Event-triggered internally stabilizing sporadic control for MIMO plants with non measurable state

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1 Event-triggered internally stabilizing sporadic control for MIMO plants with non measurable state L. Jetto V. Orsini Universitá Politecnica delle Marche, Ancona, Italy, ( Abstract: This paper proposes a new approach to the sporadic networed control problem: the output of a remotely controlled sampled plant is required to trac an external reference with a reduced communication cost. The main novelty is that internal stability conditions are derived without assuming a measurable state of the plant. The new event driven Communication Logic (CL) is composedof asensorcl (SCL) and ofacontroller CL (CCL). The SCLis basedon the explicitcomputation ofalyapunov-lie functionalofthetracingerrorandofacorresponding time-varying threshold: a networ message from the sensor to the controller is triggered only if the functional exceeds the current value of the threshold. The CCL is directly driven by the SCL: the dynamic output controller sends a feedforward message to the plant only if it has received a message from the sensor at the previous time instant. Keywords: Event-driven control, sporadic control, internal stability, reference tracing. 1. INTRODUCTION This paper considers the problem of reducing the number of triggered messages along the feedbac and feedforward paths ofaclosed loopsystem forthenetwored control of a MIMO plant. Many authors considered this and similar topics in the general framewor of event based control Astrom (2008) and proposed several CLs whose common feature is invoing a message among the control system components only if a significant event occurred. The heuristic deadband and send-on-delta sampling techniques described in Arzen (1999), Otanez et al. (2002), Vasyutynsyyet al. (2006),have been successfullyapplied totheregulation problem for SISO plants but no formal analysis of closed-loop stability is provided. Ultimate boundedness is proved in Heemels et al. (2008), Lunze et al. (2010),where the event driven state-feedbac control depends on the measured error relative to the state trajectory. A performance analysis of the class of sporadic event-based controllers is provided in Henningsson et al. (2008) where first order linear stochastic systems are considered. The methods described in Mazo et al. (2008), Wang et al. (2009a) and Wang et al. (in press), consider the more general framewor ofdecentralized multi-loop systems and propose event-triggered strategies where each node of the networ uses its local data to state when to transmit information. Reference Mazo et al. (2008) is based on the dissipation inequalities initially proposed in Tabuada (2007). Self-triggered control is strictly related to eventtriggered techniques butdiffers because the next controller update instant is determined using the last state measurement Lemmon et al. (2007), Wang et al. (2009b) and Wang et al. (2010). All the aforementioned papers are based on the assumption ofameasurable state vector. Systems with nonaccessible state are considered in Yoo et al. (2002), but only conditions for BIBO stability are stated. This paper deals with the tracing problem of an external reference for networed controlled MIMO sampled plants. Unlie all the other papers, a sporadic internally stabilizing sporadic control law is derived without assuming a measurable state vector of the plant. The new CL proposed here is based on the definition of an SCL and of a related CCL. The SCL is driven by the values assumed by a Lyapunov-lie functional of the tracing error: a feedbac networ message is triggered from the sensor to thecontroller onlyifthefunctionalexceeds atime-varying threshold. The CCL follows the SCL with a delay of one sampling period: the dynamic output controller sends a feedforward message to the plant only if it has received a message from the sensor at the previous time instant. The shift between