1. Introduction
Time-delay frequently arises in various practical control systems, such as communication systems, power systems, and chemical systems. The existence of delays can lead to instability and degrades the performances of systems. Thus, the time-delayed systems have been an active area of research for many years, and many effective methods have been developed, such as the delay-partitioning approach [
1], the free weighting matrices technique [
2,
3], the Wirtinger inequality approach [
4], and the generalized free-matrix-based integral inequality [
5]. The input-output (IO) approach is also an appropriate approach for time-delayed systems, its main idea being to transform the original systems into two interconnection subsystems. Based on the IO approach and the scaled small gain (SSG) theorem, a new time-varying delay approximation model by employing a three-term approximation was first proposed in [
6].
It is important to note that discrete systems are suitable for computer realization, and continuous systems are convenient for theoretical analysis. The unique framework that can combine some related results in both continuous and discrete domains is the delta operator approach [
7,
8]. Recently, a great deal of attention has been paid to the analysis and synthesis of delta operator systems. For example, the problem of asymptotic stability and stabilization of delta operator systems with uncertainties and time-varying delays was studied in [
9]. In Reference [
10], the authors used the delta operator approach to investigates the robust
filtering design for T-Sfuzzy systems with uncertainties and time-varying delay. In Reference [
11], the robust stability of delta operator systems was discussed via the three-term approximation approach.
On the other side, due to the fact that physical inputs are usually limited, actuator saturation is very ubiquitous in all practical control systems. The controller design without taking into account the saturation effects can lead to performance degradation, causes the instability of the closed-loop system, and could even involve security problems, such as crashes of the aircraft JAS Gripen and the meltdown of the Chernobyl power plant. Therefore, people are paying more and more attention to control systems with actuator saturation [
12,
13,
14]. A common technique to avoid the effect of saturation is the anti-windup compensator. A large amount of research results about the anti-windup compensator exists in the literature. In [
15], the authors addressed the problem of anti-windup design for sampled data with time delay and input saturation. The input delay approach was used to develop an anti-windup compensator for Active Queue Management (AQM) in Transmission Control Protocol/Internet Protocol (TCP/IP) networks in [
16]. Some efforts were devoted to the problem of delta operator systems under input limitations; see for example [
17,
18]. However, there have been no result on the delta operator systems with input limitations via the anti-windup design, so this motivates the current study.
The aim of this paper is to synthesize an anti-windup compensator for delta operator systems with saturating control and time-varying delay. By incorporating the IO approach, Wirtinger’s integral inequality, and the Lyapunov–Krasovskii functional, sufficient conditions are derived by means of linear matrix inequalities (LMIs), ensuring the asymptotic stability of the system for any initial condition inside an estimate of the domain of attraction. An optimization procedure is proposed to enlarge the region of initial conditions that ensure the stability of the system. Further, numerical simulation examples are presented to illustrate the significance of the proposed results.
Notation 1. We use the following notation to illustrate the letters and symbols in the relevant analysis. We let , where T is the sampling period of a system. represents the series connection of mappings and . denotes the maximal eigenvalue of a matrix P. denotes a block-diagonal matrix. denotes the norm of series , and represents the -induced norm of a transfer function matrix or a general operator.
2. Problem Statement and Preliminaries
As in [
7,
19], the delta operator is defined by:
where
when
,
T is a sampling period,
t is the continuous time, and
k is the time step with
. The delta operator system with time-varying delay is described as:
where
,
,
are the state vector, control vector, and measured output, respectively, with
and
constant real matrices,
is the time-varying delay that ensures
, where
and
are known constants and
T is the sampling period, and
is a real-valued initial function.
Remark 1. Note that for continuous systems , the time delay is assumed to be a bounded function. However, when discretizing this function, it becomes dependent on the sampling period, and an additional assumption is added. That is, the time delay is a multiple of T, mathematically . Note that this assumption is applied only on the discrete time domain.
To control System (2), let a dynamic stabilizing controller in the following form,where is the controller state, is the controller input, and is the controller output. , and are known matrices of appropriate dimensions. Due to the existence of saturation, the control input of the system can be expressed as:where In order to overcome the undesirable effects caused by saturation, we add the anti-windup signal such that,where is a decentralized dead-zone nonlinearity, such that .Define , then we obtain the closed-loop system as follows:where: Consider a matrix , and define the following polyhedral set:where denote the row of matrices and H, respectively. Before ending this section, we give the following preliminary results, which will be used subsequently.
Definition 1. [20] The delta operator system (2) is asymptotically stable, if the following condition holds:where is a Lyapunov functional in the delta domain. Lemma 1. [21] If , then the following inequality holds for any positive diagonal matrix : Lemma 2. [22] For a given symmetric positive definite matrix , scalars , and vector function , such that the following sum is well defined, then:where .
