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This paper presents a generalization of the Nyquist stability criterion to include general multivariable linear stationary systems subject to linear static and dynamic feedback. At the same time, a unifying proof is given for all known versions of the Nyquist criterion for finite dimensional systems.
A description is given of what are considered to be the natural multivariable analogs of concepts from classical control theory. A satisfactory generalization of the Nyquist criterion is described, and a clear analog of the asymptotic properties of the root locus is obtained in the 'multiparameter' case. An example is given, however, which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains. It is noted that this calls for a new ingredient, a compactification of the space of gains, and that what is the most interesting new feature in this circle of ideas comes in the area of pole placement.
Nonlinear feedback system stability using Nyquist theory as applied to Popov criterion
Development of a stability criterion of the Nyquist type from Michailov's criterion for a large class of distributed parameter systems, in particular, for a large class of transmission line systems with feedback. Following a generalization of Michailov's criterion, simplifying assumptions, usually valid in practice, are shown to yield a simplified test for determining whether 'encirclement-counting' constitutes a valid stability test. The results are reformulated for an open-loop analysis. Three examples show various aspects of the theoretical analysis.
A new methodology for multirate sampled-data control design based on a new generalized control law structure, two new parameter-optimization-based control law synthesis methods, and a new singular-value-based robustness analysis method are described. The control law structure can represent multirate sampled-data control laws of arbitrary structure and dynamic order, with arbitrarily prescribed sampling rates for all sensors and update rates for all processor states and actuators. The two control law synthesis methods employ numerical optimization to determine values for the control law parameters. The robustness analysis method is based on the multivariable Nyquist criterion applied to the loop transfer function for the sampling period equal to the period of repetition of the system's complete sampling/update schedule. The complete methodology is demonstrated by application to the design of a combination yaw damper and modal suppression system for a commercial aircraft.
Thirty years of control systems research has produced an enormous body of theoretical results in feedback synthesis. Yet such results see relatively little practical application, and there remains an unsettling gap between classical single-loop techniques (Nyquist, Bode, root locus, pole placement) and modern multivariable approaches (LQG and H infinity theory). Large scale, complex systems, such as high performance aircraft and flexible space structures, now demand efficient, reliable design of multivariable feedback controllers which optimally tradeoff performance against modeling accuracy, bandwidth, sensor noise, actuator power, and control law complexity. A methodology is described which encompasses numerous practical design constraints within a single unified formulation. The approach, which is based upon coupled systems or modified Riccati and Lyapunov equations, encompasses time-domain linear-quadratic-Gaussian theory and frequency-domain H theory, as well as classical objectives such as gain and phase margin via the Nyquist circle criterion. In addition, this approach encompasses the optimal projection approach to reduced-order controller design. The current status of the overall theory will be reviewed including both continuous-time and discrete-time (sampled-data) formulations.