Nonlinear Analysis: Hybrid Systems. Hybrid output regulation for nonlinear systems: Steady-state vs receding horizon formulation

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1 Nonlinear Analysis: Hybrid Systems 29 (2018) 1 19 Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: Hybrid output regulation for nonlinear systems: Steady-state vs receding horizon formulation Sergio Galeani, Mario Sassano * Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma Tor Vergata, Via del Politecnico 1, 00133, Rome, Italy a r t i c l e i n f o a b s t r a c t Article history: Received 25 July 2016 Accepted 31 December 2017 Keywords: Output regulation Nonlinear hybrid systems Periodic jumps The hybrid output regulation problem in the presence of periodic jumps is approached in this paper for a class of nonlinear hybrid systems satisfying rather mild assumptions. We provide sufficient conditions that characterize steady-state trajectories achieving output regulation that are defined, mimicking the linear case, in terms of two equations: the first one describes the solution of a flow-only output regulation problem, while the second is associated to an auxiliary output regulation problem concerning the monodromy equivalent system of the flow zero-dynamics. Such conditions are then revisited and a receding-horizon, steady-state-less, solution to the (hybrid) output regulation problem is suggested, based on the solution of a sequence of two-point boundary value problems Elsevier Ltd. All rights reserved. 1. Introduction Output regulation and tracking problems consist in enforcing a response of the controlled system in such a way that a given output of interest of the plant evolves over time following a desired profile generated by a reference system, regardless of external disturbances which are typically only known to belong to certain classes of signals [1 3]. Output regulation represents, together with stabilization, one of the fundamental problems in control theory, and as such has been an active research area for more than four decades [2 5]. Compared to stabilization, however, the fascinating feature of output regulation consists in the fact that complex steady-state evolutions may be considered, as induced by the exogenous signals. Departing from the basic formulation provided in the linear time-invariant case [2,1,3], several research directions have been pursued, culminating with the definition of the output regulation problem in the nonlinear context [4], see also the more recent monographs [6,7] for a more complete survey. More recently, motivated by the ubiquitous interaction between digital devices and continuous processes, the increasing attention devoted to the hybrid framework (see [8] for a comprehensive introduction) has encouraged the study of the output regulation problem for classes of hybrid linear and nonlinear systems. Clearly, since these systems are described by flow (continuous-time) dynamics as well as jump (discrete-time) dynamics, this problem becomes significantly more involved than the purely continuous-time (or purely discrete-time) counterpart. Hence, even in the linear setting, very few contributions are available in the literature. Output regulation for completely general classes of hybrid systems is a largely unexplored territory, due to a number of difficulties making the general hybrid regulation problem much more intricate then the classic (non-hybrid) one. However, a framework in which hybrid regulation can be addressed for hybrid systems with some sort of continuity with respect to the classic case has been singled out in [9]: this consists in focusing on a class of hybrid systems for which jump times are a priori fixed as multiples of a basic value, similarly to the classic single rate sampled-data systems. Such restriction on * Corresponding author. addresses: sergio.galeani@uniroma2.it (S. Galeani), mario.sassano@uniroma2.it (M. Sassano) X/ 2018 Elsevier Ltd. All rights reserved.

2 2 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) 1 19 the jump times avoids the above mentioned problem, meanwhile preserving most of the interesting behavior due to the interplay between flow and jump dynamics that has to be carefully taken care of in order to achieve regulation. The same class of systems has been further investigated by the same authors e.g. in [10], with partial extensions to the nonlinear case in [11]. The linear, multi-input multi-output case (possibly with more inputs than outputs) has been addressed by [12 14], removing all assumptions of minimum phaseness or on the relative degree. In this paper, mostly focusing on the preliminary structural problem of identifying motions compatible with zero regulation error, a twofold contribution is given. First, sufficient conditions of output regulation for a class of nonlinear hybrid systems in the presence of periodic jumps are provided, extending the conditions provided in [15,16] for the linear setting. In this context, it is shown that, in the nonlinear case as well, the purely continuous-time contribution to the hybrid solution, referred to as the heart of the hybrid regulator, has a crucial role, just like the presence of redundant inputs (namely, the fact that the plant has a number of inputs strictly larger than the number of outputs, which is not required in non-hybrid output regulation). The main differences with respect to the previous works [12,13,15,17,16,14] can be summarized as follows. Firstly, the case of nonlinear plant and exosystem is considered, whereas previous works were limited to the linear case. It is remarked that the techniques developed in [15] heavily rely upon the explicit characterization of solutions to the underlying (linear) ordinary differential equations, whose use is questionable in the nonlinear setting. Technically, dealing with the nonlinear case implies, among other things, that the monodromy dynamics can be expressed in terms of flows and jumps in a format reminiscent of the linear case, although it cannot be explicitly computed, and thus requires alternative approaches in order to arrive at explicitly computable solutions. Hence, as a second contribution, a rather different and unconventional path is pursued, and the problem of defining a steady-state response achieving output regulation is sidestepped by proposing to achieve output regulation via a receding horizon approach, without the need of defining a steady-state. Such an approach allows to circumvent the above mentioned difficulties in computing the flow in closed form, and apparently has never been used or studied even in the linear context. The problems solved in a receding horizon fashion are actually two-point boundary value (ode) problems, which can also be enriched with additional (state or input) constraints as well as performance index to be optimized. Note, in fact, that the receding horizon solution, involving the computation of the control input separately on each flow interval, opens the door to the possibility of achieving special features which were not even considered in previous approaches, even in the linear case. As an example, it would be possible to optimize different cost functions on different flow intervals, possibly choosing the cost of interest in real-time during operation; such a feature is especially desirable in applications, since it gives the opportunity to achieve different higher level objectives, while preserving the achievement of output regulation. Finally, the implementation of the receding horizon