the two CLs is due to the state equation of the controller. Internal stability conditions are derived usingadynamic outputfeedbaccontroller with thesame structure proposed in Desoer et al. (1980). This paper is organized in the following way. The structure of the event-driven scheme and problem statement are given in Section 2, some preliminaries are briefly recalled in Section 3. The main result is given in Section 4 where theclis describedin detailand theconditions for internal stability are stated. A numerical example taen from the literature is given in Section THE GENERAL CONTROL SCHEME The control scheme Σ l shown in Fig. 1 is an autonomous system given by the connection of an unforced, unstable, external reference generator Σ r with a logic, switched closed-loop system Σ f. The control loop elements of Σ l are implemented at two intelligent nodes communicating Copyright by the International Federation of Automatic Control (IFAC) 10225

2 Σ r Σ l r() Σ g e() Σ f CCL u() Z.O.H Σ d u(t) Σ p y(t) T s y() SCL e() e() where ẽ( ) = [e T ( ),r T ( )] T. The discretized plant Σ d is assumed to be reachable and observable, with no invariant zero in z = 1. The triplet (H,G,F) is designed so as to guarantee the internal asymptotic stability of Σ f,c, and asymptotic exact tracing of r(). Following Desoer et al. (1980), the structure of Σ g is given by the connection of a suitable internal model system Σ m, with an LTI observer Σ ob of x( ). Denoting by x m ( ) and ξ( ) the states of Σ m and Σ ob respectively, their state-space representations are x m ( + 1) = A m x m () + B m e(), (4) ξ( + 1) = (A LC)ξ() + Bu() + L(r() e()),(5) where A m and B m are as specified in Desoer et al. (1980), and the control effort is given by u() = K m x m () Kξ(). (6) Fig. 1. The logic control scheme Σ l through a common networ. The controlled output of Σ l is the tracing errore(h) =r(h) y(h), where the numerical output y(h) of Σ d is obtained by periodic sampling of y(t),t = ht s. The event based digital controller incorporates the event-based control input generator Σ g and the CCL. The SCL is endowed with the same algorithm for generating r(h) with the purpose of computing e(h) and a related a Lyapunov lie functional (defined later). The SCL transmits the current e(h) to Σ g only when thelyapunovfunctional exceeds atime-varyingthreshold. Every time e(h) is triggered, the dynamic controller Σ g updates its internal state z(h + 1) and the corresponding control action u(h + 1). At the same time instant h, Σ g transmits u(h) totheplantaccordingtotheccl, namely only if at the previous time instant h 1 it has received e(h 1). Hence thecclis implicitly definedbythescl. If for some h, the SCL does not trigger e( h), then the eventbased Σ g does not even tae into account r( h) and does not update either its state or the control law. Hence the effect of the CCL is to suspendthe triggering ofu( ) to the Z.O.H. as long as the sensor does not transmit a new value e(l), for some l > h. The effect of the two CLs is to produce a switched Σ l operating in two different modalities: openloop configuration Σ o and closed-loop configuration Σ c. Accordingly Σ f assumes the two configurations Σ f,o and Σ f,c respectively. The closed-loop configuration Σ c of Σ l is given by the connection of Σ r with Σ f,c which is the feedbac connection of the LTI controller Σ g with the discretized plant Σ d. The external input of Σ f,c is the reference r(h) to be traced. In the closed-loop configuration, e(h) is transmitted at each sampled time instant ht s and u(t) is ept constant by the ZOH over one sampling period T s. The state space representations of Σ r, Σ d and Σ g are { xr ( + 1) = A Σ r : r x r () r() = C r x r () { x( + 1) = Ax() + Bu() Σ d : y() = Cx() { z( + 1) = Gz() + Fẽ() Σ g : u() = Hz() (1) (2) (3) where [ K m, K and] L[ are] chosen so as to give matrices  = A 0 B [ K K B m C A m 0 m ], and A LC, with the desired eigenvalues. This is surely possible by the assumptions on