Multivariable Control
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Multivariable Control. Root Locus . By Kirsten Mølgaard Nielsen. Outline. Root Locus Design Method Root locus definition The phase condition Test points, and sketching a root locus Dynamic Compensation Lead compensation Lag compensation. Introduction.

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Multivariable Control

Root Locus

By Kirsten Mølgaard Nielsen


Outline
Outline

  • Root Locus Design Method

    • Root locus

      • definition

      • The phase condition

      • Test points, and sketching a root locus

    • Dynamic Compensation

      • Lead compensation

      • Lag compensation


Introduction
Introduction

Closed loop transfer function

Characteristic equation,

roots are poles in T(s)


Introduction1
Introduction

  • Dynamic features depend on the pole locations.

  • For example, time constant t, rise time tr, and overshoot Mp

(1st order)

(2nd order)


Introduction2
Introduction

Root locus

  • Determination of the closed loop pole locations under varying K.

  • For example, K could be the control gain.

The characteristic equation can be written in various ways


Introduction3
Introduction

  • Root locus of a motor position control (example)


Introduction4
Introduction

break-away point


Introduction5
Introduction

Some root loci examples (K from zero to infinity)


Sketching a root locus
Sketching a Root Locus

  • Definition 1

    • The root locus is the values of s for which 1+KL(s)=0 is satisfied as K varies from 0 to infinity (pos.).

  • Definition 2

    • The root locus is the points in the s-plane where the phase of L(s) is 180°.

      • The angle to a point on the root locus from zero number i is yi.

      • The angle to a point on the root locus from pole number i is fi.

      • Therefore,


Sketching a root locus1
Sketching a Root Locus

s0

f1

p

y1

f2

p*

A root locus can be plotted using Matlab: rlocus(sysL)


Root locus characteristics
Root Locus characteristics

  • Rule 1 (of 6)

    • The n branches of the locus start at the poles L(s) and m of these branches end on the zeros of L(s).


Root locus characteristics1
Root Locus characteristics

  • Rule 2 (of 6)

    • The loci on the real axis (real-axis part) are to the left of an odd number of poles plus zeros.

  • Notice, if we take a test point s0 on the real axis :

    • The angle of complex poles cancel each other.

    • Angles from real poles or zeros are 0° if s0 are to the right.

    • Angles from real poles or zeros are 180° if s0 are to the left.

    • Total angle = 180° + 360° l


Root locus characteristics2
Root Locus characteristics

  • Rule 3 (of 6)

    • For large s and K, n-m branches of the loci are asymptotic to lines at angles fl radiating out from a point s = a on the real axis.

fl

a

For example


Root locus characteristics3
Root Locus characteristics

n-m=1

n-m=2

n-m=3

n-m=4

For ex.,

n-m=3


Root locus
Root Locus

3 poles:

q = 3

  • Rule 4: The angles of departure (arrival) of a branch of the locus from a pole (zero) of multiplicity q.

f1,dep = 60 deg.

f2,dep = 180 deg.

f3,dep = 300 deg.


Root locus characteristics4
Root Locus characteristics

  • Rule 5 (of 6)

    • The locus crosses the jw axis at points where the Routh criterion shows a transition from roots in the left half-plane to roots in the right half-plane.

    • Routh: A system is stable if and only if all the elements in the first column of the Routh array are positive.


Root locus characteristics5
Root Locus characteristics

  • Rule 6 (of 6)

    • The locus will have multiplicative roots of q at points on the locus where (1) applies.

    • The branches will approach a point of q roots at angles separated by (2) and will depart at angles with the same separation.

Characteristic equation


Example

Root locus for double integrator with P-control

Rule 1: The locus has two branches that starts in s=0. There are no zeros. Thus, the branches do not end at zeros.

Rule 3: Two branches have asymptotes for s going to infinity. We get


Rule 2: No branches that the real axis.

Rule 4: Same argument and conclusion as rule 3.

Rule 5: The loci remain on the imaginary axis. Thus, no crossings of the jw-axis.

