Nonlinear and Adaptive Control

The following sections are included:

• PREFACE

• CONTENTS

We study the extension of the class of linear time-invariant plants that may be transformed into SPR systems introducing an observer. It is shown that for open-loop stable systems a cascaded observer achieves the result. For open-loop unstable systems an observer-based feedback is required to succeed. In general any stabilizable and observable system may be transformed into an SPR system using cascade or cascade and feedback controllers. This overcomes the old conditions of minimum phase and relative degree one. The result is illustrated with some examples.

The stabilization problem for a class of nonlinear feedforward systems is solved using bounded control. It is shown that when the lower subsystem of the cascade is input-to-state stable and the upper subsystem not exponentially unstable, global asymptotic stability can be achieved via a simple static feedback having bounded amplitude that requires knowledge of the “upper” part of the state only. This is made possible by invoking the bounded real lemma and a generalization of the small gain theorem. Thus, stabilization is achieved with typical saturation functions, saturations of constant sign, or quantized control. Moreover, the problem of asymptotic stabilization of a stable linear system with bounded outputs is solved by means of dynamic feedback. Finally, a new class of stabilizing control laws for a chain of integrators with input saturation is proposed. Some robustness issues are also addressed and the theory is illustrated with examples on the stabilization of physical systems.

The problem of global output feedback stabilization for a class of nonlinear systems whose zero dynamics are not necessarily stable is addressed in this chapter. It is shown that, using a novel observer design tool together with standard backstepping and small-gain techniques, it is possible to design a stabilizing output feedback controller which ensures robustness with respect to dynamic uncertainties. The proposed stabilization method generalizes existing tools in several directions. The method is illustrated by means of an academic example and is applied to the stabilization of the benchmark translational oscillator/rotational actuator (TORA) system with measurement of the rotational position.

This chapter 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 chapter the method is extended, transforming the matching conditions obtained in the IDA-PBC method into a set of quasi-linear and linear PDEs.

A method for the generation of attractive and neutrally stable limit cycles for nonlinear systems is presented. It consists in designing an output that, when regulated through a suitable feedback, forces the existence of a limit cycle or neutral oscillations in the zero dynamics. Conditions are then given to ensure that those characteristics of the zero dynamics translate to the whole system. A particular focus is placed on the generation of neutrally stable oscillations through that method, because it is not always easy to build an output that results in the existence of a limit-cycle in the zero dynamics. A special case where such a difficulty arises is given in the analysis of oscillations generation around the upper vertical for the Pendubot. The regulation of the output results in neutrally stable oscillations, and we present a method for ensuring that those oscillations converge towards the desired ones.

In this chapter we present two control algorithms to stabilize a small helicopter with four rotors. First, we propose a simple nonlinear controller based on Lyapunov analysis for a Planar Vertical Take Off and Landing aircraft (PVTOL). It is proved that the proposed control scheme is globally asymptotically stable. Second, we present a discrete-time controller based on state feedback and the prediction of the state. It is shown that this controller stabilizes possibly unstable continuous-time delay systems. The stability is shown to be robust with respect to uncertainties in the knowledge on the plant parameters, the system delay and the sampling period. Both results have been experimentally tested in a laboratory prototype. Real-time experiments show a satisfactory performance of the proposed controls schemes.

The problem of global stabilization of the attitude dynamics for a magnetically actuated spacecraft is considered, and the cases of inertial pointing and Earth pointing are discussed. For the inertial pointing case, an almost global solution to the problem is obtained by means of static attitude and rate feedback. A local solution based on dynamic attitude feedback is also presented. For the Earth pointing case, almost global stabilization is achieved by means of an adaptive PD-like state feedback control law. Simulation results are presented to illustrate the performance of the proposed approaches.

This chapter addresses the control of the microbial specific growth rate in fed-batch bioreactors. Practical constraints concerning the available signals and information on the process are dealt with. Then, for a large class of bioreactions, namely pure cultures with one limiting substrate and oxygen in excess, the control problem is regarded as a problem of coordinating control involving the invariance and attractivity of nontrivial sets in state space. Both monotonous and non-monotonous kinetic functions are dealt with. The dissolved oxygen signal can give valuable information about the cell culture, when the oxygen supply is in balance with the consumption. This is utilized for a probing feeding strategy, treated in Chapter 9.

This chapter addresses the control of glucose feeding in fed-batch reactors. The feeding strategy is based on a pulse technique, which does not require a priori knowledge of the process. A feedback algorithm optimizes the feed rate with respect to the constraints of aerobic conditions and overfeeding. Stability and performance analysis are performed on a simplified model of the reactor, using standard LMI optimization routines. Tuning guidelines that help the user for the design are also derived.

The wave reflection phenomenon that appears when actuator and plant are connected through long cables is studied in this chapter. In several applications, the perturbation induced by the presence of these reflected waves is non-negligible and seriously degrades the performance of the control and the operativity of the system. Standard compensation schemes are based on matching impedances at specific frequencies and are realized with the addition of linear RLC filters. Impedance matching is clearly ineffective if there is no single dominant frequency in the system and/or the plant is highly uncertain. In this work a novel compensator design framework applicable for the latter scenario is proposed. In contrast with the standard schemes the compensators are active and require for their implementation regulated sources placed either on actuator or plant side. A port-interconnection viewpoint is adopted and the cable is modeled via the transmission line equations in their scattering representation. Under the assumptions of plant linearity and piecewise approximation of the signals – which is reasonable if the line propagation delay is small – a family of current decoupling compensators, that requires only knowledge of the line parameters, and ensures stability and asymptotic tracking for all (unknown) plants with passive impedance, is proposed. An adaptive version that estimates the line characteristic impedance is also presented. Some simulation results in a benchmark example of voltage overshoot suppression in AC drives are shown.