Fuzzy Adaptive Prescribed Performance Control for Uncertain Horizontal Platform System with Unknown Control Gain

Abstract

This paper proposes a fuzzy adaptive control method for uncertain horizontal platform system with unknown control gain, which is capable of guaranteeing the prescribed performance bounds. An error transformation is introduced to transform the original constrained system into an equivalent unconstrained one. Then, based on the error transformation technique and the predefined performance technique, a fuzzy adaptive controller is designed for the unconstrained system. It is shown that all the variables of the resulting closed-loop system are bounded. Finally, an illustrative example is given to demonstrate the effectiveness and usefulness of the proposed method.

Academic Editor:Yang Tang

Department of Mathematics, Huainan Normal University, Huainan 232038, China

Received 11 August 2015; Revised 7 November 2015; Accepted 10 November 2015; 29 November 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

Over the past two decades, it has been acknowledged that several mechanical systems exhibit chaotic behavior [1-3]. The horizontal platform system (HPS) is one of the most interesting and attractive nonlinear dynamical systems. It is a mechanical device that can freely rotate around the horizontal axis. The horizontal platform devices are widely used in offshore and earthquake engineering. Recent research has found that these systems display many dynamic behaviors including chaotic. So, how to suppress chaotic phenomenon for horizontal platform systems (HPSs) is a hot research topic. Until now, a wide variety of approaches have been proposed for HPSs. For example, Wu et al. [4] have applied the Lyapunov direct method to achieve a sufficient criterion for chaos synchronization between two identical HPSs which have been coupled by a linear state error feedback controller. In [5], Pai and Chang proposed a fuzzy sliding mode control scheme to achieve generalized projective synchronization (GPS) of two horizontal chaos platform vibration systems with system uncertainty and external disturbance. By using time-delay feedback control, Ding and Han [6] discussed master-slave synchronization for two identical nonautonomous HPSs. Using a linear state error feedback controller, the robust synchronization of the chaotic HPSs with phase difference and parameter mismatches has been studied in [7]. Based on Lyapunov stability theory, Pai and Yau [8] designed an adaptive sliding mode controller such that the controlled HPS state can be driven to a desired orbit. The problem of robust finite-time synchronization of two nonautonomous HPSs is investigated in [9]. Recently, Xiang and Liu [10] proposed an adaptive fuzzy terminal sliding mode control scheme for uncertain HPS. Based on fuzzy system rules, the proposed control approach guarantees the boundedness of all the signals in closed-loop system. These considered horizontal platform systems are in the form [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . If control gain [figure omitted; refer to PDF] is unknown, how can we design the control scheme to achieve the stability for uncertain HPS?

To the author's best knowledge, there are few literatures to research the stability problem for uncertain HPS with unknown control gain. To handle unknown nonlinear function [figure omitted; refer to PDF] in HPS, several nonlinear approximators, for example, fuzzy logic systems and neural networks, have been used. Two of the main features of adaptive fuzzy approaches are as follows: (i) they can be used to deal with those nonlinear systems without satisfying the matching conditions and (ii) they do not require the unknown nonlinear functions being linearly parameterized [11-15]. In order to meet control objectives, the used controller in indirect adaptive schemes is in the form [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a new control input. [figure omitted; refer to PDF] is parameterized approximation of actual function [figure omitted; refer to PDF] . [figure omitted; refer to PDF] represents the adjustable parameter of the approximation. Since the approximation [figure omitted; refer to PDF] is generated online by estimating the parameter [figure omitted; refer to PDF] , one can notice that the above controller is not well-defined because of the singular problem of the controller. In this paper, we use the regularized inverse of [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . The regularized inverse is well-defined even when [figure omitted; refer to PDF] is singular. Compared with related works, there are four main contributions that are worth being emphasized:

(1) Compared with the results in [4-10], the uncertain HPS with unknown control gain is considered.

(2) The prescribed performance function (PPF) is incorporated into the control design.

(3) Adaptive laws are proposed to update the fuzzy parameters.

