Reset control systems with reset band: Well-posedness, limit cycles and stability analysis

Systems & Control Letters 63 (2014) 1–11

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Reset control systems with reset band: Well-posedness, limit cycles and stability analysis✩ Antonio Barreiro a,∗ , Alfonso Baños b , Sebastián Dormido c , José A. González-Prieto a a

Dept. Ing. Sist. y Automática, University of Vigo, 36200 Vigo, Spain

b

Dep. Informática y Sistemas, University of Murcia, 30071 Murcia, Spain

c

Dept. Informática y Automática, UNED, 28040 Madrid, Spain

article

info

Article history: Received 17 January 2012 Received in revised form 30 July 2013 Accepted 6 October 2013

Keywords: Reset control Poincaré maps Limit cycles Zeno solutions

abstract Reset controllers provide a simple way to improve performance when controlling strongly traded-off plants. A reset controller operates most of the time as a linear system, but when some condition holds, it performs a zero resetting action on its state. Recently, some generalizations have been proposed, for example, the anticipation of the reset condition with the so-called reset band. There is a lack of analysis tools for reset systems with reset band. In this paper we address this problem by means of Poincaré maps (PM). It is shown how PM can predict the existence and stability of limit cycles, and give also information on pathologies such as Zenoness, and provide parameter ranges where the system is guarded against those behaviors and thus global asymptotical stability (GAS) is guaranteed. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The techniques of reset control [1] date back to the Clegg integrator (CI) [2] in 1958 and to the ideas of Horowitz and coworkers [3,4] in the seventies. Since the beginning, the motivation was to overcome fundamental limitations in linear systems by means of a simple procedure: resetting to zero the state(s) of the controller when the tracking error is zero. The idea was left aside some time, until rigorous results were developed in [5,6] for general reset systems, with arbitrary dimension and partial reset. Later, other proposals for reset control were presented and analyzed [7–10], in which the reset condition was based on the sign of the input and output signals. The recent monograph [1] reviews the different reset control techniques and present analysis and design methods developed by the authors. The main advantage of reset control is its ability to overcome linear fundamental limitations, that may be specially hard in plants with right-half-plane (RHP) poles or zeros, or with time delays. These limitations can be quantified in the frequency domain [11] in the form of bounds on the crossover frequency ωgc for given

✩ This work has been supported in part by Ministerio de Ciencia e Innovación (Gobierno de España) under project DPI2010-DPI2010-20466-C02-01 (A. Barreiro, J.A. G-Prieto), DPI2010-20466-C02-02 (A. Baños), and DPI2012-31303 (S. Dormido). ∗ Corresponding author. Tel.: +34 986 812232. E-mail addresses: [email protected] (A. Barreiro), [email protected] (A. Baños), [email protected] (S. Dormido), [email protected] (J.A. González-Prieto).

0167-6911/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sysconle.2013.10.002

phase margins ϕm . In [12] it has been proven that reset controllers overcome the limitations induced by delays. This is possible because they are not restricted to the Bode gain–phase relation: the CI has slope −20 dB/dec but with phase −38° (not −90°). The case for RHP poles or zeros (like the example in this paper) can also be proven following [11,12]. To improve this advantage, we can produce some extra phase lead, for example, a modified CI with slope −20 dB/dec but with phase even less negative than −38°. This anticipation can be obtained by means of a lead block acting on the trigger signal [12] or, as chosen in this paper, in the form of a reset band defined in the error plane (e, e˙ ), that has a simpler implementation. This reset band can be applied to a general reset controller or (like the example in this paper) to the CI part of the so-called PI + CI controller, well studied and successfully tested in various process applications [13,14]. Although the performance improvements derived from these anticipative reset controllers are well recognized, there is still a need of more precise techniques and tools for addressing wellposedness, detecting limit cycles, or ensuring global asymptotical stability (GAS) of the target setpoints. In this paper we address this problem by means of Poincaré maps (PM) [15], for the following reasons. It is well known that the stability of a reset system cannot be related to the stability of its base linear system (the system without reset). Thus, Lyapunov techniques could be considered, but the available Lyapunov-based criterion (Hβ condition [6]) is only applicable to reset without band. Furthermore, Lyapunov techniques

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A. Barreiro et al. / Systems & Control Letters 63 (2014) 1–11

give sufficient stability conditions, that can be conservative, while Poincaré maps capture the exact dynamic behaviors, without conservativeness. On the other hand, the frequency domain framework is useful for dealing with design tradeoffs and adequate to show the overcoming of fundamental limitations with reset control, see related work [16,17]. Within the frequency domain, the Describing Function (DF) techniques are useful for prediction of amplitude and frequency of limit cycles, but in general the DF results are only approximate, not exact, whereas the PM, after numerical solution of fixed points and times-to-impact, produce the exact limit cycle (not an approximation based on harmonics). These are the advantages of the PM technique: it captures the exact dynamic behaviors, without conservativeness or approximations. The drawback of PM is the computational burden: the times-to-impact require numerical simulations. Furthermore, PM converts n-dimensional continuous-time dynamics to m-dimensional discrete dynamics, where m is the dimension of the after-reset manifold. Fortunately, in many practical cases, m is small. In the case m = 1 the PM is a scalar map from which a global picture of dynamics can be easily derived. For m ≥ 2, the PM can also be applied, with careful sweeping of parameter ranges and controlling accuracy of the results. Having in mind the previous comments, the objective of this paper is to analyze reset control systems with reset band, using PM. In this way the results here are complementary to the results in [18], where the analysis tool was the DF. The PM has been applied to piecewise linear systems in [19] (with switching in place of resetting) and to impulsive systems in [20] (under the condition that times-to-impact are continuous). Our approach is valid for discontinuous times-to-impact and addresses not only limit cycles but well-posedness and GAS as well. Thus, our results here are complementary to [21], which studies well posedness for reset systems but without reset band. The paper is organized as follows. Section 2 introduces the reset control system and the considered reset band that we are going to address. Next, absence of pathological dynamics is studied in Section 3. The central part is Section 4, that shows how the Poincaré map can be adapted to this particular systems and predict existence and stability of limit cycles and, additionally, zenoness or the GAS property. Finally, a numerical example and conclusions are presented in Sections 5 and 6. 2. Problem statement



x˙ p (t ) = Ap xp (t ) + Bp up (t ) yp (t ) = Cp xp (t )

(2.1)

where xp ∈ Rnp , up and yp are scalars, and the matrices Ap , Bp , Cp are of appropriate dimensions, and

 x˙ c (t ) = Ac xc (t ) + Bc uc (t ), if c (t ) false C : xc (t + ) = Aρ xc (t ), if c (t ) true y (t ) = C x (t ) + D u (t ) c c c c c

(2.2)

where xc ∈ R , uc and yc are scalars, Ac , Aρ , Bc , Cc , Dc are of matrices of appropriate dimensions and condition c (t ) is defined below. Consider also the autonomous system, without exogenous signals (r = d = n = 0), so that the closed loop connections are nc

up (t ) = yc (t ),

uc (t ) = e(t ) = −yp (t ),

Fig. 2. Reset set B = B+ ∪ B− and the two reset lines B+ , B− .

change of the state (xc → x+ c ) applied whenever some reset condition c (t ) is true. Standard reset controllers are typically triggered by the zero-crossing of the error signal c (t ) : e(t ) = 0.

