Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 116 views

    Unit-III - Root Locus

    Uploaded by

    Rachana Gade
    This document summarizes a lecture on stability analysis using root locus. It introduces root locus as a technique to analyze how the location of closed-loop poles changes with variations in system parameters. Examples are provided to illustrate how to construct root loci and use them to design controllers. The advantages of root locus include indicating how to modify system poles and zeros to meet performance specifications without detailed analysis of the closed loop system.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Unit-III - Root Locus

    Uploaded by

    Rachana Gade
    116 views67 pages
    This document summarizes a lecture on stability analysis using root locus. It introduces root locus as a technique to analyze how the location of closed-loop poles changes with variations in system parameters. Examples are provided to illustrate how to construct root loci and use them to design controllers. The advantages of root locus include indicating how to modify system poles and zeros to meet performance specifications without detailed analysis of the closed loop system.

    Original Description:

    Feedback control systems

    Original Title

    Unit-III_Root locus PPT

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    This document summarizes a lecture on stability analysis using root locus. It introduces root locus as a technique to analyze how the location of closed-loop poles changes with variations in system parameters. Examples are provided to illustrate how to construct root loci and use them to design controllers. The advantages of root locus include indicating how to modify system poles and zeros to meet performance specifications without detailed analysis of the closed loop system.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    116 views67 pages

    Unit-III - Root Locus

    Uploaded by

    Rachana Gade
    This document summarizes a lecture on stability analysis using root locus. It introduces root locus as a technique to analyze how the location of closed-loop poles changes with variations in system parameters. Examples are provided to illustrate how to construct root loci and use them to design controllers. The advantages of root locus include indicating how to modify system poles and zeros to meet performance specifications without detailed analysis of the closed loop system.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 67

    IFC course

    Feedback Control System (FCS)


    [ICE(IF)-20001]

    Unit:
    Stability Analysis using Root Locus
    Dr. S. D. Hanwate
    Faculty, Dept. of Instrumentation and Control
    College of Engineering Pune
    Introduction
    • Primary aim of a control engineer: To design a control system that
    meets the desired specifications.
    • Stability of the system is also a necessary condition.
    • Analytical approach can be followed till second order systems while
    Routh-Hurwitz criterion can be followed for higher order systems.
    • Drawbacks of Routh-Hurwitz criterion -
    (a) does not provide sufficient information about relative stability, which
    may make the system unstable,
    (b) does not help in designing a control system where tuning of parameters
    is important.
    Example:
    Root Locus Technique
    Root Locus: Example
    Root Locus: Example (contd.)
    Root Locus: Example (contd.)
    Advantages
    • The advantages of root locus technique are:
    • Clearly showing the contributions of each loop poles or zeros to the
    location of the closed loop poles.
    • Indicating the manner in which the loop poles and zeros should be
    modified so that the response meets system performance
    specifications.
    • Knowledge of the open loop system is sufficient to analyse the
    behaviour of the system, detailed study of the closed loop system is
    not required.
    System Parameters and Pole Locations
    System Parameters and Pole Locations
    System Parameters and Pole Locations
    Evans’ Conditions
    Evans’ Conditions (contd.)
    Points on Root locus
    Example
    Rules for constructing Root Loci
    • Segment of root loci can exist in the right half of s-plane. This signifies instability, the points at
    which the root loci cross the imaginary axis define the stability limits.
    • Basic application of the Routh array determines the gains at the satability limits. By substituting
    this value of gain in the auxiliary equation, the value of s=jw at the stability limit is evaluated.
    Illustrative example:
    Design problem:
    Over all Closed loop transfer function:
    Effects of Pole and Zeros on Root locus
    Example 2:
    Control system Design Problem:
    Effect of addition of zero (derivative) term
    Example:
    • For the unity feedback system in Figure,

    • Design a controller for following specification:


    a) Design rate feedback to yield a step response with no more than
    25% overshoot and settling time < 2 seconds.
    b) Plot the states, control signal u without and with controller.
    Solution:
    • The desired operating point is found from the desired specifications,

    • Thus,
    • Hence the design point is –2 +j4.5324.
    • Now, add a pole at the origin to increase system type and drive error
    to zero for step inputs.
    • Now design a PD controller. The angular contribution to the design
    point of the system poles and pole at the origin is 101.9 ⁰. Thus, the
    compensator zero must contribute 180⁰ – 101.9 ⁰ =78.1 ⁰. Using
    thegeometry below,
    • Using Hence, zc = 2.955.
    • The compensated open-loop transfer function with PD compensation
    is

    • Adding the compensator zero to the system and evaluating the gain
    for this at the point –2 + j4.5324,
    • yields K = 294.51.
    Matlab code:
    K= 294.51 %root locus gain for new compensated system
    num = [-2.995]
    den = [0 -4 -6 -10]
    sys = zpk(num,den,K)
    rlocus(sys)
    grid
    v = [-15 1 -10 10];axis(v)
    hold on
    plot(-2,0,'*')
    hold on
    plot(0,4.5324,'*')
    T=feedback(sys,1);
    figure
    step(T)
    stepinfo(T)
    • Since we have able to obtain the both the desired performance
    • Mp < 25% = 19.3%
    • Ts < 2 sec = 1.78 sec.
    References:
    1) K. Ogata, Modern Control Engineering. Prentice Hall, 2010.
    2) Norman Nise, “Control System Engineering”, Wiley India, Fifth ed., 2009
    3) G. C. Goodwin, S. F. Graebe, M. E. Salgado; “Control System Design”, PHI, First ed. 2002
    4) Friedland, “Control System Design”, Dover Publication, First ed., 2005.
    5) NPTEL Video lecture:
    ➢ Lec1
    ➢ Lec2
    ➢ Lec3

    FCS (AY 2021-22) / COEP Dr. S. D. Hanwate

    You might also like