Electrical Engineering - Control Systems - Classical Control

Classical control, a fundamental aspect of control systems within the broader realm of electrical engineering, focuses on the analysis and design of systems governed by differential equations, particularly those that can be represented through linear time-invariant (LTI) models. This sub-discipline prioritizes achieving desired behavior in dynamic systems such as electrical circuits, mechanical devices, and various industrial processes by utilizing feedback loops.

In Classical Control, we primarily deal with Single-Input Single-Output (SISO) systems, where the input and output are both scalar quantities. These systems are typically analyzed and designed using tools from linear control theory and frequency domain methods.

Fundamental Concepts

1. Transfer Functions:
A transfer function, \( G(s) \), represents the relationship between the input and output of a linear time-invariant system in the Laplace transform domain. It is defined as:

\[ G(s) = \frac{Y(s)}{U(s)} \]

where \( Y(s) \) is the Laplace transform of the output and \( U(s) \) is the Laplace transform of the input.

2. Stability:
A key objective in control design is to ensure system stability. A system is stable if, in response to a bounded input, it produces a bounded output. This is often analyzed using the location of the poles of the transfer function \( G(s) \). For a system to be stable, all poles must lie in the left half of the complex plane.

3. Time-Domain Analysis:
Time-domain methods involve analyzing the system’s transient and steady-state responses using differential equations. Key performance metrics include rise time, settling time, overshoot, and steady-state error.

4. Frequency-Domain Analysis:
Frequency-domain methods include techniques such as Bode plots, Nyquist plots, and Root Locus. These analyses provide insight into system behavior and stability over a range of frequencies.

• Bode Plot:
  A graphical representation of a system’s frequency response. It includes two plots: one showing the magnitude (in decibels) and the other showing the phase (in degrees) versus frequency (on a logarithmic scale).

• Nyquist Plot:
  A complex plot mapping the frequency response of a transfer function, used to assess system stability based on the Nyquist stability criterion.

• Root Locus:
  A plot showing the trajectories of the poles of a transfer function as a system parameter (usually gain) is varied, used to design and analyze the control system’s stability and transient response.

5. PID Control:
One of the most widely used controllers in classical control systems is the Proportional-Integral-Derivative (PID) controller. The PID controller algorithm can be described by:

\[ u(t) = K_p e(t) + K_i \int_0^t e(\tau) \, d\tau + K_d \frac{de(t)}{dt} \]

where:
- \( u(t) \) is the control input.
- \( e(t) \) is the error signal, defined as the difference between the desired setpoint and the measured output.
- \( K_p \) is the proportional gain.
- \( K_i \) is the integral gain.
- \( K_d \) is the derivative gain.

Design Techniques

Designing classical control systems often involves ensuring that the closed-loop system meets specified performance criteria. Techniques include:

• Root Locus Method:
  Used to determine the placement of poles and zeros to achieve desired transient response characteristics.

• Frequency Response Methods:
  Such as Bode plot design can be used to shape the open-loop gain and phase to ensure desired performance in terms of gain and phase margins.

• Compensators:
  Design of lead, lag, or lead-lag compensators to improve the system’s transient and steady-state performance.

Classical control remains the cornerstone of control engineering due to its simplicity and effectiveness for a wide range of practical systems. Understanding classical control concepts provides a strong foundation for more advanced control techniques, including modern control, robust control, and adaptive control.