˆ (x ; δ) εβ¯(δ) and hence x ˆN (δ) for X { ∈ | ≤ } N 0 ≤ N 0 ≤ 0 ∈ X Theorem 2. Let Assumptions 1, 2, and 3 hold true. any x ˆN (δ). By repeating this argument, the result- 0 ∈ X Consider problem (12) with the terminal cost Fˆ( ) ing closed-loop system state satisfies x(k) ˆN (δ) k 0 · ∈ X ∀ ≥ from (16) and let the set ˆN (δ) be defined according to for any x(0) ˆN (δ), which shows that all future states X ∗ ∈ X Definition 6. Then, the feedback u(k)=ˆu0(x(k)) asymp- and inputs will be strictly feasible. Moreover, due to the ˆ∗ totically stabilizes the origin of system (1) under strict design of the relaxed barrier functions, JN (x; δ) is a well- satisfaction of all input and state constraints for any defined, positive definite, and radially unbounded func- x(0) ˆN (δ). tion. Hence, in combination with (20) it can be used as ∈ X

7 a Lyapunov function, proving asymptotic stability of the sufficient and may be rather conservative. In particu- origin with a guaranteed region of attraction of at least lar, for a given δ R the actual region of attraction of ∈ ++ ˆN (δ). the closed-loop system may be considerably larger than X the set ˆN (δ). Likewise, a very small δ may be needed The following results state some useful properties of the X ˆ ◦ to achieve 0 N (δ) when 0 approaches N . How- region of attraction ˆN (δ) and show that the feasible set X ⊂ X X X X ever, despite possible practical limitations, the presented N of the corresponding non-relaxed formulation can be X results provide interesting insights and a theoretical jus- recovered by making the relaxation parameter arbitrarily tification for the use of relaxed barrier functions in the small. context of MPC. Lemma 2 (cf. [12]). Let the assumptions in Theorem 2 B2) Global stabilization with a nonrelaxed terminal set hold and let the set ˆN (δ) be defined according to Defini- X tion 6. Then, ˆN (δ) is a nonempty compact and convex Assuming controllability of system (1), we will in the fol- X ˆ ◦ ˆ set. Furthermore, N (δ) N and N (δ) N as lowing present a second approach that allows to guarantee δ 0. X ⊆ X X → X asymptotic stability for any initial condition by relaxing → the barrier functions of state and input constraints while Proof. For any δ R satisfying Assumptions 2 and 3, ++ strictly enforcing the terminal set constraint. J ∗ ( ; δ) is a positive∈ definite function that is convex and N · radially unbounded. As β¯(δ) is strictly positive (see Defi- Assumption 4. The pair (A, B) is controllable and the nition 5), and ε R++, the first part follows immediately. prediction horizon satisfies the condition N n. More- ∈ As shown in the proof of Theorem 2, any initial con- over, the terminal cost function is given as ≥ dition x0 ˆN (δ) results in strictly feasible input and ∈ X ◦ ˆ ⊤ state sequences, which implies that ˆN (δ) . We F (x)= x P x + εBf (x) , (22) X ⊆ XN now show that ˆN (δ) will contain any compact subset ◦ X where P and Bf ( ) are chosen according to Assumption 1. of N if we make the relaxation parameter arbitrarily · X ◦ ∗ ∗ ∗ small. Assume x0 and let u˜ (x0), x˜ (x0), J˜ (x0) Theorem 3. Let Assumptions 1, 2, and 4 hold true ∈ XN N denote the solution of the corresponding nonrelaxed prob- and consider problem (12) with the terminal cost Fˆ( ) · lem formulation (4). Then, there always exists δ0(x0) := from (22). Then, independently of the relaxation parame- i i j j ∗ ∗ min C x˜k(x0) + d , C u˜k(x0) + d , 1 ϕ(˜x (x0), ter δ R , the feedback u(k)=ˆu (x(k)) asymptotically {− x x − u u − N ∈ ++ 0 i = 1,...,qx, j = 1,...,qu, k = 0,...,N 1 R++ stabilizes the origin of system (1) for any initial condition − } ∈ with the property that the solutions of relaxed and non- x(0) Rn. ∈ relaxed formulation will be equivalent for all δ δ0(x0), i.e., uˆ∗(x ) = u˜∗(x ), xˆ∗(x ) = x˜∗(x ), Jˆ∗ (≤x ; δ) = Proof. Due to the above controllability assumption, there 0 0 0 0 N 0 n ˜∗ exists for any x0 R an input sequence uˆ(x0) = JN (x0). Note that δ0(x0) is a continuous function of x0 ∈ ◦ u∗ x∗ uˆ0,..., uˆN−1 such that xN (uˆ(x0), x0) . Hence, since both ˜ (x0) and ˆ (x0) are continuous due to the { } ∈ Xf u∗ ˆ∗ Rn smooth problem formulation (12). Let us further define ˆ (x0) and JN (x0; δ) are defined for any x0 . ′ R ˜∗ ¯ ∗ ∈ δ0(x0) := max δ ++ δ δ0(x0), JN (x0) εβ(δ) Since the corresponding terminal state satisfies xN (x0) { ∈ | ≤ ˆ ′ ≤ } ◦, the local controller u = Kx can again be used∈ with the property that x0 N (δ) δ δ0(x0). As f ˆ∗ ˆ∗ ∈ X˜∗ ∀ ≤ X u+ JN (x0; δ) = JN (x0; δ0(x0)) = JN (x0) for all δ δ0(x0) to construct a feasible control sequence ˆ (x0) = ¯ ≤ ∗ ∗ ∗ + and β(δ) is, on the other hand, continuous and strictly in- uˆ1,..., uˆN−1, KxN for the successor state x0 = Ax0 + ′ R { ∗ } creasing for decreasing δ, we can state that δ0(x0) ++ Buˆ0. Since all parameters are chosen according to As- ◦ ∈ exists for any x0 N . Moreover, since both δ0(x0) sumption 1, the same arguments as in part ii) of the proof ˜∗ ∈ X ′ and JN (x0) are continuous, δ0(x0) also is a continuous of Theorem 2 can be used in order to show that function of x0. Consider now an arbitrary compact set ∗ ∗ ∗ ◦ ′ ˆ + ˆ ˆ Rn ¯ R ¯ JN (x ; δ) JN (x0; δ) ℓ0(x0, uˆ (x0)) x0 . (23) 0 N and define δ0 ++ as δ0 = minx0∈X0 δ0(x0). 0 0 X ⊆ X ∈ ′ − ≤− ∀ ∈ Due to the continuity of δ0(x0) and the compactness of ˆ Moreover, due to the design of all involved barrier func- 0, this value always exists and it holds that 0 N (δ) ˆ∗ X ¯ ˆ X ⊆ X tions, JN (x; δ) is a well-defined, positive definite, and ra- for all δ δ0. This implies that N (δ) N as δ 0, ˆ∗ which completes≤ the proof. X → X → dially unbounded function. Hence, JN ( ; δ) can be em- ployed as a Lyapunov function for proving· global asymp- totic stability of the origin, see part iii) of the proof of Corollary 1. Consider the relaxed barrier function based Theorem 2. MPC problem formulation (12) with Fˆ( ) given by (16). ◦ · For any compact set 0 there exists a δ¯0 R++ such X ⊆ XN ∈ The above result illustrates how we can achieve stabi- that for any δ δ¯0 and any x(0) 0 the feedback u(k)= ∗ ≤ ∈ X lization of the origin for any initial condition, which makes uˆ0(x(k)) asymptotically stabilizes the origin of system (1) under strict satisfaction of all input and state constraints. the corresponding MPC scheme very robust against un- certainties, disturbances, or even infeasible start configu- It has to be noted that the above conditions for closed- rations. As we need to strictly enforce the terminal set loop stability and constraint satisfaction are of course only constraint, the benefit of global stabilization comes with

