Technical Report: Timing Abstraction of Perturbed LTI systems with \mathcal{L}_{2}-based Event-Triggering Mechanism

# Technical Report: Timing Abstraction of Perturbed LTI systems with L2-based Event-Triggering Mechanism

Arman Sharifi Kolarijani, Manuel Mazo Jr. and Tamás Keviczky The authors are with the Delft Center for Systems and Control, Delft University of Technology, The Netherlands.
{a.sharifikolarijani,m.mazo,t.keviczky}@tudelft.nl
###### Abstract

In networked control systems, the advent of event-triggering strategies in the sampling process has resulted in the usage reduction of network capacities, such as communication bandwidth. However, the aperiodic nature of sampling periods generated by event-triggering strategies has hindered the schedulability of such networks. In this study, we propose a framework to construct a timed safety automaton that captures the sampling behavior of perturbed LTI systems with an -based triggering mechanisms proposed in the Literature. In this framework, the state-space is partitioned into a finite number of convex polyhedral cones, each cone representing a discrete mode in the abstracted automaton. Adopting techniques from stability analysis of retarded systems accompanied with a polytopic embedding of time, LMI conditions to characterize the sampling interval associated with each region are derived. Then, using reachability analysis, the transitions in the abstracted automaton are derived.

## I Introduction

Wireless networked controlled systems (WNCS’s) represent a class of spatially distributed control systems for which the feedback loops are closed via shared communication components possessing limited bandwidth. Several advantages of WNCS’s, such as the ease of maintenance and the flexibility of implementation, make them attractive to industrial environments. Meanwhile, WNCS’s are burdened with characteristics, such as limited battery life and communication bandwidth. Under these circumstances, the resource over-utilization caused by (traditional) periodic implementations, the so-called time-driven control (TDC), makes such implementations less appealing for WNCS’s.

To address the aforementioned issues, control researchers have proposed event-driven control (EDC) strategies that are aperiodic, such as event-triggered control (ETC) [1] and self-triggered control (STC) [2]. In EDC strategies, the core idea relies on the fact that the dynamics of the control system during the inter-sample interval determine the next sampling instant to attenuate the usage of resources, particularly the communication bandwidth. In these strategies, control task executions only happen when a pre-specified condition is violated. Such condition is called the triggering mechanism (TM). It is derived based on stability and/or performance of the closed-loop system. On the other hand, the schedulability of ETC strategies, due to their aperiodic nature, is more arduous compared to TDC strategies. In fact, in TDC strategies, the control and scheduler designs are naturally decoupled via the (pre-defined) fixed sampling period. This phenomenon is called the separation-of-concerns in the real-time systems community [3]. It is worth mentioning that ETC strategies are almost always equipped with a minimum inter-execution time (MIET) to prevent the occurrence of Zeno behavior in the sampling process. This quantity can be technically used in the synthesis of task scheduling. However, it is a conservative approximation of the lower bound on all the possible generated sampling periods. Thus, such synthesis does not make use of the beneficiary characteristics of ETC strategies in an efficient manner. To address this shortcoming, researchers have proposed another class of approaches, the so-called co-design approaches. In this class, the problem of controller and scheduler synthesis for real-time systems is tackled in a unified framework, see e.g. feedback modification to task attributes [4, 5], [6, 7], anytime controllers [8, 9], and event-based control and scheduling [10, 11]. Recently, alternative to the unified frameworks mentioned above, [12, 13] have proposed a decoupling framework to capture the sampling behavior of LTI systems with ISS-based TM’s using timed safety automata (TSA’s).

Generally speaking, TSA is a simplified version of timed automaton (TA) [14, 15]. It is a powerful tool to model the timing behavior of real-time systems for scheduling purposes since its reachability analysis is decidable [16, 17]. In this study, following the same path as in [12, 13], we propose a framework to capture the sampling behavior of perturbed LTI systems with the -based TM proposed by [18]. We show that the derived TSA -approximately simulates the sampling behavior of the -based ETC system. It is evident that such characterizations can be analyzed independently for scheduling purposes, thus providing a scalable and versatile event-triggered WNCS design procedure.

