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Mathematical Problems in Engineering
Volume 2015, Article ID 160542, 10 pages
http://dx.doi.org/10.1155/2015/160542
Research Article

Design of a Backstepping Tracking Controller for a Class of Linear Systems with Actuator Delay

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received 29 April 2015; Revised 2 October 2015; Accepted 15 October 2015

Academic Editor: Alfonso Banos

Copyright © 2015 Yonglong Liao and Fucheng Liao. 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

This paper presents a method for designing a backstepping tracking controller for a class of continuous-time linear systems with actuator delay subject to a reference signal. The actuator delay can be modeled by a first-order hyperbolic PDE, and then a PDE-ODE coupled system is obtained. By applying the backstepping transformation to the coupled system, a feedback controller that includes the state of the system, the integral of the input control, and the integral of the tracking error is derived. We show that the closed-loop system is asymptotically stable at the equilibrium point and achieves complete regulation under the stabilizability assumption. The designs in this paper are illustrated with numerical simulations.

1. Introduction

Over recent decades, many researchers have explored the design of servomechanisms for linear systems with time delay [13]. They aimed to design a controller such that the outputs track the reference signals without steady-state errors and the closed-loop system is asymptotically stable at the equilibrium point. They eliminated the influence of time delay on the systems by introducing a predictor that represents the integrator of the time-delay process outputs [46]. Or they dealt with the delay in robust control methods for some systems under certain conditions by treating the delay as an uncertain factor [79] and then designing the tracking controllers. However, when the plant is unstable, the predictor may fail to achieve closed-loop stability [10], while the robust design method can only solve a portion of specific problems.

Krstic and Smyshlyaev noted that the solution of the first-order hyperbolic PDE can be used to replace the actuator delay if the boundary condition is . Then, the actuator delay systems can be modeled by the PDE-ODE coupled systems. By applying the backstepping transformation, a controller is derived [11]. This kind of controller is equivalent to the classical predictive controller. Moreover, the closed-loop system can be proved to be exponentially stable at the equilibrium point by constructing a Lyapunov-Krasovskii functional. This approach to dealing with delay systems is called backstepping control.

In this way, Bekiaris-Liberis and Krstic [12] constructed an explicit feedback law for systems with simultaneous input and state delay. Krstic and Bresch-Pietri [13] presented an adaptive control design for unstable systems with actuator delay of substantial length and completely unknown value. Other researchers [1416] have studied systems with time-varying input delay, distributed delay, and pointwise delay, respectively. In their papers, they proved that the designed closed-loop systems are exponentially stable by constructing Lyapunov functionals. Bresch-Pietri et al. [17] made a more intensive study of equilibrium regulation under partial measurements, disturbance rejection, and parameter or delay adaptation. Karafyllis and Krstic [18] provided formulas according to which one can compute estimates of the least upper bound of the magnitude of the delay perturbation. And recently, Lin and Cheng [19] proposed an adaptive block backstepping control scheme for a class of time-delay systems.

The backstepping approach shows a strong superiority in dealing with time-delay systems, but we found almost no systematic results in studying the tracking problem for delay systems with this method. This paper is a new attempt to design a tracking controller for a continuous-time linear system with actuator delay, in which we develop a type of servomechanism with PDE style and ODE style state feedback by using the backstepping technique. We formulate the control problem and offer some necessary assumptions in Section 2. A backstepping tracking controller is derived in Section 3 based on the theory of pole placement and the backstepping method. Section 4 shows that the closed-loop system is asymptotically stable at the equilibrium point and achieves complete regulation under the stabilizability assumption. In Section 5, we give a practical example and design a backstepping tracking controller; the simulation results demonstrate the effectiveness of the controller. Section 6 is a brief conclusion.

2. Problem Formulation

Consider a continuous-time linear system with the actuator delaywhere is the state vector, is the control vector delayed by units of time, and is the output vector to be controlled. , , and are constant matrices.

Let be the reference signal. The objective of the paper is to develop a feedback controller such that (i) the closed-loop system is asymptotically stable at the equilibrium point and (ii) the output vector tracks the reference signal ; namely,The following assumptions will be needed throughout the paper.

Assumption 1. is a controllable pair.

Assumption 2. The reference signal is a piecewise-continuous function satisfying where is a constant.

Assumption 3. The matrix has full row rank; that is,

3. Design of Backstepping Tracking Controller

In this section, we will design a backstepping tracking controller for system (1), so that the closed-loop system is asymptotically stable at the equilibrium point, while the output of the system asymptotically tracks the reference signal.

