A new eventdriven outputbased discretetime control ...

PUBLISHED VERSION Jetto, L., Orsini, V., Romagnoli, R. A mixed numerical-analytical stable pseudoinversion method aimed at attaining an almost exact tracking (2015) International Journal of Robust and Nonlinear Control, 25 (6), pp. 809-823. DOI: 10.1002/rnc.2921

A new event-driven output-based discrete-time control for the sporadic MIMO tracking problem L. Jetto*,† and V. Orsini Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, Ancona, Italy

SUMMARY This paper presents a new sporadic control approach to the tracking problem for MIMO closed-loop systems. An LTI sampled data plant with unmeasurable state affected by external unknown disturbances is considered. The plant is interconnected to an event-based digital dynamic output-feedback controller via a network. Both the external reference and the unknown disturbance are assumed to be generated as the free output response of unstable LTI systems. The main feature of the new event-driven communication logic (CL) is that it works without the strict requirement of a state vector available for measurement. The purpose of the CL is to reduce as much as possible the number of triggered messages along the feedback and feedforward paths with respect to periodic sampling, still preserving internal stability and without appreciably degrading the control system tracking capability. The proposed event-driven CL is composed of a sensor CL (SCL) and of a controller CL (CCL). The SCL is based on the computation of a quadratic functional of the tracking error and of a corresponding suitably computed time-varying threshold: a network message from the sensor to the controller is triggered only if the functional equals or exceeds the current value of the threshold. The CCL is directly driven by the SCL: the dynamic output controller sends a feedforward message to the plant only if it has received a message from the sensor at the previous sampled instant. Formulation of the controller in discrete-time form facilitates its implementation and provides a minimum inter-event time given by the sampling period. An example taken from the related literature shows the effectiveness of the new approach. The focus of this paper is on the stability and performance loss problems relative to the sporadic nature of the control law. Other topics such as network delay or packets dropout are not considered.

KEY WORDS:

event-driven control; sporadic control; internal stability; reference tracking

1. INTRODUCTION This paper considers the problem of reducing the number of triggered messages along the feedback and feedforward paths of a closed-loop system for the networked control of an LTI MIMO sampled plant. Many authors considered this and similar topics in the general framework of event-based control [1] and proposed several communication logics (CLs) whose common feature is invoking a message among the control system components only if a ‘significant’ event occurred. The heuristic deadband and send-on-delta sampling techniques described in [2–5] have been successfully applied to the regulation problem for SISO plants, but no formal analysis of closed-loop stability is provided. Ultimate boundedness is proved in [6, 7], where the event-driven state-feedback control depends on the measured error relative to the state trajectory. A performance analysis of the class of sporadic event-based controllers is provided in [8], where first-order linear stochastic systems are considered. *Correspondence to: L. Jetto, Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, Ancona, Italy. †E-mail: [email protected]

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L. JETTO AND V. ORSINI

The methods described in [9–12] consider the more general framework of decentralized multi-loop systems and propose event-triggered strategies where each node of the network uses its local data to state when to transmit information. References [9] and [12] are based on the dissipation inequalities initially proposed in [13]. All the aforementioned methods work in the continuous-time domain and obtain a reduction of the communication load trying to maximize the minimum inter-event time between two consecutive control tasks. To this purpose, the control effort is updated only when the error with respect to the desired state trajectory becomes too large. A more direct implementation of event-driven sampling strategies can be obtained considering the event-based control in the discrete-time framework (see, e.g., the stabilization problems considered in [14, 15]). In [14], new computational methods for event-based control are developed for minimizing an upper bound on system performance associated with the use of an approximate value function for the underlying Markov decision process. To limit state measurements and subsequent control actions, a penalty term is introduced in the quadratic functional to be minimized. In [15], a new event-driven model predictive control approach is proposed. The event-triggered formulation imposes an upper bound on the norm of a suitably defined state error. A serious drawback of all the aforementioned event-triggered techniques, both in the discrete-time and continuous-time domain, is the basic assumption of a measurable state vector. An available state vector allows the use of state-feedback control laws but is a very strict requirement, which is not assumed to hold in this paper. Recently, an output-based event-triggered control inspired by [13] has been proposed in [16]. Using an impulsive system approach, the authors show that stability and performance with respect to L1 -bounded norm disturbances can be obtained with larger minimum inter-event times than those already proposed in the literature, provided some LMI-based conditions are satisfied. However, the improvements are inversely proportional to the norm of the initial state of the plant and to the L1 norm of the disturbance. Given the preceding literature, the salient features of the proposed approach are as follows: (i) sampled data plants are considered, and an event-driven output-based control is proposed to reduce triggering both in feedback and feedforward paths. The minimum inter-event time is implicitly defined and is given by the sampling time Ts chosen for discretizing the plant; this also implies that any triggered event takes place only at times hTs , for some h 2 †ZC ; (ii) the new approach allows the presence of any external signal (reference and disturbance) generated as the free output response of autonomous unstable systems; (iii) no extra conditions are imposed besides those required for the solution of the classical tracking problem [17]. With respect to (i), it is worth mentioning that a discrete-time process originating by the sampling of a continuous-time plant corresponds to the most cases of a practical interest. Moreover, the sampled nature of the discrete-time process implies the presence of a holding device. Hence, even if triggering and update of the control effort are stopped by the controller CL (CCL), the continuoustime plant keeps to be forced by a continuous-time signal frozen on the last triggered value. This allows us to define a stabilizing controller with tracking performance and reduced communication cost both in the feedforward and feedback paths. In a purely discrete-time context, the holding device is lacking, and different control strategies have to be used. In [14], the plant exhibits an unforced output from time to time; in [15], the (possibly not updated ) control input is triggered at any time instant. These two latter references only considered the stabilization problem and do not attain a simultaneous reduction of network occupation both in the feedforward and feedback paths. Like most of the aforementioned references [2–4, 6–9, 13–16], the present contribution is focused on sporadic control; other topics such as network delay or packet dropouts fall outside the purpose of this paper. The problem is here formulated in terms of stabilization of the error system. To this purpose, the new proposed sensor CL (SCL) is based on the computation of a quadratic functional of the tracking error and of a corresponding time-varying threshold: a feedback network message from the sensor to the controller is triggered only if the functional exceeds or equals the current threshold, whose value is not based on heuristic considerations but is the result of exact computations that guarantee the internal stability of the feedback system. The CCL follows the SCL with a delay of

