EE3331C Feedback Control Systems - Course Overview & Dynamic Systems Review

Course Mechanics

  • All class information, lecture notes, and lab manuals are available on the CANVAS website.
  • Course requirements:
    • Midterm quiz (20%) (closed-book, 1 A4 help-sheet)
    • Laboratory sessions (15%) – 2 sessions (tentative: weeks 5 and 9), Matlab (online) and DC motor (physical lab).
    • Assignments – 3 Matlab assignments (15%)
    • Final exam (50%) (physical: closed-book, 1 A4 help-sheet)
    • Homework and reading assignments.
  • Textbook and references:
    • Feedback Control of Dynamic Systems by Franklin, Powell, and Emami-Naeini (FPE)
    • Modern Control Systems by Dorf and Bishop (DB)
  • Software: Matlab and its Control Systems Toolbox (download from NUS IT webpage).

Motivation

  • Control involves using algorithms and feedback in engineered systems.
  • A control system uses a sensed quantity to modify a system's behavior through computation and actuation.
  • Feedback manages uncertainty by measuring system operation, comparing it to a reference, and adjusting control variables.

Historical Context

  • 1788: James Watt's flyball governor (speed control for steam engines).
  • 1920s: Feedback amplifiers by H.S. Black (Bell Labs), laying mathematical foundations for classical control using negative feedback.

Components of feedback:

  • Process (or Plant):
    • Examples: CD player, aircraft, car engine, high-rise building, computer network, industrial process, elevator.
  • Actuator:
    • Examples: hydraulic, pneumatic, electric motors, pumps, heaters, aircraft control surfaces, voice coil, piezo-electric transducer.
  • Sensor:
    • Examples: radar altimeter, GPS, shaft encoder, LVDT, strain gauge, accelerometer, tachometer, microphone, pressure and temperature transducers.
  • Controller:
    • Examples: human operator, mechanical, electro-mechanical, analog electrical, digital processor.
  • Disturbances:
    • Examples: wind gusts, earthquakes, road surface variations, variations in feed material.

Simple Feedback Control System

A simple feedback control system consists of:

  • Process: the system to be controlled.

  • Actuator: affects the process output.

  • Sensors: measure reference and output signals.

  • Controller: logic for calculating the control signal.

    • Signals: Variables that carry information.
    • Systems: Process input signals to produce output signals.

Examples of feedback control

  • Automobile steering control: Driver uses visual and tactile feedback to adjust steering.
  • Household furnace: Thermostat controls temperature by sensing and adjusting furnace output.
  • Unmanned Aerial Vehicles:
    • Sensing: GPS, ultrasonic sonar, RPM sensor, camera.
    • Actuation: Rotor torques.
    • Computation: On-board computer system, image processing.
    • Effect: Autonomous UAVs, ground target tracking.
  • Robot Soccer:
    • Sensing: Overhead camera system, wheel angle encoders.
    • Actuation: Motor torques, kick mechanism.
    • Computation: Centralized computer, vehicle microcomputers.
    • Effect: Autonomous robot soccer platform, agile motion.
  • Congestion control and Internet
    • Sensing: Data, ACK packets via TCP
    • Actuation: Transmit rate, router paths.
    • Computation: Source, destination, router processors.
    • Effect: High-speed data transmission, tolerant to link failures.
  • Semiconductor manufacturing: Thermal processing systems to heat a heater plate to a desired temperature.

Key takeaways

  • Control engineering: Present in modern engineering systems.
  • Simple feedback control system elements: process, actuators, sensors, controller.
  • Multidisciplinary nature of control: sensors, actuators, communications, computing.

Learning Outcomes

  • Explain a linear time-invariant (LTI) system and its properties.
  • Model LTI systems using differential equations and transfer functions.
  • Describe and analyze system behavior in the time and frequency domains.
  • Apply feedback to achieve stable automatic control.
  • Evaluate the stability and performance of negative feedback systems.
  • Design simple feedback controllers.
  • Use MATLAB/Labview for simulation, design, and control.

Time-Domain Analysis of Dynamic Systems Review

  • Signals: A function of time (e.g., voltage).
  • Continuous-Time Signals: Defined for all time ( ).
  • ** Discrete-Time Signals**: Defined at discrete points in time.
  • Elementary Signals: Basic building blocks (e.g., step, impulse, ramp, sinusoidal, exponential).

