Stabilization of systems with asynchronous sensors and controllers

Automatica 81 (2017) 314–321

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Brief paper

Stabilization of systems with asynchronous sensors and controllers✩ Masashi Wakaiki a,1 , Kunihisa Okano b , João P. Hespanha c a

Department of Electrical and Electronic Engineering, Chiba University, Chiba, 263-8522, Japan

b

Graduate School of Natural Science and Technology, Okayama University, Okayama, 700-8530, Japan

c

Center for Control, Dynamical-systems and Computation (CCDC), University of California, Santa Barbara, CA 93106-9560, USA

article

info

Article history: Received 1 February 2016 Received in revised form 15 October 2016 Accepted 2 March 2017

Keywords: Networked control systems Clock offsets Parametric uncertainty

abstract We study the stabilization of networked control systems with asynchronous sensors and controllers. Offsets between the sensor and controller clocks are unknown and modeled as parametric uncertainty. First we consider multi-input linear systems and provide a sufficient condition for the existence of linear time-invariant controllers that are capable of stabilizing the closed-loop system for every clock offset in a given range of admissible values. For first-order systems, we next obtain the maximum length of the offset range for which the system can be stabilized by a single controller. Finally we illustrate the results with a numerical simulation. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In networked and embedded control systems, the outputs of plants are often sampled in a nonperiodic fashion and sent to controllers with time-varying delays. To address robust control with such imperfections, various techniques have been developed, for example, the input-delay approach (Fridman, Seuret, & Richard, 2004; Mirkin, 2007), the gridding approach (Donkers, Heemels, van de Wouw, & Hetel, 2011; Fujioka, 2009; Oishi & Fujioka, 2010), and the impulsive systems approach based on Lyapunov functionals (Naghshtabrizi, Hespanha, & Teel, 2010), on looped functionals (Briat & Seuret, 2012), and on clock-dependent Lyapunov functions (Briat, 2013); see also the surveys (Hespanha, Naghshtabrizi, & Xu, 2007; Hetel et al., 2017). In contrast to the references mentioned above, here we assume that time-stamps are used to provide the controller with information about the sampling times and the communication delays incurred by each measurement. In this approach, sensors send measurements to controllers together

✩ This material is based upon work supported by the National Science Foundation under Grant No. CNS-1329650. M. Wakaiki acknowledges Murata Overseas Scholarship Foundation for the support of this work. K. Okano is supported by JSPS Postdoctoral Fellowships for Research Abroad. The material in this paper was partially presented at the 2015 American Control Conference, July 1–3, 2015, Chicago, IL, USA. This paper was recommended for publication in revised form by Associate Editor Yasuaki Oishi under the direction of Editor Richard Middleton. E-mail addresses: [email protected] (M. Wakaiki), [email protected] (K. Okano), [email protected] (J.P. Hespanha). 1 Fax: +81 43 290 3345.

http://dx.doi.org/10.1016/j.automatica.2017.04.005 0005-1098/© 2017 Elsevier Ltd. All rights reserved.

with time-stamps, and the controllers exploit this information to mitigate the effect of variable delays and sampling periods (Garcia, Antsaklis, & Montestruque, 2014; Graham & Kumar, 2004; Nakamura, Hirata, & Sugimoto, 2008). However, when the local clocks at the sensors and at the controllers are not synchronized, the time-stamps and the true sampling instants do not match. Protocols to establish synchronization have been actively studied as surveyed in Rhee, Lee, Kim, Serpedin, and Wu (2009), and synchronization by the global positioning system (GPS) or radio clocks has been utilized in some systems. Nevertheless, synchronizing clocks over networks has fundamental limits (Freris, Graham, & Kumar, 2011), and a recent study (Jiang, Zhang, Harding, Makela, & Domíngues-García, 2013) has shown that synchronization based on GPS signals is vulnerable against attacks. In this paper, we study the stabilization problem of systems with asynchronous sensing and control. We assume that the controller can use the time-stamps but does not know the offset between the sensor and controller clocks, but we do assume that this offset is essentially constant over the time scales of interest. Our objective is to find linear time-invariant (LTI) controllers that achieve closed-loop stability for every clock offset in a given range. We formulate the stabilization of systems with clock offsets as the problem of stabilizing systems with parametric uncertainty, which can be regarded as the simultaneous stabilization of a family of plants, as studied in Vidyasagar (1985, Sec. 5.4) and Vidyasagar and Viswanadham (1982). However, we had to overcome a few technical difficulties that distinguish the problem considered here from previously published results: Infinitely many plants: We consider a family of plant models that is indexed by a continuous-valued parameter. Such a family includes

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infinitely many plants, but the approaches for simultaneous stabilization, e.g., in Shi and Qi (2009) exploit the property that the number of plant models is finite. Nonlinearity of the uncertain parameter: In this work, the uncertain parameter appears in a non-linear form. Therefore, it is not suitable to use the techniques based on linear matrix inequalities (LMIs) in de Oliveira, Bernussou, and Geromel (1999) for the robust stabilization of systems with polytopic uncertainties. Although the robust stability analysis based on continuous paths of systems with respect to the ν -gap metric was developed in Cantoni, Jönsson, and Kao (2012), controller designs based on this approach have not been fully investigated. Common unstable poles and zeros: Earlier studies on simultaneous stabilization consider a restricted class of plants. For example, the sufficient condition in Blondel, Campion, and Gevers (1993) is obtained for a family of plants with no common unstable zeros or poles. The set of plants in Maeda and Vidyasagar (1984) has common unstable zeros (or poles) but all the plants are stable (or minimum-phase). These assumptions are not satisfied for the systems in the present paper. We make the following technical contributions for multi-input systems and first-order systems: First we consider multi-input systems and obtain a sufficient condition for stabilization with asynchronous sensing and control. We construct a stabilizing controller from the solution of an appropriately defined H ∞ control problem. The above mentioned difficulties found in the simultaneous stabilization problem we consider are circumvented by exploiting geometric properties on H ∞ . For first-order systems, we obtain an explicit formula for the exact bound on the clock offset that can be allowed for stability. This result is based on the stabilization of interval systems (Ghosh, 1988; Olbrot & Nikodem, 1994), to which our problem can be reduced for first-order plants. We start by formulating the problem in the context of state feedback without disturbances and noise, but we show in Section 3.2 that the above results also apply for output feedback with disturbances and noise. The authors in the previous study (Okano, Wakaiki, & Hespanha, 2015) have considered systems with time-varying clock offsets and have proposed a stabilization method with causal controllers, based on the analysis of data rate limitations in quantized control. The stability analysis and the L2 -gain analysis of systems with variable clock offsets have been investigated in Wakaiki, Okano, and Hespanha (2015, 2016), respectively. The major difference with respect to those studies is that here we consider only constant offsets but design stabilizing LTI controllers. This paper is based on the conference paper (Wakaiki, Okano, & Hespanha, 2015), but here we extend the preliminary results for single-input systems to the multi-input case. The remainder of the paper is organized as follows. Section 2 introduces the closed-loop system we consider and presents the problem formulation. Section 3 is devoted to the discretization of the closed-loop system. In Section 4, we obtain a sufficient condition for the stabilizability of general-order systems. In Section 5, we derive the exact bound on the permissible clock offset for firstorder systems. A numerical example is presented in Section 6. Notation and definitions: We denote by Z+ the set of non-negative integers. The symbols D and T denote the open unit disc {z ∈ C : |z | < 1} and the unit circle {z ∈ C : |z | = 1}, respectively. We denote by Dc the complement of the open unit disc {z ∈ C : |z | ≥ 1}. A square matrix F is said to be Schur stable if all its eigenvalues lie in the unit disc D. We say that a discrete-time LTI system ξk+1 = F ξk + Guk , yk = H ξk is stabilizable (detectable) if there exists a matrix K (L) such that F − GK (F − LH) is Schur stable. We also use the terminology (F , G) is stabilizable (respectively, (F , H ) is detectable) to denote this same concept.

