Stability and Time Delay Tolerance Analysis Approach for Networked Control Systems

Abstract

Networked control system is a research area where the theory is behind practice. Closing the feedback loop through shared network induces time delay and some of the data could be lost. So the network induced time delay and data loss are inevitable in networked control Systems. The time delay may degrade the performance of control systems or even worse lead to system instability. Once the structure of a networked control system is confirmed, it is essential to identify the maximum time delay allowed for maintaining the system stability which, in turn, is also associated with the process of controller design. Some studies reported methods for estimating the maximum time delay allowed for maintaining system stability; however, most of the reported methods are normally overcomplicated for practical applications. A method based on the finite difference approximation is proposed in this paper for estimating the maximum time delay tolerance, which has a simple structure and is easy to apply.

Ashraf F. Khalil 1 and Jihong Wang 2

Academic Editor:Yun-Bo Zhao

1, Electrical and Electronic Engineering Department, University of Benghazi, Benghazi, Libya
2, School of Engineering, University of Warwick, Coventry CV4 7AL, UK

Received 9 August 2014; Revised 27 October 2014; Accepted 29 October 2014; 29 June 2015

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.

1. Introduction

The key feature of networked control systems (NCSs) is that the information is exchanged through a network among control system components. So the network induced time delay is inevitable in NCSs. The time delay, either constant (up to jitter) or random, may degrade the performance of control systems and even destabilize the systems. NCSs can be defined as a control system where the control loop is closed through a real-time communication network [1]. The term networked control systems first appeared in Walsh's article in 1999 [2]. A typical organization of an NCS is shown in Figure 1. The reference input, plant output, and control input are exchanged through a real-time communication network. The main advantages of NCSs are modularity, simplified wiring, low cost, reduced weight, decentralization of control, integrated diagnosis, simple installation, quick and easy maintenance [3], and flexible expandability (easy to add/remove sensors, actuators, or controllers with low cost). NCSs are able to easily fuse global information to make intelligent decisions over large physical spaces which is important for many engineering systems such as the power system.

Figure 1: A typical networked control system.

[figure omitted; refer to PDF]

As the control loop is closed through a communication network the time delay and data dropout are unavoidable. Therefore networked control system can be regarded as a special case time delay system and many authors applied the time delay theorems to study NCSs [4]. Time delay, no doubt, increases complexity in analysis and design of NCSs. Conventional control theories built on a number of standing assumptions including synchronized control and nondelayed sensing and actuation must be reevaluated before they can be applied for NCSs [5].

The main goal of the most recent work is to reduce the conservativeness of the maximum time delay by using Lyapunov-Krasovskii functional with improved algorithms for solving the linear matrix inequalities (LMIs) set but with the expense of increasing complexity and computation time. Analytical and graphical methods have been studied in the literature (see, e.g., [6]). The stability criteria for NCSs based on Lyapunov-Krasovskii functional approach have been reported in [7-9]. In [7], a Lyapunov-Krasovskii function is used to derive a set of LMIs and the stability problem is generalized to a feasibility problem for the LMIs set. In many of the previously reported works, the controller is designed in the absence of the time delay. In [10], an improved Lyapunov-Krasovskii function is used with triple integral terms. The LMI methods require the closed-loop system to be Hurwitz [8, 11, 12]. In [13], a modified cone complementary linearization algorithm based on the Lyapunov-Krasovskii approach is implemented. And the method reported in [14] is claimed to be less conservative and the computational complexity is reduced.

The authors in [15] derived an LMI-based method in the frequency domain, and then the LMI is transformed onto an equivalent nonfrequency domain LMI by applying Kalman-Yakubovich-Popov lemma. It has been reported in [16] that the ordinary Lyapunov stability analysis is linked by a suggested variable to state vectors through convolution and the stability analysis is simplified to only require solving a nonlinear algebraic matrix equation.

In [11], the hybrid system technique is used to derive a stability region. An upper bound is derived for time delay in an inequality form and the results are rather conservative. The hybrid system stability analysis technique has also been used in [17]. A simple analytical relation is derived between the sampling period, the time delay, and the controller gains. The same approach is used in [18] with more conservative stability region results. The model-based approach for deriving necessary and sufficient conditions for stability is presented in [19]. The stability criteria are derived in terms of the update time and the parameters of the model. The model-based approach is then extended to multiunits NCS in [20]. The optimal stochastic control was studied in [21] with a discrete-time system model where the random time delays are modeled using Markov chains and the controller uses the knowledge of the past state time delays by time stamping.

