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Delay-dependent data rate bounds for containability of scalar systems*
Proceedings of the 20th World Congress Proceedings of the the 20th World World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the 20th World Congress The International Federation of Automatic Control The International Federation of Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com The International Federation of Automatic Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 7875–7880
Delay-dependent data rate bounds for Delay-dependent data rate bounds Delay-dependent data ratesystems bounds for for containability of scalar containability of scalar systems containability of scalar systems Steffen Linsenmayer, Rainer Blind, Frank Allg¨ ower Steffen Linsenmayer, Linsenmayer, Rainer Blind, Blind, Frank Allg¨ Allg¨ ower wer Steffen o Steffen Linsenmayer, Rainer Rainer Blind, Frank Frank Allg¨ ower Institute for Systems Theory and Automatic Control, Institute for Systems Theory and and Automatic Control, Institute for Theory Control, University of Stuttgart, Germany. Institute for Systems Systems TheoryStuttgart, and Automatic Automatic Control, University of Stuttgart, Stuttgart, Germany. University Stuttgart, Stuttgart, {linsenmayer, blind, allgower}@ist.uni-stuttgart.de University of of Stuttgart, Stuttgart, Germany. Germany. {linsenmayer, blind, allgower}@ist.uni-stuttgart.de allgower}@ist.uni-stuttgart.de {linsenmayer, {linsenmayer, blind, blind, allgower}@ist.uni-stuttgart.de
Abstract: This paper considers the problem of containability for a scalar linear unstable system Abstract: This paper considers considers the problem of containability for a scalar unstable system Abstract: This the problem of for linear unstable system where the communication from the to the actuator is performed overlinear a network with finite Abstract: This paper paper considers thesensor problem of containability containability for aa scalar scalar linear unstable system where the theThe communication from the sensor to the actuator is performed over aa memoryless network with finite where communication from the sensor to the actuator is performed over network with finite capacity. possibilities of event-based sampling concepts in a setup with where the communication from the sensor to the actuator is performed over a network withcoding finite capacity. The possibilities possibilities ofacting event-based sampling concepts in aa setup setup with memoryless coding capacity. The of event-based sampling concepts in with memoryless coding and decoding components together with a static controller are investigated. A robust capacity. The possibilities of event-based sampling concepts in a setup with memoryless coding and decoding components acting together with a static controller controller are investigated. A robust and components acting with are A controller is designed such that thetogether finite communication control system employs an arbitrary, and decoding decoding components acting together with aa static static controller are investigated. investigated. A robust robust controller is designed such that the finite communication control system employs an arbitrary, controller is designed such that the finite communication control system employs an arbitrary, possibly average data the ratefinite in a delay free setup while guaranteeing containability. The controllervanishing, is designed such that communication control system employs an arbitrary, possibly vanishing, average data rate rate in aaofdelay delay free setup while guaranteeing containability. The possibly average data free while The controller can be applied in the delays as setup well, whereas the samplingcontainability. mechanism needs possibly vanishing, vanishing, average datapresence rate in in a delay free setup while guaranteeing guaranteeing containability. The controller can be applied in the presence of delays as well, whereas the sampling mechanism needs controller can applied in presence of well, whereas the mechanism needs to be adjusted. leads to an increasing dataas and this increase is precisely quantified. controller can be beThis applied in the the presence of delays delays asrate well, whereas the sampling sampling mechanism needs to be adjusted. adjusted. This leads to an an increasing increasing data rate rate and and this this increase increase is precisely quantified. to leads data is quantified. Indications why This the uncertainty time rather causes the increasing data to be be adjusted. This leads to to an in increasing datathan ratethe andpure thisdelay increase is precisely precisely quantified. Indications why the uncertainty in timeregarding rather than the pure delay delay causes the increasing data Indications why in pure causes increasing data rate are given as the welluncertainty as considerations the the relation transmitted bits and delay. Indications why the uncertainty in time time rather rather than than the pure of delay causes the the increasing data rate are are given given as as well well as as considerations considerations regarding regarding the the relation relation of of transmitted transmitted bits bits and and delay. delay. rate rate are given as well as considerations regarding the relation of transmitted bits and delay. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Event-based control; Quantized systems; Control under communication constraints. Keywords: Event-based control; control; Quantized systems; systems; Control under under communication constraints. constraints. Keywords: Keywords: Event-based Event-based control; Quantized Quantized systems; Control Control under communication communication constraints. 1. INTRODUCTION An alternative, aperiodic, sampling concept is given by 1. INTRODUCTION An alternative, aperiodic, sampling concept is is given by 1. An aperiodic, concept by event-based sampling. An sampling early mathematical analysis 1. INTRODUCTION INTRODUCTION An alternative, alternative, aperiodic, sampling concept is given given by event-based sampling. An early mathematical analysis ˚ event-based sampling. An early mathematical analysis its properties is given Astr¨ om and Bernhardsson event-based sampling. Anbyearly mathematical analysis The fact that the consideration of quantization signifi- of ˚ of its properties is given by A str¨ oom and Bernhardsson ˚ of is A The fact that the nature consideration of quantization signifiand the Lyapunov based framework by Tabuada ˚ of its its properties properties is given given by by Astr¨ str¨ om m and and Bernhardsson Bernhardsson The that of cantly changes of control problems goessignifiback (2002) The fact fact that the the consideration consideration of quantization quantization signifi(2002) and the Lyapunov based framework by (2002) and the Lyapunov based framework by Tabuada Tabuada cantly changes the nature of control problems goes back (2007) initiated a plenty of interesting investigations. The (2002) and the Lyapunov based framework by Tabuada cantly nature control problems to the changes study of the finite wordof representation in back digi- (2007) initiated a plenty of interesting investigations. cantly changes the nature oflength control problems goes goes back The (2007) initiated a plenty of interesting investigations. The to the study of finite word length representation in digiprimary works in event-triggered control are extended (2007) initiated a plenty of interesting investigations. The to the study of finite word length representation in digital systems. (1990) worked out primary works in event-triggered control are extended to the study In of particular, finite word Delchamps length representation in digiprimary works in event-triggered control are extended tal systems. In particular, Delchamps (1990) worked out towards the usage of quantized state information by primary works in event-triggered control are extended tal systems. In particular, Delchamps (1990) worked out that knowledge of the quantization procedure and the tal systems. In particular, Delchamps (1990) worked out towards the quantized state information by towards the usage of quantized state information by that knowledge knowledge ofcan thebequantization quantization procedure and the Lehmann and usage Lunze of (2010). A significant advance in the towards the usage of quantized state information by that of the procedure and the dynamical system used to improve the performance that knowledge of the quantization procedure and the Lehmann and Lunze (2010). A significant advance in the Lehmann and Lunze Lunze (2010). (2010). Aunder significant advance in the the dynamical system can be used to improve the performance area of event-triggered control bounded bit rates Lehmann and A significant advance in dynamical system can be used to improve the performance of the overall system. Together thethe introduction of area of event-triggered control under bounded bit rates is dynamical system can be used to with improve performance is under bounded bit is of the overall overall system. Together with the introduction of area givenof Tallapragada control and Cort´ es (2016). Therein cases ofbyevent-triggered event-triggered control under bounded bit rates rates is of Together with introduction of communication networks with finite bandwidth in conof the the overall system. system. Together with the the introduction of area given by Tallapragada and Cort´ eess (2016). Therein cases given by Tallapragada and Cort´ (2016). Therein cases communication networks with finite bandwidth in conwith disturbances and possible known delays are considgiven by Tallapragada and Cort´ e s (2016). Therein cases communication networks with finite bandwidth in control systems, this encouraged huge research efforts in the communication networks with finite bandwidth in con- with disturbances possible delays are considwith disturbances and possible known delays are trol systems,area this encouraged huge research research efforts in the with aand design for a known controller and trigger rule with together disturbances and possible known delays are considconsidtrol huge efforts in intersecting of information control theory. A ered trol systems, systems, this this encouraged encouraged hugeand research efforts in the the ered together with a design for a controller and trigger rule ered together together with a design design for aa on controller and trigger trigger rule intersecting area of information and control theory. A that respects a uniform bound the amount of data to ered with a for controller and rule intersecting area of information and control theory. A continuous-time is treated by andtheory. Brockett intersecting areasetup of information andWong control A that that respects a uniform bound on the amount of data to respects a uniform bound on the amount of data to continuous-time setup is treated by Wong and Brockett be sent at each transmission and guarantees existence of that respects a uniform bound on the amount of data continuous-time setup is treated by Wong and Brockett (1999), where sampling are by defined byand the Brockett constant be sent at each transmission and guarantees existence to continuous-time setup istimes treated Wong of be sent at each transmission and guarantees existence of (1999), where sampling times are defined by the constant a uniform lower bound on the inter-transmission intervals. be sent at each transmission and guarantees existence of (1999), where sampling times are defined by the constant delay that is assumed to come along with the finite rate (1999), where sampling times are defined by the constant aa uniform lower bound on the inter-transmission intervals. uniform lower bound on the inter-transmission intervals. delay that is assumed to come along with the finite rate Another approach using event-based methods in quantized a uniform lower bound on the inter-transmission intervals. delay that is assumed to come along with the finite rate and the amount of data to be transmitted. The influence delay that is assumed to come along with the finite rate Another Anotherproblems approachisusing using event-based methods in quantized quantized event-based in and the the amount of data to to be transmitted. The influence influence given by Pearsonmethods et al. (2015) where Another approach approach using event-based methods in quantized and of be The of information patterns on sufficient rate control anddifferent the amount amount of data data to be transmitted. transmitted. The data influence control problems is given by Pearson et al. (2015) where control problems is given by Pearson et al. (2015) where of different information patterns on sufficient data rate the sampling is