## Ii Preliminaries

denotes the -dimensional Euclidean space, denotes the positive reals. is the set of nonnegative integers, and is the set of all closed intervals such that and . For any set , denotes the set of all subsets of , i.e. the power set of . and are the set of all real-valued matrices and the set of all real-valued symmetric matrices, respectively. For a matrix , (or ) means is a negative (or positive) semidefinite matrix and () indicates is a negative (positive) definite matrix. is the cone of all symmetric positive definite matrices. indicates the largest integer not greater than . and denote the Euclidean norm of a vector and the Frobenius norm of a matrix , respectively. For a matrix , and denote the set of eigenvalues and the largest eigenvalue of . Consider two sets , their Minkowski sum is given by . We state the following known results that will be used in Subsection II-A.

###### Lemma 1

([19]) For any real matrices , and real symmetric positive definite matrix , with compatible dimensions,

 EG+GTET⪯EPET+GTP−1G.
###### Lemma 2

([20]) For all , if , then, .

###### Proposition 1

(Jensen Inequality [19]) For any matrix with constant entries, scalar , vector function such that the integrations concerned are well defined, then:

 γ∫γ0ωT(β)Mω(β)dβ≥(∫γ0ω(β)dβ)TM∫γ0ω(β)dβ.

### Ii-a L2-Based ETC System:

In this subsection, an overview of the ETC strategy proposed by [18] along with a new result (see Theorem 1) are presented. Consider a sampled-data system that is given by:

 ˙ξ(t)=Aξ(t)+Bν(t)+Eω(t),∀t∈[0,τ(x)),ξ(0)=x, (1)

where , , , denotes the sampling period associated with , and , , and have compatible dimensions. The control law is implemented in a sample-and-hold manner as follows:

 ν(t)=−Kx. (2)

Furthermore, assume that the disturbance is a vanishing type disturbance [18], i.e.,

 ∃W≥0~{}such that~{}|w(t)|2≤W|x|2,∀t∈[0,τ(x)). (3)

Denote by , the error signal endured by the system (1)-(2), where is the solution of (1). Reformulating (1), the evolution of state and error signals can be rewritten in a compact form as follows:

 ξx(t)=Λ(t)x+Ω(t) (4)

and

 ϵ(t)=(I−Λ(t))x−Ω(t) (5)

where

 {Λ(t)=I+∫t0eAsds(A−BK),Ω(t)=∫t0eA(t−s)Eω(s)ds. (6)

Assume that there exists a quadratic Lyapunov function such that is the solution to the Algebraic Riccati Equation (ARE) given by:

 PA+ATP−Q+R=0 (7)

where

 Q=PBBTP,R=1γ2PEETP,γ>0. (8)

The existence of guarantees that the system (1) with the full-state feedback is finite-gain stable from to with an induced gain less than [18]. Then, the state-dependent TM, proposed by [18], is given by:

 τ(x):=inf{t>0|ϵT(t)Mϵ(t)≥xTNx} (9)

where

 M=(1−β2)I+PBBTP,N=12(1−β2)I+PBBTP, (10)

and is a user-defined scalar related to the TM (9).

###### Theorem 1

Consider the system (1)-(2) with the triggering mechanism (9). Assume there exist a scalar and a symmetric matrix such that

 μ≥0,Ψ≻0,M+Ψ⪯μI, (11)
 Φ(t)⪰0, (12)

where

 Φ(t)=[Φ1(t)Φ2(t)Φ3(t)Φ4(t)], (13)
 Φ1(t)=(Λ(t)−I)TM(Λ(t)−I)+tWμλmax(ETE)dA(t)I−N,Φ2(t)=ΦT3(t)=(Λ(t)−I)TMT,Φ4(t)=−Ψ,

are satisfied. Then, the sampling period generated by (9) is lower bounded by:

 τ′(x):=inf{t>0|Φ(t)⪰0}. (14)
{proof}

Substitute (5) into (9). Then, the TM (9) can be rewritten by:

 τ(x)=min{t>0|Fω(x,t)≥0},∀x∈Rn, (15)

where

 Fω(x,t)=xT[(Λ(t)−I)TM(Λ(t)−I)−N]x+xT(Λ(t)−I)TMΩ(t)+ΩT(t)M(Λ(t)−I)x+ΩT(t)MΩ(t). (16)

