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Event–Triggered Observation of Nonlinear Lipschitz Systems via Impulsive Observers
10th IFAC Symposium on Nonlinear Control Systems 10th IFAC Symposium on Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA 10th IFAC Symposium on Control Systems 10th IFAC Symposium on Nonlinear Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA Available August 23-25, 2016. Monterey, California, USA August 23-25, 2016. Monterey, California, USA online at www.sciencedirect.com
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of of of of
Event–Triggered Observation Event–Triggered Observation Event–Triggered Observation Event–Triggered Observation Nonlinear Lipschitz Systems via Nonlinear Lipschitz Systems Nonlinear Lipschitz Systems via Nonlinear Lipschitz Systems via via Impulsive Observers Impulsive Observers Impulsive Observers Impulsive Observers ∗,∗∗ ∗,∗∗
Lucien Etienne ∗,∗∗ , and Stefano Di Gennaro ∗,∗∗ ∗,∗∗ Lucien Etienne ∗,∗∗ , and Stefano Di Gennaro Lucien Lucien Etienne Etienne ∗,∗∗ ,, and and Stefano Stefano Di Di Gennaro Gennaro ∗,∗∗ ∗ ∗ Department of Information Engineering, Computer Science ∗ Department of Information Engineering, Computer Science ∗ Department Information Engineering, Computer and of Mathematics, of L’Aquila Department of InformationUniversity Engineering, Computer Science Science and Mathematics, University of L’Aquila and Mathematics, University of L’Aquila Via Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. and Mathematics, University of L’Aquila Via Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. ∗∗Via Via Vetoio, Vetoio, Loc. Loc. Coppito, Coppito, 67100 67100 – – L’Aquila, L’Aquila, Italy. Italy. ∗∗ Center of Excellence DEWS, University of L’Aquila ∗∗ Center of Excellence DEWS, University of L’Aquila ∗∗ Via Center of Excellence DEWS, University of L’Aquila Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. Center of Excellence DEWS, University of L’Aquila Via Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. Via Vetoio, Loc. Coppito, 67100 – L’Aquila, E.mail: [email protected], [email protected]. Via Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. Italy. E.mail: [email protected], [email protected]. E.mail: E.mail: [email protected], [email protected], [email protected]. [email protected]. Abstract: In this paper we investigate the observation problems for a class of nonlinear Abstract: In thissubject paper investigate the observation for aa class of nonlinear Abstract: In paper we investigate the problems for of nonlinear Lipschitz systems, towe network constraints. In order toproblems address this problem, an impulsive Abstract: In this thissubject paper to wenetwork investigate the observation observation problems forproblem, a class class an of impulsive nonlinear Lipschitz systems, constraints. In order to address this Lipschitz systems, subject to network network constraints. In order order to to address this this problem, an impulsive impulsive observer is designed, making use of constraints. the event–triggered technique in order to diminish the Lipschitz systems, subject to In address problem, an observer is designed, making use of the event–triggered technique in order to diminish the observer is designed, making use of the event–triggered technique in order to diminish the network usage. The proposed observer ensures practical state estimation. The output sampling observer is designed, making use of the event–triggered technique in order to diminish the network usage. The proposed practical state estimation. sampling network usage. The observer ensures practical estimation. The output sampling is done on a periodic basis, observer but the ensures data transmission is regulated byThe an output event–triggering network usage. The proposed proposed observer ensures practical state state estimation. The output sampling is done on a periodic basis, but the data transmission is regulated by an event–triggering is done on a periodic basis, but the data transmission is regulated by an event–triggering mechanism. The performance of the observer is tested in simulations of a flexible joint. is done on a periodic basis, but the data transmission is regulated by an event–triggering mechanism. The The performance performance of of the the observer is is tested tested in in simulations simulations of of aa flexible flexible joint. joint. mechanism. mechanism. performance of theofobserver observer is Control) tested inHosting simulations of a Ltd. flexible joint.reserved. © 2016, IFACThe (International Federation Automatic by Elsevier All rights Keywords: Impulsive observer, Event triggered sampling, Nonlinear systems. Keywords: Keywords: Impulsive Impulsive observer, observer, Event Event triggered triggered sampling, sampling, Nonlinear Nonlinear systems. systems. Keywords: Impulsive observer, Event triggered sampling, Nonlinear systems. 1. INTRODUCTION make the sampling time dependent on the output value, 1. make the dependent on value, 1. INTRODUCTION INTRODUCTION make the sampling sampling time dependent on the the output output value, one possibility is thetime use of the event–triggered policy. 1. INTRODUCTION make the sampling time dependent on the output value, one possibility is the use of the event–triggered policy. one possibility is the use of the event–triggered policy. one possibility is sampling the use ofhas thebeen event–triggered policy. in Event–triggered already considered sampling has been already considered in State observation, or estimation, is one of the central prob- Event–triggered Event–triggered sampling has been already considered in other contexts, e.g. those inhas which some specific tasks have Event–triggered sampling been already considered in State observation, or estimation, is one of the central probother contexts, e.g. those in which some specific tasks have State observation, or estimation, is one of the central problem in system theory. Since the seminal work of Luenberger other contexts, e.g. those in which some specific tasks have to be completed before sampling variable. More recently, State observation, or estimation, is one work of the central prob- other contexts, e.g. those in whichaa some specific tasks have lem in theory. Since the Luenberger be before More lem in system system theory.have Since the seminal seminal work of ofboth Luenberger (1966), many works treated this problem, for lin- to to be completed completed before sampling sampling a variable. variable. More recently, recently, driven by the necessity of reducing the bandwidth used in lem in system theory. Since the seminal work of Luenberger to be completed before sampling a variable. More recently, (1966), works have this both for by necessity of the used (1966), many workssystems. have treated treated this problem, problem, both for linlin- driven ear andmany nonlinear Recently, impulsive observers driven by the thechannel, necessity of reducing reducing the bandwidth bandwidth used in in the wireless such an event–triggering mechanism (1966), many works have treated this problem, both for lindriven by the necessity of reducing the bandwidth used in ear and nonlinear systems. Recently, impulsive observers the wireless channel, such an event–triggering mechanism ear and nonlinear systems. Recently, impulsive observers for continuous–time systems have been proposed (Raff, the wireless channel, such an event–triggering mechanism has been reconsidered to reduce the amount of data ear and nonlinear systems. Recently, impulsive observers the wireless channel, such an event–triggering mechanism for continuous–time systems been proposed been to reduce of for continuous–time systems have been proposed (Raff, (Raff, 2007; Chen, 2011, 2013) and,have using discontinuous Lya- has has been reconsidered reconsidered tothe reduce the amount of data data transmitted. In fact, while outputthe canamount be continuously for continuous–time systems have been proposed (Raff, has been reconsidered to reduce the amount of data 2007; Chen, 2011, 2013) and, using discontinuous Lyatransmitted. In fact, while the output can be continuously 2007; Chen, 2011,sufficient 2013) and, and, using for discontinuous Lyapunov Chen, functions, conditions the existence of transmitted. transmitted. In fact, while the output can be continuously or, rather, periodically or aperiodically sampled, the data 2007; 2011, 2013) using discontinuous LyaIn fact, while the output can be continuously punov functions, sufficient conditions for existence of rather, periodically or aperiodically sampled, the data punov functions, sufficient conditions for the the existence of or, these observers have been derived in terms of linear matrix or, rather, periodically or aperiodically sampled, the data transmission is decided by an event–triggering mechanism. punov functions, sufficient conditions for the existence of or, rather, periodically or aperiodically sampled, the data these observers have derived in of is event–triggering mechanism. these observers have been been derived in terms terms of linear linear matrix matrix inequalities (LMIs). Further recent developments in the transmission transmission is decided decided by by an an event–triggering mechanism. Moreover, event–triggered techniques allow the execution these observers have been derived in terms of linear matrix transmission is decided by an event–triggering mechanism. inequalities (LMIs). Further recent developments in the the Moreover, event–triggered techniques allow the execution inequalities (LMIs).observers Further can recent developments in design of impulsive be find in Khaled (2013) Moreover, event–triggered techniques allow so theminimizing execution of computation tasks as rarely as possible, inequalities (LMIs). Further recent developments in the Moreover, event–triggered techniques allow the execution design of impulsive observers can be find in Khaled (2013) of computation tasks as rarely as possible, so minimizing design of impulsive observers can be find in Khaled (2013) for linear systems and in Andrieu (2010), Guillen (2013), of computation tasks as rarely as possible, so minimizing the energy consumption and leaving the digital processor design of impulsive observers can be find in Khaled (2013) of computation tasks as rarely as possible, so minimizing for linear systems and energy consumption and leaving the digital processor for linear systems and in in Andrieu Andrieu (2010), Guillen Guillen (2013), (2013), the Dinh (2015) for nonlinear systems.