solution is permitted by the interesting geometric characterization of the required conditions. Essentially, this point amounts to provide, and analyze in terms of suitable relations among subsets of the state space, the conditions under which an output regulation problem which involves the evolution on a complete (infinitely long) time domain can be decomposed and solved in a sequential fashion, simply by computing the solution on each single flow interval [t k, t k+1 ] and disregarding other flow intervals, as far as suitable conditions are satisfied. The rest of the paper is organized as follows. After establishing some preliminaries in Section 2, the aim of Section 3 consists in formalizing the definition of the output regulation problem under examination together with some basic notation and preliminary results. The derived sufficient conditions are then presented in the full information case in Section 4, in which similarities and (crucial) differences with respect to the linear case are discussed. In Section 5 an interesting interpretation of the hybrid output regulation problem is provided and the task is formulated in terms of a sequence of twopoint boundary value problems, thus introducing a receding-horizon solution to output regulation. The underlying geometric picture is sketched in Section 6. The paper is concluded by numerical simulations on two academic examples in Section Notation and preliminaries In this paper we focus on a special class of hybrid systems considered in [10] for the linear case, characterized by having all solutions defined on the same hybrid time domain 1 { 0, if h = 0, T := {(t, k) : t [t k, t k+1 ], k N}, t h := (1) ϕ + (h 1)τ M, if h N 1, with τ M > 0 given, where t denotes the current value of continuous time and k denotes the number of jumps already occurred. For a function χ(, ) defined on T, denote as usual χ(t, k) := d χ(t, k) and χ + (t, k) := χ(t, k+1), provided that t = dt t k. Solutions to the considered hybrid systems are piecewise absolutely continuous functions χ(, ) that satisfy a differential equation (flow dynamics) χ = f (χ) almost everywhere in T, and moreover they satisfy a difference equation (jump dynamics) χ + = g(χ) when (t, k) T is such that t = t k, k N 1. Note that, while for general hybrid systems [8] it is necessary to explicitly define the flow set (in which the solution is allowed to flow) and the jump set (in which the solution is allowed to jump), having the fixed time domain (1) implies that there is no ambiguity about when the solution is flowing and when it is 1 An equivalent way to introduce such time domains (see [8,10]) consists in introducing clocks described by the hybrid dynamics τ = 1 for τ [0, τm ] and τ + = 0 for τ = {τ M } by choosing of τ (0, 0) = ϕ. Since such clocks are not exploited in this paper, a relevant notational simplification is achieved by using the equivalent, explicit definition (1).

3 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) jumping, and so for brevity flow and jump sets will not be explicitly indicated; for the same reason, it is generally not necessary to explicitly write the arguments (t, k). For brevity, the notation x [k], x k may be adopted in place of x(t k, k) and x(t k+1, k) to refer to the value at the beginning and at the end, respectively, of the kth flow time interval. For any function v(, ) defined on T, v 0 means v(t, k) = 0, for all (t, k) T. The variable σ := t t k, σ [0, τ M ], is used to measure the time elapsed since last jump; clearly, v may be used to denote derivative with respect to both t and σ. Throughout the paper, we use the term monodromy system to refer to a purely discrete-time system associated to a hybrid system with trajectories defined on T, which describes the evolution of the state (and possibly other quantities) of the hybrid system at the time instants (t k, k). In addition, we briefly recall some notions and results about differential geometry that are extensively used in the paper. Let M be a smooth manifold. A vector field f on M is a mapping assigning to each point p M a tangent vector f (p) in the tangent space T p M. Given a vector field f on R n R m, φ f (t t 0, u, x 0 ) denotes the solution (integral curve) of ẋ(t) = f (x(t), u(t)) at time t under the action of the control input u(τ ), τ [t 0, t], and with x(t 0 ) = x 0. If an autonomous system is considered, namely ẋ(t) = f (x(t)), then its integral curve is defined by φ f (t t 0, x 0 ). Let λ be a continuously differentiable real-valued function and f a vector field, then the real-valued function L f λ(x) dλ(x)f (x) is the Lie derivative of λ along f. Let f and g be continuously differentiable vector fields, then the vector field [f, g] g f f (x) g(x) is the Lie bracket of f and g. A distribution = span{f 1,..., f d } assigns a vector subspace of the tangent space to each point p M and its dimension at a point p x x is the dimension of the subspace (p). A distribution, defined on an open set, is said nonsingular if there exists an integer κ such that dim( (x)) = κ for all x in the set and involutive if f 1 and f 2 imply that [f 1, f 2 ]. Moreover, given a vector f and a distribution, is invariant with respect to f if [f, ]. Finally, a distribution of constant dimension κ is said to be completely integrable if there exist n κ real-valued functions λ 1,..., λ n κ such that span{dλ 1,..., dλ n κ } =, where denotes the collection of row vectors (covector fields) that annihilate the vector fields in. 3. Problem formulation Consider a class of nonlinear hybrid systems with hybrid arcs defined over the time domain (1) and described by flow dynamics of the form ẋ 1 = f 1 (x) + b 11 (x)u c,1 + b 12 (x)u c,2 + p 1 (x)w, ẋ 2 = f 2 (x 2, x 3 ) + b 22 (x)u c,2 + p 2 (x)w, ẋ 3 = A 33 x 3 + B 32 (f 3 (x) + b 32 (x)u c,2 + p 3 (x)w), e = C 3 x 3 + q(w), with x = [x 1, x 2, x 3 ], and jump dynamics described by x + = ϵ(x, u d, w), where x R n, x 1 R n 1, x 2 R n 2, x 3 R n 3 n = n 1 + n 2 + n 3, denotes the state of the system, u c R mc, u c = [u c,1, u c,2 ] with u c,1 R mc p and u c,2 R p, u d R m d and e R p are the continuous-time inputs, impulsive inputs and regulated outputs, respectively, and w R ns is the state of the nonlinear hybrid exosystem ẇ = s(w), w + = j(w). Note that, in addition to the input u c,2 having the same size of the regulated output, the additional redundant inputs u c,1 and u d are considered, whose key role to ensure solvability of the problem in the hybrid setting has been clarified in [15]; such inputs are named redundant from the point of view of non-hybrid output regulation, since such additional inputs play no role in non-hybrid output regulation. The mappings involved in (2) (4) are assumed to be sufficiently smooth and such that f i (0) = 0, ϵ i (0, 0, 0) = 0, i = 1, 2, 3, q(0) = 0, s(0) = 0 and j(0) = 0, while det(b 32 (0)) = 0, which implies that the matrix-valued function b 32 is square of dimension p p, as well as locally nonsingular. Moreover, C 3 R p n 3, C 3 = [blockdiag{[1, 0 1 (r1 2)],..., [1, 0 1 (rp 2)]} 0 p p ], the matrix A 33 R n 3 n 3 is defined as 2 [ ] blockdiag{ml } blockcolumn{n A 33 = l }, (5) 0 p (n3 p) 0 p p with all the entries of N l R rl 1 p equal to zero apart from the element of position (r l 1, l) which is equal to 1 and M l R (r l 1) (r l 1), M l = , (6) and B 32 = [0 p (n3 p) I p p ]. Note that the r i are such that p i=1 (r i 1) + p = n 3. (2a) (2b) (2c) (2d) (3a) (4a) (4b) 2 The notation blockcolumn {Ai } describes in compact form the matrix [A 1, A 2,..., A n ].