Define the error vector asLet be the integral of the tracking errorIn order to ensure that the system’s output tracks the reference signal asymptotically, we need to obtain the integral of the tracking error as part of the state variables. Differentiating both sides of (6) derives Let be the augmented vectorThen, it follows from (1) and (7) thatwhere , , and are , , and constant matrices defined by

Consider the following first-order hyperbolic PDE: In (11), the delay-free control signal acts as a boundary condition. is the initial condition. For convenience, we let the history control over the time interval be , . It is well known that the hyperbolic PDE (11) has the following explicit solution:Then, the outputis the seconds delayed input. For this reason, the delay system (9) can be modeled by the following PDE-ODE coupled system:The input signal acts both as a controller and as a boundary condition.

So far, we have converted the tracking problem of an actuator delay system into a regulation problem of a PDE-ODE coupled system. Next, we will design an appropriate state feedback controller for system (14) such that the closed-loop system is asymptotically stable at the equilibrium point. We now introduce the backstepping transformation of system (14) and the inverse of this transformation in a theorem form.

Theorem 4. Suppose that is controllable. Then,
(i) the backstepping transformationmaps system (14) into the following target system:where is a stabilizing gain vector such that the matrix is Hurwitz.
(ii) The transformation (15) is invertible and the inverse transformation has the following form:

Proof. Let us prove the first part of the theorem. Considering the second equation in (15) with , we getSubstituting this expression into the first equation in (14), we get that is, the first equation in (16). Let us calculate the time derivatives of the second equation in (15):Using , we haveAfter a simple calculation,Calculating the spatial derivatives of the second equation in (15) yieldsSubtracting (23) from (22) leads toWith moved to the right side of the equation, we get the second equation in (16). Thus, we prove that the backstepping transformation (15) maps system (14) into the target system (16).
We will offer a straightforward method to prove the second part of this theorem. Note that if we substitute (17) into (15) and get an identity, then we demonstrate that (17) and (15) are inverse.
Substituting (17) into (15), we haveThen, we only need to demonstrate that (26) is an identity. Taking a collection of the similar items in and , we get For the second part on the right hand of (27), we add to the first integral and then subtract it. By integral calculation, we get And for the third part, we add to the third integral and then, subtracting it, we have Then, exchanging integral order and calculating it, we get Thus, (26) is an identitywhich illustrates that (15) and (17) are inverse transforms with each other. The proof is complete.

In the proof of Theorem 4, we have not mentioned the third equation in (16). In fact, the boundary condition of (16) depends not only on the transformation of (15), but also on the boundary input for (14). In order to get the desired results, we let Then, the boundary input for (14) should beEquation (33) is the feedback controller for (14). Noting that , we give the backstepping tracking controller for (1) by the following theorem.

Theorem 5. Suppose that is controllable and the reference signal satisfies Assumption 2. Then, the backstepping tracking controller is given bywhere is a stabilizing gain vector such that the matrix is Hurwitz. and are the feedback gains defined by

Remark 6. We can find that the backstepping tracking controller consists of three parts. The first is the feedback in partial differential form, and its essence is the feedback of the input delay integral, which is used to offset the lag input’s effect on the system. The second part is the feedback of the present state. And the third part is the integral action on the tracking error: it guarantees that the output tracks the reference signal asymptotically.

Remark 7. Specifically, if , then (34) iswhich consists of the feedback of the state and integral of error but does not consist of the PDE section. This indicates that the effect of introducing PDE is to eliminate the input delay’s impact formally. On the other hand, when , and then is stable. This is consistent with the results of general control theory and is equivalent to the controller design with the pole placement method.

In fact, with the backstepping coordinate transformation of (15), the closed-loop system constituted by (14) and (33) is (16). If we can prove that the closed-loop system (16) is asymptotically stable at the equilibrium point, then we have demonstrated that (34) is the exact controller we want to design.

4. Stability of the Closed-Loop System

In this section, we will consider the stability and tracking property of the closed-loop system. Two lemmas are needed.

Lemma 8 (see [20]). The pair is controllable if and only if is controllable and the matrix has full row rank.

Lemma 9 (see [21]). Consider the linear continuous-time system where is Hurwitz and is a bounded measurable function on . And . Then for any the state vector satisfies

Now we present the stability theorem of the closed-loop system (16).