(2010), the energy consumption and leaving the digital processor available for other tasks. for linear systems and in Andrieu (2010), Guillen (2013), the energy consumption and leaving the digital processor Dinh available Dinh (2015) (2015) for for nonlinear nonlinear systems. systems. available for for other other tasks. tasks. Dinh systems. for other In this(2015) work for thisnonlinear kind of observers is used to estimate the available In this paper this tasks. mechanism is considered as key techIn this this of is to the this paper this is as key In thisofwork work this kind kindLipschitz of observers observers is used used to estimate estimate the In state a nonlinear system, where the sensor In this along paper with this mechanism mechanism is considered considered as design key techtechnique, the impulsive observer to an In this work this kind of observers is used to estimate the In this paper this mechanism is considered as key techstate of a nonlinear Lipschitz system, where the sensor nique, along with the impulsive observer to design an state of aacommunicated nonlinear Lipschitz Lipschitz system,channel. where the the sensor nique, data are via a wireless Regarding along with the impulsive observer to design an observer ensuring practical state estimation. The outstate of nonlinear system, where sensor nique, along with practical the impulsive observer to design an data are via channel. Regarding ensuring state outdata are communicated communicated via a a wireless wireless channel. Regarding observer this latter aspect, the theoretic assumption of continuous observer ensuring practical state estimation. estimation. The output sampling is done on a periodic basis, but The the data data are communicated via a wireless channel. Regarding observer ensuring practical state estimation. The outthis latter aspect, the theoretic of sampling done basis, data this latter aspect, theverified theoretic assumption of continuous continuous output sensing is not in assumption practice. In fact, with the put put sampling isis is regulated done on on aaabyperiodic periodic basis, but but the theevent– data transmission the aforementioned this latter aspect, the theoretic assumption of continuous put sampling is done on periodic basis, but the data output sensing is not verified in practice. In fact, with the transmission is regulated by the aforementioned event– output sensing is not verified in practice. In fact, with the use of digital devices, the output is sampled at with discrete transmission is regulated by the aforementioned event– triggering mechanism. This mechanism is usually called output sensing is not verified in practice. In fact, the transmission is regulated bymechanism the aforementioned event– use digital the output sampled at mechanism. is called use ofinstants. digital devices, devices, the sampling output is iscan sampled at discrete discrete timeof The output be performed on triggering triggering mechanism. This This mechanism is usually usuallyensures called periodic event–triggered sampling. The periodicity use of digital devices, the output is sampled at discrete triggering mechanism. This mechanism is usually called time instants. The output sampling can be performed on periodic event–triggered sampling. The periodicity ensures time instants. The output sampling can be performed on the basis of a periodic/aperiodic sampling, orperformed on an event– periodic event–triggered sampling. The periodicity ensures implicitly the existence of a minimum iter–event time, time instants. The output sampling can be on periodic event–triggered sampling. The periodicity ensures the basis aa periodic/aperiodic on the existence of aa between minimum iter–event time, the basis of of periodic/aperiodic sampling, orapproach on an an event– event– triggered basis. The advantage ofsampling, the classicor with implicitly implicitly the existence of minimum iter–event time, i.e. a nonzero time interval two transmission the basis of a periodic/aperiodic sampling, or on an event– implicitly the existence of a between minimumtwo iter–event time, triggered basis. of the with aa nonzero transmission triggered basis. The Theis advantage advantage of closed–loop the classic classic approach approach with periodic sampling to allow the system to be i.e. i.e. nonzero time interval betweenthe twoimpact transmission events. In this time work,interval in particular, of the triggered basis. The advantage of the classic approach with i.e. a nonzero time interval between two transmission periodic sampling is to allow the closed–loop system to be In work, the of periodic sampling is to toofallow allow the closed–loop closed–loop system to be be events. analyzedsampling on the basis sampled–data formalism (Astrom events. In this this technique work, in in particular, particular, the ofimpact impact of the the event–triggered on the design an impulsive periodic is the system to events. In this work, in particular, the impact of the analyzed on the basis of sampled–data formalism (Astrom event–triggered technique on the design of an impulsive analyzed on the basis of sampled–data formalism (Astrom (1997) for linear systems, and Karafyllis (2009), Postoyan event–triggered technique on the design of an impulsive observer is analyzed. Someonstudies are available on this analyzed on the basis of sampled–data formalism (Astrom event–triggered technique the design of an impulsive (1997) systems, and (2009), analyzed. studies are available this (1997) for linear systems, and Karafyllis (2009),isPostoyan Postoyan (2012)).for Onlinear the other hand, theKarafyllis main drawback the fact observer observer issubject analyzed. Some studies are Lehmann available on on this particularis (seeSome Donkers (2010), (2011), (1997) for linear systems, and Karafyllis (2009), Postoyan observer is analyzed. Some studies are available on this (2012)). On the other hand, the main drawback is the fact particular subject (see Donkers (2010), Lehmann (2011), (2012)). On the other hand, the main drawback is the fact that the sampling instant does not depend on the output particular subject (see Donkers (2010), Lehmann (2011), Tallapradaga (2012), Tallapradaga (2013) and the(2011), refer(2012)). On the other hand, the main drawback is the fact particular subject (seeTallapradaga Donkers (2010), Lehmann that sampling does depend on and referthat the sampling instant does not not dependalso on the the output value,the and thereforeinstant the output is sampled if itsoutput value Tallapradaga Tallapradaga (2012), Tallapradaga (2013) and the the references therein).(2012), The vast majority of (2013) the existing literature that the sampling instant does not depend on the output Tallapradaga (2012), Tallapradaga (2013) and the refervalue, and output is also its value therein). The majority existing literature value, and therefore therefore the output is sampled sampled also ifthis its sense, value ences is not varied sensiblythe since the last sampling. In if ences therein). The vast vast majority of the existingthe literature is dedicated to linear systems. On of thethe contrary, field of value, and therefore the output is sampled also if its value ences therein). The vast majority of the existing literature is not varied sensibly since the last sampling. In this sense, is dedicated to linear systems. On the contrary, the field of is not varied sensibly since the last sampling. In this sense, the classic sampling is done in “open–loop”. In order to is dedicated to linear systems. On the contrary, the field of is not variedsampling sensibly since theinlast sampling. InInthis sense, is dedicated to linear systems. On the contrary, the field of the classic is done “open–loop”. order to the classic sampling is done in “open–loop”. In order to the classic sampling is done in “open–loop”. In order to Copyright © 2016, 2016 IFAC 678Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 678 Copyright © 2016 IFAC 678 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 678Control. 10.1016/j.ifacol.2016.10.242
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observation of nonlinear system with event triggered communications has been subject of little investigation (Etienne (2015), Abdelrahim (2015)). The event–triggered sampling technique has been developed in recent years (Arzen, 1999), as substitute of the time–periodic sampling, see for instance Heemels (2012) for an introduction to the topic and, for further details, Astrom (2003), Heemels (2008), Lunze (2010), Tabuada (2007), Wang (2011). In the case of the event–triggering the sampling is performed in order to ensure that some desired properties established in the continuous–time design, primarily the asymptotic convergence to zero of the observation error, can be maintained under sampling. Conceptually, this means to introduce a feedback in the sampling process, with a constant monitoring of the output to determine if the desired properties is ensured. The structure of the article is the following. In Section 2, the observation problem is formulated and the structure of the impulsive observer is given. In Section 3, this problem is solved for a class of Lipschitz nonlinear systems, in terms of LMIs. In Section 4, this impulsive observer is applied to a one–link manipulator with flexible joint. Finally, in Section 5, some conclusions are drawn.
h→0,h>0
Remark 2.1. For the sake of simplicity we considered a global Lipschitz constant and Rn as the domain of definition of the system. However, under appropriate technical assumptions, the results exposed here remain valid for compact sets. Output sensors sample the outputs periodically, but the data are transmitted to the observer for elaboration at time instants that are a subset of the sampling instants. More precisely, while Ss = {tk = kδ}k∈N is the periodic sampling sequence for the sensor, with δ = tk+1 − tk > 0 the sensor sampling period, the data transmission sequence is {ts }∈N ⊆ Ss . The situation is depicted in Fig. 1. The data communications between the plant and the observer, assumed instantaneous, take place only at the discrete time instants ts .
ts1 ts2
ts3 ts4
ts5 kδ
δ
u(t)
System
y(t)
Sensor
t ℓs
y(t ℓs ) Observer
Data transmission
Notations: x , M denote the Euclidean norms when applied to a vector x and to a matrix M , while x P = xT P x is the norm induced by a symmetric positive definite P matrix P . Moreover, λP min , λmax are the smallest and the biggest eigenvalues of a square matrix P . Furthermore, R + , R+ 0 will denote the set of positive real numbers and the set of positive real numbers including zero, and N = {0, 1, 2, · · · } the set of natural numbers including zero. With I we denote the identity matrix. Finally, t+ = lim (t + h), and t+ lim (tk + h), k ∈ N. When k =
667
x ˆ (t)
h→0,h>0
not necessary, the dependence on time t will be omitted.