4 4 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) 1 19 Remark 1. As discussed in detail in the Appendix, the structure of the flow dynamics (2) may be obtained from a generic nonlinear system under mild assumptions, which are reasonable in the context of output regulation, i.e. a well-defined vector relative degree at the origin and a partial disturbance decoupling assumption, namely Assumptions 3 and 4, respectively. Mimicking the classic (non-hybrid) case, the following standing assumptions are instrumental for the derivation of sufficient conditions for output regulation in the considered class of nonlinear hybrid systems. In the assumption, φ s (τ M, w [k] ) denotes an integral curve of the exosystem after τ M time units, from initial condition w [k], which describes the state of the exosystem at the beginning of each flow interval. Assumption 1 (Poisson Stability). The monodromy system w [k+1] = j(φ s (τ M, w [k] )) is (positively) Poisson stable. 3 Assumption 1 ensures, on one hand, that the trajectories of the exosystem are bounded for all (t, k) T, while avoiding, on the other hand, trivialities for the (hybrid) output regulation task (which correspond to the case when the exosystem generates trajectories asymptotically convergent to zero). Moreover, it appears evident that, since the following statements provide only sufficient conditions for output regulation, such assumption may in fact be relaxed. Consider, in addition, the following standing assumption, which is not restrictive by the arguments derived in the discussion of the Appendix, concerning the controllability properties of the x 1 subsystem (2a). Assumption 2 (Controllability). The x 1 -subsystem (2a) is locally accessible from the origin by the control input u c,1, namely the controllability rank condition [18] is satisfied. Remark 2. A decomposition similar to (2) has been considered in [15] in the linear setting, firstly with respect to observability properties of the system via the regulated output, hence defining the weakly unobservable subspace {x R n : x 3 = 0}, and subsequently with respect to controllability properties by means of redundant inputs of the unobservable part, namely determining the controllable weakly unobservable subspace {x R n : x 2 = 0, x 3 = 0}. In the nonlinear setting, the former can be related to the transformations involved in the characterization of the (vector) relative degree while the latter can be expressed in terms of a suitably defined invariant distribution, namely the controllability distribution (whose construction is detailed in the next remark). Interestingly, precisely as in the linear case, there is exactly one evolution for the input u c,2 and the state x 3 of system (2c) compatible with zero error and the parameterization of such steady-state trajectories with respect to w can be provided in closed-form, as shown in the following section. Remark 3. Consider a nonlinear system described by the equations ẋ = f (x) + g(x)u, with x R n and u R m. The controllability distribution = {f, g 1,..., g m } span{g 1,..., g m } may be constructed by means of the following nondecreasing sequence of distributions 0 = span{g 1,..., g m } k = k 1 + [f, k 1 ] + m [g j, k 1 ]. j=1 It can be shown that the sequence converges to {f, g 1,..., g m } span{g 1,..., g m } in a finite number of steps, provided each k of the sequence is nonsingular. We consider here the following definition of Full-Information Hybrid Output Regulation Problem with periodic jumps, in which the information available to the feedback controller consists in the entire state of the plant and the exosystem. Problem 1. Consider the nonlinear hybrid system (2) (3) driven by the exosystem (4), defined on time domain T in (1). The Full-Information Nonlinear Hybrid Output Regulation problem consists in finding, if possible, a full-information static feedback (7) u = α(x, w), (8) such that there exists a neighborhood V R n R ns such that, for each initial condition (x(0, 0), w(0, 0)) V the corresponding trajectories of the closed-loop hybrid system (2) (8) are bounded and such that lim e(t, k) = 0. (t+k) + The above problem actually entails two main aspects, that is the characterization of the responses of the plant ensuring regulation (namely, associated with identically zero regulated output) and the stabilization of the trajectories of the closedloop system towards such responses (so that the regulated output is guaranteed to converge to zero at least asymptotically). In classical (non-hybrid) output regulation for nonlinear systems, the emphasis is typically on the former task, while the (9) 3 A point w0 is said to be positively Poisson stable if and only if for each neighborhood V of w 0 and each integer N 0 there exists some integer k > N such that w [k] V. The system is said to be positively Poisson stable if there exists a neighborhood of the origin W such that any w W is positively Poisson stable. The interested reader is referred to [6, Rmk 3.2] for a similar definition in the continuous time setting.

