Brief paperRobust adaptive control of a class of nonlinear systems with unknown dead-zone☆
Introduction
Generally, each industrial motion control system has the structure of a dynamical system, usually of the Lagrangian form, preceded by some nonsmooth nonlinearities in the actuator, either dead-zone, backlash, saturation, etc. Furthermore, these nonsmooth nonlineárities in such actuators (e.g. hydraulic servo valves, electric servomotors) are often unknown and time-variant. For example, a common source of nonsmooth nonlinearities arises from friction, which vary with temperature, speed and wear, or even differ significantly between mass-produced components. Thus, the study of nonsmooth nonlinearities involved has been of great interest to control researchers for a long time. The control of such systems needs to be robust, in order to compensate the time-variant effects of these nonlinearities. The problems are particularly important when the expected accuracy of the motion system is high.
Dead-zone, which can severely limit system performances, is one of the most important nonsmooth nonlinearities arisen in actuators, such as servo valves and DC servo motors. In most practical motion systems, the dead-zone parameters are poorly known, and robust control techniques are being sought. Proportional-derivative (PD) controllers have been observed to result in limit cycles. Due to the nonanalytic nature of dead-zone in actuators and the fact that the exact parameters (e.g. width of dead-zone) are unknown, systems with dead-zones present a challenge for the control design engineers. An immediate method for the control of dead-zone is to construct an adaptive dead-zone inverse. This approach was pioneered by Tao and Kokotovic 1994, Tao and Kokotovic 1995. Continuous- and discrete-time adaptive dead-zone inverses for linear systems with unmeasurable dead-zone outputs were built by Tao and Kokotovic 1994, Tao and Kokotovic 1995, respectively. Simulations indicated that the tracking performance is greatly improved by using dead-zone inverse. The work by Cho and Bai (1998) continued the above research and a perfect asymptotical adaptive cancellation of an unknown dead-zone was achieved analytically to systems in which the output of a dead-zone is measurable. To simplify the controller design, Kim, Park, Lee, and Chong (1994) proposed a two-layered fuzzy logic controller for the control of systems with dead-zones. In which, a fuzzy precompensator and a normal PD type fuzzy controller were introduced to control systems with dead-zones. Most recently, Lewis, Tim, Wang, and Li (1999) proposed a fuzzy precompensator in nonlinear industrial motion system and Selmic and Lewis (2000) employed neural networks to construct a dead-zone precompensator. Such approaches promise to improve the tracking performance of motion system in presence of unknown dead-zones.
A common feature for the above mentioned approaches is the construction of an inverse dead-zone nonlinearity to minimize the effects of dead-zone. However, different approaches may also be pursued. Based only on the intuitive concept and piece-wise description of dead-zones (Section 2), in this paper, a new approach for adaptive control of linear or nonlinear systems with dead-zones is introduced without constructing the inverse of the dead-zone. The new control law ensures a global stability of the entire adaptive system and asymptotical tracking (Section 4). Computer simulations were carried out to illustrate the effectiveness of the approach (Section 5).
Section snippets
Dead-zone model and its intuitive properties
The dead-zone with input v(t) and output w(t), as shown in Fig. 1, is described by
As stated by Tao and Kokotovic (1994), this dead-zone model is a static simplification of diverse physical phenomena with negligible fast dynamics. Eq. (1) is a good model for a hydraulic servo valve or a servo motor.
The key features of the dead-zone in the control problems investigated in this paper are
- (A1)
The dead-zone output w(t) is not
Control problem statement
In this paper, the system to be controlled consists of nonlinear plants preceded by actuators with dead-zone. That is, the dead-zone is present in series as the input of the nonlinear plant.
A dead-zone nonlinearity can be denoted as an operatorwith v(t) as input and w(t) as output. The operator D(v(t)) has been described in the previous section. The nonlinear dynamic system preceded by the above dead-zone is described as
We have the
Adaptive controller design
In this section, we shall propose an adaptive controller for plants of the form in (6), preceded by a dead-zone described in (1), which will guarantee global system stability and yields the system output tracking to a desired trajectory within a desired accuracy.