8 the cost of possible violations of the state and input con- to the relaxation, v(x) is a valid but possibly suboptimal straints. However, using the arguments of the previous input sequence that steers the state from x(0) = x to the ◦ ˆ section, we can for any 0 N still recover strict sat- origin in a finite number of steps. Consequently, F ( ) is isfaction of all state andX input⊆ X constraints by making the an upper bound for the infinite-horizon cost-to-go, which· relaxation parameter δ arbitrarily small. allows us to state the following stability result. Theorem 4. Let the Assumptions 2 and 5 hold and con- C. Closed-Loop Stability: Terminal Set Free Approaches sider problem (12) with the terminal cost Fˆ( ) from (24). · In the previous section, we have seen how asymptotic sta- Then, independently of the relaxation parameter δ R++, ∗ ∈ bility and even strict constraint satisfaction can be guar- the feedback u(k)=ˆu0(x(k)) asymptotically stabilizes the anteed in the context of relaxed logarithmic barrier func- origin of system (1) for any initial condition x(0) Rn. ∈ tion based MPC by making use of suitable terminal set Proof. Due to Assumption 5, there exist for any x Rn formulations. As in conventional MPC schemes, the ter- 0 and any input sequence uˆ(x ) with resulting terminal∈ minal set is on the one hand used for ensuring the ex- 0 state x = x (uˆ(x ), x ) suitable tail sequences v(x ) istence of a feasible local control law and, thus, the re- N N 0 0 N and z(x ). Moreover, as all barrier functions for the cursive feasibility of the corresponding open-loop optimal N state and input constraints are relaxed, see Assumption 2, control problem. On the other hand, only restricting the both the stage and terminal cost are always well defined. terminal state to a compact set around the origin allowed Hence, also uˆ∗(x ) and Jˆ∗ (x ; δ) are defined for any us to derive a quadratic upper bound for the barrier func- 0 N 0 x Rn, which shows that problem (12) always admits a tion B ( ), see Assumption 1 and the proof of Theorem 1. 0 K feasible∈ solution. In the following,· we will show that the use of relaxed log- We now want to show that the value function Jˆ∗ (x(k); δ) arithmic barrier functions in fact allows us to circumvent N decreases under the applied feedback for all x(0) these two problems and to design novel MPC approaches Rn u∗ ∗ ∗ x∗ ∈ . Let ˆ (x0) = uˆ0,..., uˆN−1 and (x0) = which not only eliminate the need for an explicit terminal ∗ ∗ { } x0, x1,...,xN denote the optimal open-loop input and set constraint but also to prove global asymptotic stabil- { } n state sequences for a given initial condition x0 R ity of the origin. v∗ v ∗ z∗ z ∗∈ and let (x0) := (xN (x0)) and (x0) := (xN (x0)) C1) Tail sequence based terminal cost function be the corresponding tail sequences for the resulting pre- It is a well-known result that closed-loop stability of both dicted terminal state. Consider now the successor state + ∗ uˆ+ ∗ ∗ ∗ linear and nonlinear MPC schemes may be ensured by x0 = Ax0 + Buˆ0. Clearly, (x0) = uˆ1,..., uˆN−1, v0 v+ ∗ ∗ { } and (x0)= v1 ,...,vT −1, 0 are suboptimal input se- choosing the terminal cost as a suitable CLF that is an up- { } + per bound for the infinite-horizon cost-to-go, see e.g. [18]. quences which steer the state from x0 to the origin in In the presence of input and state constraints, deriving a finite number of N + T 1 steps. The resulting state x+− ∗ ∗ ∗ such a function in global form is generally not possible, sequences are given by (x0) = x1,...,xN ,z1 and z+(x ) = z∗,...,z∗ , 0 . Using{ the above subop-} which directly motivates the use of a local CLF in com- 0 { 1 T −1 } bination with a corresponding terminal set constraint. timal input and state sequences and the fact that the However, when considering the relaxed problem formu- tail sequences within the terminal cost are simply ap- lation (12), any input sequence which steers the state to pended by zero values, it is straightforward to show that ˆ u+ + ˆ∗ ˆ ∗ Rn the origin in a finite number of steps can be used to derive JN (ˆ (x0), x0 ; δ) JN (x0; δ)= ℓ(x0, uˆ0(x0)) x0 . − u+ − ∀ ∈ an upper bound on the infinite-horizon cost-to-go. From the suboptimality of ˆ (x0) it follows immediately that Assumption 5. Let v(x) := v0(x),...,vT −1(x) be an { } Jˆ∗ (x+; δ) Jˆ∗ (x ; δ) ℓˆ(x , uˆ∗(x )) x Rn . (25) input sequence which steers the state of system (1) to the N 0 − N 0 ≤− 0 0 0 ∀ 0 ∈ origin in a finite number of T n steps for any x Rn ≥ ∈ By the design of the relaxed barriers and the assumption and assume that vl(x)=0 l = 0,...,T 1 x = ∀ − ⇔ that vl(x)=0 l = 0,...,T 1 x = 0, the function 0. Furthermore, let z(x) := z (x),...,z − (x) with ∗ 0 T 1 Jˆ ( ; δ) is well-defined,∀ positive− definite,⇔ and radially un- z (x) = x, z (x) = Az (x)+{Bv (x), l = 0,...,T} 1, N 0 l+1 l l bounded.· Thus, it can be used as a Lyapunov function and z (x)=0 be the corresponding state sequence. − T for the closed-loop system, proving global asymptotic sta- Based on Assumption 5 we propose to choose the ter- bility of the origin. minal cost as Note that in order to ensure convexity of the termi- T −1 nal cost function Fˆ( ), the elements of the parametrized ·v Fˆ(x)= ℓˆ(zl(x), vl(x)) , (24) tail input sequence (x) should be affine in the argu- l=0 ment x. In combination with the condition that vl(x) = X 0 l = 1,...,T 1 x = 0, this in fact limits v( ) where ℓˆ: Rn Rm R refers to the already introduced to∀ contain a sequence− ⇔ of linear state feedback laws, i.e.· × → + modified stage cost based on relaxed logarithmic barrier v(x) = K0x,...,KT −1x . In the following, we briefly functions for the state and input constraints, cf. (12). Due discuss two{ different design} approaches for v(x) that meet

9 this requirement and thus allow to guarantee stability of present explicit solutions for more general start and end the closed-loop system as well as convexity of the result- point constraints based on a parametrization of all the ing overall cost function. solutions of the extended symplectic system. However, as discussed in the Appendix, the optimal input sequence The easiest way to design suitable tail sequences v( ) can also be computed directly in vector form as V ∗(x)= and z( ) is by making use of a linear dead-beat con-· ∗⊤ ∗⊤ ⊤ Tm×n · v (x) v (x) = KV x with KV R . In troller. To this end, we may choose a terminal control gain 0 ··· T −1 ∈ Rm×n both cases, the resulting terminal cost function can be K in such a way that the matrix AK = A + BK   is nilpotent,∈ i.e., that it satisfies Ar = 0 for some r n. formulated as K ≤ Under the assumption of controllability, this can for ex- T −1 ⊤ ample be achieved by a suitable pole placement procedure Fˆ(x)= x P x + ε Bˆx(zl(x)) + Bˆu(vl(x)) , (28) which ensures that all eigenvalues of the matrix A are l=0 K X located at the origin of the complex plane. Based on these ideas, we may set T = n and choose the tail input where zl(x) and vl(x) denote the elements of the re- sequence as spective state and input tail sequences and the matrix Sn P ++ can be constructed by inserting these sequences − ∈ v(x)= Kx,...,K (A + BK)n 1x (26) into the quadratic stage cost ℓ( , ), see Appendix. As for the previous approach, global· asymptotic· stability of n o which results in the corresponding state sequence z(x)= the closed-loop system can be concluded from Theorem 4. n−1 x,...,AK x . Due to the design of the matrix K, it ob- However, the possibility to choose the length of the tail { } n viously holds that zn = A x = 0. Thus, v( ) and z( ) sat- sequences within the construction of the terminal cost K · · isfy the conditions in Assumption 5, and Theorem 4 can typically leads to an overall improved closed-loop perfor- be used to conclude stability of the closed-loop system. mance. Especially if no constraints are active or violated Note that, depending on the algorithm, the pole place- by the open-loop state and input tail sequences, the ter- ment problem may become numerically ill-conditioned minal cost function Fˆ( ) based on (27) may give a quite when assigning all poles to exactly the same location. good approximation of the· real infinite-horizon cost-to-go. Hence, for practical implementations it may be meaning- Remark 2. ful to distribute the poles of AK in an ε-ball around the Note that the discussed approaches force the origin. predicted state to the origin in N + T steps and are hence similar to MPC approaches that are based on an explicit While the above approach allows for a rather simple de- zero terminal state constraint, see e.g. [19]. However, by v z sign of the tail sequences ( ) and ( ), and thus of the using the proposed terminal cost function, this behavior terminal cost Fˆ( ), the implicit· requirement· that the pre- · is enforced implicitly, i.e., without introducing an explicit dicted terminal state is steered to the origin in at most n equality constraint within the optimization problem. Fur- steps might be restrictive, leading to suboptimal or even thermore, stability can be guaranteed for a global region of aggressive behavior of the overall closed-loop system. In attraction and the tail sequence horizon T may be made the following, we present a second approach that elimi- arbitrary large without increasing the number of optimiza- nates this restriction by allowing for tail sequences with tion variables. T n elements. In order to enforce in addition a certain ≥ optimality with respect to the underlying performance C2) Quadratic terminal cost criterion, we propose to choose the parametrized tail in- In the previous section, we exploited the fact that the re- put sequence v( ) as the solution to the finite-horizon · laxation of input and state constraint allows to apply any LQR problem with zero terminal state constraint, i.e., sequence of inputs at the end of the prediction horizon. T −1 In the following, we will show that in the presence of re- v(x) = arg min ℓ(z , v ) (27a) laxed barrier functions we can in addition derive a global v l l l quadratic upper bound for the combined state and input X=0 s. t. zl+1 = Azl + Bvl, l =0,...,T 1 , (27b) constraint barrier function, which makes it possible to use − a purely quadratic terminal cost function term without z =0, z = x (27c) T 0 the need for a corresponding terminal set constraint. for the standard quadratic stage cost ℓ(z, v) = z 2 + Q Assumption 6. Let Bˆx( ) and Bˆu( ) be relaxed gradient 2 v∗ k k · · v R. It can be shown that the solution (x) to prob- or weight recentered logarithmic barrier functions accord- lemk k (27) can be expressed as a sequence of static lin- ∗ ing to Assumption 2, see (13) and (14), respectively. Let ear state feedbacks of the form v (x) = Klx with l = l the relaxing function be quadratic and given by β2( ; δ) 0,...,T 1. Explicit expressions for the optimal con- from (10). · trol law− and the corresponding state (and costate) tra- jectories were given in [26] based on the solution of two Lemma 3. Let (A, B) be stabilizable and let K Rm×n unconstrained infinite-horizon LQR problems and a suit- be a corresponding stabilizing linear control gain.∈ Fur- able iteration scheme. Furthermore, in [14] the authors thermore, let Assumption 6 hold and consider BˆK (x) =

