Research Article | Open Access
Stabilization of Discrete-Time Delayed Systems in Presence of Actuator Saturation Based on Wirtinger Inequality
This paper examines the stability analysis of discrete-time control systems particularly during the event of actuator saturation and time-varying state delay. With the help of Wirtinger inequality along with Lyapunov-Krasovskii functional gain of state feedback controller is determined for stabilization of above system. The saturation nonlinearity is represented in the terms of convex hull. A new linear matrix inequality (LMI) criterion is settled with reciprocally convex combination based inequality which is dependent on delay. The proposed criterion is less conservative in concern to increase the delay bound and a controller is also simulated for real time problem of missile control system in this paper. It is also attained that projected stability criterion is less conservative compared to other outcomes. Furthermore, an optimization procedure together with LMI constraints has been proposed to maximize the attraction of domain.
Practically, every system is distressed due to common non-linearities like saturation, deadzone, backlash, etc. Among them actuator saturation is more general in industrial control systems since there is always limitation in inputs of control in the form of amplitude as well as rate saturation. Saturation nonlinearity leads to degradation of system stability and is responsible for poor performance of closed loop system. In most of the research works, sector based conditions  and convex hull approach  are selected to work on the actuator or input saturation nonlinearity. The problem of control system stability with actuator saturation has received extensive attention [3–10].
Time delay is another phenomenon which exists in dynamical systems due to communication between data processing elements in computerised control systems, e.g., processes controlled via SCADA in chemical industries , optimal communication network systems , metrology systems , power systems, and neural network control system [14, 15]. Performance deterioration and instability are other issues created by this phenomenon. An adequate amount of work has been carried out on delayed systems [[16–21] and the references therein]. There are two common and popular ways to deal with time delayed systems, i.e., dependent on delay and not dependent on delay. The information about delay is utilized in delay-dependent method which yields less conservative results for limited delay range. Presently, many researchers are devoting their attention in stability analysis of time delayed processes using various approaches (such as model transformation, free weighting matrix, free matrix based integral inequality, Park’s inequality, Jensen’s inequality, Convex combination method and Wirtinger-based integral inequality, etc.) for achieving less conservative results.
Numerous versions of Wirtinger-based integral inequality have been applied for stability of open loop system in presence of delay, which provides less conservative result as compared to well-known Jensen’s inequality [22–28]. In the same direction, the above cited new advancements are tested on delayed discrete systems [29–31]. In , an accurate integral inequality  is considered to reduce the conservatism and it is dependent on as well as on . A new summation inequality is utilized for discrete-time delay systems in , which is extension of integral inequality given in . The discrete version of Wirtinger’s inequality is derived in  and also applied for stabilization of delayed discrete systems along with interval like time-varying delay. A novel summation inequality has been projected for delayed discrete systems by considering Lyapunov-Krasovskii functional with two additional terms in difference of forward calculation in . An advanced sequence of summation inequality has been obtained in , which includes Jensen, Wirtinger and Free matrix summation based inequality ones as specific case. Moreover, advancement in previous results has been obtained by Zhang et al. in 2018  by extending the Wirtinger’s inequality. In , discrete inequalities for single summation and double summation have been proposed which are based on multiple auxiliary functions and include the Jensen discrete inequality and the discrete Wirtinger-based inequality as special cases. In , a new augmented Lyapunov-Krasovskii functional with single and double summation terms is used to derive the stability conditions for discrete-time neural networks by using Wirtinger-based inequality and decomposing the time-delay interval into two non-equidistant subintervals. Delay-dependent stability criteria are established by using Wirtinger inequality approach, and based upon the memoryless state feedback controller is designed for uncertain linear system with time-varying delay and disturbance . They have utilized all possible configurations of the delay such as its, upper bounds, lower bounds and derivative of upper bounds. A new method of delay partitioning is introduced by , in which tractable conditions for stability are deduced for discrete time- delayed systems and the obtained conditions are also dependent on the partitioning part.
Further, the problems reported in the above literature are related to stability analysis of considered system in the absence of controller design part, which is essentially required to stabilize the practical systems. For stabilization of closed loop systems, controller design is a very challenging task in industrial processes inflicted to different nonlinearities in which what is more challenging is the input saturation along with time-varying state delay. Very few results are obtained for controller design for time delayed systems . Stability analysis and control synthesis problems incorporated with both nonlinearities (i.e., saturation and delay) have appeared in [8, 21, 38, 40] using LMI technique.
Inspired by the above literature, especially , in this manuscript a controller based on feedback of state is invented to stabilize discrete systems along with input saturation and time-varying state delays through novel summation inequality in LKF. A new and less conservative stability criterion is established by using Lyapunov-Krasovskii functional based on reciprocal convex approach along with Wirtinger inequality. In this work, novelty is to synthesize a feedback controller for discrete-time systems in presence of nonlinearities like actuator saturation and time-varying state delays, and maximized domain of attraction is estimated. The anticipated criterion is simulated for stabilization of missile control system.
The manuscript is arranged as follows. In Section 2, problem is formalized and necessary lemmas are recalled. A delay-dependent approach is utilized for deriving asymptotic stability conditions in Section 3. To show the adequacy of the projected work numerical examples are demonstrated in Section 4. Finally, the paper ends with general remarks.
