Dynamic feedback triggering fuzzy control for Takagi–Sugeno discrete systems

Author’s Accepted Manuscript Dynamic Feedback Triggering Fuzzy Control for Takagi-Sugeno Discrete Systems Magdi S. Mahmoud, Nezar M. Alyazidi

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S0016-0032(16)30487-2 http://dx.doi.org/10.1016/j.jfranklin.2016.12.016 FI2841

To appear in: Journal of the Franklin Institute Received date: 8 April 2016 Revised date: 14 December 2016 Accepted date: 22 December 2016 Cite this article as: Magdi S. Mahmoud and Nezar M. Alyazidi, Dynamic Feedback Triggering Fuzzy Control for Takagi-Sugeno Discrete Systems, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.12.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Dynamic Feedback Triggering Fuzzy Control for Takagi-Sugeno Discrete Systems Magdi S. Mahmoud , Nezar M. Alyazidi

Abstract We study the design problem of dynamic event-triggered (ET) fuzzy control methodology for discretetime Takagi-Sugeno (T–S) systems in this work. The observed-state error is utilized to derive the event condition. Sufficient condition for the asymptotic stability of the closed-loop fuzzy system and the observer error system under the event-triggered fuzzy control is provided by linear matrix inequality (LMI). Moreover, the corresponding self-triggered (ST) fuzzy control is developed where the following control action is estimated in terms of the current sampled data. Furthermore, simulations are given to represent the feasibility and efficiency of the achieved results. Index Terms Event-based control, self-triggered, discrete-time fuzzy systems.

I. I NTRODUCTION Recent employments of artificial intelligence (AI) approaches aimed to imitate human practice into a implementable code for processing by computers [1]. AI techniques paved the way to intelligent control by generating efficient advanced control methods. In general, intelligent systems are often characterized by: 1) Analogies with biological systems to build up dynamic models, 2) By exploring at how human actuality carries out control tasks to characterize course of actions, 3) Recognize patterns or make decisions to reach prescribed goals The disparity between humans and machines are readily obvious: humans persuade normally in uncertain, complicated methods. However, computers are mainly performing in terms of binary persuading. In this regard, the basic concept of fuzzy logic is an attempt to mimic human control logic thereby making machines immense intelligent, thus permissive them to persuade based on a fuzzy technique such humans. Fuzzy logic developed to be a powerful tool to handle uncertain, and inaccurate problems [2]. Dynamic systems based on fuzzy-logic representations are often called Takagi-Sugeno (T–S) systems. This class of fuzzy dynamic models consists of a group of models that carefully linked with fuzzy membership functions [3]. In particular, fuzzy logic method can be performed in the field of a real implementation using: 1) Fuzzification: adapt data into fuzzy data or membership functions (MFs). 2) Fuzzy Inference Process: integrate MFs with control laws to acquire the fuzzy output. 3) Defuzzification: calculate every related output and generate a look-up table. 4) Next, catch up the output from the look-up list by means of the present input. Prominent features of fuzzy-logic controllers (FLC) attributed to robustness properties, ease of modification and cheap to implement. A short survey on study and architecture methodologies of the T–S models is demonstrated in [4]. Stability and dissipativity conditions were obtained for the fuzzy schemes with nonuniform uncertain sampling MsM-KFUPM-EventTrig-DiscreteFuzzy-JFI.tex Distributed Control Research Lab, Systems Engineering Department, KFUPM, Dhahran 31261, Saudi Arabia. e-mail: msmahmoud,[email protected].