(2.3)

where e(t ) stands for the error signal. The second equation for the controller (2.2) is the reset action, an instantaneous or impulsive

(2.4)

But this paper will consider reset controllers with reset band. This means that the reset condition is activated when the error hits the reset lines shown in Fig. 2, in the plane (e, e˙ ), where B+ = {(e, e˙ ) ∈ R2 : e = δ, e˙ ≤ −ϵ} and B− is symmetric to B+ . The number δ > 0 is the width of the band, or simply the band, and ϵ ≥ 0 is a small threshold. In this way, the reset condition becomes c (t ) : (e(t ), e˙ (t )) ∈ B := B+ ∪ B− .

Consider a reset control system as shown in Fig. 1, formed by a plant P and a reset controller C given by P :

Fig. 1. Reset control system.

(2.5)

When no threshold is specified, it is assumed by default ϵ = 0. The case δ = ϵ = 0 recovers the standard reset (2.4), without band. To achieve a more compact representation of (2.1)–(2.3), one   can stack the state vectors in x = xp , xc and form the closedloop matrices. Motivated by this, consider the general reset control system, with reset band, given by x˙ (t ) = Ax(t ), when (e(t ), e˙ (t )) ̸∈ B x(t +) = Ar x(t ), when (e(t ), e˙ (t )) ∈ B e(t ) = C x(t ),

(2.6)

where x ∈ Rnp +nc , A and Ar are the square matrices for the continuous and impulsive modes, respectively, and C is a row vector that defines the reset signal e. Given the reset condition c (t ) in (2.5), any system in the form (2.1)–(2.3) can be written as in (2.6), but not every system (2.6) comes from (2.1)–(2.3). In this way, the last equations provide more generality to the problem formulation. This paper addresses the problem of characterizing the limit cycles and related dynamic behaviors possibly appearing in reset systems with reset band as described by (2.6).

A. Barreiro et al. / Systems & Control Letters 63 (2014) 1–11

3. Well posedness and basic results Before proceeding any further with the limit cycles analysis, some care has to be taken regarding well-posedness of the problem. The Eqs. (2.6) define a particular class of hybrid systems, and it is well known that hybrid systems may exhibit complex dynamic patterns. Notice that the condition (e, e˙ ) ∈ B in (2.6) amounts to (e, e˙ ) = (C x, C x˙ ) = (C x, CAx) ∈ B, so it can be translated into x ∈ M0 , where the base reset manifold M0 is given by: M0 = M0 ∪ M0 , +

−

M0 = x ∈ Rn : C x = δ, CAx ≤ −ϵ , +





(3.1)

with M0− symmetric to M0+ . Now, define the after-reset manifold N as:

  N = Ar · M0 = x+ ∈ Rn : x+ = Ar x, for some x ∈ M0 .

(3.2)

For the reset system (2.6), any initial condition x0 ∈ Rnp +nc will produce reset instants tk = τk (x0 ), k = 1, 2, . . ., that are recursively defined by

τk (x0 ) = min{t ∈ R+ : CeA(t −tk−1 ) x(tk−1 ) ∈ M0 }

(3.2b)

where by definition τ0 (x0 ) = 0. That is, τk (x0 ) are the times when the trajectory hits the manifold. These hits are also called crossings. Note that for a particular initial condition there may exist no crossings, a finite or a infinite number of crossings, and in a finite or infinite time interval. A function x : [0, ∞) → Rn is a solution of the reset system (2.6) on [0, ∞), with initial condition x(0) = x0 , if x(·) is leftcontinuous and x(t ) satisfies (2.6) for all t ∈ [0, ∞). In general, the reset system (2.6) will have well-posed reset instants if for any initial condition x0 ∈ Rn the reset instants tk = τk (x0 ), k = 0, 1, 2, . . ., are well-defined and are distinct, that is 0 = t0 < t1 < t2 < · · ·. Finally, for a reset system (2.6) with well-posed reset instants, a Zeno-type solution x(·) exists on Ix0 = [0, τZ (x0 )) for some initial condition x0 ∈ Rn if there exists an infinite sequence of reset instants (tk = τk (x0 ))∞ k=0 such that tk+1 − tk → 0 as k → ∞, and in addition x(t ) → xZ as t → τZ (x0 ) for some limit point xZ ∈ Rn . Note that this definition of Zeno-type solution may include solutions in which the limit point is reached after a finite or infinite time, that is in general τZ (x0 ) ∈ R+ ∪ {∞}. A reset system is Zeno-free or does not have Zeno behavior if it does not have Zenotype solutions. Note that in addition the pathological behaviors referred to as beating and deadlock (in the sense given in [20]) would be present if reset instants are not well-posed. The three pathological behaviors that should be avoided are: (i) Beating To avoid beating or re-reset, the reset manifold M0 (where reset condition c (t ) is true) must be revised in order that the state x+ , immediately after reset, do not satisfy the reset condition again. (ii) Deadlock To avoid deadlock, if the state after reset x+ is in the boundary of the reset manifold M , then the continuous flow after reset (Ax+ ) should not move the state towards M (the vector Ax+ should not point towards M ). (iii) Zenoness The reset system is Zeno-free if the reset time interval 1tk = tk − tk−1 , between any two consecutive reset instants tk−1 , tk , is lower bounded 1tk ≥ 1tmin , at least in some working domain Ω . For item (i), notice that the set M0 ∩ N is the re-reset manifold. Such a set might be problematic, because if x+ ∈ M0 ∩N then x+ ∈ M0 , so it has to be reset again, to x++ . This might induce an infinite sequence of re-resets (x+ , x++ , x+++ , . . .) that have to be applied

3

instantaneously, which is a formal problem. Eliminating this reresetting is very easy if one simply redefines the reset manifold M as M = M0 \ (M0 ∩ N ) = M0 \ N = {x : x ∈ M0 , x ̸∈ N } .

