2002/03
Undergraduate Module Catalogue
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ELEC3150
Control
20 credits
Taught
Semester 2,
Year running
2002/03
Pre-requisites
ELEC2100
Co-requisites
None
Objectives
On completion of this module, students should be able to: System Compensation (5 lectures) (i) Know the definition of phase margin, gain margin and crossover frequency of a system and obtain these from Bode and Nyquist plots (ii) understand the relationship between crossover frequency and closed loop bandwidth and between phase margin and closed loop transient response (iii) know the characteristics and circuit realisation of phase lead and phase lag compensators, and be able to carry out simple compensator design to achieve given closed loop performance specifications. State Space (4 lectures) (i) Be able to derive the state space description of simple electrical and electro- mechanical systems, and to convert a single-input, single-output transfer functions into state space form, (ii) be able to determine system stability by finding eigenvalues. Controllability and Observability (7 lectures) (i) Understand the what is meant by controllability and carry out the rank test on the controllability matrix (ii) know that in a controllable system the closed loop poles may be placed arbitrarily using state feedback, and to carry out simple pole placement problems (v) understand what is meant by observability and carry out the rank test on the observability matrix (vi) know that it is possible to estimate the states of an observable system using either a full or reduced order observer having arbitrary poles, and to carry out simple observer design problems. Observer-Based Controller Design (4 lectures) (i) Understand that for a controllable and observable system is possible to design an observer-based controller such that the closed loop poles of the system are those of the observer together with those which would have been obtained using sate feedback. (ii) be able design observer-based controllers for simple systems and to appreciate that these have a structure similar to that of classical compensators. Introduction to Digital Control (1 lecture)Describe the structure of a computer-control system, with particular emphasis to the function of the digital controller as a recursive rule, the Zero-Order-Hold and the sampler; Explain the relevance of digital control to industrial applications. Z-Transforms and Discrete Linear Systems (5 lectures) Define the Z-transform as an infinite sum and derive the transforms of simple signals including step, ramp, exponential and sinusoid signals. Explain the properties of the transform, with particular emphasis on linearity, time-shift, the initial and final value theorem, and the convolution/multiplication duality; Recover time-sequences from their Z-transforms, via partial-fraction expansion and polynomial division. Explain the properties of Linear Time-Invariant (LTI) Systems, their unit-pulse response and their transfer function, and how these properties are related to one-another. Explain the concept of causality for LTI systems, and give realisability conditions of their transfer functions; Calculate Zero-Order-Hold equivalent discrete-time systems from continuous-time models. Explain what is the frequency-response of a discrete-time system, and how it can be used to obtain the steady-state output of stable systems driven by sinusoidal excitations; Sketch the magnitude and phase plots of simple systems. Z-Plane Methods (4 lectures) Explain how the transient response of a discrete-time system is related to the location of its poles in the Z-plane (stability region, number of samples per oscillation, constant damping/natural frequency contours, etc). State an appropriate transformation to map between the s- and z-planes, explain its properties, and describe how the poles of the closed-loop system should be selected to satisfy rise-time, overshoot and settling-time specifications, using second-order system approximations. Define the Nyquist frequency, explain the phenomena of aliasing and hidden oscillations, and describe how the sampling frequency of a digital control system shoul d be selected. Stability (2 lectures) Define stability in practical and mathematical terms (BIBO stability). Give conditions of stability in terms of (i) the location of system poles, (ii) the system?s impulse response, and (iii) the coefficients of the characteristic polynomial; Explain how Routh?s array test may be adapted to assess the stability of discrete-time systems using an appropriate bilinear transformation, and apply this method to simple examples involving parametric uncertainty, gain margin calculations, etc. Control Design (8 lectures) Explain the main objectives of control design, including stability, performance and robustness objectives. Calculate steady-state errors using the final value theorem, define system type and explain how it is related to steady-state errors for polynomial-type inputs (steps, ramps, etc.). Root Locus methods: Explain what is the root locus and describe the properties used for its construction including asymptotic behaviour, angles of departure/arrival, symmetry, double points, root-locus calibration, etc. Explain how the method can be extended to non-standard systems and how approximations can be made for its construction. Use the Root-locus to calculate stability margins, and explain how it can be used as a compensation tool for simple designs, including pole/zero cancellations and dipole addition. Frequency-domain methods: Explain how Nyquist?s stability criterion may be adapted to the discrete-time case; define gain/phase margins; describe how the closed-loop properties of a feedback system can be inferred from its open-loop Nyquist plot (steady-state error, damping, bandwidth, etc.) and carry out simple designs in the Nyquist plane via loop shaping. Explain the properties of Tustin?s-transformation and apply phase lead-lag techniques to design controllers for simple discrete-time systems.
Syllabus
Classical frequency domain design: Nyquist stability criterion. Gain and phase margins; relationship between gain and phase margins and closed loop response. Design of single loop control systems using pase lead and phase lag networks. State-variable analysis: State-variable description of continuous systems; conversion of transfer functions to state-space forms; conversion between continuous and discrete systems; canonical forms and pole-placement; system stability using eigenvalues; controllability, observability; state-feedback; state reconstruction with observers; reduced-order observers. Discrete systems: Sampled signals, the z-transform and relation between the s and z-planes; Discrete-time transfer functions and the unit pulse response; conversion between continuous and discrete systems; Frequency response; Reconstruction of signals from their samples; the zero order hold; Stability analysis: The bilinear transform and its use in extending the Routh-Hurwitz criterion to discrete-time systems; Root locus methods. Sampled-data systems: spectrum of sampled signals, aliasing and hidden oscillations Controller design and implementation: Closed-loop transfer function specification and pole-placement for desired transient response; The w-plane; discrete Bode and Nyquist plots and revision of gain and phase margins; Discrete-time compensation; dipole addition. Realisation of discrete-time controllers as difference equations. Digital controller implementation, choice of sampling rate and anti-aliasing filters.
Form of teaching
Lectures 44 x 1 hour; Practical Classes: 9 x 3 hours CAD labs.
Form of assessment
1 x 3 hour examination 70% and 4 x reports 30%.
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