Fig. 1. Closed-loop system with a time-stamp aware estimator.

We denote by RH ∞ the space of all bounded holomorphic realrational functions in D. The field of fractions of RH ∞ is denoted by RF ∞ . For a commutative ring R, M(R) denotes the set of matrices with entries in R, of any order. For M ∈ M(C), ∥M ∥ denotes the induced 2-norm. For G ∈ M(RH ∞ ), the H ∞-norm is defined as

G12 11 ∥G∥∞ = supz ∈D ∥G(z )∥. For G = GG21 ∈ M(RF ∞ ) and G22 ∞ Q ∈ M(RF ), we define a lower linear fractional transformation of G and Q as Fℓ (G, Q ) := G11 + G12 Q (I − G22 Q )−1 G21 . A pair (N , D) in M(RH ∞ ) is said to be right coprime if the Bezout identity XN + YD = I holds for some X , Y ∈ M(RH ∞ ). P ∈ M(RF ∞ ) admits a right coprime factorization if there exist D, N ∈ M(RH ∞ ) such that P = ND−1 and the pair (N , D) is right ˜ , N˜ ) in M(RH ∞ ) is left coprime if the coprime. Similarly, a pair (D ˜ Y˜ = I holds for some X˜ , Y˜ ∈ M(RH ∞ ). Bezout identity N˜ X˜ + D ∞ P ∈ M(RF ) admits a left coprime factorization if there exist ˜ N˜ ∈ M(RH ∞ ) such that P = D˜ −1 N˜ and the pair (D˜ , N˜ ) D,

is left coprime. If P is a scalar-valued function, then we use the expressions coprime and coprime factorization. 2. Problem statement Consider the following LTI plant:

ΣP : x˙ (t ) = Ax(t ) + Bu(t ),

(1)

where x(t ) ∈ R and u(t ) ∈ R are the state and the input of the plant, respectively. As shown in Fig. 1, this plant is connected through a sampler and a zero-order hold (ZOH) to a time-stamp aware estimator and a controller, which will be described soon. Let s1 , s2 , . . . be sampling instants from the perspective of the controller clock. A sensor measures the state x(sk ) and sends it to a controller together with a time-stamp. However, since the sensor and the controller may not be synchronized, the time-stamp determined by the sensor typically includes an unknown offset with respect to the controller clock. In this paper, we assume that the clock offset is constant. Although clock properties are affected by environments such as temperature and humidity, the change of such properties is slow for the time scales of interest. Furthermore, the difference of clock frequencies can be ignored. This is justified by noting that time synchronization techniques, like the one proposed in He, Cheng, Shi, Chen, and Sun (2014), can achieve asymptotic convergence of the clock frequencies (in the mean-square sense), even in the presence of random network delays. We thus assume that the time-stamp sˆk reported by the sensor is given by n

sˆk = sk + ∆

(k ∈ N)

m

(2)

for some unknown constant ∆ ∈ R. Let h > 0 be the update period of the ZOH. The control signal u(t ) is assumed to be piecewise constant and updated periodically at times tk = kh (k ∈ N) with values uk computed by the controller: u(t ) = uk for t ∈ [tk , tk+1 ). We place a basic assumption for stabilization of sampled-data systems.

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3.1. Discretized system and its basic properties

Fig. 2. Sampling instants sk , reported time-stamps sˆk , and updating instants tk of the zero-order hold.

Assumption 1 (Stabilizability and Non-pathological Control Update). The plant (A, B) is stabilizable and the update period h is non-pathological, that is, (λp − λq )h ̸= 2π jℓ (ℓ = ±1, ±2, . . .) for each pair (λp , λq ) of eigenvalues of A. While the ZOH updates the control signal u(t ) periodically, the true sampling times sk and the reported sampling times sˆk may not be periodic. However, we do assume that both sk and sˆk do not fall behind tk by more than the ZOH update period h. This assumption is formally stated as follows. Assumption 2 (Bounded Clock Offset). For every k ∈ Z+ , sk , sˆk ∈ [tk , tk+1 ). This assumption implies that the clock offset ∆ is smaller than the control update period h, which holds in most mechatronics systems. In fact, control update periods for mechatronics systems generally take values from 100 µs to 10 ms, while recent clock synchronization algorithms such as the IEEE 1588 Precision Time Protocol (PTP) (IEEE Standard, 2008) make clock offsets smaller than a few tens of microseconds. Fig. 2 shows the timing diagram of the sampling instants sk , the reported time-stamps sˆk , and updating instants tk of the control inputs. The controller side is comprised of a time-stamp aware estimator and a controller as in the model-based or emulationbased control of networked control systems (Garcia et al., 2014). The time-stamp aware estimator generates the state estimate xˆ (tk+1 ) ∈ Rn from the data (x(sk ), sˆk ) according to the following dynamics:

ΣE :



x˙ˆ (t ) = Axˆ (t ) + Bu(t ) xˆ (ˆsk ) = x(sk ),

(tk < t ≤ tk+1 ) (k ∈ Z+ ).