utilizing the time-dependent Lyapunov-Krasovskii functional technique [5]. The scope of dissipativity study and dissipative controller layout have been introduced in [6]. Recent results on fuzzy control and stabilization can be found in [7]–[12]. The foregoing results considered the control loops are closed all the time, which is provides a burden on control cost. The authors of [13] investigated a fuzzy observer-based controller for discrete-time systems with intelligence Lyapunov functions utilizing the single-step LMIs. Very recent, [14] has been effectively applied adaptive sliding mode control to cope the uncertain nonlinear systems and to ensure the closed loop system is bounded. Thus, uncertainties is performed by type-2 fuzzy system. [15] has implemented a powerful fuzzy adaptive observer to evaluate the unknown parameters for MIMO model. Furthermore, an output feedback controller get benefits from the former adaptive observer to partially tracking errors. [16] have proposed a robust output feedback H∞ control for a sort of nonlinear systems subject to actuator faults. In the sense, the nonlinear behaviors are mainly illustrated by TS fuzzy models while the fault actions are described by a Markov process. Advanced H∞ methodologies have been designed for Markovian fuzzy models utilizing a partial information of transition rates [17] [18]. On another research avenue, the classical feedback control approach incorporates periodic sampling in which the control signal is fixed with zero-order hold. This is called time-triggered control (TTC) in [19]–[23] and it is generally heads to conservativeness in the utilization of computation, communication agents and limited bandwidth. Therefore, to seek for a better overall system performance, it is preferred to drop the periodic constraint. Confronted with this demand, event-triggered control (ETC) was successfully investigated in the place of the TTC. Under ETC, the control responsibility is only achieved when needed as described by an event condition [20]–[24]. Events comprise of satisfied logical statuses. Thus statuses rely on sensor measurements, variables, and the amount of time. Furthermore, these sequence of are utilized closed-loop system that can sustain stability, and appropriate performance. The ETC is being extensively proposed [25]-[33] to explore higher task intervals by means of applicable performances. Nevertheless, to address the problem of continually monitor the state or the output systems to establish the event triggered or not. Related problems filtering and control of stochastic systems are treated recently in [34]–[36], In [37], control of singular systems is dealt with under networked environment and using event-triggered scheme. One standing issue remains as the plants state or output must be constantly monitored in order to determine if an event should be triggered or not. To handle this issue, ST control procedures are extremely applied in [39]–[42]. Additionally, the issue of regularly testing a triggering status can be tackled by employing an event scheduler. This scheduler is primarily responsible for performing a new sampling event. In this sense, the decision is basically taken place to sample by means of awareness about the system and on either the current sampled state or an estimate of it, Counting on or in case the whole dynamic of states is available or not. From the reported results, it appeared that few studies have focused on discrete-time systems. In this study, we establish an improved technology to jointly design the ET condition and feedback control law for discrete (T–S) systems. The main contributions of our work are: 1) Novel results are developed on dynamic event-triggering fuzzy control. 2) LMI-based sufficient conditions are provided to guarantee the stability of both the observer-based control fuzzy system and the dynamic error under event-triggering strategy. 3) Extending the results to self-triggering fuzzy control. 4) A detailed lab-scale system is simulated and the ensuing results support the analytical developments.

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II. S YSTEM D ESCRIPTION Reference is labeled to Fig. 1, where we examine an nonlinear discrete plants illustrated by the state-space representation: IF θ1 is Mℓ1 and ... and θg is Mℓg , T HEN xp (k + 1) = Aℓp xp (k) + Bpℓ up (k), yp (k) = Cpℓ xp (k)

(1)

where x(k) ∈ ℜn is the state vector, u(k) ∈ ℜm is the control input vector and y(k) ∈ ℜp is the output vector. The matrices Aℓp ∈ ℜn×n , Bpℓ ∈ ℜn×m and Cpℓ ∈ ℜp× n are system matrices of rule-ℓ with † suitable dimensions. Moreover, Cpℓ is of full row rank C ℓ p demonstrates the right-inverse. By hypothesis, {θ, Mℓq }, q = 1, 2, ..., g, demonstrate the figure of the fuzzy rules in system Σ and the linguist fuzzy sets of the rule ℓ, respectively, and θ = [θ1 (t), θ2 (t), ..., θg (t)] are certain measurable based on variables for system (1) which may be equal to x(t) or a function of x(t) and Mℓq = [Mℓ1 , Mℓ2 , ..., Mℓg ].

