A set-theoretic generalization of dissipativity with applications in Tube MPC

This paper introduces a framework for analyzing a general class of uncertain nonlinear discrete-time systems with given state-, control-, and disturbance constraints. In particular, we propose a set-theoretic generalization of the concept of dissipativity of systems that are affected by external disturbances. The corresponding theoretical developments build upon set based analysis methods and lay a general theoretical foundation for a rigorous stability analysis of economic tube model predictive controllers. Besides, we discuss practical prodecures for verifying set-dissipativity of constrained linear control systems with convex stage costs.


Introduction
Dissipativity theory can be regarded as one of the most fundamental tools for analyzing the stability of control systems [8]. The origins of dissipativity theory can be traced to the work by Willems [28,29], who analyzed the theoretical properties of dissipative systems as well as formalized the concepts of energy supply and energy storage for general control systems.
Recent work on dissipativity theory has focused on its application to optimally operated control systems. For example, [1] established a link between dissipativity of a control system and the existence of optimal steady-states. In [10], a thorough review of economic model predictive control (MPC) schemes is presented. Unlike standard tracking problems, economic MPC controllers are based on objective functions which are, in general, not positive definite. For such controllers a number of stability conditions are available [1,16,17,30], which all rely on dissipativity theory.
In order to understand why one may wish to develop a generalization of dissipativity for set-valued systems, one must be aware of of set-valued analysis [2] and its importance in the development and analysis of robust control methods [5,6]. Among the various set-theoretic control methodologies, Tube model predictive control strategies have been analyzed exhaustively during the past two decades [14,15]. Here, the main idea is to replace trajectories by robust forward invariant tubes (RFITs), i.e., set-valued functions in the state space enclosing all future system states, independently of the uncertainty realization. A great variety of methods for Tube MPC synthesis can be found in the overview article [19].
In this context, one of the main contributions of this paper is the development of a rigorous mathematical framework for the stability analysis of a rather general class of set-valued control systems. Towards this aim, Section 2 introduces a set-based generalization of cost-to-travel functions, which have originally been developed for certainty-equivalent control systems [12]. Section 3 builds on this construction to propose a set-theoretic generalization of dissipativity for a particular class of storage functions. The practical applicability of these rather abstract concepts is discussed in Section 4, which establishes set-theoretic stability conditions for a large class of Tube MPC controllers with possibly economic objectives ad no assumptions on the feedback structure. These controllers can be based, in the most general case, on parameterizations where the set-valued cross-sections of the tube itself are free optimization variables. The theoretical developments of this paper are illustrated throughout the paper using a series of academic examples. Section 5 concludes the paper.

Notation and preliminaries
We use the symbols K n and K n C to denote the sets of compact and compact convex subsets of R n , respectively. The Hausdorff distance between two sets A, B ∈ K n is denoted by Notice that (K n , d H ) is a metric space [26].
As this paper uses functions whose arguments are sets in K n , we introduce the following definitions.
Definition 1 Let D ⊆ K n be a given domain. A function L : D → R is called (1) continuous on D if there exists for every A ∈ D and every > 0 a δ > 0 such that lower semi-continuous on D if there exists for every A ∈ D and every > 0 a constant δ > 0 such that Moreover, we also introduce the generalized Hausdorff distance, which is defined for any D, E ⊆ K n . The symbol K n is used to denote the topological space of all nonempty subsets of K n that are compact in 2 K n -the power set of K n . Recall that d H induces a metric in K n , using this one can show that the generalized Hausdorff distance H induces a metric in K n [25].
The following definition is useful for analyzing difference inclusions, as needed in the context of Tube MPC. Definition 2 Consider the function F : K n → K n . It is called continuous if there exists for every > 0 a δ > 0 such that 2 Set-based cost-to-travel functions The main goal of this paper is to analyze uncertain discrete-time control systems of the form Here, x k ∈ R nx , u k ∈ R nu , and w k ∈ R nw denote the state, control, and disturbance vectors at time k. The disturbance sequence w is unknown, but assumed to take values in the given set W ∈ K nw . Associated state-and control constraint sets, X ∈ K nx and U ∈ K nu , are also assumed to be given.
Since (1) depends on an uncertain disturbance sequence, its reachable set is, in general, not a singleton. Hence, Section 2.1 briefly reviews some concepts from robust forward invariance [6], used for the analysis. Section 2.2 introduces a novel set-theoretic generalization of cost-to-travel functions [12], whose properties are analyzed in Section 2.3.