Using expression (2), system (6) becomesIn which, the state variables of the control problem become linear to the input signal v(t). It is very important to note that d(v(t)) is
Simulation studies
In this section, we will illustrate the above method on a nonlinear systems described as (Zhang & Feng, 1997)where w(t) is an output of a dead-zone. The parameters to be simulated are b=1 and a1=a2=a3=1. Without control, i.e., w(t)=0, as stated by Zhang and Feng (1997), the system (19) is unstable (i.e. the linearization of the system at the origin is unstable).This also has been verified by simulation. The control objective is to let the system
Conclusion
In practical control systems, dead-zones with unknown parameters in physical components may severely limit the performance of control. In this paper, a robust adaptive control architecture is proposed for a class of continuous-time nonlinear dynamic systems preceded by a dead-zone. By using a new description of a dead-zone and by showing the properties of this dead-zone model intuitively, this robust adaptive control scheme is developed without constructing a dead-zone inverse. The new control
Acknowledgements
The authors wish to acknowledge the support of the Natural Science and Engineering Research Council of Canada, the Institute for Robotic and Intelligent Systems (IRIS), and the Chinese National Scientific Foundation (Project No. 59885002). The first author would also like to acknowledge the financial support of the Canadian International Development (CIDA) Bureau.
X.-S. Wang received his B.S. and M.S. degrees from Zhejiang University, Hangzhou, China, in 1988 and 1991, respectively, and the Ph.D. degree from Southeast University, Nanjing, China, in 2000, all in mechanical engineering.
Dr. Wang was a professor-assistant, lecturer, associate professor, all with the Department of Mechanical Engineering at Southeast University in 1991, 1993, 1998, respectively. Currently, he is a professor in the same department. From July 2000 to December 2000 and September
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Cited by (0)
X.-S. Wang received his B.S. and M.S. degrees from Zhejiang University, Hangzhou, China, in 1988 and 1991, respectively, and the Ph.D. degree from Southeast University, Nanjing, China, in 2000, all in mechanical engineering.
Dr. Wang was a professor-assistant, lecturer, associate professor, all with the Department of Mechanical Engineering at Southeast University in 1991, 1993, 1998, respectively. Currently, he is a professor in the same department. From July 2000 to December 2000 and September 2001 to February 2002, he was a visiting scientist in the Department of Mechanical and Industrial Engineering at Concordia University, Montreal, Canada.
His current research interests include control theory with application in precision manufacturing system design, engineering measurement and instrument design, robotics and mechatronic system design. In these areas he has finished two NSF projects and one 863 project as well as several industrial projects in China.
Chun-Yi Su received his B.E. degree in control engineering from Xian University of Technology in 1982, his M.S. and Ph.D. degrees in control engineering from South China University of Technology, China, in 1987 and 1990, respectively. His Ph.D. study was jointly directed at Hong Kong Polytechnic (now Hong Kong Polytechnic University), Hong Kong.
After a long stint at the University of Victoria, he joined Concordia University in 1998, where he is currently an Associate Professor and holds the Concordia Research Chair (Tier II) in intelligent control of non-smooth dynamic systems. He is Guest Professor at the South China University of Technology, P. R. China, and at the Nanjing Normal University, P. R. China. He has also held several short-time visiting positions in Japan, Singapore, China and New Zealand.
Dr. Su's main research interests are in nonlinear control theory and robotics. He is the author or co-author of over 100 publications, which have appeared in journals, as book chapters and in conference proceedings. Aside from his teaching and research duties, Dr. Su is also heavily involved in other activities such as reviewing papers for a number of journals and conferences on a regular basis. He was the General Co-Chair of the Fourth International Conference on Control and Automation (ICCA’03). Currently, he is the Chair for Invited Sessions for the 2004 IEEE International Symposium on Intelligent Control.
Henry Hong received his Ph.D. degree in mechanical engineering in 1995, from Concordia University, Montreal, Quebec. He is presently Assistant Professor in the Department of Mechanical and Industrial Engineering at Concordia. His present research includes solenoid and voice-coil actuated alternative fuel diesel injectors and variable intake/exhaust valves. The electromagnetic solenoid has a permanent magnet as its core. Feedback is used for infinite variable injector and valve position control. Control schemes are used for the compensation of magnetic hysteresis. Dr. Hong has been the Faculty Advisor to Concordia University's Collegiate Chapter of the Society of Automotive Engineers since 1996. He was Concordia's Faculty Advisor to the FutureCar (1996-1999) and FutureTruck (1999-2001) student competition projects. Dr. Hong received the SAE Faculty Advisors Award in 2000 and the Ralph R. Teetor Educational Award in 2001, from the Society of Automotive Engineers.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associated Editor Jan Willem Polderman under the direction of Editor Robert R. Bitmead.