periodic but the decision on what symbols control problems is given by Pearson et al. (2015) where of different information patterns on sufficient data rate bounds is explored by Tatikonda andsufficient Mitter (2004) af- the sampling is periodic but the decision on what symbols of different information patterns on data rate the sampling is periodic but the decision on what symbols bounds is explored by Tatikonda and Mitter (2004) afto send is made in an event-based fashion. the sampling is periodic but the decision on what symbols bounds is explored by Tatikonda and Mitter (2004) after stating a general necessary bound, dependent on the bounds is explored by Tatikonda and Mitter (2004) af- to send is made in an event-based fashion. send is in fashion. ter stating aaof general general necessary bound, dependent on the to send is made made in an an event-based event-based fashion. ter necessary dependent on eigenvalues the system matrix,bound, for a linear discrete-time ter stating stating a general necessary bound, dependent on the the to The great possibilities of event-based sampling concepts eigenvalues of the system matrix, for a linear discrete-time eigenvalues of the system matrix, for a linear discrete-time The great possibilities of event-based sampling system. Theof general study in Nair al. (2004) relates in The great possibilities of event-based sampling concepts eigenvalues the system matrix, for aetlinear discrete-time data rate limited applications is highlighted byconcepts Kofman The great possibilities of event-based sampling concepts system. The general study in Nair et al. (2004) relates system. The general study in Nair et al. (2004) relates in data rate limited applications is highlighted by Kofman minimal datageneral rates for stabilization, for relates linear and in data rate limited applications is highlighted by system. The study in Nair etnot al. only (2004) Braslavsky (2006), where it is constructively shown data rate limited applications is highlighted by Kofman Kofman minimalsystems, data rates rates for stabilization, stabilization, not ofonly only for linear in minimal data for not for linear and Braslavsky (2006), where it is constructively shown control directly to a property the dynamiand Braslavsky (2006), where it is constructively shown minimal data rates for stabilization, not only for linear that the average data rate can be made arbitrary small in a and Braslavsky (2006), where it is constructively shown control systems, directly to feedback a property of theFor dynamicontrol systems, directly of that the average data rate can be made arbitrary small in a a cal system, the topological entropy. linear that the average data rate can be made arbitrary small in control systems, directly to to aa property property of the the dynamidynamidelay free setup. The investigation of possible time delays that the average data rate can be made arbitrary small in a cal system, the topological feedback entropy. For linear cal topological feedback entropy. For free setup. The investigation of possible time delays systems it isthe then shown that this entropy rate given delay delay free setup. The investigation of possible time delays cal system, system, the topological feedback entropy. Foris linear linear in event-triggered control with quantized information is delay free setup. The investigation of possible time delays systems it is then then shown shown that that this characterization. entropy rate is is given systems it event-triggered with quantized information is by the previously The in in control with information is systems it is is thenknown showneigenvalue that this this entropy entropy rate rate is given given accounted for in Licontrol et al. (2012) by the notion stabilizing in event-triggered event-triggered control with quantized quantized information is by the previously known eigenvalue characterization. The by the previously known eigenvalue characterization. The accounted for in Li et al. (2012) by the notion stabilizing investigation of Ishii and Francis (2003) provides insight accounted for in Li et al. (2012) by the notion stabilizing by the previously known eigenvalue characterization. The bit-rate that relates the amount of information to the accounted for in Li et al. (2012) by the notion stabilizing investigation of Ishii and Francis (2003) provides insight investigation and (2003) provides bit-rate that that relatesdelay. the The amount of work information to the the in the role of of theIshii sampling times in the quantizer design bit-rate relates of to investigation of Ishii and Francis Francis (2003) provides insight insight allowable recent by Khojasteh bit-rate that relates the the amount amount of information information to the in the role of the the sampling times in theusing quantizer design maximal in role sampling in design allowable delay. The recent work by Khojasteh for quantized sampled-data systems, an uniform maximal allowable delay. The recent work by Khojasteh in the the role of of the sampling times times in the the quantizer quantizer design maximal et al. (2016) investigates the influence of time-delay on allowable delay. The recent work by Khojasteh for quantized sampled-data systems, using an an uniform maximal for sampled-data systems, using al. (2016) investigates the influence of time-delay on sampling approach with constant sampling et al. (2016) investigates the influence of time-delay on for quantized quantized sampled-data systems, usingtimes. an uniform uniform et necessary and sufficient data rates in scalar systems and et al. (2016) investigates the influence of time-delay on sampling approach approach with with constant constant sampling sampling times. times. sampling necessary and sufficient data rates in scalar systems and necessary and sufficient data rates in scalar systems and sampling approach with constant sampling times. is thus closely to data the work this paper necessary and related sufficient ratespresented in scalar in systems and is thus thus closely related to the the work presented presented in this this paper is closely related to work that also considers scalar systems. is thus closely related to the work presented in in this paper paper that also considers scalar systems. that also considers scalar systems. The authors would like to thank the German Research Foundation that also considers scalar systems. The just mentioned work investigates the possibilities The authors would like to thank the German Research Foundation would like the German Research (DFG) for financial of the theFoundation Cluster of The The just mentioned work the possibilities The authors authors would support like to to thank thank theproject Germanwithin Research Foundation The just work investigates the possibilities of controllers andinvestigates the goals are Thedynamic just mentioned mentioned work investigates the asymptotic, possibilities (DFG) for financial support of the project within the Cluster of (DFG) financial support of project within of Excellence Simulation Technology at the the Cluster University of dynamic controllers and the goals are asymptotic, (DFG) for for in financial support of the the (EXC project310/2) within the Cluster of of dynamic controllers and the goals are asymptotic, respectively exponential, observability and stabilizability, Excellence in Simulation Technology (EXC 310/2) at the University of dynamic controllers and the goals are asymptotic, Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart and the project AL 316/13-1. Excellence in Simulation Technology (EXC 310/2) at the University respectively exponential, observability and stabilizability, respectively of Stuttgart Stuttgart and the the project AL AL 316/13-1. respectively exponential, exponential, observability observability and and stabilizability, stabilizability, of of Stuttgart and and the project project AL 316/13-1. 316/13-1. Copyright © 2017, 2017 IFAC 8151Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 8151 Copyright © 2017 IFAC 8151 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 8151Control. 10.1016/j.ifacol.2017.08.742
Proceedings of the 20th IFAC World Congress 7876 Steffen Linsenmayer et al. / IFAC PapersOnLine 50-1 (2017) 7875–7880 Toulouse, France, July 9-14, 2017
Zero-order hold u(t) Plant x(t) Sampling (7) (1) (3) uk xk Decoder/ Network with sk Coder Controller finite data rate (4) (6) and possible delay Fig. 1. Control over a communication network. whereas here a setup, where only one bit is transmitted at every sampling instant and then used in a memoryless decoder and controller, is considered. Such a binary control has been shown to be the most robust control quantization under time-varying data-rate constraints in a scalar setup in Li and Baillieul (2004). Due to the memoryless setup, the problem of containability, introduced in Wong and Brockett (1999), is investigated in this work. It is shown that the result of Kofman and Braslavsky (2006), that the sufficient bound on the average data rate can be made arbitrarily small, holds for containability as well. A robust controller design is provided that has the inherent benefit to not only show the possibility of a vanishing average data rate but also directly provides a control design for arbitrary, not necessarily vanishing, average data rates. The second contribution is a sufficient data rate bound in a setup with unknown transmission delay. The delay value is restricted to the fundamental delay bounds in memoryless one bit settings. This is obviously a shortcoming of the static controller setup in contrast to the setup in Khojasteh et al. (2016) that investigates the whole possible delay spectrum. On the other hand, besides requiring less complex components, the approach provides the possibility to draw conclusions about the stabilizing bit-rate as introduced in Li et al. (2012) and it indicates that rather time uncertainty than time delay is the crucial point that increases the data rate bounds. In the next section we precisely formulate the problem that is investigated, followed by the delay free investigation in Section 3. The case with delays is investigated in Section 4 and the paper closes with a numerical example illustrating the results in Section 5 and a conclusion in Section 6.
The motivation for the introduction of this notion was the use of memoryless finite communication control laws in a setup with transmission delays. This is the reason why we investigate this problem here as well, since we also assume the coder, decoder, and controller to be memoryless. Although the concept was introduced due to the presence of time delay we want to first have a look on the classical results in the delay free setup with periodic sampling and sampling time h, i.e., xk := x(tk ) = kh. In this case the strong invariance conditions in Nair et al. (2004) can be directly transferred to containability. Furthermore, with a periodic sampling and a zero-order hold input with the same sampling time it holds that h xk+1 = eah xk + beah 0 e−as dsuk . Thus one can derive from Nair et al. (2004) that an average transmission data rate of log2 eah bits per sampling instant divided by h seconds per sampling instant is necessary and sufficient for strong invariability and thus for containability. Having a closer look one observes that the bound log2 eah Rh := = log2 ea h is actually independent of the chosen sampling time and is the known fundamental bound for periodic sampled-data systems. To be able to analyze the potential of eventbased sampling on containability we define the average transmission data rate under event-based sampling, that corresponds to the information transmission rate in Khojasteh et al. (2016), as R = lim inf k→∞
k−1 1 #bits transmitted at tj . k j=0 tj+1 − tj
In the remainder of this section we will precisely define the elements of the finite communication control system in Fig. 1. The sampling times tk are defined by the function σ : R2 × R2≥0 → R>0 , i.e., tk = σ(x(t), xk−1 , t, tk−1 )
The problem considered in this paper is the control of a linear, unstable scalar system, d x(t) = ax(t) + bu(t) (1) dt for t > 0 with x(t) ∈ R, a ∈ R>0 , b ∈ R \ {0}, and x(0) = x0 ∈ R, over a communication network with finite data rate and possible transmission delays as depicted in Fig. 1. The control goal is to guarantee containability of the finite communication control system as introduced in Wong and Brockett (1999). Definition 1. (Wong and Brockett (1999)). A finite communication control system on Rn is containable if for any sphere N centered at the origin there exists an open neighborhood of the origin M and coding and feedback control laws such that any trajectory started in M remains in N for all time.
