sensors Article

On Event-Triggered Adaptive Architectures for Decentralized and Distributed Control of Large-Scale Modular Systems Ali Albattat 1 , Benjamin C. Gruenwald 2 and Tansel Yucelen 2, * 1 2

*

Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA; [email protected] Laboratory for Autonomy, Control, Information, and Systems (LACIS), Department of Mechanical Engineering, University of South Florida, Tampa, FL 33620, USA; [email protected] Correspondence: [email protected]; Tel.: +1-813-974-5656

Academic Editor: Dan Zhang Received: 24 June 2016; Accepted: 5 August 2016; Published: 16 August 2016

Abstract: The last decade has witnessed an increased interest in physical systems controlled over wireless networks (networked control systems). These systems allow the computation of control signals via processors that are not attached to the physical systems, and the feedback loops are closed over wireless networks. The contribution of this paper is to design and analyze event-triggered decentralized and distributed adaptive control architectures for uncertain networked large-scale modular systems; that is, systems consist of physically-interconnected modules controlled over wireless networks. Specifically, the proposed adaptive architectures guarantee overall system stability while reducing wireless network utilization and achieving a given system performance in the presence of system uncertainties that can result from modeling and degraded modes of operation of the modules and their interconnections between each other. In addition to the theoretical findings including rigorous system stability and the boundedness analysis of the closed-loop dynamical system, as well as the characterization of the effect of user-defined event-triggering thresholds and the design parameters of the proposed adaptive architectures on the overall system performance, an illustrative numerical example is further provided to demonstrate the efficacy of the proposed decentralized and distributed control approaches. Keywords: large-scale modular systems; networked control systems; uncertain dynamical systems; event-triggered control; decentralized control; distributed control; system stability and performance

1. Introduction The design and implementation of decentralized and distributed architectures for controlling complex, large-scale systems is a nontrivial control engineering task involving the consideration of components interacting with the physical processes to be controlled. In particular, large-scale systems are characterized by a large number of highly coupled components exchanging matter, energy or information and have become ubiquitous given the recent advances in embedded sensor and computation technologies. Examples of such systems include, but are not limited to, multi-vehicle systems, communication systems, power systems, process control systems and water systems (see, for example, [1–6] and the references therein). This paper concentrates on an important class of large-scale systems; namely, large-scale modular systems that consist of physically-interconnected and generally heterogeneous modules.

Sensors 2016, 16, 1297; doi:10.3390/s16081297

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1.1. Motivation and Literature Review Two sweeping generalizations can be made about large-scale modular systems. The first is that their complex structure and large-scale nature yield to inaccurate mathematical module models, since it is a challenge to precisely model each module of a large-scale system and the interconnections between these modules. As a consequence, the discrepancies between the modules and their mathematical models, that is system uncertainties, result in the degradation of overall system stability and the performance of the large-scale modular systems. To this end, adaptive control methodologies [7–13] offer an important capability for this class of dynamical systems to learn and suppress the effect of system uncertainties resulting from modeling and degraded modes of operation, and hence, they offer system stability and desirable closed-loop system performance in the presence of system uncertainties without excessively relying on mathematical models. The second generalization about large-scale modular systems is that these systems are often controlled over wireless networks, and hence, the communication costs between the modules and their remote processors increase proportionally with the increase in the number of modules and often the interconnection between these modules. To this end, event-triggered control methodologies [14–16] offer new control execution paradigms that relax the fixed periodic demand of computational resources and allow for the aperiodic exchange of sensor and actuator information with the remote processor to reduce overall communication cost over a wireless network. Note that adaptive control methodologies and event-triggered control methodologies are often studied separately in the literature, where it is of practical importance to theoretically integrate these two approaches to guarantee system stability and the desirable closed-loop system performance of uncertain large-scale modular systems with reduced communication costs over wireless networks, which is the main focus of this paper. More specifically, the authors of [6,17–23] proposed decentralized and distributed adaptive control architectures for large-scale systems; however, these approaches do not make any attempts to reduce the overall communication cost over wireless networks using, for example, event-triggered control methodologies. In addition, the authors of [24–30] present decentralized and distributed control architectures with event triggering; however, these approaches do not consider adaptive control architectures and assume perfect models of the processes to be controlled; hence, they are not practical for large-scale modular systems with significant system uncertainties. Only the authors of [31–36] present event-triggered adaptive control approaches for uncertain dynamical systems. In particular, the authors of [31,32] consider data transmission from a physical system to the controller, but not vice versa, while developing their adaptive control approaches to deal with system uncertainties. On the other hand, the adaptive control architectures of the authors in [33–36] consider two-way data transmission over wireless networks; that is, from a physical system to the controller and from the controller to this physical system. However, none of these approaches can be directly applied to large-scale modular systems. This is due to the fact that large-scale modular systems require decentralized and distributed architectures, and direct application of the results in [31–36] to this class of systems can result in centralized architectures, which is not practically desired due to the large-scale nature of modular systems. To summarize, there do not exist resilient adaptive control architectures for large-scale systems in the literature to deal with system uncertainties while reducing the communication costs between the models and their remote processors. 1.2. Contribution The contribution of this paper is to design and analyze event-triggered decentralized and distributed adaptive control architectures for uncertain large-scale systems controlled over wireless networks. Specifically, the proposed decentralized and distributed adaptive architectures of this paper guarantee overall system stability while reducing wireless network utilization and achieving a given system performance in the presence of system uncertainties that can result from modeling and degraded modes of operation of the modules and their interconnections between each other. From a theoretical viewpoint, the proposed event-triggered adaptive architectures here can be viewed

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as a significant generalization of our prior work documented in [35,36] to large-scale modular systems, which consider a state emulator-based adaptive control methodology with robustness against high-frequency oscillations in the controller response [10,13,37–42]. In this generalization, we also adopt necessary tools and methods from [6,23] on decentralized and distributed adaptive controller construction for large-scale modular systems. In addition to the theoretical findings including rigorous system stability and boundedness analysis of the closed-loop dynamical system and the characterization of the effect of user-defined event-triggering thresholds, as well as the design parameters of the proposed adaptive architectures on the overall system performance, an illustrative numerical example is further provided to demonstrate the efficacy of the proposed decentralized and distributed control approaches. 1.3. Organization The contents of the paper are as follows. In Section 2, we consider an event-triggered decentralized adaptive control approach for large-scale modular systems, where the considered approach assumes that physically-interconnected modules cannot communicate with each other for exchanging their state information. Specifically, Theorem 1 and Corollaries 1–4 show the main results of Section 2 subject to some structural conditions on the parameters of the large-scale modular systems and the proposed event-triggered decentralized control architecture (see Assumptions 4 and 5). In Section 3, we consider an event-triggered distributed adaptive control approach in Theorem 2 and Corollaries 5–7 for getting rid of such structural conditions, where the considered approach assumes that physically-interconnected modules can locally communicate with each other for exchanging their state information. Finally, the illustrative numerical example is presented in Section 4, and conclusions are summarized in Section 5. 1.4. Notation The notation used in this paper is fairly standard. Specifically, R denotes the set of real numbers; denotes the set of n × 1 real column vectors; Rn×m denotes the set of n × m real matrices; R+ denotes the set of positive real numbers; Rn+×n denotes the set of n × n positive-definite real matrices; Sn×n denotes the set of n × n symmetric real matrices; Dn×n denotes the set of n × n real matrices with diagonal scalar entries; (·)T denotes transpose; (·)−1 denotes inverse; tr(·) denotes the trace operator; diag( a) denotes the diagonal matrix with the vector a on its diagonal; and “,” denotes equality by definition. In addition, we write λmin ( A) (respectively, λmax ( A)) for the minimum and respectively maximum eigenvalue of the Hermitian matrix A, k · k for the Euclidean norm and k · kF for the Frobenius matrix norm. Furthermore, we use “∨” for the “or” logic operator and “(·)” for the “not” logic operator. We adopt graphs [43] to encode physical interactions and communications between modules. In particular, an undirected graph G is defined by VG = {1, · · · , N } of nodes and a set EG ∈ VG × VG of edges. If (i, j) ∈ EG , then the nodes i and j are neighbors, and the neighboring relation is indicated with i ∼ j. The degree of a node is given by the number of its neighbors, where di denotes the degree of node i. Lastly, the adjacency matrix of a graph G , A(G) ∈ R N × N , is given by: ( 1, if (i, j) ∈ EG (1) [A(G)]ij , 0, otherwise

Rn

2. Event-Triggered Decentralized Adaptive Control In this section, we introduce an event-triggered decentralized adaptive control architecture, where it is assumed that physically-interconnected modules cannot communicate with each other. For organizational purposes, this section is broken up into two subsections. Specifically, we first briefly overview a standard decentralized adaptive control architecture without event-triggering and then present the proposed event-triggered decentralized adaptive control approach, which includes rigorous

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stability and performance analyses with no Zeno behavior and generalizations to the state emulator case for suppressing the effect of possible high-frequency oscillations in the controller response. 2.1. Overview of a Standard Decentralized Adaptive Control Architecture without Event-Triggering Consider an uncertain large-scale modular system S consisting of N interconnected modules Si , i ∈ VG , given by: " #

Si :

x˙ i (t) = Ai xi (t) + Bi Λi ui (t) + ∆i ( xi (t)) + ∑ δij ( x j (t)) ,

xi (0) = xi0

(2)

i∼ j

where xi (t) ∈ Rni is the state of Si , ui (t) ∈ Rmi is the control input applied to Si , Ai ∈ Rni ×ni , m ×m Bi (t) ∈ Rni ×mi are known matrices and the pair ( Ai , Bi ) is controllable. In addition, Λi ∈ R+i i ∩ Dmi ×mi is an unknown module control effectiveness matrix; ∆i : Rni → Rmi represents matched module bounded uncertainties; and δij : Rn j → Rmi represents matched unknown physical interconnections with respect to module j, j ∈ VG , such that (i, j) ∈ EG . Assumption 1. The unknown module uncertainty is parameterized as: T ∆i ( xi (t)) = Woi β i ( xi (t)),

x i ∈ Rn i

(3)

where Woi ∈ Rgi ×mi is an unknown weight matrix, which satisfies kWoi kF ≤ ωi∗ , ωi∗ ∈ R+ , and β i ( xi (t)) : Rni → Rgi is a known Lipschitz continuous basis function vector satisfying:

k β i ( x1i ) − β i ( x2i )k ≤ L βi k x1i − x2i k

(4)

with L βi ∈ R+ . Assumption 2. The function δij ( x j (t)) in Equation (2) satisfies:

kδij ( x j (t))k ≤ αij k x j (t)k,

αij > 0,

x j ∈ Rn j

(5)

Next, consider the reference model Sri capturing a desired closed-loop performance for module i, i ∈ VG given by:

Sri :

x˙ ri (t) = Ari xri (t) + Bri ci (t),

xri (0) = xri0

(6)

where xri (t) ∈ Rni is the reference state vector of Sri , ci (t) ∈ Rmi is a given bounded command of Sri , Ari ∈ Rni ×ni is the reference system matrix and Bri ∈ Rni ×mi is the command input matrix. Assumption 3. There exist K1i ∈ Rmi ×ni and K2i ∈ Rmi ×mi , such that Ari = Ai − Bi K1i and Bri = Bi K2i hold with Ari being Hurwitz. Using Assumptions 1 and 3, Equation (2) can be equivalently written as: h i x˙ i (t) = Ari xi (t) + Bri ci (t) + Bi Λi ui (t) + WiT σi ( xi (t), ci (t)) + Bi ∑ δij ( x j (t))

(7)

i∼ j

h iT T , Λ −1 K T , Λ −1 K T where Wi , Λi−1 Woi ∈ R( gi +ni +mi )×mi is the unknown weight matrix and 1i 2i i i 
 T T σi xi (t), ci (t) , β i ( xi (t)) , xiT (t) , cTi (t) ∈ Rgi +ni +mi . Motivated from the structure of the uncertain terms appearing in Equation (7), let the decentralized adaptive feedback controller of Si , i ∈ VG , be given by:

Ci :

ˆ i (t)T σi ( xi (t), ci (t)) u i ( t ) , −W

(8)

