Event-Triggered Output-Feedback Control for Disturbed Linear Systems

In the last few decades, event-triggered control received considerable attention, because of advantages in reducing the resource utilization, such as communication load and processor. In this paper, we propose an event-triggered output-feedback controller for disturbed linear systems, in order to achieve both better resource utilization and disturbance attenuation properties at the same time. Based on our prior work on state-feedbackH∞ control for disturbed systems, we propose an approach to design an outputfeedback H∞ controller for the system whose states are not completely observable, and a sufficient condition guaranteeing the asymptotic stability and robustness of the system is given in the form of LMIs (Linear Matrix Inequalities).


Introduction
The implementation of the feedback law is typically done by the time-triggered scheme, in which sampling the state and computing and transmitting the control input are executed in an equidistant time interval called sampling period.The main problem is to determine the frequency of the execution so that the desired system performance is achieved.In the traditional way, the control task is executed at equidistant sampling time intervals, for example, [1,2].However, it is not smart enough to update the control input in a periodic way regardless of the state of the system, especially for systems in which significant changes could happen to the plant rapidly.Meanwhile, the sampling period is chosen according to the worst situation, which can lead to an insufficient utilization of system resource [3].
The event-triggered fashion is such an alternative to the time-triggered paradigm, in which the control task execution is triggered by a so-called "event-condition" usually according to the plant states.The event-triggered control can lead to a remarkable reduction of the system resource, so it was widely used in the networked systems [4,5] and state estimation [6][7][8][9].Event-triggered control can also improve the overall system performance, which should be attributed to a better use of the state information, because the control input is transmitted to the plant only when the event condition is violated, which indicates the plant states are in an unexpected condition.In the last decade, research on event-triggered control is successful in both theoretical analysis [10] and applications [11].A state-feedback approach was applied to event-triggered control in [12].The study [13] extends its work to the disturbance rejection of input-output linearizable systems with a relative degree.In [14] an event-triggered control for linear disturbed system was studied in which the disturbance is bounded by a linear function of the system state.Reference [15] relaxes the assumption and only assumes that the induced  2 norm of the disturbance is finite.Other works also drive event-triggered control into disturbed systems meeting with other problems, such as a class of stochastic systems [16], simple nonlinear components [17], and time delay [18].Some other researches focus on the event-triggered outputfeedback control for systems in which the full state information is not observable.In [19] the observer-based eventtriggered control is proposed in the continuous-time systems, although in the analysis and examples the full state information is available.In [20] the stability of an event-triggered output-feedback control system with event-triggered state observer was also studied.But the effect of disturbance was not taken into account in those researches.The observerbased output-feedback approach is studied in [21,22].In [23,24] the problem of output-based event-triggered control is considered in discrete-time framework, from an optimal control perspective in line with the classical Linear Quadratic Gaussian (LQG) setup [25].In the above papers, the authors either design the controller simply making the closed-loop system matrix Hurwitz [18,21] or design a robust  ∞ controller in a continuous feedback scheme and achieve the  2 stability of the system by adjusting the conservatism of the event condition [14,15].However, these methods will bring in very large conservatism when designing the controller by solving an optimal program.In this paper, we draw the disturbance into the system and investigate the robust  ∞ outputfeedback controller in an event-triggered paradigm and propose a codesign method, in which the event condition and the event-triggered controller can be optimized at the same time.
The outline of the paper is as follows.We review our prior work on state-feedback  ∞ control system based on the event-triggered scheme for the disturbed linear system in Section 2. Based on that, we develop an approach to design an output-feedback  ∞ controller for the system whose states are not completely observable in Section 3.And a sufficient condition guaranteeing the asymptotic stability and robustness of the system is given in the form of LMIs.By solving these LMIs, we can determine an event condition implying the longest sample period on the premise that the stability and the expected disturbance attenuation properties are satisfied.Finally, in Section 4 we illustrate our findings with a numerical example and provide conclusive remarks in Section 5.

Problem Statement and Preliminaries
2.1.Notations.We shall use the notation R  to denote the dimensional Euclidean space and R × denotes the set of all  × -dimensional real matrices.The notation | ⋅ | denotes the Euclidean norm of the matrix or vector if there is no special description.Notation  > 0 ( < 0) is used to say that the matrix  is a positive (negative) matrix.Denote by   and  −1 the transpose and the inverse of , respectively. represents the identity matrix of appropriate dimensions.