Fig. 1. Observer-based fuzzy control system

By incorporating the nominal fuzzy inference approach, hence, an individual fuzzifier, minimum fuzzy inference, and central-average defuzzifier, model (1), are able to be cast as [2], [3] :

xp (k + 1) = yp (k) =

r X

ℓ=1 r X

ℓ



Aℓp xp (k)

ℓ



Cpℓ xp (k)

µ (θq (k)) µ (θq (k))

ℓ=1

+



Bpℓ up (k)



= Ap (µ)xp (k) + Bp (µ)up (k)

= Cp (µ)xj (k)

(2)

where µℓ (θq (k)) =

Y ω ℓ (θq (k)) ℓ Pr , ω (θ (k)) = (Mℓq (θq (k))) q ℓ (θ (k)) ω q ℓ=1

(3)

q=1

and Mℓq (θq (k)) represents the category of membership of (θq (k)) in Mℓq . It i not difficult to see that the normalized fuzzy weights µℓ (θqj (k)) are governed by µℓ (θq (k)) ≥ 0, ℓ = 1, 2, ..., r;

We will use

µℓ (θ

q (k))

r X

µℓ (θq (k)) = 1

ℓ=1

≡ µ from now onwards for the sake of brevity.

3

(4)

III. DYNAMIC F EEDBACK F UZZY C ONTROLLER In the sequel, we develop the issue of implementation of a dynamic feedback fuzzy controller with regular sampling. We propose the following observer-based controller: IF θ1 is Mℓ1 and ... and θg is Mℓg , T HEN x ˆ(k + 1) = Aℓp x ˆ(k) + Bp (µ)up (k) + Lℓ (µ)(yp (k) − yˆ(k)),

yˆ(k) = Cpℓ x ˆ(k),

up (k) = K ℓ x ˆ(k)

(5)

Employing the parallel distributed compensation method, the fuzzy dynamic output-feedback controller can be described as x ˆ(k + 1) = Ap (µ)ˆ x(k) + Bp (µ)up (k) + L(µ)(µ)(yp (k) − yˆ(k)), yˆ(k) = Cp (µ)ˆ x(k),

up (k) = K(µ)ˆ x(k)

(6)

ℜn

ℜn

where x ˆ(k) ∈ denotes the the observer state-vector, yˆ(k) ∈ is the observer output. In addition, m×n n×p K(µ) ∈ ℜ is the controller gain matrix and L(µ) ∈ ℜ is the observer gain matrix are definitely picked to to sustain that the closed-loop fuzzy system, comprises of (2) and (6), is asymptotically stable. In terms of the observation error ep (k) = xp (k)− x ˆ(k), it follows from (1) and (6) that the error dynamics cab be deduced as: e(k + 1) = (Ap (µ) − L(µ)Cp (µ))e(k), (7) which shows that the error system (7) can be asymptotically stable with the choice of gain matrix L(µ) under regular sampling. IV. S YSTEMS

E VENT- TRIGGERING

WITH

We now direct attention to systems with event-triggering in which the event-driven control system is essentially utilized to provide a new the input regulation once the state is fluctuated from the desired value. In the sequel, we let sk be the k -th sampling time and Tk = sk+1 − sk represents the k -th sampling period. During Tj , the control input is kept constant by zero-order hold method. Hence, the control law becomes, up (k) := u(sk ) = K(µ)ˆ x(sk ),

k ∈ [sk , sk+1 ).

(8)

The corresponding observer state-error is defined by, ǫ(k) = x ˆ(sk ) − x ˆ(k),

k ∈ [sk , sk+1 ).