(3)
for k ∈ N and t0 = σ0 (x(t), t) with σ0 : R × R≥0 → R≥0 . The successive coder employs the binary alphabet S = {0, 1} with fixed size 2 and is defined as a memoryless coder using the function γ : R → S, i.e., sk = γ(xk )
2. PROBLEM SETUP
(2)
(4)
for all k ∈ N0 . Note that, due to the definition of S at every sampling instant the fixed amount of 1 bit is sent over the network. Thus the definition of the average transmission data rate given in (2) can be given directly as R = lim inf k→∞
k−1 1 1 . k j=0 tj+1 − tj
(5)
The memoryless and finite decoder and controller can jointly be represented by the function δ : S → R, i.e., uk = δ(sk )
(6)
for k ∈ N≥0 . Depending on the communication network it is possible that symbols that are sent over the network arrive with a possibly time-varying transmission delay τk ∈ R≥0 and thus one defines the arrival times rk = tk + τk . In all cases that we investigate the sampling times are defined such that the condition tk ≤ rk < tk+1 is satisfied. Using the arrival times just defined, the zero-order hold unit is given as
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u(t) =
uk 0
if t ∈ [rk , rk+1 ), k ∈ N≥0 if t < r0
(7)
for t > 0 where the initial input 0 until the first symbol arrives resembles an initial open-loop phase. As mentioned before our control goal is to derive bounds on the average transmission data rate that are sufficient to guarantee containability. In our scalar event-based setup this means that for every compact interval K, centered around the origin, i.e., K := [−κ, κ] (whose boundary ∂K coincides with the sphere mentioned in Definition 1) we need to provide a sampling mechanism (3), coder (4), decoder/controller (6), and an open interval M containing the origin, that x0 is restricted to, such that x(t) stays in K for all time, i.e., K is rendered invariant. Thus we intend to design these components while reducing the average transmission data rate as far as possible.
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to this observation it is quite natural to choose our 1 bit coder such that the sign of the current state is coded, i.e., 1 if xk > 0 sk = γ(xk ) = . (10) 0 if xk < 0
Due to the just given definitions and the delay free setup, the controller knows xk exactly at tk . Furthermore it has to be designed such that the state is prevented from leaving K. Thus we introduce a controller that pushes the state towards the interior of K. The parameter that decides how fast the state moves back to the interior is introduced as
> 0. As stated before, the limit case of = 0 will be commented on in the separate Remark 4. The controller is defined as 1 (11) uk = δ(sk ) = (−1)sk (a + )κ b for all k ∈ N0 with ∈ R>0 .
3. EVENT-BASED SAMPLING WITHOUT DELAY
We continue with the computation of tk+1 − tk . A relation that follows from (8), (9), and (10) is
This section is devoted to the analysis of the delay free case, i.e., τk = 0 and thus rk = tk for all k ∈ N0 . The idea of the sampling concept that we derive is to detect an event as soon as the state touches the boundary of the given compact interval K. Therefore the controller design aims for pushing the state from the boundary towards the interior of the interval. A robust approach will be used that could be applied to systems with uncertainties as well. A special choice of this parameter is related to the deadbeat controller presented in Kofman and Braslavsky (2006), which will be explained later in this section. We will now state and prove the first result. Theorem 2. The scalar, event-based, finite communication control system composed of (1), (3), (4), (6), and (7) without transmission delay is containable if and only if the average transmission data rate R is nonnegative.
xk = κ(−1)sk +1 . Furthermore we know, due to (8) and (9), that tk+1 is detected as soon as (12) x(t) = −xk = −κ(−1)sk +1 = κ(−1)sk for t > tk . To use this we compute the explicit solution of x(t) for t ≥ tk as t a(t−tk ) x(t) = e xk + ea(t−s) buk ds
Proof. The necessity of R being nonnegative is directly given by the definition of R in (5). Thus it remains to show the sufficiency. To do so, we start with a given compact interval K centered around the origin. Dependent on K we derive the open interval M where we allow our initial conditions to lie in, as well as formal definitions of the sampler, coder and decoder/controller following the ideas given in the beginning of the section. Using these definitions we compute the average transmission data rate R. Furthermore, we show that trajectories starting in M will never leave K. The explicit case of R = 0 is possible as well and will be commented on separately in Remark 4. The derivation of the open interval M is done by allowing it to be the whole interior of the compact interval K, i.e., M = int K. Correspondingly, a sampling instant is detected as soon as the state reaches the boundary of K. Formally the sampling mechanism is defined by the functions σ0 and σ as σ0 (x(t), t) = inf t : x(t) ∈ ∂K = {−κ, κ} (8) t≥0
and σ(x(t), xk−1 , t, tk−1 ) = inf t : x(t) ∈ ∂K ∧ x(t) = xk−1 . t>tk−1
(9) From the definition in (8) one knows that x0 ∈ {−κ, κ} and according to (9) xk = −xk−1 holds for all k ∈ N. Due
tk
(a + )κ (−1)sk 1 − ea(t−tk ) a (a + )κ a(t−tk ) sk − (−1)sk . κ(−1) (13) =e a a Since we are interested in tk+1 − tk we derive, according to the previous considerations in (12), (a + )κ
κ(−1)sk = ea(tk+1 −tk ) κ(−1)sk − (−1)sk a a 1 2a + (14) ⇔tk+1 − tk = ln a
for all k ∈ N0 . Using (14) we can now compute R according to (5) as a(t−tk )
=e
κ(−1)
R = lim inf k→∞
sk +1
−
k−1 a 1 1 . = 2a k j=0 tj+1 − tj ln + 1
(15)
Employing continuity of the function that maps s ∈ (1, ∞) a , with respect to s, we conclude on ln(s) a =0 lim R = >0→0 ln lim>0→0 2a +1 and thus the average transmission data rate can be made arbitrarily small using the given sampler, coder and controller. It remains to show that K is invariant under this control. First, recall that xk ∈ {−κ, κ} ⊂ K. Furthermore the explicit solution in between two event times is given by (13) and one can see, that x(t) ∈ K for tk ≤ t ≤ tk+1 for all k ∈ N0 . It remains to show that x(t) ∈ K for 0 ≤ t < t0 . Since this is guaranteed by the fact that x(0) ∈ M = int K, the continuity of the solution x(t) under the constant input 0, and the fact that x(t0 ) ∈ ∂K ⊂ K the proof is complete.
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The proof of Theorem 2 presented above is constructive in the sense that it provides a sampler, coder and controller rendering K invariant with arbitrarily small data rate for every given K. In the following corollary we summarize that it further allows to explicitly compute a suitable controller for any arbitrary, not necessarily small, average transmission data rate. This possibility is directly linked to the robustness parameter . Corollary 3. Consider a compact interval K centered around the origin, system (1), and a network providing a maximal average transmission data rate Rmax > 0. Then the sampler (8), (9), coder (10), and decoder/controller (11) with 2a (16) ≤ a R max e −1 render K invariant using an average transmission data rate R ≤ Rmax , where equality corresponds to equality in (16). Proof. The proof of this statement follows directly from the proof of Theorem 2 since we use the sampler, coder, and controller presented therein with a special choice of , where we can guarantee due to the assumption a > 0 that is strictly positive. Thus we can conclude that K is rendered invariant and R can be computed due to (15) −1 as R = a ln 2a ≤ Rmax and the statement is +1 proven. Remark 4. Up to now the controller (11) is parametrized with > 0, i.e., the limit case = 0 is excluded. Nevertheless this limit provides some interesting insight. It represents a controller that needs only one transmission of 1 bit information at the first time the boundary of K is reached. With this knowledge the limit case controller u0, =0 = (−1)s0 aκ b renders the point at the boundary x0 ∈ ∂K as an equilibrium for all time and thus we know that K is rendered invariant with the total amount of 1 bit that is communicated. The limit case just presented can be seen as a finite, memoryless communication controller that corresponds to the deadbeat control strategy presented in Kofman and Braslavsky (2006) that drives the state to the origin in finite time with a total amount of 2n + 2 bits being communicated. It obviously stresses the fact that the key to reduce the necessary data rate is precise time information. Indeed, in this scalar case the precise time information gives the possibility to code the boundary of the compact interval and thus we transmit only one bit but completely specify the current state at the event time with infinite precision with this information and knowledge about the sampling concept that is applied. Thus, as mentioned in Kofman and Braslavsky (2006) and done in Khojasteh et al. (2016) using a dynamic controller, it is inevitable to study the case with nonvanishing transmission delays. This is done in the following section. 4. EVENT-BASED SAMPLING WITH TRANSMISSION DELAY
well as from a theoretical point of view since we code state information as time information. To analyze this delayed setup we impose an assumption. It introduces a uniform upper bound on the transmission delays. The bound restricts the delay to the area where our 1 bit setup can be used. The value will become more intuitive when we derive the sampling mechanism. Assumption 5. The transmission delays are uniformly upper bounded by τ ∈ (0, lna2 ), i.e., for all k ∈ N0 0 ≤ τk ≤ τ <
ln 2 . a
This gives an indication that in the presence of very large delays one should consider using, if possible, more advanced components, as actuators that can run dynamic controllers as in Khojasteh et al. (2016). Using this assumption it is now possible to derive a delaydependent sufficient condition on the average transmission data rate for containability. The bound has the desired property that for τ → 0 the bound approaches the arbitrarily small data rate derived in Theorem 2 and that it only depends on a and τ . Theorem 6. The scalar, event-based, finite communication control system composed of (1), (3), (4), (6), and (7) with transmission delays satisfying Assumption 5 is containable if a + R R≥ ln eaτ1−1 with R being an arbitrarily small positive constant.
