Output Feedback Tracking Controller for a Class of Nonlinear Systems

Abstract

In this research project, a linear, output feedback control strategy is developed for a nonlinear class of single-input, single-output (SISO) systems satisfying a linear growth function. Specifically, the proposed output feedback controller couples a linear observer structure with a corresponding linear control design in order to exponentially drive the output tracking error signal to within an arbitrarily small region about zero (i.e., global exponential uniformly ultimately bounded tracking). Simulation results of two representative systems are provided to illustrate the performance and mechanics of implementation

Introduction

Though sensorless control appears to be the pinnacle target within the theoretical development cycle of control design, the output feedback problem still offers the more viable alternative due to the increase in system robustness in exchange for the addition of a single output sensor. As evidenced by the abundance of literature, the output feedback problem still offers incentive for investigation and can be largely classified into two broad categories: i) the regulation problem and ii) the tracking problem.

Since regulation is considered to be a subset of tracking, the focus of the literature review is biased towards thhat of the output feedback tracking control problem. For example, Chen et al. [3] utilized a backstepping based control of uncertain nonlinear systems. In [3], the proposed controller utilizes nonlinear damping terms to ensure overall system stability. In [1], Besancon et al. developed an output feedback approach under the assumption of output constraining for a class of nonlinear systems in which a semi-global with respect to the output variable was achieved (Besancon et al. also illustrates in [1] global tracking with respect to the unmeasurable states). Marino et al. [8] addressed the development of a global adaptive output feedback controllers for single-input, single-output nonlinear systems whose states are linear with respect to the control input and linear with respect to an unknown parameter vector. Though growth conditions are not required in [8], the nonlinearities of the systems under consideration are restricted to be a function of the output signal only. Jiang et al. [6] utilized a nonlinear output feedback obserber/controller to globally asymptotically track a time-varying time signal for the nonlinear benchmark TORA (translational oscillator with a rotational actuator) problem. Canbolat et al. [2] proposed a near output feedback control strategy (i.e., the controller required dynamic measurements of the output state but did require knowledge of other system state initial conditions) that achieved semi-global uniformly ultimately bounded link position tracking for a class of robotic systems. The globally bounded output feedback controller of Oh et al. [10] utilized a variable structure control approach in order to drive the tracking error in an arbitrarily small region about zero. Lee et al. [7] utilized high gain observers for the estimation of joint velocities to design an adaptive output feedback controller for rigid robots to achieve link position tracking. In [5], a singularity-free, adaptive, output-feedback controller, which does not require measurements of the rotor flux or the stator current, was proposed to provide for global asymptotic rotor velocity and global exponential rotor flux tracking despite parametric uncertainty associated with the mechanical subsystem.

As seen from the literature above, a global, exponential, tracking result represents the ultimate target of the output feedback control design; however, this achievement usually comes with a price tag of implementation complexity. To this end, a linear observer/controller approach is investigated to achieve global, exponential uniformly ultimately bounded tracking of the output variable. The linear structure is considered in an effort to provide a less complex control methodology for a class of nonlinear system while maintaining an acceptable performance margin. Though originally developed for the output feedback stabilizing problem, the previous work of Qian et al. [11] serves as a sprind board for the proposed output feedback tracking compensator. Specifically, Qian et al. [11] developed a global stablizing controller for a class of nonlinear systems subjected to a linear growth condition in which a linear output dynamic compensator to exponentially regulate the system states. Thus, the linear observer/controller approach of Qian et al. [11] is extended to achieve global exponential bounded tracking of the output variable. In addition to utilizing only output feedback measurements the observer/controller design compensated for the "Lipschitz-like" behavior of the nonlinear state functions that includes a non-zero condition at the system origin (i.e., when all system states equal zero). In addition, the observer structure allows for the tracking error bound to be forced arbitrarily small through the utilization of high-gain observer feedback.

Simulation Results

Two systems were utilized in order to demonstrate the performance of the proposed observer and the output feedback controller. Specifically, the first system was originally posed in [11] in an effort to illustrate the difficulty of designing an output feedback stabilizing controller for a system that contains unknown parameters multiplied by non-measurable states; therefore, we will utilize this system, under the same assumptions and constraints, to illustrate the tracking capabilities of the proposed observer/output feedback controller. In addition, a mechanical system is also considered to provide more applications oriented simulation result.

Fig.1 Tracking Error for System # 1

Fig.2 Control Input for System # 1

Fig.3 Tracking Error for system # 2

Fig.4 Control Input for system # 2

Conclusion

In this research, a linear observer structure is coupled with a Lyapunov-based control design to achieve global, exponential, bounded output tracking for a nonlinear class of single-input, single-output systems satisfying a triangular type linear growth function. Specifically, the controller exploits a "Lipschitz-like" assumption on the nonlinear state functions in order to design the linear cascading observer/control design that allows the output tracking error signal to be exponentially driver into an arbitrarily small region about zero. To offer an attractive scheme to mechanical applications, further research effort will focus on extending the current linear observer/controller approach to achieve tracking in at least two state variables with only output feedback. Currently, the proposed observer/controller scheme promotes tracking in only the output variable while high-order states are "regulated" which would obviously generate poor position tracking as accurate velocity control is needed for precise positioning applications.

Conference Papers

For more details on this research, please refer the followingconference paper:

  • B. Xian, M. Feemster, and D. Dawson "Output Feedback Tracking Controller for a Class of Nonlinear Systems". (Submitted to the 2004 Conference on Control Applications)

E-MailRAMAL HomePage