NEW EVENT-DRIVEN OUTPUT-BASED DISCRETE-TIME CONTROL

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one sampling period, namely, the CCL allows the dynamic output controller to send a feedforward message to the plant only if it has received a message from the sensor at the previous sampled time instant. This shift between the two CLs is due to the state equation of the controller. The dynamic output-feedback controller used here is based on [17]. This paper is organized as follows. The structure of the event-driven scheme is described in Section 2, Section 3 briefly recalls some preliminaries and formally states the sporadic control problem, and Section 4 defines the triggering rules (TRs) of the CL and details the resulting switched closed-loop dynamics. The proof of internal stability is given in Section 5. A numerical example taken from the literature is reported in Section 6. Concluding remarks and an appendix on some technicalities end the paper. 2. THE GENERAL CONTROL SCHEME For simplicity of notation, the explicit dependence of sampled signals on Ts will be omitted throughout the paper. The control scheme †l shown in Figure 1 is a hybrid autonomous system whose control loop elements are implemented at two intelligent nodes communicating through a common network. The controlled output of †l to be zeroed is the tracking error e.h/ D r.h/ y.h/, where r.h/ is the free output response of the known external reference generator †r .Ar , Cr /, starting from a given xr .0/ and y.h/ is the numerical output obtained by periodic sampling of y.t /, t D hTs , 8h 2 †ZC . As shown in Figure 1, †d is the sampled version of the series connection †p of †w .Aw , Cw / with †p .Cp , Ap , Bp / and is given by ² x.h Q C 1/ D Ad x.h/ Q C Bd u.h/ (1) †d W Q y.h/ D Cd x.h/ where x.h/ Q , Œx.h/T , xw .h/T T , xw .h/ 2 Z w , x.h/ 2 Z n , u.h/ 2 Z p , y.h/ 2 Z c , p > c. The autonomous unstable system †w generates the external disturbance w./ 2 IRd affecting †p as a free output response. The actual disturbance signal w./ is unknown (although Aw is known), both because it originates from an unknown initial internal state xw .0/ of †w and because it is assumed to be unmeasurable. The following is also assumed: (A1) the triplet .Cp , Ap , Bp / is reachable and observable; (A2) the extended plant †p is observable; (A3) the chosen sampling period Ts preserves the assumed structural properties of †p and †p ; and (A4) no transmission zero of †d coincides with an eigenvalue of Ar .

Figure 1. The logic control scheme †l .

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The block SCL contains the same algorithm for generating r.h/ with the purpose of computing e.h/ and a related quadratic functional (defined later). The SCL transmits the current e.h/ to †g only when the related quadratic functional equals or exceeds a time-varying threshold. At the same sampled instant hTs , †g transmits u.h/ to the plant according to the CCL, namely, only if at the previous sampled instant .h 1/Ts it has received e.h 1/. Hence, the CCL is implicitly defined by the SCL. N s , the quadratic functional of the tracking error is smaller than the current threshold, If for some hT N The effect of the CCL is to suspend the triggering of u.h/ and the then SCL does not trigger e.h/. continuous-time control input u.t / is kept constant by the zero-order hold (Z.O.H.) on the last comN t 2 ŒhT N s , .` C 1/Ts /, as long as the sensor does not transmit a puted frozen value u.t / D u.h/, N new value e.`/ for some `Ts > hTs . The value e.`/ is triggered as soon as the functional equals or exceeds the current threshold. Hence, the effect of the two CLs is to produce a switched †l operating in two different modalities: closed-loop configuration †c and open-loop configuration †o whose respective inter-event times N s. between two control updates result to be Ts and .` C 1 h/T Remark 1 It should be noted that endowing the SCL with the same algorithm for computing r.h/ is not a strict requirement because this information is known in advance. At most, in the case of a discontinuous external reference, it could be necessary to transmit from time to time a possible state resetting of †r if the discontinuity time instants are not programmed ‘a priori’. 2.1. The controller structure The controller †g is given by the connection of a suitable internal model system †m .Im , Q [17]. If †l were always kept Am , Bm / of both r.h/ and w.h/, with an LTI observer †ob of x.h/ frozen in the closed-loop configuration †c , †g would yield an internally asymptotically stable closed-loop configuration of †f , with asymptotic disturbance rejection and null tracking error over a sufficiently small compact set containing the nominal parameters of the plant [17]. Denoting by xm .h/ 2 IRm and .h/ D Œx .h/T , w .h/T T , the states of †m and †ob , respectively, the state space form of †g results to be 0 Am xm .h/ xm .h C 1/ Bm 0 e.h/ D C , (2) .h C 1/ .h/ L L r.h/ Bd Km Ad LCd Bd KN N u.h/ D ŒKm , K

xm .h/ .h/

,

(3)

