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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

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  • Control system design essentialsThe root-locus method to controller design

    Spacemaster Preparation CourseControl system design basics

    Lei Ma

    Department of Computer Science VII: Robotics and TelematicsUniversity of Würzburg

    10.10.2007

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    Outline

    1 Control system design essentialsObjectivesThe standard control loopBehaviors of the control loopThe controllers

    2 The root-locus method to controller designThe ideaConstructing root lociController design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe 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 2004Stroud, K.A., Booth, D.J. Engineering Mathematics.Industrial Press 2001Dettman, J.W. Introduction to Linear Algebra andDifferential Equations. Dover Publications 1986.Golub, G.H., Van Loan, C.F. Matrix Computations (JohnsHopkins Studies in Mathematical Sciences). The JohnsHopkins University Press 1996

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe controllers

    Outline

    1 Control system design essentialsObjectivesThe standard control loopBehaviors of the control loopThe controllers

    2 The root-locus method to controller designThe ideaConstructing root lociController design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe controllers

    The principal goal of control

    The fundamental goal of control engineering is to findtechnically, environmentally, and economically feasible ways ofacting 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 essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe 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 tdRise time tr (underdamped 0~100%, overdamped 10%~90%)

    Peak time tpMaximum 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 essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe controllers

    Outline

    1 Control system design essentialsObjectivesThe standard control loopBehaviors of the control loopThe controllers

    2 The root-locus method to controller designThe ideaConstructing root lociController design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe 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 essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe 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 essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe controllers

    Outline

    1 Control system design essentialsObjectivesThe standard control loopBehaviors of the control loopThe controllers

    2 The root-locus method to controller designThe ideaConstructing root lociController design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe 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

    11+G0(s)

    .

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

    G0(s) = kslbqsq+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 = kb0a0

    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 essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe controllers

    Outline

    1 Control system design essentialsObjectivesThe standard control loopBehaviors of the control loopThe controllers

    2 The root-locus method to controller designThe ideaConstructing root lociController design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe 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 tdRise time tr (underdamped 0~100%, overdamped 10%~90%)

    Peak time tpMaximum overshoot Mp

    Settling time ts

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    ObjectivesThe standard control loopBehaviors of the control loopThe controllers

    The PID controller

    kp

    kI/s

    kDs

    kp 1/sTI

    TDs

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

    KPID(s) = kP +kIs

    + kDs = kP(1 +1

    sTI+ sTD)

    Rise time Overshoot Settling time S-S errorkP Decrease Increase Small change DecreasekI Decrease Increase Increase EliminatekD 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 essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Outline

    1 Control system design essentialsObjectivesThe standard control loopBehaviors of the control loopThe controllers

    2 The root-locus method to controller designThe ideaConstructing root lociController design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController 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 essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Idea: Use the poles and zeroes of the open loop system todetermine the properties of the closed loop system when ONEparameter is changing.

    In control theory, the root locus is the locus of the poles of atransfer function as the system gain K is varied on someinterval. The root locus is a useful tool for analyzing single inputsingle output (SISO) linear dynamic systems. A system isstable if all of its poles are in the left-hand side of the s-plane.(Wikipedia)

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Angle and magnitude conditions

    the characteristic equation indicates that G(s)H(s) = −1.Therefore the values of s that fulfill both the angle andmagnitude conditions are the roots of the characteristicequation, or the closed-loop poles.

    Angle condition:∠G(s)H(s) = ±180◦(2k + 1), k = 0, 1, 2, . . ..

    Magnitude condition: |G(s)H(s)| = 1.

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Outline

    1 Control system design essentialsObjectivesThe standard control loopBehaviors of the control loopThe controllers

    2 The root-locus method to controller designThe ideaConstructing root lociController design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Characters of root loci

    Symmetry: As all roots are either real or complexconjugate pairs so that the root locus is symmetrical to thereal axis.

    Number of branches: The number of branches of the rootlocus is equal to the number of poles of the open-looptransfer function.

    Locus start and end points: The locus starting points are atthe open-loop poles and the locus ending points are at theopen-loop zeros. n − m branches end at infinity. Thenumber of starting branches from a pole and endingbranches at a zero is equal to the multiplicity of the polesand zeros, respectively. A point at infinity is considered asan equivalent zero of multiplicity equal to n − m.

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    General rules of constructing root loci (Ogata)

    Locate the poles and zeros of G(s)H(s) on the s plane.Determine the root loci on the real axis.Determine the asymptotes of root loci.Find the break-away and break-in points.Determine the angle of departure (angle of arrival) of theroot locus from a complex pole (at a complex zero).Find the points where the root loci may cross the imaginaryaxis.Taking a series of test points in the broad neighborhood ofthe origin of the s plane.Determine closed-loop poles.

    For details see Ogata!

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Example

    G(s)

    H(s)

    R(s) Y(s)

    G(s) = Ks(s+1)(s+2) , H(s) = 1

    Angle condition: ∠G(s) = ∠ Ks(s+1)(s+2) =

    −∠s − ∠(s + 1) − ∠(s + 2) = ±180◦

    (2k + 1), k = 0, 1, 2, . . .

    Magnitude condition: |G(s)| =∣

    Ks(s+1)(s+2)

    ∣= 1.

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Example

    Locate the poles and zeros of G(s)H(s) on the s plane: 3poles, no zeros. Root loci have 3 branches.

    Determine the root loci on the real axis using anglecondition.

    Determine the asymptotes of root loci

    Angles of asymptotes = ±180◦

    (2k+1)3 , k = 0, 1, 2, . . .;

    Finding the point where the asymptotes intersect the realaxis:s = −

    ∑ni=1 pi−

    ∑qi=1 zi

    n−q .

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Example

    Find the break-away point where the two branches startfrom s = 0 and s = −1 leave the real axis and go to thecomplex plane: dkds = 0.

    Determine the points where the root loci cross theimaginary axis by setting s = jω.

    Choose a test point in the broad neighborhood of the jωaxis and the origin.

    Draw the root loci.

    Determine a pair of dominant complex-conjugateclosed-loop poles such that ζ = 0.5.

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Typical distributions of open-loop poles and zeros andthe root loci

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Typical distributions of open-loop poles and zeros andthe root loci

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Outline

    1 Control system design essentialsObjectivesThe standard control loopBehaviors of the control loopThe controllers

    2 The root-locus method to controller designThe ideaConstructing root lociController design based on root locus

    Lei Ma Pre-Course: Control system design basics

  • Control system design essentialsThe root-locus method to controller design

    The ideaConstructing root lociController design based on root locus

    Typical design procedure using root locus

    Derive locations of the dominant pare of poles based onthe design criteria;

    Choose a controller kK̂ (s);

    Draw the root loci for the open-loop systemkK̂ (s)G(s)H(s);

    Determine value of the gain k which gives the dominantpare of poles. Determine locations of other poles;

    Simulate the system. Choose complexer K̂ (s) if the designcriteria are not satisfied.

    controller: kK̂ (s).

    Lei Ma Pre-Course: Control system design basics

    Control system design essentialsObjectivesThe standard control loopBehaviors of the control loopThe controllers

    The root-locus method to controller designThe ideaConstructing root lociController design based on root locus