Let denote for the sake of compactness. Using Lemma 1 the terms that are dependent on both and in can be decoupled into:

 xT(Λ(t)−I)TMΩ(t)+ΩT(t)M(Λ(t)−I)x≤ΩT(t)ΨΩ(t)+xT(Λ(t)−I)TMΨ−1M(Λ(t)−I)x, (17)

where . Then, it follows that:

 (18)

where

 dA(t)=⎧⎨⎩1λAmax(eλAmaxt−1),λAmax≠0t,λAmax=0. (19)

Based on the aforementioned procedure, one concludes that:

 Fω(x,t)≤xTΘ(t)x (20)

where

 Θ(t)=(Λ(t)−I)T(M+MΨ−1M)(Λ(t)−I)+tWμλmax(ETE)dA(t)I−N. (21)

Then, we employ the Schur complement in order to transform (21) into (13). Note that (21) is not linear in while (13) is linearly dependent on . Considering (20), since implies by the Schur complement, it follows that . This concludes the proof.

Thus, Theorem 1 enables us to avoid the unknown behavior of perturbation in analyzing the sampling begavior of (9). However, it is still intractable to use (14) for the analysis since it has to be checked for an infinite number of instants and it clearly lacks any insight on how the state at the sampling instant affects the sampling period .

### Ii-B Systems and Relations

In what follows, we review some notions from the field of system theory to formally characterize the outcome of the proposed framework. Let be a set and be an equivalence relation on . Then, denotes the equivalence class of and denotes the set of all equivalence classes. A metric (or a distance function) on satisfies, : i) , ii) , and ii) . The ordered pair is said to be a metric space.

###### Definition 1

(Hausdorff Distance [21]) Assume and are two non-empty subsets of a metric space . The Hausdorff distance is given by:

 max{supx∈Xinfy∈Yd(x,y),supy∈Yinfx∈Xd(x,y)}.

It follows that the ordered pair is a metric space. Now, we introduce some concepts from system theory and particularly a modified notion of quotient system adopted from [12] (see e.g. [22] for the traditional definition).

###### Definition 2 (System [22])

A system is a sextuple consisting of:

• a set of states ;

• a set of initial states ;

• a set of inputs ;

• a transition relation ;

• a set of outputs ;

• an output map .

When the set of outputs of a system is endowed with a metric, it is called a metric system. An autonomous system is a system for which the cardinality of its input set is at most one.

###### Definition 3 (Approximate Simulation Relation [22])

Consider two metric systems and with , and let , where represents the set of nonnegative real numbers. A relation is an -approximate simulation relation from to if the following three conditions are satisfied:

1. ;

2. ;

3. in in satisfying .

We say that -approximately simulates , denoted by , if there exists an -approximate simulation relation from to .

###### Definition 4 (Power Quotient System [12])

Let be an autonomous system and be an equivalence relation on . The power quotient of by , denoted by , is the autonomous system consisting of:

• ;

• ;

• if with and ;

• ;

• .

###### Lemma 3 ([12])

Let be an autonomous metric system, be an equivalence relation on , and let the autonomous metric system be the power quotient system of by . For any

 ε≥maxx∈x/Rx/R∈X/Rd(H(x),H/R(x/R)),

with the Hausdorff distance over the set , -approximately simulates , i.e. .

Now, we appropriately modify Definition 4 and Lemma 3 for the case that one can construct an over approximation of the power quotient system, namely .

###### Definition 5

(Approximate Power Quotient System [13]) Let be a system, be an equivalence relation on , and be the power quotient of by . An approximate power quotient of by , denoted by , is a system such that, , , and , .

###### Corollary 1 ([13])

Let be a metric system, be an equivalence relation on , and let the metric system be the approximate power quotient system of by . For any

 ε≥maxx∈x/Rx/R∈X/Rd(H(x),¯H/R(x/R)),

with the Hausdorff distance over the set , -approximately simulates , i.e. .