Proof. The idea behind the derivation of M in this case is the following. We compute M such that, assuming the initial zero input for the worst case delay time the state stays in K when starting in M . Using Assumption 5 it can be shown that it is always possible to find such a set of initial conditions M . If one has no specific knowledge about τ0 one still knows due to Assumption 5 that τ0 < lna2 and thus eaτ0 < 2. Thus one can conclude that x(τ0 ) ∈ K for all trajectories starting in M = (− κ2 , κ2 ). The sampling concept explicitly takes this set into account, since it needs to guarantee that at the first sampling instant t0 the state is not outside the boundary of M . Thus t0 is defined by (17) σ0 (x(t), t) = inf t : x(t) ∈ ∂M . t≥0
Due to the possible delay we need to tighten the bounds for detection of the further triggering instants as well. This will be taken into account by the introduction of the ˜ = [−˜ compact set K κ, κ ˜ ], where κ ˜ ∈ R>0 is chosen such ˜ guarantees that the state does not that triggering at ∂ K leave K for the maximal possible delay. The concrete value for κ ˜ will be derived after the controller is introduced, but before we complete the representation of the sampling mechanism with ˜ σ(x(t), xk−1 , t, tk−1 ) = inf t : x(t) ∈ ∂ K t>tk−1
The above mentioned points make it necessary to analyze how the results change when transmission delays are present, both from an engineering point of view since it is not possible to transmit the information in zero time as
∧ sgn(x(t)) = sgn(xk−1 ). (18) The same coder as in the proof of Theorem 2 is used, i.e., 1 if xk > 0 , (19) sk = γ(xk ) = 0 if xk < 0
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guaranteeing again that exactly one bit is sent at every transmission. Additionally also the controller stays the same compared to the delay free case. This is due to the fact that although we detect events before the state reaches the boundary the tightened bound will be chosen such that the state can still reach the boundary due to the transmission delay. Thus the controller is given as 1 (20) uk = δ(sk ) = (−1)sk (a + )κ b for all k ∈ N0 with ∈ R>0 .
We continue now with the computation of the tightened bounds for the sampling mechanism that are represented ˜ such that the state never by κ ˜ . The idea is to compute K ˜ More formally, leaves K when an event is detected at ∂ K. due to (17), (18), (19), and (20) we know xk = (−1)sk +1 κ ˜ for all k ∈ N. Furthermore uk is given as in (20) and due to the sign condition in the sampler we know uk+1 = −uk for all k ∈ N0 . Thus one can now compute κ ˜ such that it still holds that x(t) ∈ K for all t ∈ [tk , rk ] and k ∈ N. For this interval we have t
x(t) = ea(t−tk ) xk + buk−1
ea(t−s) ds
tk
a+ κ(1 − ea(t−tk ) ). a To reduce the analysis here we will now analyze the case sk = 1. The computations for sk = 0 follow vice versa. Thus, a+ x(t) = ea(t−tk ) κ κ(ea(t−tk ) − 1) ˜+ a and one concludes that x(t) is monotonically increasing with t. Therefore the condition to derive κ ˜ is x(rk ) ≤ κ. Thus the computation a+ κ(ea(rk −tk ) − 1) ˜+ x(rk ) = ea(rk −tk ) κ a a+ κ(eaτ − 1) ˜+ (21) ≤ eaτ κ a delivers aτ − 1) 1 − a+ a (e =: κρ (22) κ ˜≤κ eaτ and we choose κ ˜ = ρκ for the remainder. Note that due to Assumption 5 one can always compute a value crit > 0 such that for all ∈ (0, crit ), κ ˜ is guaranteed to be positive. This is the reason why we announced the bound to be necessary for our 1 bit coding scheme. We will assume to be chosen according to this restriction in the remainder. = ea(t−tk ) (−1)sk +1 κ ˜ + (−1)sk
For computation of the average transmission data rate, at first we know due to sgn(x1 ) = sgn(x0 ) and continuity of the solution that t1 − t0 is lower bounded by a positive constant τ˜0 , i.e., t1 − t0 ≥ τ˜0 > τ0 ≥ 0. The next step is devoted to compute a uniform lower bound for the intervals tk+1 −tk for all k ∈ N, i.e., to compute τ˜ such that ˜ tk+1 − tk ≥ τ˜. Due to (21) one knows that x(t) ∈ K \ int K for t ∈ [tk , rk ]. Furthermore, due to the given sampler, tk+1 is determined only after a sign change. Thus it becomes clear that τk > 0 prolongs tk+1 − tk , i.e., for a worst case analysis we assume τk = 0. For this case we know for all t ∈ [tk = rk , tk+1 ] t ea(t−s) ds x(t) = ea(t−tk ) xk + buk tk
a(t−tk )
=e
(−1)
sk +1
ρκ − (−1)sk
a+ κ(1 − ea(t−tk ) ), a
7879
and we know that x(tk+1 ) = −xk = (−1)sk κ ˜ = (−1)sk ρκ. Thus, with these two equations one can compute a+ a+ a(tk+1 −tk ) +ρ =e ρ− − a a and conclude that tk+1 − tk ≥ τ˜ for all k ∈ N with a+ ρ+ a ln a+ −ρ a . τ˜ ≥ a Using the definition of ρ from (22) we can write 2+ a ln 2 a+ eaτ −(1+ ln 2(a+)e2a+ a+ aτ −(2a+) ) a a τ˜ ≥ = a a and proceed with the computation of R as a 1 1 1 . = + (k − 1) R ≤ lim inf 2a+ k→∞ k τ˜0 τ˜ ln aτ 2(a+)e
−(2a+)
Employing again continuity of the function that maps a , with respect to s, we conclude s ∈ (1, ∞) on ln(s) a lim R= ∈(0,crit )→0 ln eaτ1−1 and thus the average transmission data rate satisfies the given bound.
The invariance of K for trajectories starting in M until the first control input is received at t = r0 is guaranteed by the derivation of M in combination with the fact that the first triggering instant is detected at the boundary of M . Afterwards invariance is guaranteed by the design of the controller as in the delay free case and the tightened ˜ bounds for triggering represented by K. The result just presented gives a direct trade-off between transmission delay and the average transmission data rate that is sufficient for containability. As stated in the beginning of the section this has two interpretations. From an engineering point of view one can consider the relation #bit/transmission , introduced as the stabilizing bit rate in Li τ et al. (2012), as a measure for the maximal data rate that needs to be provided by the network and thus this gives now the opportunity to reduce this maximal data rate. From a theoretical point of view we introduced uncertainty in the timing. This uncertainty immediately results in an increase of the data rate, since we have an unstable system and the data rate corresponds to the growth of uncertainty due to the instability of the system. The role of this uncertainty is even more highlighted in the following remark. Remark 7. As just mentioned, not the delay itself forces to employ a non-vanishing data rate, but the uncertainty that is introduced in the system. In the scenario we investigate, the delay is characterized as 0 ≤ τk ≤ τ and thus the uncertainty corresponds directly to the maximum delay. One could now also consider the special case, when no jitter occurs and all messages are delayed with the maximal delay, i.e., τk = τ for all k. With this specification the delay introduces no uncertainty anymore since it is completely deterministic. It is possible to show that in this configuration, again K can be rendered invariant with arbitrarily small data rate, since this would again be a scenario where the state moves, although triggering is demanded at the tightened bounds, from one boundary
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6. CONCLUSION
κ
−κ
In this paper we presented an analysis of minimal data rates using event-based sampling for containability of scalar systems. The first outcome is that in a setup without transmission delays containability can be achieved with arbitrarily small average data rate. In addition the case with unknown time-varying transmission delays was analyzed and it is shown that uncertainty in the time information generates an increasing data rate that needs to be provided. Thus the main statements of the paper confirm the claim that with event-based sampling precise time information is used to reduce the average data rate and thus timing uncertainty relates to necessary data rates, providing interesting trade-offs. Since this rather exemplary study is far from being complete there are many further directions that seem to be interesting, such as the scalability to higher dimensions and a deeper insight in the different effects of pure delay and uncertainty in time.
=0.068 =0 0
2
4 6 Time t (s)
8
10
8
10
Data (bit)
Fig. 2. Evolution of the state without delay. 10 8 6 4 2 0
=0.068 =0 0
2
4 6 Time t (s)
State x
Fig. 3. Amount of transmitted data without delay. κ κ ˜ −˜ κ −κ
=0.01
0
2
REFERENCES
4 6 Time t (s)
8
10
Fig. 4. Evolution of the state with delay. of K to the other. Since we also use the same controller as in the delay free case the data rate can be made arbitrarily small with the parameter going to zero. 5. NUMERICAL EXAMPLE In this section the results are illustrated given a dynamical system as in (1) with a = 5, b = 1 and the compact interval K = [−κ, κ] with κ = 2. In a first step the undelayed setup is illustrated. Therefore, we sample and control the system according to (8), (9), (10), and (11) with different values for . A simulation result with initial condition x(0) = 0.9 is given in Fig. 2. The specific choice of the values for illustrate Corollary 3, since = 0.068 was computed to guarantee that Rmax = 1 is not violated, and Remark 4, since = 0 gives the limit case where the boundary of K is rendered as an equilibrium. The amount of data that is transmitted over that, depicted in Fig. 3, shows that the statements hold true, i.e. the constructive design leads to the desired average data rate and the case = 0 needs only one 1 bit transmission. The corresponding results in a setup with a maximal delay of τ = 0.1s are given in Fig. 4. In this setup the controller is designed with = 0.01 and the parameter κ ˜ for the reduced region ˜ = [−˜ K κ, κ ˜ ] is computed according to (22) as κ ˜ = 0.424. In the simulation results one can see that K is still rendered invariant, but the transmission frequency is larger than in the undelayed setup. This confirms the derivations given in the previous section on the trade-off between average data rate and maximal delay. Furthermore the trade-off between maximal data rate and average data rate is stressed, since we know that the amount of data communicated per transmission equals 1 bit and thus the maximal data rate corresponds to the relation 1bit/τ .