(3.3)

Notice that M = M ∪ M = (M0 \ N )∪(M0 \ N ). In this way, the problem in (i) is solved if the system (2.6) is slightly modified to the re-reset free system +



x˙ = Ax x+ = Ar x,

−

+

−

x ̸∈ M = M + ∪ M − x ∈ M = M + ∪ M −.

(3.4)

The conditions that arrange (ii)(iii), and some basic results, will be given now. First, some terminology will be fixed. Given an m-dimensional subspace S ⊂ Rn , an affine subspace is a set in the form x0 + S = {x0 + x : x ∈ S }. One can always find a matrix U = (U1 | · · · |Um ) with Ui ∈ Rn such that x0 + S = x0 + Imag U = {x0 + U1 λ1 +· · ·+ Um λm : λi ∈ R}. Now, let us call an m-dimensional half manifold to a set of the form

{x0 + U1 λ1 + U2 λ2 + · · · Um λm : λ1 ≥ ϵ, λ2 , . . . , λm ∈ R}. That is, the coordinates λ2 , . . . , λm are free, but the coordinate λ1 is ’halved’. The number ϵ can be assumed zero by including U1 ϵ in x0 . The inequality can be reversed changing U1 by −U1 . For example, the set B+ (and B− ) in Fig. 2 is an 1-dimensional half-manifold, it can be written in the form (e, e˙ ) = (δ, −ϵ) + λ1 (0, −1) with λ1 ≥ 0. The main result in this section (see the Appendix for a proof) is the following proposition. Proposition 3.1. Consider the system (3.4), with M given by (3.3). Assume that C , CA are linearly independent. Then (a) The system is symmetric with respect to the origin 0 ∈ Rn , that is, given any solution x(t ), the symmetric trajectory −x(t ) is also a solution. (b) Inherited from the two reset lines in Fig. 2, the base reset manifold, + − M0 , is formed by two components, M0 , M0 , each of them is an (n − 1)-dimensional half manifold. The after-reset manifold, N , is formed by two components, N + , N − , each of them is mdimensional, with m ≤ (n − 1). They may be affine subspaces or half manifolds. (c) If the threshold is positive, ϵ > 0, then the system is free of deadlock. (d) Assume that Ar = diag(1, . . . , 1, 0, . . . , 0) with rank(Ar ) = n − nr > 0 (reset of the last nr states). Partition accordingly C = (C1 , C2 ) with C2 ∈ R(1,nr ) . Then, if C2 = 0 and ϵ > 0, and working in a bounded region Ω (given by ∥x∥ ≤ RΩ with RΩ large but finite), there is no Zeno behavior. Remark 3.1. The conditions on Ar , C in (d) are typical in reset control systems, as the reset condition for the controller usually depends only on the plant states. 4. Limit cycles and Poincaré maps This section presents a study on the application of Poincaré maps [15] to the detection and stability analysis of limit cycles in reset systems with a reset band. Although the objective is the characterization of limit cycles, it will be shown that other undesired dynamics, as Zeno-type behaviors, can also be captured with Poincaré maps. Let us start with the reset system (3.4), where M + , M − are (n − 1) dimensional manifolds, disjoints and symmetric: M + ∩ M − = ∅, M + = −M − , see Fig. 3. The reset maps or jumps J + , J − are given by the reset matrix Ar : J + : M + −→ N + x −→ Ar x

J − : M − −→ N − x −→ Ar x.

(4.1)

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A. Barreiro et al. / Systems & Control Letters 63 (2014) 1–11

Fig. 4. Limit cycle x0 → z0 → . . . → x0 and nearby trajectory. Fig. 3. Reset manifolds M − , M + ; after-reset manifolds N − , N + ; continuous − flows F+ , F++ , F−− , F−+ ; discrete resets or jumps J − , J + .

The after reset manifolds are the images J + (M + ) ⊂ N + and J − (M − ) ⊂ N − . In the Fig. 3, J + , J − are dashed arrows to indicate that they are instantaneous. The continuous maps or flows − F+ , F++ , F−− , F−+ are the solid arrows in Fig. 3, related to the dynam+ − ics x˙ = Ax. The flow F− (and similarly F+ , F++ , F−− ) is defined by:

  + F− : dom F−+ ⊂ N − −→ M + x −→ eAt x

(4.2)

where the elapsed time to impact from any x ∈ N − is t = τ1 (x). + The domain dom(F− ) is the set of all x ∈ N − such that exists a fiAτ1 (x) nite τ1 (x) and e x ∈ M + . All the conditions on well-posedness listed in Proposition 3.1 are assumed in what follows, unless otherwise explicitly stated. The interest now is on the existence of periodic solutions x(t ) to (3.4), the period is the minimum T such that x(t + T ) = x(t ) for all t. Although it might be possible, in principle, that the periodic solutions follow, inside one period interval, complex patterns of impacts with M + , M − , we will be interested here in simple limit cycles and by this we mean a periodic solution that in each period impacts once on M + and once on M − . Thus, more formally (see Fig. 4) a simple limit cycle is a solution x(t ) to (3.4) which is periodic with period T and such that at some time t0 belongs to, say, N + , that is, x(t0 ) = x0 ∈ N + , and verifies (see Fig. 4): − x(t0 + T /2) = eAT /2 x0 = F+ (x0 ) =: z0 ∈ M − , + x(t0 + T ) = eAT /2 w0 = F− (w0 ) =: y0 ∈ M + ,

x+ (t0 + T ) = Ar y0 = J + (y0 ) = x0 ∈ N + . Thus, that precise concatenations of flows and jumps, when presenting a fixed point x0 , characterizes a simple limit cycle. Consider first the half-cycle Poincaré map:

+− and, if range(Phalf ) ∩ dom F−+ Poincaré map:





(4.3)

̸= ∅, define also the (full cycle)

P ± : dom F+ ⊂ N + −→ N +     x −→ J + F−+ J − F+− (x) .



+− +− P ± (x) = −Phalf −Phalf (x) .