Lemma 4. Define

ξk :=

x(tk ) − xˆ (tk ) . xˆ (tk )





The dynamics of the discretized system Σd comprised of the plant ΣP , the estimator ΣE , the ZOH, and the sampler can be described by the following equations:

Σd : ξk+1 = F∆ ξk + G∆ uk ,

ηk = H∆ ξk ,

(5)

where Λ := eAh , Θ := e−A∆ − I, and

   −ΛΘ −ΛΘ , H∆ := 0 I Λ(I + Θ ) Λ(I + Θ )    h−∆   h e−Aτ dτ B e −Aτ d τ − ( I + Θ ) Λ   0 0 . G∆ :=   h−∆   −Aτ Λ(I + Θ ) e dτ B 

F∆ :=

(6)

0

Proof. Using Λ = eAh , we have from the state equation (1) that x(tk+1 ) = Λx(tk ) + Λ

h



e−Aτ dτ · Buk .

(7)

0

We compute xˆ (tk+1 ) in terms of x(tk ) and uk . It follows from the dynamics of the estimator ΣE in (3) that xˆ (tk+1 ) = eA(tk+1 −ˆsk ) xˆ (ˆsk ) +

tk+1



eA(tk+1 −τ ) Bdτ · uk

(8)

sˆk

(3)

Note that if the time-stamp is correct, i.e., sk = sˆk , then this estimator consistently produces xˆ (t ) = x(t ) for all t, perfectly compensating transmission delays. Time-stamp aware estimators have been used to compensate for network-induced imperfections, e.g., in Garcia et al. (2014), Graham and Kumar (2004) and Nakamura et al. (2008). The controller is a discrete-time LTI system and generates the control input uk based on the state estimate xˆ k := xˆ (tk ):

 ζk+1 = Ac ζk + Bc xˆ k ΣC : uk = Cc ζk + Dc xˆ k ,

The following lemma provides a realization for the discretized system:

(4)

where ζk ∈ Rnc is the state of the controller. The objective of the present paper is to find a discrete-time LTI controller ΣC as in (4) that achieves closed-loop stability for every clock offset in a given range of admissible values. Specifically, we want to solve the following problem: Problem 3. Given an offset interval [∆, ∆], determine if there exists a controller ΣC as in (4) such that x(t ), xˆ (t ) → 0 as t → ∞ and ζk → 0 as k → ∞ for every ∆ ∈ [∆, ∆] and for every initial states x(0) and ζ0 . Furthermore, if one exists, find such a controller ΣC .

and xˆ (ˆsk ) = x(sk ) = eA(sk −tk ) x(tk ) +

sk



eA(sk −τ ) Bdτ · uk .

(9)

tk

Since tk+1 − tk = h and sˆk = sk + ∆, it follows that eA(tk+1 −ˆsk ) · eA(sk −tk ) = eA(h−∆) ,

(10)

and also that eA(tk+1 −ˆsk )



sk tk

eA(sk −τ ) dτ =



sˆk tk + ∆

eA(tk+1 −τ ) dτ .

(11)

Using Λ = eAh and Θ = e−A∆ − I, we conclude from (8)–(11) that xˆ (tk+1 ) = Λ(I + Θ )x(tk ) + Λ(I + Θ )



h−∆

e−Aτ Bdτ · uk .

(12)

0

From (7) and (12), we obtain the F∆ and G∆ in (6). Moreover, we have H∆ = [0 I ] by the definition of the extended state ξk .  Next we show that if the extended state ξk and the controller state ζk converge to the origin, then the intersample values of x and xˆ also converge to the origin.

3. Discretization of the closed-loop system To solve Problem 3, we discretize the system comprised of the plant ΣP , the estimator ΣE , the ZOH, and the sampler. In this section, we obtain a realization for the discretized system and describe its basic properties related to stability, stabilizability, and detectability. Moreover, we extend the discretized system to scenarios with disturbances/noise and output feedback.

Proposition 5. For the discretized system Σd in Lemma 4, we have that ξk , ζk → 0 as k → ∞ if and only if x(t ), xˆ (t ) → 0 as t → ∞ and ζk → 0 as k → ∞. Proof. The statement that x(t ), xˆ (t ) → 0 as t → ∞ and ζk → 0 as k → ∞ imply ξk , ζk → 0 as k → ∞, follows directly from the definition of ξk .

M. Wakaiki et al. / Automatica 81 (2017) 314–321

To prove the converse statement, assume that ξk , ζk → 0 as k → ∞. Then xˆ (tk ) = H∆ ξk → 0 and x(tk ), uk → 0 as k → ∞. Since

∥x(tk + τ )∥ ≤ e

∥A∥h

∥x(tk )∥ +

h



e∥A∥h ∥B∥dt · ∥uk ∥ 0

for all k ∈ Z+ and all τ ∈ [0, h), we derive x(t ) → 0 (t → ∞). Similarly, we see from the dynamics of the estimator ΣE that xˆ (t ) → 0 as t → ∞. This completes the proof.  This proposition allows us to conclude Problem 3 can be solved by finding LTI controllers ΣC achieving ξk , ζk → 0 (k → ∞) for every ∆ ∈ [∆, ∆] and for every initial states ξ0 and ζ0 . The following result allows us to conclude that the discretized system Σd is detectable and stabilizable for all ∆ and almost all h if the plant (A, B) is stabilizable.