10 Bˆx(x)+ Bˆu(Kx) for a given δ R++. Then, it holds that from (32). Then, independently of the relaxation parame- ∈ ∗ ter δ R++, the feedback u(k)=ˆu (x(k)) asymptotically ⊤ ⊤ n 0 BˆK (x) x M + K M K x x R , (29) ∈ ≤ x u ∀ ∈ stabilizes the origin of system (1) for any initial condition Rn n×n m×m x(0) . where the matrices Mx R
and Mu R are ∈ 1 ∈⊤ 1 ∈ u∗ ˆ∗ defined as Mx := 2δ2 Cx diag ( + wx) Cx and Mu := Proof. Again, both ˆ (x0) and JN (x0; δ) are defined for 1 ⊤ 1 Rqx Rn 2δ2 Cu diag ( + wu) Cu, respectively. Here, wx + and any x0 due to the relaxed state and input con- qu ∈ w R are suitable weighting vectors when consider- straints.∈ Based on the quadratic terminal cost func- u ∈ + ing weight recentered barrier functions, whereas wx = 0, tion above, it is now straightforward to show that the ˆ∗ wu =0 in the gradient recentering case. value function JN (x(k); δ) decreases under the applied n n MPC feedback for all x(k) R . For any x0 R Proof. We exemplary consider the state constraints and ∈ ∗ ∈ there exist optimal input and state sequences uˆ (x0) = ˆ 2 Rn ∗ ∗ ∗ ∗ ∗ show that Bx(x) x Mx x . Based on Tay- x ≤ k k ∀ ∈ uˆ0,..., uˆN−1 and (x0) = x0, x1,...,xN as well lor’s Theorem (see [25, Theorem 2.1]) we know that { } { u}+ ˆ ˆ ˆ ⊤ 1 T 2 ˆ as the suboptimal input and state sequences ˆ (x0) = Bx(x)= Bx(0) + [ Bx(0)] x + 2 x Bx(λx)x for some ∗ ∗ ∗ + ∗ ∗ ∗ ∇ ∇ uˆ ,..., uˆ − , Kx and x (x0)= x ,...,x , AK x λ (0, 1). Due the recentering of the barrier func- { 1 N 1 N } { 1 N N } ∈ + ∗ ˆ∗ + tions, the first two terms vanish and we get Bˆ (x) = for the successor state x0 = Ax0 + Buˆ0 with JN (x0 ; δ) x ∗ + ∗ − T ˆ ˆ u + ˆ 1 2 ˆ JN (x0; δ) JN (ˆ (x0), x0 ,δ) JN (x0; δ). This gives us 2 x Bx(λx)x for some λ (0, 1). In particular, it holds ≤ − ∇ˆ ⊤ R∈n Sn that Bx(x) x Mx x for any M ++ satisfying ˆ∗ + ˆ∗ ∗ 2 ∗ 2 2 ≤ ∀n ∈ ∈ JN (x0 ; δ) JN (x0; δ) AK xN P xN P ... (33) Bx(x) M x R . When considering the quadratic − ≤k k −k k ∇  ∀ ∈ ∗ 2 ∗ 2 ∗ ∗ Rn relaxing function β ( ; δ) from (10), the Hessian of Bˆ ( ) + xN Q + KxN R + εBK (xN ) 0 xN , 2 x k k k k ≤ ∀ ∈ is given by · · where we used the global quadratic bound on BK ( ) from · 2 ˆ ⊤ Lemma 3 and the choice of the terminal cost matrix P , Bx(x)= C diag D1(x),...,Dqx (x) Cx , (30a) ∇ x ˆ∗ i see (31b). Thus, JN ( ; δ) can again be used as a Lyapunov 1+wx i i i i 2 C x + d >δ
· (−Cxx+dx) x x function for proving global asymptotic stability. where Di(x)= i − (30b) 1+wx i i ( 2 C x + d δ . δ − x x ≤ In summary, we can conclude that the concept of re- laxed logarithmic barrier functions allows us to design Note that wx = 0 in the case of gradient recentered barrier i 1+wx n globally stabilizing MPC schemes without making use of functions. Since D (x) 2 x R , it follows im- i δ an explicit terminal set or state constraint. Instead, the 2 ˆ ≤ 1 ⊤∀ ∈ 1 mediately that Bx(x) δ2 Cx diag ( + wx) Cx x presented approaches are based on a suitable design of Rn ∇  ∀ ∈ . Combining this upper bound on the Hessian the respective terminal cost function term that heavily ˆ with the previous arguments, it follows that Bx(x) exploits the properties and advantages of the underlying ⊤ Rn 1 ⊤ 1 ≤ x Mx x x with Mx = 2δ2 Cx diag ( + wx) Cx. barrier function relaxation. Thus, the presented results ∀ ∈ ˆ Similarly, it is straightforward to show that Bu(Kx) may be seen as a barrier function based counterpart to ⊤ ⊤ n ≤ x K MuKx x R with Mu as defined above. existing terminal set free MPC approaches relying on suit- ∀ ∈ able terminal cost functions ([18, 21]), a sufficiently large Sn prediction horizon ([2, 7, 32]), or particular controllability Let now the controller matrix K and the matrix P ++ be solutions to the modified Riccati equation ∈ assumptions ([4, 15, 16]). However, with the exception of Theorem 2, no guaran- ⊤ −1 ⊤ K = R + B PB + εMu B P A (31a) tees on the satisfaction of state and input constraints have −⊤ ⊤ been discussed so far. In fact, the presented global stabil- P = AKP AK + K (R + εM
u)K + Q + εMx , (31b) ity results can only be achieved since the relaxed barrier functions allow for in principle arbitrarily large constraint where M and M are defined according to Lemma 3. We x u violations. In the next section, we will show that the max- then propose to choose the terminal cost function as imal violation of state and input constraints in closed-loop Fˆ(x)= x⊤P x , (32) operation is always bounded, and we discuss how we can compute and control it a priori for a given set of initial where P Sn is the solution to (31b). Note that the conditions by adjusting the relaxation parameter δ R . ++ ∈ ++ controller∈ gain K is in principle arbitrary. However, the above choice results in a minimal value of the terminal D. Maximal Constraint Violation Guarantees cost function and ensures that for ε 0 or in the absence of constraints, K and P will reduce→ to the solution of As outlined above, one of the main advantages of model the unconstrained LQR problem. We can now state the predictive control is given by its ability to deal with in- following stability result. put and state constraints. On the other hand, possible violations of the corresponding constraints seem to be in- Theorem 5. Let Assumption 6 hold true and con- herent to the concept of relaxed barrier functions. In sider problem (12) with the terminal cost function Fˆ( ) the following, we will show how guarantees on the strict ·