(9)

Simple computations show that the closed-loop event-triggered system is described as, xp (k + 1) = Φ1 (µ)(k) + Bp (µ)K(µ)ǫ(k), Φ1 (k) = (Ap (µ) + Bp (µ)K(µ))xp (k) − Bp (µ)K(µ)ep (k),

x ˆ(k + 1) = Φ2 (µ)(k) + Bp (µ)K(µ)ǫ(k),

(10)

Φ2 (k) = (Ap (µ) + Bp (µ)K(µ))ˆ x(k) + L(µ)Cp (µ)ep (k), ep (k + 1) = (Ap (µ) − L(µ)Cp (µ))ep (k).

In the sequel, we define the event-triggering condition by

ǫ(k)ǫ(k) < β x ˆT (k)ˆ x(k), k ∈ [sk+1 , sk ).

(11)

This implies that the (j + 1)-th control task is implemented whenever condition (7) is defied. The coming result provides LMI-based conditions to realize a proper β to sustain closed-loop system (6) is asymptotically stable subject to event condition (7). 4

Theorem 1: System (6) subject to condition (7) is asymptotically stable if there exist fuzzy-basis-dependent matrices 0 < X(µ), Y (µ), Z(µ), Q(µ), W (µ), M (µ), N (µ), R(µ) and scalar parameters β > 0, σ > 1 satisfying the following LMIs,        

−X(µ) 0 0 0 Π(µ) 0 • −S(µ) − βI 0 0 0 Π(µ) • • −N(µ) 0 −RT (µ) ZT (µ) • • • −(σ − 1)I RT (µ) RT (µ) • • • • −X(µ) 0 • • • • • −Q(µ) + 2I

Π(µ) = X(µ)ATp (µ) + YT (µ)BpT (µ)   −W(µ) − M(µ) ATp (µ) − ZT (µ) ≤ 0 • −W(µ) + 2I



    ≤ 0   

(12)

(13)

Moreover, the observer and feedback gain matrices are given by K(µ) = Y (µ)X −1 (µ), L(µ) = Z(µ)X −1 (µ)Cp† (µ), where Cp† is the pseudo-inverse of Cp . Proof 1: Consider the fuzzy-basis-dependent Lyapunov-Krasovskii functional (LKF), V (µ) = xTp (k)P(µ)xp (k) + x ˆT (k)Q(µ)ˆ x(k) + eTp (k)W(µ)ep (k),

(14)

where 0 < P, 0 < Q, 0 < W are weighting matrices of appropriate dimensions such that V (k) > 0. The first-forward difference of V (k) along the trajectories of system (6) can be expressed as ∆V (µ) = xTp (k + 1)P(µ)xp (k + 1) − xTp (k)P(µ)xp (k) +x ˆT (k + 1)Q(µ)ˆ x(k + 1) − x ˆT (k)Q(µ)ˆ x(k)

+ eTp (k + 1)W(µ)ep (k + 1) − eTp (k)W(µ)ep (k),

= ΦT1 (k)P(µ)Φ1 (k) + 2ΦT1 (k)P(µ)Bp Kǫ(k) − xTp (k)P(µ)xp (k)

+ ǫT (k)K T (µ)BpT (µ)P(µ)Bp (µ)K(µ)ǫ(k) − x ˆT (k)Q(µ)ˆ x(k)

(15)

+ ΦT2 (k)Q(µ)Φ2 (k) + 2ΦT2 (k)Q(µ)Bp (µ)K(µ)ǫ(k)

+ ǫT (k)K T (µ)BpT (µ)Q(µ)Bp (µ)K(µ)ǫ(k)   + eTp (k) −W(µ) + (Ap (µ) − L(µ)Cp (µ))T W(µ)(Ap (µ) − L(µ)Cp (µ)) ep (k).

