In recent years, many automatic control systems for guided...

In recent years, many automatic control systems for guided vehicles in factories have been installed. One system uses a guidance cable embedded in the floor to guide the vehicle along the desired lane $[10$, $15]$. An error detector, composed of two coils mounted on the front of the cart, senses a magnetic field produced by the current in the guidance cable. An example of a guided vehicle in a factory is shown in Figure P7.19(a). We have $$ G(s)=\frac{s^2+3.6 s+81}{s(s+1)(s+5)} $$ and $K_a$ is the amplifier gain. (a) Sketch a root locus and determine a suitable gain $K_a$ so that the damping ratio of the complex roots is 0.707 . (b) Determine the root sensitivity of the system for the complex root $r_1$ as a function of (1) $K_a$ and (2) the pole of $G(s)$ at $s=-1$.

In recent years, many automatic control systems for guided vehicles in factories have been installed. One system uses a guidance cable embedded in the floor to guide the vehicle along the desired lane $[10$, $15]$. An error detector, composed of two coils mounted on the front of the cart, senses a magnetic field produced by the current in the guidance cable. An example of a guided vehicle in a factory is shown in Figure P7.19(a). We have
$$
G(s)=\frac{s^2+3.6 s+81}{s(s+1)(s+5)}
$$
and $K_a$ is the amplifier gain. (a) Sketch a root locus and determine a suitable gain $K_a$ so that the damping ratio of the complex roots is 0.707 . (b) Determine the root sensitivity of the system for the complex root $r_1$ as a function of (1) $K_a$ and (2) the pole of $G(s)$ at $s=-1$.

Key Concepts

Gain Selection

Gain selection involves choosing an appropriate value for the system gain to achieve desired performance specifications such as stability, damping (transient response), and steady-state error. This process often involves analyzing how varying the gain affects the system poles, which is frequently explored through root locus techniques.

Root Sensitivity

Root sensitivity refers to the way in which the locations of the closed-loop poles change in response to variations in system parameters, such as gain or pole locations of the open-loop transfer function. It is crucial for understanding how robust a control system is to parameter variations, and is useful for design and tuning purposes.

Root Locus

The root locus is a graphical method used in control systems to illustrate how the roots of the closed-loop characteristic equation change with variation in a system parameter, typically the gain. It is a vital tool for assessing the stability and transient response of the system by examining the trajectories of the poles in the s-plane.

Transfer Function

A transfer function is a mathematical representation of the relationship between the input and output of a linear time-invariant system in the Laplace domain. It captures the dynamics of the system through its poles and zeros and is fundamental in analyzing and designing control systems.

Damping Ratio

The damping ratio is a parameter that describes the rate at which oscillations in a system decay following a disturbance. It is a key performance metric in control system design, indicating the degree of overshoot, settling time, and overall system stability, especially in systems with complex conjugate poles.

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