Rule 6: Easy to see, no further multiple poles. Verification:


Example

Root locus for satellite attitude control with PD-control


Selecting the parameter value
Selecting the Parameter Value

  • The (positive) root locus

    • A plot of all possible locations of roots to the equation 1+KL(s)=0 for some real positive value of K.

    • The purpose of design is to select a particular value of K that will meet the specifications for static and dynamic characteristics.

    • For a given root locus we have (1). Thus, for some desired pole locations it is possible to find K.


Selecting the parameter value1
Selecting the Parameter Value

Example

Selection of K using Matlab: [K,p]=rlocfind(sysL)


Dynamic compensation
Dynamic Compensation

  • Some facts

    • We are able to determine the roots of the characteristic equation (closed loop poles) for a varying parameter K.

    • The location of the roots determine the dynamic characteristics (performance) of the closed loop system.

    • It might not be possible to achieve the desired performance with D(s) = K.

  • Controller design using root locus

    • Lead compensation (similar to PD control)

      • overshoot, rise time requirements

    • Lag compensation (similar to PI control)

      • steady state requirements


  • Dynamic compensation1
    Dynamic Compensation

    PD

    Lead

    PI

    Lag


    Lead compensation
    Lead Compensation

    PD

    P

    Lower damping ratio with PD !


    Lead compensation1
    Lead Compensation

    • PD control

      • Pure differentiation

      • Output noise has a great effect on u(t)

      • Solution: Insert a pole at a higher frequency

    r(t)

    e(t)

    controller

    D(s)

    u(t)

    plant

    G(s)

    y(t)

    +

    -

    +

    +

    noise



    Lead compensation3
    Lead Compensation

    • Selecting p and z

      • Usually, trial and error

      • The placement of the zero is determined by dynamic requirements (rise time, overshoot).

      • The exact placement of the pole is determined by conflicting interests.

        • suppression of output noise

        • effectiveness of the zero

      • In general,

        • the zero z is placed around desired closed loop wn

        • the pole p is placed between 5 and 20 times of z


    Lead compensation4
    Lead Compensation

    • Design approach (A)

      • Initial design z=wn, p=5z

      • Noise: additional requirement, max(p)=20

      • Iterations

        • First design: z=wn, p=5z

          (neglecting the add. req.)

        • Second design: z=wn, p=20

        • Third design: new z, p=20


    Lead compensation5
    Lead Compensation

    Possible

    pole location

    Matlab – design 1

    sysG = tf([1],[1 1 0])

    sysD = tf([1 7],[1 5*7])

    rlocus(sysD*sysG)


    Lead compensation6
    Lead Compensation

    No possible

    pole location !

    Matlab – design 2

    sysG = tf([1],[1 1 0])

    sysD = tf([1 7],[1 20])

    rlocus(sysD*sysG)


    Lead compensation7
    Lead Compensation

    Possible

    location

    New zero

    location

    Matlab – design 3

    sysG = tf([1],[1 1 0])

    sysD = tf([1 4],[1 20])

    rlocus(sysD*sysG)


    Lead compensation8
    Lead Compensation

    • Design approach (B)

      • Pole placement from the requirements

      • Noise: Additional requirement, max(p)=20. So, p=20 to minimize the effect of the pole.



    Lead compensation10
    Lead Compensation

    Pole location

    Matlab – design B

    sysG = tf([1],[1 1 0])

    sysD = tf([1 5.4],[1 20])

    rlocus(sysD*sysG)


    Lead compensation11
    Lead Compensation

    • Exercise

      • Design a lead compensation D(s) to the plant G(s) so that the dominant poles are located at s = -2 ± 2j

    You can use, triangle

    a/sin(A) = b/sin(B)

    r(t)

    e(t)

    controller

    D(s)

    u(t)

    plant

    G(s)

    y(t)

    +

    -


    Lead compensation12
    Lead Compensation

    • Answer

      • we have three poles and one zero

    f1

    y

    f2

    Notice, triangle

    a/sin(A) = b/sin(B)


    Dynamic compensation2
    Dynamic Compensation

    PD

    Lead

    PI

    Lag


    Lag compensation
    Lag Compensation

    • Lag compensation

      • Steady state characteristics (requirements)

        • Position error constant Kp

        • Velocity error constant Kv

      • Let us continue using the example system

      • and the designed controller (lead compensation)


    Lag compensation1
    Lag Compensation

    • Calculation of error constants

      • Suppose we want Kv≥ 100

      • We need additional gain at low frequencies !


