Research Article  Open Access
Wangyan Li, Guoliang Wei, Hamid Reza Karimi, Xiaohui Liu, "Nonfragile GainScheduled Control for DiscreteTime Stochastic Systems with Randomly Occurring Sensor Saturations", Abstract and Applied Analysis, vol. 2013, Article ID 629621, 10 pages, 2013. https://doi.org/10.1155/2013/629621
Nonfragile GainScheduled Control for DiscreteTime Stochastic Systems with Randomly Occurring Sensor Saturations
Abstract
This paper is devoted to tackling the control problem for a class of discretetime stochastic systems with randomly occurring sensor saturations. The considered sensor saturation phenomenon is assumed to occur in a random way based on the timevarying Bernoulli distribution with measurable probability in real time. The aim of the paper is to design a nonfragile gainscheduled controller with probabilitydependent gains which can be achieved by solving a convex optimization problem via semidefinite programming method. Subsequently, a new kind of probabilitydependent Lyapunov functional is proposed in order to derive the controller with less conservatism. Finally, an illustrative example will demonstrate the effectiveness of our designed procedures.
1. Introduction
In reality, virtually almost all dynamic systems are subject to stochastic perturbation, and stochastic model has been successfully established to describe many practical systems, such as economic systems, process control systems, networked control systems (NCSs), and sensor network. For several decades, the study of stabilization, control, and filtering problem has drawn many researchers’ attention; some results can be found in [1–16]. On the other hand, time delays also serve as one of the main sources for poor performance and instability. Consequently, the stochastic control issue for timedelay systems has also been intensively investigated; see, for example, [2, 4, 7, 8, 10, 11, 13, 15].
The randomly occurring phenomenon is a newly emerged research topic which has drawn many researchers' attention; see, for example, [1–3, 5, 6, 8, 9, 12–16]. It refers to these phenomena appearing in a random way based on a certain kind of probabilistic law including randomly occurring nonlinearities (RONs), missing measurements, randomly occurring actuator faults, randomly varying sensor delays (RVSDs), and randomly occurring sensor saturations (ROSSs), and so on. For more details about randomly occurring phenomena, the reader is referred to [9]. If not handled appropriately, these phenomena could cause a reduction of performance and/or launch a threat to the safety and reliability of the plant. Therefore, it is not surprising that various filtering and control techniques have been developed to deal with such randomly occurring phenomena, in addition to control [16]/filtering [12] and state estimation [1] methods. In [2], a robust sliding mode control has been designed for system with mixed timedelays, randomly occurring uncertainties, and RONs; while gainconstrained recursive filter approach has been used in [5] for system with probabilistic sensor delays, the extended Kalman filtering and quantized recursive filtering problem for system with missing measurements have been studied in [3, 6], respectively. Therefore, in this paper, the ROSS (one of the important randomly occurring phenomena) is studied by exploiting gainscheduling method, which is another motivation of this paper.
Sensor saturation phenomenon is very common in practical engineering. It means that sensors cannot provide signals of unlimited amplitude due mainly to the physical or technological constraints. In another aspect, because of random occurrences of networked induced phenomena in networked control systems (NCSs), such as random sensor failures leading to intermittent saturation and sensor aging resulting in changeable saturation level, sensor saturation may occur in a random way. We consider this phenomenon as randomly occurring sensor saturation, which has received increasing attention, for instance, [1, 12]. Reference [1] discussed the state estimation problem for discretetime complex networks with ROSSs and RVSDs, while [12] turned to design an filter for system with ROSSs and missing measurements. However, to the best of authors' knowledge, rare published literature has dealt with ROSSs; therefore, this paper tries to flourish the research on this phenomenon by designing a nonfragile gainscheduled controller.