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ˆ i (t) is an estimate of Wi satisfying the update law: where W h i ˆ˙ i (t) , γi Proj W ˆ i (t) , σi ( xi (t), ci (t)) ( xi (t) − xri (t))T Pi Bi , W m

ˆ i (0) = W ˆ i0 W

(9)

where Projm denotes the projection operator defined for matrices [10,35,44,45], γi ∈ R+ being the n ×n learning rate and Pi ∈ R+i i ∩ Sni ×ni being a solution of the Lyapunov equation:

n × ni

with Ri ∈ R+i

0 = ATri Pi + Pi Ari + Ri

(10)

ei (t) , xi (t) − xri (t)

(11)

∩ Sni ×ni . Now, letting:

˜ i (t) , W ˆ i (t) − Wi W

(12)

and using Equations (6) and (7), the module-level closed-loop error dynamics are given by: ˜ T (t)σi ( xi (t), ci (t)) + Bi ∑ δij ( x j (t)), e˙i (t) = Ari ei (t) − Bi Λi W i

ei (t) = ei0

(13)

i∼ j

2.2. Proposed Event-Triggered Decentralized Adaptive Control Architecture We now present the proposed event-triggered decentralized adaptive control architecture for large-scale modular systems, which reduces wireless network utilization and allows a desirable command tracking performance during the two-way data exchange between the module Si , i ∈ VG , and its local controller Ci , over a wireless network. For this objective, we utilize event-triggering control theory to schedule the data exchange dependent on errors exceeding user-defined thresholds. Specifically, the module sends its state signal to its local adaptive controller only when a predefined event occurs. The k i -th time instants of the state transmission of the module are represented by the  ∞ monotonic sequence ski k =1 , where ski ∈ R+ . The local controller uses this triggered module state i signal to compute the control signal using adaptive control architecture. In addition, the local controller sends the updated feedback control input to the module only when another predefined event occurs. The ji -th time instants of the feedback control transmission are then represented by the monotonic  ∞ sequence r ji j =1 , where r ji ∈ R+ . As depicted in Figure 1, each module state signal and its local i control input are held by a zero-order-hold operator (ZOH) until the next triggering event for the corresponding signal takes place. The delay in sampling, data transmission and computation is not considered in this paper. Consider the uncertain dynamical module i given by: "

#

x˙ i (t) = Ai xi (t) + Bi Λi usi (t) + ∆i ( xi (t)) + ∑ δij ( x j (t)) ,

Si :

xi (0) = xi0

(14)

i∼ j

where usi (t) ∈ Rmi is the sampled control input vector. Using Assumptions 1 and 3, Equation (14) can be equivalently written as: x˙ i (t)

=

i h Ari xi (t) + Bri ci (t) + Bi Λi usi (t) + WiT σi ( xi (t), xsi (t), ci (t)) + Bi ∑ δij ( x j (t)) i∼ j

+ Bi Λi (usi (t) − ui (t)) + Bi K1i ( xsi (t) − xi (t))

(15)

where xsi (t) ∈ Rni is the sampled state vector, σi ( xi (t), xsi (t), ci (t)) , βTi ( xi (t)) , xsiT (t) , cTi (t) Now, let the adaptive feedback control law be given by: 

Ci :

ˆ i (t)T σi ( xsi (t), ci (t)) u i ( t ) = −W

T

∈ R gi + n i + m i .

(16)

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where σi ( xsi (t), ci (t)) = update law:



 T ( t ) , cT ( t ) T ∈ R gi +ni +mi , and W ˆ i (t) satisfies the weight βTi ( xsi (t)) , xsi i

h i ˆ˙ i (t) = γi Proj W ˆ i (t) , σi ( xsi (t), ci (t)) eT (t) Pi Bi , W si m

ˆ i (0) = W ˆ i0 W

(17)

with esi (t) , xsi (t) − xri (t) ∈ Rni being the error of the triggered module state vector. Note that using Equation (16), Equation (15) can be rewritten as: ˜ T (t)σi ( xsi (t), ci (t)) − Bi Λi gi (·) + Bi ∑ δij ( x j (t)) x˙ i (t) = Ari xi (t) + Bri ci (t) − Bi Λi W i i∼ j

+ Bi Λi (usi (t) − ui (t)) + Bi K1i ( xsi (t) − xi (t))

(18)

where gi (·) , WiT [σi ( xsi (t), ci (t)) − σi ( xi (t), xsi (t), ci (t))], and using Equations (18) and (6), we can write the module error dynamics as: ˜ T (t)σi ( xsi (t), ci (t)) − Bi Λi gi (·) + Bi ∑ δij ( x j (t)) + Bi Λi (usi (t) − ui (t)) e˙i (t) = Ari ei (t) − Bi Λi W i i∼ j

+ Bi K1i ( xsi (t) − xi (t))

(19)

The proposed event-triggered decentralized adaptive control algorithm is based on the two-way data exchange structure depicted in Figure 1, where the local controller generates ui (t) and the uncertain dynamical module is driven by the sampled version of its local control signal usi (t) depending on an event-triggering mechanism. Similarly, the local controller utilizes xsi (t) that represents the sampled version of the uncertain dynamical module state xi (t) depending on an event-triggering mechanism. For this purpose, let exi ∈ R+ be a given, user-defined sensing threshold to allow for data transmission from the uncertain dynamical system to the controller. In addition, let eui ∈ R+ be a given, user-defined actuation threshold to allow for data transmission from the local controller to the uncertain dynamical module. Similar in fashion to [33,35], we now define three logic rules for scheduling the two-way data exchange: E1i :

k xsi (t) − xi (t)k ≤ exi

(20)

E2i :

kusi (t) − ui (t)k ≤ eui

(21)

E3i :

The controller receives xsi (t)

(22)

Specifically, when the inequality in Equation (20) is violated at the ski moment of the k i -th time instant, the uncertain module triggers the measured state signal information, such that xsi (t) is sent to its local controller. Likewise, when Equation (21) is violated or the local controller receives a new transmitted module state from the uncertain dynamical system (i.e., when E2i ∨ E3i is true), then the local controller sends a new control input usi (t) to the uncertain dynamical module at the r ji moment of the ji -th time instant. We now analyze the system stability and performance of the proposed event-triggered decentralized adaptive control algorithm introduced in this section using the error dynamics given by Equation (19), as well as the data exchange rules E1i , E2i , and E3i respectively given by Equations (20)–(22). For organizational purposes, the rest of this section, is divided into four subsections. Specifically, we analyze the uniform ultimate boundedness of the resulting closed-loop dynamical system in Section 2.2.1, compute the ultimate bound and highlight the effect of user-defined thresholds and the adaptive controller design parameters on this ultimate bound in Section 2.2.2, show that the proposed architecture does not yield to a Zeno behavior in Section 2.2.3 and generalize the decentralized event-triggered adaptive control algorithm using a state emulator-based framework in Section 2.2.4.

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Uncertain Large-Scale Module

ZOH

Event Triggering Mechanism

ZOH Adaptive Controller

Figure 1. Event-triggered adaptive control for large-scale modular systems.

2.2.1. Stability Analysis and Uniform Ultimate Boundedness We now present the first result of this paper, where the following assumption is needed. Assumption 4. D1i , λmin ( Ri ) − 2λmax ( Pi )k Bi kF ∑i∼ j αij − ∑i∼ j λmax ( Pj )k Bj kF α ji is positive by suitable selection of the design parameters. Theorem 1. Consider the uncertain large-scale modular system S consisting of N interconnected modules Si described by Equation (14) subject to Assumptions 1–4. Consider, in addition, the reference model given by Equation (6), and the module feedback control law given by Equations (16) and (17). Moreover, let the data transmission from the uncertain dynamical module to the local controller occur when E1i is true and the data transmission from the controller to the uncertain dynamical system occur when E2i ∨ E3i is true. Then, the ˜ i (t)) is uniformly ultimately bounded for all i = 1, 2, ..., N. closed-loop solution (ei (t), W Proof. Since the data transmission from the uncertain modules to their local controllers and from the local controllers to the uncertain modules occur when E1i and E2i ∨ E3i are true, respectively, note that k xsi (t) − xi (t)k ≤ exi and kusi (t) − ui (t)k ≤ eui hold. Consider now the Lyapunov-like function given by:   1 1 ˜ i ) = eT Pi ei + γ−1 tr (W ˜ i Λ 2 ) T (W ˜ iΛ 2 ) V i ( ei , W (23) i i i i ˜ i ) > 0 for all (ei , W ˜ i ) 6= (0, 0). The time-derivative of Equation (23) Note that Vi (0, 0) = 0 and Vi (ei , W is given by:

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˜ i (t)) V˙ i (ei (t), W   ˜ T ( t )W ˜˙ i (t)Λi = 2eiT (t) Pe˙i (t) + 2γi−1 tr W i ˜ T (t)σi ( xsi (t), ci (t)) − Bi Λi gi (·) + Bi ∑ δij ( x j (t)) ≤ 2eiT (t) Pi Ari ei (t) − Bi Λi W i i∼ j

!



˜ T (t)Λi σi ( xsi (t), ci (t)) eT (t) Pi Bi + Bi Λi (usi (t) − ui (t)) + Bi K1i ( xsi (t) − xi (t)) + 2tr W i si



≤ −eiT (t) Ri ei (t) − 2eiT (t) Pi Bi Λi gi (·) + 2eiT (t) Pi Bi ∑ δij ( x j (t)) + 2eiT (t) Pi Bi Λi (usi (t) − ui (t)) i∼ j



˜ T (t)Λi σi ( xsi (t), ci (t)) ( xsi (t) − xi (t))T Pi Bi +2eiT (t) Pi Bi K1i ( xsi (t) − xi (t)) + 2tr W i



≤ −λmin ( Ri )kei (t)k2 + 2kei (t)kλmax ( Pi )k Bi kF kΛi kF k gi (·)k + k2ei (t) Pi Bi ∑ δij ( x j (t))k i∼ j

˜ i (t)kF kΛi kF +2kei (t)kλmax ( Pi )k Bi kF kΛi kF eui + 2kei (t)kλmax ( Pi )k Bi kF kK1i kF exi + 2kW

(24)

·kσi ( xsi (t), ci (t)) kexi λmax ( Pi )k Bi kF It follows from Assumption 1 that an upper bound for k gi (·)k in Equation (24) can be given by:



k gi (·)k = WiT [σi ( xsi (t), ci (t)) − σi ( xi (t), xsi (t), ci (t))]

≤ kΛi−1 kF ωi∗ L βi k xsi (t) − xi (t)k ≤ K gi exi {z } |

(25)

K gi

where K gi ∈ R+ . Equation (24) as:

In addition, one can compute an upper bound for kσi ( xsi (t), ci (t)) k in

kσi ( xsi (t), ci (t)) k ≤ k β i ( xsi (t))k + k xsi (t)k + kci (t)k ≤ L βi k xsi (t)k + k xsi (t)k + kci (t)k = ( L βi + 1)exi + ( L βi + 1)kei (t)k + ( L βi + 1) xri∗ + kci (t)k

(26)

where k xri (t)k ≤ xri∗ . Then, using the bounds given by Equations (25) and (26) in Equation (24), one can write: ˜ i (t)) V˙ i (ei (t), W  ≤ −λmin ( Ri )kei (t)k2 + 2λmax ( Pi )k Bi kF kΛi kF K gi exi + 2λmax ( Pi )k Bi kF kΛi kF eui + 2λmax ( Pi )  ˜ i (t)kF kΛi kF ( L βi + 1)λmax ( Pi )k Bi kF exi kei (t)k + 2kW ˜ i (t)kF kΛi kF ·k Bi kF kK1i kF exi + 2kW
· ( L βi + 1)exi + ( L βi + 1) xri∗ + kci (t)k λmax ( Pi )k Bi kF exi + k2ei (t) Pi Bi ∑ δij ( x j (t))k i∼ j