T N D ŒK, 0.h/ D Kx .h/. Every with L D LT1 , LT2 , Am , and Bm are as specified in [17], K.h/ time e.h/ is triggered, the dynamic controller †g updates its internal state and the corresponding N N s , the SCL does not trigger e.h/ control action according to (2) and (3), respectively. If for some hT then †g updates its state assuming e.h/ D 0, while stops updating and triggering its output u.h/. Therefore, in the open-loop configuration, the equations defining the controller become 32 2 3 2 3 2 3 0 0 xm .h/ xm .h C 1/ Am 0 4 .h C 1/ 5 D 4 0 Ad LCd Bd 5 4 .h/ 5 C 4 L 5 r.h/, (4) 0 0 Ip u.h C 1/ u.h/ 0 2

3 xm .h/ u.h/ D Œ0, 0, Ip 4 .h/ 5 , u.h/

(5)

N .h/, N u.h/. N The control effort has been introduced in the state variables of the starting from Œxm .h/, open-loop controller for convenience in view of future manipulations (see Appendix A). .


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Remark 2 The assumption e.h/ D 0 in (4) is justified by the fact that if the quadratic functional of the tracking error is small enough, also e.h/ can be considered to be nearly zero. 3. SOME PRELIMINARIES AND THE PROBLEM STATEMENT Consider a generic unforced linear time varying (LTV) †s .Cs .h/, As .h// . Let xs .h/, ys .h/, and ˆs .h, k/, .h > k/ be the state, the free output response, and the state transition matrix of †s , respectively. The observability gramian Ws .h0 , h0 C hN / of †s is defined as Ws .h0 , h0 C hN / D Ph0 ChN 1 T ˆs .j , h0 /CsT .j /Cs .j /ˆs .j , h0 /. If †s is uniformly completely observable, there exist j Dh0 a positive integer hN and positive constants 1 , 2 , such that, 8h0 2 †ZC , Ws .h0 , h0 C hN / is bounded as 1 I 6 Ws .h0 , h0 C hN / 6 2 I .

(6)

Let the functional V .h, h / be defined as V .h, h / D

h X

ysT .k/ys .k/,

h > ,

(7)

kDh

it is easily seen that the definition of observability gramian and (7) imply V .h, h / D xs .h /T Ws .h , h/xs .h / whence, by (6), kxs .h /k2 1 6 V .h, h / 6 kxs .h /k2 ,

(8)

for some finite overbounding Ws .h , h/. Lemma 1 [18] The uniformly completely observable system †s is internally exponentially stable if and only if, for any initial state xs .0/ and 8N 2 .0, 1/, a sufficiently large integer N > hN can be found, such that, 8 > N , the functional V .h, h / given by (7) satisfies V .h, h / 6 N V .h , h 2 /,

8h > 2 .

(9)

As plays an important role in the definition of the CL, it is here briefly recalled how it can be computed. For more details, see the proof of the aforementioned lemma [18]. 1 1 Defining .h, h / , kWs .h , h/ 2 ˆs .h , h 2 /Ws .h 2 , h / 2 k2 , where > hN , h > 2 , some algebraic manipulation show that V .h, h / 6 .h, h /V .h , h 2 /. Exploiting the properties of Ws .h , h/ and taking into account that kˆs .h, 0/k 6 mh for some m > 1, 0 < < 1, and 8h 2 †ZC , it can be shown that .h, h / is decreasing as .h, h / 6 kWs .h , h C 1/kkWs .h 2 , h /1 km2 2 6 3 2 , for some 3 > 0. For any arbitrarily fixed N 2 .0, 1/, condition (9) holds if is chosen such that .h, h / 6 . N Definition 1 A dynamic unforced time-varying system †0s is said state-output stable if its free output response ys0 .h/ originating from any initial bounded state xs0 .0/ is uniformly bounded, namely, kys0 .h/k 6 My < 1, 8h 2 †ZC , and 8xs0 .0/, such that kxs0 .0/k 6 Mx < 1. If †0s is state-output stable and kys0 .h/k ! 0 for h ! 1, then †0s is said asymptotically state-output stable. 3.1. The problem statement Recalling that the controlled output of †l is the tracking error e.h/, the considered control problem is stated as follows. Problem statement: the sporadic control problem consists in defining a CL, acting in the control system of Figure 1 according to the scheme outlined in Section 2, with the purpose of attaining a sporadic feedback control action still guaranteeing the state-output stability of the switched unforced system †l and the internal stability of the switched system †f .

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4. THE EVENT-BASED COMMUNICATION LOGIC The event-driven CL solving the sporadic control problem will be defined in this section, exploiting Lemma 1. As it refers to observable systems, the following has to be intended as referred to the m observable realizations †m c and †l of †c and †l , respectively, (see Appendix A). 4.1. The Lyapunov-like functional V .h, h / A functional V .h, h / like (7) is defined with reference to the free response of the unforced system †m (which gives the same unforced output of †l ), l V .h, h / D

h X

e T .k/e.k/.