Elementary Signals

  • Constant (DC) Signal: u(t) = k (where k is constant).
  • Unit Step Signal: u(t) = \begin{cases} 0, & t < 0 \ 1, & t \geq 0 \end{cases}
  • Unit Ramp Signal: u(t) = \begin{cases} 0, & t < 0 \ t, & t \geq 0 \end{cases}
  • Sinusoidal Signal: u(t) = A \cos(\omega t \pm \phi) or A \sin(\omega t \pm \phi)
  • Exponential Signal: u(t) = e^{at} (decays if a < 0, grows if a > 0).
  • Complex Exponential Function: Ae^{zt}, where z = \sigma + j\omega.

Complex Exponential Function Details:

  • Euler’s Identity: e^{j\phi} = \cos \phi + j \sin \phi
  • A \cos(\omega t + \phi) = \Re{A e^{j\omega t} e^{j\phi}}
  • A \sin(\omega t + \phi) = \Im{A e^{j\omega t} e^{j\phi}}
  • Multiplying a sinusoidal signal with an exponential signal: y(t) = Ae^{(\sigma + j\omega)t} = Ae^{\sigma t} e^{j\omega t} = Ae^{\sigma t} [\cos(\omega t) + j \sin(\omega t)]
    • Real part: Re{Ae^{zt}} = Ae^{\sigma t} \cos(\omega t)
    • Imaginary part: Im{Ae^{zt}} = Ae^{\sigma t} \sin(\omega t)

Impulsive Signals

  • Dirac's Delta Function (Impulse): \delta(t), is very large near t = 0, very small away from t = 0, the area under the curve equals 1.

  • Properties:

    • \int_{a}^{b} f(t) \delta(t) dt = f(0), provided a < 0 and b > 0 and f(t) is continuous at t = 0.
    • \delta(t) = 0 for t \neq 0
    • \delta(0) is not defined.
    • Plotted as a solid arrow.

  • Scaled Impulses: \alpha \delta(t - T)\n * Is an impulse at time T with magnitude \alpha

    • \int_{a}^{b} \alpha \delta(t - T) f(t) dt = \alpha f(T), for a < T < b and f continuous at T.

Physical Interpretation:

-Impulse functions model physical signals that act over short time intervals.
-Effect depends on the integral of signal.
-Example: Rapid Charging of Capacitor Circuit.

Relationship Between The Unit Step and The Unit Impulse

Systems

  • A system transforms input signals into output signals.
  • Examples: Hi-fi system, RC circuit, automobile, image enhancement system.
  • Notation: y = Gu or y = G(u), system G acts on input signal u to produce output signal y.

System Properties

  • Static vs. Dynamic:
    • Static (Memoryless): Output y(t) depends only on the input at time t. Example: Resistor, V(t) = i(t)R.
    • Dynamic (Memory): Output at time t depends on past or future values of input. Examples: Capacitor, Inductor.
  • Causal vs. Non-Causal:
    • Causal (Non-Anticipative): Output at any time depends only on values of input at the present time and in the past, i.e., the voltage only responds to present and past values of source voltage.
    • All real-time physical systems are causal.
  • Linearity: A system G is linear if:
    • Homogeneity: G(au) = aG(u)
    • Additivity: G(u1 + u2) = Gu1 + Gu2
    • Scaling before or after the system is the same & summing before or after the system is the same
  • Time Invariance: A time delay/advance of the input signal leads to an identical time shift in the output signal.

System properties summary

Static vs Dynamic
Causal vs. non-causal system
Linearity
Time Invariance

Modeling of Physical Systems

  • Differential Equations for Electrical Circuits:
    • Resistor: vR(t) = RiR(t)
    • Capacitor: vC(t) = \frac{1}{C} \int{0}^{t} iC(\tau) d\tau, or iC(t) = C \frac{dv_C(t)}{dt}
    • Inductor: vL(t) = L \frac{diL(t)}{dt}
      Linear motions (force-displacement relationship)
    • Mass: f(t) = M \frac{d^2 x(t)}{dt}
    • Spring: f(t) = Kx(t)
    • Damper: f(t) = f_v \frac{dx(t)}{dt}
      Angular motions (torque-angular displacement relationship)
    • Inertia: T(t) = J \frac{d^2 \theta(t)}{dt}
    • Spring: T(t) = K\theta(t)
    • Damper: T(t) = D \frac{d\theta(t)}{dt}

Partial List of Tentative Topics

  • Review of signals, systems, and Laplace Transform
  • Modeling of dynamic systems, Transfer Functions
  • Time domain analysis of dynamic system’s response
  • Open-loop versus Closed-loop control: Transient performance, steady-state error analysis
  • Stability, Root Locus Analysis
  • Proportional, Integral, and Derivative (PID) control
  • Frequency response, Bode plot and polar plot, Nyquist Stability Criterion
  • Robust stability – gain and phase margins
  • Lead and Lag compensator, design of lead compensator (both s-plane method and Bode plot method)
  • Digital implementation of feedback control (via lab)