Nyquist stability criterion

In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer Felix Strecker at Siemens in 1930 and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, is a graphical technique for determining the stability of a dynamical system. Because it only looks at the Nyquist plot of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system (although the number of each type of right-half-plane singularities must be known). As a result, it can be applied to systems defined by non-rational functions, such as systems with delays. In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. In addition, there is a natural general

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A new method for robust ﬁxed-order H∞ controller design by convex optimization for multivariable systems is investigated. Linear Time-Invariant Multi-Input Multi- Output (LTI-MIMO) systems represented by a set of complex values in the frequency domain are considered. It is shown that the Generalized Nyquist Stability criterion can be approximated by a set of convex constraints with respect to the parameters of a multivariable linearly parameterized controller in the Nyquist diagram. The diagonal elements of the controller are tuned to satisfy the desired performances, while simultaneously, the oﬀ-diagonal elements are designed to decouple the system. Multimodel uncertainty can be directly considered in the proposed approach by increasing the number of constraints. A simulation example illustrates the eﬀectiveness of the proposed approach. by a simulation example on an unstable system.

Elsevier2010

H-infinity Controller Design for Spectral MIMO Models by Convex Optimization

Gorka Galdos Sanz de Galdeano, Alireza Karimi, Roland Longchamp

A recent method for robust fixed-order H-infinty controller design by convex optimization proposed in [1] is investigated in this paper for the tuning of multivariable controllers. Linear Time-Invariant Multi-Input Multi-Output (LTI-MIMO) systems represented by a finite set of complex values in the frequency domain are considered. It is shown that the Generalized Nyquist Stability criterion can be approximated by a set of convex constraints with respect to the parameters of a multivariable linearly parameterized controller in the Nyquist diagram. Simultaneously, diagonal elements of the controller are tuned to satisfy the desired performances, while the off-diagonal elements decouple the system. Multimodel uncertainty can be directly considered in the proposed approach by increasing the number of constraints. An application example illustrates the effectiveness of the proposed approach.

2009

Fixed-order H-infinity Controller Design for Nonparametric Models by Convex Optimization

Gorka Galdos Sanz de Galdeano, Alireza Karimi

A new approach for robust fixed-order H-infinity controller design by convex optimization is proposed. Linear time-invariant single-input single-output systems represented by nonparametric models in the frequency domain are considered. It is shown that the H-infinity robust performance condition can be represented by a set of linear or convex constraints with respect to the parameters of a linearly parameterized controller in the Nyquist diagram. Multimodel and frequency-domain uncertainty can be considered straightforwardly in the proposed approach. The proposed method is compared with the standard H-infinity control problem. Moreover, a solution to an international benchmark problem is given that meets all specifications with the lowest order controller.

2010

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