2

= −c1i kei (t)k + c2i kei (t)k + c3i + k2ei (t) Pi Bi δij ( x j (t))k

(27)

where c1i , λmin ( Ri ), c2i , 2λmax ( Pi )k Bi kF kΛi kF K gi exi + 2λmax ( Pi )k Bi kF kΛi kF eui + 2λmax ( Pi )k Bi kF kK1i kF · exi
+ 2w˜ i∗ kΛi kF ( L βi + 1)λmax ( Pi )k Bi kF exi and c3i , 2w˜ i∗ kΛi kF ( L βi + 1)exi + ( L βi + 1) xri∗ + kci (t)k λmax ( Pi )k Bi kF exi ˜ i (t)||F ≤ w˜ ∗ due to utilizing the projection operator in the weight update law given by with ||W i Equation (9). ∗ , it follows from Assumption 2 that: Since x j (t) = e j (t) + xrj (t) with k xrj (t)k ≤ xrj

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k ∑ δij ( x j (t))k ≤ i∼ j

h i ∗ α k e ( t )k + x ∑ ij j rj

(28)

i∼ j

Furthermore, using Equation (28) in the last term of Equation (27) results in:

k2ei (t) Pi Bi δij ( x j (t))k ≤ 2λmax ( Pi )kei (t)kk Bi kF k ∑ δij ( x j (t))k i∼ j

h i ∗ ≤ 2λmax ( Pi )kei (t)kk Bi kF ∑ αij ke j (t)k + xrj i∼ j

h

∗ ≤ λmax ( Pi )k Bi kF ∑ αij 2kei (t)kke j (t)k + 2kei (t)k xrj

i

i∼ j

i h ∗2 ≤ λmax ( Pi )k Bi kF ∑ αij 2kei (t)k2 + ke j (t)k2 + xrj

(29)

i∼ j

where Young’s inequality [46] is considered in the scalar form of 2xy ≤ νx2 + y2 /ν, where x, y ∈ R and ∗ with ν = 1. Hence, Equation (27) becomes: ν > 0, and applied to terms kei (t)kke j (t)k and kei (t)k xrj h i ˜ i (t)) ≤ − c1i − 2λmax ( Pi )k Bi kF ∑ αij kei (t)k2 + λmax ( Pi )k Bi kF ∑ αij ke j (t)k2 V˙ i (ei (t), W {z } i∼ j | i∼ j fi | {z } d1i

+ c2i kei (t)k + ϕi

(30)

∗ 2. where ϕi , c3i + λmax ( Pi )k Bi kF ∑i∼ j αij xrj Introducing: N

V (·) =

∑ Vi (ei (t), W˜ i (t))

(31)

i =1

for the uncertain system S results in:

V˙ (·) ≤

N

∑

h

i =1 N

=

∑

i =1

− d1i kei (t)k2 + f i ∑ αij ke j (t)k2 + c2i kei (t)k + ϕi

i

i∼ j

h





− d1i − ∑ f j α ji kei (t)k2 + c2i kei (t)k + ϕi

i

(32)

i∼ j

|

{z

D1i

}

 T  where D1i > 0 is defined in Assumption 4. Letting ea (t) , ke1 (t)k, . . . , ke N (t)k , D1 , diag D11 , . . . , 
 
D1N , C2 , diag c21 , . . . , c2N and ϕ a , ∑iN=1 ϕi , Equation (32) can equivalently be written as:

V˙ (·) ≤ −eTa (t) D1 ea (t) + C2 ea (t) + ϕ a ≤ −λmin ( D1 )kea (t)k2 + λmax (C2 )kea (t)k + ϕ a When kea (t)k > ψ, this renders V˙ (·) < 0, where ψ , are uniformly ultimate bounded for all i = 1, 2, ... , N.

2

λmax (C2 ) √ + λmin ( D1 )

√

r

λ2 max (C2 ) + ϕ a 4λmin ( D1 )

λmin ( D1 )

(33)

˜ i (t) . Hence, ei (t) and W

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2.2.2. Computation of the Ultimate Bound for System Performance Assessment For revealing the effect of user-defined thresholds and the event-triggered feedback adaptive controller design parameters to the system performance, the next corollary presents a computation of the ultimate bound for the system S . For this purpose, we define the  
following, Pmin , diag ([λmin ( P1 ), . . . , λmin ( PN )]), Pmax , diag λmax ( P1 ), . . . , λmax ( PN ) , h i   1 ˜ a ( t ) , kW ˜ 1 (t)kF , . . . , kW ˜ N (t)kF T . γa , diag γ1−1 , . . . , γ− , Λ a , diag ([kΛ1 kF , . . . , kΛ N kF ]), W N Corollary 1. Consider the uncertain dynamical system S consisting of N interconnected modules Si described by Equation (14) subject to Assumptions 1–4. Consider, in addition, the reference model given by Equation (6), and the module feedback control law given by Equations (16) and (17). Moreover, let the data transmission from the uncertain modules to their local controllers occur when E1i is true and the data transmission from the controllers to the uncertain modules occur when E2i ∨ E3i is true. Then, the ultimate bound of the system error between the uncertain dynamical system and the reference model is given by: 1

˜ − 2 ( Pmin ), ||ea (t)|| ≤ Φλ min

t≥T

(34)

where:  1 ˜ a (t)k2 2 ˜ , λmax ( Pmax )ψ2 + λmax (γa )λmax (Λ a )kW Φ

(35)

˜ a (t)) ≤ 0 outside the compact set Proof. It follows from the proof of Theorem 1 that V˙ (ea (t), W given by:

S , {ea (t) : kea (t)k ≤ ψ}

(36)

˜ a (t)) cannot grow outside S , the evolution of V (ea (t), W ˜ a (t)) is upper That is, since V (ea (t), W bounded by: ˜ a (t)) ≤ V ( e a ( t ), W

˜ a (t)) max V (ea (t), W

ea (t)∈S

˜ a (t)k2 = λmax ( Pmax )ψ2 + λmax (γa )λmax (Λ a )kW ˜2 = Φ ˜ a ) that kea (t)k2 ≤ It follows from eTa Pmin ea ≤ V (ea , W

˜2 Φ , λmin ( Pmin )

(37)

and Equation (34) is immediate.

2.2.3. Computation of the Event-Triggered Inter-Sample Time Lower Bound We now show that the proposed event-triggered decentralized adaptive control architecture does not yield to a Zeno behavior, which implies that it does not require a continuous two-way data exchange and reduces wireless network utilization. For the following corollary presenting the result of this

k subsection, we consider rqii ∈ ski , ski +1 to be the qi -th time instant when E2i is violated over ski , ski +1 , n o  ∞  ∞  ∞  ∞ S ki ∞,mki and since ski k =1 is a subsequence of r ji j =1 , it follows that r ji j =1 = ski k =1 r qi , i i i i k i =1,qi =1
where mki ∈ N is the number of violation times of E2i over ski , ski +1 . Corollary 2. Consider the uncertain dynamical system S consisting of N interconnected modules Si described by Equation (14) subject to Assumptions 1–4. Consider, in addition, the reference model given by Equation (6), and the module feedback control law given by Equations (16) and (17). Moreover, let the data transmission from the uncertain dynamical module to the local controller occur when E1i is true and the data transmission from the

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controller to the uncertain dynamical system occur when E2i ∨ E3i is true. Then, there exist positive scalars α xi , Φexi and αui , Φeui such that: 1i

2i

s k i +1 − s k i

> α xi ,

− rqkii

> αui ,

k r q i +1 i

∀k i ∈ N  ∀qi ∈ 0, ..., mki ,

(38)

∀k i ∈ N

(39)


Proof. The time derivative of k xsi (t) − xi (t)k over t ∈ ski , ski +1 , ∀k i ∈ N, is given by: d k x (t) − xi (t)k dt si

≤ k x˙ si (t) − x˙ i (t)k = k x˙ i (t)k h ≤ k Ari kF [kei (t)k + xri∗ ] + k Bri kF kci (t)k + k Bi kF kΛi kF w˜ i∗ L βi exi + kei (t)k i
+ xri∗ + kK1i kF (exi + kei (t)k + xri∗ ) + kK2i kF kci (t)k + k Bi kF kΛi kF K gi exi   ∗ + k Bi kF kΛi kF eui + k Bi kF kK1i kF exi (40) +k Bi kF ∑ αij ke j (t)k + xrj i∼ j

Since the closed-loop dynamical system is uniformly ultimately bounded by Theorem 1, there exists an upper bound to Equation (40). Letting Φ1i denote this upper bound and with the initial condition satisfying limt→s+ k xsi (t) − xi (t)k = 0, it follows from Equation (40) that: ki

k xsi (t) − xi (t)k ≤ Φ1i (t − ski ), Therefore, when E1i is true, then limt→s−

k i +1

that ski +1 − ski ≥ α xi .

∀ t ∈ ( s k i , s k i +1 )

(41)

k xsi (t) − xi (t)k = exi , and it then follows from Equation (41)

  k k Similarly, the time derivative of kusi (t) − ui (t)k over t ∈ rqii , rqi +1 , ∀qi ∈ N, is given by: i

d ku (t) − ui (t)k dt si

≤ ku˙ si (t) − u˙ i (t)k = ku˙ i (t)k

ˆ˙ T ˆ T ˙ i ( xsi (t), ci (t)) = W

i ( t ) σi ( xsi ( t ), ci ( t )) + Wi ( t ) σ ≤ γi k Bi kF λmax ( Pi ) kesi (t)k kσi ( xsi (t), ci (t))k2 + kΛi−1 kF kK2i kF kc˙i (t)k h ≤ γi k Bi kF λmax ( Pi ) (kei (t)k + exi ) L βi (exi + kei (t)k + xri∗ ) + kK1i kF (exi i2 + kei (t)k + xri∗ ) + kK2i kF kci (t)k + kΛi−1 kF kK2i kF kc˙i (t)k (42)

Once again, since the closed-loop dynamical system is uniformly ultimately bounded by Theorem 1, there exists an upper bound to Equation (42). Letting Φ2i denote this upper bound, and with the initial condition satisfying lim ki+ kusi (t) − ui (t)k = 0, it follows from Equation (42) that: t →r qi

k

kusi (t) − ui (t)k ≤ Φ2i (t − rqii ), Therefore, when E¯ 2i ∨ E3i is true, then lim k

k

k

t →r q i − +1

  ∀t ∈ rqkii , rqkii +1

(43)

kusi (t) − ui (t)k = eui , and it then follows from

i

Equation (43) that rqi +1 − rqii ≥ αui . i

Corollary 2 shows that the inter-sample times for the module state vector and decentralized feedback control vector are bounded away from zero, and hence, the proposed event-triggered adaptive control approach does not yield to a Zeno behavior. As discussed earlier, this implies that the proposed event-triggered decentralized adaptive control methodology does not require a continuous two-way data exchange, and it reduces wireless network utilization.

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2.2.4. Generalizations to the Event-Triggered Decentralized Adaptive Control with State Emulator We now generalize our framework to a state emulator-based design, since this framework has the capability to suppress possible high-frequency oscillation in the control signal of the uncertain module Si [10,13,37–42]. Consider the (modified) reference system, so-called the state emulator of Si , given by: xˆ˙ i (t) = Ari xˆi (t) + Bri ci (t) + Li ( xsi (t) − xˆi (t)) ,

xˆi (0) = xˆi0

(44)

n ×n

where Li ∈ R+i i ∩ Dni ×ni is the state emulator gain. Letting eˆi (t) , xˆi (t) − xri (t) ∈ Rni , the reference model error dynamics capturing the difference between the ideal reference model in Equation (6) and the state emulator-based (modified) reference model in Equation (44) is given by: eˆ˙i (t) = Ari eˆi (t) + Li ( xsi (t) − xˆi (t))

(45)

In addition, letting x˜i (t) , xi (t) − xˆi (t) ∈ Rni to denote the system state error vector, the (state emulator-based) system error dynamics follows from Equations (18) and (44) as: x˜˙ i (t)

=

˜ T (t)σi ( xsi (t), ci (t)) − Bi Λi gi (·) + Bi δij ( x j (t)) + Bi Λi (usi (t) − ui (t)) A Li x˜i (t) − Bi Λi W i

+( Bi K1i − Li )( xsi (t) − xi (t)),

x˜i (0) = x˜i0

(46)

where A Li , Ari − Li ∈ Rni ×ni is Hurwitz by a suitable selection of the state emulator gain Li (e.g., A Li is Hurwitz with Li = κi I, κi ∈ R+ , since Ari is Hurwitz). To maintain system stability, we utilize the adaptive controller given by Equation (16) with the update law described by: h i ˆ i (t) , σi ( xsi (t), ci (t)) ( xsi (t) − xˆi (t))T Pi Bi , ˆ˙ i (t) , γi Proj W W m n × n i ∩Sn i × n i

where Pi ∈ R+i

(47)

is the unique solution of the algebraic Riccati equation: 0 = ATLi Pi + Pi A Li − Pi Bi Ri−1 BiT Pi + Qi

m ×m

ˆ i (0) = W ˆ i0 W

(48)

n ×n

with Ri ∈ R+i i ∩ Sni ×ni and Qi ∈ R+i i ∩ Sni ×ni . Note from [10,42] that the state emulator-based adaptive control framework achieves stringent transient and steady-state system performance specifications by judiciously choosing the learning rate γi and the state emulator gain Li without causing high-frequency oscillations in the controller response, unlike standard model reference adaptive controllers overviewed earlier in this section. We also note that if one selects Li = 0, then the results of this paper hold for standard model reference adaptive controllers, and hence, there is no loss in generality in using a state emulator-based adaptive control framework for the main results of this paper. Consider a parameter-dependent Riccati equation [23,47] given by: 0 ˜ Qi

= =

ATri P˜i + P˜i Ari + Q˜ i µi P˜i Li LT P˜i + Q˜ oi , i

(49) (50)

n ×n n ×n where P˜i ∈ R+i i is a unique solution with Q˜ oi ∈ R+i i and µi > 0.