317

where d(t ) ∈ Rn and n(t ), y(t ) ∈ Rp are the disturbance, measurement noise, and output of the plant, respectively. As in Garcia et al. (2014, Chap. 3), Xu and Hespanha (2005), and the references therein, we assume that a smart sensor is co-located with the plant and that the sensor has the following observer to generate the state estimate, which is sampled and sent to the controller side:

ΣO : x˙¯ (t ) = Ax¯ (t ) + Bu(t ) + L(y(t ) − C x¯ (t )), where x¯ (t ) ∈ Rn is the state estimate and L is an observer gain such that A − LC is Hurwitz. The sampler sends the state estimate x¯ , and the resulting dynamics of the time-stamp aware estimator ΣE′ is provided by

ΣE : ′



x˙ˆ (t ) = Axˆ (t ) + Bu(t ) xˆ (ˆsk ) = x¯ (sk ) + wk ,

(tk ≤ t < tk+1 ) (k ∈ Z+ ),

Proposition 6. The discretized system Σd in (5) is detectable for all ∆ and h. Moreover, Σd is stabilizable for all ∆ if Assumption 1 holds.

where wk ∈ Rn is the quantization noise. A calculation similar to the one performed in the proof of Lemma 4 can be used to show that the dynamics of the discretized system Σd′ is given by

¯ ∆ , H¯ ∆ ) of the Proof. Let us first obtain another realization (F¯∆ , G discretized system Σd in (5). We can transform F∆ into

Σd′ : ξk+1 = F∆ ξk + G∆ uk + dk ,

 ¯F∆ := T −1 F∆ T = Λ

−Θ where T := I +Θ





0 , 0

0

−I I



where dk :=

.



h

J1 :=

h−∆

J2 :=

0

e−Aτ dτ ,



0



0 0

.

(13)

−Λ(I + Θ )−1



,

0

(14)

¯ ∆ is Schur stable. Therefore, the discretized and clearly F¯∆ − L∆ H system Σd is detectable for all h and ∆. To show stabilizability, we use the well-known rank conditions (see, e.g., Zhou, Doyle, & Glover, 1996, Sec. 3.2). We have that [zI − F¯∆ G¯ ∆ ] is full row rank for all z ∈ Dc if and only if zI − Λ



  ΛJ1 B = zI − eAh

h



Aτ

e Bdτ

is full row rank for all z ∈ Dc . Hence, the discretized system Σd is stabilizable for all ∆ if Assumption 1 holds. 3.2. Extension to the output feedback case with disturbances and noise Instead of ΣP in (1), consider a plant ΣP′ with disturbances, noise, and output feedback:

ΣP :

x˙ (t ) = Ax(t ) + Bu(t ) + d(t ) y(t ) = Cx(t ) + n(t ),







e(A−LC )(sk −τ ) (d(τ ) − Ln(τ )) − eA(sk −τ ) d(τ ) dτ



The only difference from the original idealized system Σd in (5) is that Σd′ has the disturbance dk . Hence, for the output feedback case with bounded disturbances and noise, solutions of Problem 3 achieve the boundedness of the closed-loop state. Proposition 7. Assume that ξk , ζk → 0 as k → ∞ for the idealized system Σd in Lemma 4 (in the context of state feedback without disturbances and measurement noise). If d(t ), n(t ), and wk are bounded for all t ≥ 0 and all k ∈ Z+ , then the states x(t ), x¯ (t ), xˆ (t ), and ζk are also bounded for all t ≥ 0 and all k ∈ Z+ . Moreover, if d(t ) = n(t ) = wk = 0 for all t ≥ 0 and all k ∈ Z+ , then x(t ), x¯ (t ), xˆ (t ), and ζk converge to the origin. Proof. Since dk is bounded for every k ≥ 0 and every sk , sˆk ∈ [tk , tk+1 ), it follows that ξk and ζk are also bounded for all k ≥ 0. The rest of the proof follows the similar lines as that of Proposition 5, and hence it is omitted. See also Wakaiki et al. (2016) for the L2 -gain analysis of systems with time-varying offsets.



0

′

eA(tk+1 −τ ) d(τ )dτ − d2,k

ek := x(tk ) − x¯ (tk ).

Then we have that

¯∆ = F¯∆ − L∆ H

and

tk

 ΛJ1 B G∆ = −Λ(J1 − (I + Θ )J2 )B − ΘΛJ1 B 

¯ ∆ := H∆ T = [I + Θ I ]. We have thus another realization and H (F¯∆ , G¯ ∆ , H¯ ∆ ) for Σd . Next we check detectability and stabilizability by using the ¯ ∆ , H¯ ∆ ). Define realization (F¯∆ , G L∆ :=

sk

+

  Λ(I + Θ )−1

(15)



then we obtain

¯ ∆ := T G

ηk = H∆ ξk ,

d2,k := −eA(tk+1 −ˆsk ) e(A−LC )(sk −tk ) ek − wk

0

−1

⊤ d⊤ 2 ,k ]

tk



e −Aτ d τ ,

tk+1

d1,k :=

Furthermore, if we define



[ d⊤ 1,k

4. Controller design via simultaneous stabilization 4.1. Preliminaries We first consider a general simultaneous stabilization problem not limited to the system introduced in Section 2. The transfer function P of the system ξk+1 = F ξk + Guk , yk = H ξk is usually defined by the Z-transform of the system’s impulse response, i.e., H (zI − F )−1 G, but in this paper, we define the transfer function P by P (z ) := H (1/z · I − F )−1 G for consistency of the Hardy space theory; see Vidyasagar (1985, Sec. 2.2) for details. Hence the transfer function of a causal system is not proper. We say that C ∈ M(RF ∞ ) stabilizes P ∈ M(RF ∞ ) if (I + PC )−1 ,

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M. Wakaiki et al. / Automatica 81 (2017) 314–321

C (I + PC )−1 , and (I + PC )−1 P belong to M(RH ∞ ). We recall that when these three transfer functions belong to M(RH ∞ ), they will have no poles in the closed unit disk. Consider the family of plants Pθ ∈ M(RF ∞ ) parameterized by θ ∈ S, where S is a nonempty parameter set, and assume that we have a doubly coprime factorization of Pθ over RH ∞



Yθ −N˜ θ

Xθ ˜θ D



Dθ Nθ

 −X˜ θ = I, Y˜θ

(16)

1 ˜ −1 ˜ where Pθ = Nθ D− θ and Pθ = Dθ Nθ are a right coprime factorization and a left coprime factorization, respectively. We explicitly construct the matrices in (16) using a stabilizable and detectable realization of Pθ ; see, e.g., Vidyasagar (1985, Theorem 4.2.1). The following theorem provides a necessary and sufficient condition for simultaneous stabilization:

Theorem 8 (Vidyasagar, 1985; Vidyasagar & Viswanadham, 1982). Given a nonempty set S, consider the plant Pθ having a doubly coprime factorization (16) for each θ ∈ S. Fix θ0 ∈ S and define



Uθ





˜θ Vθ := D

N˜ θ

  Y˜θ 0 X˜ θ0

−Nθ0



Dθ0

(θ ∈ S ).