(4.5) ±

The interest of the Poincaré map P (4.4) is that it characterizes the existence and stability of simple limit cycles, as stated next (see Appendix for a proof). Proposition 4.1. Consider the system (3.4), with the conditions for well-posedness in Proposition 3.1, namely: C , CA independent, ϵ > 0 and CAr = C . Then (a) The system has a simple limit cycle if and only if P ± has a fixed point. (b) The linearization of the Poincaré map P ± (x) about its fixed point x = x0 is given by ∆xk+1 = (Ar · E · Ar · E ) ∆xk , where ∆xk = x(tk ) − x0 and E = eAτ0 − A eAτ0 x0 (C eAτ0 )/(C A eAτ0 x0 ) with τ0 the half-period of the limit cycle. (c) The limit cycle of the hybrid, continuous-time system is locally asymptotically stable (l.a.s.) if and only if the fixed point of the discrete-time Poincaré map is l.a.s. Now it will be shown that Zeno behavior can also be predicted by means of adequate Poincaré maps. First, as the conditions in Proposition 3.1 ensure well-posedness and no Zenoness, some of the conditions have to be relaxed. From the proof of part (d) in Proposition 3.1 it is clear that ϵ > 0 is critical for avoiding zenoness, so let us suppose that ϵ = 0, that is, suppose that the reset band in Fig. 2 has no threshold. Proposition 4.2. Consider the system (3.4), with C , CA independent and CAr = C , but with ϵ = 0. Then (see Appendix for a proof):

x+ (t0 + T /2) = Ar z0 = J − (z0 ) =: w0 ∈ N − ,

  +− Phalf : dom F+− ⊂ N + −→ N −  x −→ J − F+− (x)

Notice that, from symmetry, the full cycle map depends on the half cycle map:

 −

(4.4)

(a) If the system presents Zeno-type behavior, that is for some initial condition there exist a sequence of reset times (tk )∞ k=0 such that 1tk = tk+1 − tk → 0 as k → ∞, then the limit point xZ = x0 belongs to x0 ∈ N + ∩ ∂ M + or x0 ∈ N − ∩ ∂ M − , where ∂ M + , ∂ M − denote the boundaries of the sets M + , M − (see Fig. 5). (b) The approximation of the Poincaré map around thelimit point xZ = x0 of the Zeno behavior is given by xk+1 = I − 2Fk + 2Fk2 xk , where Fk (xk ) = A(CAxk )/(CA2 xk ).



5. Examples Consider a reset control system as shown in Fig. 1, formed by a strictly proper plant given by (Ap , Bp , Cp ) and by a PI + CI with band

A. Barreiro et al. / Systems & Control Letters 63 (2014) 1–11

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Fig. 5. Zeno behavior and nearby trajectories.

δ , used as a reset controller. Since we are interested in stability and limit cycles, and not in tracking behaviors, let us assume that the reference r (t ) and disturbance d(t ) take constant steady-state values and remove their effect by state coordinate change. Thus the closed-loop Eq. (2.6) give:

 A=

Ap − Bp kp Cp −Cp −Cp

 Ar =

In p 0 0

0 1 0

Bp ki 0 0

0 0 , 0

Bp kc 0 0

 ,

t− (x1 ) = min t > 0 : e(t ) = CeAt x1 = −δ, e˙ (t ) > 0 ≤ ∞.





C = (−Cp , 0, 0).

⊤ ⊤ The whole state vector is x = (x⊤ p , xc ) , of dimension n = np + nc = np + 2 and (kp , ki , kc ) are the gains of the PI + CI controller. Next, we consider several cases.

Example 5.1 (Third Order Plant. Case Without Limit Cycles). Consider the plant P (s) = b0 /(a0 + a1 s + a2 s2 + s3 ). Using the observable canonical form we reach:



0 1  A = 0 0 0

0 0 1 0 0

−a0 − kp b0 −a 1 −a 2 −1 −1

k i b0 0 0 0 0



kc b0 0   0 , 0  0



There are three possibilities (see Fig. 6): (i) the first impact is again on N + (ii) first impact on N − and (iii) If both t+ , t− are infinite, the trajectory never resets again. This means that the state follows the base linear dynamics eAt x1 . Consider for example the plant P (s) = 1/(25 + 15s + 15s2 + s3 ) and the controller tuning kp = 5, ki = 10, kc = 10, with unity band, δ = 1. In the linear mode the controller is a PI (10+20/s). The closed loop error evolution is fourth-order, with poles −14.079, −0.7476, and −0.0868 ± j1.3757. This means that the closed loop base linear system is stable. This is a necessary condition for the origin 0 to be locally attractive. The after-reset manifold N + is: N + = {(x1 , x2 , −1, x4 , 0)⊤ : x2 > −15, x1 , x4 free}. To compute the times-to-impact and Poincaré maps, since PM is a numerical method, we must fix a mesh D + within N + , for example D + = {(x1 , x2 , x4 ) ∈ [−40, 40]41 × [−14.9, 14.9]41

× [−2, 2]31 },

Ar = diag(1, 1, 1, 0), C = (0, 0, −1, 0, 0) . Other relative degrees could be treated in a similar way, by using b0 + b1 s + b2 s2 as plant numerators. The reset manifold M0+ is defined by the hitting of the error on the reset line B+ given by e = δ, e˙ ≤ −ϵ (see Fig. 2). Since e = Cx = δ and e˙ = CAx ≤ −ϵ , taking ϵ → 0 very small, and renaming the states (x1 , x2 , . . . , x5 )⊤ , the manifold is: M0 = {(x1 , x2 , −δ, x4 , x5 )⊤ : x2 > −a2 δ, x1 , x4 , x5 free} +

which defines M0+ as a 4-dimensional half manifold (a halfhyperplane in R5 ). In the same way, M0− is the symmetric of M0+ . Regarding the after reset manifold N + = {Ar x : x ∈ M0+ }, whenever a state hits M0+ and is affected by the reset x+ 5 = 0, then it takes the form (x1 , x2 , −δ, x4 , 0)⊤ . Thus N + = {(x1 , x2 , −δ, x4 , 0)⊤ : x2 > −a2 δ, x1 , x4 free}.

Finally, N − is symmetric to N + . Both are 3-dimensional manifolds in R5 . For computing the flows, jumps and ’timesto-impact’, see Fig. 3. Notice that, from symmetry, it suffices to consider that the initial condition x1 is in N + . The error signal is given by e(t ) = CeAt x1 . The related ‘times-to-impact’ are given by: t+ (x1 ) = min t > 0 : e(t ) = CeAt x1 = +δ, e˙ (t ) < 0 ≤ ∞



Fig. 6. Error trajectories. Solid: first impact at N + . Dotted: first impact at N − . Dashed: no impacts.