11 satisfaction or the maximal violation of input and state for i = 1,...,qx and j = 1,...,qu, whereα ˆ(x0; δ) is de- constraints can be given for the globally stabilizing re- fined according to (36). Based on this, we can formulate laxed logarithmic barrier function based MPC schemes the following result on the maximal constraint violations from the previous section. We start with the following that can occur in closed-loop operation. Lemma which allows to upper bound the values of the relaxed barrier functions in closed-loop operation by an Theorem 6. Let problem (12) be formulated based on one expression that depends on the initial condition. of the globally stabilizing MPC schemes discussed above (Theorems 3, 4, 5) and let xcl = x(0), x(1),... and Lemma 4. Let problem (12) be formulated based on u { ∗ } cl = u(0),u(1),... with u(k)=u ˆ0(x(k)) for k 0 one of the globally stabilizing MPC schemes discussed denote{ the resulting closed-loop} state and input trajecto-≥ above (Theorems 3, 4, 5). Furthermore, let xcl = n ries. Then, for any x0 = x(0) R and any k 0 it x(0), x(1),... and ucl = u(0),u(1),... with u(k) = ∈ ≥ {∗ } { } holds that uˆ0(x(k)) for k 0 denote the resulting state and input trajectories of the≥ closed-loop system. Then, for any ini- Cxx(k) dx +ˆzx(x0,δ), Cuu(k) du +ˆzu(x0,δ), (38) tial condition x = x(0) Rn and any k 0 it holds that ≤ ≤ 0 ∈ ≥ ˆ 1 ˆ∗ ⊤ ∗ where the elements of the maximal constraint violation Bx(x(k)) JN (x0; δ) x0 Puc x0 , (34a) Rqx Rqu ≤ ε − vectors zˆx(x0; δ) and zˆu(x0; δ) are given ∈ ∈ 1   by (37). Bˆ (u(k)) Jˆ∗ (x ; δ) x⊤P ∗ x , (34b) u ≤ ε N 0 − 0 uc 0 ∗   Proof. Follows directly from Lemma 4 and the definition where P Sn is the solution to the discrete-time alge- uc ++ ofα ˆ(x ; δ) andz ˆi (x ; δ),z ˆj (x ; δ) in (36) and (37). braic Riccati∈ equation related to the infinite-horizon LQR 0 x 0 u 0 problem. Note that the optimization problems in (37) are con- Proof. Based on the respective Theorems we know that vex as Bˆx( ) and Bˆu( ) are convex functions. Moreover, ˆ∗ ˆ∗ ˆ · · JN (x(k + 1); δ) JN (x(k); δ) ℓ(x(k),u(k)) for any when considering gradient recentered logarithmic barrier − n ≤ − k 0 and any x(0) R . In particular, this ensures functions, we can exploit the fact that Bˆ ( ) and Bˆ ( ) ≥ ∈ x u that the closed-loop system is asymptotically stable and consist only of positive definite terms. In this· case, upper· ˆ∗ that limk→∞ JN (x(k); δ) = 0. Summing up over all future bounds for the maximal constraint violations are given by i i i sampling instants and using a telescoping sum on the left zˆ (x0; δ) max z¯ , 0 , wherez ¯ is given as ˆ∗ ∞ ˆ x ≤ {− x } x hand side, we get that JN (x(0); δ) k=0 ℓ(x(k),u(k)). ∞ ˆ ≥ ∞ Furthermore, k=0 ℓ(x(k),u(k)) = k=0 ℓ(x(k),u(k)) + i i z αˆ(x0; δ) ∞ ∞P z¯ = min z β(z; δ) + ln(d )+ 1= ε Bˆ (x(k)) + Bˆ (u(k)) and ℓ(x(k),u(k)) x x di − ε k=0 x P u Pk=0  x  x(0)⊤P ∗ x(0) due to the optimality of the unconstrained≥ (39) P uc P infinite-horizon LQR solution. In combination this yields for i = 1,...,qx and similar for the input con- straints, cf. (13). If the relaxing function β( ; δ) is chosen ∞ · Jˆ∗ (x(0); δ) x(0)⊤P ∗ x(0) + ε Bˆ (x(k)) + Bˆ (u(k)) as βe( ; δ) or βk( ; δ) with k > 2, a nonlinear equation N ≥ uc x u · · i k solver can be used to find suitable solutionsz ¯x. However, X=0 (35) in the case of β( ; δ) = βk( ; δ) with k = 2, the equality · · and finally, since all terms in the sum on the right hand constraint in (39) reduces to a quadratic equation in z ˆ ˆ∗ side are positive definite, εBx(x(k)) JN (x(0); δ) and a closed form expression of the maximal constraint ⊤ ∗ ˆ ≤ ˆ∗ − violation can be given as x(0) Pucx(0) as well as εBu(u(k)) JN (x(0); δ) ⊤ ∗ ≤ − x(0) Pucx(0) k 0. ∀ ≥ i 2 z¯x = δ γi,1 γi,1 γi,2 , i =1,...,qx , (40) For ease of notation, let now the scalarα ˆ(x ; δ) R be − − 0 ∈ +  q  defined as i δ dx 2 ˆ∗ ⊤ ∗ where γi,1 := 2 di and γi,2 := 1 + 2 ln δ ε αˆ(x0; δ), αˆ(x0; δ) := JN (x0; δ) x0 Puc x0 . (36) − x − − see also Lemma 21 in [12]. Note that the  values for the As the relaxed barrier function are positive definite and maximal constraint violations obtained above may also be radially unbounded, Lemma 4 implies that also the vio- negative. In this case, the corresponding constraints will lations of the corresponding constraints are bounded. In not be violated but rather satisfied with the respective R R particular, for any ε ++, δ ++ and any initial con- safety margin. An important consequence of Theorem 6 dition x = x(0) R∈n, upper∈ bounds for the maximal 0 ∈ is that, as stated in the following Lemma, there always violations of state and input constraints are given by exists a set of initial conditions for which the input and

i i i αˆ(x0; δ) state constraints will not be violated at all. zˆ (x0; δ) =max C x d Bˆx(x) (37a) x x x − x ≤ ε   Lemma 5. Let problem (12) be formulated based on one

j j j ˆ αˆ(x0; δ) of the globally stabilizing MPC schemes discussed above zˆu(x0; δ) =max C x d Bu(u) (37b) u u − u ≤ ε x   (Theorems 3, 4, 5) and let cl = x(0), x(1),... and { }