(17)

Then (Vθ , Uθ ) is right coprime for every θ ∈ S. Moreover, there exists a controller that stabilizes Pθ for every θ ∈ S if and only if there exists Q ∈ M(RH ∞ ) such that for all θ ∈ S,

(Uθ + Vθ Q )−1 ∈ M(RH ∞ ).

(18)

Such a stabilizing controller is given by C := (X˜ θ0 + Dθ0 Q )(Y˜θ0 − Nθ0 Q )−1 .

(19)

Remark 9. Although the simultaneous stabilization of a finite family of plants is considered in Vidyasagar (1985, Sec. 5.4) and Vidyasagar and Viswanadham (1982), generalization to an arbitrary family of plants is readily apparent, as mentioned in the last paragraph of Section 3 in Vidyasagar and Viswanadham (1982). Remark 10. A left coprime factorization of stabilizing controllers is used in Vidyasagar (1985, Sec. 5.4) and Vidyasagar and Viswanadham (1982), whereas we represent controllers by a right coprime factorization in (19). Therefore, Theorem 8 is slightly different from its counterpart in Vidyasagar (1985, Sec. 5.4) and Vidyasagar and Viswanadham (1982). 4.2. Robust controller design It is generally not easy to verify in a computationally efficient fashion that a transfer function Q satisfying (18) exists. In the next theorem, we develop a simple sufficient condition for (18) to hold, by exploiting geometric properties on H ∞ inspired by results on strong stabilization (Zeren & Özbay, 2000). Theorem 11. Given a nonempty set S, assume that each plant Pθ (θ ∈ S) has a doubly coprime factorization (16) such that there exist θ0 ∈ S, W ∈ M(RH ∞ ), and R(θ ) ∈ M(R) satisfying D˜ θ = D˜ θ0 and N˜ θ (z ) − N˜ θ0 (z ) = R(θ )W (z ),

(20)

for all θ ∈ S. If there exists Q ∈ M(RH ) satisfying the following H ∞ -norm condition: ∞

∥W (X˜ θ0 + Dθ0 Q )∥∞ <

1 sup ∥R(θ )∥

,

(21)

θ∈S

then Q satisfies (18), and hence the controller C in (19) stabilizes Pθ for every θ ∈ S.

˜ θ = D˜ θ0 , it follows Proof. We define Uθ and Vθ as in (17). Since D ˜ ˜ ˜ from (17) and the Bezout identity Dθ0 Yθ0 + Nθ0 X˜ θ0 = I in (16) that ˜ θ Y˜θ0 + N˜ θ X˜ θ0 = I + (N˜ θ − N˜ θ0 )X˜ θ0 . Uθ = D −1 1˜ ˜− Moreover, since D θ0 Nθ0 = Nθ0 Dθ0 , we obtain

˜ θ Nθ0 + N˜ θ Dθ0 = (N˜ θ − N˜ θ0 )Dθ0 . Vθ = −D Hence Uθ + Vθ Q = I + (N˜ θ − N˜ θ0 )(X˜ θ0 + Dθ0 Q ). Since (I + Φ )−1 ∈ M(RH ∞ ) for all Φ ∈ M(RH ∞ ) satisfying ∥Φ ∥∞ < 1, it follows that if

∥(N˜ θ − N˜ θ0 )(X˜ θ0 + Dθ0 Q )∥∞ < 1 (θ ∈ S ),

(22)

then (18) holds for all θ ∈ S. From the assumption (20),

∥(N˜ θ − N˜ θ0 )(X˜ θ0 + Dθ0 Q )∥∞ ≤ ∥R(θ )∥ · ∥W (X˜ θ0 + Dθ0 Q )∥∞ . Hence if Q satisfies (21) for all θ ∈ S, then (22) holds, and consequently Pθ is simultaneously stabilizable by C in (19) from Theorem 8.  The proposition below shows that our discretized system Σd in ˜ θ and N˜ θ that appear in (5) always satisfies the assumptions on D Theorem 11. This result also provides the matrices R and W in (20) without explicitly calculating a coprime factorization of Pθ for all θ ∈ S. Proposition 12. Define the transfer function P∆ (z ) := H∆ (1/z · I − F∆ )−1 G∆ . For all ∆ ∈ (−h, h), there exists a doubly coprime ˜ ∆ (z ) = D˜ 0 (z ) = I − zeAh , and (20) holds factorization (16) such that D with R(∆) :=

∆



eA(h−τ ) Bdτ ∈ Rn×m

(23)

0

W (z ) := z (z − 1) ∈ RH ∞ .

(24)

¯ ∆ , H¯ ∆ ) in the proof of Proof. Consider the realization (F¯∆ , G Proposition 6. For every ∆ ∈ (−h, h), the matrix L∆ in (13) ¯ ∆ as shown in (14). From achieves the Schur stability of F¯∆ − L∆ H ˜ ∆ , e.g., in Vidyasagar (1985, Theorem 4.2.1), we the realization of D ˜ ∆ as can write D ˜ ∆ (z ) = I − H¯ ∆ (1/z · I − (F¯∆ − L∆ H¯ ∆ ))−1 L∆ = I − z Λ. D Noticing that the far right-hand side of the equation above does not ˜ ∆ (z ) = D˜ 0 (z ) = I − z Λ. depend on ∆, we have D ˜ 0 (P∆ − P0 ). From the realization It follows that N˜ ∆ − N˜ 0 = D (F¯∆ , G¯ ∆ , H¯ ∆ ), we see that P∆ (z ) = z (I + Θ )(I − z Λ)−1 ΛJ1



 − Λ(J1 − (I + Θ )J2 ) − ΘΛJ1 B. Since Θ = 0 and J2 = J1 for ∆ = 0, it follows that P∆ (z ) − P0 (z ) = z Θ (I − z Λ)−1 ΛJ1



 − Λ(J1 − (I + Θ )J2 ) − ΘΛJ1 B. (25)  ∆ −Aτ ¯, On the other hand, we have J1 − (I + Θ )J2 = 0 e dτ =: Θ

and

¯A=− Θ

∆

 0



d −Aτ e dτ



dτ = −Θ .