where [a, b]n denotes a linearly spaced set of n values from a to b. The total number of points is n1 n2 n4 = 41 × 41 × 31 = 52, 111. The resulting times to impact t+ (x1 , x2 , x4 ) and t− (x1 , x2 , x4 ) were computed from x0 = (x1 , x2 , −1, x4 , 0)⊤ for all the mesh values (x1 , x2 , x4 ) ∈ D + by simulations for t ∈ [0, tmax ]—since A is stable and ∥x0 ∥ bounded, we can fix a large enough tmax such that the detection of the first reset e = ±δ = ±1 can be restricted to t ∈ [0, tmax ]. In our case, tmax = 10 s. is a valid choice. The step size has been set to 0.0002 = 2 · 10−4 s. The computation time was about fifteen minutes. The Fig. 7 shows the values t+ (x1 , x2 , x4 ), t− (x1 , x2 , x4 ), in seconds, as a function of the initial point x1 for four initial values of x2 , x4 . For some values of (x1 , x2 , x4 ) neither t+ nor t− are defined, corresponding to the case shown in Fig. 6 when the trajectory never resets again (and thus tends to 0). In the same way as t+ , t− , the new,+ ,+ Poincaré maps have the form: x1 (x1 , x2 , x4 ), xnew ( x1 , x2 , x4 ) , 2 new,+ and x4 (x1 , x2 , x4 ) in the case of the first reset at N + and new,− xi (x1 , x2 , x4 ) for i = 1, 2, 4, in case of a reset at N − . Once the times-to-impact and PM are computed for D + , they can be stored as nD look-up tables and can be used to consult other initial conditions (other meshes D0 ), and recursively iterate the PM, obtaining the discrete-time mesh evolution D1 , D2 , . . . , Dk . For brevity, given the amount of data, we only report a choice of

6

A. Barreiro et al. / Systems & Control Letters 63 (2014) 1–11

Fig. 7. Times-to-impact, in seconds, as a function of the initial value of x1 , for the four combinations of x2 = ±14.9, x4 = ±2. Solid: t+ (x1 , x2 , x4 ). Dotted: t− (x1 , x2 , x4 ).

Fig. 8. PM iterated x1 × x2 submesh for x4 = 0 at k = 0, 1, 2, 3. Label ‘·’: Trajectory with reset at step k; Label ‘0’. No reset and linear evolution towards 0; No label: inconclusive points.

possible illustrative graphics and numerical results. If, for example, we choose a submesh: D0 = {(x1 , x2 , x4 ) ∈ [−40, 40]21 × [−14.9, 14.9]21

× [−2, 2]11 }, (it only has 21 · 21 · 11 = 4, 851 points) and compute some PM iterations D1 . . . D3 , we can examine their entries and draw conclusions on the system. After the first iterate, in D1 = PM(D0 ) we obtain nr = 2769 trajectories that reset and n0 = 2082 that do not reset (and enter in the linear stable mode). For k = 2 we obtain nr = 561 (second reset), n0 = 4163, nx = 127 where nx is the number of trajectories that fall out-of-range (outside the nD table limits). Finally for k = 3 we obtain nr = 0, n0 = 4724, nx = 127. Thus only 127/4851 ≈ 3% initial points are inconclusive, the remaining 97% reset once or twice, but eventually they enter the linear mode (and tend to 0). The Fig. 8 shows part of the results. It is based on submeshes of Dk (for x4 = 0) at several steps k = 0, 1, 2, 3, of the PM iterations.

Fig. 9. Simulation of x3 (t ), x5 (t ) starting from (40, −14.9, −1, 0, 0).

At k = 0 we see all the 21 × 21 initial values (x1 , x2 ). At k = 1, 2, 3, and in the position (x1 , x2 ) of the initial condition, we label with ‘0’ the trajectories entering linear stable mode, with a dot ‘·’ the trajectories that reset, and we do not display the inconclusive trajectories. If in Fig. 8, we choose for example the bottom right corner position, this corresponds to (x1 , x2 ) = (40, −14.9). At this lower right corner for k = 1, 2 we have a reset (‘·’) and for k = 3 we have (‘0’) no more resets and linear stable evolution. The simulation of the trajectory starting at (40, −14.9, −1, 0, 0) is shown in Fig. 9, confirming this behavior (2 resets and stability) and the predicted reset values. In summary, what Fig. 8 illustrates (in its last subplot) is a section or projection of the domain of attraction (DoA) to the origin: all the initial conditions (except the small inconclusive part at the lower left corner) tend asymptotically to 0. If we were interested in the inconclusive points, or in other initial conditions, we could use the same Matlab code, adjust the settings and/or enlarge the initial mesh of points D + to cover the target regions. For example, for initial conditions in the form (x1 , 0, 0, 0, 0) we easily reach this result: there are no resets for |x1 | < 1, there is one single reset for 1 < |x1 | < 11.32, there are two resets for 11.32 < |x1 | < 14.8. For 14.8 < |x1 | the mesh D + is inconclusive. Example 5.2 (Third Order Plant. Case With Limit Cycles). Consider now the plant P (s) = 1/(25 + 20s + 20s2 + s3 ) with the controller kp = 15, ki = 30, kc = 30. Now the closed loop poles are −19.05, −1.309 and +0.1803 ± j1.5407. This means that the closed loop base linear system is unstable. One important difference with the previous example is that now all the considered trajectories either impact on N + or on N − . That is, either t+ or t− are always defined for any initial (x1 , x2 , x4 ). This is related to the fact that the base system is unstable and all trajectories grow unbounded until a resetting appears. The Fig. 10 shows the impact times t+ (x1 , x2 ) as a function of the initial values (x1 , x2 ), for x4 = 0. In Fig. 10 (left) we see the region where t+ (x1 , x2 ) is defined (the next impact is on N + ). The complementary region is where t− (x1 , x2 ) is defined (impact on N − ). The Fig. 10 (right) shows the values of the times in seconds. Thus from Fig. 10 (left) we see that the ’domains of attraction’ of N + or N − have a ’spiral’ shape. This is related to the unstable linear oscillations, dominated by a law e(t ) ≈ edom (t ), with edom (t ) = e+σ t (a sin ωt + b cos ωt ), with growing maxima and minima.

A. Barreiro et al. / Systems & Control Letters 63 (2014) 1–11

7

Fig. 10. Impact-times t+ (x1 , x2 ) for x4 = 0. Left: region where t+ is defined. Right: Impact-times in seconds. Fig. 12. Simulation of x3 (t ), x5 (t ) from (0, 0, −5, 0, 0).

Fig. 11. PM iterations k = 10, 20, 30, from a mesh (x1 , x2 , x4 ) in N + .

