There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

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Flatness and defect of non-linear systems: introductory theory and examples

A direct adaptive tracking control architecture is proposed and evaluated for a class of continuous-time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture uses a network of Gaussian radial basis functions to adaptively compensate for the plant nonlinearities. Under mild assumptions about the degree of smoothness exhibit by the nonlinear functions, the algorithm is proven to be globally stable, with tracking errors converging to a neighborhood of zero. A constructive procedure is detailed, which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy. A stable weight adjustment mechanism is determined using Lyapunov theory. The network construction and performance of the resulting controller are illustrated through simulations with example systems. >

Gaussian networks for direct adaptive control

Barrier Lyapunov Functions for the control of output-constrained nonlinear systems

An observer for nonlinear systems is constructed under rather general technical assumptions (the fact that some functions are globally Lipschitz). This observer works either for autonomous systems or for nonlinear systems that are observable for any input. A tentative application to biological systems is described. >

A simple observer for nonlinear systems applications to bioreactors

STATE FEEDBACK. Feedback Linearization. Adaptive Feedback Linearization. Output Tracking. OUTPUT FEEDBACK. Adaptive Observers. Stabilization and Exponential Tracking. Robust Regulation and Adaptive Tracking.

Nonlinear control design: geometric, adaptive and robust

A nonlinear adaptive state feedback input-output linearizing control is designed for a fifth-order model of an induction motor which includes both electrical and mechanical dynamics under the assumptions of linear magnetic circuits. The control algorithm contains a nonlinear identification scheme which asymptotically tracks the true values of the load torque and rotor resistance which are assumed to be constant but unknown. Once those parameters are identified, the two control goals of regulating rotor speed and rotor flux amplitude are decoupled, so that power efficiency can be improved without affecting speed regulation. Full state measurements are required. >

Adaptive input-output linearizing control of induction motors

For pt.I, see ibid., p.17-32 (1993). The problem of designing global output-feedback robust stabilizing controls for a class of single-input single-output minimum-phase uncertain nonlinear systems with known and constant relative degree is addressed. They are assumed to be linear with respect to the input and nonlinear with respect to an unknown constant parameter vector. The nonlinearities depend on the output only. The nonlinearities may be uncertain and are only required to be bounded by known smooth functions. The order of the robust compensator is equal to the relative degree minus one and is static when the relative degree is one. A self-tuning version of the robust control capable of achieving set point regulation is developed in which the control gains are tuned by an output-feedback adaptive algorithm. When the parameter vector enters linearly, the self-tuning regulator does not require the knowledge of parameter bounds and guarantees set point regulation for the same class of systems considered in Part I. >

Global adaptive output-feedback control of nonlinear systems. II. Nonlinear parameterization

Motivated by technological advances in power electronics and signal processing, and by the interest in using direct drives for robot manipulators, we investigate the control problem of high-performance drives for switched reluctance motors (SRM's). SRM's are quite simple, low cost, and reliable motors as compared to the widely used dc motors. However, the SRM presents a coupled nonlinear multivariable control structure which calls for complex nonlinear control design in order to achieve high dynamic performances. We first develop a detailed nonlinear model which matches experimental data and establish an electronic commutation strategy. Then, on the basis of recent nonlinear control techniques, we design a state feedback control algorithm which compensates for all the nonlinearities and decouples the effect of stator phase currents in the torque production. The position dependent logic of the electronic commutator assigns control authority to one phase, which controls the motion, while the remaining phase currents are forced to decay to zero. Simulations for a direct drive, single link manipulator with the SRM are reported, which show the control performance of the algorithm we propose in nominal conditions and test its robustness versus the most critical parameter uncertainties of payload mass and stator resistance.

Riccardo Marino

Papers

Nonlinear control design: geometric, adaptive and robust

Adaptive input-output linearizing control of induction motors

Global adaptive output-feedback control of nonlinear systems. II. Nonlinear parameterization

Feedback linearizing control of switched reluctance motors

Nonlinear control design