Remark 1 [23]. Let 0 < µi < µ¯ i define the largest set within which there is a positive-definite solution for P˜i . Since P˜i > 0 for µi = 0 and P˜i > 0 depends continuously on µi , the existence of P˜i (µi ) > 0 for 0 < µi < µ¯ i is assured. The next lemma shows that for µi < µ¯ i , Equations (49) and (50) can reliably be solved for P˜i > 0 using the Potter approach given in [48]. This also implies that µ¯ i can be determined by searching for the boundary value, µ¯ i . We employ notation ric(·) and dom(·) as defined in [48].

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Lemma 1 [23,48]. Let P˜i > 0 satisfy the parameter dependent Riccati equation given by Equations (49) and (50), and let the modified Hamiltonian be given by: " Hi =

Ari − Q˜ oi

µi Li LTi − ATri

# (51)

Then, for all 0 < µi < µ¯ i , Hi ∈ dom(ric) and P˜i = ric( Hi ). λmin ( Qi ) − λmin ( Ri−1 )λ2max ( Pi )k Bi k2F − µli − 3λmax ( Pi )k Bi kF ∑i∼ j αij − i ∑i∼ j λmax ( Pj )k Bj kF α ji and D2i , li λmin ( Q˜ oi ) − ∑i∼ j λmax ( Pj )k Bj kF α ji , li > 0, are positive by suitable selection of the design parameters.

Assumption 5.

D1i

,

Corollary 3. Consider the uncertain dynamical system S consisting of N interconnected modules Si described by Equation (14) subject to Assumptions 1–3 and 5. Consider in addition, the ideal reference model given by Equation (6), the state emulator given by Equation (44) and the module feedback control law given by Equations (16) and (47). Moreover, let the data transmission from the uncertain dynamical module to the local controller occur when E1i is true and the data transmission from the controller to the uncertain dynamical ˜ i (t), eˆi (t)) is uniformly ultimately system occur when E2i ∨ E3i is true. Then, the closed-loop solution ( x˜i (t), W bounded for all i = 1, 2, ..., N. Proof. Consider the Lyapunov-like function given by: 1

1

˜ i , eˆi ) = x˜ T Pi x˜i + γ−1 tr(W ˜ i Λ 2 ) + li eˆT P˜i eˆi ˜ i Λ 2 ) T (W Vi ( x˜i , W i i i i i

(52)

where li > 0 and P˜i > 0 satisfies the parameter dependent Riccati equation in Equations (49) and (50). ˜ i , eˆi ) > 0 for all ( x˜i , W ˜ i , eˆi ) 6= (0, 0, 0). The time-derivative of Note that Vi (0, 0, 0) = 0 and Vi ( x˜i , W Equation (52) is given by: ˜ i (t), eˆi (t)) V˙ i ( x˜i (t), W 1 1 ˜ i ( t ) Λ 2 ) T (W ˜˙ i (t)Λ 2 ) + 2li eˆT (t) P˜i eˆ˙i (t) = 2x˜iT (t) Pi x˜˙ i (t) + 2γi−1 tr(W i i i h
˜ T (t)σi ( xsi (t), ci (t)) − Bi Λi gi (·) + Bi δij ( x j (t)) + Bi Λi usi (t) − ui (t) ≤ 2x˜iT (t) Pi A Li x˜i (t) − Bi Λi W i i ˜ T (t)σi ( xsi (t), ci (t)) ( xsi (t) − xˆi (t))T Pi Bi Λi +( Bi K1i − Li )( xsi (t) − xi (t)) + 2trW i   T +2li eˆi (t) P˜i Ari eˆi (t) + Li ( xsi (t) − xˆi (t))

≤ − x˜iT (t) Qi x˜i (t) + x˜iT (t) Pi Bi Ri−1 BiT Pi x˜i (t) − 2x˜iT (t) Pi Bi Λi gi (·) + 2x˜iT (t) Pi Bi δij ( x j (t))
˜ i ( t )T +2x˜iT (t) Pi Bi Λi usi (t) − ui (t) + 2x˜iT (t) Pi ( Bi K1i − Li )( xsi (t) − xi (t)) + 2trW ·σi ( xsi (t), ci (t)) ( xsi (t) − xi (t))T Pi Bi Λi − li eˆiT (t) Q˜ oi eˆi (t) − li eˆiT (t)µi P˜i Li LTi P˜i eˆi (t)

(53)

+2li eˆiT (t) P˜i Li ( xsi (t) − xi (t)) + 2li eˆiT (t) P˜i Li x˜i (t) Young’s inequality [46] applied to the last term in Equation (53) produces: 2li eˆiT (t) P˜i Li x˜i (t) ≤ µi li eˆiT (t) P˜i Li LTi P˜i eˆi (t) +

li T x˜ (t) x˜i (t) µi i

(54)

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Using Equation (54) in Equation (53) yields: ˜ i (t), eˆi (t)) V˙ i ( x˜i (t), W

≤ − x˜iT (t) Qi x˜i (t) + x˜iT (t) Pi Bi Ri−1 BiT Pi x˜i (t) − 2x˜iT (t) Pi Bi Λi gi (·) + 2x˜iT (t) Pi Bi δij ( x j (t))
˜ T (t)σi ( xsi (t), ci (t)) +2x˜iT (t) Pi Bi Λi usi (t) − ui (t) + 2x˜iT (t) Pi ( Bi K1i − Li )( xsi (t) − xi (t)) + 2trW i l ·( xsi (t) − xi (t))T Pi Bi Λi − li eˆiT (t) Q˜ oi eˆi (t) + 2li eˆiT (t) P˜i Li ( xsi (t) − xi (t)) + i x˜iT (t) x˜i (t) µi ≤ −λmin ( Ri )k x˜i (t)k2 + λmin ( Ri−1 )λ2max ( Pi )k Bi k2F k x˜i (t)k2 + 2λmax ( Pi )k Bi kF kΛi kF k gi (·)kk x˜i (t)k +k2x˜i (t) Pi Bi δij ( x j (t))k + 2k x˜i (t)kλmax ( Pi )k Bi kF kΛi kF eui + 2k x˜i (t)kλmax ( Pi ) k Bi K1i kF
˜ i (t)kF kσi ( xsi (t), ci (t)) kexi λmax ( Pi )k Bi kF kΛi kF − li λmin ( Q˜ oi )keˆi (t)k2 +k Li kF exi + 2kW +2li keˆi (t)kλmax ( P˜i )k Li kF exi +

(55)

li k x˜ (t)k2 µi i

Using Equations (25) and (26), Equation (55) can be written: ˜ i (t), eˆi (t)) V˙ i ( x˜i (t), W    l ≤ − λmin ( Qi ) − λmin ( Ri−1 )λ2max ( Pi )k Bi k2F − i k x˜i (t)k2 − li λmin ( Q˜ oi )keˆi (t)k2 + 2λmax ( Pi ) µi ˜ i (t)kF kΛi kF · k Bi kF kΛi kF K gi exi + 2λmax ( Pi )k Bi kF kΛi kF eui + 2λmax ( Pi )k Bi kF kK1i kF exi + 2kW 
˜ i (t)kF kΛi kF ( L βi + 1)exi + ( L βi + 1) x ∗ + kci (t)k · ( L βi + 1)λmax ( Pi )k Bi kF exi k x˜i (t)k + 2kW ri

· λmax ( Pi )k Bi kF exi + k2x˜i (t) Pi Bi δij ( x j (t))k + 2li λmax ( P˜i )k Li kF exi keˆi (t)k = −c1i k x˜i (t)k2 − c2i keˆi (t)k2 + c3i k x˜i (t)k + c4i keˆi (t)k + c5i + k2x˜i (t) Pi Bi δij ( x j (t))k where c1i , λmin ( Qi ) − λmin ( Ri−1 )λ2max ( Pi )k Bi k2F −

(56)

li µi ,

c2i , li λmin ( Q˜ oi ), c3i , 2λmax ( Pi )k Bi kF kΛi kF K gi exi + 2λmax ( Pi )k Bi kF kΛi kF eui + 2λmax ( Pi )k Bi kF kK1i kF exi + 2w˜ i∗ kΛi k( L βi + 1)λmax ( Pi )k Bi kF exi , c4i , 2li λmax ( P˜i )
k Li kF exi and c5i , 2w˜ i∗ kΛi kF ( L βi + 1)exi + ( L βi + 1) xri∗ + kci (t)k λmax ( Pi )k Bi kF exi . Since x j (t) = x˜ j (t) + eˆj (t) + xrj (t), it follows from Assumption 2 that:

kδij ( x j (t))k ≤

h i ∗ ˜ ˆ α k x ( t )k + k e ( t )k + x j ∑ ij j rj

(57)

i∼ j

Furthermore, using Equation (57) in the last term of Equation (56) results in:

k2x˜i (t) Pi Bi δij ( x j (t))k ≤ 2λmax ( Pi )k x˜i (t)kk Bi kF kδij ( x j (t))k h i ∗ ≤ 2λmax ( Pi )k x˜i (t)kk Bi kF ∑ αij k x˜ j (t)k + keˆj (t)k + xrj i∼ j

h

∗ ≤ λmax ( Pi )k Bi kF ∑ αij 2k x˜i (t)kk x˜ j (t)k + 2k x˜i (t)kkeˆj (t)k + 2k x˜i (t)k xrj

i

i∼ j

h i ∗2 ≤ λmax ( Pi )k Bi kF ∑ αij 3k x˜i (t)k2 + k x˜ j (t)k2 + keˆj (t)k2 + xrj

(58)

i∼ j

where Young’s inequality [46] is considered in the scalar form of 2xy ≤ νx2 + y2 /ν, with x, y ∈ R ∗ with ν = 1. Hence, and ν > 0, and applied to terms k x˜i (t)kk x˜ j (t)k, k x˜i (t)kkeˆj (t)k and k x˜i (t)k xrj Equation (56) becomes:

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˜ i (t), eˆi (t)) V˙ i ( x˜i (t), W i h ≤ − c1i − 3λmax ( Pi )k Bi kF ∑ αij k x˜i (t)k2 − c2i keˆi (t)k2 + c3i k x˜i (t)k + c4i keˆi (t)k i∼ j

{z

|

}

d1i

+ λmax ( Pi )k Bi kF ∑ αij k x˜ j (t)k2 + λmax ( Pi )k Bi kF ∑ αij keˆj (t)k2 + ϕi | {z } i∼ j | {z } i∼ j fi

(59)

fi

∗ 2 . Introducing: where ϕi , c5i + λmax ( Pi )k Bi kF ∑i∼ j αij xrj N

V (·) =

∑ Vi (x˜i (t), W˜ i (t)eˆi (t))

