Tracking control for underactuated non-minimum phase multibody systems

Abstract

We consider tracking control for multibody systems which are modeled using holonomic and non-holonomic constraints. Furthermore, the systems may be underactuated and contain kinematic loops and are thus described by a set of differential-algebraic equations that cannot be reformulated as ordinary differential equations in general. We propose a control strategy which combines a feedforward controller based on the servo-constraints approach with a feedback controller based on a recent funnel control design. As an important tool for both approaches, we present a new procedure to derive the internal dynamics of a multibody system. Furthermore, we present a feasible set of coordinates for the internal dynamics avoiding the effort involved with the computation of the Byrnes–Isidori form. The control design is demonstrated by a simulation for a nonlinear non-minimum phase multi-input, multi-output robotic manipulator with kinematic loop.

Output feedback control with prescribed performance via funnel pre-compensator

“…In particular, output tracking with unknown output derivatives is possible for the class of linear minimum phase systems (4); and moreover, for a class of linear non-minimum phase systems (single-input, single output systems as well as multi-input, multioutput systems). Since the investigations in the recent works [9] and [6] show applicability of existing control techniques to nonlinear non-minimum phase systems we are confident that an integration of the funnel pre-compensator into this particular context will also be fruitful.…”

“…Remark 3. 6 The somewhat arcane condition T ∈ T rm,q σ,1 reflects the intuition that in order to have the conjunction of the system with the funnel pre-compensator being a minimum phase system, we must be able to conclude from the available information (only the output y) that the internal dynamics stay bounded. Note that this, however, does not mean that the operator T does not act on the output signal's derivatives but on y only, see Example 5.2.…”

Input-constrained funnel control of nonlinear systems

“…As soon as the saturation becomes inactive again, the performance funnel recovers its desired shape exponentially fast. The concept of funnel control was developed in the seminal work [20] (see also the recent survey in [6]) and proved advantageous in a variety of applications such as control of industrial servo-systems [15] and underactuated multibody systems [5,8], control of electrical circuits [11,29], control of peak inspiratory pressure [26], adaptive cruise control [10] and even the control of infinitedimensional systems such as a boundary controlled heat equation [27], a moving water tank [9] and defibrillation processes of the human heart [3].…”

Funnel control of linear systems under output measurement losses

“…The concept of funnel control goes back to the seminal work [14], see also the survey in [15]. The funnel controller proved to be the appropriate tool for tracking problems in various applications such as control of industrial servosystems [16] and underactuated multibody systems [17], [18],…”