The Fig. 11 shows the result of performing k = 10, 20, 30 iterations of the PMs. We simulate 21 × 21 × 11 trajectories starting from a mesh covering the range (x1 , x2 , x4 ) ∈ [−40, 40] × (−20, 20) × [−2, 2] in N + . Next, we apply recursively the PM and plot in 3D the (x1 , x2 , x4 ) position of the k-th iterate of the mesh. All initial conditions (except a small fraction that falls out of range) are captured by the iterated map and as k increases we see that all the trajectories tend to two symmetric fixed points ∞ ∞ (±x∞ 1 , ±x2 , ±x4 ) ≈ (±6.3, ±18.8, ±0.44) that represent the after reset coordinates of an induced limit cycle, according to Proposition 4.1. The PM predicts a period T ≈ 4.1 s. The PM iterations prove numerically that the limit cycle is attractive, but if we compute the map linearization we obtain a Schurstable matrix, and thus confirm that the limit cycle is attractive (see Proposition 4.1). The simulation of different trajectories, like those shown in Fig. 12, starting from (0, 0, −5, 0, 0), confirms the existence of the predicted limit cycle in steady state, with the period T and after-reset values predicted by the PM. Example 5.3 (Third Order Plant. Effect of Parameter Variations). Consider the previous third order plant P (s) = 1/(25 + 20s + 20s2 + s3 ), but with the controller gains k0 = (kp0 , ki0 , kc0 ) = (15, 30, 30)

Fig. 13. Steady-state PM values of x4 as a function of λ.

scaled by an attenuation factor λ ≥ 1:

(kp , ki , kc ) = (15, 30, 30)/λ,

λ ≥ 1.

The PM is a numerical method, thus for studying the effect of

λ one has to discretize the range and repeat a PM procedure for each discretized value. The case λ = 1, addressed in the previous subsection, gives an unstable base system and trajectories tending to a simple limit cycle. For λ large and a stable base system we obtain behaviors similar to the one in the first subsection, with an attractive equilibrium at 0. Interestingly, some ranges of λ produce more complex behaviors. We discretize the range λ ∈ [1.45, 1.9] and (piecewisely for subsets of λ’s) apply the PM iterations for a long time, more than 150 s, corresponding to more than 25 oscillation periods and at least k = 50 half-cycle PM iterations. Then, to obtain the steadystate or limit behavior, we discard the transients and store the last reset values (x1 , x2 , x4 ) (see Fig. 13). We see that, for λ < 1.5, x4 takes a single value: the steadystate value x4 (tk ) at the reset instants tk of the simple limit cycle. But, for example, for λ = 1.6 the graph gives two steady state

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A. Barreiro et al. / Systems & Control Letters 63 (2014) 1–11

Fig. 14. Subharmonics in steady-state of x4 (t ), x5 (t ) for λ = 1.6.

Fig. 16. Poincaré maps f (solid) and g (dashed). The fixed point LC defines a stable limit cycle.

details in [22]):

−K p A = −1 −1 

Ki 0 0



Ar =

1 0 0

0 1 0

Kc 0 0

0 0 , 0

 ,



C = (−1, 0, 0),

where Kp = (kkp − p), Ki = kki and Kc = kkc . The reset manifold + M0 is now: +

M0 : x1 = −δ,

−Kp δ + Ki x2 + Kc x3 > 0,

that is, it is a 2-dimensional half manifold (a half-plane in R3 ). In the same way, M0− is the symmetric of M0+ . Next, the after-reset manifold takes the form: N + : x1 = −δ,

Fig. 15. Chaotic steady-state oscillations of x4 (t ), x5 (t ) for λ = 1.8.

values x4 (tk ) ≈ ±0.51, x4 (tk ) ≈ ±0.18 (from symmetry, the negative values, not shown, also appear). The Fig. 14 shows the related time response of the controller states x4 (t ) and x5 (t ). We see that at the reset times of x5 (t ) the values of x4 are precisely ±0.51, ±0.18, predicted by PM. We see also that there exists an underlying subharmonic of third order, proving that more complex periodicity patterns appear. The inspection of the bifurcation diagram in Fig. 13 shows a dispersion of the limit points for λ ∈ [1.75, 1.9] that suggest lack of periodicity and possible emergence of chaos. The figure Fig. 15 plots x4 (t ), x5 (t ) in steady-state (for t ∈ [100, 160] s), for λ = 1.8. The apparent 3rd order subharmonic pattern of the first 9 cycles is broken at the 10th cycle in an aperiodic intermittent pattern, as suggested by the PM bifurcation diagram. A deeper study of these complex behaviors is out of the scope of this paper. Example 5.4 (First Order Plant). Consider now a first order plant P (s) = k/(s − p) and a PI + CI with band δ and gains (kp , ki , kc ). Let us call x1 to the plant output, and x2 , x3 to the outputs of the integrator I and the CI. We reach (2.6) with matrices (see more

x2 free,

x3 = 0,

where x2 is free, for Kc ̸= 0, which defines N + as an affine 1dimensional line. Finally, N − is symmetric to N + . The times-toimpact and the various PMs are functions of the initial condition x0 = (−δ, x20 , 0)⊤ ∈ N + , with x20 ∈ R. Consider the parameters Kp = −0.5, Ki = 5, Kc = 2.5, δ = 1. Then the characteristic polynomial is s2 − 0.5s + 7.5, and the linear mode has two complex unstable poles. Applying the PM procedure, we reach the Poincaré maps in Fig. 16. The horizontal axis is the value x20 of the initial point in N + . The map g (dashed) is the next value of x2 in N + reached directly. The map f (solid) is the next value of x2 in N + but is reached passing once through N − . The Zeno point Z is actually off the diagonal (if ϵ > 0, no matter how small) thus there is no fixed point of g and from Proposition 3.1(d) there is no Zeno behavior (the iterations close to Z escape from it to the right). The only fixed point, x2 = f (x2 ), is the point marked by LC . The slope of f at this point is smaller than 1, so according to Proposition 4.1, it defines a limit cycle, locally asymptotically stable. Furthermore, from the graphical information in the Fig. 16, it can be deduced that all possible iterations (x2 → f (x2 ) or x2 → g (x2 )) tend eventually to LC , so the limit cycle is globally asymptotically stable. The simulation results are given in Fig. 17, showing a limit cycle in steady-state. Zooming the values in steady-state, it can be seen that the limit cycle starts at the after-reset value xLC ≈ 0.365. 2 It is the same value obtained independently for the fixed point

A. Barreiro et al. / Systems & Control Letters 63 (2014) 1–11

9

be possible with the same conceptual approach (the PM technique) but would require more involved manipulations. Future work will also consider frequency-domain analysis (forced response) of reset control systems with band. Appendix. Proofs Proof of Proposition 3.1. The property (a) follows from the fact that the continuous flow, the reset action, and the reset conditions are invariant with respect to the symmetry change x → −x. The property (b) states more precisely what follows: (b) Assume that the rows C , CA are linearly independent. Build

⊤

row-wise an invertible (n × n) matrix W = C ⊤ |(CA)⊤ | . . . , and compute the inverse W −1 = (V1 | . . . |Vn ), with Vi ∈ Rn . Then M0 and N are as follows. (b1) The base reset manifold, M0 , is formed by two components, + − M0 , M0 , each of them is a half (n − 1)− dimensional affine manifold (half affine hyperplane) in the form



Fig. 17. Response with steady-state limit cycle, x1 (t ) (solid), x2 (t ) (dashed), x3 (t ) (dash–dot).

x0 + V2 λ2 + V3 λ3 + · · · + Vn λn

λ2 ≥ 0, λ3 , . . . , λn free.