12 u ∗ cl = u(0),u(1),... with u(k)=ˆu0(x(k)) for k 0 de- When considering the approach based on Theorem 5 the note the{ resulting closed-loop} state and input trajectorie≥ s. problem occurs that the matrix P from (31), and hence R ˆ′ ˆ T Furthermore, for δ ++ let the set N (δ) be defined as the quadratic terminal cost function F (x) = x P x, will ∈ X grow without bound for δ 0. Also, in this case it ′ ′ ˆ (δ) := x αˆ(x, δ) εβ¯ (δ) , (41) does not necessarily hold that→ ˆ′ (δ¯) as the re- XN ∈ X | ≤ N N sulting optimal state and inputX sequences⊂ X for a given ¯′ ¯ ¯ ¯ ¯ with β (δ) := min βx(δ), βu(δ) , where βx(δ ) and βu(δ) ˆ′ { } x0 N (δ) will in general not satisfy the additional are defined according to Definition 5. Then, for any ini- ∈ X ◦ constraint xN = 0. However, for any x0 N there tial condition x(0) ˆ′ (δ) and any k 0 it holds that ∈ X N exists input and state sequences u¯(x0) and x¯(x0) which C x(k) d as well∈ asX C u(k) d . ≥ x ≤ x u ≤ u strictly satisfy the conditions specified in (44), in par- ˆ′ ticular xN (u¯(x0), x0) = 0. For these sequences, we can Proof. For any x0 = x(0) N (δ) and all k 0 it ∈ X ¯′ ≥ always find a δ0(x0) R++ such that JˆN (u¯(x0), x0; δ) = holds due to Lemma 4 and the definition of β (δ) that ∈ J˜N (u¯(x0), x0) for all δ δ0(x0), where J˜N (u¯(x0), x0) de- εBˆx(x(k)) αˆ(x0; δ) εβ¯x(δ) and εBˆu(u(k)) αˆ(x, δ) ≤ ≤ ≤ ≤ notes the value function≤ of a nonrelaxed problem formu- εβ¯u(δ). In combination with the definition of β¯x(δ) and lation with the additional constraint xN = 0, cf. the β¯u(δ), it follows directly that x(k) , u(k) k 0, cf. Lemma 1. ∈ X ∈ U ∀ ≥ proof of Lemma 2. Furthermore, due to the suboptimal- u ˆ∗ ity of the input sequence ¯(x0), it holds that JN (x0; δ) ˆ′ ˆ u ˆ∗ ˜ u ≤ One may now ask how large we can make the set N (δ) of JN (¯(x0), x0; δ). Hence, JN (x0; δ) JN (¯(x0), x0) for X ≤ ◦ initial conditions for which strict satisfaction of all input all δ δ0(x0), which shows that for any x0 N the and state constraints is guaranteed. In the following, we ≤ ˆ∗ ∈ X value function JN (x0; δ), and hence also the expression give an answer to this question for each of the different αˆ(x ; δ), will stay bounded as δ 0. As, on the other 0 → MPC approaches discussed above. In particular, we show hand, β¯(δ) increases without bound and both δ0(x0) and that it is always possible to recover the feasible set of a ˆ∗ JN (x0; δ) are continuous functions, there exists for any suitable corresponding nonrelaxed MPC formulation. ◦ ¯ R compact set a δ minx0∈X0 δ (x ) X0 ⊆ XN 0 ∈ ++ ≤ 0 0 such that ˆ′ (δ) δ δ¯ , see also the proof of Theorem 7. Let problem (12) be formulated based on one X0 ⊆ XN ∀ ≤ 0 of the globally stabilizing MPC schemes discussed above Lemma 2. (Theorems 3, 4, 5). Moreover, let N be defined as X Note that for a general stabilizable system, the set N X N = x : u s.t. xk ,uk , xN f (42) in (44) may be restricted to a lower-dimensional subspace X { ∈ X ∃ ∈ X ∈U ∈ X } of Rn or it may even be empty. However, in case of a when considering the approach based on a nonrelaxed ter- controllable system it is straightforward to show that N minal set constraint (Theorem 3), as is a nonempty polytope. X In general, the parameter δ¯ that is sufficient for ensuring N = x : u s.t. xk ,uk , 0 X { ∈ X ∃ ∈ X ∈U (43) strict constraint satisfaction based on the above results zl(xN ) , vl(xN ) ∈ X ∈U} may be very small. For practical applications, it might when considering the approach based on auxiliary tail se- therefore be reasonable to enforce the satisfaction of input and state constraints only with a predefined tolerance. quences zl( ) and vl( ) with l =1,...,T 1 (Theorem 4), and as · · − Based on the above arguments, we can easily state the following result. N = x : u s.t. xk ,uk , xN =0 (44) X { ∈ X ∃ ∈ X ∈U } Corollary 2. Let problem (12) be formulated based on when considering the approach based on a purely quadratic one of the globally stabilizing MPC schemes discussed above (Theorems 3, 4, 5) and let the respective sets N be terminal cost function (Theorem 5), where in all three X cases x0 = x and xk = xk(u, x) for k =1,...,N. Then, given according to Theorem 7. Then, for any given con- ◦ R for any compact set of initial conditions there straint violation tolerance zˆtol + and any given set of 0 N ◦ ∈ ′ X ⊆ X initial conditions 0 there exists a δ¯0 R++ such exists a δ¯0 R++ such that 0 ˆ (δ) for all δ δ¯0. N ∈ X ⊆ XN ≤ that for all δ δ¯X and⊆ X any initial condition∈x(0) ≤ 0 ∈ X0 Proof. We first consider the approaches based on Theo- the maximal possible constraint violation is less than zˆtol, rem 3 and Theorem 4. The proof is closely related to that i.e., that for all k 0 of Lemma 2 and only a sketch is given here. In particular, ≥ ˆ′ tol1 tol1 it can be shown that N (δ) is a nonempty and compact Cxx(k) dx +ˆz , Cuu(k) du +ˆz . (45) ˆ′ ◦ X ˆ′ ≤ ≤ set with N (δ) N and N (δ) N as δ 0. The result is againX based⊆ X an theX fact that→ X the existence→ of a Based on the above results, we can formulate the follow- ◦ ing iterative algorithm that allows to determine a priori a strictly feasible solution implies that for any x0 ∈ XN sufficiently small relaxation parameter for ensuring satis- there exists a δ0(x0) such thatα ˆ(x0; δ) will stay constant faction of a given maximal constraint violation tolerance for all δ δ0, whereas β¯(δ) can be made arbitrarily large as δ 0.≤ for a given set of initial conditions. →

13 for the respective terminal set free approaches. Consider ∗ ◦ ∗ ∗ Algorithm 1 (Compute δ¯0 for 0 and zˆtol R+) Jˆ (x ; δ) for a given x and let J˜ (x ) and J (x ) X ∈ N 0 0 N N 0 N 0 denote to the value functions∈ X for the corresponding nonre- Input: problem formulation, set 0, tolerancez ˆtol X laxed problem formulation and a conventional MPC for- Output: parameter δ¯0 R++ s.t. the maximal con- straint violation is bounded∈ byz ˆ for any x(0) mulation, both based on the constraints that define the tol ∈ X0 ◦ 1: choose initial ε,δ R and set up problem (12) set N . By definition, there exists for any x0 N an ∈ ++ X ∈ X 2: repeat feasible state and input sequences that result in strict 3: x decrease relaxation parameter: δ γδ, γ (0, 1) satisfaction of all inequality constraints in the respec- x ← ∈ tive description of N , which implies that there exists a 4: determineα ˆ( 0,δ) = maxx∈X0 αˆ(x, δ) based X δ (x ) R suchX that Jˆ∗ (x ; δ)= J˜∗ (x ) δ δ (x ), on (36) 0 0 ∈ ++ N 0 N 0 ∀ ≤ 0 0 5: x computez ˆi ( ,δ),z ˆj ( ,δ) from (37) with see the proof of Lemma 2. In particular, for any compact x X0 u X0 ◦ ¯ R αˆ( ,δ) set of initial conditions 0 N there exists a δ0 ++ X0 X ⊆ X ∈ 6: until zˆi ( ; δ) zˆ andz ˆj ( ; δ) zˆ holds i, j such that for any δ δ0, any x(0) 0, and any k 0 it x 0 tol u 0 tol ˆ∗ ≤ ˜∗ ∈ X ≥ X ≤ X ≤ ∀ holds that JN (x(k); δ) = JN (x(k)). This shows that the performance of a corresponding nonrelaxed formulation Remark 3. Up to now, it has not been clarified whether can always be recovered within the interior of the respec- the function αˆ(x; δ) that is maximized in step 4 of Al- tive feasible set. Furthermore, it is well known that the gorithm 1 is convex. However, for all tested parameter solution of the nonrelaxed barrier function based prob- configurations and examples with convex , the maximal X0 lem (4) converges to the solution of the corresponding value of αˆ(x; δ) was in fact always attained at on of the conventional problem when the barrier function weight- vertices of the set . Moreover, a conservative solution X0 ing parameter ε approaches zero [6, ch. 11]. Thus, for can always be found by evaluating a convex upper bound arbitrary small ε,δ R , the presented relaxed bar- ˆ∗ ∈ ++ of αˆ(x; δ), e.g., the value function JN (x; δ) itself, at the rier function based MPC schemes recover the closed-loop vertices of the set . X0 behavior and performance of related conventional MPC Summarizing, we can state that, despite the use of the schemes. The question whether and under which circum- relaxed barrier functions, the maximal possible violation stances the relaxed barrier function based setup may also of input and state constraints in closed-loop operation is lead to an overall improved performance of the closed-loop bounded and depends directly on the choice of the re- system, e.g., a decreased cumulated cost, can be consid- laxation parameter δ. Furthermore, the presented results ered as possible future work. To the experience of the au- allow to compute an estimate for the maximal constraint thors, the resulting closed-loop behavior is already very violation a priori or to determine a suitable relaxation pa- good for moderate values of the barrier parameters, i.e. −2 rameter that guarantees satisfaction of a given constraint ε,δ in the order of 10 , and typically close to the so- violation tolerance. Note that certain constraints can be lution of the original MPC problem for ε,δ in the order −4 prioritized by making use of different δi in the relaxation. of 10 . This important fact may for example be used in order Concerning the robustness of the closed-loop system, it to enforce satisfaction of physically motivated hard input has to be noted that, due to the relaxation of the under- constraints. lying state and input constraints, the resulting overall cost function is defined for any x Rn. Hence, the presented E. Closed-Loop Performance and Robustness MPC schemes are robust against∈ arbitrary effects caused by disturbances, uncertainties, or measurement errors in In this section, we briefly discuss some aspects concern- the sense that the corresponding open-loop optimal con- ing the performance and robustness properties of the pre- trol problem always admits a feasible solution. However, sented relaxed barrier function based MPC approaches. can we say something about the qualitative robust stabil- Of course, a thorough investigation of these issues is well ity properties of the closed-loop system? For example, we beyond the scope of this paper and can be considered might consider the disturbance affected dynamics as future work. We begin with the following arguments, which show that the presented relaxed barrier function x(k +1)= Ax(k)+ Buˆ∗(x(k)) + w(k) , (46) based MPC schemes will always recover the closed-loop 0 performance of a related MPC scheme based on nonre- where w(k) Rn denotes an unknown but bounded addi- laxed barrier functions if the relaxation parameter δ is tive disturbance∈ at time instant k 0. When considering small enough as well as that of a conventional linear MPC the approach based on Theorem 2,≥ any stability or conver- scheme when the barrier weighting ε goes to zero. gence guarantees of the closed-loop system will in general Let the relaxed barrier function based MPC problem (12) be lost as the disturbance might cause the system state to be formulated based on one of the stabilizing design ap- leave the set ˆN (δ). On the other hand, the approaches proaches discussed above (Theorem 2, 3, 4, or 5), and let based on TheoremsX 3, 4, and 5 allow to guarantee global the set N be defined according to (42) for the termi- asymptotic stability of the undisturbed closed-loop sys- nal set basedX approaches and according to (43) and (44) tem. Interestingly, based on the main result of [1], this