Theorem 10. Suppose that is controllable and the matrix has full row rank, and the reference signal is asymptotically stable. Then, the closed-loop system (16) is asymptotically stable at the equilibrium point. And the output of system (1) asymptotically tracks the reference signal under the controller (34); namely,

Proof. We first show that the closed-loop system is asymptotically stable at the equilibrium point. Under the conditions Assumptions 1 and 3, it follows from Lemma 8 that is controllable and there exists a matrix such that is Hurwitz. Let be the equilibrium point of (16): it follows thatSolving (40), we getLetting we transform (16) intoNow, we give a detailed demonstration of the asymptotic stability of (43) through direct calculation.
We consider the partial differential equation in (43) first. According to and () in Section 3, we have (). That is, the initial condition of the partial differential equation is determined by the initial input on . Hence, is determined by and the initial input . For briefness in calculation, we let by giving a suitable initial input. It should be pointed out that the proof is also correct when .
Thus, the partial differential equation in (43) with initial and boundary condition is described as follows: Solving (44) and letting (), we obtain Taking the norm on both sides of (45), we have According to Assumption 2, we have . Thus, we haveOn the other hand, according to (45) we have Substituting (48) into the first equation of (43), we getLet Equation (49) is written as For (50), the following inequality holds:Combining , we have Using Lemma 9, we obtainCombining (47) and (54), we have which indicates that (43) is asymptotically stable.
Obviously, the target system (16) is also asymptotically stable at the equilibrium point because (42) is just a coordinate moving in parallel with (16). And the equilibrium point is Hence, the statement in Section 1 is proved.
From the discussions above, we havewhere , are constant vectors and satisfyDenoting , (58) can also be written asWe havethat is,Namely,Equation (62) suggests the system’s output asymptotically tracks the reference signal. Thus, the proof of Theorem 10 is complete.

Summarizing the results above, we give the following tracking controller theorem.

Theorem 11. Suppose that Assumptions 1 to 3 are satisfied; then, there exists such that is stable, and the closed-loop system of (1) isThe control input is determined by the following PDE-ODE coupled system:Theorem 11 illustrates that can be obtained through solving the PDEThen, is obtained from (64), and finally we get the control input .

5. A Simulation Example

We will give the actuator delay RLC network [22] described in Figure 1 to illustrate the effectiveness of the controller.

Figure 1: An RLC actuator delay network.

In Figure 1, is the current through the inductor , is the voltage across the capacitor and is also the output of the system, is the input voltage which has a time delay to the network, and , , , and . Now, we make the definition and ; applying Kirchhoff’s current and voltage laws to the RLC network, we get the following mathematical model:where , , and . The output equation is given bywhere . Let and , .

We give the reference output of as the following four types.

Step Signal

Periodic Signal

Ramp Signal

Fading Signal

The control system can be described asAnd the control input is given by the following PDE-ODE coupled system:where , , and . Prescribing the poles to the nominal system (14) results inand hence we have

The system response under the backstepping tracking controller is depicted in Figures 2, 3, 4, and 5. It can be observed that the backstepping tracking controller can indeed guarantee that the output tracks the reference signals asymptotically. That is to say, the design method of the controller in this paper is effective.

Figure 2: Closed-loop responses of step signal (68).
Figure 3: Closed-loop responses of periodic signal (69).
Figure 4: Closed-loop responses of ramp signal (70).
Figure 5: Closed-loop responses of fading signal (71).

6. Conclusion

In this paper, we have proposed a backstepping tracking control scheme for a class of linear systems with actuator delay. By constructing a PDE-ODE coupled system based on the augmented system constructed, we converted the tracking problem into a regulation one. The proposed control scheme can effectively track the reference signal and guarantee the asymptotic stability of the closed-loop system. The numerical simulations show the efficiency of the backstepping tracking controller.

Appendix

The backstepping transformation (15) can be obtained by this method. Let the transformation have the following form:where and are kernels to be determined and satisfying is Hurwitz. This converts plant (14) into the following target system:

Calculating the time derivatives of the second equation in (A.1),and calculating the spatial derivatives of the second equation in (A.1), by the target system (A.2) we getThis equation should be valid for all , , and , so we haveOn the other hand, let in the second equation in (A.1), which givesSubstituting (A.7) into the first equation in (14) and (A.2), we haveSoThe kernels and should satisfyWe can easily obtain by (A.11) that . Substituting this expression into (A.10), we get . Thus, we get the transformation (A.1):And that is (15).

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61174209) and the Oriented Award Foundation for Science and Technological Innovation, Inner Mongolia Autonomous Region, China (2012).

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