Most of the previously developed approaches require excessive load of computations, and also, for higher order systems, the load of computations will increase dramatically. In practice, engineers may find it difficult to apply those available methods in control system design because of the complexity of the methods and lack of guideline in linking between the design parameters and the system performance. Almost all the design procedures highly depend on the postdesign simulation to determine the design parameters. So there is a demand for a simple design approach with clear guidance for practical applications. In this paper, a new stability analysis and control design method is proposed, in which the design approach is simple and a clear design procedure is given.

The paper starts from the mathematical model of NCS and then the proposed method for estimating the maximum allowable delay bound is briefly described. A few examples are illustrated and the results are compared with those previously published in the literature. The cart and inverted pendulum problem is used to study the effect of the parameters on the maximum allowable delay bound.

2. Mathematical Analysis

Although the issues involved with time delays in control systems have been studied for a long time, it is difficult to find a method simple enough to be accepted by control system design engineers. It is found that the most previously reported methods rely on LMI techniques and they are generally too complicated for practical engineers to use and also involve heavy load of numerical calculations and computing time. The paper proposes a new method which has a simple structure and is used for estimating the maximum time delay allowed while the system stability can still be maintained. In most control systems the sampling time is preferred to be small [22]. The maximum allowable delay bound (MADB) can be defined as the maximum sampling period that guarantees the stability even with poor system performance. A continuous time-invariant linear system is shown in Figure 2 and given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the system state vector, [figure omitted; refer to PDF] is the system control input, [figure omitted; refer to PDF] is the system output, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are constant matrices with appropriate sizes.

Figure 2: A networked control system with the time delay both from the sensor to the controller and from the controller to the actuator.

[figure omitted; refer to PDF]

Suppose that the control signals are connected to the control plant through a kind of network, so the time delay is inevitable to be involved in the feedback loop. The state feedback is therefore can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the time delay between the sensor and the controller, [figure omitted; refer to PDF] is the time delay in the controller, and [figure omitted; refer to PDF] is the time delay from the controller to the actuator. [figure omitted; refer to PDF] represents the feedback control gains with appropriate size. From 2 the networked control system can be modeled where the time delay is lumped between the sensor and the controller as shown in Figure 3.

Figure 3: A simplified model of the networked control system.

[figure omitted; refer to PDF]

The time delay may be constant, variable, or even random. In NCSs, the time delay is composed of the time delay from sensors to controllers, time delay in the controller, and controllers to actuators time delay which is given by [figure omitted; refer to PDF] For a general formulation the packet dropouts can be incorporated in 3 as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the number of dropouts and [figure omitted; refer to PDF] the sampling period. And by 4 the data dropouts can be considered as a special case of time delay [23, 24]. It is supposed that the following hypotheses hold.

Hypothesis 1 (H.1).

(i) The sensors are clock driven. (ii) The controllers and the actuators are event driven. (iii) The data are transmitted as a single packet. (iv) The old packets are discarded. (v) All the states are available for measurements and hence for transmission.

Hypothesis 2 (H.2).

The time delay [figure omitted; refer to PDF] is small to be less than one unit of its measurement.

Definition 1 (D.1).

For a function [figure omitted; refer to PDF] , the [figure omitted; refer to PDF] th order reminder for its Taylor's series expansion is defined by [figure omitted; refer to PDF] Applying the state feedback proposed in 2 to the system 1, we have [figure omitted; refer to PDF] From 6, the following can be derived: [figure omitted; refer to PDF]

Theorem 2.

Suppose that (H.1) and (H.2) hold. For system 1 with the feedback control of 2, the closed-loop system is globally asymptotically stable if [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] and all the state variables' 2nd order reminders are small enough for the given value of [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF]

Proof.

The expression for [figure omitted; refer to PDF] can be obtained by Taylor expansion as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] th order derivative. The first order approximation of the delay term is given by [figure omitted; refer to PDF] From 10 it can be seen that [figure omitted; refer to PDF] depends on the time delay, [figure omitted; refer to PDF] , and the higher order derivatives of [figure omitted; refer to PDF] which can be neglected if the time delay and the norm of [figure omitted; refer to PDF] are small. Then we have [figure omitted; refer to PDF]

The assumption in 11 can be used without significant error, and this can be true for the following reasons. Firstly, the time delay in a computer network is very small in order of milli- or microseconds and at the worst few tenths of the second. Secondly, in most of the real control system applications the linearized model is used and the higher order terms are already neglected. Additionally, the higher order derivatives will be multiplied by [figure omitted; refer to PDF] which is much more smaller than [figure omitted; refer to PDF] because [figure omitted; refer to PDF] . Substituting 11 into 7, the following can be derived: [figure omitted; refer to PDF] The system 13 will be globally asymptotically stable if [figure omitted; refer to PDF]

Corollary 3.