By Lyapunov stability theory, a sufficient stability condition is V (µ) > 0, ∆V (µ) ≤ 0. Note that ˆ ∆V (µ) ≤ ζ T (k)Ξ(µ)ζ(k) , ζ(k) = [xTp (k) x ˆT (k) eTp (k) ǫt (k)]T and   0 −P(µ) 0 0 0 ATp (µ) + K T (µ)BpT (µ)  • −Q(µ) 0 0 0 ATp (µ) + K T (µ)BpT (µ)    T BT T (µ)LT (µ)   • • −M(µ) 0 −K C p p ˆ (16) Ξ(µ) =  T T T T   • • • I K (µ)Bp (µ) K (µ)Bp (µ)   −1   • • • • −P (µ) 0 −1 • • • • • −Q (µ) −M(µ) = −W(µ) + (Ap (µ) − L(µ)Cp (µ))T W(µ)(Ap (µ) − L(µ)Cp (µ))

(17)

ˆ We seek to establish that Ξ(µ) ≤ 0. Incorporating condition (7) into (12), resorting to the S-procedure [38] for some σ > 1 and applying the congruence transformation T (µ) = diag[X(µ), X(µ), I, X(µ), I, I], X(µ) = P −1 (µ)

5

ˇ ˆ then Ξ(µ) = T (µ)Ξ(µ)T (µ) ≤ 0 via S(µ) = X(µ)Q(µ)X(µ) can be expressed in the form (8)-(9) using the algebraic inequality −Q−1 ≤ −Q + 2I . Therefore we conclude that system (6) is asymptotically stable in the sense that limk→∞ ζ(k) = 0. Remark 1: It must be observed that Theorem 1 entails a co-design criteria since the feasible solution guarantees both an event-triggering condition and stabilizing controller. Additionally, the maximum value of β can be determined during the course of solving the LMIs (9). In turn, this implies that both the feedback controlled system and the observer error system converge asymptotically with the maximized task periods. Remark 2: Another consideration of the event condition (7) means that there will be no accumulation point in the sampling time sequence. Thus, at any sampling time sk , ||ǫ(sk )|| = 0 holds. When ||ˆ x(k)|| 6= 0, 2 2 it implies that ||ǫ(k)|| < β||ˆ x(k)|| . On the other hand, if ||ˆ x(k)|| = 0, then condition (7) is violated at sk which means a zero sampling periods appears. However, in view of system (6), ||ˆ x(k)|| = 0 leads to ||xp (k)|| = 0, ||ep (k)|| = 0. In conclusion, the asymptotic stability of system (6) is attained and the zero sampling period occurs when the desired control objective has been reached. Remark 3: Given β > 0 and specify an alternative event-condition as, ǫT (k)ǫ(k) < αˆ xT (sk )ˆ x(sk ), k ∈ [sk+1 , sk ), α = (β − 1)/2(β + 1)

(18)

Simple algebra shows that ǫT (k)ǫ(k) < −βǫT (k)ǫ(k) + (β − 1)/2ˆ xT (sk )ˆ x(sk )

< −βǫT (k)ǫ(k) + (β/2)ǫT (k)ǫ(k) + (β/2)ˆ xT (sk )ˆ x(sk )

+ (1/2)ˆ xT (sk )ˆ x(sk ) − (1/2)ˆ xT (sk )ˆ x(sk )

< −(β/2)ǫT (k)ǫ(k) + (β/2)ˆ xT (sk )ˆ x(sk )

(19)

+ (β/2)ǫT (k)ǫ(k) + (β/2)ˆ xT (sk )ˆ x(sk ) = βx ˆT (sk )ˆ x(sk ),

which is equivalent to (7). By Theorem 1, it follows that system (6) subject to condition (13) is asymptotically stable. V. S YSTEMS

