Stabilization of Systems with Asynchronous Sensors and Controllers

Stabilization of Systems with Asynchronous Sensors and Controllers††thanks:

Masashi Wakaiki, Kunihisa Okano, and João P. Hespanha This paper was partially presented at the 2015 American Control Conference, July 1-3, 2015, the USA. 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. Corresponding author M. Wakaiki. Tel. +1 805 893 7785. Fax +1 805 893 3262. M. Wakaiki is with the Department of Electrical and Electronic Engineering, Chiba University, Chiba, 263-8522, Japan (e-mail:wakaiki@chiba-u.jp). K. Okano is with the Graduate School of Natural Science and Technology, Okayama University, Okayama, 700-8530, Japan (e-mail:kokano@okayama-u.ac.jp). J. P. Hespanha is with the Center for Control, Dynamical-systems and Computation (CCDC), University of California, Santa Barbara, CA 93106-9560 USA (e-mail:hespanha@ece.ucsb.edu).
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, this bound is compared with the offset bounds that would be allowed if we restricted our attention to static output feedback controllers.

I 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 [11, 22], the gridding approach [12, 25, 7], and the impulsive systems approach based on Lyapunov functionals [23], on looped functionals [4], and on clock-dependent Lyapunov functions [3]; see also the surveys [17, 18]. 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 with time-stamps, and the controllers exploit this information to mitigate the effect of variable delays and sampling periods [15, 24, 13]. 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 [28], 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 [10], and a recent study [19] 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 [32, Sec. 5.4] and [33]. 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 infinitely many plants, but the approaches for simultaneous stabilization e.g., in [30] 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 [6] for the robust stabilization of systems with polytopic uncertainties. Although the robust stability analysis based on continuous paths of systems with respect to the -gap metric was developed in [5], 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 [2] is obtained for a family of plants with no common unstable zeros or poles. The set of plants in [21] 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 control problem. The above mentioned difficulties found in the simultaneous stabilization problem we consider is circumvented by exploiting geometric properties on . 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 [14, 27], 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 [26] 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 -gain analysis of systems with variable clock offsets have been investigated in [34] and [36], 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 [35], 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 first-order systems. In Section 6, we discuss stabilizability with static controllers and the comparison of the offset bounds obtained for LTI controllers and static controllers.

Notation and definitions:  We denote by the set of non-negative integers. The symbols , , and denote the open unit disc , the closed unit disc , and the unit circle , respectively. We denote by the complement of the open unit disc .

A square matrix is said to be Schur stable if all its eigenvalues lie in the unit disc . We say that a discrete-time LTI system is stabilizable (detectable) if there exists a matrix () such that () is Schur stable. We also use the terminology is stabilizable (respectively, is detectable) to denote this same concept.

We denote by the space of all bounded holomorphic real-rational functions in . The field of fractions of is denoted by . For a commutative ring , denotes the set of matrices with entries in , of any order. For , denotes the induced 2-norm. For , the -norm is defined as . For and , we define a lower linear fractional transformation of and as .

A pair in is said to be right coprime if the Bezout identity holds for some , . admits a right coprime factorization if there exist , such that and the pair is right coprime. Similarly, a pair in is left coprime if the Bezout identity holds for some , . admits a left coprime factorization if there exist , such that and the pair is left coprime. If is a scalar-valued function, then we use the expressions coprime and coprime factorization.

Ii Problem Statement

Consider the following LTI plant:

 ΣP: ˙x(t)=Ax(t)+Bu(t), (1)

where and 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 be sampling instants from the perspective of the controller clock. A sensor measures the state 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 environment 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 [16], 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 reported by the sensor is given by

 ^sk=sk+Δ(k∈N) (2)

for some unknown constant .

Let be the update period of the ZOH. The control signal is assumed to be piecewise constant and updated periodically at times () with values computed by the controller: for . We place a basic assumption for stabilization of sampled-data systems.

Assumption II.1

(Stabilizability and non-pathological control update) The plant is stabilizable and the update period is non-pathological, that is, () for each pair of eigenvalues of .

While the ZOH updates the control signal periodically, the true sampling times and the reported sampling times may not be periodic. However, we do assume that both and do not fall behind by more than the ZOH update period . This assumption is formally stated as follows.

Assumption II.2

(Bounded clock offset) For every , .

This assumption implies that the clock offset is smaller than the control update period , which holds in most mechatronics systems. In fact, control update periods for mechatronics systems generally take values from 100 s to 10 ms, while recent clock synchronization algorithms such as the IEEE 1588 Precision Time Protocol (PTP) [1] make clock offsets smaller than a few tens of microseconds.

Fig. 2 shows the timing diagram of the sampling instants , the reported time-stamps , and updating instants of the control inputs.