(10)

kDh

Lemma 1 is applied supposing that †l is always in the LTI closed-loop configuration †c , whose minimal realization †m c is internally stabilized by †g given by (2)–(3). Hence, the value of needed m to define V .h, h / is computed with reference to the LTI †m c †l . Lemma 1 implies that V .h, h / given in (10) would converge to zero for h ! 1 according to (9) if †l were always frozen in the LTI closed-loop configuration †c . Nevertheless, in the actual sporadic control, †l switches from †c to †o and vice versa. As long as †l operates in the closedloop configuration, V .h, h / obeys (9), but when the triggering of e.h/ is suspended by the SCL, †l operates in the open-loop configuration †o . As a consequence, V .h, h / stops obeying (9), and increasing jumps of its values could be observed. The proposed SCL is based on the idea of stopping the feedback triggering of e.h/ only if V .h, h / is smaller than a suitable threshold .h/, whose value has to be chosen so as to ensure that V .h, h / remains bounded for h ! 1. By the way V .h, h / is defined in (10), it is clear that this is enough to guarantee the state-output stability of the unforced switched system †l according to Definition 1. It will be shown later that this also implies the internal stability of †f . How to choose the time-varying threshold and how to define the TRs are reported in the following sections 4.2. The time-varying threshold (minimal realization of †l ). Let 1 and be Let Wlm ., / be the gramian of the time-varying †m l ., / the finite lowerbound and overbound of Wlm .h , h/ respectively. Moreover let xlm ./ and ˆm l , respectively. be the state and the state transition matrix of †m l The time-varying threshold .h/ is the piecewise constant function defined as V hQ i , hQ i 1 , h 2 Ti D ŒhQ i , hQ iC1 /, (11) .h/ D iC1 , 12 (12) with iC1 < i , m where 1 , max ˆl .h C 1, h/, hQ 0 D , and hQ i , i > 0 is the first time instant such that h

V hQ i 1, hQ i 1 < i , V hQ i , hQ i > i .

(13)

Remark 3 It will be shown in the theorem that choosing .h/ according to (11) surely guarantees the fulfillment of (12). 4.3. The triggering rules Recalling that e.h/ is the actual controlled output of †l and u.h/ is the actual output of †g , the variables e.h/ O and u.h/ O denote the actual inputs of †g and of zero-order hold (Z.O.H.), respectively.


The TRs can be briefly summarized as follows: 8 for h 2 Œ0, /, TR1 W 8e.h/ O D e.h/, u.h O C 1/ D u.h C 1/, ˆ ˆ < O D e.h/, if V .h, h / > iC1 then e.h/ < Q Q u.h O C 1/ D u.h C 1/, for h 2 Ti , Œhi , hiC1 /, i D 0, 1, .. TR2 W ˆ ˆ : : otherwise e.h/ O D 0, u.h O C 1/ D u.h/. O

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.a/ .b/

From (11) and TR2, it follows that the value of i , i > 1, is updated at each switching instant of †l from †o to †c . Given the aforementioned TRs, a detailed description of how the proposed event-based CL drives the switching of †l between †c and †o is reported as follows. 4.4. System dynamics under the CL (1) At time h D 0, †l starts operating in the ‘LTI’ closed-loop configuration. †c , u.h/, and e.h/ are triggered at each sampled time instant according to TR1. The SCL starts computing and storing e.h/, keeping the last computed 2 C 1 samples, for h > 2 . (2) At time h D , hQ 0 , TR1 is replaced by TR2. At the same instant, the SCL starts computing V .h, h /, 8h > . On the basis of the measured V . , 0/, it also computes the threshold 1 through (11). As long as V .h, h / > 1 , the closed-loop configuration †c is kept acting, according to TR2 (a). By Lemma 1, for h > 2 , V .h, h / is surely decreasing according to (9). Hence, there exists an h1 2 T0 D ŒhQ 0 , hQ 1 / (not necessarily greater than 2 ) such that V .h1 , h1 / < 1 . (3) At time h D h1 , †l starts to operate in the open-loop configuration †o ; hence, according to TR2 (b), the reachable part of †p , namely, †p starts being forced by the continuous-time control input u.t / frozen by the Z.O.H. on the last transmitted value u.h O 1 /. In general, this is not a stabilizing control law, so that V .h, h / stops obeying (9). As a consequence, for some h D hQ 1 > h1 , it may happen that conditions (13) hold. (4) At time h D hQ 1 , a new threshold 2 is computed according to (11). By (12), one has V .hQ 1 , hQ 1 / > 2 so that †l switches from †o to †c according to TR2 (a). The closed-loop configuration is maintained on the basis of the same decision logic used over Œ0, h1 /. Using the aforementioned CL, one has †l †c over the intervals I0 , Œ0, h1 /, Ii , ŒhQ i , hiC1 / Ti , i > 0, while †l †o over the intervals Ii0 D Œhi , hQ i / Ti1 , i > 0. 4.5. Performance of the proposed CL For any h > 0, the communication cost C.h/ of the aforementioned CL is C.h/ D K.h/= h, where K.h/ is defined as the total number of discrete-time instants belonging to the time intervals Ii wholly or partially occurred up to h. Hence, the reduction of communication cost R.h/ gained through the aforementioned CL with respect to periodic triggering up to any time instant h > 0 is given by R.h/ D 1 C.h/ D K 0 .h/= h, where K 0 .h/ is the total number of discrete-time instants belonging to the time intervals Ii0 wholly or partially occurred up to h. The index R.h/ is surely positive because, as evident by point 2, the CL implies the existence of time intervals Ii0 over which V .h, h / becomes smaller than the last computed threshold. However, the exact reduction of communication cost depends on the actual behavior of V .h, h /, which in turn depends on many elements that, in general, are a priori unknown, such as, for example, the initial states of †p or the possible presence of step changes in the disturbance dynamics. Hence, the numerical computation of R.h/ can be only performed ‘a posteriori’ by simulation or through real application. 5. STABILITY CONDITIONS The TRs define a nonlinear system †l switching between the two LTI configurations †o and †c . The nonlinearity originates from the dependence of the switching sequence on the initial internal state xl .0/ of †l . In fact, xl .0/ affects the values of V .h, h / and of the thresholds; as a consequence, it also affects the switching time instants hi and hQ i . Nevertheless, as for any fixed initial