(26)

Since A(I − z Λ) = A − zAeAh = A − zeAh A = (I − z Λ)A, it follows that A(I − z Λ)−1 = (I − z Λ)−1 A. Therefore we derive from (25)

¯ (I − z Λ)−1 (−I + z Λ(I − AJ1 )) B. P∆ (z ) − P0 (z ) = z ΛΘ

M. Wakaiki et al. / Automatica 81 (2017) 314–321

Similarly to (26), we have I − AJ1 = Λ−1 , and hence P∆ (z ) − ¯ (I − z Λ)−1 B. Since λ, Θ ¯ , and (I − z Λ)−1 are P0 (z ) = z (z − 1)ΛΘ commutative, we derive

¯ B, P∆ (z ) − P0 (z ) = z (z − 1)(I − z Λ)−1 ΘΛ ¯ B. Thus (20) holds with R in (23) and N˜ ∆ (z ) − N˜ 0 (z ) = z (z − 1)ΘΛ and W in (24).  Define 1

γ :=

  ∆

max 

∆∈[∆,∆]

0

. 

(27)

eA(h−τ ) Bdτ 

W X˜ 0 Φ := −I

Consider an unstable first-order plant: x˙ = ax + bu with a scalar a > 0. If a < 0, the stabilization problem is trivial because a zero control input u(t ) = 0 leads to the stability of the closedloop system. Hence in the remainder of this section, we focus our attention on the case a > 0. The following theorem provides the exact bound on clock offsets for first-order systems: Theorem 14. Assume −h < ∆ < 0 < ∆ < h. There exists a controller that stabilizes the discretized system Σd in (5) for all ∆ ∈ [∆, ∆] if and only if 2 log(eah + 1) − log(eah − 1)



From Theorem 11, to obtain a controller ΣC as in (4), it is enough to solve the following suboptimal problem: Find Q ∈ M(RH ∞ ) satisfying ∥W (X˜ 0 − D0 Q )∥∞ < γ . This problem is equivalent to a standard suboptimal H ∞ control problem (Zhou et al., 1996, Chaps. 16, 17): Find Q ∈ M(RH ∞ ) such that ∥Fℓ (Φ , Q )∥∞ < γ , where Φ is defined by



319



WD0 . 0

(28)

The results of this section can be summarized through the following controller design algorithm: Algorithm 1. (1) Using the realization

  Ah e (F¯0 , G¯ 0 , H¯ 0 ) =  0





0 ,  0



h

e Bdτ  , Ah

0



I

  I ,

0

∆−∆<

the matrix L0 = [e 0] , and an arbitrary matrix K0 such ¯ 0 K0 is Schur stable, set that Φ := F¯0 − G ⊤

¯0 D0 (z ) := I − K0 (1/z · I − Φ )−1 G ¯ 0 (1/z · I − Φ )−1 G¯ 0 N0 (z ) := H X˜ 0 (z ) := K0 (1/z · I − Φ )

−1

L0

Y˜0 (z ) := I + H0 (1/z · I − Φ )−1 K0

P∆ ( z ) =

b

Remark 13. We have from Proposition 12 that P∆ = P0 + 1 WD− 0 R(∆) for constant ∆, where P∆ is expressed as the nominal 1 component P0 plus the uncertainty block WD− 0 R(∆). If we obtain a similar formula for the case of time-varying offsets as studied for systems with aperiodic sampling in Fujioka (2009), we can deal with the stabilization problem of systems with time-varying offsets through a small gain theorem. Although the uncertainty part of the discretized system Σd may be non-causal, the small gain theorem for systems with non-causal uncertainty in Ünal and Iftar (2008) can be used. This extension is a subject for future research. 5. Exact bound on offsets for first-order systems In this section, the plant class is restricted to first-order systems, and we provide an explicit formula for the exact bound on the clock offset that LTI controllers can allow.

.

(29)

 (eah − 1)z eah z (z − 1) −a ∆ − ( e − 1 ) , 1 − eah z 1 − eah z



a

(30)

which is the so-called interval system (Ghosh, 1988; Olbrot & Nikodem, 1994). From the stabilization results in Ghosh (1988) and Olbrot and Nikodem (1994), there exists a controller stabilizing P∆ for all ∆ ∈ [∆, ∆] if and only if the associated Pick matrix is positive definite, i.e.,



1



1 − φ2

1

  > 0,

1 − 1/

(31)

e2ah

where φ := 1 − ea(∆−∆)/2 / 1 + ea(∆−∆)/2 . From the Schur complement formula, (31) is equivalent to

 



1 − φ2 1 − 1/e2ah

> 1.



(32)

We see that (32) is

W (z ) := z (z − 1). (2) For a given offset interval [∆, ∆], set γ as in (27), and solve the H ∞ control problem (Zhou et al., 1996, Chaps. 16, 17): Find Q ∈ M(RH ∞ ) such that ∥Fℓ (Φ , Q )∥∞ < γ , where Φ is defined by (28). (3) If the H ∞ control problem is not solvable, then the algorithm fails. Otherwise the transfer function C of the controller ΣC is given by C = (X˜ 0 + D0 Q )(Y˜0 − N0 Q )−1 .

a

Proof. As in Section 3, taking the Z-transform and then mapping z → 1/z, we obtain the transfer function P∆ of the discretized system Σd :

1

A⊤ h



ea(∆−∆) <



eah + 1

2

eah − 1

.

Taking the logarithm function of both sides gives the desired conclusion.  Remark 15. Although here we use the results of Ghosh (1988) and Olbrot and Nikodem (1994), an alternative proof based on Theorem 8 is presented in Wakaiki, Okano, and Hespanha (2017). Remark 16. In the case a = 0, the transfer function of the discretized system, P∆ , is given by P∆ ( z ) =

hz 1−z

− 1z .

Similarly to the case a > 0, one can show that there exists a controller stabilizing P∆ for all ∆ ∈ (−h, h). This result is consistent with that in the case when a → 0 in Theorem 14, but we omit the proof for brevity. 6. Numerical example Consider the unstable batch reactor studied in Rosenbrock (1974), where the system matrices A and B in (1) are