Suppose (H.1) and (H.2) hold. For the control system 1 with the control law 2, the closed-loop system is globally asymptotically stable if [figure omitted; refer to PDF]

Proof.

For system 1, suppose that the state feedback has been designed to ensure [figure omitted; refer to PDF] . Therefore, for a chosen positive definite matrix [figure omitted; refer to PDF] , it will find a positive definite matrix [figure omitted; refer to PDF] to have [figure omitted; refer to PDF] Choose a Lyapunov functional candidate as [figure omitted; refer to PDF] The objective for the next step is to find the range of [figure omitted; refer to PDF] that will ensure ( [figure omitted; refer to PDF] ) [25-27]. Taking the derivative of 18, [figure omitted; refer to PDF] Rearranging the terms in the above equation, then [figure omitted; refer to PDF] If [figure omitted; refer to PDF] then 20 will become [figure omitted; refer to PDF] Move the last term to the right hand side; the following will be derived: [figure omitted; refer to PDF] So [figure omitted; refer to PDF] .

Assuming that we can find a positive number to make the following hold: [figure omitted; refer to PDF] then [figure omitted; refer to PDF] can be considered as the norm of [figure omitted; refer to PDF] . Therefore, we have [figure omitted; refer to PDF] Choose [figure omitted; refer to PDF] Use Neumann series formula for the inverse of the sum of two matrices: [figure omitted; refer to PDF] For small time delays [figure omitted; refer to PDF] 26 can be given as [figure omitted; refer to PDF] Applying 27 into 25 then we have [figure omitted; refer to PDF] And finally we get [figure omitted; refer to PDF] That is, for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the system will be globally asymptotically stable.

Theorem 2 and Corollary 3 give us a simple tool in estimating the maximum allowable time delay for NCSs. Further analysis in the frequency domain is described below. Taking Laplace transform of 12, we have [figure omitted; refer to PDF] The characteristics equation is defined as [figure omitted; refer to PDF] For a stable system the roots of the characteristics equation 31 must lie in the left hand side of the [figure omitted; refer to PDF] -plane. From the characteristics equation, it is clear that the term [figure omitted; refer to PDF] influences the system performance and the stability as the term of [figure omitted; refer to PDF] may push the closed-loop system poles toward the right hand side of the [figure omitted; refer to PDF] -plane.

As we have seen the system characteristic is determined by the term [figure omitted; refer to PDF] in a certain level. This term can be regarded as a differentiator in the feedback loop, so it will introduce extra zeros to the closed-loop system and the time delay can be considered to have resulted in a variable gain to the feedback path. For more accurate estimation the second or third-order difference approximation can be used as follows: [figure omitted; refer to PDF] In the following a simple corollary for estimating the MADB in single-input-single-output NCS will be derived.

Corollary 4.

Suppose that (H.1) and (H.2) hold. The system 2 with the controller 3 is asymptotically stable if [figure omitted; refer to PDF]

Proof.

The main assumption is that the eigenvalues of the compensator, [figure omitted; refer to PDF] , are all negative, [figure omitted; refer to PDF] , and are given by [figure omitted; refer to PDF] The characteristic equation is the determinant of 34. Assume that the eigenvalues are given by [figure omitted; refer to PDF]

Preliminary 1 (inverse eigenvalues theorem [28]).

Given a matrix [figure omitted; refer to PDF] that is nonsingular, with eigenvalues [figure omitted; refer to PDF] are eigenvalues of [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] are eigenvalues of [figure omitted; refer to PDF] .

The eigenvalues of [figure omitted; refer to PDF] are given by [figure omitted; refer to PDF] Replacing [figure omitted; refer to PDF] by [figure omitted; refer to PDF] in 37 we get [figure omitted; refer to PDF] Solving 38 the eigenvalues are given as [figure omitted; refer to PDF] If [figure omitted; refer to PDF] then all the eigenvalues are positive and the system is asymptotically stable, and if [figure omitted; refer to PDF] at least one of the eigenvalues will be negative then.

If [figure omitted; refer to PDF] and (H.1) and (H.2) hold then the system is asymptotically stable.