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state xl .0/, also the corresponding switching sequence between the two LTI systems †o and †c is fixed; it follows that for any given xl .0/, the whole state trajectory xl .h/, h 2 †ZC , and the observcan be computed as in the usual LTV discrete-time case. As a ability gramian Wlm .h , h/ of †m l consequence, all the preliminaries of Section 3 can be exploited for proving the following Theorem. Theorem 1 The CL, described in Section 4, guarantees the state-output stability of the switched unforced system †l and the internal stability of the switched system †f provided the thresholds i are chosen according to (11). Moreover, the sequence ¹i º is monotonically decreasing as stated in (12). Proof of Theorem 1 The CCL is such that the stabilizing †g given by (2)–(3) restarts acting at each hQ i C 1, namely, one time instant after the triggering of e.h/ has been restarted by the SCL. By Lemma 1, †m V .h, h / surely restarts to decrease according to (9) for h > hQ i C 1 C 2 , because †m c . l Q Q Nevertheless, possible increasing jumps with respect to V .hi , hi / may occur over the interval Ki , ŒhQ i C1, hQ i C1C2 /. The worst case occurs when V .h, h / is increasing over all the interval Ki . This means that hQ i C 1 C 2 < hiC1 or equivalently Ki Ii . Denoting by Mi the overbound of V .h, h / in the worst case, it can be computed in the following way. By (8), for any initial state , one has for j D 0, , 2 xlm .0/ of †m l 2 V hQ i C j , hQ i C j 6 xlm hQ i C j 2 2 m Q Q i C j , hQ i 6 ˆm h h x i l l 6 22 V hQ i , hQ i =1 , Mi , i > 0, (14) where 2 D max ˆm hQ i C j , hQ i . l j D0,:::,2

Moreover, (8) also implies 2 2 V hQ i , hQ i 6 xlm hQ i 6 xlm hQ i 1 12 V hQ i 1, hQ i 1 6 12 , i > 0. 1

(15)

As V .hQ i 1, hQ i 1 / < i by (13), it readily follows that if i is chosen according to (11), then (16) V hQ i , hQ i < V hQ i1 , hQ i1 , i > 0. By (14), (16) implies Mi < Mi1 . Moreover, by (11), (16) implies (12). Hence, in the worst case, , the maximum value assumed by V .h, h /, for one has that for any initial state xlm .0/ of †m l C h 2 Ki , i 2 †Z , is upper bounded by the monotonically decreasing sequence ¹Mi º, and for h 2 ŒhQ i C 1 C 2 , hiC1 /, V .h, h / is decreasing according to (9). Besides, over each Ii0 , i 2 †ZC , V .h, h / is upper bounded by the monotonically decreasing sequence ¹i º. Two different asymptotic behaviors of †l are possible: (i) an infinite switching of †l between †c and †o takes place; and (ii) the switching stops after a finite time hiN , for some iN > 0. In the first case, V .h, h / is converging to zero for h ! 1, and exact asymptotic tracking is follows by its attained, because e.h/ ! 0 for h ! 1. The internal asymptotic stability of †m l minimality. By Definition 1, it also follows that †l is asymptotically state-output stable and that †f is externally stable. In the second case, one has V .h, h / < iN , 8h > hiN , but no asymptotic convergence to zero of e.h/ is guaranteed. Condition (10) only assures that ke.h/k2 < iN =. C 1/ 8h > hiN . In this case, the as well as the state-output stability of †l and the external stability of simple internal stability of †m l †f are guaranteed.


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The internal stability of the switched system †f is proved, showing that for any uniformly bounded external reference and disturbance signals, also x.h/, xm .h/, .h/, and u.h/ are uniformly bounded for h ! 1 in both the possible asymptotic behaviors of †l . Suppose at first that the switching stops after a finite time hiN . For any h > hiN , †l is permanently frozen in the open-loop configuration †o , and the controller †g is evolving according to (4)–(5). As Am provides an internal model of uniformly bounded signals, it follows that also xm .h/ is uniformly bounded. The uniform boundedness of x.h/ Q follows by the assumed observability of †d and by the boundedness of y.h/, consequence of the already proved external stability of †f . Let e.h/ Q D x.h/ Q .h/ be the state observation error; recalling (4), easy calculation shows that Q 1/ Le.h/. e.h/ Q D .Ad LCd /e.h

(17)