Fig. 1. Sensor sampling and transmission time instants. 2. PROBLEM STATEMENT The class of systems under study is characterized by the following equations x˙ = Ax + Bu + Dφ(Hx) (1) y = Cx with x ∈ Rn the state, u ∈ Rm the input, y ∈ Rq the output, A ∈ Rn×n , B ∈ Rm×n , C ∈ Rq×n , D ∈ Rn×ν , H ∈ Rp×n are constant matrices. In the following one assumes that the pair (A, C) is observable. Note that milder results can be easily expressed in terms of detectability. In (1), Dφ(Hx) gives the structure of the nonlinearity acting on the system, with φ : Rp → Rν a nonlinear function satisfying the following condition φ(Hχ1 ) − φ(Hχ2 ) ≤ γ H(χ1 − χ2 ) (2) ∀ (χ1 , χ2 ) ∈ Rn × Rn . It is clear that (2) implies that (1) is Lipschitz with respect to x. 679
The observation problem consists of determining an event– triggering condition, fixing the time instants in which the sampled output data are sent to the observer, and an impulsive state observer having the structure x ˆ˙ = Aˆ x + Bu + Dφ(H x ˆ) ˆ(kδ) + δG y(ts ) − yˆ(kδ) x ˆ(kδ + ) = x (3) = (I − δGC)ˆ x(kδ) + δGCx(ts )
such that x ˆ(t) tends asymptotically to x(t) in a practical ˆ(t), and G ∈ Rn×q sense. In (3), x ˆ(kδ + ) is the left limit of x is the observer gain matrix. Note that the right limit is x ˆ(kδ − ) = x ˆ(kδ). The observer dynamics correspond to a copy of the system dynamics between sampling instants kδ, (k + 1)δ, while it undergoes a jump in the state at the sampling instants. 3. AN EVENT–TRIGGERED IMPULSIVE OBSERVER The impulsive observer (3) will be implemented making use of a triggering mechanism, determining when the
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sensor transmits the sampled data to the plant. More precisely, these data are sent to the plant at the time instants t = ts , ∈ N, such that the following event– triggering condition is satisfied (4) ts +1 = min kδ > ts | y(kδ) − y(ts ) ≥ εs k
where εs > 0 is a threshold value on the output error y(kδ) − y(ts ).
Given the system and the observer dynamics (1), (3), one has to consider both the continuous dynamics of the observation error e(t) = x(t) − x ˆ(t), given by e˙ = Ae + D φ(Hx) − φ(H x ˆ) (5)
and the error discrete dynamics, due to the impulses on the observer state. These latter have the expression e(kδ + ) = x(kδ + ) − x ˆ(kδ + )
= (I − δGC)e(kδ) + δGC x(kδ) − x(ts )
(6)
since x(kδ + ) = x(kδ − ) = x(kδ). It is worth noting that, at the triggering instants ts in which (4) is satisfied and the system output sensor sends the sampled data to the observer, this expression reduces to e(ts + ) = e(kδ + ) = (I − δGC)e(kδ) = (I − δGC)e(ts )
since x(kδ) = x(ts ). In all the other discrete time instants, the term δGC x(kδ)−x(ts ) = δG y(kδ)−y(ts ) appears, which can be seen as a perturbation induced by the absence of communications from the sensor. As stated by the following result, the event–triggering condition (4) along with an appropriate choice of the observer gain ensure the exponential practical stability of the observation error (Lakshmikantham, 1990). It is worth noting that this is true also when the system dynamics (1) are not stable. Theorem 3.1. Let us consider the system (1), with (A, C) observable and φ satisfying (2). If, for a fixed sampling time δ > 0, the following LMIs where N1 =
N1 ≤ −εI,
N2 ≤ −εI,
T 2 T P1 A + A P1 + γ H H +
N3 ≤ 0 P2 − P1 δ
P1 D
P2 − P1 P A + A P2 + γ H H + N2 = 2 δ P2 D P1 − δP3 C −P2 N3 = (P1 − δP3 C)T −P1 T
2
T
(7) P1 D −I
Remark 3.2. It is worth noting that in (7) the sampling period δ is fixed a priori. Therefore, (7) constitute a set of classic LMIs. Proof. The error dynamics (5), (6) can be rewritten as follows e˙ = Ae + Dd2 (9) + e(kδ ) = (I − δGC)e(kδ) + δGCd3
with d2 = φ(Hx) − φ(H x ˆ) and d3 = x(kδ) − x(ts ). Following Suykens (1998), Raff (2007), let us consider the Lyapunov candidate Vo (e, t) = e 2P (t) , where P (t) is a time–varying matrix given by the following convex combination of the matrices P1 , P2 t − kδ (P2 − P1 ) = λP1 + (1 − λ)P2 P (t) = P1 + δ (10) (k + 1)δ − t ∈ [0, 1) λ= δ defined for t ∈ (kδ, (k + 1)δ]. This combination allows constructing an appropriate Lyapunov function which, under the hypotheses (7), can show the practical exponential stability. For, first note that P (kδ + ) = P1 and P ((k + 1)δ) = P2 . Considering that P (t) is periodic with period δ, its definition can be extended for all t ≥ 0. Note also that λmin e 2 ≤ Vo ≤ λmax e 2 . Using (9), and P2 − P1 P˙ (t) = δ obtained from (10), one works out P 2 − P1 e V˙ o = eT P A + AT P + δ + eT P Dd2 + dT2 DT P e + dT2 d2 − dT2 d2 ∀ t ∈ kδ, (k + 1)δ . Using (2)
d2 2 ≤ γ 2 eT H T He and considering the definition of P (t) one gets P 2 − P1 T T 2 T ˙ +γ H H e PA + A P + Vo ≤ e δ + eT P Dd2 + dT2 DT P e − dT2 d2 ¯ξ = ξ T N1 ξ + (t − kδ)ξ T N
P2 D −I
P2 1 where ε¯ = ε/λmax , and λmin = min λP min , λmin , λmax = P2 1 max λP max , λmax .
(k + 1)δ − t T t − kδ T ξ N1 ξ + ξ N2 ξ δ δ T T ¯ = (N2 − for t ∈ (kδ, (k +1)δ], where ξ = e dT2 and N N1 )/δ. Hence, using (7), one finally obtains (k + 1)δ − t t − kδ ξ 2 − ε ξ 2 = −ε ξ 2 V˙ o ≤ −ε δ δ ≤ −ε e 2 ≤ −¯ εVo i.e. V˙ o is bounded by a negative definite function for all t ∈ (kδ, (k + 1)δ]. Therefore, =
have solutions P1 , P2 , P3 , with Pi = PiT > 0, i = 1, 2, for an ε > 0, then the observer (3), with the event–triggering condition (4), and the gain G = P1−1 P3 , ensures that the origin of error dynamics (5), (6) is globally practically exponentially stable, with attractive set λmax δ G Iεs = e ≤ ρs , ρs = εs (8) λmin 1 − e−δε¯/2 680
Vo (t) = e−¯ε(t−t0 ) Vo (t0 ) ∀ t0 , t ∈ (kδ, (k + 1)δ], t0 ≤ t
(11)
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and, in particular, Vo (k + 1)δ ≤ e−δε¯Vo (kδ + ). Note also that in the inter–sampling λmax −¯ ε(t−t0 )/2 e(t0 ) e(t) = e λmin ∀ t0 , t ∈ (kδ, (k + 1)δ], t0 ≤ t.
