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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

  • Root Locus Design Method
    • Root locus
      • definition
      • The phase condition
      • Test points, and sketching a root locus
    • Dynamic Compensation
      • Lead compensation
      • Lag compensation

Closed loop transfer function

Characteristic equation,

roots are poles in T(s)

  • Dynamic features depend on the pole locations.
  • For example, time constant t, rise time tr, and overshoot Mp

(1st order)

(2nd order)


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

  • Root locus of a motor position control (example)

break-away point


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







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.



For example

root locus characteristics3
Root Locus characteristics





For ex.,


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



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:



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


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
lead compensation
Lead Compensation



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














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


pole location

Matlab – design 1

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

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


lead compensation6
Lead Compensation

No possible

pole location !

Matlab – design 2

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

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


lead compensation7
Lead Compensation



New zero


Matlab – design 3

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

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


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])


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)











lead compensation12
Lead Compensation
  • Answer
    • we have three poles and one zero




Notice, triangle

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

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




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


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)


Root locus for some example system (heat exchanger)

Padé (2,2)

(p,p) Approx. in Matlab :


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 !!!