Robust Nonlinear Regulation: Continuous-time Internal Models and Hybrid Identifiers

Transcription

1 53rd IEEE Conference on Decision and Control December 15-17, Los Angeles, California, USA Robust Nonlinear Regulation: Continuous-time Internal Models and Hybrid Identifiers Francesco Forte, Lorenzo Marconi 1 and Andrew R. Teel 2 Abstract The paper deals with the problem of robust output regulation for minimum-phase nonlinear systems in a semiglobal setting. We present a different perspective to the problem of adaptive regulation in which prediction error identification methods, which are routinely used in other control contexts, can be adopted to design robust nonlinear regulators. The proposed control structure combines continuous-time dynamics and hybrid identifiers, the latter specifically designed to estimate the actual steady state control law. Besides presenting the main idea and a general framework, the paper addresses the specific case in which a linear regression law is used as model structure for the steady state control law and a least square optimization criterion is adopted as estimation method. The proposed framework encompasses existing frameworks proposed so far in the nonlinear continuous-time literature. I. INTRODUCTION Although the nonlinear regulation theory has reached a maturity stage, there are some crucial aspects still open as far as the design of robust regulators are concerned. In particular, a systematic design of robust regulators having the so-called internal model property in presence of steady state laws affected by parametric or structural uncertainties is definitely an open research field. So far, many researchers dealt with this problem using adaptive techniques as in [4] and [11], while others faced the problem using techniques typically adopted in the context of adaptive observers design, achieving interesting results both in the linear and in the nonlinear case and in global and semi-global context, see [1], [10] and [20]. More recently, some authors have proposed regressionlike methods by developing adaptive and learning algorithms for nonlinear internal models to deal with uncertainties in the steady state control law, see among others [21]. Relying on the same philosophy, in [3], the authors have shown how to design regression-based internal model regulators using static adaptation laws, instead of standard dynamical estimation schemes, to offset parametric uncertainties in the steady state control law. Something inherent to the field of hybrid systems has been developed in [5], in which the interconnection of a feed-forward model of the exosystem with a hybrid adaptive law is presented. In this paper we present a different perspective to the problem of adaptive 1 Francesco Forte and Lorenzo Marconi are with Department of Electrical, Electronic and Information Engineering DEI, University of Bologna, Bologna, Italy f.forte@unibo.it, lorenzo.marconi@unibo.it. Research framed in the integrated project SHERPA G.A supported by the European Community under the 7th Framework Programme. 2 Andrew R. Teel is with the Department of Electrical and Computer Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA teel@ece.ucsb.edu. Research supported in part by AFOSR grant FA and NSF grant ECCS regulation in which prediction error identification methods, which are routinely used in other control contexts [15], can be adopted to design robust nonlinear regulators. The point of departure is the design procedure presented in [8] in which the steady state control law and its time-derivatives up to a certain order are assumed to satisfy a regression formula with known regression vector by which internal model regulators can be designed by means of high-gain tools. The regression formula in our context is thought of as a prediction model relating the next time derivative of the steady state control law to the previous derivatives through a unknown regression vector. The proposed control structure combines continuous-time dynamics and hybrid identifiers, the latter specifically designed to estimate the actual regression vector. The fact of considering hybrid systems as identifiers is essentially motivated by the goal of setting up a general framework where many design strategies can be cast. In fact, on one hand, the proposed approach aims to capture continuous-time adaptive regulator design procedures, proposed so far in literature, as particular cases. On the other hand, we aim to open the doors to identification tools that typically rely on sampling the available data set and to update the prediction model in a discrete-time fashion. In this context the kind of result that we are able to claim is practical output regulation with an asymptotic error that depends on the prediction