### Ii-C Timed Safety Automaton

In what follows, we present a formal definition for TSA. A TSA [23] is a directed graph extended with real-valued variables (called clocks) that model the logical clocks. We define as a set of finitely many clocks. Clock constraints are used to restrict the behavior of the automaton. A clock constraint is a conjunctive formula of atomic constraints of the form or for , and . We use to denote the set of clock constraints.

###### Definition 6

(Timed Safety Automaton [15]) A timed safety automaton is a sextuple where:

• is a set of finitely many locations (or vertices);

• is the initial location;

• is the set of actions;

• is a set of finitely many real-valued clocks;

• is the set of edges;

• assigns invariants to locations.

The location invariants are restricted to constraints of the form: or where is a clock and is a natural number.

### Ii-D Problem Statement

Consider the system :

• ;

• ;

• iff given by (1)-(2), and (9);

• ;

• where .

The output of the above system generates all possible sequences of inter-sample intervals of the concrete system (1)-(2) with the TM (9).

###### Problem 1

Provide a construction of power quotient systems of systems as defined above.

Based on Definition 4, we propose to construct the system where

• ;

• ;

• if , such that as determined by (1)-(2);

• , where represents the set of closed intervals such that ;

• .

The equivalence relation on partitions the state space of (i.e. the ETC system) into the set with a finite cardinality. However, since the exact construction of is in general impossible, we construct instead (see Definition 5). Later on, it will be shown that the constructed is equivalent to a TSA.

## Iii Abstractions of event-triggered LTI systems

In this section, we introduce the framework to solve Problem 1 in the following order: 1) a proper definition of an equivalence relation on , 2) a tractable approach to compute the output map and its corresponding output set , and 3) a reachability-based analysis to derive the discrete transitions among abstract states .

### Iii-a State set

The type of state set construction approach mainly relies on an intuitive observation from (20).

###### Remark 1

Consider that the right-hand side of (20) is used to analyze the sampling behavior of (15) instead of . Then, the sampling periods of all states, located on a line that passes through the origin excluding the origin itself, are lower bounded by the same sampling period, i.e. , .

It follows that a proper approach to abstract the state space is via partitioning it into a finite number of convex polyhedral cones (pointed at the origin) where and . This state space abstraction technique is proposed by [24], dividing each of the angular spherical coordinates of : , into (not necessarily equidistant) intervals resulting in conic regions. Furthermore, since the term is quadratic in , it is sufficient to only analyze half of the state space (e.g. by taking ). Thus, the equivalence relation to construct the abstraction is given by:

 (x,x′)∈P⇔∃s∈{1,…,q}s% .t.x,x′∈Rs,

where is the number of equivalence classes. Hence, the equivalence classes of are defined by polyhedral cones pointed at the origin given by whenever or , otherwise.

### Iii-B Output Map

In this subsection, we present how to construct and . For all , the output is equal to the time interval indicating . We make use of the polytopic embedding technique proposed by [25]. In the space of real matrices, a sequence of convex polytopes is constructed around the matrix . Doing so replaces the evaluation of (14) at infinitely many instants by the evaluation of at finitely many vertices in the sequence of polytopes generated by . Assume a scalar denoting a time instant for which the TM (9) is enabled in the whole state space, i.e. . Consider is the number of vertices employed to define the polytope containing in a given time interval, and denotes the number of time subdivisions considered in the time interval .

###### Lemma 4

Let . Consider a time instant , a scalar and a symmetric matrix satisfying (11). If holds , then, it follows that , with defined in (13) and

 \lx@converttounder¯Φ(i,j),s=~\lx@converttounder¯Φ(i,j),s+ηI
 ~\lx@converttounder¯Φ(i,j),s=⎧⎪⎨⎪⎩Σik=0^\lx@converttounder¯Φ(i,j),s(σl)k,j<⌊\lx@converttounder¯τslσ⌋Σik=0^\lx@converttounder¯Φ(i,j),s(\lx@converttounder¯τs−jσl)k,j=⌊\lx@converttounder¯τslσ⌋, (22)
 (23)

and

 \lx@converttounder¯L0,j=ˇΠTjMˇΠj−N+~L0,j (24)

