Matching, linear systems, and the ball and beam

Andreev, F. V.; Auckly, Dave; Gosavi, Shekhar V.; Kapitanski, Lev; Kelkar, Atul G.; White, Warren J.

doi:10.1016/s0005-1098(02)00145-0

Cited by 44 publications

(25 citation statements)

References 14 publications

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“…Remark 4: Writing out (6) and using (9) it is easy to show that the control laws defined in (6) and (13) are the same. Notice that the control law is a state feedback law, depending only on q andq.…”

Section: Propositionmentioning

confidence: 99%

Matching in the Method of Controlled Lagrangians and IDA-Passivity Based Control

Blankenstein

Schaft

2005

Nonlinear and Adaptive Control

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This paper reviews the method of controlled Lagrangians and the interconnection and damping assignment passivity based control (IDA-PBC) method. Both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an Euler-Lagrange system, respectively Hamiltonian system, by searching for a stabilizing structure preserving feedback law. The conditions under which two Euler-Lagrange or Hamiltonian systems are equivalent under feedback are called the matching conditions (consisting of a set of nonlinear PDEs).Both methods are applied to the general class of underactuated mechanical systems and it is shown that the IDA-PBC method contains the controlled Lagrangians method as a special case by choosing an appropriate closed-loop interconnection structure. Moreover, explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms.The λ-method as introduced in recent papers for the controlled Lagrangians method transforms the matching conditions into a set of linear PDEs. In this paper the method is extended, transforming the matching conditions obtained in the IDA-PBC method into a set of quasi-linear and linear PDEs.

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Section: Propositionmentioning

confidence: 99%

Matching in the Method of Controlled Lagrangians and IDA-Passivity Based Control

Blankenstein

Schaft

2005

Nonlinear and Adaptive Control

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“…They are frequently used to realize experimental models, validate the efficiency of emerging control techniques and verify their implementation. The most common robotic benchmarks are Acrobot [7], Pendubot [8], the Furuta Pendulum [9], the inverted pendulum [10], the Reaction Wheel Pendulum [11], the bicycle [12], the VTOL aircraft [13], the Beam-and-Ball system [14] and TORA [15].…”

Section: Introductionmentioning

confidence: 99%

The Inverted Pendulum Benchmark in Nonlinear Control Theory: A Survey

Boubaker

2013

International Journal of Advanced Robotic Systems

125

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For at least fifty years, the inverted pendulum has been the most popular benchmark, among others, in nonlinear control theory. The fundamental focus of this work is to enhance the wealth of this robotic benchmark and provide an overall picture of historical and current trend developments in nonlinear control theory, based on its simple structure and its rich nonlinear model. In this review, we will try to explain the high popularity of such a robotic benchmark, which is frequently used to realize experimental models, validate the efficiency of emerging control techniques and verify their implementation. We also attempt to provide details on how many standard techniques in control theory fail when tested on such a benchmark. More than 100 references in the open literature, dating back to 1960, are compiled to provide a survey of emerging ideas and challenging problems in nonlinear control theory accomplished and verified using this robotic system. Possible future trends that we can envision based on the review of this area are also presented.

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“…Sliding mode controller can overcome the problem associated with singular states [10]. The λ -method matching control law for the ball and beam was discussed by [1]. An embedded single chip microprocessor is fabricated with backstepping control algorithm for a ball and beam system with magnetic suspension actuators [11].…”

Section: Introductionmentioning

confidence: 99%

Synchronization of ball and beam systems with neural compensation

2010

Int. J. Control Autom. Syst.

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Ball and beam system is one of the most popular and important laboratory models for teaching control system. It is a big challenge to synchronize ball and beam systems. There are two problems for ball and beam synchronized control: 1) many laboratories use simple controllers such as PD control, and theory analysis is based on linear models, 2) nonlinear controllers for ball and beam system have good theory results, but they are seldom used in real applications. In this paper we first use PD control with nonlinear exact compensation for the cross-coupling synchronization. Then a RBF neural network is applied to approximate the nonlinear compensator. The synchronization control can be in parallel and serial forms. The stability of the synchronization is discussed. Real experiments are applied to test our theory results.

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Matching, linear systems, and the ball and beam

Cited by 44 publications

References 14 publications

Matching in the Method of Controlled Lagrangians and IDA-Passivity Based Control

Matching in the Method of Controlled Lagrangians and IDA-Passivity Based Control

The Inverted Pendulum Benchmark in Nonlinear Control Theory: A Survey

Synchronization of ball and beam systems with neural compensation

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