The controller side is comprised of a time-stamp aware estimator and a controller as in the model-based or emulation-based control of networked control systems [13]. The time-stamp aware estimator generates the state estimate from the data according to the following dynamics:

 ΣE: {˙^x(t)=A^x(t)+Bu(t)(tk

Note that if the time-stamp is correct, i.e., , then this estimator consistently produces for all , perfectly compensating transmission delays. Time-stamp aware estimators have been used to compensate for network-induced imperfections, e.g., in [15, 24, 13].

The controller is a discrete-time LTI system and generates the control input based on the state estimate :

 ΣC: {ζk+1=Acζk+Bc^xkuk=Ccζk+Dc^xk, (4)

where is the state of the controller.

The objective of the present paper is to find a discrete-time LTI controller 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 II.3

Given an offset interval , determine if there exists a controller as in (4) such that , as and as for every and for every initial states and . Furthermore, if one exists, find such a controller .

Iii Discretization of the Closed-loop System

To solve Problem II.3, we discretize the system comprised of the plant , the estimator , 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.

Iii-a Discretized system and its basic properties

The following lemma provides a realization for the discretized system:

Lemma III.1

Define

 ξk:=[x(tk)−^x(tk)^x(tk)].

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

 Σd: ξk+1=FΔξk+GΔuk,ηk=HΔξk, (5)

where , , and

 FΔ GΔ :=⎡⎢⎣Λ(∫h0e−Aτdτ−(I+Θ)∫h−Δ0e−Aτdτ)BΛ(I+Θ)∫h−Δ0e−AτdτB⎤⎥⎦. (6)

Proof: Using , we have from the state equation (1) that

 x(tk+1) =Λx(tk)+Λ∫h0e−Aτdτ⋅Buk. (7)

We compute in terms of and . It follows from the dynamics of the estimator in (3) that

 ^x(tk+1)=eA(tk+1−^sk)^x(^sk)+∫tk+1^skeA(tk+1−τ)Bdτ⋅uk (8)

and

 ^x(^sk)=x(sk)=eA(sk−tk)x(tk)+∫sktkeA(sk−τ)Bdτ⋅uk. (9)

Since and , it follows that

 eA(tk+1−^sk)⋅eA(sk−tk)=eA(h−Δ), (10)

and also that

 eA(tk+1−^sk)∫sktkeA(sk−τ)dτ=∫^sktk+ΔeA(tk+1−τ)dτ. (11)

Using and , we conclude from (8)–(11) that

 ^x(tk+1) =Λ(I+Θ)x(tk)+Λ(I+Θ)∫h−Δ0e−AτBdτ⋅uk. (12)

From (7) and (12), we obtain the and in (6). Moreover, we have by the definition of the extended state .  \QED

Next we show that if the extended state and the controller state converge to the origin, then the intersample values of and also converge to the origin.

Proposition III.2

For the discreteized system in Lemma III.1, we have that as if and only if as and as .

Proof: The statement that as and as imply as , follows directly from the definition of .

To prove the converse statement, assume that as . Then and , as . Since

 ∥x(tk+τ)∥≤e∥A∥h∥x(tk)∥+∫h0e∥A∥h∥B∥dt⋅∥uk∥

for all and all , we derive (). Similarly, we see from the dynamics of the estimator that as . This completes the proof.  \QED

This proposition allows us to conclude Problem II.3 can be solved by finding LTI controllers achieving , () for every and for every initial states and .

The following result allows us to conclude that the discretized system is detectable and stabilizable for all and almost all if the plant is stabilizable.

Proposition III.3

The discretized system in (5) is detectable for all and . Moreover, is stabilizable for all if Assumption II.1 holds.

Proof: Let us first obtain another realization of the discretized system in (5). We can transform into

 ¯FΔ :=T−1FΔT=[Λ000],~{}~{}where~{}~{}T:=[−Θ−II+ΘI].

Furthermore, if we define

 J1:=∫h0e−Aτdτ,J2:=∫h−Δ0e−Aτdτ,

then we obtain

 ¯GΔ:=T−1GΔ=[ΛJ1B−Λ(J1−(I+Θ)J2)B−ΘΛJ1B]

and We have thus another realization for .

Next we check detectability and stabilizability by using the realization . Define

 LΔ:=[Λ(I+Θ)−10]. (13)

Then we have that

 ¯FΔ−LΔ¯HΔ=[0−Λ(I+Θ)−100], (14)

and clearly is Schur stable. Therefore, the discreteized system is detectable for all and .