(b2) The after-reset manifold, N, is formed by two components, N + , N − , each of them is m-dimensional, where m = rank (Ar (V2 | . . . |Vn )) ≤ (n − 1). If Ar V2 ∈ Imag Ar (V3 | . . . |Vn ) then N + , N − are m-dimensional affine subspaces. Otherwise, they are half m-dimensional affine manifolds Thus, property (b), shows the structure of M0 and N. Each component of M0 , say M0+ , is defined by C x = δ (an affine hyperplane) and CAx ≤ −ϵ (a half space). If C , CA are independent, the intersection defines a half affine hyperplane. More explicitly,

⊤

using W = C ⊤ |(CA)⊤ | . . . , the conditions C x = δ, CAx ≤ −ϵ , amount to W x = (λ1 , . . . , λn )⊤ with λ1 = δ, λ2 ≤ −ϵ and λ3 , . . . , λn free. Introducing V = W −1 , this implies x = V1 δ + V2 λ2 + · · ·, which shows the explicit form of M0+ as a half affine hyperplane, thus proving (b1). As N is the image of M0 by the linear application Ar , (b2) easily follows. In particular, if Ar V2 ̸∈ Imag Ar (V3 | . . . |Vn ) the half affine nature is preserved (λ2 ≤ −ϵ yields an active restriction). Otherwise, N + , N − become simple affine subspaces (not halved). Regarding (c), for avoiding deadlock as in (ii), one has to suppose that the state after reset x+ is in the closure of M . More generally, suppose that x+ is in the closure of M0 , which from (3.1) coincides with itself, cl M0 = M0 , that is, let x+ ∈ M0 . At this point, the error e˙ = C x˙ = CAx+ ≤ −ϵ < 0 is nonzero. This fact,  +velocity  C Ax ̸= 0, means, as M0+ , M0− are contained in the hyperplanes C x = ±δ , that the continuous flow after reset Ax+ points outside + − M0 = M0 ∪ M0 ⊃ M = M + ∪ M − so that the state x+ moves outside M and there is no deadlock. Finally, for avoiding Zeno behavior in (d), notice that from the conditions on C = (C1 , 0) and Ar , it holds that CAr = C . Consider an impact at t = tk on, say M + , with x(tk ) = xk ∈ M + , so that ek := e(tk ) = C xk = +δ . Consider also the subsequent instantaneous reset on N, at x(tk+ ) = x+ (tk ) = x+ k = Ar xk ∈ N. + + Then e+ k := e(tk ) = C xk = CAr xk = C xk = +δ . Now, it will be shown that it takes some minimum time 1tmin ≤ (tk+1 − tk ) to satisfy again, at t = tk+1 , a reset condition. As e+ k = +δ , there are two possibilities for ek+1 = e(tk+1 ) = C x(tk+1 ) = C xk+1 with xk+1 ∈ M = M + ∪ M − , namely xk+1 ∈ M − , implying ek+1 = −δ , or xk+1 ∈ M + , implying ek+1 = +δ . In the first case, it takes some time to pass from e+ = +δ to k ek+1 = −δ . More precisely,



Fig. 18. Family of Poincaré maps for 20 discretized values of λ ∈ [0, 1].

in Fig. 16, thus confirming the correctness of the procedure. If we introduce now an uncertain parametric variation within the interval (Kp , Ki , Kc ) = λ(1, 5, 1) + (1 − λ)(−0.5, 5, 2.5), with 0 ≤ λ ≤ 1, the family of Poincaré maps is displayed in Fig. 18. The fixed point of f (solid), related to the limit cycle, appears for λ ∈ [0, 1/3). In the other cases, when λ ∈ (1/3, 1], there is no fixed point and, furthermore, there exists an interval where neither f nor g are defined, meaning that the related trajectories never reset. Since the base system has a stable polynomial, s2 + Kp s +(Ki + Kc ) = 0, it can be concluded that for λ ∈ (1/3, 1] the reset system has a global equilibrium at 0. 6. Conclusions and future work This paper develops a systematic approach, based mainly on Poincaré maps, for dynamical analysis of reset systems with reset band. The approach gives procedures to ensure well-posedness, characterize reset manifolds and obtain PMs. From the computed PMs it is shown how to detect limit cycles, zenoness and, otherwise, global asymptotical stability GAS. Extensions of these results to more general classes of trajectories (e.g., nonlinear flows) would

1t ≥

2δ max |˙e|

=

2δ max |C x˙ |

=

2δ max |CAx|

≥

2δ

∥CA∥ RΩ

,

10

A. Barreiro et al. / Systems & Control Letters 63 (2014) 1–11

as x ∈ Ω , that is, ∥x∥ ≤ RΩ . The second case, xk+1 ∈ M + and ek+1 = +δ , admits two subpossibilities, depending on the close+ + + + ness of x+ k ∈ N to M , namely dist(xk , M ) ≥ ϵ1 or dist(xk , + M ) < ϵ1 , for some small ϵ1 > 0 to be fixed later. + In the first case, dist(x+ k , M ) ≥ ϵ1 , the continuous flow from + + xk to xk+1 ∈ M takes at least a duration:

1t ≥

ϵ1 max ∥˙x∥

ϵ1

=

max ∥Ax∥

≥

ϵ1 . ∥A∥ RΩ

+ ˙+ ˙ (tk+ ) = In the last case, dist(x+ k , M ) ≤ ϵ1 , notice that e k := e + + CAxk = CAx + CA(xk − x) ≤ −ϵ + ∥CA∥ϵ1 , for some x ∈ M + . We ϵ can always make the choice ϵ1 = 2∥CA , ensuring that e˙ + k ≤ −ϵ/2 ∥ for all k. Then the fastest way to reach from the after-reset condi˙+ tion e+ k = δ, e k ≤ −ϵ/2 < 0 to the new reset condition ek+1 = δ is achieved when e˙ + k = −ϵ/2 and when the maximum acceleration e¨ is applied to the error signal e(t ), yielding a uniformly accelerated trajectory in the form e(t ) = δ−(ϵ/2) (t − tk )+¨emax (t − tk )2 /2. Imposing the new reset condition e(tk+1 ) = δ gives the lower bound for 1t = (tk+1 − tk ):

