Nyquist stability criterion

Nyquist stability criterion
Name	Nyquist stability criterion
Caption	A typical Nyquist plot for a feedback system
Field	Control theory
Discoveredby	Harry Nyquist
Year	1932
Relatedto	Bode plot, Routh–Hurwitz stability criterion, Cauchy's argument principle

Nyquist stability criterion. The Nyquist stability criterion is a fundamental graphical technique in control theory used to assess the stability of a feedback system. Developed by Bell Labs engineer Harry Nyquist in 1932, it provides a method to determine the stability of a closed-loop system by analyzing the frequency response of the corresponding open-loop transfer function. The criterion is particularly powerful because it can be applied to systems for which only experimental frequency response data is available, and it directly indicates the degree of stability or instability.

Introduction

The development of the Nyquist stability criterion was a pivotal advancement in the field of telecommunications and early analog computing. Prior to its formulation, stability analysis for complex systems, such as those involving vacuum tube amplifiers in transcontinental telephone lines, was challenging. Harry Nyquist's work, building upon the mathematical foundation of the Cauchy's argument principle from complex analysis, provided a practical graphical tool. This criterion allowed engineers at institutions like Bell Labs and the Massachusetts Institute of Technology to design stable feedback amplifiers and control systems, influencing later work by figures like Hendrik Bode and Walter Evans.

Mathematical formulation

The mathematical foundation of the criterion rests on complex analysis, specifically the mapping of contours in the complex plane. Consider an open-loop transfer function \( L(s) \), which is a ratio of polynomials resulting from the Laplace transform of system equations. The criterion involves evaluating \( L(s) \) as the complex variable \( s \) traverses a specific closed contour, known as the Nyquist contour, that encloses the entire right-half of the s-plane. The behavior of the resulting mapped contour, called the Nyquist plot, is analyzed using the principle of the argument from Augustin-Louis Cauchy. The key relationship involves the poles and zeros of the characteristic equation \( 1 + L(s) = 0 \).

Nyquist plot and encirclements

A Nyquist plot is a parametric plot of the frequency response of \( L(s) \), where the real part is plotted against the imaginary part as frequency varies from negative to positive infinity. The critical point for stability analysis is the point \( -1 + j0 \) in the complex plane. The stability is determined by the number of net clockwise encirclements, \( N \), of this critical point by the Nyquist plot. These encirclements are counted after the plot is completed for the entire Nyquist contour. The direction and number of these encirclements are directly related to the number of open-loop poles in the right-half plane, denoted \( P \), through the equation \( Z = N + P \), where \( Z \) represents the number of closed-loop poles in the right-half plane.

Stability criterion statement

The formal Nyquist stability criterion states that a closed-loop system is stable if and only if the number of net clockwise encirclements \( N \) of the point \( -1 + j0 \) by the Nyquist plot of \( L(s) \) is equal to the negative of the number of open-loop poles \( P \) located in the right-half of the s-plane. That is, \( Z = N + P = 0 \), ensuring no closed-loop poles reside in the right-half plane. For a system that is open-loop stable (\( P = 0 \)), this simplifies to the requirement that the Nyquist plot must not encircle the \( -1 \) point. This criterion is a direct application of the Cauchy's argument principle to the control theory problem.

Examples and applications

A classic example is the stability analysis of a feedback amplifier designed for the AT&T long-distance network. Engineers would plot the frequency response data from the open-loop system to generate the Nyquist plot. The criterion has been extensively applied in the design of autopilots for aircraft like those developed during World War II, in process control systems for chemical plants like those operated by Dow Chemical, and in the stability margins of power system networks managed by entities like the Tennessee Valley Authority. It is also a foundational concept taught in engineering curricula at universities like the California Institute of Technology and is used alongside tools like the Bode plot and Nichols chart.

Limitations and considerations

While powerful, the criterion has specific limitations. Its application assumes the open-loop transfer function is known or can be measured, which can be difficult for highly nonlinear systems or those with significant time delays, such as those found in some European Space Agency satellite controls. Accurate plotting requires care near frequencies where the plot passes near the critical point, as in conditionally stable systems. Furthermore, the standard criterion does not directly quantify robustness measures like gain margin and phase margin, though these can be inferred from the plot's proximity to the \( -1 \) point. It is also less intuitive for systems with many poles and zeros compared to methods like the root locus technique developed by Walter Evans.

Category:Control theory Category:Electrical engineering Category:Stability theory Category:Engineering theorems