Time‐varying linear controllers for exponential tracking of non‐holonomic systems in chained form

Abstract

SUMMARYIn this paper, the authors address the tracking problem for non-holonomic systems in chained form with target signals that may exponentially decay to zero. By introducing a time-varying co-ordinate transformation and using the cascade-design approach, smooth time-varying controllers are constructed, which render the tracking-error dynamics globally K-exponentially stable. The result shows that the popular condition of persistent excitation or not converging to zero for the reference signals is not necessary even for the globally K-exponential tracking of the chained-form system. The effectiveness of the proposed controller is validated by simulation of two benchmark mechanical systems under non-holonomic constraints.

Robust formation tracking control of mobile robots via one-to-one time-varying communication

“…The main condition is that both the control gains and the reference angular velocity are persistently exciting. It must be recognised, however, that one of the interesting contributions of Cao and Tian (2007) and Tian and Cao (2007) consists of relaxing the persistency of excitation condition on the angular reference velocity to the case of functions converging exponentially to zero; then, to maintain the necessary excitation in the controller, the authors use an exponentially increasing gain on the reference.…”

“…We employ and extend the cascades-based control design method originally proposed in Panteley et al (1998) and followed by other authors, including Cao and Tian (2007), Tian and Cao (2007), Lefeber (2000), Sadowska et al (2011), and some of the references therein. Our main results ensure uniform global exponential stability in closed loop; in particular, one has Each controller, reminiscent of that proposed in Panteley et al (1998), is linear time varying and local to each robot; it uses information of one (leader) vehicle only.…”

“…(2007) and Tian and Cao (2007), the persistency of excitation on the reference angular velocity is relaxed to allow for exponentially decaying functions by multiplying the given exponentially decaying reference trajectory by a growing exponential and using this new reference, instead, in the control law. Proof.…”

“…Each local controller is linear time varying as we assume that position measurements may be lost on intervals of time. From an analytical viewpoint, our controllers are inspired by the cascades-based persistently exciting control method originally introduced in Panteley, Lefeber, Loría and Nijmeijer (1998) and subsequently applied and extended by other authors, including Cao and Tian (2007), Lefeber (2000), Sadowska et al (2011), and Tian and Cao (2007). The significance of the proof method relies on avoiding the use of graph theory, eigenvalue analysis, and other tools difficult to extend to the realm of nonlinear systems.…”

Robust Linear Control of Nonlinear Flat Systems

“…An excellent book, dealing with some appropriate control techniques for this class of systems, is that of Dixon et al Dixon et al (2001). A useful approach to control non-holonomic mechanical systems is based on linear time-varying control schemes (see Pomet (1992); Tian & Cao (2007)). …”