Corollary 5.

Suppose that (H.1) and (H.2) hold. For system 1 with the control law 2, the closed-loop system is globally asymptotically stable if [figure omitted; refer to PDF] From Preliminary 1, the signs of the eigenvalues of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the same. For a single-input-single-output control system the matrix [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF] The interesting property of [figure omitted; refer to PDF] is that it is singular. The eigenvalues of [figure omitted; refer to PDF] are given by [figure omitted; refer to PDF] The characteristics equation of [figure omitted; refer to PDF] is the determinant of 42 and is given by [figure omitted; refer to PDF] Because [figure omitted; refer to PDF] is singular [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] Substituting 44 into 43, then 43 becomes [figure omitted; refer to PDF] Finally the eigenvalues of [figure omitted; refer to PDF] are [figure omitted; refer to PDF] Equation 46 shows that the minimum eigenvalue of [figure omitted; refer to PDF] equals [figure omitted; refer to PDF] . If the eigenvalues of [figure omitted; refer to PDF] are [figure omitted; refer to PDF] , then the eigenvalues of [figure omitted; refer to PDF] are [figure omitted; refer to PDF] . The eigenvalues of [figure omitted; refer to PDF] are given by [figure omitted; refer to PDF] By solving 47 it can be found that [figure omitted; refer to PDF] if [figure omitted; refer to PDF] .

For single-input-single-output NCS we have [figure omitted; refer to PDF] if [figure omitted; refer to PDF] and both (H.1) and (H.2) hold then the system is asymptotically stable.

This inequality can be used as a simple and fast tool for estimating the MADB in NCS and involves only single calculation.

3. Stability Analysis Case Studies

In general, two approaches are applied to controller design for NCSs. The first approach is to design a controller without considering time delay and then to design a communication protocol that minimizes the effects caused by time delays. The second approach is to design the controller while taking the time delay and data dropouts into account [11, 29]. The proposed method in this paper is used to estimate the MADB for predesigned control system. In this section, a number of examples are studied to demonstrate the proposed method and compare its results with the previously published cases in the literature. In particular, the results derived using the method proposed in this paper have been compared with the results using the LMI method given in [7] and with the fourth-order Pade approximation. The fourth-order Pade approximation [6] is used for the delay term in the [figure omitted; refer to PDF] -domain and is defined as [figure omitted; refer to PDF] The coefficients are given by [figure omitted; refer to PDF] With the fourth-order Pade approximation, the truncation error in the time delay calculation is less than 0.0001. The LMI-based method which has been used for the comparisons is based on using Lyapunov-Krasovskii functional and can be summarized as follows.

Corollary 6 (see [7]).

For a given scalar [figure omitted; refer to PDF] and a matrix [figure omitted; refer to PDF] , if there exist matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) of appropriate dimension such that [figure omitted; refer to PDF] then the system 1-2 is exponentially asymptotically stable. With a given controller gain [figure omitted; refer to PDF] , solving the LMI in Corollary 6 using the LMI Matlab Toolbox the maximum time delay can be computed.

Example 7.

The system in this example is the most widely used example in the literature and is described by the following equation: [figure omitted; refer to PDF] In previous reports [1, 7], the feedback control is chosen to be [figure omitted; refer to PDF] From Corollary 3, [figure omitted; refer to PDF] , so the MADB is estimated to be 0.8695 s. Using Theorem 2 and Corollary 5 the MADB is 0.8695 s. The same result can be obtained using the LMI method as reported in [7, 23, 24, 30]. In [11, 17], the value reported for MADB is [figure omitted; refer to PDF] s and in [22] it is 0.0538 s. In [29], the MADB is 0.785 s. It has been reported in [10], where an improved Lyapunov-Krasovskii approach has been used, that the MADB is 1.0551 s and also 1.05 s reported in [23] with improved algorithm for solving the LMI. In [1], the MADB is 1.0081 s. Using the proposed method with second order finite difference approximation we can obtain 1.13 s as the MADB. The system response with 0.8695 s time delay and [figure omitted; refer to PDF] is shown in Figure 4 which proves the system is stable with the estimated MADB.

Figure 4: The response of the system in Example 7 with 0.8695 s delay.

[figure omitted; refer to PDF]

Example 8 (see [31]).

Consider [figure omitted; refer to PDF]

For this third-order system both the LMI and our method give 0.0909 s as the MADB. Also with Corollary 5 the MADB is 0.0909 s.

