A single-loop negative feedback system has a loop transfer...

A single-loop negative feedback system has a loop transfer function $$ G_c(s) G(s)=\frac{K(s+2)^2}{s\left(s^2+1\right)(s+8)} . $$ (a) Sketch the root locus for $0 \leq K \leq \infty$ to indicate the significant features of the locus. (b) Determine the range of the gain $K$ for which the system is stable. (c) For what value of $K$ in the range $K \geq 0$ do purely imaginary roots exist? What are the values of these roots? (d) Would the use of the dominant roots approximation for an estimate of settling time be justified in this case for a large magnitude of gain $(K>50)$ ?

A single-loop negative feedback system has a loop transfer function
$$
G_c(s) G(s)=\frac{K(s+2)^2}{s\left(s^2+1\right)(s+8)} .
$$
(a) Sketch the root locus for $0 \leq K \leq \infty$ to indicate the significant features of the locus. (b) Determine the range of the gain $K$ for which the system is stable. (c) For what value of $K$ in the range $K \geq 0$ do purely imaginary roots exist? What are the values of these roots? (d) Would the use of the dominant roots approximation for an estimate of settling time be justified in this case for a large magnitude of gain $(K>50)$ ?

Key Concepts

Dominant Pole Approximation

The dominant pole approximation is a simplification technique where the dynamic behavior of a system is approximated by the poles closest to the imaginary axis (the dominant poles), neglecting the influence of faster (non-dominant) dynamics. This approximation is justified when a pair of poles is significantly closer to the imaginary axis compared to the other poles, thereby primarily governing the transient response such as the settling time. However, when multiple poles contribute comparably to the dynamics, or when the system gain causes significant changes in pole positions, this approximation may not accurately predict the system's time response.

Imaginary Axis Crossing and Pure Imaginary Roots

In analysis of control systems, pure imaginary roots indicate a boundary condition between stability and instability, typically corresponding to sustained oscillations (neutrally stable behavior). Determining the gain value at which the root locus crosses the imaginary axis can be done by substituting s = j? into the characteristic equation, leading to conditions that unveil the existence of purely imaginary roots and help predict or control oscillatory phenomena.

Open-Loop Poles and Zeros

Open-loop poles and zeros are the roots of the denominator and numerator, respectively, of the loop transfer function. In control system analysis, these points are crucial because they define the starting points (poles) and termination points (zeros) of the root locus branches. Their locations in the complex plane largely affect the system's overall dynamics and stability properties.

Root Locus Analysis

Root locus analysis is a graphical technique used in control systems to show how the roots of the characteristic equation (which determine the closed?loop system poles) vary as a system parameter, typically the gain K, is varied. This method helps in understanding system stability, transient response, and how changes in gain affect the dynamic behavior of the system by plotting the trajectories of the poles in the complex plane.

System Stability in the Context of Feedback

The stability of a negative feedback system is determined by the location of the closed-loop poles which must all lie in the left half of the complex plane. Techniques such as the root locus method and Routh-Hurwitz criterion are used to determine the range of gain values for which the closed-loop system is stable. The investigation of these stability limits is critical in control system design to ensure desired performance without oscillations or divergence.

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