2.004
Dynamics and Control II
Modeling, analysis, and control of dynamic systems. System modeling: lumped parameter models of mechanical, electrical, and electromechanical systems; interconnection laws; actuators and sensors. Linear systems theory: linear algebra; Laplace transform; transfer functions, time response and frequency response, poles and zeros; block diagrams; solutions via analytical and numerical techniques; stability. Introduction to feedback control: closed-loop response; PID compensation; steady-state characteristics, root-locus design concepts, frequency-domain design concepts. Laboratory experiments and control design projects. Enrollment may be limited due to laboratory capacity; preference to Course 2 majors and minors.
Spring 2008
Topics
- Introduction
- Example: cruise control
- Laplace transform definition, properties
- Block diagram algebra
- Modeling: impedance of electrical components
- Kirchoff’s laws, circuit equations
- Transfer functions, loop/mesh currents
- Modeling: real components as Thevenin and Norton sources
- Modeling: one-dimensional mechanical components
- Modeling: impedance of mechanical components
- Transfer functions in MATLAB and Maple
- Operational amplifiers
- Generalized system modeling
- Modeling: rotational systems
- Example: rotational systems
- Modeling: two-port components
- LTI system response
- Standard input functions: delta, step, ramp, sinusoid
- Poles and zeros
- Standard 1 st and 2 nd order system responses
- Higher order systems, LTI system properties
- Example: finding system responses
- Effects of poles and zeros
- Closed-loop systems, steady-state errors
- System stability, Routh-Hurwitz criterion
- Stability of closed-loop systems, root locus plots
- Root locus development
- Root locus development (cont.)
- Root locus summary, MATLAB
- Sinusoidal system response
- Frequency response and pole-zero plots
- Bode plots
- Poles and zeros on bode plots
- Bode plots, conclusion
Readings
- Nise, Norman S. Control Systems Engineering . 5th ed. New York, NY: John Wiley & Sons, 2007. ISBN: 9780471794752. (The 4th ed. will work just as well.)
- Rowell and Wormley. System Dynamics: An Introduction . Upper Saddle River, NJ: Prentice Hall, 1996.
- Dorf and Bishop. Modern Control Systems . 7th ed. Reading, MA: Addison-Wesley, 1995.
- Ogata. Modern Control Engineering . 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1996.
Fall 2007 · Systems, Modeling, and Control II
Topics
- Introduction; mechanical elements
- Solving ODEs; cruise control
- Laplace transforms; transfer functions; translational and rotational mechanical transfer functions
- Electrical and electro-mechanical system transfer functions
- DC motor transfer function
- Poles and zeros; 1st order systems
- 2nd order systems
- 2nd order systems (cont.)
- More than 2 poles; zeros; nonlinearities and linearization
- Examples of modeling and transfer functions
- Block diagrams; feedback
- Analysis of feedback systems
- Stability; Routh-Hurwitz criterion
- Stability analysis
- Steady state error analysis
- Root locus introduction
- Root locus example
- Design of transient response using root locus
- Positive feedback
- Examples of design via root locus
- Steady-state error compensation
- Transient response compensation; transient and steady-state error compensation
- Compensation examples
- Feedback compensation and its physical realization
- Feedback design examples
- Frequency response; bode plots
- Bode plot examples
- Gain margin and phase margin
- Design using the frequency response; lead, lag, lead-lag compensators
- The state-space representation
- Solving the state equations in the time and space domains
- State equation examples
- Stability and steady-state error in state space; controllability and observability
- Optimal control; the minimum time problem
- Review: modeling and transfer functions
- Review: root locus, feedback design
- Review: frequency domain and design
Readings
- Nise, Norman S. Control Systems Engineering . 4th ed. Hoboken, NJ: John Wiley, 2003. ISBN: 9780471445777.