5 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) latter task is accomplished by means of more basic tools, e.g. linearization techniques around an equilibrium point [18]. A similar analysis is carried out here. Therefore, as it will become more evident in the sequel, in this paper we are essentially interested in characterizing, for instance in terms of existence and uniqueness, steady-state trajectories achieving hybrid output regulation, namely hybrid arcs along which the error e is identically equal to zero. It has to be stressed, in fact, that the explicit description of error-zeroing steady-state trajectories is a fundamental step (representing a somewhat necessary condition, which in the linear case corresponds to establishing the solvability of the Francis equations) towards the comprehensive solution to the problem. To further substantiate the above analysis, we provide a separate statement for the problem of interest. Problem 2. Consider the nonlinear hybrid system (2) (3) driven by the exosystem (4), defined on time domain T in (1). Find, if possible, a full-information static feedback (8) such that for each initial condition w(0, 0) W R ns there exists an initial condition x(0, 0) R n such that e(0, 0) = 0 implies that e(t, k) = 0, for all (t, k) T, along the bounded trajectories of the closed-loop hybrid system (2) (8). 4. Sufficient conditions for the existence of error-zeroing trajectories In this section we discuss sufficient conditions that guarantee existence of trajectories achieving hybrid output regulation, namely solving Problem 2. Interestingly, such conditions may be interpreted as a direct extension of those presented in [15] for the linear setting, which are briefly recalled here for completeness. Towards this end, in the linear case, (2) (4) are replaced by the equations ẋ = Ax + Bu c + Pw, ẇ = Sw, (10a) x + = Ex + Fu d + Rw, w + = Jw, (10b) e = C 3 x 3 + Q w, (10c) respectively, with [ A11 A 12 ] A 13 [ B11 ] B 12 [ E11 E 12 ] E 13 A = 0 A 22 A 23, B = 0 B 22, E = E 21 E 22 E 23, P = 0 0 A 33 0 B 32 E 31 E 32 E 33 [ P1 P 2 P 3 ], (11) and F and R matrices without any specific structure. The internal partition of the matrices in (11), namely the presence of structural blocks of zeros, describes the decomposition already discussed in Remark 2. With such data, the sufficient in fact also necessary conditions of [15] are given by a pair of matrix (Francis-like) equations reminiscent of the classical, non-hybrid case, namely the reduced flow Francis equation ΠS = A 33 Π + B 32 Γ + P 3, 0 = C 3 Π + Q, in the unknowns Π and Γ, and the monodromy Francis equation: Π S = Ã Π + B Γ + P, 0 = C Π + D Γ + Q, with unknowns Π and Γ, and coefficient matrices given by [ ] A11 A A 0 = 12, S = Je Sτ M (14a) 0 A 22 [ ] [ ] E11 E Ã = 12 e A 0τ E11 M F, B = 1, (14b) E 21 E 22 E 21 F 2 C = [ ] [ ] E 31 E 32 e A 0 τ M, D = E31 F 3, (14c) [ ] [ ] τm R1 + E P = 13 Π 3c e R 2 + E 23 Π Sτ E11 M E + 12 e A 0(τ M τ ) Sτ ˆPe dτ, (14d) 3c E 22 E 21 Q = (R 3 + E 33 Π 3c Π 3c J)e Sτ M + [ ] τ M E 31 E 32 e A 0(τ M τ ) Sτ ˆPe dτ, 0 ] [ ] [ ] [ ] [ˆP1 P1 A13 B12 ˆP = = + Π ˆP 2 P 2 A 3c + Γ 23 B 2c A quick look at (14) will convince the reader that a crucial role in defining the parameters of the monodromy Francis equation is played by the description of the (free and forced) response of the plant over one interval [t k, t k+1 ] followed by a single jump of the hybrid system, namely of the monodromy system. In the nonlinear case studied in this paper, the above conditions are replaced by the following ones, yielding a solution to Problem 2 in the nonlinear hybrid setting. (12a) (12b) (13a) (13b) (14e) (14f)

6 6 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) 1 19 Proposition 1. Consider the nonlinear hybrid system (2) (3) driven by the exosystem (4), defined on time domain T in (1). Suppose that there exist π 1 : R ns R n 1, π 2 : R ns R n 2, π 3 : R ns R n 3, defined in a neighborhood W R ns of the origin, α c,1 : R n R ns R mc p, α c,2 : R n R ns R p and α d : R n R ns R m d such that 4 (i) (reduced flow FBI equation) π 3 w s(w) = A 33π 3 (w) + B 32 (f 3 (x 1, x 2, π 3 (w)) + b 32 (x 1, x 2, π 3 (w))α c,2 + p 3 (x 1, x 2, π 3 (w))w), 0 = C 3 π 3 (w) + q(w), for all (x 1, x 2 ) R n 1 R n 2 and w R ns ; (ii) (monodromy Francis equation) ) π i (j(φ s (τ M, w 0 ))) = ϵ i ([φ 1 (τ M, α c, π(w 0 )), φ 2 (τ M, α c,2, π(w 0 )), π 3 (φ s (τ M, w 0 )) ], α d, φ s (τ M, w 0 ) (15a) (15b) (16) i = 1, 2, 3, for any w 0 W; with π [π, π, π ] and α c = [α, c,1 α c,2 ]. Then, for any w(0, 0) W the trajectories of the closed-loop extended system (2) (4) with initial conditions x i (0, 0) = π i (w(0, 0)), i = 1, 2, 3, are bounded and such that e(t, k) = 0 for all (t, k) T. Proof. To begin with, note that, by the structure of (2c), during flows of the hybrid plant, namely in the time intervals t (t k, t k+1 ), k N 1 the regulated output e is not affected by the components of the flow zero-dynamics, i.e. x 1 and x 2. Therefore, following arguments identical to those employed for a purely continuous-time nonlinear plant, sufficient conditions for regulation are obtained by rendering a suitable error-zeroing submanifold controlled-invariant. Since the manifold can be expressed as the graph of a mapping x 3 = π 3 (w), condition (15a) ensures controlled-invariance of the submanifold while (15b) implies zero error along trajectories evolving on the submanifold. To deal then with the hybrid behavior of system (2) (3), we should guarantee that the error e remains zero after jumps if it is equal to zero before the jump. This is achieved by requiring that x 3 = π 3 (w) implies x + = 3 π(w+ ), which is precisely condition (16) with i = 3 that exploits the flows φ 1, φ 2 and φ s to relate the value of the states at the end of the flowing interval, namely x 1, k, x 2, k and w k, to the same initial condition w 0 W. The two conditions (16) with i = 1, 2, on the other