Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 195120, 11 pages
http://dx.doi.org/10.1155/2015/195120

Research Article

Discrete-Time Sliding Mode Control for Uncertain Networked System Subject to Time Delay

Saulo C. Garcia, Marcelo C. M. Teixeira, José Paulo F. Garcia, Uiliam N. L. T. Alves, and Jean M. S. Ribeiro

Control Research Laboratory, Department of Electrical Engineering, São Paulo State University (UNESP), 15385-000 Ilha Solteira, SP, Brazil

Received 6 June 2014; Revised 4 August 2014; Accepted 6 August 2014

Academic Editor: Zhan Shu

Copyright © 2015 Saulo C. Garcia 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.

Abstract.

We deal with uncertain systems with networked sliding mode control, subject to time delay. To minimize the degenerative effects of the time delay, a simpler format of state predictor is proposed in the control law. Some ultimate bounded stability analyses and stabilization conditions are provided for the uncertain time delay system with proposed discrete-time sliding mode control strategy. A numerical example is presented to corroborate the analyses.

1. Introduction

Networked control system (NCS) is a very convenient strategy of control for industrial plants, where actuators and machines are located in rustic setting [1]. In those environments, a digital device, which computes the control signals, is susceptible to degeneration. Many other advantages, such as simple installation, great flexibility, and low cost, make the NCS a strategy widely used and researched [2]. However, despite these advantages, NCS implies the need of data acquisition, conversion of analog signal in digital form, and processing the data to generate the control signal. These steps may cause time delays of several sampling periods, which tend to deteriorate the system performance [3]. Many control strategies that are robust with respect to parametric variations and nonlinearities of the plant exhibit great sensitivity when delays are present, losing all their features of robustness. Due to this and other problems and high control utilization, NCS has significantly attracted research communities with important results for numerous control methods such as fuzzy control, neural control, adaptive control, sliding mode control, optimal control techniques, and many more techniques [4–11].

In particular, the strategy sliding mode control (SMC) [12, 13] is more sensitive to time delay. The main advantage of SMC is its robustness with respect to matched uncertainties [14]. By using a high speed switching control law in order to take the states trajectory to a sliding surface, if the states used are delayed, the control law may not direct the states to this surface, which can also generate performance loss or even lead system to instability. The damage caused by delays to sliding mode control motivates several studies with important results, including SMC performed by NCS [15–18].

In this paper we deal with uncertain system with discrete-time sliding mode control (DSMC) performed with NCS subject to time delay. To minimize the degenerative effects of delay, a discrete-time state predictor in simplified form used in conjunction with DSMC is proposed. Some ultimate bounded stability analyses and stabilization conditions are provided for the uncertain time delay system with proposed discrete-time sliding mode control strategy. A numerical example is presented to corroborate the results shown in the analyses.

2. Systems with Time Delay due to NCS

Consider the following uncertain discrete-time system:

where is the state vector, is the control vector, and is the bounded uncertainties vector. The matrices and are constants with nominal values of the plant, which is controllable.

The discrete-time law is given by

where is the sampling period and is the state feedback gain matrix, whose design can be done through various methods. For the design of matrix , in this paper, we analyze the stability robustness of the technique discrete-time sliding mode control (DSMC) performed by NCS.

As represented in Figure 1, when the control of the system is performed by a network in direct structure, the presence of delay due to the time required for analog and digital conversion is very common for receiving, processing, and transmission of data.

Figure 1: NCS representation in direct structure.

When the delay is directly related to the time required for state vector signal transmission, we have

where is the number of periods of delayed sampling state vector. Thus, we have the equivalence to a system with delayed control signal as

where .

When the delay is due to the time required for control signal transmission we have

where is the number of sampling periods of the delayed control signal.

Usually these time delays occur simultaneously in the NCS so that

where . So, we have the following system:

If the sampling time is , the control time delay in the continuous time system is . It is known that time delays in control signals degrade system performance [6–9]. Specifically, sliding mode control is more sensitive to these types of failures. In the next section, the use of a simpler format of state predictor to minimize the damaging effects of the time delay is proposed.