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IFAC PapersOnLine 50-1 (2017) 7875–7880
Delay-dependent data rate bounds for Delay-dependent data rate bounds Delay-dependent data ratesystems bounds for for containability of scalar containability of scalar systems containability of scalar systems Steffen Linsenmayer, Rainer Blind, Frank Allg¨ ower Steffen Linsenmayer, Linsenmayer, Rainer Blind, Blind, Frank Allg¨ Allg¨ ower wer Steffen o Steffen Linsenmayer, Rainer Rainer Blind, Frank Frank Allg¨ ower Institute for Systems Theory and Automatic Control, Institute for Systems Theory and and Automatic Control, Institute for Theory Control, University of Stuttgart, Germany. Institute for Systems Systems TheoryStuttgart, and Automatic Automatic Control, University of Stuttgart, Stuttgart, Germany. University Stuttgart, Stuttgart, {linsenmayer, blind, allgower}@ist.uni-stuttgart.de University of of Stuttgart, Stuttgart, Germany. Germany. {linsenmayer, blind, allgower}@ist.uni-stuttgart.de allgower}@ist.uni-stuttgart.de {linsenmayer, {linsenmayer, blind, blind, allgower}@ist.uni-stuttgart.de
Abstract: This paper considers the problem of containability for a scalar linear unstable system Abstract: This paper considers considers the problem of containability for a scalar unstable system Abstract: This the problem of for linear unstable system where the communication from the to the actuator is performed overlinear a network with finite Abstract: This paper paper considers thesensor problem of containability containability for aa scalar scalar linear unstable system where the theThe communication from the sensor to the actuator is performed over aa memoryless network with finite where communication from the sensor to the actuator is performed over network with finite capacity. possibilities of event-based sampling concepts in a setup with where the communication from the sensor to the actuator is performed over a network withcoding finite capacity. The possibilities possibilities ofacting event-based sampling concepts in aa setup setup with memoryless coding capacity. The of event-based sampling concepts in with memoryless coding and decoding components together with a static controller are investigated. A robust capacity. The possibilities of event-based sampling concepts in a setup with memoryless coding and decoding components acting together with a static controller controller are investigated. A robust and components acting with are A controller is designed such that thetogether finite communication control system employs an arbitrary, and decoding decoding components acting together with aa static static controller are investigated. investigated. A robust robust controller is designed such that the finite communication control system employs an arbitrary, controller is designed such that the finite communication control system employs an arbitrary, possibly average data the ratefinite in a delay free setup while guaranteeing containability. The controllervanishing, is designed such that communication control system employs an arbitrary, possibly vanishing, average data rate rate in aaofdelay delay free setup while guaranteeing containability. The possibly average data free while The controller can be applied in the delays as setup well, whereas the samplingcontainability. mechanism needs possibly vanishing, vanishing, average datapresence rate in in a delay free setup while guaranteeing guaranteeing containability. The controller can be applied in the presence of delays as well, whereas the sampling mechanism needs controller can applied in presence of well, whereas the mechanism needs to be adjusted. leads to an increasing dataas and this increase is precisely quantified. controller can be beThis applied in the the presence of delays delays asrate well, whereas the sampling sampling mechanism needs to be adjusted. adjusted. This leads to an an increasing increasing data rate rate and and this this increase increase is precisely quantified. to leads data is quantified. Indications why This the uncertainty time rather causes the increasing data to be be adjusted. This leads to to an in increasing datathan ratethe andpure thisdelay increase is precisely precisely quantified. Indications why the uncertainty in timeregarding rather than the pure delay delay causes the increasing data Indications why in pure causes increasing data rate are given as the welluncertainty as considerations the the relation transmitted bits and delay. Indications why the uncertainty in time time rather rather than than the pure of delay causes the the increasing data rate are are given given as as well well as as considerations considerations regarding regarding the the relation relation of of transmitted transmitted bits bits and and delay. delay. rate rate are given as well as considerations regarding the relation of transmitted bits and delay. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Event-based control; Quantized systems; Control under communication constraints. Keywords: Event-based control; control; Quantized systems; systems; Control under under communication constraints. constraints. Keywords: Keywords: Event-based Event-based control; Quantized Quantized systems; Control Control under communication communication constraints. 1. INTRODUCTION An alternative, aperiodic, sampling concept is given by 1. INTRODUCTION An alternative, aperiodic, sampling concept is is given by 1. An aperiodic, concept by event-based sampling. An sampling early mathematical analysis 1. INTRODUCTION INTRODUCTION An alternative, alternative, aperiodic, sampling concept is given given by event-based sampling. An early mathematical analysis ˚ event-based sampling. An early mathematical analysis its properties is given Astr¨ om and Bernhardsson event-based sampling. Anbyearly mathematical analysis The fact that the consideration of quantization signifi- of ˚ of its properties is given by A str¨ oom and Bernhardsson ˚ of is A The fact that the nature consideration of quantization signifiand the Lyapunov based framework by Tabuada ˚ of its its properties properties is given given by by Astr¨ str¨ om m and and Bernhardsson Bernhardsson The that of cantly changes of control problems goessignifiback (2002) The fact fact that the the consideration consideration of quantization quantization signifi(2002) and the Lyapunov based framework by (2002) and the Lyapunov based framework by Tabuada Tabuada cantly changes the nature of control problems goes back (2007) initiated a plenty of interesting investigations. The (2002) and the Lyapunov based framework by Tabuada cantly nature control problems to the changes study of the finite wordof representation in back digi- (2007) initiated a plenty of interesting investigations. cantly changes the nature oflength control problems goes goes back The (2007) initiated a plenty of interesting investigations. The to the study of finite word length representation in digiprimary works in event-triggered control are extended (2007) initiated a plenty of interesting investigations. The to the study of finite word length representation in digital systems. (1990) worked out primary works in event-triggered control are extended to the study In of particular, finite word Delchamps length representation in digiprimary works in event-triggered control are extended tal systems. In particular, Delchamps (1990) worked out towards the usage of quantized state information by primary works in event-triggered control are extended tal systems. In particular, Delchamps (1990) worked out that knowledge of the quantization procedure and the tal systems. In particular, Delchamps (1990) worked out towards the quantized state information by towards the usage of quantized state information by that knowledge knowledge ofcan thebequantization quantization procedure and the Lehmann and usage Lunze of (2010). A significant advance in the towards the usage of quantized state information by that of the procedure and the dynamical system used to improve the performance that knowledge of the quantization procedure and the Lehmann and Lunze (2010). A significant advance in the Lehmann and Lunze Lunze (2010). (2010). Aunder significant advance in the the dynamical system can be used to improve the performance area of event-triggered control bounded bit rates Lehmann and A significant advance in dynamical system can be used to improve the performance of the overall system. Together thethe introduction of area of event-triggered control under bounded bit rates is dynamical system can be used to with improve performance is under bounded bit is of the overall overall system. Together with the introduction of area givenof Tallapragada control and Cort´ es (2016). Therein cases ofbyevent-triggered event-triggered control under bounded bit rates rates is of Together with introduction of communication networks with finite bandwidth in conof the the overall system. system. Together with the the introduction of area given by Tallapragada and Cort´ eess (2016). Therein cases given by Tallapragada and Cort´ (2016). Therein cases communication networks with finite bandwidth in conwith disturbances and possible known delays are considgiven by Tallapragada and Cort´ e s (2016). Therein cases communication networks with finite bandwidth in control systems, this encouraged huge research efforts in the communication networks with finite bandwidth in con- with disturbances possible delays are considwith disturbances and possible known delays are trol systems,area this encouraged huge research research efforts in the with aand design for a known controller and trigger rule with together disturbances and possible known delays are considconsidtrol huge efforts in intersecting of information control theory. A ered trol systems, systems, this this encouraged encouraged hugeand research efforts in the the ered together with a design for a controller and trigger rule ered together together with a design design for aa on controller and trigger trigger rule intersecting area of information and control theory. A that respects a uniform bound the amount of data to ered with a for controller and rule intersecting area of information and control theory. A continuous-time is treated by andtheory. Brockett intersecting areasetup of information andWong control A that that respects a uniform bound on the amount of data to respects a uniform bound on the amount of data to continuous-time setup is treated by Wong and Brockett be sent at each transmission and guarantees existence of that respects a uniform bound on the amount of data continuous-time setup is treated by Wong and Brockett (1999), where sampling are by defined byand the Brockett constant be sent at each transmission and guarantees existence to continuous-time setup istimes treated Wong of be sent at each transmission and guarantees existence of (1999), where sampling times are defined by the constant a uniform lower bound on the inter-transmission intervals. be sent at each transmission and guarantees existence of (1999), where sampling times are defined by the constant delay that is assumed to come along with the finite rate (1999), where sampling times are defined by the constant aa uniform lower bound on the inter-transmission intervals. uniform lower bound on the inter-transmission intervals. delay that is assumed to come along with the finite rate Another approach using event-based methods in quantized a uniform lower bound on the inter-transmission intervals. delay that is assumed to come along with the finite rate and the amount of data to be transmitted. The influence delay that is assumed to come along with the finite rate Another Anotherproblems approachisusing using event-based methods in quantized quantized event-based in and the the amount of data to to be transmitted. The influence influence given by Pearsonmethods et al. (2015) where Another approach approach using event-based methods in quantized and of be The of information patterns on sufficient rate control anddifferent the amount amount of data data to be transmitted. transmitted. The data influence control problems is given by Pearson et al. (2015) where control problems is given by Pearson et al. (2015) where of different information patterns on sufficient data rate the sampling is periodic but the decision on what symbols control problems is given by Pearson et al. (2015) where of different information patterns on sufficient data rate bounds is explored by Tatikonda andsufficient Mitter (2004) af- the sampling is periodic but the decision on what symbols of different information patterns on data rate the sampling is periodic but the decision on what symbols bounds is explored by Tatikonda and Mitter (2004) afto send is made in an event-based fashion. the sampling is periodic but the decision on what symbols bounds is explored by Tatikonda and Mitter (2004) after stating a general necessary bound, dependent on the bounds is explored by Tatikonda and Mitter (2004) af- to send is made in an event-based fashion. send is in fashion. ter stating aaof general general necessary bound, dependent on the to send is made made in an an event-based event-based fashion. ter necessary dependent on eigenvalues the system matrix,bound, for a linear discrete-time ter stating stating a general necessary bound, dependent on the the to The great possibilities of event-based sampling concepts eigenvalues of the system matrix, for a linear discrete-time eigenvalues of the system matrix, for a linear discrete-time The great possibilities of event-based sampling system. Theof general study in Nair al. (2004) relates in The great possibilities of event-based sampling concepts eigenvalues the system matrix, for aetlinear discrete-time data rate limited applications is highlighted byconcepts Kofman The great possibilities of event-based sampling concepts system. The general study in Nair et al. (2004) relates system. The general study in Nair et al. (2004) relates in data rate limited applications is highlighted by Kofman minimal datageneral rates for stabilization, for relates linear and in data rate limited applications is highlighted by system. The study in Nair etnot al. only (2004) Braslavsky (2006), where it is constructively shown data rate limited applications is highlighted by Kofman Kofman minimalsystems, data rates rates for stabilization, stabilization, not ofonly only for linear in minimal data for not for linear and Braslavsky (2006), where it is constructively shown control directly to a property the dynamiand Braslavsky (2006), where it is constructively shown minimal data rates for stabilization, not only for linear that the average data rate can be made arbitrary small in a and Braslavsky (2006), where it is constructively shown control systems, directly to feedback a property of theFor dynamicontrol systems, directly of that the average data rate can be made arbitrary small in a a cal system, the topological entropy. linear that the average data rate can be made arbitrary small in control systems, directly to to aa property property of the the dynamidynamidelay free setup. The investigation of possible time delays that the average data rate can be made arbitrary small in a cal system, the topological feedback entropy. For linear cal topological feedback entropy. For free setup. The investigation of possible time delays systems it isthe then shown that this entropy rate given delay delay free setup. The investigation of possible time delays cal system, system, the topological feedback entropy. Foris linear linear in event-triggered control with quantized information is delay free setup. The investigation of possible time delays systems it is then then shown shown that that this characterization. entropy rate is is given systems it event-triggered with quantized information is by the previously The in in control with information is systems it is is thenknown showneigenvalue that this this entropy entropy rate rate is given given accounted for in Licontrol et al. (2012) by the notion stabilizing in event-triggered event-triggered control with quantized quantized information is by the previously known eigenvalue characterization. The by the previously known eigenvalue characterization. The accounted for in Li et al. (2012) by the notion stabilizing investigation of Ishii and Francis (2003) provides insight accounted for in Li et al. (2012) by the notion stabilizing by the previously known eigenvalue characterization. The bit-rate that relates the amount of information to the accounted for in Li et al. (2012) by the notion stabilizing investigation of Ishii and Francis (2003) provides insight investigation and (2003) provides bit-rate that that relatesdelay. the The amount of work information to the the in the role of of theIshii sampling times in the quantizer design bit-rate relates of to investigation of Ishii and Francis Francis (2003) provides insight insight allowable recent by Khojasteh bit-rate that relates the the amount amount of information information to the in the role of the the sampling times in theusing quantizer design maximal in role sampling in design allowable delay. The recent work by Khojasteh for quantized sampled-data systems, an uniform maximal allowable delay. The recent work by Khojasteh in the the role of of the sampling times times in the the quantizer quantizer design maximal et al. (2016) investigates the influence of time-delay on allowable delay. The recent work by Khojasteh for quantized sampled-data systems, using an an uniform maximal for sampled-data systems, using al. (2016) investigates the influence of time-delay on sampling approach with constant sampling et al. (2016) investigates the influence of time-delay on for quantized quantized sampled-data systems, usingtimes. an uniform uniform et necessary and sufficient data rates in scalar systems and et al. (2016) investigates the influence of time-delay on sampling approach approach with with constant constant sampling sampling times. times. sampling necessary and sufficient data rates in scalar systems and necessary and sufficient data rates in scalar systems and sampling approach with constant sampling times. is thus closely to data the work this paper necessary and related sufficient ratespresented in scalar in systems and is thus thus closely related to the the work presented presented in this this paper is closely related to work that also considers scalar systems. is thus closely related to the work presented in in this paper paper that also considers scalar systems. that also considers scalar systems. The authors would like to thank the German Research Foundation that also considers scalar systems. The just mentioned work investigates the possibilities The authors would like to thank the German Research Foundation would like the German Research (DFG) for financial of the theFoundation Cluster of The The just mentioned work the possibilities The authors authors would support like to to thank thank theproject Germanwithin Research Foundation The just work investigates the possibilities of controllers andinvestigates the goals are Thedynamic just mentioned mentioned work investigates the asymptotic, possibilities (DFG) for financial support of the project within the Cluster of (DFG) financial support of project within of Excellence Simulation Technology at the the Cluster University of dynamic controllers and the goals are asymptotic, (DFG) for for in financial support of the the (EXC project310/2) within the Cluster of of dynamic controllers and the goals are asymptotic, respectively exponential, observability and stabilizability, Excellence in Simulation Technology (EXC 310/2) at the University of dynamic controllers and the goals are asymptotic, Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart and the project AL 316/13-1. Excellence in Simulation Technology (EXC 310/2) at the University respectively exponential, observability and stabilizability, respectively of Stuttgart Stuttgart and the the project AL AL 316/13-1. respectively exponential, exponential, observability observability and and stabilizability, stabilizability, of of Stuttgart and and the project project AL 316/13-1. 316/13-1. Copyright © 2017, 2017 IFAC 8151Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 8151 Copyright © 2017 IFAC 8151 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 8151Control. 10.1016/j.ifacol.2017.08.742
Proceedings of the 20th IFAC World Congress 7876 Steffen Linsenmayer et al. / IFAC PapersOnLine 50-1 (2017) 7875–7880 Toulouse, France, July 9-14, 2017
Zero-order hold u(t) Plant x(t) Sampling (7) (1) (3) uk xk Decoder/ Network with sk Coder Controller finite data rate (4) (6) and possible delay Fig. 1. Control over a communication network. whereas here a setup, where only one bit is transmitted at every sampling instant and then used in a memoryless decoder and controller, is considered. Such a binary control has been shown to be the most robust control quantization under time-varying data-rate constraints in a scalar setup in Li and Baillieul (2004). Due to the memoryless setup, the problem of containability, introduced in Wong and Brockett (1999), is investigated in this work. It is shown that the result of Kofman and Braslavsky (2006), that the sufficient bound on the average data rate can be made arbitrarily small, holds for containability as well. A robust controller design is provided that has the inherent benefit to not only show the possibility of a vanishing average data rate but also directly provides a control design for arbitrary, not necessarily vanishing, average data rates. The second contribution is a sufficient data rate bound in a setup with unknown transmission delay. The delay value is restricted to the fundamental delay bounds in memoryless one bit settings. This is obviously a shortcoming of the static controller setup in contrast to the setup in Khojasteh et al. (2016) that investigates the whole possible delay spectrum. On the other hand, besides requiring less complex components, the approach provides the possibility to draw conclusions about the stabilizing bit-rate as introduced in Li et al. (2012) and it indicates that rather time uncertainty than time delay is the crucial point that increases the data rate bounds. In the next section we precisely formulate the problem that is investigated, followed by the delay free investigation in Section 3. The case with delays is investigated in Section 4 and the paper closes with a numerical example illustrating the results in Section 5 and a conclusion in Section 6.
The motivation for the introduction of this notion was the use of memoryless finite communication control laws in a setup with transmission delays. This is the reason why we investigate this problem here as well, since we also assume the coder, decoder, and controller to be memoryless. Although the concept was introduced due to the presence of time delay we want to first have a look on the classical results in the delay free setup with periodic sampling and sampling time h, i.e., xk := x(tk ) = kh. In this case the strong invariance conditions in Nair et al. (2004) can be directly transferred to containability. Furthermore, with a periodic sampling and a zero-order hold input with the same sampling time it holds that h xk+1 = eah xk + beah 0 e−as dsuk . Thus one can derive from Nair et al. (2004) that an average transmission data rate of log2 eah bits per sampling instant divided by h seconds per sampling instant is necessary and sufficient for strong invariability and thus for containability. Having a closer look one observes that the bound log2 eah Rh := = log2 ea h is actually independent of the chosen sampling time and is the known fundamental bound for periodic sampled-data systems. To be able to analyze the potential of eventbased sampling on containability we define the average transmission data rate under event-based sampling, that corresponds to the information transmission rate in Khojasteh et al. (2016), as R = lim inf k→∞
k−1 1 #bits transmitted at tj . k j=0 tj+1 − tj
In the remainder of this section we will precisely define the elements of the finite communication control system in Fig. 1. The sampling times tk are defined by the function σ : R2 × R2≥0 → R>0 , i.e., tk = σ(x(t), xk−1 , t, tk−1 )
The problem considered in this paper is the control of a linear, unstable scalar system, d x(t) = ax(t) + bu(t) (1) dt for t > 0 with x(t) ∈ R, a ∈ R>0 , b ∈ R \ {0}, and x(0) = x0 ∈ R, over a communication network with finite data rate and possible transmission delays as depicted in Fig. 1. The control goal is to guarantee containability of the finite communication control system as introduced in Wong and Brockett (1999). Definition 1. (Wong and Brockett (1999)). A finite communication control system on Rn is containable if for any sphere N centered at the origin there exists an open neighborhood of the origin M and coding and feedback control laws such that any trajectory started in M remains in N for all time.