14 directly implies that system (46) is integral input-to-state B. Numerical Example stable (iISS) with respect to the disturbance w. As an im- portant consequence, the state remains bounded and con- In the following, we briefly illustrate the outlined design verges asymptotically to the origin for any disturbance se- procedure and the closed-loop behavior of the proposed quence with bounded energy, see [1] for more details. Fur- MPC schemes by means of an academic numerical exam- ther investigations, e.g., proving stronger input-to-state ple. We consider a discrete-time double integrator system stability results for system (46), are the subject of possi- of the form ble future work. 2 1 Ts Ts x(k +1)= x(k)+ u(k) ,Ts =0.1 . (47) 0 1 Ts V. Example and Numerical Aspects     with the input and state constraint sets = u R : 2 U { ∈ In this section, we outline a selection of steps which can be 2 u 1 and = x R : 2 x1 3, x2 used to get from a given control problem to a stabilizing −0.8 ≤. ≤ } X { ∈ − ≤ ≤ | | ≤ relaxed logarithmic barrier function based MPC scheme, Following} Step 1 of our design procedure, we choose the and illustrate both the design and the behavior of the basic problem parameters to N = 10, Q = diag(1, 0.1), closed-loop system by means of a numerical example. In and R = 1. In a second step, we decide on using weight addition, we also briefly comment on some interesting nu- recentered logarithmic barrier functions with a quadratic merical aspects of the resulting open-loop optimal control relaxation based on β( ; δ)= β2( ; δ), see Definition 1 and problem. Eq. (10). We fix the barrier· function· weighting parameter to ε = 10−2 and set up the different MPC schemes from A. Overall MPC Design Procedure Table 1 for varying values of the relaxation parameter δ. For the terminal set based approaches from Section B, we Step 1: Basic problem setup. Choose suitable values employ a contractive polytopic terminal set as discussed for the problem parameters that are not related to the in [11]. barrier functions, i.e., the weighting matrices Q Sn and + Inspired by our results above, we are in particular inter- R Sn as well as the prediction horizon N N∈ . ∈ ++ ∈ + ested in the region of attraction of the locally stabilizing Step 2: Relaxed barrier functions. Decide on proce- approach from Section B1 as well as in the regions with dures for relaxing and recentering the logarithmic barrier guaranteed strict or approximate constraint satisfaction functions and choose suitable (initial) values for the bar- when considering the globally stabilizing approaches from R R rier function parameters ε ++, δ ++. In general, the the Sections B2 and C1/C2. Furthermore, we would like quadratic relaxation β( ; δ)∈ = β ( ;∈δ) seems to work well · 2 · to compare the closed-loop behavior of the corresponding in practice. different MPC schemes and illustrate how the maximal Step 3: Terminal cost function. Choose a suitable constraint violation can be controlled by adjusting the approach that allows to design the terminal cost Fˆ( ) in relaxation parameter. such a way that asymptotic stability of the closed-loop· Fig. 2 depicts and compares some of the δ-dependent sets system is guaranteed. The different approaches presented that have been discussed in the results above, i.e., regions in this paper are summarized in Table 1 together with ad- of initial conditions for which we can guarantee properties ditional information regarding the necessity of a terminal like asymptotic stability or satisfaction of input and state set f , the underlying assumptions, and the respective constraints. Note that the superscript in each case in- regionX of attraction (ROA). dicates the respective MPC approach by referring to the Step 4: Parameter tuning. Formulate the open-loop corresponding theorem number (we use extra indices a optimal control problem (12) based on Steps 1-3. The and b to distinguish between the two tail sequence based ˆ2 ˆ′# relevant problem parameters may be adjusted in order to approaches). While the sets N (δ) and N (δ) are given X ˆX∗ achieve a desired closed-loop performance, e.g. based on as sublevel sets of the value function JN (x; δ), the sets # closed-loop simulations. In addition to the usual MPC 0 are sublevel sets of the resulting maximal constraint parameters, the parameters ε and δ as well as the chosen violationX based on Theorem 6, both for a given δ = 10−6. terminal cost function will have a major impact on the In this case, the functions were evaluated over a fine grid resulting behavior. To the experience of the authors, and and Matlab’s contour function was used for plotting. in accordance with the above results, the barrier func- Whereas the sets for which strict constraint satisfaction tion weighting parameter ε may be used to influence the can be ensured may be very small, approximate constraint −3 closed-loop performance while the relaxation parameter δ satisfaction with a meaningful tolerance ofz ˆtol = 10 primarily allows to control the occurrence and the max- is achieved for much larger regions of initial conditions. imal amount of state and input constraint violations. In The approach based on a purely quadratic terminal cost particular, Algorithm 1 from Section D may be used to Fˆ(x) = x⊤P x results in very small sets which is due to adapt the relaxation parameter in such a way that sat- the conservative quadratic upper bound (see Lemma 3), isfaction of a given constraint violation tolerance can be that causes P to grow rather fast depending on the rela- guaranteed. tion ε/δ2.

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Table 1: Summary of the presented MPC schemes based on relaxed logarithmic barrier functions.

Approach Section Terminal cost Fˆ(x) f (A, B) Relaxing function Theorem ROA constraint violation X ⊤ Relaxed f B1 x P x + εBˆ (x) yes stabilizable βk( ; δ)/βe( ; δ) Thm. 2 ˆN (δ) none X f · · X ⊤ n ′ Nonrelaxed f B2 x P x + εB (x) yes controllable βk( ; δ)/βe( ; δ) Thm. 3 R adjustable, none in ˆ (δ) X f · · XN T −1 n ′ Tailsequences C1 ℓˆ(zl(x), vl(x)) no controllable βk( ; δ)/βe( ; δ) Thm. 4 R adjustable, none in ˆ (δ) l=0 · · XN Quadratic bound C2 x⊤P x no stabilizable β ( ; δ) , quadratic Thm. 5 Rn adjustable, none in ˆ′ (δ) P 2 · XN

In Fig. 3, the behavior of the resulting closed-loop sys- tems is illustrated for different initial conditions and a 0.8 varying relaxation parameter δ. It can be seen how con- Xˆ′3 N (δ) vergence to the origin is achieved even for infeasible initial X 2 0.4 f Xˆ (δ) conditions and how strict or approximate constraint sat- N 5 isfaction may be enforced by making δ sufficiently small. X0 (δ)