(13)
and
ζ = GCd3 = G y(kδ) − y(ts )
It is well–known that for Hermitian matrices, −N3 ≥ 0 is equivalent to P2 > 0 and S2 ≥ 0 (see Zhang (2005), Theorem 1.12, pag. 34). Hence, since −N3 ≥ 0 and P2 > 0, then S2 ≥ 0. Therefore, −eT (kδ)S2 e(kδ) ≤ 0. Furthermore, note that eT (kδ)(I − δGC)T P1 (I − δGC)e(kδ) = eT (kδ)(P1 − δP3 C)T P1−1 (P1 − δP3 C)e(kδ)
= eT (kδ)(P2 − S2 )e(kδ) ≤ eT (kδ)P2 e(kδ) = Vo (kδ). This last observation allows writing ζ T (P1 − δP3 C)e(kδ) = ζ T P1 (I − δGC)e(kδ) ≤ ζ T P1 ζ eT (kδ)(I − δGC)T P1 (I − δGC)e(kδ) 1 ≤ λP λmax G εs Vo (kδ) max ζ Vo (kδ) ≤
k→∞
c
λmax
δ G εs 1−e 1 − e−δε¯/2 where the series sum exists since a < 1. Hence, lim sup e(kδ + ) ≤ ρs . Considering that (13) ensures that −δ ε¯/2
=
k→∞
e(t) decreases exponentially between kδ + and (k + 1)δ, one can conclude that the error dynamics (5), (6) converges exponentially to the attractive set Iεs . The convergence is global since all the passages do not depend on the initial state. This concludes the proof.
Corollary 3.3. Let us consider the system (1), with (A, C) observable and φ satisfying (2). Under the same hypotheses and notations of Theorem 3.1, the observer (3), with the event–triggering condition (4), and the gain G = P1−1 P3 , ensures that the origin of error dynamics (5), (6) globally practically converge to the set Iεb = e ≤ (1 + εb )ρs
ρs given by (8), in a time 2 d − (1 + d)e−δε¯/2 1 T ≤ T,εb = ln , d= ε¯ εb δ G εs for any fixed εb , > 0, and for any observer initial condition such that e(0) ≤ . Proof. Since 2 2 2 Vo (0) = e(0) 2P2 ≤ λP max ≤ λmax one has Vo (0+ ) ≤ Vo (0) + c ≤ λmax + c and, from (14), k−1 λmin e(kδ + ) ≤ Vo (kδ + ) ≤ a Vo (0+ ) + c ai
where (4) has been used. Finally, ζ T P1 ζ = G y(kδ) − y(ts ) 2P1
2 s 2 1 ≤ λP max G y(kδ) − y(t )
k
2 2 2 2 1 ≤ λP max G εs ≤ λmax G εs .
λmax G εs Vo (kδ) 2 Vo (kδ) + c + δ 2 λmax G 2 ε2s = 2 ≤ a Vo ((k − 1)δ + ) + c √ a = e−δε¯/2 , c = λmax δ G εs , where (12) has been used, so that Vo (kδ + ) ≤ a Vo ((k − 1)δ + ) + c. This linear discrete–time dynamics is exponentially stable to the origin since the dynamic matrix is Schur. Moreover, its solution is given by Vo (kδ + ) ≤ Vo (kδ) + 2δ
k→∞
c 1−a
It is worth noting that the convergence ensured by Theorem 3.1 to Iεs is asymptotic. If one requires a finite–time convergence one needs to enlarge Iεs . This is stated in the following result.
S2 = P2 − (P1 − δP3 C)T P1−1 (P1 − δP3 C) the Schur complement of the element (2,2) of the matrix −N3 .
Therefore,
and for k →∞ lim sup λmin e(kδ + ) ≤ lim sup Vo (kδ + ) ≤ =
+ 2δζ T (P1 − δP3 C)e(kδ) + δ 2 ζ T P1 ζ
G = P1−1 P3 ,
(14)
j=1
∆Vo := Vo (kδ + ) − Vo (kδ) = −eT (kδ)S2 e(kδ)
k−1 aj Vo (kδ + ) ≤ a Vo (0+ ) + c
(12)
Let us now analyze the stability of the discrete error dynamics, i.e. the error dynamics in the discontinuity, considering the same Lyapunov candidate Vo (t) = e(t) 2P (t) and recalling that P (kδ + ) = P1 , P (kδ) = P ((k + 1)δ) = P2 as already noted. Using (9), one gets
with
669
681
i=0
1 − ak λmax + c + c 1−a √ δ G εs . Dividing by λmin and
≤a √ a = e−δε¯/2 , c = λmax imposing c λmax + k + + (1 − ak )ρs e(kδ ) ≤ a λmin λmin 1 c = ρs = (1 + εb )ρs , 1 − a λmin one gets the bound of kδ for the time T in which the error trajectory enters Iεb . Remark 3.4. It is worth noting that in the proposed scheme it is not necessary that the system is stable in order to ensure the observer convergence. However, in practice numerical error may influence the observer performance.
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4. SIMULATION RESULTS FOR A ROBOT WITH A FLEXIBLE JOINT
10
In this section a fourth–order model of the form (1) is considered, taken from Spong (1987), Rajamani (1998), Howell (2002), Raff (2007) and representing the dynamics of a one–link manipulator, with a DC motor as actuator T 0 1 0 0 B = 0 21.6 0 0 −48.6 −1.25 48.6 0 , A= 1 0 0 0 0 0 0 1 C= 0 1 0 0 19.5 0 −19.5 0 T D= 0 0 0 1 H= 0 0 1 0 ,
with φ = 3.3 sin x3 , which is a nonlinear term due to the gravity, acting on the dynamics of x4 . As commented in Spong (1987), the joint elasticity is described by a linear torsional spring, and x1 represents the rotation of the motor, x2 = x˙ 1 is the corresponding angular velocity, x3 is the angular position of the link, x4 = x˙ 3 is its angular velocity. Physically, one measures the motor position and velocity, while the measurement of the other variables is non–trivial. The input u = sin t is applied to the system. The performance of the impulsive observer (3) is analyzed making use of this benchmark. Solving (7), with a sensor sampling period of δ = 0.15 s, one worksout 28.6269 −0.3675 −11.6820 1.0816 −0.3675 0.5296 −1.5176 0.0455 P1 = −11.6820 −1.5176 18.8721 −1.7201 1.0816 0.0455 −1.7201 0.7153 22.9349 −0.5313 −11.6047 1.0195 −0.5313 0.4136 −1.9257 0.0810 P2 = −11.6047 −1.9257 20.9671 −1.8715 1.0195 0.0810 −1.8715 0.7393 9.3334 1.0001 −48.7804 22.3665 G= . −0.0524 3.3194 19.4066 −0.3167 where λmin = 0.06, λmax = 37, ε = 1. Then, in order to impose a ball of convergence for the observation error of ρs = 0.012, from (8) one obtains εs 3.64 × 10−7 . This theoretical value is highly conservative and, as a matter of fact, it is possible to fix less restrictive values. In fact, as shown by the simulation in Figs. 2, 3 obtained with δ = 0.05 s and εs = 10−3 , one can fix well bigger event–triggering thresholds still obtaining the desired balls of convergence of dimensions ρs . As expected, at the beginning the communications are more frequent, and become less frequent when the steady–state is reached after about 1.75 s, with an average sampling time of δav = 0.31 s.