hand, ensure that x 1,[k] = π 1 (w [k] ) and x 2,[k] = π 2 (w [k] ), for all k N 1 hence, by Assumption 1, guaranteeing boundedness of the trajectories by smoothness of the vector fields involved in (2a) (2b). Remark 4. The monodromy Francis equation (16) provides a characterization of the error-zeroing steady-state trajectories. However, it may represent a daunting task from the computational point of view, since it demands the a priori construction of the maps π i, i = 1, 2, 3, α c, α d, defined on the entire state-space, that characterize the evolution of the state and the input of the system as a function of the state of the exosystem. Hence alternative, more numerically-oriented, approaches may be envisioned, in particular based on considering the current value of the state and determining a specific (portion of the) trajectory satisfying (15) and (16) and originating from such value. This approach is pursued in Proposition 3 and commented upon in Remark 7. Despite the fact that Eqs. (15) (16) are inspired by the continuous-time counterparts, i.e. the so-called FBI equations [18], only the essence of such equations, namely of ensuring invariance of a specific subset on which regulation is achieved, is replicated. Interestingly, the fact that some conditions apply during the whole flow but only on the x 3 dynamics, while the complementary conditions apply only in a monodromy (that is, on the one-period discrete time equivalent system) fashion on the whole dynamics, is not evident from the FBI conditions. Such comparison is further expanded in the following remark. Remark 5. The reduced flow Francis Byrnes Isidori (FBI) equation (15) and the monodromy Francis equation (16) can be given the following interesting interpretation. The former is related to a purely continuous-time output regulation problem associated to the part of system (2) observable via the regulated output and its (unique) solution has been referred to as the heart of the hybrid regulator in [15]. The monodromy Francis equation, on the other hand, provides conditions for the solution of an auxiliary output regulation problem associated to a discrete-time equivalent system that describes the hybrid evolution of (x 1, x 2 ) over one period between two consecutive jumps, hence combining (2a) (2b) with (3) with i = 1, 2. The reason why it is referred to as a Francis equation, apart from the heritage of the linear scenario in [15] in which (16) are explicitly recast into a Francis equation (see (13)), becomes indeed evident if one interprets the first two equations in (16), for i = 1, 2, as controlled-invariance conditions on the monodromy system for x 1 and x 2 and the last Eq. (16), for i = 3, as an (auxiliary) output-zeroing condition at the jump times. Therefore, Eqs. (15) (16) extend the FBI conditions at least in two non-trivial directions for the hybrid setting with periodic jumps: (i) Eq. (15) entails the maybe expected but yet to be proved in the nonlinear hybrid setting feature that at the heart of the hybrid regulator lies the solution to the purely continuous-time regulation problem that one would obtain in the absence of jumps; (ii) Eq. (16) recognizes the subset of the extended state-space that should be rendered invariant, in order to solve an auxiliary monodromy output regulation problem that must be solved to obtain the solution to overall genuinely hybrid problem. This aspect is suitably exploited in the following to provide a computationally viable approach to solve (16). 4 The dependence of the control inputs αc and α d on the state components x and the exosystem w is dropped for compactness.

7 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) Remark 6. A solution to Problem 1 can be provided by refining the above analysis in order to introduce, possibly only sufficient, conditions that guarantee (local) attractivity of the output-zeroing submanifold for the nonlinear hybrid system. Towards this end, let A l, E l, B l and F l be defined as and f 1 A l 0 x 1 (x) f 1 x 2 (x) f 1 f 2 x 2 (x) f A 33 (x) x 3 (x) x 3 x=(0,0,0), E l ϵ x b 11 (0) b 12 (0) B l 0 b 22 (0), F l ϵ (0, u d, 0) u d 0 B 32 b 32 (0) u d =0 (x, 0, 0) x=(0,0,0), (17). (18) Then, stabilizability of the linearized monodromy system, namely of the pair (E l e A lτ M, [El R l F l ]), with R l denoting the reachability matrix of the pair (A l, B l ), is a sufficient condition to imply that a solution to Problem 2 solves also Problem 1, at least in a neighborhood of the origin of R n R ns, see [17] for more detailed discussions. In order to render the sufficient conditions of Proposition 1 less abstract, the analysis is carried out in two parallel directions, focusing separately on the reduced flow FBI equation (15) and the monodromy Francis equation (16). As for the latter, in the following section, we show that a receding-horizon approach can be envisaged to efficiently formulate and solve the equations. The following result, on the other hand, which is derived as a minor adjustment of Corollary in [18], shows that the solution to (15) is indeed unique and can be explicitly provided in closed-form, thus characterizing the structure of the solution π 3 and u c,2 of (15). Proposition 2. Consider the nonlinear hybrid system (2) (3) driven by the exosystem (4), defined on time domain T in (1). Then and α c,2 = b 1 32 (x)(f 3(x) + p 3 (x)w) x 3 =π 3 (w) π 3,1 (w) π 3 (w)=. π 3,0(w)= 3,p (w) π 3,0 (w) q h (w) L s q h (w) π 3,h (w)=. L r h 2 s q h (w) describe the unique solution of (15). L r 1 1 s., h=1,..., p, (20b) q 1 (w) L rp 1 s q p (w) (19), (20a) Proof. The claim is proved by considering the definition of the matrix C 3 and the structure of the matrix A 33 in (5). In fact, Eq. (15b) combined with the first n 3 p equations of (15a) uniquely identifies the components of the mapping π 3. The remaining p equations in (15a) are satisfied by the suitable selection of u c,2, which is unique by (local) invertibility of the matrix-valued function b Receding-Horizon formulation of hybrid output regulation As discussed in the previous section the solution to the hybrid output regulation problem for nonlinear systems hinges essentially upon the solution of the monodromy Francis equation (16), since the reduced flow FBI equation can be explicitly solved by Proposition 2. In this section, exploiting the interpretation of (16) given in Remark 5, the task is recast into that of solving a specific multi-point boundary value problem (MPBVP), which possesses the critical advantage of not requiring explicit knowledge of the integral curves φ 1, φ 2 and φ s, referring to the dynamics (2a), (2b) and (4), respectively. While this reformulation is still hard to deal with, it will be shown later that it provides the basis for a receding horizon approach to the hybrid output regulation problem, which may be appealing from the computational point of view. Proposition 3. Consider the nonlinear hybrid system (2) (3) driven by the exosystem (4), defined on time domain T in (1). Let π 3 and u c,2 be such that (15) is satisfied (see Proposition 2). Then, all the trajectories satisfying x 3 (0, 0) = π 3 (w(0, 0)) and the

8 8 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) 1 19 following ODE s ẇ = s(w), ẋ 1 = (f 1 + b 11 u c,1 + b 12 u c,2 + p 1 w) x 3 =π 3 (w), ẋ 2 = (f 2 + b 22 u c,2 + p 2 w) x 3 =π 3 (w), (21) with multi-point boundary conditions (MPBC) w [k+1] = j(w k ) x i,[k+1] = ϵ i ([x 1, k, x 2, k, π 3(w k ) ], α d, w k ), (22) i = 1, 2, together with the auxiliary output constraint ϵ 3 ([x 1, k, x 2, k, π 3(w k ) ], α d, w k ) π 3 (j(w k )) = 0, (23) k = 1, 2,..., are such that e(t, k) = 0 for all (t, k) T. Proof. The result is proved by noting that the (explicit) conditions (16) are replaced by the sequence of (implicit) conditions (21) (23). Remark 7. Despite similarities in Propositions 1 and 3, it turns out the first is more of an analysis result, while the second yields more constructive conditions. In fact, the conceptual difference between the conditions in Proposition 1 and those in Proposition 3 is tantamount to the difference between the solution to an optimal control problem via the Hamilton Jacobi Bellman Equation versus the solution via the Pontryagin s Minimum Principle (PMP), respectively. In the former, the optimal control law is computed in terms of a feedback map defined on the entire state-space, whereas in the latter, the optimal control input is computed as a feedforward function of time (which in certain cases may be also expressed as a synthesis, namely transformed to a feedback form); the implication being that the former solution is valid for any initial state at the price of much harder computations, while the latter solution is much more amenable for numerical implementation since it focuses on a specific trajectory originating from the initial state of interest. A similar scenario is encountered when comparing Explicit Model Predictive Control (EMPC) with classic Model Predictive Control (MPC). In particular, the numerical advantages of trajectory-based computations, like those in PMP and MPC, stem from the fact that only discretization with respect to time, rather than that with respect to the entire state-space, is required; hence techniques that approximate the solution of a differential periodic equation by means of functions having piecewise constant derivatives are particularly viable. Remark 8. On each interval [t k, t k+1 ], system (21) consists of a set of n s + n 1 + n 2 nonlinear ordinary differential equations with n s + n split boundary conditions relating the values at the beginning of each flow time interval to those to be taken before jumps in order to ensure zero-error jumps on x 3. These terminal values are in particular provided by the condition (23). The presence of a number of boundary constraints greater than the number of equations may be compensated for by the presence of the redundant input u c,1, possibly together with impulsive inputs u d, and the property of subsystem (2a) of being controllable from u c,1, as stated by Assumption 2. To obtain a computationally viable approach to derive motions ensuring zero output regulation error, the multi-point boundary value problem introduced in Proposition 3 can be interpreted as an infinite sequence of two-point boundary value subproblems. Each subproblem is defined for t [t k, t k+1 ] with boundary data x [k] at t = t k given, and a suitably defined final constraint on x k at t = t k+1. Since values of x k and w k are related to values of x [k+1] and w [k+1] (at the beginning of the following interval [t k+1, t k+2 ]) according to (22) and (23), the basic idea is that to decouple the computations on each subinterval [t k, t k+1 ], k N 1 it is necessary to impose some recursive feasibility condition, namely x k must be such that for the ensuing x [k+1] the same problem just solved on [t k, t k+1 ] starting from x [k] will still be solvable on [t k+1, t k+2 ] starting from x [k+1]. The purpose of Proposition 3 consists in rephrasing (16), which describes a property of the complete solutions of the hybrid system, in terms of a property satisfied by the solutions separately on each interval [t k, t k+1 ]; in this way, instead of having to solve a priori the whole, infinite time horizon problem (perhaps in closed form, in terms of the flows φ i, which is a daunting task) it is possible to find a solution by solving a sequence of two-point boundary value problems with initial data computed by (22) (or given when k = 0) in terms of the states at the end of the previous interval, and with final constraint given by (23). Note that, compared to the above mentioned task of solving the infinite time horizon problem in terms of the flows φ i, the solution of a two-point boundary value problem is a simple and routinely solved problem. Under the above conditions, the determination of trajectories achieving identically zero regulated output can be performed in a receding horizon fashion. An explicit algorithm can be described as follows: Step 0 Compute (offline) the unique solutions α c,2 and π 3 of (15) as in Proposition 2. Step 1 At hybrid time (t k, k), solve the two point boundary value problem with boundary conditions given by the initial values x 1,I, x 2,I, w I at hybrid time (t k, k) and the final values x 1,E, x 2,E, w E at hybrid time (t k+1, k), subject to the dynamic constraints:

9 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) ẇ = s(w), ẋ 1 = (f 1 + b 11 α c,1 + b 12 u c,2 + p 1 w) x 3 =π 3 (w), u c,2 =α c,2, ẋ 2 = (f 2 + b 22 u c,2 + p 2 w) x 3 =π 3 (w), u c,2 =α c,2, and the boundary constraints: x 1,I := x 1,[k], x 2,I := x 2,[k], w I := w [k], ϵ 3 ([x 1,E, x 2,E, π 3(w E ) ], α d, w E ) π 3 (j(w E )) = 0, thus determining functions α c,1 and α d. Step 2 During the interval [t k, t k+1 ], apply the flow input u c,2 := α c,2 precomputed at Step 0 and the flow input u c,1 := α c,1 computed at Step 1. Step 3 At hybrid time (t k+1, k), apply the jump input u d := α d computed at Step 1, and go to Step 1 (for the next value of k). Note that while α c,2 is a feedback from the current state (computed as explained in Proposition 2), α c,1 is a function of the initial condition (x 1,[k], x 2,[k], w [k] ) at hybrid time (t k, k) as well as of elapsed time σ = t t k, and finally α d is a function of (x 1,[k], x 2,[k], w [k] ) alone. The two-point boundary value problem described in the algorithm can be solved by several efficient and publicly available numerical routines, mainly developed (and widely used) by the optimal control and aerospace communities. Note also that, using the same numerical routines, it is also possible to find, among the several possible solutions, the one that optimizes a suitable performance criterion on the interval [t k, t k+1 ] (possibly, a different criterion for different values of k). Clearly, two main concerns naturally arise. First, the proposed approach does not necessarily guarantee bounded trajectories for x 1 and x 2 along solutions that achieve the regulation task. Moreover, it is not obvious that recursive feasibility holds. In the following section we provide a geometric interpretation of the solutions to (21) (23), interesting per se, that allows us to simultaneously answer the two previous concerns. 6. A geometric interpretation of the MPBVP To provide a geometric characterization of the MPBVP in (21) we need a preliminary discussion to recall definitions and results concerning the so-called reachability sets. Definition 1 (Reachability Set). R(X, T ) denotes the set of states reachable at time t = T starting from any x 0 X at time t = 0 under the action of piecewise constant input functions. Consider system (2), together with a solution to the reduced flow FBI equation (15), and define the manifold E {(x 1, x 2, w) R n 1 R n 2 R n s : ϵ 3 ([x, 1 x, π 2 3(w) ], α d, w) π 3 (j(w)) = 0, α d R m d }. It is straightforward to notice that E describes the set of states that ensure a zero-error jump for the component x 3, hence, for any value w k of the state of the exosystem (4) before jumps, x 1 (t, k) and x 2 (t, k) must belong to E for t = t k+1. Then, the MPBVPs in Proposition 3 can be given the following interpretation: solving Eqs. (21) with the boundary conditions (22) and (23) requires a non-empty intersection of R(X, τ M ) and E, for some set of initial conditions X, where R(X, τ M ) denotes the set of states reachable at time t = τ M for the system (2a) (2b) under the action of the (redundant) input u c,1. Finally, in order to guarantee the boundedness of trajectories, consider the following set ϵ 1 ([x, 1 x, π 2 3(w) ], α d, w) I Ω = (x 1, x 2, w) : ϵ 2 ([x, 1 x, π 2 3(w) ], α d, w) Ω, α d R m d. (24) j(w) The above discussion is formalized in the following result in which a condition ensuring existence of bounded solutions to (21) (23) is added. It is stressed that asking that (25) holds for all χ Ω is not the same as asking (25) with R(χ, τ M ) replaced by R(Ω, τ M ): in the former case, each initial state in the set must be able to reach the considered set, whereas in the latter case it would be enough that a single initial state in the set be able to do so. Proposition 4. Consider the nonlinear hybrid system (2) (3) driven by the exosystem (4), defined on time domain T in (1). Let π 3 and u c,2 be such that (15) is satisfied. Let Ω R n 1 R n 2 R n s be a compact set such that R(χ, τ M ) E I Ω = for all χ (x 1, x 2, w ) Ω and some α d R m d. Then the MPBVP (21) (23) admits bounded solutions for all (t, k) T. (25)

10 10 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) 1 19 Proof. To begin with note that the fact that R(χ, τ M ) and E have non-empty intersection for all χ in a compact set Ω guarantees that all the trajectories of the plant flow with zero regulation error, by the properties of u c,2, and are ready to jump with zero error at t = t k, for all k N 1. In addition, the fact that these two sets non-trivially intersect I Ω implies that, after each jump, the components x 1 and x 2 are again reset into Ω, allowing to iterate the above reasoning for the following two-point boundary value problems. Note that the condition (25) implies controlled-invariance of the compact set Ω with respect to the discrete-time equivalent representation of system (21) (23). As a byproduct, the above condition guarantees bounded trajectories, assuming that the exosystem s trajectories are bounded as well, which is guaranteed by Assumption 1. Remark 9. Interestingly, the above statement entails that the MPBVP (21) (23) may be solved essentially by determining solutions to a single TPBVP, provided that condition (25) holds. In fact, the time histories of control laws enforcing zero-error trajectories, for all (t, k) T, can be constructed by suitable composition