To show stabilizability, we use the well-known rank conditions (see, e.g., [40, Sec. 3.2]). We have that is full row rank for all if and only if

 [zI−ΛΛJ1B]=[zI−eAh∫h0eAτBdτ]

is full row rank for all . Hence, the discretized system is stabilizable for all if Assumption II.1 holds.  \QED

Iii-B Extension to the output feedback case with disturbances and noise

Instead of in (1), consider a plant with disturbances, noise, and output feedback:

 Σ′P: {˙x(t)=Ax(t)+Bu(t)+d(t)y(t)=Cx(t)+n(t),

where and , are the disturbance, measurement noise, and output of the plant, respectively. As in [13, Chap. 3], [38], 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)=A¯x(t)+Bu(t)+L(y(t)−C¯x(t)),

where is the state estimate and is an observer gain such that is Hurwitz. The sampler sends the state estimate , and the resulting dynamics of the time-stamp aware estimator is provided by

 Σ′E: {˙^x(t)=A^x(t)+Bu(t)(tk≤t

where is the quantization noise. A calculation similar to the one performed in the proof of in Lemma III.1 can be used to show that the dynamics of the discretized system is given by

 Σ′d: ξk+1=FΔξk+GΔuk+dk,ηk=HΔξk, (15)

where and

 d1,k :=∫tk+1tkeA(tk+1−τ)d(τ)dτ−d2,k d2,k :=−eA(tk+1−^sk)(e(A−LC)(sk−tk)ek−wk +∫sktk(e(A−LC)(sk−τ)(d(τ)−Ln(τ))−eA(sk−τ)d(τ))dτ) ek :=x(tk)−¯x(tk).

The only difference from the original idealized system in (5) is that has the disturbance . Hence, for the output feedback case with bounded disturbances and noise, solutions of Problem II.3 achieve the boundedness of the closed-loop state.

Proposition III.4

Assume that as for the idealized system in Lemma III.1 (in the context of state feedback without distubances and measurement noise). If , , and are bounded for all and all , then the states , , , and are also bounded for all and all . Moreover, if for all and all , then , , , and converge to the origin.

Proof: Since is bounded for every and every , it follows that and are also bounded for all . The rest of the proof follows the similar lines as that of Proposition III.2, and hence it is omitted.  \QED

See also [36] for the -gain analysis of systems with time-varying offsets.

Iv Controller Design via Simultaneous Stabilization

Iv-a Preliminaries

We first consider a general simultaneous stabilization problem not limited to the system introduced in Section 2.

The transfer function of the system is usually defined by the Z-transform of the system’s impulse response, i.e., , but in this paper, we define the transfer function by for consistency of the Hardy space theory; see [32, Sec. 2.2] for details. Hence the transfer function of a causal system is not proper. We say that stabilizes if , , and belong to . We recall that when these three transfer functions belong to , they will have no poles in the closed unit disk.

Consider the family of plants parameterized by , where is a nonempty parameter set, and assume that we have a doubly coprime factorization of over

 [YθXθ−~Nθ~Dθ][Dθ−~XθNθ~Yθ]=I, (16)

where and 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 ; see, e.g., [32, Theorem 4.2.1].

The following theorem provides a necessary and sufficient condition for simultaneous stabilization:

Theorem IV.1 ([33, 32])

Given a nonempty set , consider the plant having a doubly coprime factorization (16) for each . Fix and define

 [UθVθ]:=[~Dθ~Nθ][~Yθ0−Nθ0~Xθ0Dθ0](θ∈S). (17)

Then is right coprime for every . Moreover, there exists a controller that stabilizes for every if and only if there exists such that for all ,

 (Uθ+VθQ)−1∈M(RH∞). (18)

Such a stabilizing controller is given by

 C:=(~Xθ0+Dθ0Q)(~Yθ0−Nθ0Q)−1. (19)
Remark IV.2

Although the simultaneous stabilization of a finite family of plants is considered in [32, Sec. 5.4] and [33], generalization to an arbitrary family of plants is readily apparent, as mentioned in the last paragraph of Section 3 in [33].

Remark IV.3

A left coprime factorization of stabilizing controllers is used in [32, Sec. 5.4] and [33], whereas we represent controllers by a right coprime factorization in (19). Therefore, Theorem IV.1 is slightly different from its counterpart in [32, Sec. 5.4] and [33].

Iv-B Robust Controller Design

It is generally not easy to verify in a computationally efficient fashion that a transfer function satisfying (18) exists. In the next theorem, we develop a simple sufficient condition for (18) to hold, by exploiting geometric properties on inspired by results on strong stabilization [39].

Theorem IV.4

Given a nonempty set , assume that each plant () has a doubly coprime factorization (16) such that there exist , , and satisfying and

 ~Nθ(z)−~Nθ0(z)=R(θ)W(z), (20)

for all . If there exists satisfying the following -norm condition:

 ∥W(~Xθ0+Dθ0Q)∥∞<1supθ∈S∥R(θ)∥, (21)

then satisfies (18), and hence the controller in (19) stabilizes for every .