Over the past decades, gainscheduling method is one of the most popular methods of controller designing and has been extensively studied from theoretical and practical viewpoints; see, for example, [8, 14, 15, 17–19]. The gainscheduling method is to design controller gains as functions of the scheduling parameters, which can update the controller with a set of tuning parameters in order to optimize the closedloop performance when outside environment changes (e.g., the occurrences of a variety of randomly occurring phenomena). It should be noted that the designed gainscheduling controller has not only the constant part but also timevarying part which can be scheduled online according to the corresponding timevarying parameters; see [8, 14, 15]. Therefore, it will naturally lead to less conservatism than the conventional ones with fixed gains only.
On the other hand, it is well known that in order to get better performance of the system, an accuracy controller is needed to resist the impact by the uncertainties occurring in the course of the implementation of a designed controller. Such uncertainties can be due to the existence of parameter drift, roundoff errors in numerical computation during controller implementation, and the safetuning margins provided for engineering application. In these cases, the nonfragile controller is a good choice, as it can tolerate some level of controller parameter variations; see [7, 20–22]. However, the controller with uncertainties and outside environment changes often occur simultaneously; unfortunately, few papers have tackled this phenomenon, and therefore, we proposed a nonfragile gainscheduled controller in this paper to fill the gap by making a few first attempts to deal with this problem.
The main contributions of this paper are summarized as follows: a new nonfragile gainscheduled control problem is addressed for a class of discretetime nonlinear stochastic systems with randomly occurring phenomenon; a sequence of stochastic variables satisfying Bernoulli distribution is introduced to describe the timevarying features of the ROSSs; a timevarying Lyapunov functional dependent on the saturation probability is proposed and applied to improve the performance of system; the parameters of the nonfragile gainscheduled controller can be adjusted online according to the saturating probability estimated through statistical tests.
Notation. In this paper, , , and denote, respectively, the dimensional Euclidean space, the set of all real matrices, and the set of all positive integers. refers to the Euclidean norm in . denotes the identity matrix of compatible dimension. The notation (resp., ), where and are symmetric matrices, means that is positive semidefinite (resp., positive definite). For a matrix , and represent its transpose and inverse, respectively. The shorthand denotes a block diagonal matrix with diagonal blocks being the matrices . In symmetric block matrices, the symbol is used as an ellipsis for terms induced by symmetry. Matrices, if they are not explicitly stated, are assumed to have compatible dimensions. In addition, and will, respectively, mean expectation of and probability of .
2. Problem Formulation
Consider the following discretetime nonlinear stochastic systems: where is the state, is a constant delay and , is a onedimensional Gaussian white noise sequence satisfying and , and is the initial state of the system. , , , , , , and are constant real matrices of appropriate dimensions and is of full column. The nonlinear function with is assumed as nonlinear disturbance and satisfies the following sectorbounded condition: where belongs to the sector , and are given constant real matrices.
For the technique convenience, the nonlinear function can be decomposed into a linear part and a nonlinear part as then, from (3), we have where .
The measurement output with sensor saturation is described as where is a constant real matrix of appropriate dimensions and . Here, the notation of “sign” means the signum function, and we use the notation to denote saturation functions. Note that, without loss of generality, the saturation level is taken as unity.
According to the definition of the saturation function, we can get that the nonlinear function satisfies and , where is a positive scalar satisfying , so the nonlinear function satisfies , while and satisfies .
The variable is a random white sequence characterizing the probabilistic sensor saturation, which obeys the following timevarying Bernoulli distribution: where is a timevarying positive scalar sequence and belongs to with and being the lower and upper bounds of , respectively. Throughout the paper, for simplicity, we assume that , and are uncorrelated.
Remark 1. In many practical systems, especially in NCSs, the measurement output is often subject to ROSSs, and the Bernoulli distribution model has been proven to be a very flexible and effective way to model randomly occurring phenomenon; see, for example, [1–3, 5, 6, 8, 13–15]. Furthermore, in practical engineering, the occurring probability of sensor saturation phenomenon usually changes with time. Therefore, in this paper, the occurrence of sensor saturation is described by a random variable sequence satisfying a timevarying instead of timeinvariant Bernoulli distribution model, which will reduce the conservatism when used to deal with the systems with timevarying ROSSs.