3. State Predictor

For the system (7) the following control law is proposed:

where the predictive state vector is an estimate of actual state vector at sample time , that is, .

In Xia et al. [19], the authors present a predictor of states for discrete-time systems with control time delay given by

By (9), we observe that to obtain the predicted states, several past samples of the control signal are needed and, for each sampling period, many calculations are required. We therefore propose predictors of states that require neither sampling control signal nor calculations that cause large processing time.

Consider the system (7) with the control law (8) and assume uncertainties are zero; that is, . So, and . Then,

and for the sample time we have

By (11), we define the predicted state vector as

where is a stable matrix and is an estimate value of the actual delay . Note that

It should be noted that, in the presence of uncertainties, that is, and , the predictor (12) is not accurate.

The following proposition deals with the dynamics of the proposed predictor (12).

Proposition 1. Consider the uncertain time delay system (7) with the control law (8) and the state predictor (12). Then the dynamics of predictive states are given by

where , , , , and .

Proof. Due to uncertainties, we have . We define the error vector as

In (12), at sampling period , it follows that

Substituting (7) into (16), we have

Using (8) and (13) into (17) we obtain

Substituting (15) into (18) and rearranging, we get

where , , , and .

Note 1. The predictive state vector of (19) has the same closed-loop dynamics of the actual state vector . It can be seen by substituting (8) and (12) into the system (7):

Considering (15) and the matrix , we obtain the closed-loop dynamics of the actual state vector

Also, by (19), the closed-loop dynamics of the predictive state vector are

Comparing (21) with (22) it can be noted that the matrices and are both equal to . Thus, we conclude that the dynamics of the predictive state vector are the same as actual state vector .

Note 2. In (21) and (22) it can be seen that, with the control (8) and state predictor (12), the influence of uncertainties in the dynamics of the system increases with the presence of the time delay due to the prediction error . According to control strategy adopted for the calculation of gain matrix , there will be greater or lesser influence on system performance.

In the next section, the stability robustness will be analyzed specifically for discrete-time sliding mode control (DSMC).

4. Discrete-Time Sliding Mode Control (DSMC) via NCS with Delay

In this work, the proposed DSMC law can be of the form

where is the equivalent control vector, which establishes the system dynamics in sliding mode, and is the control signal vector that takes the trajectory of states to the sliding surface. In sliding mode, the system becomes less sensitive to certain classes of disturbances and uncertainty [14].

4.1. Design of DSMC with Predictive States

The equivalent control is primarily designed. Later, the design of is performed.

A linear discrete-time sliding surface is defined as

where the gain matrix is designed such that the sliding trajectory has desired dynamics. The equivalent control is obtained with the following sliding condition:

Disregarding the uncertainties and substituting (7) and (24) into (25) we have

Thus, for the sample period we have

Using the predictor defined in (12), we obtain

where is a nonsingular matrix.

Now, the control law is designed. Consider the following Lyapunov function:

The control law should provide to the system the condition of attractiveness to the sliding mode. For this to occur, the following condition of attractiveness must be satisfied:

By (29), it follows that (30) becomes

Defining [20],

Disregarding the uncertainties and substituting (7) and (24) into (32), we have

By replacing , (26), into (33) comes

Replacing in condition (31) we find

where is the Euclidian norm of vector .

A control law that satisfies the attractiveness condition (35) is defined as

where , , and is a diagonal matrix with , . Thus, the control law becomes

with .

4.2. Robustness Analysis

This section analyzes the stability robustness of the system (7) in the following proposition.

Proposition 2. Consider the uncertain time delay system (7), with control law (37) and (12). The attractiveness condition (30) is satisfied if , with .

Proof. Taking into account the uncertainties, that is, and , (32) becomes

Substituting (37), (28), and (36) into (38), one comes to

where . Substituting (39) into (30), using (32) and considering , , we have

If , so . Thus, it is possible to rearrange condition (40) as follows:

with and . Considering that , so . With this, it can be seen, by (41), that the condition that satisfies the cases where and is the following:

Therefore, to satisfy (41) the norm must be greater than .