the plant. By (4)-(6), it readily follows that the state-space representation of Σ g is z( ) = xm ( ),G = ξ( ) [ F = Bm 0 L L Am 0, BK m A BK LC ],H = [ K m K ], whence it is apparent that the dynamical matrix of the LTI observer Σ ob is A ob = A BK LC. The above controller yields an internally asymptotic stable Σ f,c with null steady-state tracing error over a sufficiently small compact set containing the nominal parameters of the plant Desoer et al. (1980). System Σ l is said to operate in the open loop configuration Σ o as long as the triggering of e( ) is suspended. In this configuration also Σ f becomes an open-loop system Σ f,o where the continuous-time control input u(t) is ept constant by the ZOH on the last computed frozen value u(t) = u(h), t [ht,lt), until a new control action u(l), is triggered by the event-driven algorithm for some l > h. Recalling that a system is said state-output stable if its unforced output response originating from any initial state is uniformly bounded, and that the controlled output of Σ l is the tracing error e( ), the considered control problem is stated as follows. Problem statement: the sporadic control problem (SCP) consists in defining a CL whose purpose is to attain a sporadic feedbac control action still guaranteeing the state-output stability of theswitched unforced system Σ l, and the internal stability of the switched closed loop system Σ f. It will be shown that the state-output stability of Σ l can be attained through a proper CL without any further assumption. To also attain the internal stability of Σ f the following assumption is also needed. Assumption 1 The matrices K, K m and L of Σ g can be chosen such that, besides A LC and Â, also A ob = A BK LC result to be stable. The state transition matrix generated by A ob is denoted by Φ ob (, )

3 3. SOME PRELIMINARIES Consider the LTV system Σ s (C s ( ),A s ( )). Let x s ( ), y s ( ) and W s ( 0, 0 + N ) be the state, the free output response and the gramian of Σ s respectively. If Σ s is uniformly completely observable, there exist a positive integer N and positive constants γ 1, γ 2, such that, 0 Z +, W s ( 0, 0 + N ) is bounded as γ 1 I W s ( 0, 0 + N ) γ 2 I. (7) Let the functional V (, τ) be defined as V (, τ) = ys T (h)y s (h), τ, (8) h= τ it is easily seen that the definition of observability gramian and (8) imply V (, τ) = x s ( τ) T W s ( τ,)x s ( τ), (9) whence, by (7), x s ( τ) 2 γ 1 V (, τ) x s ( τ) 2 γ τ. (10) for some finite γ τ overbounding W s ( τ,). Lemma 1. Jetto et al. (2008) System Σ s is internally exponentially stable if and only if, for any initial state x s (0) of Σ s and σ (0,1), a sufficiently large integer ( σ) > N can be found, such that, τ ( σ), the functional V (, τ), given by (8) is converging to zero for as V (, τ) σv ( τ, 2τ), 2τ. (11) 4. THE EVENT BASED COMMUNICATION LOGIC The event driven CL solving the SCP will be defined in this section exploiting theprevious lemma. As the lemma refers to observable systems, the following has to be intended as referred to the minimal realizations Σ m c and Σ m l of Σ c and Σ l respectively. The idea is to define a functional V (, τ) lie (8) with reference to the free response e( ) of the unforced system Σ m l (which gives the same output e( ) of Σ l ). The lemma is applied supposing that Σ l is always in the closed loop configuration Σ c, whose minimal realization Σ m c is internally stabilized by Σ g. Hence the value of τ needed to define V (, τ) is computed as explained in Jetto et al. (2008) with reference to the LTI Σ m c Σ m l. The above lemma implies that V (, τ) = τ et (h)e(h) would converge to zero for according to (11) if Σ l were always frozen in the configuration Σ c. Nevertheless, in the actual sporadic control case, Σ l switches from Σ c to Σ o and viceversa. As long as Σ l operates in the closed loop configuration, V (, τ) obeys (11), but when the triggering of e( ) is suspended by the SCL, Σ l operates in the open loop configuration Σ o. As a