Event-triggering stabilization of complex linear systems with disturbances over digital channels

Transcription

1 Event-triggering stabilization of comlex linear systems with disturbances over digital channels Mohammad Javad Khojasteh, Mojtaba Hedayatour, Jorge Cortés, Massimo Franceschetti Abstract As stos and auses for searating arts of a sentence in language hel to convey information, it is also ossible to communicate information in communication systems not only by data ayload, but also with its timing We consider an event-triggering strategy that exloits timing information by transmitting in a state-deendent fashion to stabilize a continuous-time, comlex, time-invariant, linear system over a digital communication channel with bounded delay and in the resence of bounded system disturbance For small values of the delay, we show that by exloiting timing information, one can stabilize the system with any ositive transmission rate However, for delay values larger than a critical threshold, the timing information is not enough for stabilization and the sensor needs to increase the transmission rate Comared to revious work, our results rovide a novel encoding-decoding scheme for comlex systems, which can be readily alied to diagonalizable multivariate system with comlex eigenvalues Our results are illustrated in numerical simulation of several scenarios I INTRODUCTION An imortant asect of a cyber-hysical system [1] is the existence of a finite rate digital communication channel in feedback loo between sensor and controller Data-rate theorems [] [1] quantify the effects of the digital communication channel in the feedback loo on stabilization Event-triggering control [11] [13] is also a key comonent in cyber-hysical systems where the objective, in the context of communication, is to minimize the number of transmissions and at the same time ensuring that the control goal is achieved [14] [17] While the majority of communication systems transmit information by adjusting the signal amlitude, it is also ossible to communicate information by adjusting the transmission time of a symbol [18] [1] In a general framework [] tools from information theory have been utilized to study the fundamental limitation of using timing information for stabilization Secifically, it is ossible to stabilize a lant using inherent information in the timing of the event In fact, event-triggering control techniques encode information in the timing in a state-deendent fashion In this context, a key observation made in [3] states that in absence of the delay in the communication rocess as well as absence of system disturbances and assuming the controller has knowledge of the triggering strategy, one can stabilize the M J Khojasteh and M Franceschetti are with the Deartment of Electrical and Comuter Engineering of University of California, San Diego M Hedayatour is with the Faculty of Engineering & Alied Science, University of Regina, SK, Canada J Cortés is with the Deartment of Mechanical and Aerosace Engineering, University of California, San Diego{mkhojasteh,massimo,cortes}@ucsdedu, hedayatm@ureginaca system with any ositive rate of transmission Our revious work [4] on systems without any disturbance quantifies the information contained in the timing of the triggering events as a function of the delay in the communication channel For small values of delay in the communication channel, we show that stability can be achieved with any ositive transmission rate However, as the delay increases to values larger than a critical threshold, information imlied from the triggering action itself may not stabilize the system and because of that, to ensure stability, the transmission rate must be increased These results are comared with a time-triggered imlementation subject to delay in [14] In addition to the unknown delay, system disturbances increase the degree of uncertainty in the state estimation rocess as well Therefore, to ensure stability, it is crucial to take these effects into account Our revious work [5] on systems with disturbances derives a sufficient bit rate for stabilization of a scalar linear, real, time-invariant system subjected to bounded disturbance over a digital communication channel with bounded delay In this aer, for a system with comlex oen-loo gain subject to disturbances we derive a sufficient information transmission rate to guarantee stability More recisely, we design an encoding-decoding scheme that, together with the roosed event-triggering strategy, rules out Zeno behavior" an infinite amount of triggering events in a finite-time interval and ensures that the norm of the state remains bounded as time grows We show that for small values of delay and using only imlicit information, stability can be achieved with an arbitrary ositive transmission rate However, as the delay increases, the information gets old and also corruted by the system disturbances, therefore higher and higher communication rates are required to ensure stability In addition, this result sets the basis for the generalization of event-triggered control strategies that meet the bounds on the information transmission rate for the stabilization of vector systems with any real oen-loo gain matrix with comlex eigenvalues under disturbance Finally, we numerically validate our result in a series of simulations For reasons of sace, roofs are omitted and will aear in full elsewhere Notation: Let R, C denote the set of real and comlex numbers, resectively We let log and ln denote the logarithm with bases and e, res We denote by and the absolute value of a real number and the norm of a comlex number res Also, any Q C can be written as Q = ReQ + i ImQ or Q = Q e iφ Q, and for any y R

