Nyquist stability criterion

Nyquist stability criterion
Name	Nyquist stability criterion
Field	Control theory, Electrical engineering
Introduced	1932
Inventor	Harry Nyquist
Notable for	Frequency-domain stability test for closed-loop systems

Nyquist stability criterion The Nyquist stability criterion is a fundamental frequency-domain method for assessing the closed-loop stability of linear time-invariant feedback systems. Developed from the work of Harry Nyquist and formalized within the context of Norbert Wiener's and Hendrik Wade Bode's contemporaneous developments, it links complex analysis results used by Gustav Kirchhoff and Augustin-Louis Cauchy to practical design tasks encountered at institutions such as Bell Laboratories and Massachusetts Institute of Technology. The criterion provides a mapping between encirclements of critical points in the complex plane and the location of system poles associated with stability in systems studied by engineers at General Electric and RCA.

Introduction

The Nyquist stability criterion arose from the needs of telephone and control systems engineered by Harry Nyquist at Bell Telephone Laboratories and later expanded by theoreticians at Harvard University and Imperial College London. It builds on complex-plane techniques originally developed by Augustin-Louis Cauchy and uses concepts also exploited by James Clerk Maxwell in feedback analysis. The method connects open-loop transfer functions analyzed in the tradition of Lord Rayleigh and Oliver Heaviside to closed-loop pole locations whose movement was later studied by researchers at Stanford University and Caltech. Prominent figures in control, including Rudolf E. Kálmán and Karl Åström, incorporated Nyquist ideas into modern control curricula at ETH Zurich and University of Cambridge.

Mathematical formulation

Let L(s) denote the open-loop transfer function introduced in analyses by Harry Nyquist and further used by scholars at Princeton University and Yale University. The Nyquist stability criterion employs the argument principle of Augustin-Louis Cauchy to relate the number of zeros and poles of 1+L(s) inside a contour in the complex s-plane to the winding number of the image of that contour under L(s) about the critical point -1. The formal statement relies on residue techniques popularized by Bernhard Riemann and leverages analytic continuation ideas associated with Carl Friedrich Gauss. In practical terms, let P be the number of right-half-plane poles of L(s) (notably analyzed by practitioners at Siemens and Westinghouse Electric Company), N be the number of clockwise encirclements of -1 by the Nyquist plot of L(s), then the number of right-half-plane zeros Z of 1+L(s) satisfies Z = P - N, a relation applied in textbooks from McGraw-Hill to courses at University of California, Berkeley.

Nyquist plot construction and interpretation

Constructing the Nyquist plot follows a procedure refined in laboratories at Bell Labs and classrooms at Massachusetts Institute of Technology: sweep the frequency ω from 0 to ∞, evaluate L(jω) where j denotes the imaginary unit used in work by Oliver Heaviside, and plot the trajectory in the complex plane, linking techniques found in the studies of Heinrich Hertz and Michael Faraday. For systems with poles on the imaginary axis, contour deformations inspired by Cauchy and used in analyses at Johns Hopkins University and Cornell University are employed to avoid singularities. Interpretation of encirclements about -1 uses topological intuition developed in the tradition of Henri Poincaré and applied by control engineers at NASA and European Space Agency for stability margins. Engineers trained at Imperial College London and Delft University of Technology read gain margin and phase margin off the Nyquist plot, correlating to robustness metrics discussed by Zadeh and Bellman.

Applications and examples

The Nyquist criterion has been applied widely across industries represented by firms such as General Electric, Siemens, and Toyota in contexts ranging from automatic regulation studied by James Watt predecessors to flight control systems developed by Boeing and Airbus. Classic examples include analysis of proportional–integral–derivative controllers popularized in manuals from Honeywell and in courses at University of Michigan, design of amplifier feedback networks as in works by Claude Shannon, and stability assessment of power-system controllers at Edison Electric Institute utilities. Case studies in academic literature from Stanford University and University of Cambridge illustrate use with time delays encountered in teleoperation systems researched at MIT Media Lab and process control plants referenced in publications by Shell and BP.

Extensions and generalizations

Extensions of the Nyquist approach have been developed by researchers associated with Rudolf E. Kálmán, Zadeh, and Lotfi Zadeh for multivariable systems, leading to the multivariable generalizations employed at Lockheed Martin and Raytheon. The small-gain theorem and passivity criteria, advanced by scholars at Princeton University and University of Illinois Urbana-Champaign, provide alternative robustness tests related to Nyquist-based reasoning. Frequency-domain methods inspired by Hendrik Bode and expanded by F. L. Lewis encompass structured singular value (μ) analysis developed at AT&T Bell Labs and mathematical generalizations using algebraic geometry techniques taught at University of Oxford and University of Cambridge.

Practical considerations and limitations

In practice, engineers at ABB and Schneider Electric must account for noise, model uncertainty, and nonlinearities when applying the Nyquist criterion—issues also encountered in studies at Sandia National Laboratories and Los Alamos National Laboratory. The criterion assumes linear time-invariant models like those used in classical texts from McGraw-Hill and labs at ETH Zurich; when models violate these assumptions, complementary methods developed at Carnegie Mellon University and Imperial College London—such as Lyapunov analyses introduced by Aleksandr Lyapunov—are required. Computational tools from vendors like MathWorks and academic implementations at Wolfram Research provide numerical Nyquist plotting, but care from practitioners at NIST and IEEE standards committees is needed to handle discretization and sampling effects in digital control designs applied by National Instruments and Texas Instruments.

Category:Control theory