consequence, V (, τ) stops obeying (11) and increasing jumps of its values could be observed. The proposed SCL is based on the idea of stopping the feedbac triggering of e() only if V (, τ) is smaller than a suitable threshold, whose value has to be chosen so as to ensure that V (, τ) remains bounded for. By the way V (, τ) is defined, it is clear that this is enough to guarantee the state output stability of Σ l. It will be shown later that the internal stability of Σ f follows under Assumption 1. Giventheabove generalconsiderations, adetailed description of how the proposed event driven CL operates is reported in the following. 1 At time = 0, Σ l starts operating in the closed-loop configuration Σ c, u() and e() are triggered at each time instant. The SCL starts computing and storing e(), eeping the last computed 2τ + 1 samples, for 2τ. The controller Σ g internally stabilizes Σ f Σ f,c and would give an exact asymptotic tracing of r() in the case it were always ept acting. 2 At time = τ the SCL starts computing V (, τ), τ. On the basis of the measured V (τ,0), it also computes a fixed threshold σ 1 as explained in the following theorem. For 2τ, V (, τ) is surely decreasing according to (11). Hence there exists a 1 (not necessarily greater than 2τ) such that V ( 1, 1 τ) < σ 1. Moreover Assumption 1 implies that as long as Σ g is acting one has Φ ob (,0) m ob λ ob, for some m ob 1, and λ ob = max i λ i {A ob } < 1. Hence there exists a such that Φ ob (i+,i) m ob λ ob < α < 1,, i Z +. The SCL stops the feedbac triggering of e() for max( 1, ) = 1, while the CCL stops the feedforward triggering of u() for At the same time 1, Σ l starts operating in the open loop configuration Σ o, the event-based Σ g stops updating its state which remains frozen in the value z( 1 ), and Σ d starts being controlled by the frozen control effort u( 1 ) = Hz( 1 ). In general, this is not a stabilizing control law, so that V (, τ) stops obeying (11). As a consequence, for some = 1 > 1 it may happen that V ( 1, 1 τ) σ 1. 4 At time 1 (if any) the triggering of e() is restarted by the SCL, e( 1 ) and r( 1 ) are used to compute u( 1 + 1) according to (3). Recalling that the suspensions of feedbac and feedforward messages are shifted of one time instant, the CCL restarts the triggering of u() for At time = 1, Σ l switches from Σ o to Σ c and a new threshold σ 2 is suitably computed (see thetheorem). The closed-loop configuration is maintained on the basis of the same decision logic used over [0, 1 ) (see steps 1 and 2). More precisely, one has Σ l Σ c, for [ 1, 2 ), where 2 = max( 2, 1 + ) and 2 is such that V (, τ) < σ 2, 2. For 2, Σ l is again switched in open-loop configuration Σ o (see step 3). The whole procedure is repeated every time V (, τ) exceeds the last computed threshold. Using the above CL, one has Σ l Σ c over the intervals I 0 = [0, 1 ), I i = [ i, i+1 ), i > 0, while Σ l Σ o over the intervals I i = [ i, i ), i > 0. The sequence of thresholds is computed on the basis of the first available value of V (, τ) over each interval I i when Σ l Σ c, namely for = τ over I 0 and for = i, for i > 0. For any > 0, the communication cost C() of the above CL is C() = K()/, where K() is the total number of time instants belonging to the time intervals I i wholly or partially occurred up to. Hence, the reduction of 10227

4 communication cost R() gained through the above CL with respect to periodic triggering up to any time instant > 0 is given by R() = 1 C() = K ()/, where K () is the total number of time instants belonging to the time intervals I i wholly or partially occurred up to. The index R() is surely positive because, as evidenced by points 2 and 4, the CL implies the existence of time intervals I i over which V (, τ) becomes smaller than the last computed threshold. Hence, the exact reduction of communication cost depends on the actual behavior of V (, τ), which in turn depends on many elements that, in general, are a priori unnown, lie the initial