Note 3. From Proposition 2 it follows that, when the uncertain time delay system (7) is in steady state, its states remain in the vicinity of the ideal equilibrium point (origin). This fact implies the concept “ultimate bounded stability” [21].

Proposition 3. Consider the uncertain time delay system (7) with the control law (37) and that attractiveness condition (30) is satisfied. Also, the actual number of delayed samples is unknown and the uncertainties are bounded; that is, for all values of . So, in steady state the norm will be also bounded; that is, , with .

Proof. Consider the system (7) with and the control law (37) with predictive state vector (12). So

With we have

For and we have

and for the sample time we have

Thus, by (46), the predictive state vector is accurately obtained by the equation

The actual state vector is obtained as . So by (47) we have

We defined the predictor (not exact) as and the error is given by . Then, with (48) and considering that , we have

where . For the sampling period (49) becomes

It is easy to deduce that . So for sampling period we have

Thus, by (51), (50) can be rewritten as

The norm of is bounded. Suppose for simplicity, without loss of generality, that this vector is constant, , such that . So, (52) becomes

The closed-loop matrix is designed to be stable, that is, all its eigenvalues are inside in the unit circle. Then in the second member of (53) we have

So for (53) to be satisfied, in the first member of this equation, must be also a constant. Thus, it can be concluded that

Note 4. With , by Proposition 3, we have that and . For stability, the attractiveness condition is maintained until the system enters in the steady state, that is, until the state trajectory enters the neighborhood of the origin. The value of norm is established, as can be seen in (53) and (54), by

Then, the higher the value of , the higher the value of . This imposes a limit on the maximum number of delayed samples , in which it is possible to satisfy the attractiveness condition of sliding mode control.

Proposition 4. Consider the uncertainties with bounded norm; that is, , for all and , where . If the attractiveness condition (30) is satisfied, the system (7) will be uniformly ultimately bounded with control law (37), (8), and (12).

Proof. This proof is a version of result and terminology presented in [19]. Consider a linear transformation for the uncertain time delay system (7) of the form

where , , is the unmatched uncertainty vector;   is the matched uncertainty vector; , , ,, and are constant matrices with appropriate dimensions; and is the zero matrix. Note that, because the uncertainties has bounded norm, so and for all .
Now, let us use the following Lyapunov function:

where . So, the stability condition is

The sliding surface is

So,

Substituting (61) into (57) we have

where and .
Also,

with , .
Substituting (62) and (63) into (58)

where denote the minimum singular value of . Note that and because the attractiveness condition (30) is satisfied we have for all .
After some algebraic manipulation, (64) becomes

where

Therefore [19],

that is, the trajectory of will enter into ball with center at the origin and radius and will converge in finite time to the quasisliding mode band . At steady state it stays in the neighborhood of the origin.

Proposition 5. Consider the uncertain system without delay of the form

in which the sliding mode control law does not use predictors, that is,

where , , and , . So the attractiveness condition (30) is satisfied if .

Proof. Taking into account the uncertainties , (32) becomes

Substituting (69) into (70),

In this case, there is no need for using predictor, and the proof is similar to the proof of Proposition 2, (40)–(42), with .

Note 5. Because and Propositions 2,3, and 5, it can be concluded that the use of proposed control law (37) with state predictor (12) stabilizes the delayed system, however, increases the limits around the ideal equilibrium point. It means that uncertainties affect more the performance of the system when time delay is present.

In the following section, the presented propositions and notes are validated through a numerical example.

5. Numerical Example: System of Second Order

In this section, a system of order 2 to validate the propositions and notes presented in the previous section is utilized. Three cases are simulated and graphical results are presented through phase plans.