The uniform boundedness of e.h/ Q follows by the stability of Ad LCd (guaranteed by †g ) and by the uniform boundedness of the tracking error. As a consequence, also .h/ is uniformly bounded, as well as u.h/ by (3). In the case of infinite switching, one has that e.h/ is converging asymptotically to zero. The Q can be proved exactly as before. The uniform bounduniform boundedness of xm .h/ and x.h/ edness of .h/ is a direct consequence of (17) and (2), which imply that in the case of infinite Q 1/ M.h/e.h/, where M.h/ D 0, 8h 2 Ii , i D 0, , switching, one has e.h/ Q D .Ad LCd /e.h Q The uniform and M.h/ D L, 8h 2 Ii0 . As e.h/ ! 0, for h ! 1, by (17), one has .h/ ! x.h/. boundedness of u.h/ follows once again by (3). The following corollary allows the definition of a simplified CL (SICL). Corollary 1 Assume to stop the updating of thresholds i according to (11) for some i , and let i be the corresponding value. Accordingly, assume to modify the CL defined by TR1 and TR2 comparing V .h, h / with the fixed value i , 8h > hQ i 1 . The SICL defined in this way still guarantees the state-output stability of †l and the internal stability of †f . Proof of Corollary 1 By (15), 8` > 0, one has V hQ i C` 1, hQ i C` 1 2 2 12 < i C` 1 D i 1 , V hQ i C` , hQ i C` 6 1 1 1 so that, by (11) written for i D i , it follows that V hQ i C` , hQ i C` < V hQ i 1 , hQ i 1 .

(18)

(19)

Moreover, (14) and (18) directly imply V hQ i 1 , hQ i 1 22 Mi C` < D Mi 1 . (20) 1 Hence, the SICL yields sequences of V hQ i , hQ i and Mi , i 2 †ZC that are uniformly bounded. Therefore, by arguing as in the proof of Theorem 1, it directly follows that the state-output stability of †l and the internal stability of †f are still guaranteed. Remark 4 The SICL carries a significant advantage: it avoids the possibility of numerical and truncation errors arising when comparing monotonically decreasing quantities .i and V .hQ i1 , hQ i1 // that, in the case of very long switching sequences, could approach the zero machine. However, the SICL does not assure the fulfillment of the monotonically decreasing condition (16), so that, even in the case of

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infinite switching of †l between †c and †o , an exact asymptotic tracking is not guaranteed. In any case, an arbitrarily accurate tracking can be obtained 8h > h1 provided to slightly modify the SICL as follows. More precisely, for any arbitrarily small e > 0, the following condition can be satisfied ke.h/k 6 e,

8h > h1 ,

(21)

provided to stop the threshold updating at i D 1 and not to choose 1 according to .11/ but as explained in the following. Comparing V .h, h / with the unique threshold 1 , over the intervals Œhi , hQ i / and ŒhQ i , hiC1 /, i > 1, one has V .h, h / < 1 , 8h 2 Œhi , hQ i /, i > 1,

V .h, h / 6 M1 , 8h 2 ŒhQ i , hiC1 / i > 1,

(22)

where M1 is the overbound of V .h, h /, 8h > hQ 1 and is given by (14) putting i D 1. Hence, recalling the definition (10) of V .h, h /, fulfillment of (21) requires V .h, h / 6 M1 6 e 2 . C 1/,

8h > h1 .

(23)

As V hQ 1 1, hQ 1 1 < 1 , (14) and (15) imply M1 6

1 22 12 2 , 12

(24)

so that (21) is satisfied if 1 6

e 1 2 . C 1/12 . 22 2 12

Remark 5 In many operating conditions, the periodically sampled output y.h/ can be required to track a reference r.h/ with sharp discontinuities at some isolated points h0i (e.g., a piecewise constant signal or a linear ramp with piecewise constant slope). This is equivalent to a state resetting of †r , which has to be communicated to the algorithm generating r.h/ in the SCL block. This situation can be easily managed by considering each h0i point as a new time origin and applying the aforementioned CCL over each interval Œh0i , h0iC1 /, provided it is so long to guarantee the practical attainment of steady state. 6. NUMERICAL SIMULATIONS For a detailed description of the numerical example, it is useful to briefly recall the continuous-time state-space representations of the disturbance generator †w and of the extended plant †p given by the series connection of †w with †p : 8 ² P Ap M Cw Bp x.t/ < xQP p .t/ , x.t/ D C u.t/, xP w .t/ D Aw xw .t/, 0 Aw 0 xP w .t/ xw .t/ †p W †w W w.t/ D Cw xw .t/ : y.t/ D ŒCp , 0xQ p .t/.

(25) Example 1 This example concerns the same SISO plant †p considered in [16]. See this reference for the explicit expression of the triplet .Cp , Ap , Bp /, with c D 1, n D 2, p D 1. Unlike [16], here also an external constant disturbance w.t / of amplitude 10 premultiplied by M D Œ1, 1T is assumed to affect the state of †p . The reference signal r.h/ consists of the piecewise constant signal (dotted line) shown

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in Figure 2. Choosing Ts D 0.01 s, the state-space representation of the discretized extended plant †d is given by the following triplet .Cd , Ad , Bd /:

Moreover, †d is observable with no transmission zero in ´ D 1 (eigenvalue of Ar ). The matrices defining the controller equations (2)–(3) are as follows: Am D 1, Bm D 1, Km D 1.6336, K D Œ94.9474, 16.4476 T D Œ0.7748 310.1679 j 656.4194T . L D LT1 LT2 30 r(h) y(h)

25 20 15 10 5 0 −5 −10 −15

0

500

1000

1500

2000

2500

3000

3500

4000

h

Figure 2. Behaviors of r.h/ (dotted line) and y.h/ (continuous line), respectively. 30

a

20 10 0 −10

0

500

1.000

1500

2.000

3.500

4.000

h 20

b

10 0 −10 −20 2.000

2.500

3.000

h

Figure 3. Split of the behavior of y.h/ reported in Figure 2, 0 6 h < 2000, (diagram a), 2000 6 h 6 4000, (diagram b). Red points refer to closed-loop configuration †c ; blue dashes refer to open-loop configuration †o .