conditions of the plant, etc. Hence the numerical computation of R()can be only performed a posteriori by simulation or through real application. Let Wl m (, ) and γ τ be the gramian of the time-varying Σ m l and a finite overbound of Wl m (, τ) respectively. Moreover let x m l ( ), and Φ m l (, ) be the state and the state transition matrix of Σ m l respectively. The following theorem shows how to choose the thresholds σ i for the above CL solve the SCP. Theorem The CL described at steps 1-4 guarantees the state-output stability of the switched unforced system Σ l and the internal stability of the switched closed loop system Σ f provided Assumption 1 holds and the thresholds σ i are chosen according to V ( i 1, i 1 τ) σ i ρ 2 1 γ γ 1 = ωi, i = 1,2,, (12) τ where 0 = τ, ρ 1 = max Φ m l ( + 1,). Moreover the sequence {ω i } is monotonically decreasing. Proof: The CCL is such that the stabilizing Σ g restarts acting at each i + 1, namely one time instant after the triggering of e(h) has been restarted by the SCL. By the lemma, V (, τ) surely restarts to decrease according to (11) for i τ, but possible increasing jumps with respect to V ( i, i τ) may occur over the interval K i = [ i + 1, i τ). The worst case occurs when V (, τ) is increasing over all the interval K i. This means that i τ < i+1, or equivalently K i I i. Denoting by M i the overbound of V (, τ) in the worst case, it can be computed in the following way. By (10), for any initial state x m l (0) of Σ m l, one has for j = 0,,2τ 1 V ( i j, ) i j τ) x m 2 l ( i j τ γτ Φ m l ( i j τ, ) 2 ) x m 2 i τ l ( i τ γτ ρ 2 2(V ( i, i τ)γ τ )/γ 1 = Mi, i > 0, (13) where ρ 2 = max Φ m l ( i j τ, i τ). j=0,,2τ 1 Moreover (10) also implies V ( i, i τ) x m l x m l ) 2 ( i τ γτ ) 2 V ( i 1 τ ρ 2 1 γ τ ( i 1, i 1 τ) γ 1 ρ 2 1γ τ. As V ( i 1, i 1 τ) < σ i (by definition of i ), it readily follows that if σ i is chosen according to (12), then V ( i, i τ) < V ( i 1, i 1 τ), i > 0. (14) By (13), (14) implies M i < M i 1 and (12) implies ω i < ω i 1. Hence, in the worst case, one has that for any initial state x m l (0) of Σ m l, the maximum value assumed by V (, τ), for K i, i Z +, is upper bounded by the monotonically decreasing sequence {M i }, and for [ i τ, i+1 ), V (, τ) is decreasing according to (11). Besides, over each I i, i Z +, V (, τ) is upper bounded by the monotonocally decreasing sequence {ω i }. Two different asymptotic behaviors of Σ l are possible: 1) an infinite switching of Σ l between Σ c and Σ o taes place, 2) the switching stops after a finite time ī, for some ī > 0. In the first case V (, τ) is converging to zero for and exact asymptotic tracing is attained, because e() 0 for. The internal asymptotic stability of Σ m l follows by its minimality. It also follows that Σ l is asymptotically state-output stable and that Σ f is externally stable. In the second case one has V (, τ) < σ ī, ī, but no asymptotic convergence to zero of e( ) is guaranteed. In this case the simple internal stability of Σ m l as well as the state-output stability of Σ l and the external stability of Σ f are guaranteed. The internal stability of the switched closed loop system Σ f is proved showing that for any uniformly bounded external reference, also x( ), z( ) = [x T m( ),ξ T ( )] T and u( ) are uniformly bounded for in both the possible asymptotic behaviors of Σ l. Suppose at first that the switching stops after a finite time ī. For any ī, Σ l is permanently frozen in the openloop configuration Σ o. This means that ξ( ), x m ( ) and u( ) remain frozen in ξ( ī ), x m ( ī ) and u( ī ) respectively, over the interval [ ī, ). The uniform boundedness of x( ) follows by the observability of Σ d and by the boundedness ofy( ), consequence of the already proved external stability of Σ f. In the case of infinite switching one has that e( ) is converging asymptotically to zero so that the uniform boundedness of x m ( ) follows because Σ m contains the internal model of the bounded external reference Desoer