**International Journal of Modern Nonlinear Theory and Application** Vol.1 No.3(2012), Article ID:23082,8 pages DOI:10.4236/ijmnta.2012.13010

Adaptive Output Tracking for Nonlinear Network Control Systems with Time-Delay

College of Automation, Chongqing University of Post and Telecommunications, Chongqing, China

Email: zdyjm@163.com, zenghy99@163.com

Received August 25, 2012; revised September 11, 2012; accepted September 17, 2012

**Keywords:** Time-Delay; Network Control Systems; Backstepping Design; Adaptive Control; Output Tracking

ABSTRACT

The problem of adaptive output tracking is researched for a class of nonlinear network control systems with parameter uncertainties and time-delay. In this paper, a new program is proposed to design a state-feedback controller for this system. For time-delay and parameter uncertainties problems in network control systems, applying the backstepping recursive method, and using Young inequality to process the time-delay term of the systems, a robust adaptive output tracking controller is designed to achieve robust control over a class of nonlinear time-delay network control systems. According to Lyapunov stability theory, Barbalat lemma and Gronwall inequality, it is proved that the designed state feedback controller not only guarantees the state of systems is uniformly bounded, but also ensures the tracking error of the systems converges to a small neighborhood of the origin. Finally, a simulation example for nonlinear network control systems with parameter uncertainties and time-delay is given to illustrate the robust effectiveness of the designed state-feedback controller.

1. Introduction

Network control system is a real-time closed-loop feedback control system composed of sensors, controllers, actuators, etc. The advantages of network control systems are its easy installation and maintenance, and its high reliability and flexibility [1,2]. In recent decades, there are lots of progresses in the study of stability of the network control systems [3-6].

However, in the closed-loop control of the network control system, the data transmission process is often produce time-delay. The time-delay of network control system often affects the stability and performance of the system, and may even cause the instability of the entire system [7]. Therefore, the impact of the time-delay on network control system needs to be considered when studying network control systems and designing controllers. In [8], the authors analyzed the source of the time-delay of network control system. For the time-delay problem of network control system, a maximum allowable delay bound satisfying the requirement of stability was proposed in [9], and the maximum delay caused by the network was estimated in [10]. For designing controllers of network control systems, in [11], the authors discussed a class of uncertain systems’ adaptive control scheme, and in [12] authors analysis robust stability of networked control systems with uncertainty. Although some progresses are made in linear network control systems, nonlinear network control systems with parameter uncertainties and time-delay needs to be studied. For example, in [13-17], the authors study the problems of adaptive robust control for uncertain systems and highorder uncertain nonlinear systems, and analyze the stability of the systems by Lyapunov stability theory. But these papers did not consider the situation of the systems with time-delay.

Therefore, in this paper, the system is modeled as a class of nonlinear network control system with parameter uncertainties and time-delay. A new program is proposed to design controller for this system, and a robust controller is designed by using the backstepping method. According to Lyapunov stability theory, Barbalat lemma and Gronwall inequality, it is proved that the designed controller not only guarantees the state of nonlinear network control systems with parameter uncertainties and time-delay is uniformly bounded, but also ensures the tracking error of the systems converges to a small neighborhood of the origin. The rest parts of the paper are organized as follows: in Section 2, a class of nonlinear network control system is introduced, and the assumption and lemmas are proposed. In Section 3, the controller is designed by using the backstepping method. In Section 4, a simulation example is presented. Finally, a conclusion is given in Section 5.

2. Problem Description

In this paper, we consider a class of nonlinear network control systems with parameter uncertainties and timedelay, this system is described as

(1)

where, , and are respectively the states, the control input and system output, is a vector of unknown constant parameters, d_{i}(·) ≠ 0, ψ_{i}(·) and are unknown smooth functions, h_{i}(0) = 0 (1 ≤ i ≤ n) is also an unknown smooth functions, τ is time-delay, and τ ≥ 0.

The objective of this paper is to design an adaptive feedback controller. The designed controller ensures the state of the closed-loop systems is bounded and the trajectory of output y(t) can asymptotic track reference signal y_{r}(t).

Assumption 1 For smooth function d_{i}(t, x, u), i = 1, ···, n there exist functions and satisfies 0 < c_{i}(x_{1}, ···, x_{i}) ≤ d_{i}(t, x, u) ≤ c_{i}(x_{1}, ···, x_{i}_{ + 1}), x_{n}_{+1} = u.

Assumption 2 Because we have h_{i}(0) = 0, then the h_{i}(x_{1}(t)) can be expressed as h_{i}(x_{1}(t)) = γ_{i}(x_{1}(t)), and γ_{i}(x_{1}(t)) satisfies the following assumption

where p_{i}(x_{1}(t)) is a known and smooth enough function.

Lemma 1 If the real number a ≥ 0, b ≥ 0, m ≥ 1, then there exist the following inequality

.

Proof for any real number x ≥ 0, y > 0, n > 0, by Young inequality, we have

.

Let a = x, , m = n + 1 then we can release to Lemma 1.

Barbalat lemma [18] If x(t) is a uniformly continuous function, and

exists and is bounded, then.