error. Regarding hybrid systems, this work uses the framework and results of [19] from which also the notation is taken. II. NONLINEAR OUTPUT REGULATION We consider the following class of nonlinear systems in normal form and with relative degree equals to one 1 ẇ = sw 1 ż = fz, w, e 2 ė = qz, w, e + bz, w, eu. 3 In the previous system one can recognise two main subsystems: the first, described by 1, is the so-called exosystem with state w W R s generating possible references signals to be tracked and/or possible disturbances that must be rejected. The set W is a compact set that is assumed to be invariant for the exosystem dynamics 1. The second subsystem is the controlled plant given in 2-3 in which z, e R n R is the state, u R is the control input, and e is the regulation error. All functions in the overall 1 All the forthcoming results can be easily extended to systems with higher relative degree by means of standard tools /14/$ IEEE 4703

2 system, i.e. s, f,,, q,, and b,, are smooth in their arguments, with the function b,,, the so-called highfrequency gain of the system, that is assumed to be bounded from below by a positive number b, i.e bz, w, e b, for all z, w, e R n W R. In this framework, the problem of semiglobal asymptotic output regulation can be formulated as follows: given the sets W R s, Z R n and E R of initial conditions for the system 1 3, design an error-feedback controller with state ξ R d, for some positive d, and initial condition in a compact set Ξ R d such that all trajectories of the closed-loop system starting from W Z E Ξ are bounded and lim t et = 0 uniformly in the initial conditions. We shall approach the previous problem under assumptions that are customary in the literature of output regulation. In particular we assume the existence of a smooth function π : R s R n that solves the regulator equation L sw πw = fπw, w, 0, 4 for all w W. This implies the existence of a compact set A := {w, z W R n : z = πw that is invariant for the dynamics ẇ = sw, ż = fz, w, 0. 5 The previous system is easily recognized to be the zero dynamics of system 1-3 relative to the input u and to the output e. As in most of the literature about output regulation, we make a minimum-phase assumption on system 5 that is formalized as follows. Assumption Minimum Phase The set A is globally asymptotically and locally exponentially stable for 5 with a domain of attraction of the form W R n. In the design of the regulator a crucial role is played by the function c : W R n R defined as cw, z = qw, z, 0/bw, z, 0. 6 This function is readily seen to be the friend associated to the zero dynamics of system 1 3 see [2]. In the context of output regulation, the output signals generated by system 5 with output 6 with initial conditions ranging in A are the steady state control inputs that must be generated by the controller in order to keep the regulation error identically to zero. It is thus apparent that system 5 restricted to the set A with output 6 plays a crucial role in the design of the regulator. As a matter of fact it is a well-known fact [16] that the output regulation problem is solved by a continuoustime regulator if one is able to design smooth functions M : R d R d, G : R d R d 1, and γ : R d R, such that, for some smooth function τ : R s R d and with T the compact set defined as T := {w, ξ W R d : ξ = τw, the set A T is locally asymptotically stable for the system ẇ = sw, ż = fw, z, 0, ξ = Mξ + Gξcw, z 7 with a domain of attraction W R n C with C an open set of R d satisfying C Ξ, and, in addition, γτw = cw, πw w W. 8 In this context, in fact, the continuous-time controller that solves the problem at hand is a system of the form ξ = Mξ + Gγξ + v u = γξ + v, v = sgne κ e where κ is a properly defined class-k function. As a matter of fact, the closed loop system given by 1 3 and 9 is a system that has relative degree one relative to the input v and output e and has a zero dynamics precisely given by 7. Under these circumstances, standard highgain arguments can be used to show that an high-gain function κ succeeds in making the set A T {0 locally asymptotically stable with a domain of attraction containing the compact set of initial conditions. Now, by letting u : W R be the restriction of 6 to the set A defined as 9 u w = qw, πw, 0/bw, πw, 0, 10 there is a relevant context, originally presented in [8], where a constructive design procedure can be given in which 10 fulfills the following regression formula L d su w = ϕu w, L sw u w,..., L d 1 sw u w+νw, 11 w W, for some positive d, for some continuous function ν : W R and some known locally Lipschitz function ϕ : R d R. In this case, in fact, the theory of high-gain observers [12] can be used to show that the above properties are fulfilled with Gξ = G := colλ 