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m m h Denoting by ˆm c ., / the state transition matrix of †c , one has kˆc .h, 0/k 6 9851 .0.9/ , whence following the procedure reported after Lemma 1 with N D 0.98, it is found that the functional V .h, h /, corresponding to †m c , is converging to zero according to (9) with D 123. By definition of observability gramian, it is numerically found that the scalars 1 and satisfying 1 6 Wlm .h, h C / 6 are 1 D 5, D 1.4278 107 . Moreover, it is also numerically found 1 D 1.4696 103 . The simulation has been performed over 4000 samples, starting from the initial Q D Œx.0/T , xw .0/T T D Œ1, 1, 10T , .0/ D Œ0, 0, 0T . conditions: xm .0/ D 0, xr .0/ D 10, x.0/ The plant †p considered in this example is unstable, and the control u.t / effort could not be kept constant by the Z.O.H. over a too long time interval. As a consequence, the CL would give a frequent switching of †l between †c and †o , and the inconvenience mentioned in Remark 4 could occur. For this reason, the SICL has been applied, in this case stopping the computation of the timevarying thresholds to the first value 1 over each interval Œh0i , h0iC1 /. The graphics of y.h/ is reported in Figure 2, (continuous line). The behavior of y.h/ in closed-loop (red points) and open-loop (blue dashes) configurations is reported in Figure 3. Tables I and II summarize the most relevant data of the SICL relative to the simulation with reference to the two intervals Œh00 , h01 / D Œ0, 2000/ and Œh01 , 4000 D Œ2000, 4000, respectively.

Table I. Relevant data of the CL for the considered example over the interval Œh00 , h01 / D Œ0, 2000/. Œh00 , h01 / D Œ0, 2000/ V .h00 C , h00 / D V .123, 0/ D 4.669 103 I10

I0 Œh00 , h1 / D Œ0, 297/ †l †c

1 D 7.572 1010 I20

I1

I2 Q Q Q Q Œh1 , h1 / D Œ297, 1186/ Œh1 , h2 / D Œ1186, 1337/ Œh2 , h2 / D Œ1337, 1914/ Œh2 , 2000/ †l †o †l †c †l †o †l †c V .h, h / < 1 , 8h 2 I10 V .h, h / < 1 , 8h 2 I20

Table II. Relevant data of the CL for the considered example over the interval Œh00 , h01 / D Œ0, 2000/. Œh01 , 4000 D Œ2000, 4000 V .h01 C , h01 / D V .2123, 2000/ D 7.5267 103

1 D 1.2204 109

I10 I1 I20 I2 Q Q Q Q h1 / D Œ2000, 2294/ Œh1 , h1 / D Œ2294, 3212/ Œh1 , h2 / D Œ3212, 3365/ Œh2 , h2 / D Œ3365, 3942/ Œh2 , 4000 †l †c †l †o †l †c †l †o †l †c 0 0 V .h, h / < 1 , 8h 2 I1 V .h, h / < 1 , 8h 2 I2 I0

Œh01 ,

1 R(h)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

500

1000

1500

2000

2500

3000

3500

4000

h

Figure 4. Reduction of communication cost R.h/.


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The performance of the proposed SICL has been evaluated in terms of reduction of commu PI.h/ nication cost R.h/ and performance cost P C defined as P C D max r.h/ with PI.h/ D h

y.h/ y .h/, where y.h/ and y .h/ represent the numerical output of † d with reduced and 100% −5

1

x 10

PI(h)/r(h)

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

500

1000

1500

2000

2500

3000

3500

4000

h

Figure 5. Behavior of PI.h/=r.h/. 20 x (h) 1

x (h)

10

2

0 −10 −20 −30 −40 −50 0

100

200

300

500

400

h

(a) 1 R(h)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

500

1000

1500

2000

2500

3000

3500

4000

h

(b) Figure 6. Example 2: (a) Transient behavior of state variables and (b) reduction of the communication cost R.h/.