320

M. Wakaiki et al. / Automatica 81 (2017) 314–321

given by 1.38 −0.5814 A :=  1.067 0.048

−0.2077 −4.29 4.273 4.273 





0

5.679 B :=  1.136 1.136

6.715 0 −6.654 1.343

 −5.676 0.675  5.893  −2.104

0 0  . −3.146 0

This example has been developed over the years as a benchmark example for networked control systems, and its data were transformed by a change of basis and time scale (Rosenbrock, 1974). Here we compare the proposed method with the robust stabilization method in Doyle and Stein (1981) and Vidyasagar (1985, Chap. 7) based on the following fact: Consider a family of plants P∆ ∈ M(RF ∞ ) with ∆ ∈ [∆, ∆]. Assume that P∆ has no poles on T and the same number of unstable poles for every ∆ ∈ [∆, ∆] and that a function r ∈ RH ∞ satisfies jω

jω

jω

∥P∆ (e ) − P0 (e )∥ < |r (e )|

(33)

for all ∆ ∈ [∆, ∆] and all ω ∈ [0, 2π ]. If the controller C ∈ M(RF ∞ ) stabilizes P0 and satisfies

  rC (I − P0 C )−1 

∞

≤ 1,

(34)

then C stabilizes P∆ for all ∆ ∈ [∆, ∆]. The order of such a controller is typically equal to the order of the following transfer function:





0 I

rI . P0

We compute the length of the allowable offset interval [∆, ∆] obtained from the sufficient condition (21) for each h ∈ [0.2, 3.6], which is shown as the solid line in Fig. 3. On the other hand, the dashed line in the figure represents the length of the offset interval [∆, ∆] obtained from the robust control approach that leads to the condition (34) with an appropriate function r ∈ RH ∞ satisfying (33). For example, we use r (z ) = 0.1766(z − 1)/(0.9389z − 1) for h = 1 and [∆, ∆] = [−0.02, 0.02], and this r satisfies (33) and 2π



1 2π



 |r (ejω )| − ∥P∆ (ejω ) − P0 (ejω )∥ dω ≤ 8.5 × 10−3

0

for all ∆ ∈ [∆, ∆] = [−0.02, 0.02]. The solid line is obtained by finding the maximum and minimum of ∆ that satisfies the condition

   

∆

A(h−τ )

e 0

  Bdτ  ≤

1 min ∥Fℓ (Φ , Q )∥∞

,

Q ∈RH ∞

whereas to derive the dashed line, we first calculate r satisfying (33) for a fixed [∆, ∆] and then check the existence of a controller C that stabilizes P0 and achieves the H ∞ -norm condition (34). We see from Fig. 3 that the proposed sufficient condition (21) is less conservative than (34). Consider the case h = 1, and let C1 and C2 be controllers that are obtained from the sufficient conditions (21) and (34) with the maximum offset length, respectively. The order of the controller C1 is 7, but applying balanced model truncation (Zhou et al., 1996, Chap. 6) to the controller C1 , we can obtain an approximated controller Capp with order 5, which satisfies ∥Capp −C1 ∥∞ /∥C1 ∥∞ = 0.023. From iterative calculations of the eigenvalues of the discretized closed-loop system for each ∆, we find that both C1 and Capp stabilize the discretized system Σd in (5) for all ∆ ∈ (−1, 1). The controller C2 has order 5 and allows the offsets ∆ ∈ [−0.054, 0.068] without compromising the closed-loop stability. Approximated controllers with any order obtained by applying balanced model truncation to C2 do not achieve the closed-loop

Fig. 3. Allowable offset length ∆ − ∆ versus control update period h.

stability even in the case ∆ = 0. For comparison, a linear quadratic regulator whose state weighting matrix and input weighting matrix are identity matrices with appropriate dimension stabilizes the discretized system Σd in (5) only for ∆ ∈ [−0.029, 0.062]. From this numerical result, we see that the derived controller achieves better robust performance against clock offsets than a linear quadratic regulator designed without regard to the clock offset and also than the robust controller based on (34). 7. Concluding remarks We studied the problem of stabilizing systems in which the sensor and the controller have a constant clock offset. We formulated the problem as the stabilization problem for systems with parametric uncertainty. For multi-input systems, we derived a sufficient condition that is numerically testable, based on the results of simultaneous stabilization. For first-order systems, we obtained the maximum offset length that can be allowed by an LTI controller. However, a full investigation of the problem for generalorder systems and systems with model uncertainty is still an open area for future research. References Blondel, V., Campion, G., & Gevers, M. (1993). A sufficient condition for simultaneous stabilization. IEEE Transactions on Automatic Control, 38, 1264–1266. Briat, C. (2013). Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints. Automatica, 49, 3449–3457. Briat, C., & Seuret, A. (2012). Convex dwell-time characterizations for uncertain linear impulsive systems. IEEE Transactions on Automatic Control, 57, 3241–3246. Cantoni, M., Jönsson, U. T., & Kao, C.-K. (2012). Robustness analysis for feedback interconnections of distributed systems via integral quadratic constraints. IEEE Transactions on Automatic Control, 57, 302–317. de Oliveira, M. C., Bernussou, J., & Geromel, J. C. (1999). A new discrete-time robust stability condition. Systems & Control Letters, 37, 261–265. Donkers, M. C. F., Heemels, W. P. M. H., van de Wouw, N., & Hetel, L. (2011). Stability analysis of networked control systems using a switched linear systems approach. IEEE Transactions on Automatic Control, 56, 2101–2115. Doyle, J. C., & Stein, G. (1981). Multivariable feedback design: Concepts for a classical/modern synthesis. IEEE Transactions on Automatic Control, 26, 4–16. Freris, N. M., Graham, S. R., & Kumar, P. R. (2011). Fundamental limits on synchronizing clocks over network. IEEE Transactions on Automatic Control, 56, 1352–1364. Fridman, E., Seuret, A., & Richard, J.-P. (2004). Robust sampled-data stabilization of linear systems: An input delay approach. Automatica, 40, 1441–1446. Fujioka, H. (2009). A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices. IEEE Transactions on Automatic Control, 54, 2440–2445. Garcia, E., Antsaklis, P. J., & Montestruque, A. (2014). Model-based control of networked systems. Springer. Ghosh, B. K. (1988). An approach to simultaneous system design. Part II: Nonswitching gain and dynamic feedback compensation by algebraic geometric methods. SIAM Journal on Control and Optimization, 26, 919–963. Graham, S., & Kumar, P.R. (2004). Time in general-purpose control systems: The Control Time Protocol and an experimental evaluation. In Proc. 43rd IEEE CDC.