1 g, λ 2 g 2,..., λ d g d, where g is a design parameter and the λ i s that are coefficients of an Hurwitz polynomial, Mξ := colξ 2,...,, ξ d,, ϕ s ξ Gξ 1, where ϕ s is a uniformly bounded and locally Lipschitz function, and γξ = ξ 1. By choosing M, G, and γ in this way, by letting τw = colu w,..., L d 1 sw u w, and by choosing ϕ s so that it agrees with ϕ for all ξ = τw, w W, it turns out that there exists g > 1, only dependent on the Lipschitz constant of ϕ, such that for all g g the same regulator presented above guarantees that the closedloop trajectories originating from the given compact sets are bounded and the regulation error fulfills lim sup et c t g max νw 12 w W where c > 0. Practical, instead of asymptotic, regulation is thus achieved with a residual error that depends on the entity of the residual bias νw. III. OUTPUT REGULATION AND IDENTIFICATION TOOLS A. Main Idea The previous high-gain framework and relation 11 are at the basis of our robust regulator design. The main idea developed in the paper is to regard the function ϕ in 11 as unknown and to estimate it on line by adopting prediction error identification methods, [15]. In particular, relation 11 is regarded as a prediction model of the d-th time derivative of the signal u wt at time t using the regression vector 4704

3 u wt, L s u wt,..., L d 1 s u wt, with the identification objective that is to estimate the function ϕ best fitting the data associated to the actual u wt. The goal is to design a practical regulator in which the asymptotic bound on the closed-loop regulation error is a function of the asymptotic value of the prediction error between the actual value of the d-th time derivative of the signal u wt and its estimated value obtained by processing the regression vector. Since u wt,..., L d su wt are not measurable in our output regulation context, the idea that is pursued in the paper is to estimate their value by employing the dirty derivative using the terminology in [6] features of the internal model of the form indicated in the previous section. Namely, the ability of the ξ-system in 7, to roughly estimate asymptotically the function u t and its time derivative up to the order d 1, with an estimation error that can be arbitrarily decreased by increasing g, regardless the specific form of ϕ s provided that a bound on the Lipschitz constant is fixed. Since the identification problem potentially requires the knowledge also of L d su w, the regulator that is presented later has dimension d + 1, namely one more with respect to the one presented above. The extra state variable ξ d+1, that is redundant as far as the internal model property is concerned, has precisely the role of providing a dirty estimate of L d su w that is used in the identification algorithm. In our approach the dynamical system providing the estimation of the d-th derivative according to the regression vector is an hybrid system combining continuous and discrete-time dynamics, [18]. The fact of considering hybrid systems as identifiers is essentially motivated by the goal of setting up a general framework where many design strategies can be cast. B. Regulator Structure In our design the identifier is an hybrid dynamical system whose flow dynamics and jump map are described by η c F c η c η η e = F e η e, u η c, η e, u η C c R m R d+1 η c + J c η c η e + η = J e η e, u η c, η e, u η D c R m R d+1 13 with output ˆϕ = Γ η η e, u η1 14 where η c, η e R R m, m > 0, F c : C c R and J c : D c R are outer semicontinuous and locally bounded setvalued mappings, C c and D c are closed intervals of R, u η = colu η1, u η2, with u η1 R d and u η2 R, is a vector of inputs, and F e : R m R d+1 R m, J e : R m R d+1 R m and Γ η : R m R d R are smooth functions, with J e and Γ η that are globally Lipschitz. The scalar variable η c plays the role of clock governing the length of the flow intervals and the jump times according to the definition of the flow and jump sets C c and D c. Both discrete-time and continuoustime dynamics can be captured by the previous description. With τw defined as in Section II, the hybrid identifier 13 should be ideally driven by the inputs u η1 = τw, representing the regression vector in the interpretation given in Section III, and u η2 = L d su w, representing the next derivative, yielding an estimate ˆϕt = Γ η ηt, τwt able to best predict L d su wt on the basis of the values of the regression vector. Since τw and L d su w are not accessible, the hybrid identifier 13 is fed with the state ξ e = colξ, ξ d+1, ξ R d, ξ d+1 R, of an extended internal model unit 2, namely u η1 = ξ, u η2 = ξ d+1, 15 governed by the hybrid system ξ Sξ + Bξ ξ e = = d+1 + Gv ξ d+1 Γ ηsη, ξ e + λ d+1 g d+1 v ξ ξ e + + ξ = ξ + = d+1 Γ η J e η e, ξ e, ξ u = Cξ + v 16 where