(3)
for k ∈ N and t0 = σ0 (x(t), t) with σ0 : R × R≥0 → R≥0 . The successive coder employs the binary alphabet S = {0, 1} with fixed size 2 and is defined as a memoryless coder using the function γ : R → S, i.e., sk = γ(xk )
2. PROBLEM SETUP
(2)
(4)
for all k ∈ N0 . Note that, due to the definition of S at every sampling instant the fixed amount of 1 bit is sent over the network. Thus the definition of the average transmission data rate given in (2) can be given directly as R = lim inf k→∞
k−1 1 1 . k j=0 tj+1 − tj
(5)
The memoryless and finite decoder and controller can jointly be represented by the function δ : S → R, i.e., uk = δ(sk )
(6)
for k ∈ N≥0 . Depending on the communication network it is possible that symbols that are sent over the network arrive with a possibly time-varying transmission delay τk ∈ R≥0 and thus one defines the arrival times rk = tk + τk . In all cases that we investigate the sampling times are defined such that the condition tk ≤ rk < tk+1 is satisfied. Using the arrival times just defined, the zero-order hold unit is given as
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u(t) =
uk 0
if t ∈ [rk , rk+1 ), k ∈ N≥0 if t < r0
(7)
for t > 0 where the initial input 0 until the first symbol arrives resembles an initial open-loop phase. As mentioned before our control goal is to derive bounds on the average transmission data rate that are sufficient to guarantee containability. In our scalar event-based setup this means that for every compact interval K, centered around the origin, i.e., K := [−κ, κ] (whose boundary ∂K coincides with the sphere mentioned in Definition 1) we need to provide a sampling mechanism (3), coder (4), decoder/controller (6), and an open interval M containing the origin, that x0 is restricted to, such that x(t) stays in K for all time, i.e., K is rendered invariant. Thus we intend to design these components while reducing the average transmission data rate as far as possible.
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to this observation it is quite natural to choose our 1 bit coder such that the sign of the current state is coded, i.e., 1 if xk > 0 sk = γ(xk ) = . (10) 0 if xk < 0
Due to the just given definitions and the delay free setup, the controller knows xk exactly at tk . Furthermore it has to be designed such that the state is prevented from leaving K. Thus we introduce a controller that pushes the state towards the interior of K. The parameter that decides how fast the state moves back to the interior is introduced as
> 0. As stated before, the limit case of = 0 will be commented on in the separate Remark 4. The controller is defined as 1 (11) uk = δ(sk ) = (−1)sk (a + )κ b for all k ∈ N0 with ∈ R>0 .
3. EVENT-BASED SAMPLING WITHOUT DELAY
We continue with the computation of tk+1 − tk . A relation that follows from (8), (9), and (10) is
This section is devoted to the analysis of the delay free case, i.e., τk = 0 and thus rk = tk for all k ∈ N0 . The idea of the sampling concept that we derive is to detect an event as soon as the state touches the boundary of the given compact interval K. Therefore the controller design aims for pushing the state from the boundary towards the interior of the interval. A robust approach will be used that could be applied to systems with uncertainties as well. A special choice of this parameter is related to the deadbeat controller presented in Kofman and Braslavsky (2006), which will be explained later in this section. We will now state and prove the first result. Theorem 2. The scalar, event-based, finite communication control system composed of (1), (3), (4), (6), and (7) without transmission delay is containable if and only if the average transmission data rate R is nonnegative.
xk = κ(−1)sk +1 . Furthermore we know, due to (8) and (9), that tk+1 is detected as soon as (12) x(t) = −xk = −κ(−1)sk +1 = κ(−1)sk for t > tk . To use this we compute the explicit solution of x(t) for t ≥ tk as t a(t−tk ) x(t) = e xk + ea(t−s) buk ds
Proof. The necessity of R being nonnegative is directly given by the definition of R in (5). Thus it remains to show the sufficiency. To do so, we start with a given compact interval K centered around the origin. Dependent on K we derive the open interval M where we allow our initial conditions to lie in, as well as formal definitions of the sampler, coder and decoder/controller following the ideas given in the beginning of the section. Using these definitions we compute the average transmission data rate R. Furthermore, we show that trajectories starting in M will never leave K. The explicit case of R = 0 is possible as well and will be commented on separately in Remark 4. The derivation of the open interval M is done by allowing it to be the whole interior of the compact interval K, i.e., M = int K. Correspondingly, a sampling instant is detected as soon as the state reaches the boundary of K. Formally the sampling mechanism is defined by the functions σ0 and σ as σ0 (x(t), t) = inf t : x(t) ∈ ∂K = {−κ, κ} (8) t≥0
and σ(x(t), xk−1 , t, tk−1 ) = inf t : x(t) ∈ ∂K ∧ x(t) = xk−1 . t>tk−1
(9) From the definition in (8) one knows that x0 ∈ {−κ, κ} and according to (9) xk = −xk−1 holds for all k ∈ N. Due
tk
(a + )κ (−1)sk 1 − ea(t−tk ) a (a + )κ a(t−tk ) sk − (−1)sk . κ(−1) (13) =e a a Since we are interested in tk+1 − tk we derive, according to the previous considerations in (12), (a + )κ
κ(−1)sk = ea(tk+1 −tk ) κ(−1)sk − (−1)sk a a 1 2a + (14) ⇔tk+1 − tk = ln a
for all k ∈ N0 . Using (14) we can now compute R according to (5) as a(t−tk )
=e
κ(−1)
R = lim inf k→∞
sk +1
−
k−1 a 1 1 . = 2a k j=0 tj+1 − tj ln + 1
(15)
Employing continuity of the function that maps s ∈ (1, ∞) a , with respect to s, we conclude on ln(s) a =0 lim R = >0→0 ln lim>0→0 2a +1 and thus the average transmission data rate can be made arbitrarily small using the given sampler, coder and controller. It remains to show that K is invariant under this control. First, recall that xk ∈ {−κ, κ} ⊂ K. Furthermore the explicit solution in between two event times is given by (13) and one can see, that x(t) ∈ K for tk ≤ t ≤ tk+1 for all k ∈ N0 . It remains to show that x(t) ∈ K for 0 ≤ t < t0 . Since this is guaranteed by the fact that x(0) ∈ M = int K, the continuity of the solution x(t) under the constant input 0, and the fact that x(t0 ) ∈ ∂K ⊂ K the proof is complete.
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The proof of Theorem 2 presented above is constructive in the sense that it provides a sampler, coder and controller rendering K invariant with arbitrarily small data rate for every given K. In the following corollary we summarize that it further allows to explicitly compute a suitable controller for any arbitrary, not necessarily small, average transmission data rate. This possibility is directly linked to the robustness parameter . Corollary 3. Consider a compact interval K centered around the origin, system (1), and a network providing a maximal average transmission data rate Rmax > 0. Then the sampler (8), (9), coder (10), and decoder/controller (11) with 2a (16) ≤ a R max e −1 render K invariant using an average transmission data rate R ≤ Rmax , where equality corresponds to equality in (16). Proof. The proof of this statement follows directly from the proof of Theorem 2 since we use the sampler, coder, and controller presented therein with a special choice of , where we can guarantee due to the assumption a > 0 that is strictly positive. Thus we can conclude that K is rendered invariant and R can be computed due to (15) −1 as R = a ln 2a ≤ Rmax and the statement is +1 proven. Remark 4. Up to now the controller (11) is parametrized with > 0, i.e., the limit case = 0 is excluded. Nevertheless this limit provides some interesting insight. It represents a controller that needs only one transmission of 1 bit information at the first time the boundary of K is reached. With this knowledge the limit case controller u0, =0 = (−1)s0 aκ b renders the point at the boundary x0 ∈ ∂K as an equilibrium for all time and thus we know that K is rendered invariant with the total amount of 1 bit that is communicated. The limit case just presented can be seen as a finite, memoryless communication controller that corresponds to the deadbeat control strategy presented in Kofman and Braslavsky (2006) that drives the state to the origin in finite time with a total amount of 2n + 2 bits being communicated. It obviously stresses the fact that the key to reduce the necessary data rate is precise time information. Indeed, in this scalar case the precise time information gives the possibility to code the boundary of the compact interval and thus we transmit only one bit but completely specify the current state at the event time with infinite precision with this information and knowledge about the sampling concept that is applied. Thus, as mentioned in Kofman and Braslavsky (2006) and done in Khojasteh et al. (2016) using a dynamic controller, it is inevitable to study the case with nonvanishing transmission delays. This is done in the following section. 4. EVENT-BASED SAMPLING WITH TRANSMISSION DELAY
well as from a theoretical point of view since we code state information as time information. To analyze this delayed setup we impose an assumption. It introduces a uniform upper bound on the transmission delays. The bound restricts the delay to the area where our 1 bit setup can be used. The value will become more intuitive when we derive the sampling mechanism. Assumption 5. The transmission delays are uniformly upper bounded by τ ∈ (0, lna2 ), i.e., for all k ∈ N0 0 ≤ τk ≤ τ <
ln 2 . a
This gives an indication that in the presence of very large delays one should consider using, if possible, more advanced components, as actuators that can run dynamic controllers as in Khojasteh et al. (2016). Using this assumption it is now possible to derive a delaydependent sufficient condition on the average transmission data rate for containability. The bound has the desired property that for τ → 0 the bound approaches the arbitrarily small data rate derived in Theorem 2 and that it only depends on a and τ . Theorem 6. The scalar, event-based, finite communication control system composed of (1), (3), (4), (6), and (7) with transmission delays satisfying Assumption 5 is containable if a + R R≥ ln eaτ1−1 with R being an arbitrarily small positive constant.