M. Wakaiki et al. / Automatica 81 (2017) 314–321 He, J., Cheng, P., Shi, L., Chen, C., & Sun, Y. (2014). Time synchronization in WSNs: A maximum-value-based consensus approach. IEEE Transactions on Automatic Control, 59, 660–675. Hespanha, J. P., Naghshtabrizi, P., & Xu, Y. (2007). A survey of recent results in networked control systems. Proceedings of the IEEE, 95, 138–162. Hetel, L., Fiter, C., Omran, H., Seuret, A., Fridman, E., Richard, J.-P., et al. (2017). Recent developments on the stability of systems with aperiodic sampling: An overview. Automatica, 76, 309–335. IEEE Standard for a Precision Clock Synchronization Protocol for Networked Measurement and Control Systems. (2008). IEEE Std 1588-2008. Jiang, X., Zhang, J., Harding, J. J., Makela, B. J., & Domíngues-García, A. D. (2013). Spoofing GPS receiver clock offset of phasor measurement units. IEEE Transactions on Power Systems, 28, 3253–3262. Maeda, H., & Vidyasagar, M. (1984). Some results on simultaneous stabilization. Systems & Control Letters, 5, 205–208. Mirkin, L. (2007). Some remarks on the use of time-varying delay to model sampleand-hold circuits. IEEE Transactions on Automatic Control, 52, 1109–1112. Naghshtabrizi, P., Hespanha, J. P., & Teel, A. R. (2010). Stability of delay impulsive systems with application to networked control systems. Transactions of the Institute of Measurement and Control, 32, 511–528. Nakamura, Y., Hirata, K., & Sugimoto, K. (2008). Synchronization of multiple plants over networks via switching observer with time-stamp information. In Proc. SICE annu. conf.. Oishi, H., & Fujioka, Y. (2010). Stability and stabilization of aperiodic sampleddata control systems using robust linear matrix inequalities. Automatica, 46, 1327–1333. Okano, K., Wakaiki, M., & Hespanha, J.P. (2015). Real-time control under clock offsets between sensors and controllers. In Proc. HSCC’15. Olbrot, A. W., & Nikodem, M. (1994). Robust stabilization: Some extensions of the gain margin maximization problem. IEEE Transactions on Automatic Control, 39, 652–657. Rhee, I.-K., Lee, J., Kim, J., Serpedin, E., & Wu, Y.-C. (2009). Clock synchronization in wireless sensor networks: An overview. Sensors, 9, 56–85. Rosenbrock, H. H. (1974). Computer-aided control system design. New York: Academic Press. Shi, H.-B., & Qi, L. (2009). Static output feedback simultaneous stabilisation via coordinates transformations with free variables. IET Control Theory & Applications, 3, 1051–1058. Ünal, H. U., & Iftar, A. (2008). A small gain theorem for systems with non-causal subsystems. Automatica, 44, 2950–2953. Vidyasagar, M. (1985). Control system synthesis: a factorization approach. Cambridge, MA: MIT Press, Republished in Morgan & Claypool, 2011. Vidyasagar, M., & Viswanadham, N. (1982). Algebraic design techniques for reliable stabilization. IEEE Transactions on Automatic Control, 27, 1085–1095. Wakaiki, M., Okano, K., & Hespanha, J.P. (2017). Stabilization of systems with asynchronous sensors and controllers. arXiv:1601.07888. Wakaiki, M., Okano, K., & Hespanha, J.P. (2015). Control under clock offsets and actuator saturation. In Proc. 54th IEEE CDC. Wakaiki, M., Okano, K., & Hespanha, J.P. (2015). Stabilization of networked control systems with clock offsets. In Proc. ACC’15. Wakaiki, M., Okano, K., & Hespanha, J.P. (2016). L2 -gain analysis of systems with clock offsets. In Proc. ACC’16. Xu, Y., & Hespanha, J.P. (2005). Estimation under controlled and uncontrolled communications in networked control systems. In Proc. 44th IEEE CDC. Zeren, M., & Özbay, H. (2000). On the strong stabilization and stable H ∞ -controller design problems for MIMO systems. Automatica, 36, 1675–1684. Zhou, K., Doyle, J. C., & Glover, K. (1996). Robust and optimal control. Prentice Hall.

321 Masashi Wakaiki received the B.S. degree in Engineering and the M.S. and Ph.D. degrees in Informatics from Kyoto University, Kyoto, Japan, in 2010, 2012, and 2014, respectively. He served as a research fellow of the Japan Society for the Promotion of Science from 2013 to 2015 and a visiting scholar at the University of California, Santa Barbara from 2014 to 2016. Since 2016, he has been with the Department of Electrical and Electronic Engineering, Chiba University, where he is currently an Assistant Professor. His research interests include time-delay systems and

hybrid systems.

Kunihisa Okano received the B.Eng. degree in systems science from Osaka University, Toyonaka, Japan, in 2006, the M.I.Sc.T. degree in information physics and computing from the University of Tokyo, Tokyo, Japan, in 2008, and the D.Eng. degree in computational intelligence and systems science from Tokyo Institute of Technology, Yokohama, Japan, in 2013. He was with Research and Development Headquarters, NTT DATA Corp., Tokyo, Japan, from 2008 to 2010 and a Visiting Scholar at the University of California, Santa Barbara, CA, USA from 2013 to 2016. From April 2016 to September 2016, he was an Assistant Professor at Tokyo University of Science, Tokyo, Japan. He is currently an Assistant Professor with the Department of Intelligent Mechanical Systems, Okayama University, Okayama, Japan. His research interests are in networked control systems.

João P. Hespanha received his Ph.D. degree in electrical engineering and applied science from Yale University, New Haven, Connecticut in 1998. From 1999 to 2001, he was Assistant Professor at the University of Southern California, Los Angeles. He moved to the University of California, Santa Barbara in 2002, where he currently holds a Professor position with the Department of Electrical and Computer Engineering. Prof. Hespanha is the Chair of the Department of Electrical and Computer Engineering and a member of the Executive Committee for the Institute for Collaborative Biotechnologies (ICB). His current research interests include hybrid and switched systems; multiagent control systems; distributed control over communication networks (also known as networked control systems); the use of vision in feedback control; stochastic modeling in biology; and network security. Dr. Hespanha is the recipient of the Yale University’s Henry Prentiss Becton Graduate Prize for exceptional achievement in research in Engineering and Applied Science, a National Science Foundation CAREER Award, the 2005 best paper award at the 2nd Int. Conf. on Intelligent Sensing and Information Processing, the 2005 Automatica Theory/Methodology best paper prize, the 2006 George S. Axelby Outstanding Paper Award, and the 2009 Ruberti Young Researcher Prize. Dr. Hespanha is a Fellow of the IFAC and the IEEE and he was an IEEE distinguished lecturer from 2007 to 2013.