1t ≥

ϵ e¨ max

=

ϵ max |C x¨ |

=

ϵ max |CA2 x|

≥

ϵ . ∥CA2 ∥RΩ

Taking the minimum of the three previous lower bounds for 1t = tk+1 − tk gives the universal lower bound required, ensuring no Zeno behavior, and this concludes the proof. Proof of Proposition 4.1. The Fig. 4 shows a typical trajectory passing initially through x1 and approaching the limit cycle. Property (a) is immediate. For proving (b), notice that the hybrid flow in continuous time induces a discrete-time map involving several variables: the after-reset states xk , wk ; the pre-reset states yk , zk and the elapsed times for each half cycle τk , σk . The nonlinear equations for this discrete-time map are: xk = Ar yk ,

zk = eAτk xk ,

wk = Ar zk ,

yk+1 = e

Aσk

−δ = C zk , wk , +δ = C yk+1 .

This nonlinear discrete-time map admits a linearization around the trajectory defined by the limit cycle, that is, around the values x0 , z0 , . . . satisfying x0 = Ar y0 ,

z0 = eAτ0 x0 ,

w0 = Ar z0 ,

y0 = e

Aσ0

w0 ,

−δ = C z0 , +δ = C y0 ,

where several symmetries hold: w0 = −x0 , z0 = −y0 , τ0 = σ0 . Introducing deviation variables: 1xk = xk − x0 , 1yk = yk − y0 , etc., and after a little algebra, the linearized system is given by:

1zk = eAτ0 1xk + AeAτ0 x0 1τk ,

1xk = Ar 1yk ,

0 = C 1zk ,

and by 1wk = Ar 1zk , 1yk+1 = eAσ0 1wk + AeAσ0 w0 1σk , and 0 = C 1yk+1 . Left-multiplying by C the second equation, using 0 = C 1zk , and eliminating 1τk gives:



Aτ 0

1zk = e

Aτ 0

−Ae

x0

C eAτ0 C A eAτ0 x0



1xk =: E 1xk .

In the same way, from symmetries: 1yk+1 = (eAτ0 − A eAτ0 x0

C eAτ0 C A eAτ0 x0

)1wk =: E 1wk . Putting everything together gives 1xk+1 = (Ar EAr E ) 1xk , thus proving (b). Notice that this expression is the explicit linearization of the map P ± (x) in (4.4) at the fixed point x = x0 . Finally, for proving (c), recall that the limit cycle is locally asymptotically stable when there exists some ϵ0 defining a neighborhood Nϵ0 (x0 ) such that:

∀x1 ∈ Nϵ0 (x0 ),

lim dist (x(t ; x1 ), LC ) = 0,

t →∞

(A.1)

where x(t ; x1 ) stands for the trajectory starting at x = x1 , and ‘LC ’ is the limit cycle. In the same way, the fixed point x0 of the map P ± (xk ) = xk+1 is locally asymptotically stable when some ϵ0 exists such that

∀x1 ∈ Nϵ0 (x0 ),

lim |xk − x0 | = 0.

(A.2)

k→∞

First, to prove that (A.1) implies (A.2) suppose, by contradiction, that (A.2) is false, that is, some ϵmin exists such that ∀k0 , ∃k > k0 such that |xk − x0 | ≥ ϵmin . Then, the two continuous trajectories x(t ; xk ) and x(t ; x0 ) would start for t = tk at distance greater than ϵmin which means that dist (x(t ; xk ), LC ) would be lower bounded by some positive small constant, hence (A.1) would be false. Finally, to prove that (A.2) implies (A.1), let us assume that for all ϵ > 0, ∃k0 = k0 (ϵ) such that ∀k ≥ k0 we have |xk − x0 | ≤ ϵ . This implies that |eAt xk − eAt x0 | ≤ |eAt | |xk − x0 | ≤ |eAt |ϵ , and, as the time t between reset impacts is upper bounded t ≤ t¯, working inside appropriate bounded subsets, then putting α = max{|eAt | : t ≤ t¯} gives, for the first xk to zk   half-cycle (from in Fig. 4), dist (x(t ; xk ), LC ) ≤ dist x(t ; xk ), eAt x0 ≤ αϵ , and similarly for the second half-cycle. As ϵ can be made arbitrarily small, this proves (A.1). Proof of Proposition 4.2. Only the case of Zenoness at M + , N + will be treated, the case at M − , N − follows from symmetry. The Fig. 5 shows such Zeno behavior. What characterizes Zenoness is that the reset intervals tend to zero: 1tk = tk+1 − tk → 0. This + (xk ) = amounts to the fact that the time elapsed by the flows F+ yk = eA1tk xk tend to zero, which implies that yk → xk . As every xk belongs to N + and every yk belongs to M + , then the approaching sequences {xk }, {yk } tend to the set N + ∩ M + . Recall that M + is a hyperplane given by C y = δ and CAy ≤ −ϵ = 0. Its boundary is thus given by C y = 0 and CAy = 0. Then, from Proposition 3.1(d), + the sequence (yk )∞ k=0 must tend to the boundary ∂ M . Otherwise, + if (yk )∞ would be kept away from ∂ M , it would be the same k=0 behavior that the system would present for some ϵ ̸= 0 (take −ϵ = max{k = 0, 1, . . . : CAyk < 0}), but from Proposition 3.1(d), ϵ > 0 would imply no Zenoness, so necessarily yk tends to ∂ M + . ∞ Hence, the common limit point of the sequences (xk )∞ k=0 , (yk )k=0 is some x0 ∈ N + ∩ ∂ M + . To prove (b), notice that the continuous flow approaching to the Zeno limit x0 induces a nonlinear discrete-time Poincaré map ∞ for the sequences (xk )∞ k=0 , (yk )k=0 and τk (elapsed time), given implicitly by: xk+1 = Ar yk , yk = eAτk xk , δ = C yk . This nonlinear map admits the fixed point (the Zeno point) given by x0 = y0 and τ0 = 0. The second order approximation I + Aτk + A2 τk2 /2 of the exponential eAτk , valid for τk → τ0 = 0, gives yk = xk + Axk τk + A2 xk τk2 /2. Left multiplying by C and since C yk = δ and C xk = CAr yk−1 = CAx C yk−1 = δ (see Fig. 5) gives τk = −2 CA2 xk . Define the square k

2 matrices Fk (xk ) = A(CAxk )/(  CA xk ) and substitute τk in the formula for yk , this gives yk = I − 2Fk + 2Fk2 xk , which combined with xk+1 = Ar yk proves (b).

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