In this paper, we are interested in designing the following nonfragile gainscheduled static output feedback controller: where is the controller gain sequence to be designed and assumed as the following structure: for every time step , is the timevarying parameter of the controller gain, and and are the constant parameters of the controller gain to be designed, while is an unknown matrix of appropriate dimensions and represents the uncertainty in the controller, which is assumed to be of the form where and are known constant matrices with the structured information of the uncertainty, and is an unknown, real, and timevarying matrix with Lebesguemeasurable elements satisfying
Remark 2. Instead of using the information of system states, static output feedback control directly makes use of system outputs to design controllers, which has proven to be much simpler and easier to implement and has been extensively used in various kinds of engineering fields; for more details, we recommend some papers such as [23–27].
Remark 3. Owing to the pervasive existence of the uncertainties during controller implementation, an accuracy controller is needed to resist such an impact by the uncertainties, and the nonfragile controller has been proven to be an effective one; see, for example, [7, 20–22]. In another aspect, ROSSs are ubiquitous during the process of measurement, especially in NCSs, and gainscheduling method has been successfully utilized to tackle with randomly occurring phenomenon in [8, 14, 15]. Therefore, in this paper, we design a nonfragile gainscheduled static output feedback controller for nonlinear stochastic systems to deal with uncertainties and ROSSs simultaneously.
From the aforementioned, the closedloop system with the nonfragile gainscheduled controller is
Before formulating the problem to be investigated, we first introduce the following stability concepts.
Definition 4. The closedloop system (12) is said to be exponentially meansquare stable if, with , there exist constant and such that
In this paper, our purpose is to design a probabilitydependent nonfragile gainscheduled controller of the form (8) for the system (1) by exploiting a probabilitydependent Lyapunov functional and LMI method such that, for all admissible sensor saturations and exogenous stochastic noise, the closedloop system (12) is exponentially meansquare stable.
3. Main Results
The following lemmas will be used in the proofs of our main results in this paper.
Lemma 5 ((Schur complement) [28]). Given constant matrices , , , where and , then if and only if
Lemma 6 (see [13]). Let the matrix be of fullcolumn rank. There always exist two orthogonal matrices and such that and . If matrix has the following structure: , where and , then there exists a nonsingular matrix such that .
Lemma 7 ((procedure) [28]). For given matrices , , and with appropriate dimensions, holds for all satisfying if and only if there exists such that
For convenience of presentation, we first consider the desired controller without uncertainty (i.e., ), and the result will be shown in Theorem 8. Then, we design the nonfragile gainscheduled controller in Theorem 10 based on the conclusion in Theorem 8.
Theorem 8. Consider the discretetime nonlinear stochastic systems with ROSSs (12). If there exist positivedefinite matrices and , slack matrix , and nonsingular matrices and , such that the following LMIs hold: where in this case, the constant gains of the desired controller can be obtained as follows: and the closedsystem (12) is then exponentially meansquare stable for all .
Proof. Define the Lyapunov functional
noting that , , and , we can get
Denote the following matrix variables:
then, it is obvious that
where
If , we can conclude the following matrix by Schur complement: where
by preforming the congruence transformation to (25), we havewhere
From inequality
we can get By using Lemma 6, we have , and denoting that , then (30) can be written as (17); furthermore, we can know from Lemma 5 that and, subsequently,
where is the minimum eigenvalue of . Finally, we can confirm from Lemma 1 in [13] that the closedloop system is exponentially meansquare stable; then the proof of this theorem is complete.
Remark 9. The ROSSs have been studied in [1, 12] by constructing a concise and effective timeinvariant Bernoulli distribution model; however, in many practical systems, ROSSs sometimes appear with timevarying probability. Therefore, in this case, we considered ROSSs satisfying timevarying Bernoulli distribution which is more reasonable in reality. On the other hand, unlike other timevarying parameters discussed in gainscheduling technique or parameterdependent Lyapunov functional; see, for example, [17–19], the parameter considered in this paper is the timevarying occurrence probability of ROSSs, based on which a new kind of controller is designed and a novel probabilitydependent Lyapunov functional is proposed to reduce the potential conservatism.
