Nyquist and Routh Stability Criteria Overview

The Nyquist stability criterion and the Routh criterion are fundamental tools used in control systems engineering to assess the stability of feedback controllers. While both serve the purpose of determining system stability, they operate on different principles and have distinct advantages and limitations. The question asks us to identify the statement that is FALSE regarding these two powerful analysis tools.

Routh Criterion for System Stability

The Routh-Hurwitz stability criterion is an algebraic method that determines the stability of a linear time-invariant (LTI) system by examining the coefficients of its characteristic polynomial. It checks for the presence of roots in the right half of the s-plane, which would indicate instability. Here's a breakdown:

• Application: It is applied to the characteristic equation, \(1 + G(s)H(s) = 0\), which must be a polynomial in \(s\).
• Limitations: The Routh criterion is limited to systems whose characteristic equations are polynomials. It cannot directly handle systems with non-polynomial terms, such as those arising from transport lag (dead time), which introduces terms like \(e^{-sT}\).
• Gain Range: By leaving a system gain, K, as a variable in the characteristic equation, the Routh array can be used to determine the range of K for which the system remains stable.

Nyquist Criterion for Control System Stability

The Nyquist stability criterion is a graphical method that analyzes the open-loop frequency response, \(G(j\omega)H(j\omega)\), to determine the stability of the closed-loop system. It provides information about both absolute and relative stability (gain margin and phase margin).

• Application: It involves plotting the Nyquist contour of \(G(j\omega)H(j\omega)\) in the complex plane and observing its encirclements of the critical point (-1, 0).
• Advantages: Unlike the Routh criterion, the Nyquist criterion can effectively handle systems with transport lag because the term \(e^{-j\omega T}\) simply introduces a phase shift proportional to frequency without affecting magnitude, which is readily incorporated into the frequency response plot.
• Gain and Phase Margins: The proximity of the Nyquist plot to the (-1, 0) point directly yields the gain margin and phase margin, which are crucial indicators of relative stability and the range of stable gain.

Analyzing Each Statement

Let's evaluate each given statement based on our understanding of the Nyquist and Routh stability criteria:

1. Information on Stable Gain Range

The statement is: "Both the criteria provide information relative to the stable gain range of the system."

• Routh Criterion: By keeping the system gain (K) as an unknown in the characteristic equation, the Routh array can be constructed. The conditions for stability (all elements in the first column having the same sign) can then be used to determine the range of K for which the system is stable.
• Nyquist Criterion: The Nyquist plot directly shows how close the open-loop frequency response is to the critical point (-1, 0). The gain margin and phase margin, which are derived from the Nyquist plot, directly indicate how much the gain or phase can change before instability occurs, thus defining the stable gain range.

Therefore, this statement is TRUE.

2. Nyquist Plot from Bode Magnitude for Minimum-Phase Systems

The statement is: "The general shape of the Nyquist plot is readily obtained from the Bode magnitude plot for all minimum-phase systems."

• Minimum-Phase Systems: For minimum-phase systems (systems with no poles or zeros in the right half of the s-plane), there is a unique and direct relationship between the magnitude and phase response (via the Hilbert transform).
• Bode and Nyquist: If the Bode magnitude plot is known for a minimum-phase system, its phase plot can be inferred. Since the Nyquist plot is a polar plot representing magnitude and phase at various frequencies, its general shape can indeed be derived or estimated from the Bode magnitude plot for such systems.

Therefore, this statement is TRUE.

3. Transport Lag Handling

The statement is: "The Routh criterion is not applicable in the condition of transport lag, which can be readily handled by the Nyquist criterion."

• Routh and Transport Lag: Transport lag (dead time) in a system transfer function is represented by \(e^{-sT}\). When this term is present, the characteristic equation becomes transcendental (not a finite polynomial), rendering the standard Routh-Hurwitz criterion inapplicable.
• Nyquist and Transport Lag: In the frequency domain, \(e^{-j\omega T}\) represents a phase lag of \(-\omega T\) radians without any change in magnitude. This phase shift can be easily accounted for when plotting the Nyquist diagram, making the Nyquist criterion suitable for systems with transport lag.

Therefore, this statement is TRUE.

4. Closed-Loop Frequency Response from Nyquist Plot

The statement is: "The closed-loop frequency response for a unity feedback system cannot be obtained from the Nyquist plot."

• Unity Feedback System: For a unity feedback system, the closed-loop transfer function, \(T(s)\), is given by \(T(s) = \frac{G(s)}{1 + G(s)}\), where \(G(s)\) is the open-loop transfer function.
• Obtaining from Nyquist: The Nyquist plot is a plot of \(G(j\omega)\) in the complex plane. For any frequency \(\omega\), a point \(P\) on the Nyquist plot corresponds to \(G(j\omega)\). The vector from the origin to \(P\) represents \(G(j\omega)\) and the vector from the critical point \((-1, 0)\) to \(P\) represents \(1 + G(j\omega)\).
• Graphical Determination: The ratio \(\frac{G(j\omega)}{1 + G(j\omega)}\) can be graphically determined from these vectors. The magnitude of the closed-loop frequency response, \(|T(j\omega)|\), is the ratio of the length of the vector from the origin to \(P\) to the length of the vector from \((-1, 0)\) to \(P\). The phase of \(T(j\omega)\) is the difference in the angles of these two vectors. This graphical method allows the determination of the closed-loop frequency response from the open-loop Nyquist plot.

Therefore, the statement that the closed-loop frequency response cannot be obtained from the Nyquist plot is FALSE.

Summary Comparison of Criteria

The table below summarizes some key characteristics of the Routh and Nyquist stability criteria.

Conclusion on the False Statement

Based on the detailed analysis of each statement, the false statement is: "The closed-loop frequency response for a unity feedback system cannot be obtained from the Nyquist plot." This is incorrect because the closed-loop frequency response can indeed be graphically determined from the open-loop Nyquist plot by considering the vectors from the origin and the critical point (-1, 0) to points on the Nyquist curve.