et al. (1980). The uniform boundedness of ξ( ) can be proved exploiting Assumption 1. By the way the switched Σ f is defined, ξ(h) is the output of a switched system Σ s ob coinciding with Σ ob, for h I i, and with the output of an unforced system with identity dynamical matrix for h I i. Let Φs ob (, ) be the state transition matrix of Σs ob and let T i and T i be the lengths of I i and I i respectively. The CL is such that T i, so that by Assumption 1 one has Φ s ob ( i+1, i ) = Φ ob ( i+1, i ) < α < 1. Moreover Φ s ob ( i+1, i+1 ) = 1, so that Φ s ob ( i+1, i ) = Φ s ob ( i+1, i+1 ) Φ s ob ( i+1, i ) < 1 α < α < 1. As the switching sequence is infinite, there surely exists an integer T max < such that T i T max, i Z +. As a consequence, ω (0,1) and 0 0, a sufficiently large (ω) independent of 0 can be found such that Φ s ob (, 0) < ω for 0 +(ω). It can be seen that the value of (ω) is (ω) = N( +T max), where N is such that 10228

5 m ob α N < ω. This means that the switched observer Σ s ob is uniformly asymptotically stable. Hence the exponential stability follows from Rugh (1996). Recalling that both y( ) = r( ) e( ) and x m ( ) are uniformly bounded, eq. (5) implies the uniform boundedness of ξ( ). The uniform boundedness of u( ) follows by (6). The uniform boundedness of x( ) follows by the observability assumption of Σ d and bytheboundedness ofy( ), consequence of thealready proved external stability of Σ f. Remar 1 If matrices K, K m and L satisfying Assumption 1 can not be found, A ob results to be unstable and only the external stability of the switched Σ f can be inferred. In this case there is no need to eep Σ g acting over so long intervals I i to allow Φ s ob ( i+1, i ) = Φ ob ( i+1, i ) < α < 1, and the length of each I i only depends on the fulfillment of condition V (, τ) < σ i so that I i = [ i, i+1 ) = [ i, i+1 ). Hence the CL is modified switching Σ l from Σ c to Σ o as soon as V (, τ) < σ i. 5. NUMERICAL SIMULATION The proposed method has been tested on the same two degrees of freedom mass-spring-damper system considered in Yoo et al. (2002). Assuming the same nominal parameters of the above reference and choosing T s = 0.01s, the discretized plant Σ d is described by the following triplet (C,A,B) which is reachable, observable and has no invariant zero in z = 1, C = B = ,A = T ,(15) The external reference r( ) consists of two unitary step functions generated according to (1) choosing A r = C r = I 2, x r (0) T = [1,1] T. It is found that the matrices defining the controller equations (4)-(6) are: A m = B m = I 2, T L =, K m = , ], and Assumption [ K = is verified. The chosen eigenvalues of (A LC) and  are {0.3,0.2,0.15,0.1} and {0.8,0.75,0.7,0.65,0.6,0.55} respectively. It is found that Φ ob (,0) 167 (0.66), whence Φ ob (i +,i) < 1, = 13. Moreover, denoting by Φ m c (, ) the state transition matrix of Σ m c one has Φ m c (,0) 497 (0.8), whence, following the procedure given in Lemma 2 of Jetto et al. (2008), it is found that the functional V (, τ), corresponding to Σ m c, is converging to zero according to (11) with τ = 31. By definition of observability gramian, it is numerically found that the scalars γ 1 and γ τ satisfying γ 1 Wl m (,+ τ) γ τ are γ 1 = 2, γ τ = Moreover, it is also numerically found ρ 1 = As in Yoo et al. (2002), a simulation has been performed over 4,000 samples. Starting from the initial conditions: x m (0) = [0,0] T, x(0) = [0.1,0,0.1,0] T, ξ(0) = [0,0,0,0] T, the first computed value of V (, τ) is V (31,0) = , and by (12) it is found that the first threshold σ 1 must satisfy the inequality σ = ω 1. At time = 72, one has V (, τ) = V (72,41) < σ 1 so that 1 = 72. As a consequence the SCLstopsthefeedbac triggering ofe() for max ( 1, ) = 1 = 72, while the CCL stops the feedforward triggering of u() for = 73. In this case one has V (, τ) < σ 1, for 72 4,000. Hence Σ l = Σ c for I 0 = [0, 1 ) = [0;72), and Σ l = Σ o, for I 0 = [ 1,4000] = [72,4000]. The R() Fig. 2. Reduction of communication cost R() x 10 6 PI 1 ( ) time:sec x 10 5 PI 2 ( ) time:sec Fig. 3. Time behavior of PI 1 ( ) and PI 2 ( ). performance of the proposed CL has been evaluated in terms of reduction of communication cost R(), as shown in Fig. 2, and performance ( cost defined as Yoo et al. ( ) ( ) PI (2002): PC = max max 1() PI r 1(),max 2() ) r 2() with PI i () = y i () yi (), where y i() and yi () are the i-th components of the output of Σ d with reduced and 100% communication respectively and r i () is the amplitude of the i-th reference input component at time 10229