ξ e, η, v R d+1 C c R m R during flows, ξ e, η, v R d+1 D c R m R during jumps, and S, B, C R d d R d 1 R 1 d is a triplet in prime form 3, G is defined in Section II, g is a design parameter, the λ i i = 1,..., d + 1, are coefficients of an Hurwitz polynomial, v is a residual input and Γ ηs : R m R d R is a locally Lipschitz bounded function obtained by appropriately saturating the function Γ ηη e, ξ e = Γ ηη e, ξ F η η e, ξ e + η d i=1 Γ η η e, ξ ξ i ξ i Details on how the saturation level of Γ ηs has to be chosen are presented later. The regulator is thus 13, 15, 16 where v is the residual input that will be chosen as v = κe, with κ a design parameter. The flow time intervals and the times at which jumps occur are uniquely determined by the clock dynamics. The fact of modeling the latter as differential and algebraic inclusions allows one for considering a number of clock timing not necessarily uniform in time. Fast and slow clocks might be dynamically triggered according to real time information. The only constraint that will be imposed by the forthcoming analysis to the clock dynamics is to fulfill average and reverse average dwell-time conditions. In particular, to make sure that continuous-time dynamics present in the loop exhibit their asymptotic properties, the forthcoming stability analysis will rely upon a condition asking that flow intervals are persistently present and last in the average a guaranteed amount of time. From a formal viewpoint the notion of average dwell-time [14] is used to rigorously fix the required property. We recall [14] that the clock subsystem satisfies an average dwell-time if there exist N 0 > 1 and δ > 0 such that of all t, j and s, i belonging to the hybrid time domain of the clock with t + j > s + i 2 The adjective extended has to be interpreted with respect to the internal model considered in Section II of dimension d. 3 That is S is a shift matrix all 1 s on the upper diagonal and all 0 s elsewhere, B T = and C =

4 the following holds j i δt s + N In the previous relation 1/δ denotes the average dwelltime while N 0 denotes the maximum number of consecutive jumps that might occur not separated by flow intervals. The average-dwell time condition expressed above might be eventually completed with a reverse condition asking that clocks are also persistently enforced. This condition might be crucial in order to design the hybrid identifier with the desired asymptotic properties detailed in the next Section III-C in certain discrete-time identification settings as, for instance, in the case presented in Section V. From a formal viewpoint the notion of reverse average dwell-time [13] is used to rigorously fix the required property. We recall [13] that the clock subsystem satisfies a reverse average dwelltime if there exist N 0 > 1 and δ > 0 such that of all t, j and s, i belonging to the hybrid time domain of the clock with t + j > s + i the following holds C. Identifier Design Requirements t s δj i + N 0 δ. 19 A crucial role in achieving small possibly zero asymptotic regulation error will be clearly played by the design of the hybrid identifier 13, namely by the design of the functions F e, J e and Γ η, and of the sets F c, J c, C c and D c. According to the identification literature [15], the design of the identifier entails the choice of a certain model structure for the function ϕ and to choose an estimation method to select the best member in the family defined by the model structure. In our context the specific data set with respect to which optimisation is performed is given by the steady state input u wt associated to the specific exosystem trajectory wt. The requirements assumed for the design of this system are precisely presented below. The first requirement is existence of an exponentially stable steady-state for 13 driven by the ideal input u η = colτw, L d su w denoted by η e = ση c, w in the following. As 13 is not driven by the ideal input τw, L d su w but, rather, by the available dirty derivatives state colξ, ξ d+1, a robustness property of such a steady state is required. It is given in terms of input-to-state stability with respect to a disturbance, referred to as d e in the following, additive to the ideal input τw, L d su w. The previous properties are the ones playing a role in the asymptotic stability analysis. In addition, it is assumed that the output Γ η of 13 evaluated along the steady state trajectory of the identifier is the best guess of the next time derivatives L d su w, namely the function able to minimize the prediction error which will be denoted by ε. In the following we refer to Jε the functional that is behind the selection of the best guess. The expression of Jε is deliberately left unspecified at this level of the analysis since