Next, we are in a position to consider the nonfragile gainscheduled controller design for system (12) based on what we got in Theorem 8.
Theorem 10. Consider the discretetime nonlinear stochastic systems with ROSSs (12) and the nonfragile gainscheduled controller (8). If there exist positivedefinite matrices and , slack matrix , and nonsingular matrices and , scalars , , LMIs (17), equations (18), and the following LMIs hold: where in this case, the constant gains of the desired controller can be obtained as follows: and the closedsystem (12) is then exponentially meansquare stable for all .
Proof. In order to get the nonfragile gainscheduled controller, we replace the with ; then, can be written as , . Noting that , we can rewrite (17) as From Lemma 7, we know that a necessary and sufficient condition guaranteeing (35) is that there exist scalars , such that by using the knowledge of Schur complement, we can find that (36) is equivalent to (32). Now, the proof is complete.
Remark 11. In Theorem 10, a nonfragile gainscheduled controller has been designed based on a set of LMIs. However, the LMIs are actually infinite owing to the timevarying parameter . In this case, the desired controller cannot be obtained directly due to the infinite number of LMIs. To handle such a problem, in the next theorem, we have to convert this problem to a computationally accessible one by assigning a specific form to . First of all, let us set .
Theorem 12. Consider the discretetime nonlinear stochastic system with ROSSs (12). If there exist positivedefinite matrices , and , slack matrix and nonsingular matrices and , such that the following LMIs hold: where in this case, the constant gains of the desired controller can be obtained as follows: and the closedsystem (12) is then exponentially meansquare stable for all .
Proof. Firstly, set therefore, we have with and . Similarly, let and we have with , . From the pervious transformation, we can easily get On the other hand, it is easy to find that From (40)–(45), we can have that (32) in Theorem 10 is true; then the proof is now complete.
Remark 13. By using the methods proposed in the proof of Theorem 10, we choose 4 variables; then, it is easy to calculate the number of LMIs as depending on the upper and lower bound of .
4. An Illustrative Example
In this section, the nonfragile gainscheduled controller is designed for the discretetime nonlinear stochastic systems with ROSSs.
The system parameters are given as follows:
Set the timevarying Bernoulli distribution sequences as , and the sector nonlinear function is taken by which satisfies (3). Also, select the initial state .
According to Theorem 12, the constant controller parameters , can be obtained as follows:
Then, according to the measured timevarying probability parameters , the gainscheduled controller gain and parameterdependent Lyapunov matrix can be calculated at every time step as in Table 1.

Figure 1 gives the response curves of state of uncontrolled systems. Figure 2 depicts the simulation results of state of the controlled systems. The simulation results have illustrated our theoretical analysis.
5. Conclusions
In this paper, the nonfragile gainscheduled control problem for a class of discrete stochastic systems with ROSSs is tackled, and the sensor saturation phenomenon is assumed to occur in a random way based on the timevarying Bernoulli distribution with measurable probability in real time. By employing probabilitydependent Lyapunov functional, we design a nonfragile gainscheduled controller with the gain including both constant and timevarying parameters such that, for all admissible sensor saturations, timedelays and noise disturbances, the closedloop system is still exponentially meansquare stable. Furthermore, we can extend the main results to more complex and realistic systems, for instance, complex networks and systems with several kinds of randomly occurring phenomena simultaneously. Meanwhile, we can also consider corresponding control/filtering problems for timevarying systems with timevarying ROSSs, such as robust sliding mode control, quantized recursive filtering, or extended Kalman filtering. The related references can be found; see, for example, [2, 3, 6].
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 61074016, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, the Program for New Century Excellent Talents in University under Grant NCET111051, the Leverhulme Trust of the UK, the Alexander von Humboldt Foundation of Germany, and the Innovation Fund Project for Graduate Student of Shanghai under Grant JWCXSL1202.
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Copyright © 2013 Wangyan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.