of all the possible arcs solving the TPBVP defined by the ODE s (21) together with the auxiliary output constraint ϵ 3 ([x 1, x 2, π 3(w) ], α d, w) π 3 (j(w)) = 0, (26) for all (x 1, x 2, w) Ω. Proposition 4 provides a way to assess, by means a geometric interpretation of the considered conditions, the recursive feasibility of the approach proposed in Section 5. As is often the case, checking the conditions become computationally much simpler and explicit for the linear case, as shown in the following section, but nevertheless it can often be performed even in the nonlinear case by exploiting tools for checking set invariance and reachability set computations. In fact, (25) entails the intersection of three subsets of R n, hence no dynamic considerations, in terms of differential equations or time evolutions, have to be carried out a priori to perform such intersection. In particular, the first set can be characterized by exploiting the controllability distribution introduced in Remark 3. The remaining two subsets, namely I Ω and E, are related to the mapping that describes the discrete-time evolution. The latter, in particular, is simply defined as the affine variety of such mapping, namely the set of all x R n such that all the individual functions composing the mapping are simultaneously equal to zero. Finally, the set I Ω is only needed to guarantee that there exists a compact subset of R n that is mapped again into itself by the image of the jump map, thus ensuring feasibility of the two-point boundary value problems for any future time: this could be easily achieved by considering for instance level sets of specific functions, as explained in more detail in the linear setting Linear case To better visualize the previous geometric conditions, in this section we consider the geometric interpretation discussed in Proposition 4 in the presence of linear hybrid systems. Interestingly, in this scenario, the conditions of Proposition 4 may be checked by means of LMIs, which additionally provide a closed-form solution to the corresponding TPBVPs in terms of a suitable selection for the control inputs u c,1 and u d. It is shown in particular that the possibility of solving the sequence of TPBV problems is essentially related to the stabilizability property of a suitable discrete-time linear system. Towards this end, recall that the flow dynamics of plant and exosystem, i.e. (2) and (4a), become Eqs. (10a), whereas the corresponding jump dynamics reduce to Eqs. (10b). Let S R ns ns and J R ns ns be arbitrary matrices such that all the eigenvalues of J Je Sτ M have modulus equal to 1, in order to satisfy Assumption 1. Finally, let the matrices A o R (n 1+n 2 +n s) (n 1 +n 2 +n s) and B o R (n 1+n 2 +n s) m p be defined as A 11 A 12 P1 B 11 A o = 0 A 22 P2, B o = 0, (27) 0 0 S 0 with P1 = P 1 + A 13 Π 3 + B 12 Γ 2, P2 = A 23 Π 3 + B 22 Γ 2 + P 2, where Π 3, Γ 2 denote the (unique) solution of the reduced flow Francis equation Π 3 S = A 33 Π 3 + B 32 Γ 2 + P 3, 0 = C 3 Π 3 + Q. x 1 x 10 I R(χ 0, τ M ) (x 1, x 2, w) R n 1+n 2 +n s : x 2 = e Aoτ M x v, v R n 1. (29) w w 0 0 Suppose that, in this setting, the compact set Ω is defined in terms of 1-sublevel sets 5 of a suitably defined quadratic function, namely Ω l {χ : χ Sχ 1}, for some symmetric positive definite matrix S = S > 0. Under these assumptions, (28a) (28b) 5 Given a function f : Rn R, the c-sublevel set is defined as {x R n : f (x) c}.

11 S. Galeani, M. Sassano / Nonlinear Analysis: Hybrid Systems 29 (2018) the three linear subspaces of interest, namely R(χ, τ M ), E and I Ω, may be more explicitly characterized. Towards this end, the reachability set R(χ 0, τ M ), with χ 0 = (x, 10 x, 20 w 0 ) is described by (29), while x 1 x 1 E x 2 : M 1 x 2 + F 3 u d = 0, u d R m d (30) w w with M 1 [E 31 E 32 E 33 Π 3 Π 3 J + R 3 ]. Finally, the inverse image set I Ωl is defined as x 1 F 1 I Ωl (x 1, x 2, w) R n 1+n 2 +n s : y = M 2 + ud, y Ω l, u d R m d with E 11 E 12 E 13 Π 3 + R 1 M 2 E 21 E 22 E 23 Π 3 + R J x 2 w F 2 0 The following result provides a characterization of the solution to the hybrid output regulation problem similar to that of Proposition 3 in the nonlinear setting. Proposition 5. Consider the linear hybrid plant and exosystem (10a) (10b), defined on the time domain T in (1). Let Π 3 and u c,2 = Γ 2 w be such that (28) is satisfied. Then, all the trajectories satisfying x 3 (0, 0) = Π 3 w(0, 0) and the following sequence of (linear) ode s (31) (32) ẇ = Sw, ẋ 1 = A 11 x 1 + A 12 x 2 + B 11 u 1 + P1 w, ẋ 2 = A 22 x 2 + P2 w, (33) with split boundary conditions (TPBVP) w [k+1] = Jw k x i,[k+1] = E i1 x 1, k + E i2 x 2, k + (E i3 Π 3 + R i )w k + F i u d, i = 1, 2, u d R m d, together with the auxiliary output constraint x 1, k M 1 x 2, k + F 3 u d = 0, w k k = 1, 2,..., are such that e(t, k) = 0 for all (t, k) T. (34) (35) Proof. The claim is proved by adapting to the linear setting arguments identical to those employed in the proof of Proposition 3. The following statement, mimicking the results of Proposition 4, provides conditions that allow to guarantee existence and boundedness of trajectories solving the sequence of TPBVPs in Proposition 5. To streamline the presentation of the result consider the following notation. Assume that the matrix [E 31 F 3 ] R n 3 (n 1 +m d ) is such that rank ([E 31 F 3 ]) = n 3 and let N R (n 1+m d ) (n 1 +m d ) n 3 be a basis matrix of Ker([E 31 F 3 ]), namely a matrix such that [E 31 F 3 ]N = 0 and rank (N) = (n 1 + m d ) n 3. Finally, let K c = [E 31 F 3 ] ( M 1 e Aoτ M ), where [E 31 F 3 ] denotes the (right) pseudoinverse of [E 31 F 3 ], E 11 E 12 E 13 Π 3 + R 1 K c and Ẽ E 21 E 22 E 23 Π 3 + R J E 11 F 1 F E 21 F 2 N. 0 0 e A oτ M + 0 (36) 0 Proposition 6. Consider the linear hybrid plant and exosystem (10a) (10b), defined on the time domain T in (1). Let Π 3 and u c,2 = Γ 2 w be such that (28) is satisfied. Then, the following hold: (37)