- SES #: 1 · TOPICS: Introduction; mechanical elements · READINGS: N1; notes
- SES #: 2 · TOPICS: Solving ODEs; cruise control · READINGS: Notes
- SES #: 3 · TOPICS: Laplace transforms; transfer functions; translational and rotational mechanical transfer functions · READINGS: N2.1-2.3 and 2.5-2.6
- SES #: 4 · TOPICS: Electrical and electro-mechanical system transfer functions · READINGS: N2.4 and 2.7-2.8
- SES #: 5 · TOPICS: DC motor transfer function · READINGS: Notes
- SES #: 6 · TOPICS: Poles and zeros; 1st order systems · READINGS: N4.1-4.3
- SES #: 7 · TOPICS: 2nd order systems · READINGS: N4.4-4.6
- SES #: 8 · TOPICS: 2nd order systems (cont.) · READINGS: Notes
- SES #: 9 · TOPICS: More than 2 poles; zeros; nonlinearities and linearization · READINGS: N2.10, 3.7, and 4.7-4.9
- SES #: 10 · TOPICS: Examples of modeling and transfer functions · READINGS: Notes
- SES #: 11 · TOPICS: Block diagrams; feedback · READINGS: N5.1-5.2
- SES #: 12 · TOPICS: Analysis of feedback systems · READINGS: N5.3
- SES #: 14 · TOPICS: Stability; Routh-Hurwitz criterion · READINGS: N6.1-6.3
- SES #: 15 · TOPICS: Stability analysis · READINGS: N6.4
- SES #: 16 · TOPICS: Steady state error analysis · READINGS: N7.1-7.6
- SES #: 17 · TOPICS: Root locus introduction · READINGS: N8.1-8.5
- SES #: 18 · TOPICS: Root locus example · READINGS: N8.6
- SES #: 19 · TOPICS: Design of transient response using root locus · READINGS: N8.7
- SES #: 20 · TOPICS: Positive feedback · READINGS: N8.9
- SES #: 21 · TOPICS: Examples of design via root locus · READINGS: Notes
- SES #: 22 · TOPICS: Steady-state error compensation · READINGS: N9.1-9.2
- SES #: 23 · TOPICS: Transient response compensation; transient and steady-state error compensation · READINGS: N9.3-9.4
- SES #: 24 · TOPICS: Compensation examples · READINGS: Notes
- SES #: 25 · TOPICS: Feedback compensation and its physical realization · READINGS: N9.5-9.6
- SES #: 26 · TOPICS: Feedback design examples · READINGS: Notes
- SES #: 28 · TOPICS: Frequency response; bode plots · READINGS: N10.1-10.2
- SES #: 29 · TOPICS: Bode plot examples · READINGS: Notes
- SES #: 30 · TOPICS: Gain margin and phase margin · READINGS: N10.7
- SES #: 31 · TOPICS: Design using the frequency response; lead, lag, lead-lag compensators · READINGS: N11.1-11.5
- SES #: 32 · TOPICS: The state-space representation · READINGS: N3.1-3.6
- SES #: 33 · TOPICS: Solving the state equations in the time and space domains · READINGS: N4.10-4.11
- SES #: 34 · TOPICS: State equation examples · READINGS: Notes
- SES #: 35 · TOPICS: Stability and steady-state error in state space; controllability and observability · READINGS: N6.5, 7.8, 12.3, and 12.6
- SES #: 36 · TOPICS: Optimal control; the minimum time problem · READINGS: Notes
- SES #: 37 · TOPICS: Review: modeling and transfer functions · READINGS: Notes
- SES #: 38 · TOPICS: Review: root locus, feedback design · READINGS: Notes
- SES #: 39 · TOPICS: Review: frequency domain and design · READINGS: Notes
Spring 2003 · Modeling Dynamics and Control II
Topics
- Uniaxial motion, fixed-axis rotation; first and second order systems; natural and forced response; transfer functions, zeroes, poles; elementary control: open-loop, closed-loop; closing the loop alters the natural and forced response characteristics; stability and instability.
- Collisions; conservation of momentum; conservation of energy; coefficient of restitution; bouncing ball in the presence of air drag.
- Combined translation and rotation of rigid body; angular velocity; generalized coordinates; common constraints. Kinetic energy of rigid body.
- Linear and angular momentum principles for systems including rigid bodies; derivingequations of motion using momentum principles; conservation of energy; linearization in the neighbourhood of equilibrium, stability and instability; rolling cylinder inside rollinghoop; inverted pendulum on cart.
- Natural modes and the eigenvalue problem; free and forced response to harmonic excitation; damping effects; Bode plots.
- Actuators, sensors; P, D, I feedback-loop strategies; root-locus design; application to inverted pendulum on cart, automatic self-balancing scale; frequency-domain design methods, gain margin, phase margin.
Readings
- Bedford, A., and W. Fowler. Dynamics . Addison–Wesley, 1995.
- Palm III, W. J. Modeling, Analysis, and Control of Dynamic Systems . 2nd ed. John Wiley.
Prerequisites & course information
Prereq: Physics II (GIR) and 2.003[J] U (Fall, Spring) 4-2-6 units
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Spring 2008
- MIT OCW calendar ↗
- MIT OCW syllabus ↗
- MIT OpenCourseWare ↗Fall 2007
- MIT OCW calendar ↗
- MIT OCW syllabus ↗
- MIT OCW readings ↗
- MIT OpenCourseWare ↗Spring 2003
- MIT OCW readings ↗
- MIT OCW calendar ↗
- MIT OCW syllabus ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.