Linear control systems

Define a system as an isolated device of interconnected parts that follows causality, with a command input and a resultant physical output, affected by external perturbations.

Identify linear system concepts and express input-output relations with a linear differential equation, while describing continuous systems with continuous quantities over time.

Explore the nature of input and output signals as functions of time, distinguishing deterministic signals with known values from probabilistic signals like white noise, illustrated by a figure and instrument.

Define a control system as a process delivering input signal to achieve accurate, stable results, with energy supply providing high voltage and a controller amplifying the command to drive output.

Differentiate regulation from control: regulation maintains the output at the desired value despite disturbances, while control tracks the input toward the target despite perturbations, enabling autonomous tasks, accuracy, and productivity.

Explore the Laplace transformation, converting a time-domain function f(t) to a complex frequency function F(s) via the integral from 0 to infinity e^{-st} f(t) dt, and apply its inverse.

Explore Laplace transform properties, including linearity, division, integration, scaling, and delay, and apply them to obtain transforms and inverse transforms with initial and final value considerations.

Presents famous Laplace transforms in linear control systems, illustrating how exponential signals map under the transform and showing key forms like 1/(s+a) and 1/(s^2+ω^2).

Demonstrate the inverse Laplace transform of a rational function by decomposing into partial fractions and evaluating limits to recover the original function.

Get the transfer function of a linear system by transforming its differential equation, identifying the input–output ratio, and deriving the denominator polynomial and its characteristic roots to assess stability.

Derive the characteristic equation of a first-order system from its differential equations, convert the input-output relation to a transfer function, and explain how parameters K and A shape the response.

Derive the second-order system's characteristic equation from its differential form and obtain the transfer function Y(s)/U(s) = ω0^2/(s^2+2 ω0 s+ ω0^2), highlighting ω0 as a key parameter.

Explore an rc circuit with a resistor and a capacitor, derive its differential equations and transfer function from input voltage to output, illustrating how electrical laws model dynamic behavior.

Explain a mechanical system with spring, mass, and friction, derive its equation, and obtain the transfer function relating input to output.

Explore the functional diagram that simplifies a real physical process into a symbolic differential model, highlighting the input, error, sensor, and transfer functions in a closed-loop control system.

Explore the useful formalism of the transfer function, defining the system input P and the output S, with a standard summation of inputs that yields the overall output.

Explore a global view of a control system with direct channel and feedback, detailing input, comparator, sensor, controller, amplifier, and the process within a disturbance-prone environment and noise.

Explore the open loop system by deriving its transfer function from cascaded and parallel subsystems, showing how inputs combine to produce outputs and yield a single transfer function.

Analyze the closed-loop system, derive its transfer function from input to output, and show how feedback via sensor and comparator yields T(s) = P(s)/(1+F(s)P(s)); compare with the open-loop path.

Demonstrate how to simplify block diagrams by deriving transfer functions for inner and outer loops, and show closed system transfer functions using F1, F2, F3, H1, and H2.

Explore how transfer functions describe a linear system’s response to input signals, focusing on first- and second-order dynamics and metrics like final value within ±5%, peak time, and overtaking.

Learn about the impulse (Dirac) function, the unit step, and the ramp as key signals in temperature analyses. These signals illustrate system motion from equilibrium and aid stability analysis.

Analyze the first-order system response by deriving its transfer function from the differential equation, identify the time constant, and describe exponential convergence to the steady state.

Explore the second order system response through its differential equation and transfer function, analyzing how poles and damping coefficient shape the impulse and ramp input responses.

Examine the harmonic response as the permanent response to sinusoidal input and how gain and phase vary with frequency on a logarithmic scale, using first-order system insights.

Explore the Nyquist and Bode representations of the complex frequency response, using polar coordinates on the complex plane, with omega mapping real and imaginary axes, phase, and a block diagram.

explain the bode diagram of a first order system by presenting the frequency response G(Ω)=K/(1+jΩτ) and describing how magnitude and phase vary with frequency.

This lecture shows that the Nyquist plot of a first-order system forms a circle in the complex plane, with center at (K/2, 0) and diameter K.

Analyze the block diagram of a first-order system and derive its frequency-domain transfer function, G(Ω) = A^2/(1+Ω^2), including its phase from the argument, as shown.

Explore the Bode and Black diagrams of a second-order system, and analyze its transfer function, block diagram, gain, and phase across frequency.

Assess accuracy performance in linear control systems by examining how the output follows the input, and distinguish static precision at infinity from dynamic precision related to change dynamics.

Identify stability as a linear system's ability to return to equilibrium after disturbances; a stable system has transfer function parts with real negatives, while any real positive causes instability.

Apply the algebraic Routh-Hurwitz criterion to the characteristic polynomial to assess stability, forming the denominator into a characteristic equation and checking the first column for positivity.

Explore how applying a gain K to a linear control system acts as a corrective to improve closed-loop precision, while risking stability; balance precision and stability per the classic trade-off.

Explore pole placement using transfer functions to convert an open-loop system to a stable, closed-loop controller with desired performance.

Explore the phase advance corrector and its transfer function, with alpha greater than one, noting the maximum correction (alpha - 1)/(alpha + 1) and its link to stability.

Explore the pid corrector theory, combining proportional, integral, and derivative actions with Kp, Ki, and Kd to shape transfer function and stability.

Explore PID corrector parameter tuning for linear control systems, using empirical methods and Matlab-based experimental tuning to adjust K and I, and refine the derivative action for a stable system.