2 0

Note that, in order to get comparable closed-loop perfor- x Xˆ′4b mance and execution times, we chose a doubled horizon N (δ) of N = 20 for the MPC schemes based on a dead-beat − 0.4 4b X0 (δ) 4a tail sequence and a purely quadratic terminal cost. X0 (δ)

3 XN X0 (δ) −0.8 C. Numerical Aspects −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

As outlined above, one of the main advantages of relaxed x1 barrier function based MPC formulations is given by the fact that the stabilizing control input can be character- Figure 2: The discussed sets for the different MPC schemes −2 −6 ized as the minimum of a globally defined, continuously with ε = 10 and δ = 10 . Here, Xf and XN denote differentiable, and strongly convex cost function that is the polytopic terminal set and the resulting feasible set of the ˆ2 parametrized by the current system state. In fact, after terminal set based approaches. Furthermore, XN (δ) denotes the region of attraction of the locally stabilizing approach with elimination of the linear system dynamics, the open-loop ˆ′3 ˆ′4b strict constraint satisfaction, while XN (δ) and XN (δ) denote optimal control problem can be formulated as the uncon- the sets with guaranteed zero constraint violation for the glob- strained minimization of a cost function of the form ally stabilizing approaches based on Theorems 3 and 4, where 1 the LQR based tail sequence is used in the latter case. The ˆ ⊤ ⊤ ⊤ ˆ ′4a JN (U, x)= U HU + x FU + x Y x + εBxu(U, x) , (48) corresponding sets XˆN (δ) for a dead-beat controller based tail- 2 ˆ′5 sequence and XN (δ) for a purely quadratic terminal cost where ⊤ ⊤ RnU in this case too small to be plotted. In addition, the much larger where U := u0 uN−1 , nU = Nm, and 3 4a 4b 5 ··· ∈ sets X0 (δ), X0 (δ), X0 (δ), and X0 (δ) denote the regions in Bˆ : RNm Rn R is a positive definite, con- xu  +  which the respective MPC schemes result in a maximal con- × → −3 vex, and continuously differentiable relaxed logarithmic straint violation of zˆtol = 10 . barrier function for polytopic constraints of the form SnU Rn×nU GU w + Ex. The matrices H ++ , F , ≤ × ∈ ×∈ Whether such techniques could also be applied for the re- Y Sn , G Rq nU , w Rq, and E Rq n can be ∈ + ∈ ∈ ∈ laxed barrier function based MPC approaches discussed constructed from (12) and the corresponding constraints in this paper, is an interesting open question which might by means of simple matrix operations. Note that the be considered as possible future work. matrices in the above QP formulation may be rather ill- Another interesting property of the condensed formula- conditioned when considering unstable systems and long tion (48) is that it can be tackled directly by Newton- prediction horizons. This issue can, however, be resolved based continuous-time optimization algorithms as they by a suitable prestabilization, see [31]. The then resulting are, for example, discussed in [10] and [12]. In partic- mixed input and state constraints are not considered here ular, the differentiability and convexity properties of the in detail but can be handled by all discussed approaches cost function allow to design a continuous-time dynamical after some minor modifications. While the above con- system of the form densed QP formulation allows for a compact represen- tation with a minimal number of optimization variables, U˙ (t)= f(U(t), x(t)),U(t0)= U0 (49) the original formulation (12) might allow to exploit the in- herent problem structure, which may be beneficial from a whose solution asymptotically tracks the optimal input numerical point of view. Tailored structure exploiting op- vector Uˆ ∗(x(t)), where x(t) is the continuously measured timization algorithms have been presented in the context system state, see the aforementioned references for more of both conventional linear MPC, e.g.[30] and [33], and details. Such continuous-time linear MPC algorithms nonrelaxed barrier function based linear MPC, see [34]. completely eliminate the need for an iterative on-line op-

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0 10 1 x0 3 3 , −2 2.5 10 −4 0.5 10 Xf 2 −6

u 10 0 1.5 2 0 −8 δ x 1 10

− input −10 0.5 XN x0,1 0.5 10 −12 0 10 − 1 x0,2 −14 −0.5 10

−2−1.5−1−0.5 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1 time t κ ⊤ −2 −2 −3 −4 Figure 3: Left: closed-loop behavior for x0,1 = [2.15, −0.7] ∈XN and δ ∈ {0.1, 2.5 × 10 , 10 , 10 , 10 } for the approach based on a nonrelaxed terminal set [∗]: if δ is small enough, the state trajectories stay feasible. Also shown is the behavior for two −3 ∗ infeasible initial conditions x0,2 and x0,3 with δ = 10 together with the corresponding predicted terminal states xN (x(k)) [◦]. ⊤ For the infeasible initial condition x0,2 = [−1.75, −1] , also the closed-loop trajectories of the globally stabilizing terminal set free MPC schemes from Section C are depicted (dead-beat based tail sequence [⋄], LQR based tail sequence [+], purely quadratic terminal cost [△]). ⊤ Middle: control input for the different globally stabilizing MPC schemes for the initial condition x0,2 = [−1.75, −1] using the same coloring scheme. Right: The relaxation parameter δ¯0 obtained by Algorithm 1 for ensuring strict constraint satisfaction [◦], i.e. zˆtol = 0, respec- −3 tively approximate constraint satisfaction with a tolerance of zˆtol = 10 [∗] for the approach based on Theorem 3 with initial conditions in X0 = κXN where κ ∈ (0, 1). Also depicted is the δ¯0 required for ensuring asymptotic stability and strict constraint satisfaction directly based on the approach with a relaxed terminal set [⋄], cf. Theorem 2.

timization and can be implemented for in principle ar- servative constraint violation bounds, or the design and bitrary fast system dynamics. We expect that it might comparison of tailored iterative or continuous-time opti- be possible to exploit some of the advantages of relaxed mization algorithms that explicitly exploit the properties logarithmic barrier function based formulations, e.g., the of the relaxed MPC problem formulation. global definition of the cost function and a bounded cur- vature, explicitly within the underlying numerical inte- Appendix gration. In the following, we show how the finite-horizon LQR VI. Conclusion problem with zero terminal state constraint can be solved without the use of algebraic Riccati equations. Consider In summary, our investigation showed that the concept of problem (27) for a given initial state x and horizon T n. ≥ relaxed logarithmic barrier function based model predic- By eliminating the predicted states zl for l = 1 ...,T tive control is interesting both from a system theoretical using (27b), we get the following equivalent formulation and a practical point of view. In particular, while we in vectorized form: are still able to recover many of the theoretical proper- ∗ 1 ⊤ 1 ⊤ 1 ⊤ ⊤ 1 ⊤ ties of conventional or nonrelaxed barrier function based JV (x) = min V HV + x F V + V F x + x Y x MPC schemes, the use of relaxed logarithmic barrier func- V 2 2 2 2 T tions allows to characterize the stabilizing control input as s. t. A x + S1 S2 V =0 , (50) the minimizer of a globally defined, continuously differen-   tiable and strongly convex function that is parametrized where the respective matrices are given as by the current system state. As a main result, we pre- S = AT −1B AnB , S = An−1B B sented different constructive MPC design approaches that 1 ··· 2 ··· guarantee global asymptotic stability and allow to influ- H =2 R˜ +Φ⊤Q˜Φ , F = 2Ω⊤Q˜Φ, Y =2 Q +Ω⊤Q˜Ω ence the performance and constraint satisfaction proper- ties of the closed-loop system directly by adjusting the A  B 0   Q˜ = IT Q relaxation parameter. The resulting MPC schemes are . . ···. . ⊗ Ω= . , Φ= . .. . ,     not necessarily based on the construction of a suitable T T −1 R˜ = IT R. A A B B ⊗ terminal set and, due to the underlying relaxation, inher-    ···  ently robust against disturbances, uncertainties, or sensor Due to the controllability of the system, there always ex- faults. ists a feasible solution V (x) to problem (50). We parti- Tm ⊤ ⊤ ⊤ Interesting open problems may include a thorough anal- tion the vector V R as V = V V with V1 ∈ 1 2 ∈ ysis of the performance and robustness properties of the R(T −n)m and V Rnm. The terminal state constraint 2   resulting closed-loop system, the derivation of less con- can be eliminated∈ by choosing V = S+ AT x + S V , 2 − 2 1 1