8
6
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0
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0.01 −6
0 −0.01
−8
−0.02 −10 0
1
2
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5
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6
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6.5
5
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Fig. 2. Observation error e(t), with a zoom showing e ≤ ρs . 1
0.9
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0
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Fig. 3. Sensor transmission instants ts . The triggering parameter for the sensor can be computed in order the ensure an upper bound on the size of the attractive set. The simulation results show that this upper bound is not tight, and further work is necessary to obviate this over–approximation. The observer–based stabilization problem will be the subject of a forthcoming paper. ACKNOWLEDGMENTS The authors wish to thank Jean–Pierre Barbot for the fruitful discussions on impulsive observers. REFERENCES
5. CONCLUSIONS In this work an impulsive observer was presented for a Lipschitz nonlinear system. The event–triggering mechanism is periodic. In the proposed scheme, for the observer convergence it is not necessary that the system is stable. 682
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of of of of
Event–Triggered Observation Event–Triggered Observation Event–Triggered Observation Event–Triggered Observation Nonlinear Lipschitz Systems via Nonlinear Lipschitz Systems Nonlinear Lipschitz Systems via Nonlinear Lipschitz Systems via via Impulsive Observers Impulsive Observers Impulsive Observers Impulsive Observers ∗,∗∗ ∗,∗∗
Lucien Etienne ∗,∗∗ , and Stefano Di Gennaro ∗,∗∗ ∗,∗∗ Lucien Etienne ∗,∗∗ , and Stefano Di Gennaro Lucien Lucien Etienne Etienne ∗,∗∗ ,, and and Stefano Stefano Di Di Gennaro Gennaro ∗,∗∗ ∗ ∗ Department of Information Engineering, Computer Science ∗ Department of Information Engineering, Computer Science ∗ Department Information Engineering, Computer and of Mathematics, of L’Aquila Department of InformationUniversity Engineering, Computer Science Science and Mathematics, University of L’Aquila and Mathematics, University of L’Aquila Via Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. and Mathematics, University of L’Aquila Via Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. ∗∗Via Via Vetoio, Vetoio, Loc. Loc. Coppito, Coppito, 67100 67100 – – L’Aquila, L’Aquila, Italy. Italy. ∗∗ Center of Excellence DEWS, University of L’Aquila ∗∗ Center of Excellence DEWS, University of L’Aquila ∗∗ Via Center of Excellence DEWS, University of L’Aquila Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. Center of Excellence DEWS, University of L’Aquila Via Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. Via Vetoio, Loc. Coppito, 67100 – L’Aquila, E.mail: [email protected], [email protected]. Via Vetoio, Loc. Coppito, 67100 – L’Aquila, Italy. Italy. E.mail: [email protected], [email protected]. E.mail: E.mail: [email protected], [email protected], [email protected]. [email protected]. Abstract: In this paper we investigate the observation problems for a class of nonlinear Abstract: In thissubject paper investigate the observation for aa class of nonlinear Abstract: In paper we investigate the problems for of nonlinear Lipschitz systems, towe network constraints. In order toproblems address this problem, an impulsive Abstract: In this thissubject paper to wenetwork investigate the observation observation problems forproblem, a class class an of impulsive nonlinear Lipschitz systems, constraints. In order to address this Lipschitz systems, subject to network network constraints. In order order to to address this this problem, an impulsive impulsive observer is designed, making use of constraints. the event–triggered technique in order to diminish the Lipschitz systems, subject to In address problem, an observer is designed, making use of the event–triggered technique in order to diminish the observer is designed, making use of the event–triggered technique in order to diminish the network usage. The proposed observer ensures practical state estimation. The output sampling observer is designed, making use of the event–triggered technique in order to diminish the network usage. The proposed practical state estimation. sampling network usage. The observer ensures practical estimation. The output sampling is done on a periodic basis, observer but the ensures data transmission is regulated byThe an output event–triggering network usage. The proposed proposed observer ensures practical state state estimation. The output sampling is done on a periodic basis, but the data transmission is regulated by an event–triggering is done on a periodic basis, but the data transmission is regulated by an event–triggering mechanism. The performance of the observer is tested in simulations of a flexible joint. is done on a periodic basis, but the data transmission is regulated by an event–triggering mechanism. The The performance performance of of the the observer is is tested tested in in simulations simulations of of aa flexible flexible joint. joint. mechanism. mechanism. performance of theofobserver observer is Control) tested inHosting simulations of a Ltd. flexible joint.reserved. © 2016, IFACThe (International Federation Automatic by Elsevier All rights Keywords: Impulsive observer, Event triggered sampling, Nonlinear systems. Keywords: Keywords: Impulsive Impulsive observer, observer, Event Event triggered triggered sampling, sampling, Nonlinear Nonlinear systems. systems. Keywords: Impulsive observer, Event triggered sampling, Nonlinear systems. 1. INTRODUCTION make the sampling time dependent on the output value, 1. make the dependent on value, 1. INTRODUCTION INTRODUCTION make the sampling sampling time dependent on the the output output value, one possibility is thetime use of the event–triggered policy. 1. INTRODUCTION make the sampling time dependent on the output value, one possibility is the use of the event–triggered policy. one possibility is the use of the event–triggered policy. one possibility is sampling the use ofhas thebeen event–triggered policy. in Event–triggered already considered sampling has been already considered in State observation, or estimation, is one of the central prob- Event–triggered Event–triggered sampling has been already considered in other contexts, e.g. those inhas which some specific tasks have Event–triggered sampling been already considered in State observation, or estimation, is one of the central probother contexts, e.g. those in which some specific tasks have State observation, or estimation, is one of the central problem in system theory. Since the seminal work of Luenberger other contexts, e.g. those in which some specific tasks have to be completed before sampling variable. More recently, State observation, or estimation, is one work of the central prob- other contexts, e.g. those in whichaa some specific tasks have lem in theory. Since the Luenberger be before More lem in system system theory.have Since the seminal seminal work of ofboth Luenberger (1966), many works treated this problem, for lin- to to be completed completed before sampling sampling a variable. variable. More recently, recently, driven by the necessity of reducing the bandwidth used in lem in system theory. Since the seminal work of Luenberger to be completed before sampling a variable. More recently, (1966), works have this both for by necessity of the used (1966), many workssystems. have treated treated this problem, problem, both for linlin- driven ear andmany nonlinear Recently, impulsive observers driven by the thechannel, necessity of reducing reducing the bandwidth bandwidth used in in the wireless such an event–triggering mechanism (1966), many works have treated this problem, both for lindriven by the necessity of reducing the bandwidth used in ear and nonlinear systems. Recently, impulsive observers the wireless channel, such an event–triggering mechanism ear and nonlinear systems. Recently, impulsive observers for continuous–time systems have been proposed (Raff, the wireless channel, such an event–triggering mechanism has been reconsidered to reduce the amount of data ear and nonlinear systems. Recently, impulsive observers the wireless channel, such an event–triggering mechanism for continuous–time systems been proposed been to reduce of for continuous–time systems have been proposed (Raff, (Raff, 2007; Chen, 2011, 2013) and,have using discontinuous Lya- has has been reconsidered reconsidered tothe reduce the amount of data data transmitted. In fact, while outputthe canamount be continuously for continuous–time systems have been proposed (Raff, has been reconsidered to reduce the amount of data 2007; Chen, 2011, 2013) and, using discontinuous Lyatransmitted. In fact, while the output can be continuously 2007; Chen, 2011,sufficient 2013) and, and, using for discontinuous Lyapunov Chen, functions, conditions the existence of transmitted. transmitted. In fact, while the output can be continuously or, rather, periodically or aperiodically sampled, the data 2007; 2011, 2013) using discontinuous LyaIn fact, while the output can be continuously punov functions, sufficient conditions for existence of rather, periodically or aperiodically sampled, the data punov functions, sufficient conditions for the the existence of or, these observers have been derived in terms of linear matrix or, rather, periodically or aperiodically sampled, the data transmission is decided by an event–triggering mechanism. punov functions, sufficient conditions for the existence of or, rather, periodically or aperiodically sampled, the data these observers have derived in of is event–triggering mechanism. these observers have been been derived in terms terms of linear linear matrix matrix inequalities (LMIs). Further recent developments in the transmission transmission is decided decided by by an an event–triggering mechanism. Moreover, event–triggered techniques allow the execution these observers have been derived in terms of linear matrix transmission is decided by an event–triggering mechanism. inequalities (LMIs). Further recent developments in the the Moreover, event–triggered techniques allow the execution inequalities (LMIs).observers Further can recent developments in design of impulsive be find in Khaled (2013) Moreover, event–triggered techniques allow so theminimizing execution of computation tasks as rarely as possible, inequalities (LMIs). Further recent developments in the Moreover, event–triggered techniques allow the execution design of impulsive observers can be find in Khaled (2013) of computation tasks as rarely as possible, so minimizing design of impulsive observers can be find in Khaled (2013) for linear systems and in Andrieu (2010), Guillen (2013), of computation tasks as rarely as possible, so minimizing the energy consumption and leaving the digital processor design of impulsive observers can be find in Khaled (2013) of computation tasks as rarely as possible, so minimizing for linear systems and energy consumption and leaving the digital processor for linear systems and in in Andrieu Andrieu (2010), Guillen Guillen (2013), (2013), the Dinh (2015) for nonlinear systems.