2 we have e Qy = e ReQy We denote by x and x the greatest integer less than or equal to x and the smallest integer greater than or equal to x, res II PROBLEM FORMULATION We consider a networked control system consisting of a lant, sensor, communication channel and controller, cf Figure 1 The lant is described by a comlex, continuous- Controller Plant Communication Channel Fig 1 System model time, linear time-invariant model as Sensor ẋ = Axt + But + wt, 1 where the lant state xt and control inut ut are comlex numbers for t [, Here wt C reresents a system disturbance, which is uer bounded as wt M, where M R is ositive Here, A C, B C, and since we are only interested in unstable lants, we also assume We consider the next notion of stability Definition 1: The lant 1 is ractically stable if for any x < L, where L R is nonnegative, there exists an increasing function α of M, with α < ɛ for any ɛ >, such that for all Ψ > αm, there exists T such that, xt Ψ for all t T The sequence of triggering times where the sensor transmits a data ayload acket of length gt k s bits is reresented by {t k s} k N This acket is delivered to the controller without error and entirely but with unknown uer bounded delay described as follows Let {t k c } k N be the sequence of times where the controller receives the acket transmitted at time {t k s} k N and decodes it, then we have k = t k c t k s γ, with k being the k th communication delay, and γ be a non-negative real number Also, for all k 1, we define the k th triggering interval as k = t k+1 s t k s From this oint on, when referring to a generic triggering or recetion time, for convenience we ski the suer-scrit k in t k r and t k c We denote by b c t the number of bits that controller received until time t, and we define information access rate as R c = lim su t b c t t The number of bits transmitted by the sensor, u to time t is reresented by b s t and the information transmission rate is defined as follows: R s = lim su t b s t t As the sensor transmits gt s bits of information at each triggering interval, we can write N k=1 R s = lim su gtk s N N k=1 k In order to establish a common ground to comare with the information transmission rate later, we state the generalization of the classical data-rate theorem for the comlex lant 1 Theorem 1: Consider the lant-sensor-channel-controller model with lant dynamics 1 If xt remains bounded as t aroaches infinity then R c ln We also reresent the estimated state at the controller by ˆx which evolves at each inter-recetion interval as follows: ˆxt = Aˆxt + But, t [t k c, t k+1 c ], 3 starting from ˆxt k+ c with ˆx = ˆx We assume that the sensor has causal knowledge of the time of each control action to make sure it can comute ˆxt for all time t This is equivalent to establishing an instantaneous acknowledgment link between the sensor and actuator using the control inut, as in [6], [7] For instance, this can be imlemented in our system by monitoring the actuator outut, which changes at each recetion time Assuming the sensor has only access to the lant state, a narrowband signal can be used in the control inut for exciting a secific frequency of the state to inform the sensor about the exact time of each control action by the actuator That being said, we define the state estimation error as zt = xt ˆxt, where z = x ˆx We rely on this error to determine when a triggering event occurs in our controller design, as exlained next III EVENT-TRIGGERED CONTROL DESIGN In this section we roose our event-triggering design, and utilize it to find a sufficient condition on the information transmission rate Here we describe a class of event-triggered control strategies to determine the sequence of triggering times so that the lant 1 is ractically stable A triggering occurs at t k+1 s if zt k+1 s = J, 4