Example 9 (see [31]).

The last example is the fourth-order model of the inverted pendulum shown in Figure 5 which is in many papers reduced to a second order system in order to verify the stability of NCSs. The pendulum mass is denoted by [figure omitted; refer to PDF] and the cart mass is [figure omitted; refer to PDF] ; the length of the pendulum rod is [figure omitted; refer to PDF] . The open loop system is unstable. The states are defined as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The model is given by [figure omitted; refer to PDF] The parameters used are [figure omitted; refer to PDF] kg, [figure omitted; refer to PDF] kg, and [figure omitted; refer to PDF] m. Then the linear model becomes [figure omitted; refer to PDF] Using the LQR Matlab function with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the controller is given by [figure omitted; refer to PDF]

Figure 5: The inverted pendulum on a cart.

[figure omitted; refer to PDF]

Using the LMI method the MADB is 0.0978 s and our method gives 0.0978 s using Theorem 2 and Corollary 5. We noted that there is a good agreement between our method and the LMI method because [figure omitted; refer to PDF] is small enough to make the finite difference approximation hold. The system response with 0.0978 s time delay and with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is shown in Figure 6 which shows the system is stable. Many examples have been studied to compare the results obtained using the method proposed in this paper with the results obtained using the LMI method [7] and the fourth-order Pade approximation method. The calculation results are summarized in Table 1 along with the simulation based MADB.

Table 1: The MADB (seconds) using the proposed method with 1st, 2nd, and 3rd order finite difference approximation for the delay term, the LMI method, the fourth-order Pade approximation method, and the simulation based method.

	The finite difference method	The LMI	Pade approximation	Simulation based
1st order	2nd order	3rd order
1	0.8695	0.8427	1.1321	0.8696	1.1672	1.180
2	0.1000	0.0995	0.1421	0.1000	0.1475	0.149
3	0.0100	0.0099	0.0149	0.0100	0.0156	0.0157
4	0.1428	0.1385	0.1808	0.1429	0.1855	0.1860
5	0.8217	0.8489	0.9085	0.8217	0.9091	0.9140
6	0.5000	0.4816	0.6303	0.5000	0.6474	0.6510
7	0.9940	0.9940	0.9960	0.9940	0.9960	0.9970
8	0.0856	0.0854	0.1192	0.0856	0.1230	0.1230
9	0.0906	0.0919	0.1251	0.0909	0.1284	0.1285
10	0.0416	0.0400	0.0496	0.0416	0.0505	0.0505

Figure 6: The response of the system in Example 9 with 0.0978 s delay.

[figure omitted; refer to PDF]

Remarks. From Table 1, it can be seen that the proposed new method can give values of MADB similar to the values obtained using the LMI method and the other methods; however, the method proposed in this paper has a much simpler procedure, and it should have no difficulties for practical design engineers to accept this approach. Clearly, the MADB with the first-order finite difference approximation is comparable with the LMI method. Furthermore, we found good agreement between the third-order finite difference approximation and the fourth-order Pade approximation. The simulation based results for the MADB show that the estimated MADB through the proposed method sufficiently achieves the system stability. A simple controller design method has been developed by the authors based on the method presented in this paper. In the controller design method a stabilizing controller can be derived for a given network time delay. In all the case studies or examples, only linear system examples are given. The method is limited to linear systems only. The authors are now working on extending the methods to nonlinear systems, such as, multiconverter and inverter system and engine and electrical power generation systems [32, 33].

The application of the finite difference approximation for representing the time delay is not new but we found in this paper that using higher order approximations can sufficiently represent the time delay linear system. From Table 1 it can be concluded that using the first order approximation the estimated MADB is comparable with the other two methods. This is because the derivation of the linear model from the nonlinear model is based on neglecting the higher order derivative terms. In some cases we need to use the higher derivative terms for the time delay in order to achieve more accurate results for the MADB. The current research is to derive sufficient conditions for applying the method in order to find the tolerance of the estimated MADB.

4. Concluding Remarks

The main contribution of the paper is to have derived a new method for estimating the maximum time delay in NCSs. The most attractive feature of the new method is that it is a simple approach and easy to be applied, which can be easily interpreted to design engineers in industrial sectors. The results obtained in this method are compared with those obtained through the methods introduced in the literature. The method has demonstrated its merits in using less computation time due to its simple structure and giving less conservative results while showing good agreement with other methods. The method is limited to linear systems only and the work for extending the method for a class of nonlinear systems is on-going.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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