(4) The controller will not appear as singular problem.

Recently, an attempt to establish a priori specified performance control paradigm has been exploited [16], where the maximum overshoot, the convergence rate, and steady-state error are all addressed. Motivated by the above discussion, we will propose an adaptive fuzzy control for HPSs with prescribed transient and steady-state tracking performance. Inspired by [17, 18], an improved prescribed performance function (PPF) is incorporated into the control design. An error transformed system is derived by applying the PPF on the original system. Consequently, the tracking error of the original system can be guaranteed within the prescribed bound provided the transformed system is stable. For this purpose, an adaptive prescribed performance control (APPC) is designed for uncertain HPS in the presence of system uncertainties and external disturbance, which also allows proving the closed-loop stability. A comparative example is given to emphasize the effectiveness of the proposed APPC based on the PPF design.

The organization of this paper is described as follows. In the next section, system model is derived, and the assumptions are also given. In Section 3, the design of the proposed control strategies is discussed. The simulation results are presented to demonstrate the effectiveness of proposed control scheme in Section 4. Conclusion is presented in Section 5.

2. System Descriptions and Problem Formulations

The HPS is a mechanical device composed of a platform and an accelerometer located on the platform (see Figure 1). The platform can freely rotate about the horizontal axis, which penetrates its mass center. The accelerometer produces an output signal to the actuator, subsequently generating a torque to inverse the rotation of the platform to balance the HPS, when the platform deviates from horizon. The motion equations of the HPS are given by [9, 10] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the rotation of the platform relative to horizon, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are inertia moment of the platform, [figure omitted; refer to PDF] is the damping coefficient, [figure omitted; refer to PDF] is the proportional constant of the accelerometer, [figure omitted; refer to PDF] is the acceleration constant of gravity, [figure omitted; refer to PDF] is the radius of Earth, and [figure omitted; refer to PDF] is the harmonic torque. System (1) exhibits chaotic behavior with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] (see Figure 2).

Figure 1: Physical model of the horizontal platform system (1).

[figure omitted; refer to PDF]

Figure 2: Phase plane trajectory of the horizontal platform system (1).

[figure omitted; refer to PDF]

For simplicity, we introduce the following notations: [figure omitted; refer to PDF] ; then the dynamic model of (1) with unknown control gain and external disturbances can be described by the following equations: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the control input, [figure omitted; refer to PDF] is unknown control gain, and [figure omitted; refer to PDF] is unknown external disturbance. [figure omitted; refer to PDF] is assumed to be unknown, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

The objective of this paper is to construct a fuzzy adaptive controller for system (2) such that

(P1) the system state [figure omitted; refer to PDF] tracks the reference signal [figure omitted; refer to PDF] and all the signals in the closed-loop system remain bounded,

(P2) both prescribed transient and steady-state behavioral bounds on the tracking error [figure omitted; refer to PDF] are achieved.

To meet the objective, the following assumptions are made for the system (2).

Assumption 1.

The state vector [figure omitted; refer to PDF] is measurable, and the references [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are continuous and bounded.

Assumption 2.

[figure omitted; refer to PDF] and [figure omitted; refer to PDF] are unknown but bounded. And there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Remark 3.

The present control scheme in this paper can guarantee control objectives without the knowledge of the [figure omitted; refer to PDF] value in Assumption 2.

2.1. Prescribed Performance

Definition 4.

A smooth function [figure omitted; refer to PDF] is called a prescribed performance function (PPF) if [figure omitted; refer to PDF] is decreasing and [figure omitted; refer to PDF] .

In this paper, we select [figure omitted; refer to PDF] as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are design parameters.

It is sufficient to achieve the control objective (P2) if condition (4) holds [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are design constants.

Remark 5.