WITH SELF - TRIGGERING

We now direct attention to systems with self-triggering to be specified by ǫt (k)ǫ(k) < αˆ xt (k)ˆ x(k), k ∈ [sj+1 , sj ) for some α > 0 satisfying (13). From this and (5), we manipulate to obtain a condition on the observation error as, √ x(sk )|| ⇒ ||ǫ(k)|| < − α||ˆ √ ||ˆ x(sk ) − x ˆ(k)|| ≤ α||ˆ x(sk )|| ⇒ √ 0 ≤ α||ˆ x(sk )|| − ||ˆ x(sk ) − x ˆ(k)|| ⇒ (20) √ 0 ≤ ( α − 1)||ˆ x(sk )|| − ||ˆ x(k)|| ⇒ √ x(k)||. ||ep (k)|| ≤ ( α − 1)||ˆ Observe that condition is expressed in terms of measurable quantities, which makes it appropriate for self-triggering control. The main result is summarized below, Theorem 2: Consider system (6) with s0 = 0. The k -th sampling period is defined be i h x(sk )||||Ap (µ)|| ln 1 + α||ˆ η(||ˆ x(sk ),µ||) (21) , sk+1 = sk + ln(1 + ||Ap (µ)||)

6

√ where η(||ˆ x(sk ), µ||) = ||Ap (µ)+Bp (µ)K(µ)ˆ x(sk )||+( α −1)||L(µ)Cp (µ)||||ˆ x(sk )||, α = (β −1)/2(β + 1), β > 0 satisfying LMIs (8)-(9). Then system (6) under the self-triggering control (4) with (16) is asymptotically stable. √ x(sk )|| and this means over the interval [sk+1 , sk ), Proof 2: It follows from (13) that ||ǫ(k)|| < α||ˆ √ x(sk )||. ||ǫ(sk+1 )|| = α||ˆ (22)

By (5), we have ||ǫ(sk+1 )|| − ||ǫ(sk )|| = ||ˆ x(k + 1)|| − ||ˆ x(k)||. Mathematical manipulations of (6) using (5) and (15) at k = sk ||ǫ(sk+1 )|| ≤ (1 + ||Ap(µ)||)||ǫ(sk )|| + η(||ˆ x(sk )||) √ x(sk )||. η(||ˆ x(sk ), µ||) = ||Ap (µ) + Bp (µ)K(µ)ˆ x(sk )|| + ( α − 1)||L(µ)Cp (µ)||||ˆ

Solving this difference inequality with ||ǫ(sj )|| = 0 results in i η(||ˆ x(sk ), µ||) h ||ǫ(k)|| ≤ (1 + ||Ap (µ)||)k−sk − 1 . ||Ap (µ)||

(23)

(24)

Combining (17) and (19) yields √

α||ˆ x(sk )|| = ||ǫ(sk+1 )|| ≤

η(||ˆ x(sk ), µ||) [(1 + ||Ap (µ)||)sk+1 −sk − 1] ||Ap (µ)||

A lower bound of the k -th task period is given by Tk = sk+1 − sk ≥

h ln 1 +

√

α||ˆ x(sk )||||Ap (µ)|| η(||ˆ x(sk ),µ||)

ln(1 + ||Ap (µ)||)

i

.

(25)

Using (16) and (20), we can proceed to the generate the (k + 1)-th release time. Finally, the asymptotic stability of system (6) is guaranteed in view of condition (13). Remark 4: An appealing result from Theorem2 is that the (k + 1)-th release time can be derived from the k -th input x ˆ(sk ). Recall that condition (13) is generally conservative and Tk takes on the value of the lower bound, it can be inferred that smaller task periods are generated under self-triggered fuzzy control than the corresponding event-triggered fuzzy control. VI. S IMULATION E XAMPLE Inverted pendulum is obviously a nonlinear unstable model providing that is immensity utilized for learning control methodologies using experimental or analysis works. At the SoS lab 1 , an experiment was constructed using seesaw and inverted pendulum from Quanser Consulting [44]. The cart is guided by a directed current motor through a rack and pinion instruments. The cart location is calculated by a potentiometer. A pendulum has been seated on the cart with flexible movements about a horizontal axis. With in the structure, it is certainly difficult to analyze the pendulum model. For simplification, we consider a model that given in Fig. 2. In what follows, we report on the application of dynamic triggering fuzzy control to IPS. 1

System of systems lab at the distributed control research group, KFUPM.