17 + where S2 denotes the Moore-Penrose pseudoinverse of the scheme for nonlinear model predictive control. IEE pro- matrix S2. This leads to ceedings – Control Theory and Applications, 152(3):296– 308, 2005. I(T −n)m 0 [10] C. Feller and C. Ebenbauer. A barrier function based V = + V1 + + T x =ΓV V1 +Γxx . (51) S2 S1 S2 A continuous-time algorithm for linear model predictive  −  −  control. In Proc. of the 12th European Control Confer- Inserting (51) into problem (50) results in the following ence, pages 19–26, Zurich, Switzerland, 2013. unconstrained optimization problem in the variable V1 [11] C. Feller and C. Ebenbauer. Barrier function based lin- ear model predictive control with polytopic terminal sets. rd ∗ 1 ⊤ ˜ ⊤ ˜ 1 ⊤˜ In Proc. of the 53 Conf. Decision and Control (CDC), JV (x) = min V1 HV1 + x FV1 + x Y x , (52) V1 2 2 pages 6683–6688, Los Angeles, CA, USA, 2014. [12] C. Feller and C. Ebenbauer. Continuous-time linear MPC where the transformed problem matrices are given by algorithms based on relaxed logarithmic barrier functions. ˜ ⊤ ˜ ⊤ ⊤ ˜ th H = ΓV H ΓV , F = Γx H ΓV + F ΓV , and Y = Y + ⊤ ⊤ ⊤ In Proc. of the 19 IFAC World Congress, pages 2481– ΓxH Γx + ΓxF + F Γx. Note that since V (x)=Γxx 2488, Cape Town, South Africa, 2014. is a feasible solution and Im(ΓV ) = Null( S1 S2 ), the [13] C. Feller and C. Ebenbauer. Weight recentered barrier solution is invariant under this change of coordinates,   functions and smooth polytopic terminal set formulations and the problems (50) and (52) are equivalent, cf. [6, for linear model predictive control. In Proc. of the 2015 − p. 132]. Furthermore, it always holds that H˜ S(T n)m American Control Conference, Chicago, IL, USA, 2015. ∈ ++ as the matrix ΓV has full column rank. The optimal so- to appear. lution to the reduced problem can be computed easily as [14] A. Ferrante and L. Ntogramatzidis. Employing the alge- ∗ ˜ −1 ˜⊤ braic Riccati equation for a parametrization of the solu- V1 (x)= H F x which results in − tions of the finite-horizon LQ problem: the discrete-time ∗ −1 ⊤ case. System & Control Letters (54), pages 693–703, 2005. V (x)= Γx ΓV H˜ F˜ x = KV x . (53) − [15] G. Grimm, J. Messina, M. E. Tuna, S. and R. Teel, A.˙   ∗ ⊤ Model Predictive Control: For Want of a Local Control Furthermore, the optimal cost is given by JV (x)= x PV x Lyapunov Function, All is Not Lost. IEEE Trans. Autom. 1 ⊤ −1 n with PV = Y˜ + F˜ H˜ F˜ S . Control, 50(5):546–558, 2005. 2 ∈ +   [16] L. Gr¨une. NMPC without terminal constraints. In Proc. of the IFAC Conference on Nonlinear Model Predictive References Control 2012, pages 1–13. [17] J. Hauser and A. Saccon. A Barrier Function Method [1] D. Angeli. Intrinsic robustness of global asymptotic sta- for the Optimization of Trajectory Functionals with Con- th bility. System & Control Letters (38), pages 297–307, straints. In Proc. of the 45 IEEE Conf. Decision and 1999. Control (CDC), pages 864–869, San Diego, USA, 2006. [2] A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopou- [18] A. Jadbabaie and J. Hauser. Unconstrained receding hori- los. The explicit linear quadratic regulator for constrained zon control of nonlinear systems. IEEE Trans. Autom. systems. Automatica, 38:3–20, 2002. Control, 46(5):776–783, 2001. [3] A. Ben Tal, M. Tsibulevskii, and I. Yusefovich. Modified [19] S. S. Keerthi and E. G. Gilbert. Optimal infinite horizon barrier methods for constrained and minimax problems. feedback control laws for a general class of constrained Technical report, Optimization Laboratory, Israel Insti- discrete-time systems: stability and moving-horizon ap- tute of Technology, 1992. proximations. Journal of Optimization Theory and Ap- [4] A. Boccia, L. Gr¨une, and K. Worthmann. Stability and plications, 57(2):265–293, 1988. feasibility of state-constrained linear MPC without stabi- st [20] E. C. Kerrigan and J. M. Maciejowski. Soft constraints lizing terminal constraints. In Proc. of the 21 Interna- and exact penalty functions in model predictive control. tional Symposium on Mathematical Theory of Networks In UKACC International Conference (Control 2000), and Systems (MTNS 2014), pages 453–460, 2014. Cambridge, UK, 2000. [5] F. Borrelli, A. Bemporad, and M. Morari. Predictive Con- [21] D. Limon, T. Alamo, F. Salas, and F. Camacho, E.˙ trol. Book to be published, available online. On the Stability of Constrained MPC without Terminal [6] S. Boyd and L. Vandenberghe. Convex Optimization. Constraint. IEEE Trans. Autom. Control, 51(5):832–836, Cambridge University Press, Cambridge, 2004. 2006. [7] D. Chmielewski and V. Manousiouthakis. On constrained [22] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. infinite-time linear quadratic optimal control. System & Scokaert. Constrained model predictive control: Stability Control Letters (29), pages 121–129, 1996. and optimality. Automatica, 36:789–814, 2000. [8] D. DeHaan and M. Guay. A Real-Time Framework for [23] S. G. Nash, R. Polyak, and A. Sofer. Large Scale Opti- Model Predictive Control of Continuous-Time Nonlin- mization: State of the Art, chapter A numerical compari- ear Systems. IEEE Trans. Autom. Control, 52(11):2047– son of barrier and modified barrier methods for large-scale 2057, 2007. bound-constrained optimization, pages 319–338. Kluwer [9] M. Diehl, R. Findeisen, F. Allg¨ower, H. Bock, and Academic Publishers, 1994. J. Schl¨oder. Nominal stability of real-time iteration

18 [24] Y. Nesterov and A. Nemirovskii. Interior-Point Poly- nomial Algorithms in Convex Programming. SIAM, Philadelphia, 1994. [25] J. Nocedal and S. J. Wright. Numerical Optimization. Springer, 1999. [26] L. Ntogramatzidis. A simple solution to the finite- horizon LQ problem with zero terminal state. Kyber- netika, 39(4):483–492, 2003. [27] T. Ohtsuka. A continuation/GMRES method for fast computation of nonlinear receding horizon control. Au- tomatica, 40:563–574, 2004. [28] P. Patrinos and A. Bemporad. Simple and certifiable quadratic programming algorithms for embedded linear th model predictive control. In Proc. of the 4 IFAC Con- ference on Nonlinear Model Predictive Control, pages 14– 20, Noordwijkerhout, Netherlands, 2012. [29] S. J. Qin and T. A. Badgwell. A survey of industrial model predictive control technology. Control Engineering Practice, 11(7):733–764, 2003. [30] C. V. Rao, S. J. Wright, and J. B. Rawlings. Applica- tion of interior point methods to model predictive con- trol. Journal of Optimization Theory and Applications, 99(3):723–757, 1998. [31] J. A. Rossiter, B. Kouvaritakis, and M. J. Rice. A nu- merically robust state-space approach to stable-predictive control strategies. Automatica, 34(1):65–73, 1998. [32] P. O. M. Scokaert and J. B. Rawlings. Constrained lin- ear quadratic regulation. IEEE Trans. Autom. Control, 43(8):1163–1169, 1998. [33] Y. Wang and S. Boyd. Fast Model Predictive Control Us- ing Online Optimization. IEEE Trans. on Control Sys- tems Technology, 18(2):267–278, 2012. [34] A. G. Wills. Barrier function based model predictive con- trol. PhD thesis, School of Electrical Engineering and Computer Science, University of Newcastle, Australia, 2003. [35] A. G. Wills and W. P. Heath. Barrier function based model predictive control. Automatica, 40:1415–1422, 2004. [36] M. N. Zeilinger and C. N. Jones. Real-Time Subopti- mal Model Predictive Control Using a Combination of Explicit MPC and Online Optimization. IEEE Trans. Autom. Control, 56(7):1524–1534, 2011. [37] M. N. Zeilinger, C. N. Jones, and M. Morari. Robust stability properties of soft constrained MPC. In Proc. of th the 49 IEEE Conf. Decision and Control (CDC), pages 5276–5282, Atlanta, USA, 2010.

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