872


communication respectively. The behavior of R.h/ reported in Figure 4 shows that even inside each sub-interval, where no discontinuity of r.h/ occurs, there are several instants at which †l switches between †c and †o , although a unique threshold 1 has been computed over each sub-interval. This is a consequence of the instability of †p . As for the performance cost, one has P C D 9.0461106 . The behavior of PI.h/=r.h/ is shown in Figure 5. Figures 4 and 5 and the two indexes show an insignificant loss of control performance but an effective reduction of triggered network messages greater than the 60%. A further simulation has been also performed in the same exact conditions assumed in [16], where onlypthe stabilization problem is considered starting from the initial state x.0/ D Œ5 p .2/, 5 .2/T and with no external disturbance affecting the plant. For brevity, only the behavior of the plant state variables over the first 500 samples and R.h/ are reported in Figure 6. By comparing the obtained transient behavior with that reported in [16], the superiority of the present approach is apparent, both in terms of reduced oscillations and reduced settling time, which is 1 s for the SICL against about 13 s for the method proposed in [16]. The SICL also gives a greater reduction of the communication cost: the minimum inter-event time is clearly given by the sampling period Ts D 0.01 s, whereas the maximum inter-event time is equal to 4.57 s. For [16], the minimum inter-event time is 0.0001 s, and the maximum inter-event time is 0.1 s. The first part of the numerical example also shows that almost exact asymptotic tracking is obtained even in the case of large amplitude disturbances affecting the plant. 7. CONCLUSIONS This paper has proposed a new event-driven CL whose purpose is to reduce the number of triggered messages along the feedforward and feedback paths of a networked closed-loop control system. The tracking of an external reference for MIMO sampled plants with non-accessible state affected by unknown disturbances has been considered, and an internally stabilizing CL has been defined by means of a Lyapunov-like function of the tracking error, which has to be compared with a decreasing sequence of thresholds. In particular, Theorem 1 shows that internal stability can always be attained without any further condition than those required in the classical tracking problem, provided the thresholds are properly computed. The controller has been given in discrete-time form. This makes it possible the digital implementation of the control algorithm and automatically defines a minimum inter-event time given by the chosen sampling period. This avoids the possible conservatism resulting from some continuoustime methods dealing with this aspect of the problem. The numerical simulations reveal the effectiveness of the method both in terms of tracking performance and of reduced number of triggered events. It is believed the formal analysis derived in this paper will greatly enhance the applicability of event-triggered control, making it possible to deal with all those practical tracking problems with disturbance rejection, where only output-feedback controllers can be used. APPENDIX A .h, k/, h > k of This section shows how the state xlm .h/ and the state transition matrix ˆm l are defined. For brevity, the treatment is limited to the case where † is only required to †m m l provide the internal model of constant signals (as in the numerical example). Some formal variants would allow the extension to any kind of signal generated as the unforced output response of an LTI system. can be obtained, canceling As †l is an unforced system, any minimal (observable) realization †m l those subsystems with internal states indistinguishable from the origin (unobservable in the sequel for brevity). To prove the results stated in this appendix, it is necessary to make reference to the state space representation (25) of the continuous-time series connection †p of †w with †p . Classical

873


discretization techniques of †p give the sampled plant †d .Cd , Ad , Bd /, where Cd D ŒCp , 0, A1,1 A1,2 Ad D , Bd D ŒB1T , 0T T with A1,1 D exp .Ap Ts /, A2,2 D exp .Aw Ts /, 0 A2,2 RT RT A1,2 D 0 s exp.Ap .Ts //M Cw exp .Aw /d , B1 D 0 s exp.Ap /Bp d . By †r .Ar , Cr / and (2)–(3), recalling that the controlled output of †l is e.h/ D r.h/ y.h/ and adopting the usual state variable transformation, the state space representation of †l in the closed-loop form is xlc .h C 1/ D Ac xlc .h/,

e.h/ D Cc xlc .h/,

(A.1)

where xlc .h/ , Œx.h/T , xm .h/T , .x.h/ x .h//T , .xw .h/ w .h//T , xw .h/T , xr .h/T T , and

(A.2) The subsystems with internal unobservable states are those corresponding to xw .h/ and xr .h/, because the constant unforced output responses originating from any non-null xw .0/ and xr .0/ can be compensated by the unforced output response originating by a properly chosen internal state xm .0/ of the internal model †m . Hence, the minimal realization of (A.1) results to be 2 6 6 m xlc .h C 1/ , 6 4

x.h C 1/ xm .h C 1/ x.h C 1/ x .h C 1/ xw .h C 1/ w .h C 1/

3 7 7 m 7 D Am c xlc .h/, 5

m y.h/ D Ccm xlc .h/,

(A.3)

with

(A.4) By (4), the state xlo .h/ of †l in the open-loop configuration is given by xlo .h/ Œxlc .h/T , u.h/T T , and arguing as before, we obtain the following state space form xlo .h C 1/ D Ao xlo .h/, e.h/ D Co xlo .h/,

,

(A.5)

874


where

(A.6) Also in the open-loop configuration the subsystems with internal unobservable states are those corresponding to xw .h/ and xr .h/, because the constant unforced output responses originating from any non-null xw .0/ and xr .0/ can be compensated by the unforced output response originating from N Moreover, recalling that in the open-loop configuration .h/ is not transmitted to † , u.h/ D u.h/. d the minimal realization of (A.5) results to be m .h C 1/ , xlo

x.h C 1/ u.h C 1/

m D Am o xlo .h/,

m y.h/ D C0m xlo .h/,

(A.7)

where Am o D

A1,1 0

B1 Ip

,

Com D ŒCp , 0.

(A.8)

The aforementioned calculations show that the minimal realization †m of †l is characterized l by a state vector xlm .h/ with time-varying dimensions. Hence, at the transition points hi (closed loop!open loop) and hQ i (open loop!closed loop), the dynamical matrix assumes a rectangular m m .h/ and xlo .h/ are defined, at points hi , one has form. By (3) and the way xlc m m xlo .hi C 1/ D Am c,o xlc .hi /,

m y.hi / D Ccm xlc .hi /,

(A.9)

with Am c,o

D

A1,1 B1 K K

B1 Km Km

B1 K K

0 0

.

(A.10)

m When the inverse transition occurs at points hQ i , it is seen that xlc .h/ as defined in (A.3) cannot m be expressed as a function of the xlo .h/ defined in (A.7); hence, it is necessary to redefine m m .h/ D xlc .h/, for h D hQ i , obtaining xlo m Q m Q xlc .hi C 1/ D Am o,c xlo .hi /,

m Q y.hQ i / D Ccm xlo .hi /,

(A.11)

m m with Am o,c D Ac . The aforementioned considerations show that the state of †l is a vector with m m .h, k/, two possible configurations: xlo .h/ and xlc .h/, and the relative state-transition matrix ˆm l m h > k, is given by a suitable product of the aforementioned defined dynamical matrices Am c , Ao , m Ac,o ordered according to the number of transition points hi and hQ i contained in the interval .k, h.


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