7

Fig. 2. (left) Inverted pendulum

(right) Lab experiment setup.

A. Model The dynamic of the pendulum is provided by: I1 θ¨1 (t) + I2 θ¨2 (t) = mgℓsin θ2 (t) + u(t) − δw(t)

I2 θ¨2 (t) = −βd (θ˙2 (t) − θ˙1 (t)) − βs (θ2 (t) − θ1 (t)) + mgℓsin θ2 (t)

(26)

where g = 9.8m/s2 is the gravitational acceleration constant and θ1 (t) : θ2 (t) : u(t) : w(t) : I1 : I2 : m: ℓ: βs : βd :

is the angle (rad) of the pendulum f rom the vertical is the angle (rad) of the rotor f rom the vertical is the control torque (N m) is the disturbance torque (N m) is the moment of inertia (kgm2 ) of the rotor is the moment of inertia (kgm2 ) of the pendulum is the length (kg) of the pendulum is the mass (m) f rom the center of the pendulum round its center of mass is spring constant is damping coef f icient

We consider that the shaft is not rigid, hence it can be modeled as a parallel combination of a linear torsional spring of spring-constant βs > 0 and a linear torsional damper of damping coefficient βd > 0. Introducing x11 (t) = θ2 (t), x12 (t) = θ˙2 (t), x21 (t) = κ2 (θ2 (t) − θ1 (t)) and x22 (t) = κ(θ˙2 (t) − θ˙1 (t)) with κ = 0.1 into (26), we arrive at x˙ 11 (t) = x12 (t) x˙ 12 (t) = I2−1 (mgℓsin x11 (t) − βs0 x21 (t) − βd0 x22 (t)) x˙ 21 (t) = x22 (t)

x˙ 22 (t) = I2−1 mgℓsin x11 (t) − It−1 βs0 x21 (t) − It−1 βd0 x22 (t) − I1−1 u(t) + I1−1 δw(t)

(27)

where It = I1 I2 (I1 + I2 )−1 , βs0 = κ2 βs , , βd0 = κ2 βd . The source of nonlinearity is the term sin x11 (t) and using argument of fuzzy logic theory, we represent this term: sin x11 (t) = µ1 (x11 (t))ℓx˙ 11 (t) + µ2 (x11 (t))ℓx˙ 11 (t)  sin x11 (t)/x11 (t) x11 (t) 6= 0 µ1 (x11 (t)) = 1 x11 (t) = 0  1 − sin x11 (t)/x11 (t) x11 (t) 6= 0 µ2 (x11 (t)) = 0 x11 (t) = 0

8

(28)

where µ1 (x11 (t)) and µ2 (x11 (t)) are the membership functions for values of x11 (t) of ”about 0” and ”about ±π ”, respectively. For purpose of numerical simulation, we select m = 1kg, a; ℓ = 1m, βs = 3 × 104 N m and βd = 300N ms and δ = 0.5 . The T–S fuzzy models of the flexible joint inverted pendulum are of the following form: Rule1 :

If x11 (t) is about 0 then

x˙ p (t) = Ap (1)xp (t) + Bp (1)up (t) yp (k) = Cp (1)xp (t) Rule2 :

If x11 (t) is about ± π then

x˙ p (t) = Ap (2)xp (t) + Bp (2)up (t) yp (t) = Cp (2)xp (t)

The foregoing defuzzified (nominally linear) continuous models in Rules 1 and 2 have been appropriately discretized using sample period of 0.15 sec to allow using model (2) with the numerical matrices:     0 0.85 0.1 0.05 0.02   0.2 −0.05 −0.04 −0.03    , Bp (1) =  0  , Ap (1) =    0.01 0.02  0  0.2 0.05 −1 0.3 0.01 −0.06 −0.1   1 0 0 0 Cp (1) = (29) M ode2