3 rovided t k c t k+1 s for natural number k and t 1 s At each triggering time t s the acket t s of size gt s is transmitted from the sensor to the controller The acket t s consists of the quantized version of hase of zt s, denoted by φ qzts, and a quantized version of the triggering time t s By 4 we have zt s = Je iφ zts, hence using the acket, a quantized version of zt s, denoted by qzt s, at the controller is constructed as follows q zt s = Je iφ qzts Furthermore, utilizing the bound and the received acket, the controller constructs a quantized version of t s, denoted by qt s Consequently, at controller zt c can be estimated as follows zt c = e Atc qts q zt s 5 Based on this, one can formulate a rocedure, that we term jum strategy, to udate the estimate of the state maintained by the controller Then we can write ˆxt + c = zt c + ˆxt c 6 zt + c = xt c ˆxt + c = zt c zt c At the sensor, the acket size gt s is chosen to be large enough such that the following equation for all t c [t s, t s + γ] is satisfied zt + c = zt c zt c ρ J 7 where < ρ < 1 and is considered as a design arameter Under this design, the frequency with which transmission events are triggered is catured by the triggering rate R tr = lim su N N N 8 k=1 k A tyical realization of zt under the roosed eventtriggering strategy before and after one triggering is reresented in Figure A Sufficient condition on the transmission rate In this section, we derive sufficient condition on the information transmission rate to ensure the lant 1 is ractically stable In other words, we design an encodingdecoding scheme that, together with our event-triggering olicy, rules out Zeno behavior an infinite amount of triggering events in a finite-time interval and guarantees the lant 1 is ractically stable The results resented here can be readily alied to multivariate linear lants with disturbance and diagonalizable oen-loo gain matrix with comlex eigenvalues B Design of quantization olicy We devote the first λ bits of the acket t s for quantizing the hase of zt s The roosed encoding algorithm, J zt + c zt s Fig The blue grah reresents evolution of the state estimation error in time before and after a triggering event The trajectory starts with an initial state inside a circle of radius J, and continues in a disturbed fashion along a siral trajectory due to the imaginary art of A until it hits the triggering threshold radius J, then it jums back inside the circle after the udate according to 5 During inter-recetion time intervals we have żt = Azt + wt, and the overshoot from the circle observed in the trajectory is due to the unknown delay in the communication channel In this examle, A = 3+i, B =, ut = 8ˆxt, M =, γ = 9 sec, ρ = 9 and J = 147 uniformly quantizes the circle to λ ieces of π/ λ radians After recetion, the decoder finds the right hase quantization cell and selects its center oint as φ qzts By letting ω = φ zts φ qzts, as deicted in Figure 3, we deduce: Fig 3 Estimation of hase angle after triggering event and transmission of λ bits ω π λ Furthermore, we use the encoding scheme roosed in [4] to aend a quantized version of triggering time t s of length gt s λ to the acket t s As shown inf Figure 4, to determine the time interval of the triggering event, we break the ositive time line into intervals of length bγ At the controller, after receiving the acket at t c, t s could fall anywhere between t c γ and t c Also, after breaking the ositive time line into intervals, t s falls into [jbγ, j + 1bγ] or [j + 1bγ, j + bγ] with j being a natural number Therefore, we use the λ+1 th bit of the acket to determine the correct interval of t s This bit is zero if the nearest integer less than or equal to the beginning number of the interval is an even number and is 1 otherwise This can be written

4 mathematically as t s [λ + 1] = mod ts bγ, For the Fig 4 Quantization of the triggering time t s The acket t s of length λ + 4 can be generated and sent to the controller After recetion and decoding the controller choose the center of the smallest sub-interval as its estimation of t s, denoted by qt s remaining bits of the acket, the encoder breaks the interval containing t s into gts λ 1 equal sub-intervals Once the acket is comlete, it is transmitted to the controller where it is decoded and the center oint of the smallest sub-interval is selected as the best estimate of t s Therefore, we have t s qt s bγ gts λ C Sufficient information transmission rate Here, we rely on the quantization olicy designed above to establish a sufficient bound on the information transmission rate that ensures that the lant is ractically stable We start by showing that we can achieve 7 with the quantization olicy Theorem : Consider the lant-sensor-channel-controller model with lant dynamics 1, estimator dynamics 3, triggering strategy 4, and jum strategy 6 If the controller has enough information about x such that state estimation error satisfies z < J, then the quantization olicy designed above achieves 7 for all k N with acket size lower bounded as gt s g := 9 max, λ + log ln bγ 1+e γ ρ M J e γ 1 sinπ/ λ ζ rovided, cos ImA t s qt s = 1 ζ, b > 1, ρ M J, 1a e γ 1 + e γ sinπ/ λ+1 + ζ J M e γ 1, 1b χ ζe γ χ, 1c, and λ > log π 1, arcsin 1 χ χ e γ 1d where < χ + χ < 1 Next we show that, using our encoding-decoding scheme, if the sensor has a causal knowledge of the delay in the communication channel, it can calculate the state estimation for all time Proosition 1: Consider the lant-sensor-channelcontroller model with lant dynamics 1, estimator dynamics 3, triggering strategy 4, and jum strategy 6 Using 5 and the quantization olicy deicted in