The transient and steady-state performances can be designed a priori by tuning the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

To represent (4) by an equality form, we employ an error transformation as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the transformed error and [figure omitted; refer to PDF] is smooth, strictly increasing function and satisfies the following condition: [figure omitted; refer to PDF]

Note that [figure omitted; refer to PDF] are strictly increasing functions; we have [figure omitted; refer to PDF]

Note that, for any initial condition [figure omitted; refer to PDF] , if parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are selected such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] can be controlled to be bounded, then [figure omitted; refer to PDF] holds. Thus, the condition [figure omitted; refer to PDF] can be guaranteed. Now, the tracking control problem of system (2) is now transformed to stabilize the transformed system (7).

Differentiating (7) with respect to time [figure omitted; refer to PDF] yields [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a positive constant. Then (8) can be rewritten as [figure omitted; refer to PDF] Moreover, we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is known nonlinear function and [figure omitted; refer to PDF] is a nonlinear function including unknown dynamics and disturbances.

Remark 6.

In general, [figure omitted; refer to PDF] is chosen as [figure omitted; refer to PDF] . So, we can calculate that [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

2.2. Fuzzy Logic Systems

The basic configuration of a fuzzy logic system consists of a fuzzifier, some fuzzy IF-THEN rules, a fuzzy inference engine, and a defuzzifier. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input vector [figure omitted; refer to PDF] to an output [figure omitted; refer to PDF] . The [figure omitted; refer to PDF] th fuzzy rule is written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are fuzzy sets and [figure omitted; refer to PDF] is the fuzzy singleton for the output in the [figure omitted; refer to PDF] th rule. By using the singleton fuzzifier, product inference, and the center-average defuzzifier, the output of the fuzzy system can be expressed as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the degree of membership of [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the number of fuzzy rules, [figure omitted; refer to PDF] is the adjustable parameter vector, and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the fuzzy basis function. It is assumed that fuzzy basis functions are selected so that there is always at least one active rule.

3. Main Results

Define the filtered error as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a positive constant such that the tracking error [figure omitted; refer to PDF] is bounded as long as [figure omitted; refer to PDF] is bounded.

Due to the fact that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are unknown, we need to use fuzzy logic system to approximate the nonlinear unknown functions. By applying the introduced fuzzy systems, approximation of function [figure omitted; refer to PDF] and [figure omitted; refer to PDF] can be expressed as follows: [figure omitted; refer to PDF] Optimal parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] can be defined such that [figure omitted; refer to PDF]

Define the parameter estimation errors and the fuzzy approximation errors as follows: [figure omitted; refer to PDF] We assume that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are positive constants.

The adaptive prescribed performance controller can be specified as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a small positive constant, [figure omitted; refer to PDF] is a designed positive constant, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is assumed to be known.

To generate the approximations [figure omitted; refer to PDF] and [figure omitted; refer to PDF] online, we choose the following adaptive laws: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are positive constants.

So, we have the following result.

Theorem 7.

Consider the controlled HPS (2) and the error transform (7). Suppose that Assumptions 1 and 2 are satisfied. Then controller (18) with the adaptive laws given by (21) can guarantee that all signals in the closed-loop system are bounded. Furthermore, The prescribed control performance (4) is preserved.

Proof.

Consider a Lyapunov function as [figure omitted; refer to PDF]

The time derivative of [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF]

Notice that [figure omitted; refer to PDF] . One can obtain [figure omitted; refer to PDF]

According to Assumption 2 and (20), we have [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . So, by using adaptation laws (21), we obtain [figure omitted; refer to PDF]

Therefore, [figure omitted; refer to PDF] is negative semidefinite and [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are bounded. So, [figure omitted; refer to PDF] . Then, according to the properties of function [figure omitted; refer to PDF] , we know that [figure omitted; refer to PDF] . Then, one can conclude that tracking control of system (2) with prescribed error performance (4) is achieved. This completes the proof.

Remark 8.

Compared with the results in [7-10], the unknown control gain is considered in this paper.

Remark 9.

In order to avoid the chatter in controller (18), we can modify [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a design time-varying parameter defined as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a positive constant.

Remark 10.

The control law (18) can effectively avoid the singularity problem, because even when [figure omitted; refer to PDF] is singular, this controller is well-defined.