3. Adaptive Controller Design

In this section, by using the backstepping recursive method, we design a robust adaptive output tracking controller. The designed ideas of this method are described as follows: for the i-th equation of the system, constructed a suitable Lyapunov function, and designed virtual control law α_{i}, the designed α_{i} makes the subsystem consist of previous i equations is stable, therefore, in step n, the designed controller u which makes the system consist of n equations stability is the true controller that makes the closed-loop control systems globally stable.

Step 1 Reference signal y_{r} is a smooth function and bounded, and its derivative is also bounded, the output tracking error is defined by ε_{1} = x_{1} – y_{r}.

Constructed Lyapunov function as

where λ are positive, , is estimates of the unknown constant parameter θ. Calculating the derivative of V_{1} along with system (1), we have

(2)

Because is bounded, presence non-negative smooth function, satisfies

.

By Lemma 1, for any real number σ that greater than zero, let, b = σ, so that exists a smooth function, satisfies

. (3)

By using Young inequality, let constant ξ_{1} > 0 we have

, (4)

select then we have

, (5)

where ρ_{1}(x_{1}(t)) is a smooth function.

Let z_{1} = λε_{1}ψ_{1}, Substituting (3), (4), (5) into (2), we have

Designed virtual controller as

where f_{1}(·) is smooth function that is greater than zero.

So that we can release to

And because –ε_{1}α_{2} ≥ 0, by assumption 1, we have

where η_{1} = 0.

Step 2 Let ε_{2} = x_{2} – α_{2}, constructed Lyapunov function as

Calculating the derivative of V_{2} along with system (1), we have

Let

Then we have

(6)

There exists a non-negative smooth function, satisfies

By Lemma 1, let, b = σ so that there exists a smooth function, satisfies

. (7)

And because, combined with Lemma 1, there exists a smooth function satisfies

. (8)

By using Young inequality, let constant ξ_{2} > 0, μ_{2} > 0, we have

Select

then we have

Then we have

(9)

Substituting (7), (8), (9) into (6), we have

Designed virtual controller as

where f_{2}(·) is a smooth function that is greater than zero.

By assumption 1, we have

Step i After the recursive design step i-1, we can get a group of smooth virtual controller as

where smooth function f_{k}(·) > 0, k = 1, ···, i – 1.

Constructed Lyapunov function as

The derivative of V_{i}_{–}_{1} as following

(10)

Similar to step 2, we can prove (10) is also established in the step i.

Constructed Lyapunov function as

Its derivative is given by

Let

Then we have

(11)

There exists a non-negative smooth function w_{i}(·), satisfies

By the Lemma 1, there exists a smooth function β_{i}(·) ≥ 0 satisfies

(12)

Similar to step 2, there exists a smooth function (·) ≥ 0 satisfies

. (13)

By using Young inequality, let constant ξ_{i} > 0, μ_{i} > 0, we have

select

then we have

Then we have

Then we have

Designed virtual controller as

where f_{i}(·) is a smooth function that is greater than zero.

By assumption 1, we have

Step n After repeated recurrence and proof, in the step n, constructed Lyapunov function as

Its derivative is given by

(14)

From (14), we can obtain adaptive control law u and parameter following as

(15)

(16)

where f_{n}(·) is a smooth function that is greater than zero.

Then, we have

.

When n is large enough, then we have

Select then we have .

Therefore, we get.

By Barbalat lemma, we get, and then we have, j = 1, ···, n. So that we get.

So that the entire design procedure is reasonable.

Theorem 1 Considering closed-loop systems (1), under assumption and Lemma, there exist a state feedback control law u and control law parameter. The closedloop system is bounded for all allowable uncertainties and the output tracking error converges to a relatively small area, which satisfies

.

Proof

where

By Gronwall inequality, we have

And because

So that we have

.

In summary, for any real number ε_{0} > 0, in limited time T > 0, the closed-loop system satisfies

.

4. Simulation Example

In order to show the effectiveness of the design scheme, we choose the nonlinear network control system with parameter uncertainties and time-delay as following:

(17)

In the simulation, for the closed-loop system (17), we choose the reference signal y_{r}(t) = sint, time-delay τ = 0.01s, θ = 0.2, ξ_{1} = 1, ξ_{2} = 2, σ = 0.02, λ = 1, the initial conditions x_{1}(0) = 1, x_{2}(0) = 0.5, (0) = 0.1, According to (15) and (16), the control law u and the parameter of control law following as

The simulation results are shown as in Figures 1 and 2. It can be observed that the output of closed-loop system can track the reference signal well, and the tracking error converges to a small neighborhood of the origin. Therefore the robust adaptive controller is effective.

Figure 1. Output y(t) and reference signal y_{r}(t).

Figure 2. Output tracking error y(t) – y_{r}(t).

5. Conclusion

By using the backstepping method, we design a controller for nonlinear network system with parameter uncertainties and time-delay. Through theoretical analysis, it is shown that the designed robust adaptive output tracking controller is feasible. The simulation results further expressed the effectiveness of the scheme.

6. Acknowledgements

This work was supported in part by the Natural Science Foundation of Chongqing (CSTC) under Grant No. 2009BB3280, and the National Natural Science Foundation of China under Grant No. 60873200.

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