5.1. System Model and Design of Controllers

The uncertain model is used as follows:

where and are the states, is the amplitude of the parametric uncertainty of the plant, and is the amplitude of the disturbance in the control input. The time delay control input is , milliseconds is the sampling period, and is the actual number of delayed samples. It is considered that only a bounded range of is known. The estimated time delay of the predictors used in the control design is 40 milliseconds; that is, the number of estimated delayed samples is 20. The initial states are and . Note that this model without uncertainties has open-loop eigenvalues equal to “−0.4142” and “”; thus, it represents an unstable plant.

The control law is generated through feedback of the states using the strategy DSMC, as follows.

(i) DSMC-P1 Controller:

The predicted state vector is obtained by means of (9). The matrix is the same as the one of DSMC-P2 described as follows:

(ii) DSMC-P2 Controller:

The predicted states vector is obtained by (12), . The matrix is obtained according to (28), (36), and (37). The sliding surface is given by and it is designed such that the sliding pole is the equivalent discrete of “−2”, that is, “0.9960”. So

For sampling periods, that is, 8.0 seconds,

for sampling periods, that is, 20.0 seconds,

(iii) DSMC Controller:

where is the actual state vector. The matrix is the same as the ones of DSMC-P2 and DSMC-P1.

5.2. Simulations and Analysis of Results

The computational simulations were performed using Matlab/Simulink software. Three cases are discussed as shown below.

Case (i). Uncertain system without time delay, controlled by DSMC.

In this case the uncertain system has no time delay, that is, . The used controller is DSMC. The result is shown in Figure 2, where it can be noted that the controller has stabilized the system, taking the steady state trajectory in the neighborhood of the origin. Under the point of view of disturbances rejection, the system with DSMC controller had a good performance.

Figure 2: Phase plane for uncertain system without time delay, controlled by DSMC (without predictor).

Case (ii). Uncertain system with time delay, controlled by DSMC (without predictor).

In this case the actual value is different at each simulation time interval as follows:

DSMC without state predictor is used, that is, with the control law . Figure 3 shows the result. It may be noted that the system became unstable.

Figure 3: Phase plane for uncertain system with time delay, controlled by DSMC (without predictor).

Case (iii). Uncertain system with time delay, controlled by DSMC-P1 or DSMC-P2.

Figure 4 shows the result for the system with the same time delay in the intervals , , and , but now the plant is controlled by DSMC-P1 or DSMC-P2, that is, with predictor (9) or (12), respectively. Now the system is stable. This shows that this control strategy requires state predictor when time delay is present.

Figure 4: Phase plane for uncertain system with time delay, controlled by DSMC-P1 or DSMC-P2.

For each of the controllers, it can be seen that there are three different trajectories. From the smallest to the largest area, they represent the state trajectories for each time interval , , and , respectively. The part “” of (53) increases as the delay is increased, implying a steady state error norm, , also higher at each interval , , and . These results are in accordance with the Propositions 2, 3, and 4 and also with Notes 3 and 4.

The best performance was obtained using the DSMC-P1 controller, where the predictor is given by (9). However this predictor requires several past samples of the control signal, and, for each sampling period, many calculations are required. On the other hand, the DSMC-P2 controller also proved to be efficient. The main advantage is that it requires no previous samples of the control signal, and its computation, at each period, is very simple and quick.

It can be seen, by comparing Figures 2 and 4 that the delayed system controlled by DSMC-P1 or DSMC-P2 is stable, but it is more sensitive to disturbance. This fact is in accordance with Propositions 2 to 5 and Note 4.

6. Conclusion

This work approached uncertain systems with networked discrete-time sliding mode control (DSMC), subject to time delay. To minimize the degenerative effects of time delay, a simpler format of state predictor is used in the control law. The used state predictor has the advantage of fast computation, without the necessity of control signal sampling. The analyses and results from simulations showed the effectiveness of the proposed strategy with regard to the stabilization of uncertain time delay system, even in the presence of uncertainties and delays. The simulation results also confirm the analyses concerning the influence of the uncertainties in the networked control system performance with proposed DSMC strategy.

Conflict of Interests

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

Acknowledgments

The authors thank CNPq and FAPESP, Process no. 2011/17610-0, for the financial support.

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