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# Multivariable Control - PowerPoint PPT Presentation

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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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

• 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)

• 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,

s0

f1

p

y1

f2

p*

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

• 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).

• 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

• 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

n-m=1

n-m=2

n-m=3

n-m=4

For ex.,

n-m=3

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.

• 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.

• 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

• 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.

Example

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

• 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)

• PD

PI

Lag

PD

P

Lower damping ratio with PD !

• 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

• 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

• Design approach (A)

• Initial design z=wn, p=5z

• Iterations

• First design: z=wn, p=5z

• Second design: z=wn, p=20

• Third design: new z, p=20

Possible

pole location

Matlab – design 1

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

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

rlocus(sysD*sysG)

No possible

pole location !

Matlab – design 2

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

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

rlocus(sysD*sysG)

Possible

location

New zero

location

Matlab – design 3

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

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

rlocus(sysD*sysG)

• Design approach (B)

• Pole placement from the requirements

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

Pole location

Matlab – design B

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

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

rlocus(sysD*sysG)

• 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)

+

-

• we have three poles and one zero

f1

y

f2

Notice, triangle

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

PD

PI

Lag

• Lag compensation

• Position error constant Kp

• Velocity error constant Kv

• Let us continue using the example system

• and the designed controller (lead compensation)

• Calculation of error constants

• Suppose we want Kv≥ 100

• We need additional gain at low frequencies !

• 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

• 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.

Notice, a slightly

different pole

location

Controller

Step

Ramp

• 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 ?

Step response using the lead-lag controller

• 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

Cancellation

Root Locus

The lead-lag controller with notch compensation

w0 = 50

z = 0.01

Too much overshoot !!

Also, exact cancellation might cause problems due to modelling errors

Thus, we need different parameters

• 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

The overshoot has been lowered to an acceptable level !

The oscillations have almost been removed !

So, we have reached a final design !

Root Locus

The lead-lag controller with notch compensation

w0 = 60

z = 0.3

• 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

• Modifying the root locus method (direct application)

No delay

Exact

Root locus for some example system (heat exchanger)

(p,p) Approx. in Matlab :

However, Matlab does not support this approach...

In discrete systems the sampling time introduce a time delay

• Discrete (z-domain)

• Closed loop transfer function

• Characteristic equation

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

• However, different interpretation !!!