6 ( ) PI. For the considered example one has: max 1() r 1() = ( ) PI and max 2() r 2() = , so that PC = The behaviors of PI 1 ( ) and PI 2 ( ) are shown in Fig. 3. The two adopted indexes indicate an insignificant loss of control performance but an effective reduction of triggered networ messages tending to the 98%. 6. CONCLUSIONS This paper has proposed a new event-driven CL whose purpose is to reduce the number of triggered messages along the feedforward and feedbac paths of a closed-loop control system. The tracing of an external reference for MIMO sampled plants with non accessible state has been considered and internal stability conditions are stated. This, in particular appears to be the most significant novelty because, up to now, the internal stability has been studied only assuming a state feedbac controller. The numerical simulation reveals the effectiveness of the method both in terms of tracing performance and of reduced number of triggered events. REFERENCES K.E. Arzen, A simple event-based PID controller Proc. IFAC World Congress, Beijing, China, 1999 K. J. Astrom Event based Control Analysis and design of nonlinear systems A. Astolfi, & L. Marconi (Eds.), , Berlin: Springer-Verlag, C.A. Desoer, Y.T. Wang, Linear time-invariant robust servomechanism problem - A self-contained exposition Control and dynamic systems, , New Yor, Academic Press, W. P. M. H. Heemels, J.H. Sandee, P.P. J. Van Den Bosch P. P. J.(2008), Analysis of event-driven controllers for linear systems International Journal of Control, 81, , T Henningsson, E. Johannesson, A. Cervin, Sporadic event-based control of first-order linear stochastic systems Automatica, 44, , L. Jetto, V. Orsini, Supervised stabilisation of linear discrete-time systems with boundedvariation rate IET Control Theory & Applications, 2, , M. Lemmon, T. Chantem, X. Hu, and M. Zyowsy, On self-triggered full information H-infinity controllers Proc. Int. Conf. Hybrid Syst.: Comput. Control, 2007, J.Lunze, D. Lehmann, A state-feedbac approach to eventbased control Automatica 46, , M. Mazo, P. Tabuada, On event-triggered and selftriggered control over sensor/actuator networs Proc. of the 43th IEEE Conference on Decision and Control, Cancun, Mexico, P. G. Otanez, J.R. Moyne, D. M. Tilbury, Using deadbands to reduce communication in networed control systems Proc. of the American Control conference, Anchorage, W. J. Rugh, Linear System Theory, Prentice-Hall, NJ, P. Tabuada, Event-triggered real-time scheduling of stabilizing control tass IEEE Transaction Automatic Control, 52, , V. Vasyutynsyy, K. Kabitzsch, Implementation of PID controller withsend-on-deltasampling Proc. of the International Conference Control ICC 06, Glasgow, X. Wang, M.D. Lemmon, Event-triggering in distributed networed systems with data dropouts and delays Hybrid Systems: Computation and Control, San Francisco, X. Wang, M.D. Lemmon, Self-triggered feedbac control systems with finite-gain L 2 stability Trans. Autom. Contr., Vol. 54, , X. Wang, M.D. Lemmon, Self-triggering under stateindependentdisturbances IEEE Trans. Autom. Contr., Vol. 55, , X. Wang, M.D. Lemmon, Event-triggering in distributed networed control systems IEEE Trans. Autom. Contr., to appear. J.K. Yoo, D.M. Tilbury D.M., N.R. Soparar, Trading computation for bandwidth: reducing communication in distributed control systems using state estimators IEEE Transactions on Control Systems Technology, Vol. 10, ,