(60)

i =1

for the uncertain system S results in:

V˙ (·) ≤

N

∑

h

− d1i k x˜i (t)k2 − c2i keˆi (t)k2 + c3i k x˜i (t)k

i =1

+ c4i keˆi (t)k + f i ∑ αij k x˜ j (t)k2 + f i ∑ αij keˆj (t)k2 + ϕi i∼ j

N

=

∑

h

i =1

i

i∼ j

    − d1i − ∑ f j α ji k x˜i (t)k2 − c2i − ∑ f j α ji keˆi (t)k2 i∼ j

i∼ j

|

{z

D1i

}

+ c3i k x˜i (t)k + c4i keˆi (t)k + ϕi

|

{z

}

D2i

i

(61)

 T where D1i > 0 and D2i > 0 are defined in Assumption 5. Letting x˜ a (t) , k x˜1 (t)k, . . . , k x˜ N (t)k ,  T  
 
eˆa (t) , keˆ1 (t)k, . . . , keˆN (t)k , D1 , diag D11 , . . . , D1N , D2 , diag D21 , . . . , D2N , C3 ,  
 
diag c31 , . . . , c3N , C4 , diag c41 , . . . , c4N and ϕ a , ∑iN=1 ϕi , then Equation (61) can equivalently be written as:

V˙ (·) ≤ − x˜ Ta (t) D1 x˜ a (t) − eˆTa (t) D2 eˆa (t) + C3 x˜ a (t) + C4 eˆa (t) + ϕ a ≤ −λmin ( D1 )k x˜ a (t)k2 − λmin ( D2 )keˆa (t)k2 + λmax (C3 )k x˜ a (t)k + λmax (C4 )keˆa (t)k + ϕ a ˙ Either k x˜ a (t)k > ψ1 or keˆa (t)k > ψ2 renders V(·) < 0, where ψ1 , and ψ2 ,

λmax (C4 ) √ + 2 λmin ( D2 )

r

λ2 ( C ) λ2max (C3 ) + max 4 + ϕ a 4λmin ( D1 ) 4λmin ( D2 )

√

λmin ( D2 )

2

λmax (C3 ) √ + λmin ( D1 )

r

(62)

2 λ2 max (C3 ) + λmax (C4 ) + ϕ a 4λmin ( D1 ) 4λmin ( D2 )

√

λmin ( D1 )

˜ i (t) are uniformly ultimate , and hence, x˜i (t), eˆi (t) and W

bounded for all i = 1, 2, ... , N. Corollary 4. Under the conditions of Corollary 3, we can show that ei (t) is bounded for all i = 1, 2, ..., N. Proof. It readily follows from:

kei (t)k = k xi (t) − xˆ (t) + xˆ (t) − xr (t)k ≤ k xi (t) − xˆ (t)k + k xˆ (t) − xr (t)k ≤ k x˜i (t)k + keˆi (t)k and Corollary 3 that ei (t) is bounded for all i = 1, 2, ..., N.

(63)

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Remark 2. In order to obtain the closed-loop system error ultimate bound value for Equation (63) and the no Zeno behavior characterization, we can follow the same steps highlighted in Corollaries 1 and 2, respectively. 3. Event-Triggered Distributed Adaptive Control We now introduce an event-triggered distributed adaptive control architecture in this section, where it is assumed that physically-interconnected modules can locally communicate with each other for exchanging their state information. For organizational purposes, this section is broken up into two subsections. Specifically, we first briefly overview a standard distributed adaptive control architecture without event-triggering and then present the proposed event-triggered decentralized adaptive control approach, which includes rigorous stability and performance analyses with no Zeno behavior and generalizations to the state emulator case for suppressing the effect of possible high-frequency oscillations in the controller response. As shown, the benefit of using the proposed distributed adaptive control architecture versus the decentralized architecture of the previous section is that there is no need for any structural assumptions; that is, Assumptions 4 and 5, in the distributed case to guarantee overall system stability (for applications where modules are allowed to locally communicate with each other). 3.1. Overview of a Standard Distributed Adaptive Control Architecture without Event-Triggering The standard distributed adaptive control architecture overviewed in this section builds on the problem formulation stated in Section 2.1 with an important difference that the physically-interconnected modules can locally communicate with each other for exchanging their state information, as discussed above. For this purpose, we first replace Assumption 2 of Section 2.1 with the following assumption. Assumption 6. The function δij ( x j (t)) in Equation (2) satisfies: δij ( x j (t)) = QTij φij ( x j (t))

(64)

where Qij ∈ Rgi ×mi is an unknown weight matrix and φij : Rn j → Rgi is a known Lipschitz continuous basis function vector satisfying:

kφij ( x1j ) − φij ( x2j )k ≤ Lφij k x1j − x2j k

(65)

with Lφij ∈ R+ . Remark 3. We can equivalently represent Equation (64) as:

∑ QTij φij (x j (t)) , GijT Fij (x j (t))

(66)

i∼ j

where Gij ∈ R( gi ·di )×mi is the matrix combination for the ideal weight matrices of the connected graph, Fij ( x j (t)) : Rn j → R( gi ·di ) is the vector combination for basis function vectors of the connected graph and di is the degree of the i-th agent. The right hand side of Equation (66) can be given as: GijT Fij ( x j (t)) = GiT diag(Ai ) Fi

(67)

where Gi ∈ R( gi · N )×mi is the matrix combination for all modules’ ideal weight matrices of the system toward Si , Fi ( x j (t)) : Rn j → R( gi · N ) is the vector combination for all basis function vectors of the system toward Si and Ai is the i-th row of the adjacency matrix A.

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Next, using Assumptions 1, 3 and 6, Equation (2) can be equivalently written as: h
i x˙ i (t) = Ari xi (t) + Bri ci (t) + Bi Λi ui (t) + WiT σi xi (t), ci (t), x j (t)

(68)

iT T , Λ −1 K T , Λ −1 K T , Λ −1 G T Λi−1 Woi ∈ R( gi +ni +mi +( gi ·di ))×mi is an unknown 1i 2i ij i i i h iT
weight matrix and σi xi (t), ci (t), x j (t) , βTi ( xi (t)) , xiT (t) , cTi (t) , FijT ( x j (t)) ∈ Rgi +ni +mi +( gi ·di ) . Motivated from the structure of the uncertain terms appearing in Equation (68), let the distributed adaptive feedback controller of Si , i ∈ VG , be given by: where Wi

,

h

Ci :

ˆ i (t)T σi xi (t), ci (t), x j (t) u i ( t ) = −W




(69)

ˆ i (t) is an estimate of Wi satisfying the update law: where W i h
ˆ˙ i (t) = γi Proj W ˆ i (t) , σi xi (t), ci (t), x j (t) eT (t) Pi Bi , W i m

ˆ i (0) = W ˆ i0 W

n ×n

where Pi ∈ R+i i ∩ Sni ×ni is a solution of the Lyapunov Equation (10). Equations (6) and (68), the module-level closed-loop error dynamics can be given by:
˜ T (t)σi xi (t), ci (t), x j (t) , e˙i (t) = Ari ei (t) − Bi Λi W i

ei (t) = ei0

(70) Now, from

(71)

3.2. Proposed Event-Triggered Distributed Adaptive Control Architecture We now present the proposed event-triggered distributed adaptive control architecture for modular systems, where each uncertain module can exchange its state information with its interconnected neighboring modules. Consider the uncertain dynamical module i given by:

Si :

  x˙ i (t) = Ai xi (t) + Bi Λi usi (t) + ∆i ( xi (t)) + δij ( xsj (t)) ,

xi (0) = xi0

(72)

where kδij ( xsj (t))k ≤ ∑i∼ j QTij φij ( xsj (t)) and xsj (t) ∈ Rn j . Using Assumptions 1, 3 and 6, Equation (72) can be equivalently written as: h
i x˙ i (t) = Ari xi (t) + Bri ci (t) + Bi Λi usi (t) + WiT σi xi (t), xsi (t), ci (t), xsj (t)

+ Bi Λi (usi (t) − ui (t)) + Bi K1i ( xsi (t) − xi (t))

(73)

h iT
T ( t ) , cT ( t ) , FT ( x ( t )) where σi xi (t), xsi (t), ci (t), xsj (t) , βTi ( xi (t)) , xsi ∈ Rgi +ni +mi +( gi ·di ) , i ij sj and the distributed adaptive feedback control is given by:

Ci :

ˆ i (t)T σi xsi (t), ci (t), xsj (t) u i ( t ) = −W




(74)

iT h
T ( t ) , cT ( t ) , FT ( x ( t )) ˆ i (t) where σi xsi (t), ci (t), xsj (t) , βTi ( xsi (t)) , xsi ∈ Rgi +ni +mi + gi ·di , and W i ij sj satisfies the weight update law: h i
ˆ˙ i (t) = γi Proj W ˆ i (t) , σi xsi (t), ci (t), xsj (t) eT (t) Pi Bi , W si m

ˆ i (0) = W ˆ i0 W

(75)

Now, using Equation (74) in Equation (73) yields:
˜ T (t)σi xsi (t), ci (t), xsj (t) − Bi Λi gi (·) x˙ i (t) = Ari xi (t) + Bri ci (t) − Bi Λi W i

+ Bi Λi (usi (t) − ui (t)) + Bi K1i ( xsi (t) − xi (t))

(76)

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 where gi (·) , WiT σi xsi (t), ci (t), xsj (t) − σi xi (t), xsi (t), ci (t), xsj (t) , and using Equations (76) and (6), we can write the module error dynamics as:
˜ T (t)σi xsi (t), ci (t), xsj (t) − Bi Λi gi (·) + Bi Λi (usi (t) − ui (t)) e˙i (t) = Ari ei (t) − Bi Λi W i

+ Bi K1i ( xsi (t) − xi (t))

(77)

For organizational purposes, we now divide this section into four sections. Specifically, we analyze the uniform ultimate boundedness of the resulting closed-loop dynamical system in Section 3.2.1, compute the ultimate bound in Section 3.2.2, show that the proposed architecture does not yield to a Zeno behavior in Section 3.2.3 and generalize the distributed event-triggered adaptive control algorithm using the state emulator-based framework in Section 3.2.4. 3.2.1. Stability Analysis and Uniform Ultimate Boundedness Theorem 2. Consider the uncertain dynamical system S consisting of N interconnected modules Si described by Equation (72) subject to Assumptions 1, 3 and 6. Consider, in addition, the reference model given by Equation (6) and the module feedback control law given by Equations (74) and (75). Moreover, let the data transmission from the uncertain dynamical module to the local controller occur when E1i is true and the data transmission from the controller to the uncertain dynamical system occur when E2i ∨ E3i is true. Then, the ˜ i (t)) is uniformly ultimately bounded for all i = 1, 2, ..., N. closed-loop solution (ei (t), W Proof. Since the data transmission from the uncertain dynamical module to the local controller and from the local controller to the uncertain dynamical module occur when E1i and E2i ∨ E3i are true, respectively, note that k xsi (t) − xi (t)k ≤ eyi and kusi (t) − ui (t)k ≤ eui hold. Consider the Lyapunov-like function given by:   1 1 T −1 2 T 2 ˜ ˜ ˜ Vi (ei , Wi ) = ei Pi ei + γi tr (Wi Λi ) (Wi Λi ) (78) ˜ i ) > 0 for all (ei , W ˜ i ) 6= (0, 0). The time derivative of Equation (78) Note that Vi (0, 0) = 0 and Vi (ei , W is given by: ˜ i (t)) V˙ i (ei (t), W   ˜ T ( t )W ˜˙ i (t)Λi = 2eiT (t) Pe˙i (t) + γi−1 2tr W i
˜ T (t)σi xsi (t), ci (t), xsj (t) − Bi Λi gi (·) + Bi Λi (usi (t) − ui (t)) ≤ 2eiT (t) Pi Ari ei (t) − Bi Λi W i !

