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  • Control system design essentials The root-locus method to controller design

    Spacemaster Preparation Course Control system design basics

    Lei Ma

    Department of Computer Science VII: Robotics and Telematics University of Würzburg

    10.10.2007

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Outline

    1 Control system design essentials Objectives The standard control loop Behaviors of the control loop The controllers

    2 The root-locus method to controller design The idea Constructing root loci Controller design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    References

    Ogata,K. Modern control engineering. Prentice Hall. Brogan,W.L. Modern control theory. Prentice Hall. Lunze,J. Regelungstechnik 1,2. Springer. Goodwin,G.C et al. Control system design. Prentice Hall. Jeffrey, A. Mathematics for Engineers and Scientists. Chapman & Hall/CRC 2004 Stroud, K.A., Booth, D.J. Engineering Mathematics. Industrial Press 2001 Dettman, J.W. Introduction to Linear Algebra and Differential Equations. Dover Publications 1986. Golub, G.H., Van Loan, C.F. Matrix Computations (Johns Hopkins Studies in Mathematical Sciences). The Johns Hopkins University Press 1996

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    Outline

    1 Control system design essentials Objectives The standard control loop Behaviors of the control loop The controllers

    2 The root-locus method to controller design The idea Constructing root loci Controller design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    The principal goal of control

    The fundamental goal of control engineering is to find technically, environmentally, and economically feasible ways of acting on systems to control their outputs to desired values, thus ensuring a desired level of performance. (Goodwin)

    Stability

    Disturbance rejection

    Regulation and tracking

    Dynamics

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    Typical unit-step response of a second-order system

    h(t)

    t

    Mp

    tr

    tp

    ts

    td

    Allowable tolerance (5% or

    2% of the desired output)

    Delay time td Rise time tr (underdamped 0~100%, overdamped 10%~90%)

    Peak time tp Maximum overshoot Mp

    Settling time ts

    Figure: Typical unit-step response of a second-order system

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    Outline

    1 Control system design essentials Objectives The standard control loop Behaviors of the control loop The controllers

    2 The root-locus method to controller design The idea Constructing root loci Controller design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    Some popular structures of control

    r(t) K(s) G(s)

    H(s)

    e(t) u(t) y(t)

    _ controller plant

    sensor

    K(s) G(s) e(t) u(t) y(t)

    _ controller plant

    Standard control loop Unit-feedback control

    r(t)

    K1(s) G(s)

    H(s)

    e(t) u(t) y(t)

    _ Controller 1 plant

    sensor

    r(t)

    K2(s)

    _

    Controller 2

    Inner loop

    Outer loop

    K(s) G(s)

    H(s)

    e(t) u(t) y(t)

    _ controller plant

    sensor

    K(s) r(t)

    Standard control loop

    with pre-filter

    Pre-filter

    Cascad control

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    The transfer functions

    K(s) G(s) e(t) u(t) y(t)

    _ controller plant

    Unit-feedback control with disturbances

    and noisy measurement

    r(t)

    d(t)

    disturbances

    n(t)

    noise

    Open-loop TF.: G0(s) = K (s)G(s)

    Closed-loop TF.: Gr (s) = G0(s)

    1+G0(s) , (D(s) = N(s) = 0)

    Closed-loop TF. according to disturbances: Gd(s) = 11+G0(s) , (R(s) = N(s) = 0)

    Closed-loop TF. according to noise: Gn(s) = −

    G0(s) 1+G0(s)

    = −Gr (s), (R(s) = D(s) = 0)

    Y (s) = Gr (s)R(s) + Gd(s)D(s) + Gn(s)N(s)

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    Outline

    1 Control system design essentials Objectives The standard control loop Behaviors of the control loop The controllers

    2 The root-locus method to controller design The idea Constructing root loci Controller design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    The steady-state error

    Steady-state error of step response with d(t) = 0 and n(t) = 0: limt→∞ e(t) = lims→0 s 1s

    E(s) R(s) = lims→0

    1 1+G0(s)

    .

    G0(s) does play an important role! In case n ≥ q, let

    G0(s) = ksl bqsq+bq−1sq−1+...+b1s+b0

    an−l sn−l+an−l−1sn−l−1+...+a1s+a0 , the follows are true:

    For l = 0, G0(0) = k0 = k b0 a0

    and limt→∞ e(t) = 11+k0 ;

    For l > 0, lims→0 G0(s) = ∞ and limt→∞ e(t) = 0.

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    Outline

    1 Control system design essentials Objectives The standard control loop Behaviors of the control loop The controllers

    2 The root-locus method to controller design The idea Constructing root loci Controller design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    Dynamic performance of a second-order system

    h(t)

    t

    Mp

    tr

    tp

    ts

    td

    Allowable tolerance (5% or

    2% of the desired output)

    Delay time td Rise time tr (underdamped 0~100%, overdamped 10%~90%)

    Peak time tp Maximum overshoot Mp

    Settling time ts

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    Objectives The standard control loop Behaviors of the control loop The controllers

    The PID controller

    kp

    kI/s

    kDs

    kp 1/sTI

    TDs

    = E(s)E(s) U(s) U(s)

    KPID(s) = kP + kI s

    + kDs = kP(1 + 1

    sTI + sTD)

    Rise time Overshoot Settling time S-S error kP Decrease Increase Small change Decrease kI Decrease Increase Increase Eliminate kD Small change Decrease Decrease Small change

    Table: Impact of P, I, D on dynamic performances

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    The idea Constructing root loci Controller design based on root locus

    Outline

    1 Control system design essentials Objectives The standard control loop Behaviors of the control loop The controllers

    2 The root-locus method to controller design The idea Constructing root loci Controller design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    The idea Constructing root loci Controller design based on root locus

    Open-loop and closed-loop system

    G(s)

    H(s)

    R(s) Y(s)

    Open-loop system: G(s)H(s)

    Closed-loop transfer function:

    Y (s) R(s)

    = G(s)

    1 + G(s)H(s)

    The characteristic equation:

    1 + G(s)H(s) = 1 + K (s + z1)(s + z2) . . . (s + zm) (s + p1)(s + p2) . . . (s + pn)

    = 0

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentials The root-locus method to controller design

    The idea Constructing root loci Controller design based on ro