    Lag compensation2
    Lag Compensation

    • Selecting p and z

      • To minimize the effect on the dominant dynamics z and p must chosen as low as possible (i.e. at low frequencies)

      • To minimize the settling time z and p must chosen as high as possible (i.e. at high frequencies)

      • Thus, the lag pole-zero location must be chosen at as high a frequency as possible without causing any major shifts in the dominant pole locations


    Lag compensation3
    Lag Compensation

    • Design approach

      • Increase the error constant by increasing the low frequency gain

      • Choose z and p somewhat below wn

      • Example system:

        • K = 1 (the proportional part has already been chosen)

        • z/p = 3, with z = 0.03, and p = 0.01.


    Lag compensation4
    Lag Compensation

    Notice, a slightly

    different pole

    location

    Lead-lag

    Controller





    Notch compensation
    Notch compensation

    • Example system

      • We have successfully designed a lead-lag controller

      • Suppose the real system has a rather undampen oscillation about 50 rad/sec.

      • Include this oscillation in the model

      • Can we use the original controller ?


    Notch compensation1
    Notch compensation

    Step response using the lead-lag controller


    Notch compensation2
    Notch compensation

    • Aim: Remove or dampen the oscillations

    • Possibilities

      • Gain stabilization

        • Reduce the gain at high frequencies

        • Thus, insert poles above the bandwidth but below the oscillation frequency – might not be feasible

      • Phase stabilization (notch compensation)

        • A zero near the oscillation frequency

        • A zero increases the phase, and z≈ PM/100

        • Possible transfer function

    If zn < 1, complex zeros

    and double pole at w0



    Notch compensation4
    Notch compensation

    Cancellation

    Root Locus

    The lead-lag controller with notch compensation

    w0 = 50

    z = 0.01


    Notch compensation5
    Notch compensation

    Too much overshoot !!

    Also, exact cancellation might cause problems due to modelling errors

    Thus, we need different parameters


    Notch compensation6
    Notch compensation

    • Modified parameters

      • Notch compensation

      • Dynamic characteristics

        • Increase zn to obtain less overshoot

        • Make sure that the roots move into the LHP

        • In general, obtain a satisfactory dynamic behavior

      • After some trial and error


    Notch compensation7
    Notch compensation

    The overshoot has been lowered to an acceptable level !

    The oscillations have almost been removed !

    So, we have reached a final design !


    Notch compensation8
    Notch compensation

    Root Locus

    The lead-lag controller with notch compensation

    w0 = 60

    z = 0.3


    Time delay
    Time Delay

    • Notice

      • Time delay always reduces the stability of a system !

      • Important to be able to analyze its effect

      • In the s-domain a time delay is given by e-ls

      • Most applications contain delays (sampled systems)

    • Root locus analysis

      • The original method does only handle polynomials

    • Solutions

      • Approximation (Padé) of e-ls

      • Modifying the root locus method (direct application)



    Time delay2
    Time Delay

    No delay

    Padé (1,1)

    Exact

    Root locus for some example system (heat exchanger)

    Padé (2,2)

    (p,p) Approx. in Matlab :

    [num,den]=pade(T,p)


    Time delay3
    Time Delay

    However, Matlab does not support this approach...

    In discrete systems the sampling time introduce a time delay


    The discrete root locus
    The Discrete Root Locus

    • Discrete (z-domain)

      • Closed loop transfer function

      • Characteristic equation

      • Thus, same sketching techniques as in the s-domain

      • However, different interpretation !!!


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