 
˜ T (t)Λi σi xsi (t), ci (t), xsj (t) eT (t) Pi Bi + Bi K1i ( xsi (t) − xi (t)) + 2tr W i si

≤ −eiT (t) Ri ei (t) − 2eiT (t) Pi Bi Λi gi (·) + 2eiT (t) Pi Bi Λi (usi (t) − ui (t)) + 2eiT (t) Pi Bi K1i ( xsi (t) − xi (t))  
˜ T (t)Λi σi xsi (t), ci (t), xsj (t) ( xsi (t) − xi (t))T Pi Bi +2tr W i ≤ −λmin ( Ri )kei (t)k2 + 2kei (t)kλmax ( Pi )k Bi kF kΛi kF k gi (·)k + 2kei (t)kλmax ( Pi )k Bi kF
˜ i (t)kF kΛi kF kσi xsi (t), ci (t), xsj (t) k ·kΛi kF eui + 2kei (t)kλmax ( Pi )k Bi kF kK1i kF exi + 2kW

(79)

·exi λmax ( Pi )k Bi kF where the same upper bound k gi (·)k has the same result of Equation (25). In addition, one can compute
an upper bound for kσi xsi (t), ci (t), xsj (t) k in Equation (79) as:

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kσi xsi (t), ci (t), xsj (t) k ≤ k β i ( xsi (t))k + k xsi (t)k + kci (t)k + k Fij ( xsj (t))k

≤ L βi k xsi (t)k + k xsi (t)k + kci (t)k + ∑ kφij ( x j (t))k i∼ j

= ( L βi + 1)exi + ( L βi + 1)kei (t)k + ( L βi + 1) xri∗ + kci (t)k   ∗ + ∑ Lφij exj + ke j (t)k + xrj

(80)

i∼ j

∗ . Then, using the bounds given by Equations (25) and (80) in where k xri (t)k ≤ xri∗ and k xrj (t)k ≤ xrj Equation (79) yields:

˜ i (t)) V˙ i (ei (t), W

≤ −λmin ( Ri )kei (t)k2 + 2kei (t)kλmax ( Pi )k Bi kF kΛi kF K gi exi + 2kei (t)kλmax ( Pi )k Bi kF kΛi kF eui  ˜ i (t)kF kΛi kF ( L βi + 1)exi + ( L βi + 1)kei (t)k +2kei (t)kλmax ( Pi )k Bi kF kK1i kF exi + 2kW
 ∗ +( L βi + 1) xri∗ + kci (t)k + ∑ Lφij exj + ke j (t)k + xrj exi λmax ( Pi )k Bi kF i∼ j

2



≤ −λmin ( Ri )kei (t)k + 2λmax ( Pi )k Bi kF kΛi kF K gi exi + 2λmax ( Pi )k Bi kF kΛi kF eui  +2λmax ( Pi )k Bi kF kK1i kF exi + 2w˜ i∗ kΛi kF ( L βi + 1)exi λmax ( Pi )k Bi kF kei (t)k 
 ∗ exi λmax ( Pi )k Bi kF +2w˜ i∗ kΛi kF ( L βi + 1)exi + ( L βi + 1) xri∗ + kci (t)k + ∑ Lφij exj + xrj i∼ j

+2w˜ i∗ kΛi kF exi λmax ( Pi )k Bi kF ∑ Lφij ke j (t)k i∼ j

≤ −d1i kei (t)k + d2i kei (t)k + d3i + f i ∑ Lφij ke j (t)k 2

(81)

i∼ j

where d1i , λmin ( Ri ), d2i , 2λmax ( Pi )k Bi kF kΛi kF K gi exi + 2λmax ( Pi )k Bi kF kΛi kF eui + 2λmax ( Pi )k Bi kF kK1i kF exi  +2w˜ i∗ kΛi kF ( L βi + 1)exi λmax ( Pi )k Bi kF , d3i , 2w˜ i∗ kΛi kF ( L βi + 1)exi + ( L βi + 1) xri∗ + kci (t)k +
 ∗ exi λmax ( Pi )k Bi kF and f i , 2w˜ i∗ kΛi kF exi λmax ( Pi )k Bi kF . ∑i∼ j Lφij exj + xrj Introducing: N

V (·) =

∑ Vi (ei (t), W˜ i (t))

(82)

i =1

for the uncertain system S results in:

V˙ (·) ≤

N

∑

h

i =1 N

=

∑

i =1

− d1i kei (t)k2 + d2i kei (t)k + f i ∑ Lφij ke j (t)k + d3i

i

i∼ j

h



 i − d1i kei (t)k2 + d2i + ∑ f j Lφji kei (t)k + d3i

(83)

i∼ j

|

{z

D2i

}

 T  
where D1i > 0. Letting ea (t) , ke1 (t)k, . . . , ke N (t)k , D1 , diag d11 , . . . , d1N ,  
D2 , diag D21 , . . . , D2N , and D3 , ∑iN=1 d3i , then Equation (32) can equivalently be written as:

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V˙ (·) ≤ −eTa (t) D1 ea (t) + D2 ea (t) + D3 ≤ −λmin ( D1 )kea (t)k2 + λmax ( D2 )kea (t)k + D3 When kea (t)k > ψ, this renders V˙ (·) < 0, where ψ ,

2

λmax ( D2 ) √ + λmin ( D1 )

˜ i (t) are uniformly ultimate bounded for all i = 1, 2, ... , N. W

√

r

λ2 max ( D2 ) + D 3 4λmin ( D1 )

λmin ( D1 )

(84)

, and hence, ei (t) and

3.2.2. Computation of the Ultimate Bound for System Performance Assessment For revealing the effect of user-defined thresholds and the event-triggered output feedback adaptive controller design parameters to the system performance, the next corollary presents a computation of the ultimate bound. Corollary 5. Consider the uncertain dynamical system S consisting of N interconnected modules Si described by Equation (72) subject to Assumptions 1, 3 and 6. Consider, in addition, the reference model given by Equation (6) and the module feedback control law given by Equations (74) and (75). Moreover, let the data transmission from the uncertain dynamical module to the local controller occur when E1i is true and the data transmission from the controller to the uncertain dynamical system occur when E2i ∨ E3i is true. Then, the ultimate bound of the system error between the uncertain dynamical system and the reference model is given by: 1

˜ − 2 ( Pmin ), ||ea (t)|| ≤ Φλ min

t≥T

(85)

where  1 ˜ a (t)k2 2 ˜ , λmax ( Pmax )ψ2 + λmax (γa )λmax (Λ a )kW Φ

(86)

Proof. The proof is similar to the proof of Corollary 1, and hence, omitted. 3.2.3. Computation of the Event-Triggered Inter-Sample Time Lower Bound In this subsection, we show that the proposed event-triggered distributed adaptive control architecture does not yield to a Zeno behavior, which implies that it does not require a continuous two-way data exchange and reduces wireless network utilization. For this purpose, we use the same mathematical notations introduced in Section 2.2.2 and make the following assumption. Assumption 7. Each module Si holds the received triggered state information δij ( xsj (t)) from its interconnected neighboring modules S j and sends this information to its local controller Ci when the condition E1i in Equation (20) is violated. Corollary 6. Consider the uncertain dynamical system S consisting of N interconnected modules Si described by Equation (72) subject to Assumptions 1, 3, 6 and 7. Consider, in addition, the reference model given by Equation (6) and the module feedback control law given by Equations (74) and (75). Moreover, let the data transmission from the uncertain dynamical module to the local controller occur when E1i is true and the data transmission from the controller to the uncertain dynamical system occur when E2i ∨ E3i is true. Then, there exist positive scalars α xi , Φexi and αui , Φeui , such that: 1i

2i

s k i +1 − s k i

> α xi ,

k r q i +1 i

> αui ,

− rqki

∀k i ∈ N  ∀qi ∈ 0, ..., mki ,

(87)

∀k i ∈ N

(88)

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Proof. The proof is similar to the proof of Corollary 2, and hence, omitted. Corollary 6 also shows that the inter-sample times for the module state vector and distributed feedback control vector are bounded away from zero, and hence, the proposed event-triggered distributed adaptive control approach does not yield to a Zeno behavior. 3.2.4. Generalizations to the Event-Triggered Distributed Adaptive Control with State Emulator Similar to Section 2.2.4, consider the (modified) reference model, so-called the state emulator, given by Equation (44) and the reference model error dynamics capturing the difference between the ideal reference model Equation (6), and the state emulator-based (modified) reference model Equation (44) is given by Equation (45). In addition, the (state emulator-based) system error dynamics follow from Equations (76) and (44) as: x˜˙ i (t)

=


˜ T (t)σi xsi (t), ci (t), xsj (t) − Bi Λi gi (·) Ari x˜i (t) − Bi Λi W i

+ Bi Λi (usi (t) − ui (t)) + ( Bi K1i − Li )( xsi (t) − xi (t)) − Li x˜i (t),

x˜i (0) = x˜i0

(89)

where the adaptive controller Equation (74) is used and the weight update law is given by: h i
ˆ˙ i (t) = γi Proj W ˆ i (t) , σi xsi (t), ci (t), xsj (t) ( xsi (t) − xˆi (t))T Pi Bi , W m n × ni

with Pi ∈ R+i

ˆ i (0) = W ˆ i0 W

(90)

∩ Sni ×ni being a solution to the Lyapunov Equation (10).

Corollary 7. Consider the uncertain dynamical system S consisting of N interconnected modules Si described by Equation (72) subject to Assumptions 1, 3 and 6. Consider, in addition, the ideal reference model given by Equation (6), the state emulator given by Equation (44) and the module feedback control law given by Equations (74) and (90). Moreover, let the data transmission from the uncertain dynamical module to the local controller occur when E1i is true and the data transmission from the controller to the uncertain dynamical ˜ i (t), eˆi (t)) is uniformly ultimately system occur when E2i ∨ E3i is true. Then, the closed-loop solution ( x˜i (t), W bounded for all i = 1, 2, ..., N. Proof. Consider the Lyapunov-like function given by: 1

1

˜ i , eˆi ) = x˜ T Pi x˜i + γ−1 tr(W ˜ i Λ 2 ) T (W ˜ i Λ 2 ) + 2li k Li k−1 λmax ( Pi )λmax ( Ri )eˆT Pi eˆi Vi ( x˜i , W i i F i i i

(91)

˜ i , eˆi ) > 0 for all ( x˜i , W ˜ i , eˆi ) 6= (0, 0, 0). The time-derivative of Note that Vi (0, 0, 0) = 0 and Vi ( x˜i , W Equation (91) is given by: ˜ i (t), eˆi (t)) V˙ i ( x˜i (t), W 1 1 ˜ i ( t ) Λ 2 ) T (W ˜˙ i (t)Λ 2 ) + 4li k Li k−1 λmax ( Pi )λmin ( Ri )eˆT Pi eˆ˙i (t) = 2x˜iT (t) Pi x˜˙ i (t) + 2γi−1 tr(W i F i i h

T T ˜ (t)σi xsi (t), ci (t), xsj (t) − Bi Λi gi (·) + Bi Λi usi (t) − ui (t) ≤ 2x˜i (t) Pi Ari x˜i (t) − Bi Λi W i i
˜ T (t)σi xsi (t), ci (t), xsj (t) ( xsi (t) − xˆi (t))T Pi Bi Λi +( Bi K1i − Li )( xsi (t) − xi (t)) − Li x˜i (t) + 2trW i   +4li k Li kF−1 λmax ( Pi )λmin ( Ri )eˆiT (t) Pi Ari eˆi (t) + Li x˜i (t)) + Li ( xsi (t) − xi (t))
≤ − x˜iT (t) Ri x˜i (t) − 2x˜iT (t) Pi Bi Λi gi (·) + 2x˜iT (t) Pi Bi Λi usi (t) − ui (t) + 2x˜iT (t) Pi ( Bi K1i − Li )
˜ i (t)T σi xsi (t), ci (t), xsj (t) ( xsi (t) − xi (t))T Pi Bi Λi ·( xsi (t) − xi (t)) − 2x˜ T (t) Pi Li x˜i (t) + 2trW

i

−2li k Li kF−1 λmax ( Pi )λmin ( Ri )eˆiT (t) Ri eˆi (t) + 4li k Li kF−1 λmax ( Pi )λmin ( Ri )eˆiT (t) Pi Li ( xsi (t) − xi (t)) +4li k Li kF−1 λmax ( Pi )λmin ( Ri )eˆiT (t) Pi Li x˜i (t)