Figure 3 and 4, if the sensor has causal knowledge of the delay in the communication channel, then it can calculate ˆxt for all time t We continue by showing that our event-triggered scheme does not suffer from Zeno behavior Lemma 1: Consider the lant-sensor-channel-controller model with lant dynamics 1, estimator dynamics 3, triggering strategy 4, and jum strategy 6 If the acket size satisfies 7 for all k N, then for all k N, we have t k+1 s t k s 1 ln J + M M ρ J + By Lemma 1 we deduce the triggering rate 8 is uer bounded as follows R tr ln J+ M ρ J+ M which is valid for all realization of the lant disturbance, initial conditions, and delay Combining this bound with Theorem, we obtain the following result Corollary 1: Under the assumtion of Theorem, there exists a quantization olicy that achieves 7 for all k N and for all delays and lant disturbance realizations with an information transmission rate where g R s ln J+ M ρ J+ M is defined in 9, rovided cos g, 11 ImA t s qt s = 1 ζ, b > 1, 1a, 1b, 1c, and 1d Figure 5 shows the sufficient information transmission rate 11 as a function of the channel delay uer bound γ One can observe that for small value of the delay, the sufficient information transmission rate is smaller than the rate required by the extension of the data-rate result in Theorem 1 As the delay uer bounded increases, the sufficient information transmission rate increases accordingly As a by-roduct of the above discussion, we can guarantee 1 is ractically stable, as stated in the following result Theorem 3: Under the assumtion of Theorem, when the air A, B is stabilizable, using the control rule ut = K ˆxt the system 1 is ractically stable rovided that

5 Rate bits/sec sufficient data-rate theorem Channel Delay Uerbound, γ sec Fig 5 Sufficient transmission rate 11 as functions of channel delay uer bound γ We assume A = 1 + i, B = 5, M = 1, ρ = 9 and b = 11 Also λ = log π/ arcsin 7 8 eγ and J = 8M e γ 1 + In this case, the rate dictated by datarate theorem Theorem 1 is / ln = 885 information transmission rate is lower bounded by 11 and the real art of A BK is negative Remark 1: Our revious work [4] extends the results on event-triggered control for lants without disturbance from the scalar to the vector case, but is limited to lants with real eigenvalues of the oen-loo gain matrix Our discussion for comlex lants here sets the basis for generalizing this result to lants subject to disturbance and for any real oen-loo gain matrix not necessarily with real eigenvalues IV SIMULATION RESULTS This section resents simulation results for stabilization of a comlex linear time-invariant lant with disturbances Although the results are stated for continuous systems, all the simulations are done in a digital environment by samling the continuous system at a high frequency such that the resulting aroximate system is close to the continuous system We choose the samling time δ = seconds The minimum uer bound for the channel delay is equal to two samling times We consider the state and state estimation as defined in 1 and 3 where A = +5i, B = 5, and the control inut is chosen as ut = 8ˆxt Using 1b, the triggering radius J cf 4 can be found as follows: J = 8Meγ 1 + δ, Also, to quantize the hase, using 1d we calculate λ as follows: λ = log π arcsin 7/8 e γ We carry out a set of three different simulations In simulation a, we assume the lant disturbance is zero and the channel delay is uer bounded by two samling times δ In simulation b, we assume the lant disturbance is uer bounded by M = 1 and the channel delay is uer bounded by two samling times Finally, for simulation c, we assume the lant disturbance is uer bounded by M = and the channel delay is uer bounded by γ = 1 seconds Simulation results are resented in Figure 6, where the first row reresents norm of the error zt, and triggering radius J, the second row reresents the evolution of φ xt and the third row reresents the evolution of xt in time In the third column, desite having large delays and large disturbances, the controller is able to stabilize the lant As we can see in the lot, the estimate of the state at the controller tracks the norm and hase of the state In the first row, sudden changes in the norm of the state estimation error reresent recetion of the transmitted acket at the controller ACKNOWLEDGEMENTS This research was suorted by NSF award CNS The authors would like to thank Prof Pavan Tallaragada for helful discussions V CONCLUSIONS We have resented an event-triggered control scheme for stabilization of a continuous-time, comlex, time-invariant, linear system over a digital communication channel with bounded delay and in the resence of bounded system disturbances We have tested the roosed control scheme on an unstable linear system with comlex oen-loo gain and validated the results in simulation Future work will study the identification of necessary conditions on the transmission rate in comlex systems, develoing an event-triggering design for vector systems with real and comlex eigenvalues based on the comlex system