Difference inclusions and robust invariance
As the focus of this paper is on set-based methods for analyzing (1), we introduce the map F : K nx → K nx , given by for all A ∈ K nx . This transition map F is the basis for the construction of control invariant sets and tubes for (1).
Definition 3 A sequence X = (X 0 , X 1 , . . .) of compact sets is called a robust forward invariant tube (RFIT) for (1) if it satisfies the difference inclusion ∀k ∈ N, X k+1 ∈ F (X k ) . If X = (X , X , . . .) is a time-invariant RFIT, X is called a robust control invariant (RCI) set.
Notice that F maps a set to a set of sets. This notation may appear rather abstract on the first view, but it has the advantage that we do not have introduce notation for the underlying possibly set-valued feedback law and the associated closed-loop reachability sequences which are parametric on the feedback law.

Set-based cost-to-travel functions
Let D ⊆ K n be a given domain and L : D → R a given lower semi-continuous function on D. The cost-to-travel function V D : D × D × N → R ∪ {∞} of (1) on D is given by which is defined for all sets A, B ∈ D and all N ∈ N. In order to ensure that V D is well-defined, the following assumption is needed.

Assumption 1
We assume that the domain D and the functions f and L have the following properties: (1) the right-hand side function f is continuous in all its arguments, (2) the set D ⊆ K nx is closed in the metric space (K nx , d H ), and (3) the function L : D → R is lower semi-continuous and monotonous on D in the sense of Definition 1.
Proposition 1 If Assumption 1 is satisfied, then the right-hand side of (3) either admits a minimizer or has an empty feasible set.
Proof. First notice that F is continuous in the sense of Definition 2. This is a direct consequence of the definition of F in (2), the continuity of f as well as the compactness of U and W; see, [2] for details. Since X is compact and D closed, the feasible set of (7) is compact in (K nx , d H ). Since L is lower semi-continuous, the right-hand side of (3) either admits a minimizer or has an empty feasible set. 2 Example 1 Let us consider a dynamic system given by as well as the stage cost . In this case, the cost-to-travel function V D (·, ·, 1) can be constructed explicitly. In fact, it is given by Here, we have used the shorthand notation

Properties of cost-to-travel functions
The following propositions summarize basic properties of the cost-to-travel function V D .
Proposition 2 (Monotonicity) If Assumption 1 is satisfied, then for all sets A, A , C, C ∈ D with A ⊆ A and C ⊆ C and all N ∈ N.
Proof. As discussed above, Assumption 1 ensures that V D is well-defined. The definition of F implies that the implications hold for all sets A, A , C, C ∈ D with A ⊆ A and C ⊆ C . Moreover, Assumption 1 requires L to be monotonous; that is, The statement of the proposition is a direct consequence of these three implications recalling the definition of V D in (3). 2 Proof. Assumption 1 ensures that F is continuous and L lower semi-continuous. Since X is compact, it follows, from standard arguments from set-valued analysis [2], that V D is lower semi-continuous. For example, one can use an indirect argument, as follows.
If V D was not lower-semi-continuous, we could find a sequence of sets But this means that there exists a sequence of associated feasible points X i of (3) with A and B replaced by A i and B i ; and Since X is compact, this sequence must have a convergent sub-sequence, whose limit sequence X ∞ is feasible too, and satisfies This is a contradiction, as we have The set-based cost-to-travel functions V D , satisfies the following functional equation. Proof. This statement follows from the definition of V D and Proposition 3. This ensures that either a minimizer for the minimization problem over B exists or that the expressions on both sides of the functional equation are equal to ∞. 2

A set-theoretic generalization of dissipativity
This section introduces a generalization of dissipativity in the context of discrete-time set-valued inclusions. (1) is called set-dissipative on its domain X × U × W with respect to a given supply rate S : D → R on D if there exists a nonnegative storage function Λ : D → R + such that the inequality

Definition 4 System
Notice that for the special case that W is a singleton and D the set of singletons in K nx , set-dissipativity is equivalent to dissipativity for deterministic systems with control-invariant supply rates, as introduced by Willems in [28,29]. To explain how set-dissipativity relates to the ongoing developments in this paper, we introduce the following definition.
Definition 5 A set X ∈ D is called an optimal robust control invariant set if In order to ensure that V D is well-defined the following assumption is introduced.
Proposition 5 Let Assumptions 1 and 2 hold. Then, there exists at least one optimal robust control invariant set X ∈ D.
Proof. Assumption 2 implies that there exists at least one set A ∈ D with A ⊆ X and A ∈ F (A), which ensures that the domain {(A, A) ∈ D × D | V D (A, A, 1) < ∞} is non-empty. Now, the statement of this proposition is a direct consequence of Proposition 3 and Weierstrass' theorem, which can be applied here as X is compact. 2 Example 2 Consider the setting from Example 1. Here, the optimal robust control invariant set can be found by solving Notice that (5) is a strictly convex quadratic program with its unique minimizer a = b = (−1, −1, −4, 0) . Thus, the optimal robust control invariant set is given by the line segment