(2010), the energy consumption and leaving the digital processor available for other tasks. for linear systems and in Andrieu (2010), Guillen (2013), the energy consumption and leaving the digital processor Dinh available Dinh (2015) (2015) for for nonlinear nonlinear systems. systems. available for for other other tasks. tasks. Dinh systems. for other In this(2015) work for thisnonlinear kind of observers is used to estimate the available In this paper this tasks. mechanism is considered as key techIn this this of is to the this paper this is as key In thisofwork work this kind kindLipschitz of observers observers is used used to estimate estimate the In state a nonlinear system, where the sensor In this along paper with this mechanism mechanism is considered considered as design key techtechnique, the impulsive observer to an In this work this kind of observers is used to estimate the In this paper this mechanism is considered as key techstate of a nonlinear Lipschitz system, where the sensor nique, along with the impulsive observer to design an state of aacommunicated nonlinear Lipschitz Lipschitz system,channel. where the the sensor nique, data are via a wireless Regarding along with the impulsive observer to design an observer ensuring practical state estimation. The outstate of nonlinear system, where sensor nique, along with practical the impulsive observer to design an data are via channel. Regarding ensuring state outdata are communicated communicated via a a wireless wireless channel. Regarding observer this latter aspect, the theoretic assumption of continuous observer ensuring practical state estimation. estimation. The output sampling is done on a periodic basis, but The the data data are communicated via a wireless channel. Regarding observer ensuring practical state estimation. The outthis latter aspect, the theoretic of sampling done basis, data this latter aspect, theverified theoretic assumption of continuous continuous output sensing is not in assumption practice. In fact, with the put put sampling isis is regulated done on on aaabyperiodic periodic basis, but but the theevent– data transmission the aforementioned this latter aspect, the theoretic assumption of continuous put sampling is done on periodic basis, but the data output sensing is not verified in practice. In fact, with the transmission is regulated by the aforementioned event– output sensing is not verified in practice. In fact, with the use of digital devices, the output is sampled at with discrete transmission is regulated by the aforementioned event– triggering mechanism. This mechanism is usually called output sensing is not verified in practice. In fact, the transmission is regulated bymechanism the aforementioned event– use digital the output sampled at mechanism. is called use ofinstants. digital devices, devices, the sampling output is iscan sampled at discrete discrete timeof The output be performed on triggering triggering mechanism. This This mechanism is usually usuallyensures called periodic event–triggered sampling. The periodicity use of digital devices, the output is sampled at discrete triggering mechanism. This mechanism is usually called time instants. The output sampling can be performed on periodic event–triggered sampling. The periodicity ensures time instants. The output sampling can be performed on the basis of a periodic/aperiodic sampling, orperformed on an event– periodic event–triggered sampling. The periodicity ensures implicitly the existence of a minimum iter–event time, time instants. The output sampling can be on periodic event–triggered sampling. The periodicity ensures the basis aa periodic/aperiodic on the existence of aa between minimum iter–event time, the basis of of periodic/aperiodic sampling, orapproach on an an event– event– triggered basis. The advantage ofsampling, the classicor with implicitly implicitly the existence of minimum iter–event time, i.e. a nonzero time interval two transmission the basis of a periodic/aperiodic sampling, or on an event– implicitly the existence of a between minimumtwo iter–event time, triggered basis. of the with aa nonzero transmission triggered basis. The Theis advantage advantage of closed–loop the classic classic approach approach with periodic sampling to allow the system to be i.e. i.e. nonzero time interval betweenthe twoimpact transmission events. In this time work,interval in particular, of the triggered basis. The advantage of the classic approach with i.e. a nonzero time interval between two transmission periodic sampling is to allow the closed–loop system to be In work, the of periodic sampling is to toofallow allow the closed–loop closed–loop system to be be events. analyzedsampling on the basis sampled–data formalism (Astrom events. In this this technique work, in in particular, particular, the ofimpact impact of the the event–triggered on the design an impulsive periodic is the system to events. In this work, in particular, the impact of the analyzed on the basis of sampled–data formalism (Astrom event–triggered technique on the design of an impulsive analyzed on the basis of sampled–data formalism (Astrom (1997) for linear systems, and Karafyllis (2009), Postoyan event–triggered technique on the design of an impulsive observer is analyzed. Someonstudies are available on this analyzed on the basis of sampled–data formalism (Astrom event–triggered technique the design of an impulsive (1997) systems, and (2009), analyzed. studies are available this (1997) for linear systems, and Karafyllis (2009),isPostoyan Postoyan (2012)).for Onlinear the other hand, theKarafyllis main drawback the fact observer observer issubject analyzed. Some studies are Lehmann available on on this particularis (seeSome Donkers (2010), (2011), (1997) for linear systems, and Karafyllis (2009), Postoyan observer is analyzed. Some studies are available on this (2012)). On the other hand, the main drawback is the fact particular subject (see Donkers (2010), Lehmann (2011), (2012)). On the other hand, the main drawback is the fact that the sampling instant does not depend on the output particular subject (see Donkers (2010), Lehmann (2011), Tallapradaga (2012), Tallapradaga (2013) and the(2011), refer(2012)). On the other hand, the main drawback is the fact particular subject (seeTallapradaga Donkers (2010), Lehmann that sampling does depend on and referthat the sampling instant does not not dependalso on the the output value,the and thereforeinstant the output is sampled if itsoutput value Tallapradaga Tallapradaga (2012), Tallapradaga (2013) and the the references therein).(2012), The vast majority of (2013) the existing literature that the sampling instant does not depend on the output Tallapradaga (2012), Tallapradaga (2013) and the refervalue, and output is also its value therein). The majority existing literature value, and therefore therefore the output is sampled sampled also ifthis its sense, value ences is not varied sensiblythe since the last sampling. In if ences therein). The vast vast majority of the existingthe literature is dedicated to linear systems. On of thethe contrary, field of value, and therefore the output is sampled also if its value ences therein). The vast majority of the existing literature is not varied sensibly since the last sampling. In this sense, is dedicated to linear systems. On the contrary, the field of is not varied sensibly since the last sampling. In this sense, the classic sampling is done in “open–loop”. In order to is dedicated to linear systems. On the contrary, the field of is not variedsampling sensibly since theinlast sampling. InInthis sense, is dedicated to linear systems. On the contrary, the field of the classic is done “open–loop”. order to the classic sampling is done in “open–loop”. In order to the classic sampling is done in “open–loop”. In order to Copyright © 2016, 2016 IFAC 678Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 678 Copyright © 2016 IFAC 678 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 678Control. 10.1016/j.ifacol.2016.10.242
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observation of nonlinear system with event triggered communications has been subject of little investigation (Etienne (2015), Abdelrahim (2015)). The event–triggered sampling technique has been developed in recent years (Arzen, 1999), as substitute of the time–periodic sampling, see for instance Heemels (2012) for an introduction to the topic and, for further details, Astrom (2003), Heemels (2008), Lunze (2010), Tabuada (2007), Wang (2011). In the case of the event–triggering the sampling is performed in order to ensure that some desired properties established in the continuous–time design, primarily the asymptotic convergence to zero of the observation error, can be maintained under sampling. Conceptually, this means to introduce a feedback in the sampling process, with a constant monitoring of the output to determine if the desired properties is ensured. The structure of the article is the following. In Section 2, the observation problem is formulated and the structure of the impulsive observer is given. In Section 3, this problem is solved for a class of Lipschitz nonlinear systems, in terms of LMIs. In Section 4, this impulsive observer is applied to a one–link manipulator with flexible joint. Finally, in Section 5, some conclusions are drawn.
h→0,h>0
Remark 2.1. For the sake of simplicity we considered a global Lipschitz constant and Rn as the domain of definition of the system. However, under appropriate technical assumptions, the results exposed here remain valid for compact sets. Output sensors sample the outputs periodically, but the data are transmitted to the observer for elaboration at time instants that are a subset of the sampling instants. More precisely, while Ss = {tk = kδ}k∈N is the periodic sampling sequence for the sensor, with δ = tk+1 − tk > 0 the sensor sampling period, the data transmission sequence is {ts }∈N ⊆ Ss . The situation is depicted in Fig. 1. The data communications between the plant and the observer, assumed instantaneous, take place only at the discrete time instants ts .
ts1 ts2
ts3 ts4
ts5 kδ
δ
u(t)
System
y(t)
Sensor
t ℓs
y(t ℓs ) Observer
Data transmission
Notations: x , M denote the Euclidean norms when applied to a vector x and to a matrix M , while x P = xT P x is the norm induced by a symmetric positive definite P matrix P . Moreover, λP min , λmax are the smallest and the biggest eigenvalues of a square matrix P . Furthermore, R + , R+ 0 will denote the set of positive real numbers and the set of positive real numbers including zero, and N = {0, 1, 2, · · · } the set of natural numbers including zero. With I we denote the identity matrix. Finally, t+ = lim (t + h), and t+ lim (tk + h), k ∈ N. When k =
667
x ˆ (t)
h→0,h>0
not necessary, the dependence on time t will be omitted.