design, and testing of the roosed control strategies on ractical scenarios REFERENCES [1] K-D Kim and P R Kumar, Cyber hysical systems: A ersective at the centennial, Proceedings of the IEEE, vol 1 Secial Centennial Issue, , 1 [] S Tatikonda and S Mitter, Control under communication constraints, IEEE Transactions on Automatic Control, vol 49, no 7, , 4 [3] G N Nair and R J Evans, Stabilizability of stochastic linear systems with finite feedback data rates, SIAM Journal on Control and Otimization, vol 43, no, , 4 [4] J Hesanha, A Ortega, and L Vasudevan, Towards the control of linear systems with minimum bit-rate, in Proc 15th Int Sym on Mathematical Theory of Networks and Systems MTNS, [5] P Minero, M Franceschetti, S Dey, and G N Nair, Data rate theorem for stabilization over time-varying feedback channels, IEEE Transactions on Automatic Control, vol 54, no, 43, 9 [6] P Minero, L Coviello, and M Franceschetti, Stabilization over Markov feedback channels: the general case, IEEE Transactions on Automatic Control, vol 58, no, , 13 [7] V Kostina and B Hassibi, Rate-cost tradeoffs in control, in Communication, Control, and Comuting Allerton, 16 54th Annual Allerton Conference on IEEE, 16, [8] M Franceschetti and P Minero, Elements of information theory for networked control systems, in Information and Control in Networks Sringer, 14, 3 37 [9] B G N Nair, F Fagnani, S Zamieri, and R J Evans, Feedback control under data rate constraints: An overview, Proceedings of the IEEE, vol 95, no 1, , 7 [1] A S Matveev and A V Savkin, Estimation and control over communication networks Sringer Science & Business Media, 9

6 M =, γ = 4 sec, gt s = 4 bits, λ = 3 M = 1, γ = 4 sec, gt s = 4 bits, λ = 3 M =, γ = 1 sec, gt s = 13 bits, λ = J z J 3 z 5 4 J z φ x rad 4 φ x rad 4 φ x rad x ˆx 5 x ˆx 6 5 x ˆx a b c Fig 6 The first row reresents norm of state estimation error The second row reresents variations of hase angle of the comlex state x Finally, and the last row reresents evolution of the real comonent of x and ˆx in time In these simulations A = 5 + 5i, B = 5, ρ = 9, b = 11, x = 1, ˆx =, δ = second [11] W P M H Heemels, K H Johansson, and P Tabuada, An introduction to event-triggered and self-triggered control, in 51st IEEE Conference on Decision and Control CDC IEEE, 1, [1] P Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Transactions on Automatic Control, vol 5, no 9, , 7 [13] K J Astrom and B M Bernhardsson, Comarison of riemann and lebesgue samling for first order stochastic systems, in 41st IEEE Conference on Decision and Control CDC, vol IEEE,, [14] M J Khojasteh, P Tallaragada, J Cortés, and M Franceschetti, Time-triggering versus event-triggering control over communication channels, in 17 IEEE 56th Annual Conference on Decision and Control CDC, Dec 17, [15] P Tallaragada and J Cortés, Event-triggered stabilization of linear systems under bounded bit rates, IEEE Transactions on Automatic Control, vol 61, no 6, , 16 [16] J Pearson, J P Hesanha, and D Liberzon, Control with minimal cost-er-symbol encoding and quasi-otimality of event-based encoders, IEEE Transactions on Automatic Control, vol 6, no 5, 86 31, 17 [17] S Linsenmayer, H Ishii, and F Allgöwer, Containability with event-based samling for scalar systems with time-varying delay and uncertainty, IEEE Control Systems Letters, vol, no 4, 18 [18] V Anantharam and S Verdu, Bits through queues, IEEE Transactions on Information Theory, vol 4, no 1, 4 18, 1996 [19] C Rose and I S Mian, Inscribed matter communication: Part I, IEEE Transactions on Molecular, Biological and Multi-Scale Communications, vol, no, 9 7, 16 [] A Gohari, M Mirmohseni, and M Nasiri-Kenari, Information theory of molecular communication: Directions and challenges, IEEE Transactions on Molecular, Biological and Multi-Scale Communications, vol, no, 1 14, 16 [1] T J Riedl, T P Coleman, and A C Singer, Finite block-length achievable rates for queuing timing channels, in Information Theory Worksho ITW, 11 IEEE IEEE, 11, 4 [] M J Khojasteh, M Franceschetti, and G Ranade, Stabilizing a linear system using hone calls, arxiv rerint arxiv:184351, 18 [3] E Kofman and J H Braslavsky, Level crossing samling in feedback stabilization under data-rate constraints, in 45st IEEE Conference on Decision and Control CDC IEEE, 6, [4] M J Khojasteh, P Tallaragada, J Cortés, and M Franceschetti, The value of timing information in event-triggered control, arxiv rerint arxiv: , 16 [5] M J Khojasteh, M Hedayatour, J Cortés, and M Franceschetti, Event-triggered stabilization of disturbed linear systems over digital channels, in 18 5nd Annual Conference on Information Sciences and Systems CISS, March 18, 1 6 [6] A Sahai and S Mitter, The necessity and sufficiency of anytime caacity for stabilization of a linear system over a noisy communication link Part I: Scalar systems, IEEE Transactions on Information Theory, vol 5, no 8, , 6 [7] Q Ling, Bit-rate conditions to stabilize a continuous-time linear system with feedback droouts, IEEE Transactions on Automatic Control, vol 63, no 7, , July 18