Ctrl-Class-4-Summary

Subbu begins the class on control system stability, noting that most chemical process systems are inherently stable except for exothermic reactions. He explains that oscillations in control systems typically arise from adding controllers, particularly the integral part of PID controllers, or from inherent second-order dynamics like those in pneumatic controls with force balances. While oscillations cause the system to deviate from the set point temporarily, they can help the system settle faster, and the class will now examine the conditions for system stability.

MSubbu explains that the location of poles and zeros in a transfer function affects system stability. He clarifies that poles are roots of the denominator while zeros are roots of the numerator, and these can be plotted in the complex \(s\)-plane. Systems with poles in the left half of the \(s\)-plane (negative real part) are stable with decaying responses, while poles on the imaginary axis produce sustained oscillations (marginally stable), and poles in the right half-plane (positive real part) create growing, unstable responses. MSubbu emphasizes that for stability, all poles must be in the left half of the \(s\)-plane.

MSubbu explains how poles and zeros affect system response in control systems. He clarifies that adding poles slows down the response (with more negative poles having less effect), while adding zeros speeds up the response. MSubbu relates poles to time constants, noting that higher time constants result in slower settling times. He mentions that derivative control introduces zeros to the system, which can speed up response, and emphasizes that poles primarily determine system stability, with poles in the negative plane creating stable systems.

MSubbu explains the process of analyzing stability using a characteristic equation and Routh array method. He emphasizes the importance of writing the polynomial with all coefficients (including zeros) and describes how to construct the route array in a zigzag pattern for a third-order system with four coefficients. MSubbu details the calculation method for filling subsequent rows through cross-multiplication and division of elements from previous rows.

MSubbu explains that system stability is determined by examining the first column of the Routh Array, which should not contain negative contents. For the given system, \(K_c\) should be less than 8 for stability. He demonstrates how to derive the characteristic equation and use the Route Array to determine that \(\tau_D\) should be greater than 1/15 for stability with a PD controller. MSubbu also mentions that for systems with dead time, stability can be analyzed by approximating the exponential term as a polynomial using Pade approximation or Taylor series expansion, or by using frequency response methods.

MSubbu explains the Routh-Hurwitz stability criterion for determining system stability by analyzing sign changes in the first column of the Routh Array. He demonstrates that two sign changes in the array indicate two roots with positive real parts, confirming the system is unstable. MSubbu mentions that MATLAB can be used to find polynomial roots directly, and explains that roots with negative real parts lead to stable conditions while those with positive real parts cause instability.

MSubbu explains how to analyze a polynomial system to determine stability by examining sign changes in the Routh-Hurwitz array. When encountering a zero in the array, MSubbu introduces a small positive number epsilon to avoid indeterminate forms, revealing that the system has two roots with positive real parts out of five total roots. This indicates the system is unstable, and MSubbu suggests that work needs to be done to change the problematic element to negative to achieve stability.

MSubbu explains how to analyze a 5th order polynomial to determine the number of positive, negative, and pure imaginary roots. He demonstrates the Routh array method, filling in rows with coefficients and emphasizes that all rows must be completed even if zeros appear. MSubbu notes that any polynomial with negative coefficients definitely represents an unstable system, though even polynomials with all positive coefficients can be unstable.

MSubbu explains how to handle a situation where a row ends with zeros by using an auxiliary equation. He demonstrates that when encountering a row of zeros, one should form an auxiliary equation using coefficients from the previous row, then take the derivative of the acceleration to replace the zero coefficients. 

MSubbu explains how to determine the stability of a system by analyzing the number of roots with positive real parts using the Routh array. He identifies one root with a positive real part, two purely imaginary roots, and two roots with negative real parts in a fifth-order polynomial. MSubbu demonstrates that the same method can be applied to analyze zeros of a transfer function, showing that for a given third-order polynomial, all three roots lie in the left half of the \(s\)-plane.

MSubbu explains how to analyze the stability of a cascade control system by first reducing the inner loop to a single transfer function, then writing the characteristic equation for the total loop to find that \(K_c\) must be less than 10. He notes that in cascade control, the inner loop should always respond faster than the outer loop, as seen in temperature control systems where flow control (inner loop) responds quickly while temperature (outer loop) takes longer to settle. MSubbu also covers feedforward control, explaining that it provides faster response to disturbances without feedback by using a model of the disturbance, and demonstrates how to derive the feedforward controller transfer function by setting the output response to zero for any disturbance.