Definition 6
The function V D (·, ·, N ) is called separable on D if it admits a non-negative separable lower bound The following lemma establishes the link between set-dissipativity and cost-to-travel functions. for all sets A + ∈ D with A + ∈ F (A) and A + ∈ X. In particular, this inequality must hold for A + = B, which implies This inequality also holds whenever V (A, B, 1) = ∞. Thus, W = Λ is a non-negative separable lower bound of V D (·, ·, 1) on D. Therefore, if (1) is set-dissipative on X × U × W with respect to the supply rate L(·) − L(X ) on D, then V D (·, ·, 1) is separable on the domain D.
In order to establish the converse implication, we use that L(A) = V (A, B, 1) for all A, B ∈ X with A, B ⊆ X and B ∈ F (A). Hence, for all such A, B we obtain which implies that (1) is set-dissipative with storage function Λ = W , as long as V D (·, ·, 1) is separable on D with separable lower bound W .   In this sense, one may state that strict separability of V D (·, ·, 1) is equivalent to "strict dissipativity" of (1).

Tube model predictive control
Tube MPC methods proceed by solving receding-horizon optimal control problems of the form with z ∈ R nx being the current state-measurement and T ∈ D a terminal set. Here, E : D → R, L : D → R, and M : D → R denote lower semi-continuous initial, stage, and terminal costs, respectively. It is well-known [24] that this tube MPC controller (7) is recursively feasible if T ∈ F (T ) and T ⊆ X.
Remark 1 If one is interested in adding a decoupled control penalty to the objective of the MPC controller, one can always introduce discrete-time states that satisfyx k+1 = u k , and append them to the state vector, such that the next state is equal to the current control input. In this sense, it is not restrictive to assume that the objective in (7) does not explicitly depend on the control input.
Remark 2 There is a close relation between the tube MPC problem (7) and set-based cost-to-travel functions. In particular, as a direct consequence of Proposition 4, (7) can be equivalently written as

Tube MPC feedback law
Notice that, any feasible point X of (7) is an RFIT. Thus, we can construct a control law, µ[X] : N × R nx → U, associated to this RFIT such that the state of any closed-loop system satisfies the implication x k ∈ X k =⇒ x k ∈ X k for all k ≥ k with k, k ∈ {0, 1, . . . , N }. This is a direct consequence of the definition of the transition map F .

Remark 3
Consider an RFIT X = (X 0 , X 1 , . . .), and a point z ∈ X k . One can evaluate the feedback law µ[X](k, z) by solving the robust feasibility problem In particular, the signal µ[X](k, z) = u k -with u k being a solution of the above feasibility problem, will drive z to X k+1 regardless of the uncertainty realization. Now, in contrast to this control law µ[X] associated to the RFIT, the Tube MPC feedback law ν : X → U is time-invariant and given by ν(z) = µ[Ξ](0, z) .
Here, Ξ(z) denotes a minimizing sequence of (7) as a function of the current measurement z. In the following, we use y = (y 0 , y 1 , . . .) to denote the closed-loop state recursion of the Tube MPC controller (7), given by with k ∈ N. That is, we set z = y k , solve (7), update the system using feedback (8) and repeat. In the next section we present an analysis of the stability properties of this closed-loop sequence using set-dissipativity.

Stability analysis
The goal of this section is to analyze stability of Tube MPC in the enclosure sense. Our definition of stability is motivated by the fact that the closed-loop trajectory y, given by (9), depends on the uncertainty sequence w.

Definition 8
The closed-loop state sequence y is said to admit a stable enclosure, if there exists a sequence Y = (Y 0 , Y 1 , . . .) of compact sets, Y k ∈ K nx , such that then y admits an asymptotically stable enclosure Y .