In order to obtain a control law with its adaptive laws guaranteeing control objectives without the knowledge of the [figure omitted; refer to PDF] value (see Assumption 2) and the reconstruction error bounds [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in the controller, we will propose a new adaptive control scheme.

Theorem 11.

Consider the controlled HPS (2) and the error transform (7), and suppose that Assumptions 1 and 2 are satisfied. Consider the control law [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . And the adaptive laws are given as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the estimates of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively. Suppose [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and all signals of the overall closed-loop system are bounded and the tracking errors remain in a neighborhood of the origin within the prescribed performance bounds for all [figure omitted; refer to PDF] .

Proof.

Consider a Lyapunov function as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

The time derivative of [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF]

According to Assumption 2 and (29), we have [figure omitted; refer to PDF] So, by using inequality (33), we obtain [figure omitted; refer to PDF]

Using adaptive laws (30), we have [figure omitted; refer to PDF]

By following the same reasoning as in Theorem 7, we conclude that tracking control of system (2) with prescribed error performance (4) is achieved. This completes the proof.

4. Numerical Simulations

In this section, the numerical simulations are performed to verify and demonstrate the effectiveness of the proposed control scheme. Firstly, we employ sliding mode control scheme (see [5]) to control uncertain HPS (2). Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and the error dynamic system can be rewritten as follows: [figure omitted; refer to PDF] Due to the boundedness of chaotic phenomena, we assume [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is unknown positive constant. In order to eliminate the influence of [figure omitted; refer to PDF] , we still adopt the same method in this paper.

Define sliding surface: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a designed positive constant. So, the control law is designed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the estimate function of [figure omitted; refer to PDF] . In all the simulation process, the initial values of the chaotic system are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . We define seven Gaussian membership functions uniformly distributed on the interval [figure omitted; refer to PDF] . And we choose the initial values of parameters of the fuzzy systems as [figure omitted; refer to PDF] . Figure 3 shows the time response of the error state [figure omitted; refer to PDF] under the control method (38). From Figure 3, we know that the tracking error [figure omitted; refer to PDF] violates the prescribed error bounds and cannot achieve the good performances in the beginning stage.

Figure 3: Time response [figure omitted; refer to PDF] of error dynamic system (36) under the method of (38).

[figure omitted; refer to PDF]

Now, by using the present control scheme (18), we choose [figure omitted; refer to PDF] . The simulation results are shown in Figures 4 and 5. From Figures 4 and 5, we know that the present control method can guarantee that all the variables are bounded. Moreover, the error [figure omitted; refer to PDF] remains within the prescribed performance bounds for all time. All the aforementioned results clearly show that the present PPF-based control method (18) can obtain better regulation performance; that is, [figure omitted; refer to PDF] can be retained within the PPF bound and achieves faster convergence performance compared to method (38).

Figure 4: Time response [figure omitted; refer to PDF] of error dynamic system (36) under the present method of (18).

[figure omitted; refer to PDF]

Figure 5: Time response [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) of error dynamic system (36) under the present method of (18).

[figure omitted; refer to PDF]

The simulation results show that the proposed prescribed transient and steady-state performances are achieved. Thus, the numerical simulations verify theoretical analysis.

5. Conclusions

For a class of uncertain HPS with unknown control gain, the adaptive fuzzy feedback tracking control problem has been considered. By using prescribed performance functions, we transform the system into an equivalent one, and the fuzzy logic systems are used to identify the unknown nonlinear functions. It is sufficient to guarantee the boundedness of all the variables in the closed-loop system. Simulation results have shown the effectiveness of the proposed scheme.

Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (61403157), the Natural Science Foundation of Anhui Province (1508085 QA16), Anhui Province University Humanities and Social Science Research Base project (SK2015A158, SK2015A159), the Scientific Research Project of Huainan Normal University (2014xj07zd, 2015xj07zd), and the Natural Science Foundation for the Higher Education Institutions of Anhui Province of China (KJ2015A256).

Conflict of Interests

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

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