Proof. The idea behind the derivation of M in this case is the following. We compute M such that, assuming the initial zero input for the worst case delay time the state stays in K when starting in M . Using Assumption 5 it can be shown that it is always possible to find such a set of initial conditions M . If one has no specific knowledge about τ0 one still knows due to Assumption 5 that τ0 < lna2 and thus eaτ0 < 2. Thus one can conclude that x(τ0 ) ∈ K for all trajectories starting in M = (− κ2 , κ2 ). The sampling concept explicitly takes this set into account, since it needs to guarantee that at the first sampling instant t0 the state is not outside the boundary of M . Thus t0 is defined by (17) σ0 (x(t), t) = inf t : x(t) ∈ ∂M . t≥0
Due to the possible delay we need to tighten the bounds for detection of the further triggering instants as well. This will be taken into account by the introduction of the ˜ = [−˜ compact set K κ, κ ˜ ], where κ ˜ ∈ R>0 is chosen such ˜ guarantees that the state does not that triggering at ∂ K leave K for the maximal possible delay. The concrete value for κ ˜ will be derived after the controller is introduced, but before we complete the representation of the sampling mechanism with ˜ σ(x(t), xk−1 , t, tk−1 ) = inf t : x(t) ∈ ∂ K t>tk−1
The above mentioned points make it necessary to analyze how the results change when transmission delays are present, both from an engineering point of view since it is not possible to transmit the information in zero time as
∧ sgn(x(t)) = sgn(xk−1 ). (18) The same coder as in the proof of Theorem 2 is used, i.e., 1 if xk > 0 , (19) sk = γ(xk ) = 0 if xk < 0
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guaranteeing again that exactly one bit is sent at every transmission. Additionally also the controller stays the same compared to the delay free case. This is due to the fact that although we detect events before the state reaches the boundary the tightened bound will be chosen such that the state can still reach the boundary due to the transmission delay. Thus the controller is given as 1 (20) uk = δ(sk ) = (−1)sk (a + )κ b for all k ∈ N0 with ∈ R>0 .
We continue now with the computation of the tightened bounds for the sampling mechanism that are represented ˜ such that the state never by κ ˜ . The idea is to compute K ˜ More formally, leaves K when an event is detected at ∂ K. due to (17), (18), (19), and (20) we know xk = (−1)sk +1 κ ˜ for all k ∈ N. Furthermore uk is given as in (20) and due to the sign condition in the sampler we know uk+1 = −uk for all k ∈ N0 . Thus one can now compute κ ˜ such that it still holds that x(t) ∈ K for all t ∈ [tk , rk ] and k ∈ N. For this interval we have t
x(t) = ea(t−tk ) xk + buk−1
ea(t−s) ds
tk
a+ κ(1 − ea(t−tk ) ). a To reduce the analysis here we will now analyze the case sk = 1. The computations for sk = 0 follow vice versa. Thus, a+ x(t) = ea(t−tk ) κ κ(ea(t−tk ) − 1) ˜+ a and one concludes that x(t) is monotonically increasing with t. Therefore the condition to derive κ ˜ is x(rk ) ≤ κ. Thus the computation a+ κ(ea(rk −tk ) − 1) ˜+ x(rk ) = ea(rk −tk ) κ a a+ κ(eaτ − 1) ˜+ (21) ≤ eaτ κ a delivers aτ − 1) 1 − a+ a (e =: κρ (22) κ ˜≤κ eaτ and we choose κ ˜ = ρκ for the remainder. Note that due to Assumption 5 one can always compute a value crit > 0 such that for all ∈ (0, crit ), κ ˜ is guaranteed to be positive. This is the reason why we announced the bound to be necessary for our 1 bit coding scheme. We will assume to be chosen according to this restriction in the remainder. = ea(t−tk ) (−1)sk +1 κ ˜ + (−1)sk
For computation of the average transmission data rate, at first we know due to sgn(x1 ) = sgn(x0 ) and continuity of the solution that t1 − t0 is lower bounded by a positive constant τ˜0 , i.e., t1 − t0 ≥ τ˜0 > τ0 ≥ 0. The next step is devoted to compute a uniform lower bound for the intervals tk+1 −tk for all k ∈ N, i.e., to compute τ˜ such that ˜ tk+1 − tk ≥ τ˜. Due to (21) one knows that x(t) ∈ K \ int K for t ∈ [tk , rk ]. Furthermore, due to the given sampler, tk+1 is determined only after a sign change. Thus it becomes clear that τk > 0 prolongs tk+1 − tk , i.e., for a worst case analysis we assume τk = 0. For this case we know for all t ∈ [tk = rk , tk+1 ] t ea(t−s) ds x(t) = ea(t−tk ) xk + buk tk
a(t−tk )
=e
(−1)
sk +1
ρκ − (−1)sk
a+ κ(1 − ea(t−tk ) ), a
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and we know that x(tk+1 ) = −xk = (−1)sk κ ˜ = (−1)sk ρκ. Thus, with these two equations one can compute a+ a+ a(tk+1 −tk ) +ρ =e ρ− − a a and conclude that tk+1 − tk ≥ τ˜ for all k ∈ N with a+ ρ+ a ln a+ −ρ a . τ˜ ≥ a Using the definition of ρ from (22) we can write 2+ a ln 2 a+ eaτ −(1+ ln 2(a+)e2a+ a+ aτ −(2a+) ) a a τ˜ ≥ = a a and proceed with the computation of R as a 1 1 1 . = + (k − 1) R ≤ lim inf 2a+ k→∞ k τ˜0 τ˜ ln aτ 2(a+)e
−(2a+)
Employing again continuity of the function that maps a , with respect to s, we conclude s ∈ (1, ∞) on ln(s) a lim R= ∈(0,crit )→0 ln eaτ1−1 and thus the average transmission data rate satisfies the given bound.
The invariance of K for trajectories starting in M until the first control input is received at t = r0 is guaranteed by the derivation of M in combination with the fact that the first triggering instant is detected at the boundary of M . Afterwards invariance is guaranteed by the design of the controller as in the delay free case and the tightened ˜ bounds for triggering represented by K. The result just presented gives a direct trade-off between transmission delay and the average transmission data rate that is sufficient for containability. As stated in the beginning of the section this has two interpretations. From an engineering point of view one can consider the relation #bit/transmission , introduced as the stabilizing bit rate in Li τ et al. (2012), as a measure for the maximal data rate that needs to be provided by the network and thus this gives now the opportunity to reduce this maximal data rate. From a theoretical point of view we introduced uncertainty in the timing. This uncertainty immediately results in an increase of the data rate, since we have an unstable system and the data rate corresponds to the growth of uncertainty due to the instability of the system. The role of this uncertainty is even more highlighted in the following remark. Remark 7. As just mentioned, not the delay itself forces to employ a non-vanishing data rate, but the uncertainty that is introduced in the system. In the scenario we investigate, the delay is characterized as 0 ≤ τk ≤ τ and thus the uncertainty corresponds directly to the maximum delay. One could now also consider the special case, when no jitter occurs and all messages are delayed with the maximal delay, i.e., τk = τ for all k. With this specification the delay introduces no uncertainty anymore since it is completely deterministic. It is possible to show that in this configuration, again K can be rendered invariant with arbitrarily small data rate, since this would again be a scenario where the state moves, although triggering is demanded at the tightened bounds, from one boundary
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6. CONCLUSION
κ
−κ
In this paper we presented an analysis of minimal data rates using event-based sampling for containability of scalar systems. The first outcome is that in a setup without transmission delays containability can be achieved with arbitrarily small average data rate. In addition the case with unknown time-varying transmission delays was analyzed and it is shown that uncertainty in the time information generates an increasing data rate that needs to be provided. Thus the main statements of the paper confirm the claim that with event-based sampling precise time information is used to reduce the average data rate and thus timing uncertainty relates to necessary data rates, providing interesting trade-offs. Since this rather exemplary study is far from being complete there are many further directions that seem to be interesting, such as the scalability to higher dimensions and a deeper insight in the different effects of pure delay and uncertainty in time.
=0.068 =0 0
2
4 6 Time t (s)
8
10
8
10
Data (bit)
Fig. 2. Evolution of the state without delay. 10 8 6 4 2 0
=0.068 =0 0
2
4 6 Time t (s)
State x
Fig. 3. Amount of transmitted data without delay. κ κ ˜ −˜ κ −κ
=0.01
0
2
REFERENCES
4 6 Time t (s)
8
10
Fig. 4. Evolution of the state with delay. of K to the other. Since we also use the same controller as in the delay free case the data rate can be made arbitrarily small with the parameter going to zero. 5. NUMERICAL EXAMPLE In this section the results are illustrated given a dynamical system as in (1) with a = 5, b = 1 and the compact interval K = [−κ, κ] with κ = 2. In a first step the undelayed setup is illustrated. Therefore, we sample and control the system according to (8), (9), (10), and (11) with different values for . A simulation result with initial condition x(0) = 0.9 is given in Fig. 2. The specific choice of the values for illustrate Corollary 3, since = 0.068 was computed to guarantee that Rmax = 1 is not violated, and Remark 4, since = 0 gives the limit case where the boundary of K is rendered as an equilibrium. The amount of data that is transmitted over that, depicted in Fig. 3, shows that the statements hold true, i.e. the constructive design leads to the desired average data rate and the case = 0 needs only one 1 bit transmission. The corresponding results in a setup with a maximal delay of τ = 0.1s are given in Fig. 4. In this setup the controller is designed with = 0.01 and the parameter κ ˜ for the reduced region ˜ = [−˜ K κ, κ ˜ ] is computed according to (22) as κ ˜ = 0.424. In the simulation results one can see that K is still rendered invariant, but the transmission frequency is larger than in the undelayed setup. This confirms the derivations given in the previous section on the trade-off between average data rate and maximal delay. Furthermore the trade-off between maximal data rate and average data rate is stressed, since we know that the amount of data communicated per transmission equals 1 bit and thus the maximal data rate corresponds to the relation 1bit/τ .
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