with

 ~L0,j=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Wμλmax(ETE)λAmax(jσl)(eλAmaxjσl−1)Ifor~{}λAmax≠0,Wμλmax(ETE)λAmax(jσl)2Ifor~{}λAmax=0, (25)
 \lx@converttounder¯L1,j=ˇΠTjM^Πj+^ΠTjMˇΠj+~L1,j (26)

with

 ~L1,j=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩Wμλmax(ETE)λAmax[(jσl)eλAmaxjσlλAmax+eλAmaxjσl−1]I~{} for ~{}λAmax≠0,Wμ(2jσl)λmax(ETE)I~{} for ~{}λAmax=0, (27)
 \lx@converttounder¯L2,j=ˇΠTjMA2!^Πj+^ΠTjAT2!MˇΠj+^ΠTjM^Πj+~L2,j (28)

with

 (29)
 \lx@converttounder¯Lk≥3,j=ˇΠTjMAk−1k!^Πj+^ΠTj(Ak−1)Tk!MˇΠj+^ΠTj(Σk−1i=1(Ai−1)Ti!MAk−i−1(k−i)!)^Πj+~Lk,j (30)

with

 ~Lk≥3,j=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Wμλmax(ETE)λAmax[(jσl)eλAmaxjσl(λAmax)kk!+eλAmaxjσl(λAmax)k−1(k−1)!]I~{}for~{}λAmax≠0,0~{}for~{}λAmax=0, (31)
 η≥maxt′∈[0,σl],r∈{0,...,l−1}λmax(Φ(t′+rσl)−ΣNk=0^\lx@converttounder¯Φk,r(t′)k), (32)

and

 ~\lx@converttounder¯Φ(Nconv,j)(t′)=ΣNconvk=0^\lx@converttounder¯Φk,j(t′)k. (33)
{proof}

See Appendix.

Then, using the S-procedure, the following theorem provides an approach to regionally reduce the conservatism involved in the estimates obtained from Lemma 4.

###### Theorem 2 (Regional Lower Bound Approximation)

Consider a scalar , a scalar and a symmetric matrix satisfying (11), and matrices , , defined as in Lemma 4. If there exist scalars (for ) or symmetric matrices with nonnegative entries (for ) such that for all the following LMIs hold:

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩\lx@converttounder¯Φ(i,j),s+[\lx@converttounder¯α(i,j),sQs000]⪯0if n=2,\lx@converttounder¯Φ(i,j),s+[ETs\lx@converttounder¯U(i,j),sEs000]⪯0if n≥3, (34)

then, the inter-sample time (9) of the system (1)-(2) is regionally bounded from below by .

{proof}

See Appendix.

One can follow a similar approach to find the upper bounds on the inter-sample times that is outlined in Lemma 5 and Theorem 3.

###### Lemma 5

Let . Consider a time instant , a scalar and a matrix satisfying the LMI conditions given in Lemma 4. If holds , then, it follows that , with defined in (13) and

 ¯Φ(i,j),s=−¯~Φ(i,j),s−ηI,
 ¯~Φ(i,j),s=⎧⎪⎨⎪⎩∑ik=0Lk,j((j+1)σl−¯τs)kifj=⌊¯τslσ⌋,∑ik=0Lk,j(σl)kifj>⌊¯τslσ⌋,

where are given by (24)-(31) and is defined in (32).

{proof}

See Appendix.

###### Theorem 3 (Regional Upper Bound Approximation)

Consider a scalar , a scalar and a symmetric matrix satisfying (11), and matrices , , defined as in Lemma 5. If there exist scalars (for ) or symmetric matrices with nonnegative entries (for ) such that for all the following LMIs hold:

 (35)

then, the inter-sample time (9) of the system (1)-(2) is regionally bounded from above by .

{proof}

Analogous to the proof of Theorem 2.

### Iii-C Transition Relations

In order to find all the transitions in , it is required to compute the reachable set of each over the time interval . In the sequel, we present how one is able to compute over approximations of the reachable set of each cone by the Minkowski sum of two sets. The evolution of states over this time interval is given by . Denote by the reachable set of during the time interval , that is given by:

 {x′∈Rn|∃x∈X0,s,∃τ∈[\lx@converttounder¯τs,¯τs],x′=ξx(τ)}.

Furthermore, define

 X1[\lx@converttound