it does not affect the stability analysis. A possible choice is then presented in Section V when a specific hybrid identifier is designed. For the sake of compactness, we rewrite system 13 as η F η η, u η η, u η C η R d+1 η + J η η, u η η, u η D η R d+1 where η = colη c, η e, and where the set-valued mappings F η, J η, and the flow and jump sets C η, D η are suitably defined. Furthermore, we let τ e : R s R d+1 be the smooth function defined as τ e w = colτw, L d su w. Identifier Design Requirement. The hybrid system 13 with output 14 is said to satisfy an Identifier Design Requirement if the following properties hold: a there exists a smooth function σ : R R s R m such that the hybrid system ẇ = sw η F η η, τ e w + d e w, η, τ e w + d e W C η R d+1 w + = w η + J η η, τ e w + d e w, η, τ e w + d e W D η R d+1 20 is pre-iss with respect to the input d e relative to the set B = {w, η W C η D η : η e = ση c, w 21 without restrictions on the initial state and non-zero restriction on the input, and with linear asymptotic gain. That is see [9], there exists a locally Lipschitz function V η : W R m+1 R 0, such that the following holds: there exist locally linear K functions α η, ᾱ η such that for all w, η W R m+1 α η w, η B V η w, η ᾱ η w, η B ; there exist positive r, χ η and c η, such that for all w, η W C η and for all d e fulfilling d e r we have V η w, η χ η d e V o η w, η, v c η V η w, η sw v ; F η η, τ e w + d e there exists a positive constant λ η < 1 such that for all w, η W D η and for all d e fulfilling d e r we have, with the same χ η as in the previous item, V η v max{λ η V η w, η, χ η d e sw v. J η η, τ e w + d e b Let ε : R s R R be the smooth prediction error function defined as εη c, w = L d su w Γ η ση c, w, τw. 22 Then, for all η c t, j C c D c solution of the clock subsystem in 13 and for all wt, j W solution of the 4706

5 exosystem, the hybrid identifier is optimal with respect to some estimation functional Jεη c t, j, wt, j. With the function σ introduced in the item a above, the tuning of the regulator 13, 15, 16 can be then completed by specifying Γ ηsη e, ξ e as any locally Lipschitz bounded function that agrees with Γ ηη e, ξ e for all η, ξ e C η D η R d+1 such that w, η C c, w, ξ e grτ e c for some positive c, where gr τ e = {w, ξ e W R d+1 : ξ e = τ e w. IV. MAIN RESULT In this section we study the asymptotic properties of the closed-loop system. We show how, for an appropriate tuning of the regulator, the resulting closed-loop hybrid system is pre-iss relative to a compact set, whose projection on the error space is the origin, with respect to a disturbance input given by the prediction error εη c, w. The overall closedloop system is a hybrid system flowing according to ẇ = sw ż = fz, w, e ξ Sξ + Bξ ξ e = = d+1 + Gv ξ d+1 Γ ηsη e, ξ e + λ d+1 g d+1 v η F η η, ξ e ė = qz, w, e + bz, w, ecξ κe. when w, z, ξ e, η, e W R n R d+1 C η R, and jumping according to w + = w, z + = z ξ + = ξ, ξ + d+1 = Γ ηj e η e, ξ e, ξ η + J η η, ξ e e + = e when w, z, ξ e, η, e W R n R d+1 D η R. By letting x = colw, z, ξ e, η e, e such a system is rewritten in compact form as η c F c η c ẋ = F x x η c + J c η c x + = J x x η c, x C c C x, η c, x D c D x 23 where the functions F x, J x and the sets C x, D x are appropriately defined. The main result is presented in the next theorem whose proof is omitted for reasons of space claiming that the regulation error is asymptotically bounded by a linear function of the prediction error provided that the clock subsystem satisfies an average dwell-time condition. Theorem 1: Consider the closed-loop system 23 with the zero dynamics of the regulated plant fulfilling the minimumphase assumption and with system 13 fulfilling the identifier design requirement. Furthermore, for all t, j and s, i belonging to the hybrid time domain of the clock subsystem in 23 with t + j > s + i, assume that the average dwell time condition 18 is fulfilled for some δ 0 and N 0 1. Then, for any compact set X W R n R d+1 R m R, there exist δ > 0, g > 0 and κ g > 0 such that for all δ 0, δ, g g, κ κ g, and for all t, j belonging to the hybrid time domain of system 23 with flow and jump sets restricted to C c C x X and D c D x X the following holds lim t+j sup et, j ρ lim sup εη ct, j, wt, j t+j with ρ a positive constant. V. LINEARLY PARAMETERIZED MODELS AND LEAST SQUARES METHOD 24 In this section we develop the case in which the model structure relating L d su wt and the regression vector is assumed to be a linearly parametrized function of the form L d su wt = Ψ T τwθ 25 in which Ψ : R d R p, p > 0, is a locally Lipschitz known function, and θ Θ R p is a vector of uncertain parameters with Θ a known compact set. We are interested in designing a hybrid identifier of the form 13 fulfilling