Problem Statement.
We first consider an event-triggered state-feedback system for simplicity, and the system was given by ẋ =  (, , ) , (1) in which a feedback controller needs to be designed to guarantee the corresponding closedloop system ẋ =  (,  ( + ) , ) is stable.In (3)  ∈ R  represents the measurement error.The implementation of the feedback law is typically done by the time-trigged scheme.In this paper, we apply a class of aperiodic execution schedules instead, and the main problem is to design a proper controller and find the corresponding event condition determining on which time instant the control input (  ) should be updated, while making the closedloop system stable and robust.

Prior Work.
For the sake of simplicity, we will first investigate a state-feedback robust  ∞ controller for perturbed linear systems.Firstly Lemma 1 is given as follows.
Lemma 1 (Schur complement formula [26]).Consider a symmetric matrix where Consider a perturbed linear system given by where  ∈ R  represents the system state vector, (0) =  0 is the initial state,  ∈ R  denotes the bounded exogenous disturbance, and  ∈ R  and  ∈ R  are the control input and output signals, respectively.System ( 5) is stabilized by a state-feedback controller which is only computed and transmitted on discrete-time instants   .The dynamics of the closed-loop system after one control input update  = (  ) is given by where the measurement error  is defined by Thus we can rewrite system (5): We set the event condition as which indicates that if and only if this condition is violated, the feedback loop is closed.
Then the problem is to look for a suitable control law (6) so that system (9) satisfies the demanded disturbance attenuation property.Meanwhile the parameter  in (10) which decides the sampling frequency is as large as possible.
The result of this event-triggered state-feedback control problem is given by the following theorem in our prior research [27].
Theorem 2. Consider system (5).If there exist a positive definite matrix  ∈ R × and a positive scalar  > 0, satisfying the following inequality: then we have the following: (a) System ( 5) under event condition ( 10) is asymptotically stable (without regard for the influence of the external disturbance).
Proof.Define () =   , where  is one of the positive definite solutions of (11).The time derivative of () along the solution of ( 9) is Since we have |()| 2 < |()| 2 , V() can be reduced to We can see from (11) that by the Schur complement formula.Furthermore ( 14) is equivalent to which indicates that V() < 0. Therefore system ( 9) is asymptotically stable.
Then we define and we have By Schur complement formula, ( 11) is equivalent to So we get that is, Notice that the system is asymptotically stable and let  → ∞, and we have

Design of Event-Triggered Output-Feedback Controller
In last section, we propose an LMI condition that helps us to design a robust  ∞ state-feedback controller when the states of the system are completely observable.In this section, we will investigate how to design an output-feedback controller for systems in which the states of the system are not completely observable.
The system is given by where the system state  ∈ R  is not completely observable and  ∈ R  and  ∈ R  are the controllable output and observable output of the system, respectively.The main aim of this section is to find an output-feedback controller described as ẋ =   x +   , where x is the state of the controller and   ,   ,   , and   are the parameters of the controller to be determined.We will also apply an event-triggered feedback strategy which brings in the measurement error  defined in last section, and the controller should be rewritten as Apply controller (24) to system (22), and we can get the closed-loop system According to the result of Theorem 2, the expected output-feedback controller (23) can be designed if there exists a symmetric positive definite matrix   , which makes the following inequality hold: In (27) ( Now we define which is to be determined to design the controller, and Then we get