Fig. 1. Sensor sampling and transmission time instants. 2. PROBLEM STATEMENT The class of systems under study is characterized by the following equations x˙ = Ax + Bu + Dφ(Hx) (1) y = Cx with x ∈ Rn the state, u ∈ Rm the input, y ∈ Rq the output, A ∈ Rn×n , B ∈ Rm×n , C ∈ Rq×n , D ∈ Rn×ν , H ∈ Rp×n are constant matrices. In the following one assumes that the pair (A, C) is observable. Note that milder results can be easily expressed in terms of detectability. In (1), Dφ(Hx) gives the structure of the nonlinearity acting on the system, with φ : Rp → Rν a nonlinear function satisfying the following condition φ(Hχ1 ) − φ(Hχ2 ) ≤ γ H(χ1 − χ2 ) (2) ∀ (χ1 , χ2 ) ∈ Rn × Rn . It is clear that (2) implies that (1) is Lipschitz with respect to x. 679
The observation problem consists of determining an event– triggering condition, fixing the time instants in which the sampled output data are sent to the observer, and an impulsive state observer having the structure x ˆ˙ = Aˆ x + Bu + Dφ(H x ˆ) ˆ(kδ) + δG y(ts ) − yˆ(kδ) x ˆ(kδ + ) = x (3) = (I − δGC)ˆ x(kδ) + δGCx(ts )
such that x ˆ(t) tends asymptotically to x(t) in a practical ˆ(t), and G ∈ Rn×q sense. In (3), x ˆ(kδ + ) is the left limit of x is the observer gain matrix. Note that the right limit is x ˆ(kδ − ) = x ˆ(kδ). The observer dynamics correspond to a copy of the system dynamics between sampling instants kδ, (k + 1)δ, while it undergoes a jump in the state at the sampling instants. 3. AN EVENT–TRIGGERED IMPULSIVE OBSERVER The impulsive observer (3) will be implemented making use of a triggering mechanism, determining when the
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sensor transmits the sampled data to the plant. More precisely, these data are sent to the plant at the time instants t = ts , ∈ N, such that the following event– triggering condition is satisfied (4) ts +1 = min kδ > ts | y(kδ) − y(ts ) ≥ εs k
where εs > 0 is a threshold value on the output error y(kδ) − y(ts ).
Given the system and the observer dynamics (1), (3), one has to consider both the continuous dynamics of the observation error e(t) = x(t) − x ˆ(t), given by e˙ = Ae + D φ(Hx) − φ(H x ˆ) (5)
and the error discrete dynamics, due to the impulses on the observer state. These latter have the expression e(kδ + ) = x(kδ + ) − x ˆ(kδ + )
= (I − δGC)e(kδ) + δGC x(kδ) − x(ts )
(6)
since x(kδ + ) = x(kδ − ) = x(kδ). It is worth noting that, at the triggering instants ts in which (4) is satisfied and the system output sensor sends the sampled data to the observer, this expression reduces to e(ts + ) = e(kδ + ) = (I − δGC)e(kδ) = (I − δGC)e(ts )
since x(kδ) = x(ts ). In all the other discrete time instants, the term δGC x(kδ)−x(ts ) = δG y(kδ)−y(ts ) appears, which can be seen as a perturbation induced by the absence of communications from the sensor. As stated by the following result, the event–triggering condition (4) along with an appropriate choice of the observer gain ensure the exponential practical stability of the observation error (Lakshmikantham, 1990). It is worth noting that this is true also when the system dynamics (1) are not stable. Theorem 3.1. Let us consider the system (1), with (A, C) observable and φ satisfying (2). If, for a fixed sampling time δ > 0, the following LMIs where N1 =
N1 ≤ −εI,
N2 ≤ −εI,
T 2 T P1 A + A P1 + γ H H +
N3 ≤ 0 P2 − P1 δ
P1 D
P2 − P1 P A + A P2 + γ H H + N2 = 2 δ P2 D P1 − δP3 C −P2 N3 = (P1 − δP3 C)T −P1 T
2
T
(7) P1 D −I
Remark 3.2. It is worth noting that in (7) the sampling period δ is fixed a priori. Therefore, (7) constitute a set of classic LMIs. Proof. The error dynamics (5), (6) can be rewritten as follows e˙ = Ae + Dd2 (9) + e(kδ ) = (I − δGC)e(kδ) + δGCd3
with d2 = φ(Hx) − φ(H x ˆ) and d3 = x(kδ) − x(ts ). Following Suykens (1998), Raff (2007), let us consider the Lyapunov candidate Vo (e, t) = e 2P (t) , where P (t) is a time–varying matrix given by the following convex combination of the matrices P1 , P2 t − kδ (P2 − P1 ) = λP1 + (1 − λ)P2 P (t) = P1 + δ (10) (k + 1)δ − t ∈ [0, 1) λ= δ defined for t ∈ (kδ, (k + 1)δ]. This combination allows constructing an appropriate Lyapunov function which, under the hypotheses (7), can show the practical exponential stability. For, first note that P (kδ + ) = P1 and P ((k + 1)δ) = P2 . Considering that P (t) is periodic with period δ, its definition can be extended for all t ≥ 0. Note also that λmin e 2 ≤ Vo ≤ λmax e 2 . Using (9), and P2 − P1 P˙ (t) = δ obtained from (10), one works out P 2 − P1 e V˙ o = eT P A + AT P + δ + eT P Dd2 + dT2 DT P e + dT2 d2 − dT2 d2 ∀ t ∈ kδ, (k + 1)δ . Using (2)
d2 2 ≤ γ 2 eT H T He and considering the definition of P (t) one gets P 2 − P1 T T 2 T ˙ +γ H H e PA + A P + Vo ≤ e δ + eT P Dd2 + dT2 DT P e − dT2 d2 ¯ξ = ξ T N1 ξ + (t − kδ)ξ T N
P2 D −I
P2 1 where ε¯ = ε/λmax , and λmin = min λP min , λmin , λmax = P2 1 max λP max , λmax .
(k + 1)δ − t T t − kδ T ξ N1 ξ + ξ N2 ξ δ δ T T ¯ = (N2 − for t ∈ (kδ, (k +1)δ], where ξ = e dT2 and N N1 )/δ. Hence, using (7), one finally obtains (k + 1)δ − t t − kδ ξ 2 − ε ξ 2 = −ε ξ 2 V˙ o ≤ −ε δ δ ≤ −ε e 2 ≤ −¯ εVo i.e. V˙ o is bounded by a negative definite function for all t ∈ (kδ, (k + 1)δ]. Therefore, =
have solutions P1 , P2 , P3 , with Pi = PiT > 0, i = 1, 2, for an ε > 0, then the observer (3), with the event–triggering condition (4), and the gain G = P1−1 P3 , ensures that the origin of error dynamics (5), (6) is globally practically exponentially stable, with attractive set λmax δ G Iεs = e ≤ ρs , ρs = εs (8) λmin 1 − e−δε¯/2 680
Vo (t) = e−¯ε(t−t0 ) Vo (t0 ) ∀ t0 , t ∈ (kδ, (k + 1)δ], t0 ≤ t
(11)
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and, in particular, Vo (k + 1)δ ≤ e−δε¯Vo (kδ + ). Note also that in the inter–sampling λmax −¯ ε(t−t0 )/2 e(t0 ) e(t) = e λmin ∀ t0 , t ∈ (kδ, (k + 1)δ], t0 ≤ t.