Proof: We define and as in (17). Since , it follows from (17) and the Bezout identity in (16) that

 Uθ =~Dθ~Yθ0+~Nθ~Xθ0=I+(~Nθ−~Nθ0)~Xθ0.

Moreover, since , we obtain

 Vθ =−~DθNθ0+~NθDθ0=(~Nθ−~Nθ0)Dθ0.

Hence Since for all satisfying , it follows that if

 ∥(~Nθ−~Nθ0)(~Xθ0+Dθ0Q)∥∞<1(θ∈S), (22)

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

 ∥(~Nθ−~Nθ0)(~Xθ0+Dθ0Q)∥∞≤∥R(θ)∥⋅∥W(~Xθ0+Dθ0Q)∥∞.

Hence if satisfies (21) for all , then (22) holds, and consequently is simultaneously stabilizable by in (19) from Theorem IV.1.  \QED

The proposition below shows that our discretized system in (5) always satisfies the assumptions on and that appear in Theorem IV.4. This result also provides the matrices and in (20) without explicitly calculating a coprime factorization of for all .

Proposition IV.5

Define the transfer function . For all , there exists a doubly coprime factorization (16) such that , and (20) holds with

 R(Δ) :=∫Δ0eA(h−τ)Bdτ∈Rn×m (23) W(z) :=z(z−1)∈RH∞. (24)

Proof: Consider the realization in the proof of Proposition III.3. For every , the matrix in (13) achieves the Schur stability of as shown in (14). From the realization of , e.g., in [32, Theorem 4.2.1], we can write as

 ~DΔ(z)=I−¯HΔ(1/z⋅I−(¯FΔ−LΔ¯HΔ))−1LΔ=I−zΛ.

Noticing that the far right-hand side of the equation above does not depend on , we have .

It follows that . From the realization , we see that

 PΔ(z) =z((I+Θ)(I−zΛ)−1ΛJ1−Λ(J1−(I+Θ)J2)−ΘΛJ1)B.

Since and for , it follows that

 PΔ(z)−P0(z) =z(Θ(I−zΛ)−1ΛJ1−Λ(J1−(I+Θ)J2)−ΘΛJ1)B. (25)

On the other hand, we have and

 ¯ΘA=−∫Δ0(ddτe−Aτ)dτ=−Θ. (26)

Since , it follows that . Therefore we derive from (25)

 PΔ−P0 =zΛ¯Θ(I−zΛ)−1(−I+zΛ(I−AJ1))B.

Similarly to (26), we have , and hence Since , , and are commutative, we derive

 PΔ−P0=z(z−1)(I−zΛ)−1¯ΘΛB,

and Thus (20) holds with in (23) and in (24)  \QED

Define

 γ:=1　maxΔ∈[Δ––,¯¯¯¯Δ]∥∥∫Δ0eA(h−τ)Bdτ∥∥. (27)

From Theorem IV.4, to obtain a controller as in (4), it is enough to solve the following suboptimal problem: Find satisfying . This problem is equivalent to a standard suboptimal control problem [40, Chaps. 16, 17]: Find such that , where is defined by

 Φ:=[W~X0WD0−I0]. (28)

The results of this section can be summarized through the following controller design algorithm:

Algorithm IV.6
1. Using the realization

 (¯F0,¯G0,¯H0)=([eAh000], [∫h0eAhBdτ0], [II]),

the matrix and an arbitrary matrix such that is Schur stable, set

 D0(z) :=I−K0(1/z⋅I−Φ)−1¯G0 N0(z) :=¯H0(1/z⋅I−Φ)−1¯G0 ~X0(z) :=K0(1/z⋅I−Φ)−1L0 ~Y0(z) :=I+H0(1/z⋅I−Φ)−1K0 W(z) :=z(z−1).
2. For a given offset interval , set as in (27), and solve the control problem [40, Chaps. 16, 17]: Find such that , where is defined by (28).

3. If the control problem is not solvable, then the algorithm fails. Otherwise the transfer function of the controller is given by .

Remark IV.7

We have from Proposition IV.5 that for constant , where is expressed as the nominal component plus the uncertainty block . If we obtain a similar formula for the case of time-varying offsets as studied for systems with aperiodic sampling in [12], 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 may be non-causal, the small gain theorem for systems with non-causal uncertainty in [31] can be used. This extension is a subject for future research.

Example IV.8

Consider the unstable batch reactor studied in [29], where the system matrices and in (1) are given by

 A

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 [29].

Here we compare the proposed method with the robust stabilization method in [8] and [32, Chap. 7] based on the following fact: Consider a family of plants with . Assume that has no poles on and the same number of unstable poles for every and that a function satisfies

 ∥PΔ(ejω)−P0(ejω)∥<|r(ejω)| (29)

for all and all . If the controller stabilizes and satisfies

 ∥∥rC(I+P0C)−1∥∥∞≤1, (30)

then stabilizes for all <