Lemma 3. [9] Consider an interconnected system with two subsystems and :where the forward system is known and the feedback system is unknown and time-varying, and assume that is internally stable. The closed-loop system composed by and is robustly asymptotically stable for all , if there exist matrices with: non-singular,such that the following SSGcondition holds: Finally, for a scalar , the ellipsoid is given as follows, Furthermore, we will derive an estimate of the domain of initial conditions as, Remark 2. In computer science and cybernetics, the term “discrete-time delta operator” δ is generally taken to mean a difference operator,the Euler approximation of the usual derivative with a discrete sample time T. The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling. In fact, for fast sampling, the standard forward-shift representation of a discrete-time system becomes extremely sensitive to round-off errors. In addition, it has been demonstrated [20,23,24] that the implementation of delta controllers is significantly less sensitive to round-off errors, round-off noise, and limit cycles than the implementation of the more standard z-domain controllers at a high sampling rate. Furthermore, the stability of delta operator systems by the Lyapunov approach is based on Definition 1 and has been studied by many researchers; see for instance [20,24] for more details on the conditions that must respect the Lyapunov function. 3. Main Results
In this section, we will rewrite the system in (
6) as a combination of two subsystems
and
. Therefore, the SSG theorem will be applied to drive the condition for the asymptotic stability of (
6). We emphasize here that Lemma 3 serves as a tool for analyzing the stability of any system that can be transformed into a feedback interconnection formulation
and
, in which we are interested in studying only the stability of the forward subsystem
since the feedback subsystem
can be easily normalized, consequently verifying the SSG theorem. It was shown in the literature [
5,
6] that the use of this transformation and the SSG theorem leads to less restrictive results than other approaches. Here, we used this formulation with the anti-windup strategy, which represents the main novelty in this paper.
4. Anti-Windup Optimization
Since (
14) is nonlinear, it cannot be solved directly by applying the useful numerical tools (such as the LMI-toolbox of MATLAB). It follows that it is difficult to come up with a solution such that the domain of initial conditions is the largest possible. Thus, to overcome such a problem, we propose an optimization problem under which the nonlinear conditions are transformed to matrix inequalities, which can be easily solved. We let
,
,
. With this aim, consider the following auxiliary LMIs where
, and
:
The condition (
14) is verified if the following LMI is satisfied,
Then, as in [
28,
29], we formulate a feasibility problem as follows,
min tr
Based on the above conditions, the proposed controller can be designed for given and by utilizing the following cone complementarity algorithm.
- Step 1
For given
, and
, fix a sufficiently large
such that the constrained minimization (
27) is feasible. Then, set
. Fix a sufficiently small
, and set
.
- Step 2
Solve the following LMI minimization problem:
- Step 3
Substitute the new matrix variables into (
27). If the result is feasible, then set
and the new solution as
, and repeat Step 2; otherwise,
is the required estimate: Stop.
Remark 3. The initial value of in Step 1 of the above algorithm can always be found by solving (14) when the conditions (12) and (13) are feasible. The condition (27) includes the conditions (12) and (13), and the nonlinear condition (14) is transformed into linear matrix inequalities. Furthermore, the incremental step is generally chosen sufficiently small to guarantee the best accuracy. Note that inequalities (12) and (13) can be easily solved by using the LMI toolbox of MATLAB or Scilab. Remark 4. The determination of the basin of attraction for delta operator systems under input limitations has received little attention; see [17,18,30,31]. All the cited references used the polytopic description of the saturation nonlinearity, based on the convex hull representation. In [19], nested actuator saturation was studied based on the convex hull representation of the saturation nonlinearity. However, until now, the anti-windup approach was not considered in the literature for the delta operator system. Historically, the “windup” term is linked to the effects of actuator limitations in control input that contains the integral action with extra charge, which produces side effects on the transient response of the system, like slow convergence to the equilibrium point and oscillation. To overcome the undesirable effects caused by the saturation, we can add another control-loop, that we call anti-windup compensation for the closed-loop system. In this paper, this approach is used to guarantee both asymptotic stability of the delta operator and an estimate of the domain of attraction. Remark 5. It should be mentioned that all the existing delta operator systems with control saturation (see for example [17,18]) do not investigate the anti-windup approach. However, in this paper, we utilize this approach to guarantee the system stability in an estimate of the domain of attraction for the delta operator system with input limitations. Therefore, the proposed strategy of control is a new approach for controlling delta operator systems and gives insight into the design of controllers for such systems. 5. Numerical Examples
In this section, we present two numerical examples to validate the correctness and superiority of the designed control scheme.
Example 1. In order to evaluate our results and due to the fact that there are no results in the literature that deal with the ant-windup for delta operator systems, in this example, we will compare our approach with a literature result studying the stability of delta operator systems by using the polytopic representation of the saturation function. We consider a delta operator system with the following matrices [17], The dynamic controller has the following parameters, By applying the proposed algorithm with and , the estimate of the region of attraction is obtained as and , while the result of [17] is . It can be found that the obtained is larger than that obtained in [17], which shows the effectiveness of the presented method.The state response of the closed-loop system with initial conditions is depicted in Figure 1. Obviously, the curves of the state response converge to the equilibrium point. Example 2. Given a delta operator system with: The dynamic controller is given as: Applying Theorem 1 with and , we obtain the maximum upper bound and the corresponding anti-windup gain matrix according to the value of , as listed in the following Table 1. Figure 2 and
Figure 3 depict the simulation result for state trajectories of the closed-loop system with the anti-windup compensator (for the anti-windup gain corresponding to
) and without the anti-windup compensator, respectively, with the initial condition
. It is seen clearly that the the time response was greatly improved by the usage of the proposed anti-windup compensator.
Figure 4 and
Figure 5 show the control signal with the anti-windup compensator and without the anti-windup compensator. It can be observed that the time in which the control signal remained saturated was smaller with the anti-windup.