the basic requirements specified in Section III-C in which the estimation method used to select the best θ Θ is a discretetime least square criterion. Specifically, let us consider a hybrid clock subsystem such that for all initial conditions η c0 = η c 0, 0 C c D c the associated hybrid time domain E ηc0 R 0 N fulfills an average dwell-time condition of the form 18 required by the analysis in Section IV and a reverse average dwell-time condition of the form 19 for some N 0 1 and δ > 0. The reverse condition is imposed in order to have persistent jumps required by the discretetime nature of the estimator we are going to develop. With N > 1, let I ηc0 = {t j1, j 1,..., t jn, j N be an arbitrary set of N distinct hybrid times such that t ji, j i E ηc0, and jumps occur at t ji, j i, i = 1,... N. Our goal is to develop an hybrid identifier of the form 13 such that the basic hybrid requirement in Section III-C are fulfilled with estimation functional given by Jεη c t, j, wt, j = 1 εη c t j, j, wt j, j 2 2N I ηc0 26 by using 25 as prediction model structure. As usual in the context of least square identification methods we make a persistence of excitation assumption formulated as follows. Assumption Persistence of excitation There exists a ῡ > 0 such that for all η c0 C c D c, for all sequence of N distinct jump hybrid times I ηc0, and for all exosystem trajectories wt, j W with t, j E ηc0, the following holds det t j,j I ηc0 Ψτwt j, j Ψτwt j, j T ῡ. Our identifier 13 has state η e = colη 1, η 2, η 3, η 1 R N, η 2 = colη 21,..., η 2N R pn, η 2i R p, i = 1,..., N, 4707

6 η 3 R p with dynamics { η1 = 0 η = Sη 1 + Bξ d+1 { η2 = 0 η = S I p η 2 + B I p ψξ { η3 = 0 η 3 + = Lη 1 +, η and output Γ η η, ξ = Ψξ T η 3 where S = [0, I N 1 ; 0, 0], B = [0, 1] T and Lη 1, η 2 = R sat η 2 1 N i=1 η 2i η 1i with L, that is any globally Lipschitz function and R sat η 2 := N i=1 η 2iη2i T, fulfilling, detr satη 2 2ῡ. Furthermore, since the hybrid clock time domain fulfills a reverse average dwell time condition, according to [13], the clock dynamics can be thought of as flowing according to η c = 1 and jumping according to η c + = max{0, η c δ with flow and jump sets coincident and equal to C c = D c = [0, N 0 δ]. In the next proposition we show that system satisfies the identifier design requirement. Instrumental to the statement of the proposition are the forthcoming definitions. For all η c C c D c let η c0 C c D c be such that η c = η c t, N 1 for some t R 0 such that t, N 1 E ηc0 and let ϕ w t, w be the value of the trajectory of ẇ = sw at time t with initial condition w at t = 0. Furthermore, with t i, i E ηc0, i = 0,..., N 1, the hybrid jump times, let T 1 η c, w = colt 11 η c, w,..., T 1N η c, w with T 1i : C c D c W R, i = 1,..., N defined as using the fact that η c = 1 T 1i η c, w = τ d+1 ϕ w t t i 1, w Furthermore, let T 2 η c, w = [T 21 η c, w,..., T 2N η c, w] T where T 2i : C c D c W R p, i = 1,..., N are defined as T 2i η c, w = Ψτϕ w t t i 1, w and T 3 η c, w = LT 1 η c, w, T 2 η c, w. Finally, we let and W 1 η c, w, η 1 = N i=1 c i η 1i T 1i η c, w W 2 η c, w, η 2 = N i=1 c i η 2i T 2i η c, w W 3 η c, w, η 3 = c η 3 T 3 η c, w 1 V 1 η c, w, η 1 = exp µη c W 1 η c, w, η 1 V 2 η c, w, η 2 = exp µη c W 2 η c, w, η 2 V 3 η c, w, η 3 = exp µη c W 3 η c, w, η 3 where the constants c i are such that c i = 8c i 1, i = 2,..., N, c 1 > 0, and c and µ are positive constants yet to be fixed. With the previous definition at hand, the following proposition can be given. Proposition 1: Assume that the clock dynamics fulfill a reverse average dwell-time of the form 19 for some N 0 1 and δ > 0. In the definition of V 1, V 2 and V 3, let µ be taken so that expµδ < 2 and let c be a sufficiently small positive number. Then system satisfies the identifier design requirement of Section III-C with 26 as associated functional cost, with ISS-Lyapunov function V µ η, w = V 1 η c, w, η 1 + V 2 η c, w, η 2 + V 3 η c, w, η 3 and with the set B defined as in 21 with ση c, w = colt 1 η c, w, T 2 η c, w, T 3 η c, w. VI. CONCLUSIONS In this paper we presented a different perspective to the problem of robust regulation for nonlinear continuous-time systems. Our goal was to adopt prediction error identification methods, which are routinely used in other control contexts, to design robust nonlinear regulators. Our controller combines continuous-time dynamics with hybrid identifiers. Besides presenting a general framework in which different identification tools can be in principle framed, in the paper we developed the specific case in which the steady state control input fulfills a linear regression law and a least square functional is used as underlying identification method. REFERENCES [1] A. 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