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Publication# Analysis of Cascade Structure with Predictive Control and Feedback Linearization

Résumé

Abstract Predictive control and feedback linearization are two popular approaches for the control of nonlinear systems. In this paper, a cascade combination of the two metho ds is proposed, where control based on input- output feedback linearization forms the inner lo op, and predictive control the outer lo op. With this scheme, predictive control is applied to the internal dynamics instead of the system dynamics. The proposed cascade scheme is advantageous for unstable minimum-phase systems and for unstable nonminimum-phase systems with slower internal dynamics compared to the system dynamics. A stability analysis of the global scheme is provided based on singular perturbation theory. The approach is illustrated via the simulation of an inverted pendulum on a cart system.

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Predictive Control of Unstable Nonminimum-phase Systems K. Guemghar, B. Srinivasan, Ph. Mullhaupt, D. Bonvin ´ Institut d’Automatique, Ecole Polytechnique F´d´rale de Lausanne, ee CH-1015 Lausanne, Switzerland. Predictive control is a very eﬀective approach for tackling problems with constraints and nonlinear dynamics, especially when the analytical computation of the control law is diﬃcult. Standard predictive control involves predicting the system behavior over a prediction horizon and calculating the input that minimizes a criterion expressing the system behavior in the future. Only the ﬁrst part of the computed input is applied to the system, and this procedure is repeated with the advent of each new measurement. This methodology is widely used in the process industry where system dynamics are sufﬁciently slow to permit its implementation. In contrast, applications of predictive control to fast unstable dynamic systems are rather limited. Apart from computational considerations, a fundamental limitation arises from the accuracy of that prediction, which can be quite poor due to accumulation of numerical errors if the prediction horizon is large. In this paper, an upper bound on the prediction horizon based on the location of the unstable pole(s) of the linearized system will be provided. The problem becomes more acute when nonminimum-phase systems are considered. Nonminimum phase implies that the system starts in a direction opposite to its reference (inverse response). To control such systems, it is reasonable to predict the maneuvers, thus making predictive control a natural strategy. However, a large prediction horizon is required since it is necessary to look beyond the inverse response. In this context, a lower bound on the prediction horizon based on the location of the unstable zero(s) of the linearized system will be provided. The two bounds mentioned above may lead to the situation where there exist no value of the prediction horizon that can stabilize a given unstable nonmiminum-phase system. For such a case, a combination of tools from diﬀerential geometry and predictive control is proposed in this paper. The envisaged procedure has a cascade structure and is outlined below: 1. Using the input-output linearization technique, the nonlinear system is transformed into a linear subsystem and internal dynamics. However, since the original nonlinear system is nonminimum-phase, the internal dynamics are unstable. 2. The linear subsystem is made arbitrarily fast by using a stabilizing high-gain linear feedback (inner-loop). 3. A predictive control scheme is then used to stabilize the slow internal dynamics (outerloop) by manipulating the reference of the inner loop. The stability of the proposed procedure is analyzed using a singular perturbation approach. The results will be illustrated in simulation on a system consisting of an inverted pendulum on a cart. 1

2002The control of compliant robots is, due to their often nonlinear and complex dynamics, inherently difficult. The vision of morphological computation proposes to view these aspects not only as problems, but rather also as parts of the solution. Non-rigid body parts are not seen anymore as imperfect realizations of rigid body parts, but rather as potential computational resources. The applicability of this vision has already been demonstrated for a variety of complex robot control problems. Nevertheless, a theoretical basis for understanding the capabilities and limitations of morphological computation has been missing so far. We present a model for morphological computation with compliant bodies, where a precise mathematical characterization of the potential computational contribution of a complex physical body is feasible. The theory suggests that complexity and nonlinearity, typically unwanted properties of robots, are desired features in order to provide computational power. We demonstrate that simple generic models of physical bodies, based on mass-spring systems, can be used to implement complex nonlinear operators. By adding a simple readout (which is static and linear) to the morphology, such devices are able to emulate complex mappings of input to output streams in continuous time. Hence, by outsourcing parts of the computation to the physical body, the difficult problem of learning to control a complex body, could be reduced to a simple and perspicuous learning task, which can not get stuck in local minima of an error function.

2011Dominique Bonvin, Kahina Guemghar

Abstract The problem of controlling nonlinear nonminimum-phase systems is considered, where standard input-output feedback linearization leads to unstable internal dynamics. This problem is handled here by using the observability normal form in conjunction with input-output linearization. The system is feedback linearized upon neglecting a part of the system dynamics, with the neglected part being considered as a perturbation. A linear controller is designed to accommodate the perturbation resulting from the approximation. Stability analysis is provided based on the vanishing perturbation theory.

2003