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≤ −λmin ( Ri )k x˜i (t)k2 + 2λmax ( Pi )k Bi kF kΛi kF k gi (·)kk x˜i (t)k + 2k x˜i (t)kλmax ( Pi )k Bi kF kΛi kF eui
˜ i (t)kF +2k x˜i (t)kλmax ( Pi ) k Bi K1i kF + k Li kF exi − 2λmax ( Pi )k Li kk x˜i (t)k2 + 2kW
2 −1 −1 ·kσi xsi (t), ci (t), xsj (t) kλmax ( Pi )k Bi kF kΛi kF exi − 2li k Li kF λmax ( Pi )λmin ( Ri )keˆi (t)k2 +4li λmin ( Ri )exi keˆi (t)k + 4li λmin ( Ri )keˆi (t)kk x˜i (t)k

(92)

Now, using Young’s inequality [46] for the last term in Equation (92), with µi ∈ R+ , yields: ˜ i (t), eˆi (t)) V˙ i ( x˜i (t), W

≤ −λmin ( Ri )k x˜i (t)k2 + 2λmax ( Pi )k Bi kF kΛi kF k gi (·)kk x˜i (t)k + 2k x˜i (t)kλmax ( Pi )k Bi kF kΛi kF eui
˜ i (t)kF + 2k x˜i (t)kλmax ( Pi ) k Bi K1i kF + k Li kF exi − 2λmax ( Pi )k Li kk x˜i (t)k2 + 2kW
−1 · kσi xsi (t), ci (t), xsj (t) kλmax ( Pi )k Bi kF kΛi kF exi − 2li k Li kF−1 λmax ( Pi )λ2min ( Ri )keˆi (t)k2 l + 4li λmin ( Ri )exi keˆi (t)k + 2li µi λmin ( Ri )keˆi (t)k2 + 2 i λmin ( Ri )k x˜i (t)k2 (93) µi Using Equations (25) and (79), Equation (93) can be written by: ˜ i (t), eˆi (t)) V˙ i ( x˜i (t), W   h l −1 ≤ − λmin ( Ri ) − 2λmax ( Pi )k Li kF − 2 i λmin ( Ri ) k x˜i (t)k2 − 2 li k Li kF−1 λmax ( Pi )λ2min ( Ri ) µi i h −li µi λmin ( Ri ) keˆi (t)k2 + 2λmax ( Pi )k Bi kF kΛi kF K gi exi + 2λmax ( Pi )k Bi kF kΛi kF eui
i +2λmax ( Pi ) k Bi K1i kF + k Li kF exi k x˜i (t)k + 4li λmin ( Ri )exi keˆi (t)k h +2w˜ i∗ ( L βi + 1)exi + ( L βi + 1)k x˜i (t) + eˆi (t)k + ( L βi + 1) xri∗ + kci (t)k + ∑ Lφij exj i∼ j

∗ +k x˜ j (t) + eˆj (t)k + xrj


i

λmax ( Pi )k Bi kF kΛi kF exi



 h li −1 ≤ − λmin ( Ri ) − 2λmax ( Pi )k Li kF − 2 λmin ( Ri ) k x˜i (t)k2 − 2 li k Li kF−1 λmax ( Pi )λ2min ( Ri ) µi i h −li µi λmin ( Ri ) keˆi (t)k2 + 2λmax ( Pi )k Bi kF kΛi kF K gi exi + 2λmax ( Pi )k Bi kF kΛi kF eui i
+2λmax ( Pi ) k Bi K1i kF + k Li kF exi + 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi k x˜i (t)k h i + 4li λmin ( Ri )exi + 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi keˆi (t)k 
 ∗ +2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi ( L βi + 1)(exi + xri∗ ) + kci (t)k + ∑ Lφij exj + xrj i∼ j

+2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi

∑ Lφij

k x˜ j (t)k + keˆj (t)k




(94)

i∼ j

−1 ( P )k L k−1 in Equation (94) yields: then setting µi = li λmin ( Ri )λmax i i F

˜ i (t), eˆi (t)) V˙ i ( x˜i (t), W h i −1 ≤ −λmin ( Ri )k x˜i (t)k2 − 2li k Li kF−1 λmax ( Pi )λmin ( Ri ) λmin ( Ri ) − li keˆi (t)k2 h
+ 2λmax ( Pi )k Bi kF kΛi kF K gi exi + 2λmax ( Pi )k Bi kF kΛi kF eui + 2λmax ( Pi ) k Bi K1i kF + k Li kF exi i i h + 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi k x˜i (t)k + 4li λmin ( Ri )exi + 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi keˆi (t)k 
 ∗ + 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi ( L βi + 1)(exi + xri∗ ) + kci (t)k + ∑ Lφij exj + xrj i∼ j

+ 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi ∑ Lφij k x˜ j (t)k + keˆj (t)k i∼ j




(95)

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It then follows that Equation (95) can be given by: ˜ i (t), eˆi (t)) V˙ i ( x˜i (t), W

≤ −d1i k x˜i (t)k2 − d2i keˆi (t)k2 + d3i k x˜i (t)k + d4i keˆi (t)k + d5i + f i ∑ Lφij k x˜ j (t)k i∼ j

+ f i ∑ Lφij keˆj (t)k

(96)

i∼ j

h i −1 ( P ) λ where d1i , λmin ( Ri ), d2i , 2li k Li kF−1 λmax i min ( Ri ) λmin ( Ri ) − li , d3i , 2λmax ( Pi )k Bi kF k Λi kF K gi e xi
+2λmax ( Pi )k Bi kF kΛi kF eui + 2λmax ( Pi ) k Bi K1i kF + k Li kF exi + 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi , d4i ,  4li λmin ( Ri ) · exi + 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi , d5i , 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi ( L βi + 1)(exi + xri∗ ) +
 ∗ and f i , 2w˜ i∗ λmax ( Pi )k Bi kF kΛi kF exi . To ensure that d2i is positive kci (t)k + ∑i∼ j Lφij exj + xrj definite, we consider li = θi λmin ( Ri ) and θi ∈ (0, 1). Introducing: N

V (·) =

∑ Vi (x˜i (t), W˜ i (t)eˆi (t)),

(97)

i =1

for the uncertain system S results in:

V˙ i (·) ≤

N

∑

h

i =1

− d1i k x˜i (t)k2 − d2i keˆi (t)k2 + d3i k x˜i (t)k + d4i keˆi (t)k + d5i + f i ∑ Lφij k x˜ j (t)k i∼ j

+ f i ∑ Lφij keˆj (t)k

i

i∼ j

N

=

∑

i =1

h

    i − d1i k x˜i (t)k2 − d2i keˆi (t)k2 + d3i + ∑ f j Lφji k x˜i (t)k + d4i + ∑ f j Lφji keˆi (t)k + d5i i∼ j

|

{z

D3i

i∼ j

}

|

{z

}

D4i

(98)  T  T  
Letting x˜ a (t) , k x˜1 (t)k, . . . , k x˜ N (t)k , eˆa (t) , keˆ1 (t)k, . . . , keˆN (t)k , D1 , diag d11 , . . . , d1N , D2  
 
 
, diag d21 , . . . , d2N , D3 , diag D31 , . . . , D3N , D4 , diag D41 , . . . , D4N , and D5 , ∑iN=1 d5i , then Equation (98) can equivalently be written as:

V˙ (·) ≤ − x˜ Ta (t) D1 x˜ a (t) − eˆTa (t) D2 eˆa (t) + D3 x˜ a (t) + D4 ea (t) + D5 ≤ −λmin ( D1 )k x˜ a (t)k2 − λmin ( D2 )keˆa (t)k2 + λmax ( D3 )k x˜ a (t)k + λmax ( D4 )keˆa (t)k + D5 Either k x˜ a (t)k > ψ1 or keˆa (t)k > ψ2 , renders V˙ (·) < 0, where ψ1 , and ψ2 ,

λmax ( D4 ) √ + 2 λmin ( D2 )

r

λ2 ( D ) λ2max ( D3 ) + max 4 + D5 4λmin ( D1 ) 4λmin ( D2 )

√

λmin ( D2 )

2

λmax ( D3 ) √ + λmin ( D1 )

r

(99)

2 λ2 max ( D3 ) + λmax ( D4 ) + D 5 4λmin ( D1 ) 4λmin ( D2 )

√

λmin ( D1 )

˜ i (t) are uniformly ultimate , and hence, x˜i (t), eˆi (t), and W

bounded for all i = 1, 2, ... , N. Remark 4. To show that ei (t) is bounded for all i = 1, 2, . . . , N under the condition of Corollary 7, we can follow Corollary 4 to show the boundedness of ei (t) for all i = 1, . . . , N using:

kei (t)k ≤ k x˜i (t)k + keˆi (t)k

(100)

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Furthermore, in order to obtain the closed-loop system error ultimate bound value for Equation (100) and the no Zeno characterization proof, we can follow the same steps highlighted in Corollaries 5 and 6, respectively. 4. Illustrative Numerical Example In this section, the efficacy of the proposed event-triggered decentralized adaptive control approach is demonstrated in an illustrative numerical example. For this purpose, we consider the uncertain dynamical system, which consists of five masses connected serially by springs and dampers as depicted in Figure 2. We use the following equations of motion for the i-th mass: "

"

"

x˙ 1 (t) x¨1 (t)

#

x˙ i (t) x¨i (t)

#

x˙ 5 (t) x¨5 (t)

#

"

=

1

−k1 m1

−b1 m1

"

=

x1 ( t ) x˙ 1 (t) 1

−(k i−1 +k i ) mi

−(bi−1 +bi ) mi

#"

0

1

−k4 m5

−b4 m5

#

"

+

x5 ( t ) x˙ 5 (t)

#

#

0 1 m1

#"

0

"

=

#"

0

"

+

xi ( t ) x˙ i (t) 0

[Λ1 u1 (t) + ∆1 ( x1 (t)) + δ12 ( x2 (t))] #

"

+

(101)

#

0



1 mi

 Λi ui (t) + ∆i ( xi (t)) + δij ( x j (t)) ,

i = {2, 3, 4} #

1 m5

(102)

[Λ5 ui (t) + ∆5 ( x5 (t)) + δ54 ( x4 (t))]

(103)

where mi = 1Kg, k i = 1.5 N·m−1 , bi = 0.4 N·sec·m−1 , Λi = 0.7, Woi = [3 , 1]T , and we set the basis function as β i ( xi (t)) = xi (t). In addition, δ12 ( x2 (t)), δij ( x j (t)) and δ54 ( x4 (t)), which represent the effect of the system interconnections, are given by: δ12 ( x2 (t)) δij ( x j (t))

=

h

=

h

=

h

k1

b1

i

k j = i −1

"

x2 ( t ) x˙ 2 (t)

b j = i −1

i

# (104) "

x j = i −1 ( t ) x˙ j=i−1 (t)

#

+

h

k j =i

b j =i

i

"

x j = i +1 ( t ) x˙ j=i+1 (t)

# ,

i = {2, 3, 4} δ54 ( x4 (t)))

u1 (t)

k4

b4

i

"

x4 ( t ) x˙ 4 (t)

u2 (t)

m1

u4 (t)

k2 m2

k4 m4

b2 x2 (t)

u5 (t)

k3 m3

b1 x1 (t)

(106)

u3 (t)

k1

(105)

#

m5

b3 x3 (t)

b4 x4 (t)

x5 (t)

Figure 2. Connected mass-damper-spring system.