(13)
and
ζ = GCd3 = G y(kδ) − y(ts )
It is well–known that for Hermitian matrices, −N3 ≥ 0 is equivalent to P2 > 0 and S2 ≥ 0 (see Zhang (2005), Theorem 1.12, pag. 34). Hence, since −N3 ≥ 0 and P2 > 0, then S2 ≥ 0. Therefore, −eT (kδ)S2 e(kδ) ≤ 0. Furthermore, note that eT (kδ)(I − δGC)T P1 (I − δGC)e(kδ) = eT (kδ)(P1 − δP3 C)T P1−1 (P1 − δP3 C)e(kδ)
= eT (kδ)(P2 − S2 )e(kδ) ≤ eT (kδ)P2 e(kδ) = Vo (kδ). This last observation allows writing ζ T (P1 − δP3 C)e(kδ) = ζ T P1 (I − δGC)e(kδ) ≤ ζ T P1 ζ eT (kδ)(I − δGC)T P1 (I − δGC)e(kδ) 1 ≤ λP λmax G εs Vo (kδ) max ζ Vo (kδ) ≤
k→∞
c
λmax
δ G εs 1−e 1 − e−δε¯/2 where the series sum exists since a < 1. Hence, lim sup e(kδ + ) ≤ ρs . Considering that (13) ensures that −δ ε¯/2
=
k→∞
e(t) decreases exponentially between kδ + and (k + 1)δ, one can conclude that the error dynamics (5), (6) converges exponentially to the attractive set Iεs . The convergence is global since all the passages do not depend on the initial state. This concludes the proof.
Corollary 3.3. Let us consider the system (1), with (A, C) observable and φ satisfying (2). Under the same hypotheses and notations of Theorem 3.1, the observer (3), with the event–triggering condition (4), and the gain G = P1−1 P3 , ensures that the origin of error dynamics (5), (6) globally practically converge to the set Iεb = e ≤ (1 + εb )ρs
ρs given by (8), in a time 2 d − (1 + d)e−δε¯/2 1 T ≤ T,εb = ln , d= ε¯ εb δ G εs for any fixed εb , > 0, and for any observer initial condition such that e(0) ≤ . Proof. Since 2 2 2 Vo (0) = e(0) 2P2 ≤ λP max ≤ λmax one has Vo (0+ ) ≤ Vo (0) + c ≤ λmax + c and, from (14), k−1 λmin e(kδ + ) ≤ Vo (kδ + ) ≤ a Vo (0+ ) + c ai
where (4) has been used. Finally, ζ T P1 ζ = G y(kδ) − y(ts ) 2P1
2 s 2 1 ≤ λP max G y(kδ) − y(t )
k
2 2 2 2 1 ≤ λP max G εs ≤ λmax G εs .
λmax G εs Vo (kδ) 2 Vo (kδ) + c + δ 2 λmax G 2 ε2s = 2 ≤ a Vo ((k − 1)δ + ) + c √ a = e−δε¯/2 , c = λmax δ G εs , where (12) has been used, so that Vo (kδ + ) ≤ a Vo ((k − 1)δ + ) + c. This linear discrete–time dynamics is exponentially stable to the origin since the dynamic matrix is Schur. Moreover, its solution is given by Vo (kδ + ) ≤ Vo (kδ) + 2δ
k→∞
c 1−a
It is worth noting that the convergence ensured by Theorem 3.1 to Iεs is asymptotic. If one requires a finite–time convergence one needs to enlarge Iεs . This is stated in the following result.
S2 = P2 − (P1 − δP3 C)T P1−1 (P1 − δP3 C) the Schur complement of the element (2,2) of the matrix −N3 .
Therefore,
and for k →∞ lim sup λmin e(kδ + ) ≤ lim sup Vo (kδ + ) ≤ =
+ 2δζ T (P1 − δP3 C)e(kδ) + δ 2 ζ T P1 ζ
G = P1−1 P3 ,
(14)
j=1
∆Vo := Vo (kδ + ) − Vo (kδ) = −eT (kδ)S2 e(kδ)
k−1 aj Vo (kδ + ) ≤ a Vo (0+ ) + c
(12)
Let us now analyze the stability of the discrete error dynamics, i.e. the error dynamics in the discontinuity, considering the same Lyapunov candidate Vo (t) = e(t) 2P (t) and recalling that P (kδ + ) = P1 , P (kδ) = P ((k + 1)δ) = P2 as already noted. Using (9), one gets
with
669
681
i=0
1 − ak λmax + c + c 1−a √ δ G εs . Dividing by λmin and
≤a √ a = e−δε¯/2 , c = λmax imposing c λmax + k + + (1 − ak )ρs e(kδ ) ≤ a λmin λmin 1 c = ρs = (1 + εb )ρs , 1 − a λmin one gets the bound of kδ for the time T in which the error trajectory enters Iεb . Remark 3.4. It is worth noting that in the proposed scheme it is not necessary that the system is stable in order to ensure the observer convergence. However, in practice numerical error may influence the observer performance.
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4. SIMULATION RESULTS FOR A ROBOT WITH A FLEXIBLE JOINT
10
In this section a fourth–order model of the form (1) is considered, taken from Spong (1987), Rajamani (1998), Howell (2002), Raff (2007) and representing the dynamics of a one–link manipulator, with a DC motor as actuator T 0 1 0 0 B = 0 21.6 0 0 −48.6 −1.25 48.6 0 , A= 1 0 0 0 0 0 0 1 C= 0 1 0 0 19.5 0 −19.5 0 T D= 0 0 0 1 H= 0 0 1 0 ,
with φ = 3.3 sin x3 , which is a nonlinear term due to the gravity, acting on the dynamics of x4 . As commented in Spong (1987), the joint elasticity is described by a linear torsional spring, and x1 represents the rotation of the motor, x2 = x˙ 1 is the corresponding angular velocity, x3 is the angular position of the link, x4 = x˙ 3 is its angular velocity. Physically, one measures the motor position and velocity, while the measurement of the other variables is non–trivial. The input u = sin t is applied to the system. The performance of the impulsive observer (3) is analyzed making use of this benchmark. Solving (7), with a sensor sampling period of δ = 0.15 s, one worksout 28.6269 −0.3675 −11.6820 1.0816 −0.3675 0.5296 −1.5176 0.0455 P1 = −11.6820 −1.5176 18.8721 −1.7201 1.0816 0.0455 −1.7201 0.7153 22.9349 −0.5313 −11.6047 1.0195 −0.5313 0.4136 −1.9257 0.0810 P2 = −11.6047 −1.9257 20.9671 −1.8715 1.0195 0.0810 −1.8715 0.7393 9.3334 1.0001 −48.7804 22.3665 G= . −0.0524 3.3194 19.4066 −0.3167 where λmin = 0.06, λmax = 37, ε = 1. Then, in order to impose a ball of convergence for the observation error of ρs = 0.012, from (8) one obtains εs 3.64 × 10−7 . This theoretical value is highly conservative and, as a matter of fact, it is possible to fix less restrictive values. In fact, as shown by the simulation in Figs. 2, 3 obtained with δ = 0.05 s and εs = 10−3 , one can fix well bigger event–triggering thresholds still obtaining the desired balls of convergence of dimensions ρs . As expected, at the beginning the communications are more frequent, and become less frequent when the steady–state is reached after about 1.75 s, with an average sampling time of δav = 0.31 s.
8
6
4
2
0
−2 0.02
−4
0.01 −6
0 −0.01
−8
−0.02 −10 0
1
2
3
5
5.5
6
4
6.5
5
7
6
7
Fig. 2. Observation error e(t), with a zoom showing e ≤ ρs . 1
0.9
0.8
0.7
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0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
20
Fig. 3. Sensor transmission instants ts . The triggering parameter for the sensor can be computed in order the ensure an upper bound on the size of the attractive set. The simulation results show that this upper bound is not tight, and further work is necessary to obviate this over–approximation. The observer–based stabilization problem will be the subject of a forthcoming paper. ACKNOWLEDGMENTS The authors wish to thank Jean–Pierre Barbot for the fruitful discussions on impulsive observers. REFERENCES
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