Mathematical Problems in Engineering
We define and then (34) can be rewritten as According to Lemma 3,(36) is solvable if and only if where     and   are orthogonal complements of    and , respectively.Now we get two matrix inequalities including only one matrix variable   instead of a matrix inequality about two different variables   and , between which the relationship is nonlinear.
However, the variable   not only appears in    but also appears in     , so the first inequality in (37) is not an LMI.The following task is to transform it to an equivalent LMI.
Given   , we can define Then we can get the following lemma.and also notice the fact that and we can finally get According to Lemma 4, we can design an output-feedback controller (23) for system (22) if there exists a symmetric matrix   > 0 such that The first inequality in (45) is about variable  −1  while the second is about   , so verifying the existence of   satisfying both the inequalities in (45) is a nonconvex optimal problem.Next we will analyze how to transform this problem into an LMI.
As   is symmetrical, we can write   and  −1  in the following block form: where  and  are -dimensional symmetric submatrix.
The following lemma will indicate that the inequalities in (45) only have restraints of , , and  2 .Lemma 5.   > 0 is a symmetric matrix, and , , and  2 are defined in (46).Then Mathematical Problems in Engineering 7 holds if and only if where and  1 and  2 are orthogonal complements of  2 and  12 respectively.
Proof.First we will prove that         < 0 is equivalent to (48).Notice the definition of   ,  0 ,  0 ,  0 , , and  12 , and we have So we have Notice that the second row of   is zero vector; therefore Also notice that Then we can finally get LMI (48).

Mathematical Problems in Engineering
We can similarly get LMI (49); just consider  = [ 0  0 0 0 0 0  2 0   21 0 0 0 ] , Till now we can get , , and  2 from the result of Lemma 5. Furthermore, it can be easily proved that   can always be determined by , , and  2 if and only if  −  is reversible or equivalently According to the analyses above, we can obtain a sufficient condition for designing the output-feedback controller (23), as shown in the following theorem.Theorem 6.For a perturbed system (22),   and   were defined in (50); then a robust  ∞ output-feedback controller can be designed for system (22) if there exist symmetric matrices  and  and a matrix  2 , satisfying the following LMIs: (1) (2) (3)

Simulation
In this section, we propose a simulation example to illustrate the efficiency of our theoretical results.We consider a second-order perturbed system of ( 22), in which  = [ 5 10 1 0 ] , The initial state of the system is and the disturbance signal  is given as By solving LMIs in Theorem 6, we get the maximum value of  max = 0.042.And the corresponding optimal controller is Then we apply optimal controller (63) as the eventtriggered controller based on Theorem 6.The performance of the event-triggered controlled system and the sample time are shown in Figures 1 and 2. We can see from Figure 1 that the trajectory of the closed-loop system converges rapidly and smoothly by the robust  ∞ controller (63) and good capacity of disturbance attenuation property is also guaranteed.

Conclusions
In this paper, we address the use of event-triggered control for disturbed systems, and the event condition given to determine the feedback frequency is decided in order to achieve purpose of disturbance rejection (in the  ∞ sense).We first propose a state-feedback case, and a sufficient condition which guarantees the asymptotic stability and robustness of the system is given in the form of LMI (Linear Matrix Inequality).Based on that, we develop an approach to design an output-feedback  ∞ controller for the system whose states are not completely observable.The condition to design the controller is given in three LMIs.By solving these LMIs, we can determine an event condition implying the optimal feedback frequency on the premise that the stability and the expected disturbance attenuation properties are satisfied.
A codesign method for determining the event-triggered controller and event condition simultaneously is proposed in this paper.Compared with the existing research on the eventtriggered control, in which only the event conditions are to be determined while the controllers are fixed in the context of continuous-time framework, the codesign principal is less conservable and makes it possible to optimize both the controller and the event condition.The Linear Matrix Inequalities obtained in this paper provide a flexible parameter, which can be used to improve the conservatism of the method.A dynamical  ∞ controller is proposed to alleviate the effect on the system outputs caused by the external disturbance and obtain the expected disturbance attenuation property.From the simulation example, we can see that the trajectory of the closed-loop system converges rapidly and smoothly by the dynamical  ∞ controller, and good capacity of disturbance attenuation property is also guaranteed.
Following this paper, some cone optimization methods could be applied to improve the optimization program of the matrix inequality for further reduction of the conservatism of the stability condition.And other absolute-error based event condition can be studied instead of the relative-error based event condition, to improve the triggering efficiency of the event-triggered control system.Moreover, we are also considering event-triggered control for networked systems, in which time delay and package drop may occur and high efficiency communication methods like event-triggered communication are more necessary.
(27),   , and   are unknowns based on the parameters of the controllers   ,   ,   , and   , and they all appear in the inequality in nonlinear forms.In the rest of this section, we will investigate how to transform(27)into several LMIs that are all linear and algorithmically solvable.