The control objective of each module is to enforce xi (t) to track a filtered square reference input ci (t) under the effect of uncertainties and disturbances with reduced communication effort by event-triggering architecture. For our example, we choose a second-order ideal reference model that has a natural frequency of 2 rad/s and a damping ratio of 0.707 for all Si , i = 1, . . . , 5. In addition, we use a state emulator gain Li = 9I2 and set all initial conditions to zero for all Si , i = 1, . . . , 5. For the event-triggered decentralized model reference adaptive control (which is equivalent to Li = 0), we set Qi = I2 in order to compute Pi in Equation (10). The condition in Assumption 4 holds when αij ≤ 0.26 for i = {1, 5} and αij ≤ 0.13 for i = {2, 3, 4}. In this case, Assumption 2 is

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20 r1

0

xm1

−1

x1

−2

xs1 0

20

40

80

xm3

0

x3

−2

xs3 0

20

40

60

80

t [s]

xm4

0

x4

−2

xs4 0

20

40

60

80

t [s]

0 −1

x5

−2

xs5 0

20

40

20

40

60

t [s]

(A) State signals

80

100

60

80

t [s]

100 u3

10

us3

0

20

40

60

80

t [s]

100 u4 us4

10 0

20

40

60

80

t [s]

r5 xm5

100 u2 us2

−10 0 20

100

80

0

−10 0 20

100

60

t [s]

r4

−1

1

40

10

−10 0 20

100

−1

1

20

r3

u3 (t)

x3 (t)

60

t [s]

u2 (t)

−2

xs2

u4 (t)

x2 (t)

x2

40

us1

0 −10 0 20

100

xm2

20

10

r2

0

0

x4 (t)

80

−1

1

x5 (t)

60

t [s]

1

u1

u1 (t)

1

u5 (t)

x1 (t)

satisfied for the coupling terms given in Equations (104)–(106). For the purpose of event-triggered state emulator-based decentralized adaptive control, we set Ri = 3 and Qi = I2×2 in order to compute Pi in Equation (48). For li = 0.001 and Q˜ 0i = 250I2 , the condition in Assumption 5 holds when αij ≤ 4.2 for i = {1, 5} and αij ≤ 2.1 for i = {2, 3, 4}. In addition, Assumption 2 is satisfied for coupling terms given by Equations (104)–(106). For the proposed event-triggered distributed adaptive control, we set Qi = I2 in order to compute Pi in Equation (10). Note that there are no fundamental stability conditions for the case of distributed adaptive control. Lastly, for the event-triggering thresholds, we choose exi = 0.2 and eui = 0.2 for i = {1, 3, 5} and exi = 0.07 and eui = 0.07 for i = {2, 4}. For the proposed event-triggered decentralized adaptive control design of Theorem 1 and Corollary 1, Figures 3–5 represent the results for various γi and Li . In particular, we first set γi = 50 and Li = 0 in Figure 3, which results in a control response with high-frequency oscillations. In order to suppress these undesired oscillations, we set Li = 9I2 as seen in Figure 4. In this figure, even though such oscillations are reduced, the command tracking performance becomes worse as we increase Li compared to the response in Figure 3. In addition to increasing Li , we also increase γi in Figure 5, to improve command tracking performance without causing high-frequency oscillations. In general, if one picks Li to be greater than nine, then it may also be necessary to increase γi further to obtain a similar closed-loop system performance. It should also be mentioned that choosing Li and γi to produce both a control response without any significant high-frequency oscillations, and a small uniform ultimate bound can be cast as an optimization problem, as well. Figures 6–8 represent the results of the proposed event-triggered distributed adaptive control of Theorem 2 and Corollary 7 for the same γi and Li values. Specifically, we see high frequency content in the control signal in Figure 6 when γi = 50 and Li = 0, which is mitigated by increasing the state emulator gain to Li = 9I2 , as seen in Figure 7. In order to enhance the command tracking, which is degraded by increasing the state emulator gain, we increase γi as seen in Figure 8.

100 u5

10

us5

0 −10

0

20

40

60

80

100

t [s]

(B) Control signals

Figure 3. Command following performance for the proposed event-triggered decentralized adaptive control approach with γi = 50 and Li = 0.

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20 r1

0

xm1

−1

x1

−2

xs1 0

20

40

−2

xs2 40

60

80

t [s]

xs3 40

60

80

t [s]

1

x4 (t)

u3 (t)

x3

20

xm4 x4

−2

xs4 0

20

40

60

80

t [s]

xm5

0

x5

−2

xs5 0

20

40

20

40

60

80

80

100 u3 us3

0

20

40

60

80

t [s]

100 u4 us4

10 0

20

40

60

80

t [s]

100 u5

10

us5

0 −10

100

60

t [s]

r5

−1

100 u2

10

−10 0 20

100

80

0

r4

0

60

t [s]

us2

−10 0 20

100

−1

1

40

10

−10 0 20

100

xm3

0 −1

0

20

r3

−2

x5 (t)

u2 (t)

x2

20

us1

0 −10 0 20

100

xm2

0

0

10

r2

u4 (t)

x2 (t)

80

−1

1

x3 (t)

60

t [s]

1

u1

u1 (t)

1

u5 (t)

x1 (t)

Sensors 2016, 16, 1297

0

20

40

t [s]

60

80

100

t [s]

(A) State signals

(B) Control signals

20 r1

0

xm1

−1

x1

−2

xs1 0

20

40

xs2 80

xm3

0

x3

−2

xs3 0

20

40

60

80

t [s]

x4

−2

xs4 20

40

60

80

t [s]

xm5

0

x5

−2

xs5 0

20

40

20

40

60

t [s]

(A) State signals

80

100

100 u2

60

80

t [s]

100 u3

10

us3

0

20

40

60

80

t [s]

100 u4 us4

10 0

20

40

60

80

t [s]

r5

−1

80

us2

−10 0 20

100

60

t [s]

0

−10 0 20

100

xm4

0

0

40

r4

−1

1

20

10

−10 0 20

100

−1

1

0

r3

u3 (t)

x3 (t)

60

t [s]

u2 (t)

−2

u4 (t)

x2 (t)

x2

40

us1

−10 0 20

100

xm2

20

10

r2

0

0

x4 (t)

80

−1

1

x5 (t)

60

t [s]

1

u1

u1 (t)

1

u5 (t)

x1 (t)

Figure 4. Command following performance for the proposed event-triggered decentralized adaptive control approach with γi = 50 and Li = 9.

100 u5

10

us5

0 −10

0

20

40

60

80

100

t [s]

(B) Control signals

Figure 5. Command following performance for the proposed event-triggered decentralized adaptive control approach with γi = 200 and Li = 9.

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20 r1

0

xm1

−1

x1

−2

xs1 0

20

40

−2

xs2 40

60

80

t [s]

xs3 40

60

80

t [s]

1

x4 (t)

u3 (t)

x3

20

xm4 x4

−2

xs4 0

20

40

60

80

t [s]

xm5

0

x5

−2

xs5 0

20

40

20

40

60

80

80

100 u3 us3

0

20

40

60

80

t [s]

100 u4 us4

10 0

20

40

60

80

t [s]

100 u5

10

us5

0 −10

100

60

t [s]

r5

−1

100 u2

10

−10 0 20

100

80

0

r4

0

60

t [s]

us2

−10 0 20

100

−1

1

40

10

−10 0 20

100

xm3

0 −1

0

20

r3

−2

x5 (t)

u2 (t)

x2

20

us1

0 −10 0 20

100

xm2

0

0

10

r2

u4 (t)

x2 (t)

80

−1

1

x3 (t)

60

t [s]

1

u1

u1 (t)

1

u5 (t)

x1 (t)

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0

20

40

t [s]

60

80

100

t [s]

(A) State signals

(B) Control signals

20 r1

0

xm1

−1

x1

−2

xs1 0

20

40

−2

xs2 40

60

80

t [s]

xs3 40

60

80

t [s]

1

x4 (t)

u3 (t)

x3

20

xm4 x4

−2

xs4 0

20

40

60

80

t [s]

xm5

0

x5

−2

xs5 0

20

40

20

40

60

t [s]

(A) State signals

80

100

60

80

t [s]

100 u3 us3

0

20

40

60

80

t [s]

100 u4 us4

10 0

20

40

60

80

t [s]

r5

−1

100 u2

10

−10 0 20

100

80

0

r4

0

60

t [s]

us2

−10 0 20

100

−1

1

40

10

−10 0 20

100

xm3

0 −1

0

20

r3

−2

x5 (t)

u2 (t)

x2

20

us1

0 −10 0 20

100

xm2

0

0

10

r2

u4 (t)

x2 (t)

80

−1

1

x3 (t)

60

t [s]

1

u1

u1 (t)

1

u5 (t)

x1 (t)

Figure 6. Command following performance for the proposed event-triggered distributed adaptive control approach with γi = 50 and Li = 0.

100 u5

10

us5

0 −10

0

20

40

60

80

100

t [s]

(B) Control signals

Figure 7. Command following performance for the proposed event-triggered distributed adaptive control approach with γi = 50 and Li = 9.

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0

xm1

−1

x1

−2

xs1 0

20

40

−2

xs2 40

60

80

t [s]

u3 (t)

xs3 40

60

80

t [s]

1

x4 (t)

100

x3

20

xm4 x4

−2

xs4 0

20

40

60

80

t [s]

xm5

0

x5

−2

xs5 0

20

40

20

40

60

80

80

100 u3 us3

0

20

40

60

80

t [s]

100 u4 us4

10 0

20

40

60

80

t [s]

100 u5

10

us5

0 −10

100

60

t [s]

r5

−1

100 u2

10

−10 0 20

100

80

0

r4

0

60

t [s]

us2

−10 0 20

100

−1

1

40

10

−10 0 20

xm3

0 −1

0

20

r3

−2

x5 (t)

u2 (t)

x2

20

us1

0 −10 0 20

100

xm2

0

0

10

r2

u4 (t)

x2 (t)

80

−1

1

x3 (t)

60

t [s]

1

u1

u1 (t)

1

u5 (t)

x1 (t)

Sensors 2016, 16, 1297

0

t [s]

20

40

60

80

100

t [s]

(A) State signals

(B) Control signals

Figure 8. Command following performance for the proposed event-triggered distributed adaptive control approach with γi = 200 and Li = 9.

From these results, we observe from the decentralized adaptive control case that the state emulator-based approach not only gives stringent performance without causing high frequencies in the controller response, but also tolerates the interconnection uncertainties of the modules. In addition, the performance of the distributed adaptive controller is better than the decentralized adaptive controller with the corresponding design parameter setting. The total number of the state and control event triggers of the whole system for the cases in Figures 3–8 is given in Figure 9A,B, respectively. Figure 9 shows the drastic decrement of the triggering number using the event-triggering approach and also the further triggering number decrement due to utilizing the state emulator-based approach. 4

10

4

x 10

10 No event-triggering γ = 50, L = 0 γ = 50, L = 9 γ = 200, L = 9

8

8

7

7

6 5 4

6 5 4

3

3

2

2

1 0

No event-triggering γ = 50, L = 0 γ = 50, L = 9 γ = 200, L = 9

9

Number of triggers

Number of triggers

9

x 10

1

State signal

Control signal

(A) Decentralized adaptive control

0

State signal

Control signal

(B) Distributed adaptive control

Figure 9. Number of triggers with respect to the controller design parameters.

5. Conclusions The design and analysis of event-triggered decentralized and distributed adaptive control architectures for uncertain networked large-scale modular systems were presented. For the decentralized case, it was shown in Section 2 that the proposed event-triggered adaptive control

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architecture guarantees system stability and performance with no Zeno behavior under some structural conditions stated in Assumptions 4 and 5 that depend on the parameters of the large-scale modular systems and the proposed architecture. For the distributed case, it was shown in Section 3 that the proposed event-triggered adaptive control architecture guarantees the same system stability and performance with no Zeno behavior without such structural conditions under the assumption that physically-interconnected modules can locally communicate with each other for exchanging their state information. In addition to the presented theoretical findings, the efficacy of the proposed event-triggered decentralized and distributed adaptive control approaches is demonstrated on an illustrative numerical example in Section 4, where significant reduction on the overall communication cost was obtained for large-large modular systems in the presence of system uncertainties resulting from modeling and degraded modes of operation of the modules and their interconnections between each other. For the future work, sampling, data transmission and computation delays will be considered along with the proposed results of this paper, since they also play an important role in the performance of networked control systems. Furthermore, we will also consider the cases when a set of diagonal elements of the control effectiveness matrix is zero and generalize the results of this paper to cover these so-called loss of control cases. Author Contributions: The research documented in this paper was conducted by Ali Albattat, where he received periodical support and guidance in theory development and simulations from Benjamin Gruenwald and Tansel Yucelen on theory development. Conflicts of Interest: The authors declare no conflict of interest.

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24. 25.

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