Remark 4
Notice that Y is not necessarily an RFIT, since the set sequence Y is only required-under the above definition-to contain the actual closed-loop sequence y.
The following theorem establishes a stability result for the Tube MPC controller (7) under the assumption that the initial cost function E is a strictly separable lower bound of V D (·, ·, 1). Equivalently, E must be a storage function that establishes strict dissipativity of (1) on D with respect to the supply rate S(A) = L(A) − L(X ). The statement is based on the additional assumption that the strictly separable lower bounding function E is also lower semicontinuous. At this point it has to be mentioned that a precise characterization of dissipative systems for which such a lower semi-continuous storage function exists, is still an open problem. However, there exist sufficient conditions under which one can assert the existence of continuous storage functions [18]-at least for nominal (not set-valued) systems. Moreover, in the following section we will discuss a variety of cases, where one can construct continuous functions E explicitly in order to arrive at a practical implementation.
Theorem 2 Let Assumption 1 and 2 be satisfied. Let the terminal region be an optimal robust control invariant set, T = X , and let y 0 be such that (7) is feasible for y = y 0 . If (1) is strictly set-dissipative on X × U × W with respect to the supply rate S(·) = L(·) − L(X ) on D with E being an associated lower semi-continuous storage function and M = 0, then the closed-loop sequence y of the tube MPC controller (7) admits an asymptotically stable enclosure.
For all j ∈ N: (a) Measure the state, y j (b) Set X j = Ξ(y j ), where Ξ(y j ) is the optimal solution sequence of the j-th tube MPC problem (c) Set Y j = X j 0 (d) Evaluate ν(y j ), cf. Remark 3, send the feedback signal to the system and go to (a).
For the construction in Step (b), we recall the relation between the tube MPC problem (7) and cost-to-travel functions in Remark 2.
Since we have y j ∈ X j 0 , the relation y j ∈ Y j also holds by construction. In order to show that the sets Y j are well defined, we introduce the shifted sequence Since the inclusion y j+1 ∈ X j 1 holds independently of the uncertainty realization, X j is a feasible point of the (j + 1)-th Tube MPC problem. Thus, recursive feasibility holds and Y j is well defined.
Let R D : D × D → R denote the rotated cost-to-travel function that is defined by for all A, B ∈ D such the tube MPC problem in Step (b) can be written in the equivalent form The key idea of this proof is to establish the claim that the function L D : D N +1 → R, given by can be used as a Lyapunov function for the iterates X j of the tube MPC controller.
Our first goal is to show that the sequence X j is stable and converges to the limit point X = (X , X , . . . , X ) ∈ D N +1 .
Let us establish the following properties of the candidate Lyapunov function L D . The sequence X j satisfies L D X j+1 < L D X j for all j whenever X j 0 = X .
Notice that P1 follows from Proposition 3. Moreover, P2 follows from the definition of L D and the assumption that E is a strict separable lower bound of V D (·, ·, 1). Thus, it remains to establish P3. As discussed above, the proposed tube MPC controller is recursively feasible. This implies that Here, we have used our assumption that V D (·, ·, 1) is strictly dissipative, which implies that R D X j 0 , X j 1 > 0 whenever X j 0 = X .
These properties are sufficient to conclude that L D is a Lyapunov function proving asymptotic stability of X j toX with respect to the Hausdorff metric. This implies that that the sequence Y is an asymptotically stable enclosure of y, converging to X * . 2 Similar to existing results for economic MPC schemes (see [10] and references therein) Theorem 2 establishes asymptotic stability for the proposed Tube MPC controller under a dissipativity condition. But-in contrast to nominal, certainty-equivalent, economic MPC schemes-here, the storage function E is not only needed for analysis purposes. In fact, the proposed Tube MPC controller makes explicit use of the initial cost E, as the initial tube is not fixed but an optimization variable.
Remark 5 Theorem 2 specializes-for simplicity of presentation-on the case T = X and M (A) = 0. However, a generalization of this stability result for any terminal region T ∈ D with T ⊆ X is possible under the additional assumption that the function M is lower semi-continuous and satisfies the condition see also [1] for details. An in-depth discussion on how to construct such set-based terminal costs is, however, beyond the scope of this paper.
Example 4 Let us return to the setting from Examples 1 and 2-recalling that the optimal RCI set is given by . Let us attempt to set up a robust MPC controller without initial cost and N = 2, i.e. (10) Using the notation established in Examples 1 and 2, the set optimization problem (10) can be formulated as the strictly convex parametric quadratic program with (x ) = (−1, −1, −4, 0) . Having Remark 3 in mind, we can introduce a decision variable u 0 ∈ [−5, 5] and augment (10) with the constraints hold.
Now, the parametric optimizer of (10) (augmented with (12)) is a piecewise linear function defined on 22 critical regions (non-overlapping interval boxes).
This instability issue can be fixed by adding the initial cost term E = W from Example 3. Now, the robust MPC formulation is given by Again, we can formulate this as the quadratic program augmented with the decision variable u 0 ∈ [−5, 5] and the constraints (12). The optimizer is, again, a piecewise affine function defined over 24 critical regions. Figure 1 shows the component u 0 of the parametric optimizer.  for all y ∈ [−5, 5] × [− 5,5]. This feedback law is not only recursively feasible, but also asymptotically stable in the enclosure sense.
Notice that the region [−5, 5] × [−4, 0]-depicted with a hatched pattern in Figure 1-is forward reachable in at most one step, for any initial feasible initial condition and any w 0 ∈ W. Moreover, any closed-loop sequence satisfies y k+1 ∈ X , whenever y k ∈ [−5, 5] × [−4, 0]-irrespective of w k . Since we have Y k ⊆ X for all k ≥ 0, any closed loop sequence admits a stable enclosure. In addition, the associated optimal set sequence is given by