Notations. Throughout this paper, the following representations are used: - real matrices; - real matrices; -identity matrix of suitable dimension; - null matrix or null vector; - transpose matrix of ; - maximum eigenvalue of any given matrix ; - diagonal matrix with diagonal elements ; - symmetricity in matrices; - Euclidean norm He-.
2. Problem Formulation and Prefaces
A Discrete-time system is considered in which the effect of actuator saturation and delay is present,where and specify non-negative integers. State vector is represented by , control vector as whereas output vector is . Matrices , , , are known value matrices, is the initial condition and is time-varying delay.
The saturation function is illustrated as below for asA is time-varying delay satisfying the following equation (3) as for positive lower delay bounds and upper delay bounds , respectively.
A state feedback controller is considered as for stabilization of closed loop system (1a), (1b), and (1d) In this work the saturation nonlinearity is tackled by convex hull approach which is stated in Lemma 1.
Lemma 1 (see ). The set of convex hull is described as for matrices and so that all satisfying
where is row of the matrix and represents convex hull. In this case there exist , with where .
Taking a set make up of elements either 1 or 0 which are diagonal, such as for , For , ellipsoid is described asA set is deduced as follows: When , it follows from Lemma 1 that Next, using (5)–(7), equation (4a) and (4b) becomes as with The initial condition for closed loop system (8a) and (8b) is given as
Lemma 2 (see [29, 30]). For a given symmetric positive definite matrix , any sequence of discrete-time variable in , the following inequality holds:where , and Since the factor is difficult to handle in some practical systems with time-varying delay, the following Lemma 3 is given to make this term disappear from the inequality.
Lemma 3 (see ). For a given symmetric positive definite matrix , any sequence of discrete-time variable in , where , then the following inequality holds:where and
3. Main Results
The main outcomes of the research are as follows:
Theorem 5. For the scalars , fulfilling the condition and any time-varying delay , if there exist symmetric matrices , , , , controller gain matrix , , matrix with pertinent dimension fulfilling the following LMIs (16)–(18)where then, the closed loop system defined by (8a) and (8b) is asymptotically stable using feedback controller of gain . The predicted domain-attraction for (8a) and (8b) is represented by
Proof. Following quadratic Lyapunov-Krasovskii functional is considered wherewhere .
An augmented vector is defined aswhereFor system (8a) and (8b) the difference of above LKF will bewhereis obtained by following relationwhere whereThe summation limit in the last term of (35) can be split into two parts: first one containing the terms to and next one from to On applying Lemma 2 on first term and Lemma 3 to the last three terms of (36), we get following relation: whereFurther, if there exists matrix such that (16) holds then based on Lemma 4, the above term can be modified aswhereEmploying (30)–(39), can be deduced as whereOn applying Schur’s complement on (43), we get For symmetric matrices , we have
Remark 6. For finding the values the method discussed in [38, 42] is adopted. These values can be determined iteratively.
It may be perceived that for along with (16)-(17) are LMI conditions for asymptotic stability of closed system given by (8a) and (8b). The condition can be written as .
Equation (17) indicates that the set is contained in polyhedral set . It can be seen that is equivalent to  The relation of can be found by pre and post multiply of (50) with and for all . Finally, (17) may be obtained by using Schur’s complement of (50).
3.1. Maximization the Domain of Attraction
The domain of attraction for , set of initial state , is defined as .
Next, theorem is illustrating the procedure for finding the estimate of domain of attraction.
Theorem 7. The maximized estimate of domain of attraction may be found by the following convex optimization problem.
withsubject to (16)–(18) andhas a feasible solution for the weighting parameters , positive definite symmetric matrices , , , , , , , ,, matrix , matrix and matrix .
In this situation, state feedback controller gain matrix provides a maximized estimation of domain of attraction given by , where
Proof. If the conditions given by (52) hold true, then , , , , ,,.
From (20), we get . Thus, if we minimize the value of which is given by (51), estimate of domain of attraction is being maximized. In other words, the optimization procedure in Theorem 7 orients the solutions of (16)–(18) in order to get the domain of attraction as large as possible.
Remark 8. By using LKF to deal with delay, conservative results are obtained because in this approach complication arises in model transformation and bound of some cross terms for finding the difference of above function.
3.2. Robust Stabilization
Consider the discrete time delayed system (1a), (1b), and (1d) in presence of actuator saturation and uncertainty as follows: where is the initial condition and time varying belongs to unit simplex with .
Theorem 9. For given scalars , fulfilling the condition and any time-varying delay , if there exist symmetric matrices , , , , matrix , , matrix with pertinent dimension fulfilling the conditions (16), (17) and the following set of LMI where then, the closed loop system given by (8a) and (8b) is asymptotically stable by feedback controller gain .
Proof. The proof of the above theorem is a direct use of Schur’s complement with ensuring the convexity of LMI (18) with respect to the system matrices and .
4. Numerical Examples
The numerical example are given to demonstrate the main results.
Example 1. Consider a discrete-time delayed systems The LMI conditions (16)–(18) and (52) in Theorem 7 are found to be feasible for delay range by using LMI Toolbox , and maximized estimated domain of attraction is 0.0647.
The other unknown parameters and controller which stabilizes the above system are obtained as-In Table 1, the obtained result is compared with the existing one.
It is seen that the delay bounds in  are obtained as but in the proposed method the delay bounds are obtained as .