:



  0.83 0.09 0.08 0.04  0.3 −0.07 −0.09 −0.03    , Bp (2) =  Ap (2) =   0.03 0.06  0.4 0.11  0.4 0.02 −0.08 −0.4   1 0 0 0 Cp (2) =

 0 0  , 0  −1

(30)

B. Results Implementation of Theorem 2 gives the feasible solution as β = 1.495 and  −0.0854 −2.6245 −2.2153 −0.1708 K(1) =  T −0.1533 −2.1939 −0.0902 −1.9409 L (1) =  −0.1123 −2.4789 −2.3411 −0.1824 K(2) =  −0.1487 −2.3454 −0.1321 −2.1107 LT (2) =



,



,





.

(31)

In addition, α = 0.0992. Simulation of the closed-loop subsystems under self-triggering strategy is depicted in Figs. 3–4 for the states and control signal, respectively. Smooth stable behavior is noted in all of the states. The average task periods illustrated in Figs. 5–6 are 0.0866 and 0.0461 for the event- and selftriggering strategies, respectively. This result concurs with Remark 4 and therefore supports the developed theoretical analysis. C. Comparisons To further demonstrate the effectiveness of our design approach, we employ MATLAB/Simulink environment to help in the implementation of the self-triggered controller in comparison to the linear-quadratic design used by Ref. [44]. The ensuing results are depicted in Figs. 7 and 8 for the angle of rotor θ2 and the rate of the angle of rotor θ˙2 , respectively. It is easy to see that the self-triggered controller gave better results. It is admitted that the reason for the difference between the generated plots is attributed to the different simulation methods. 9

4 x1 x2 x3 x4

3

x(t)

2

1

0

−1

−2

0

2

4

6

8

10

12

14

16

18

t (sec)

Fig. 3. State trajectories of the closed-loop system.

0.03 u(t)

0.02 0.01 0

u(t)

−0.01 −0.02 −0.03 −0.04 −0.05 −0.06

0

2

4

6

8

10 t (sec)

12

14

16

18

20

Fig. 4. Control signal of the closed-loop system.

VII. C ONCLUSIONS This paper has investigated dynamic feedback triggering fuzzy control for a class of linear discrete-time T–S systems. An approach to co-design event-triggered condition and controller gain matrices has been developed yielding the admissibility of the system. A sufficient condition has been obtained and presented in terms of LMI. We have established that the inter execution times are strictly positive which is essential to a practical control. The results of this paper have been supported by a simulation example. Future research work is to can focus on other attributes including L2 stability, model-based event-triggered fuzzy control and self-triggered fuzzy control. ACKNOWLEDGMENT The authors would like to thank the AE and unanimous reviewers for their constructive comments that help in producing the final version of the manuscript. The authors would like to thank the deanship of scientific research (DSR) at KFUPM for support through distinguished professorship project IN 141003. R EFERENCES [1] M. S. Mahmoud, Computer-Operated Systems Control, Marcel Dekker Inc., New York, 1991.

10

0.25

0.2

Time (sec)

0.15

0.1

0.05

0

0

1

2

3

4 Time (sec)

5

6

7

8

Fig. 5. The task periods under event-triggered control.

Fig. 6. The task periods under self-triggered control.

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Angle of Rotor 0.35 Step Response (Present method) Step Response (Simulation of Ref. [44] )

0.3 0.25

x1 (rad/s)

0.2 0.15 0.1 0.05 0 −0.05

0

5

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15

Time (sec)

Fig. 7. Comparison of angle of rotor

Rate of Rotor Angle 0.2 Step Response (Present method) Step Response (Simulation of Ref. [44] )

0.15 0.1

x2 (rad/sec)

0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.3

0

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Time (sec)

Fig. 8. Comparison of rate of angle of rotor

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