Multivariable Control

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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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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
• Lag compensation
Introduction

Closed loop transfer function

Characteristic equation,

roots are poles in T(s)

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

(1st order)

(2nd order)

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

Introduction
• Root locus of a motor position control (example)
Introduction

break-away point

Introduction

Some root loci examples (K from zero to infinity)

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 Locus

s0

f1

p

y1

f2

p*

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

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 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 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 characteristics

n-m=1

n-m=2

n-m=3

n-m=4

For ex.,

n-m=3

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 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 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
• 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 Value

Example

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

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)

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)

Lag Compensation
• Lag compensation
• Position error constant Kp
• Velocity error constant Kv
• Let us continue using the example system
• and the designed controller (lead compensation)
Lag Compensation
• Calculation of error constants
• Suppose we want Kv≥ 100
• We need additional gain at low frequencies !
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 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 Compensation

Notice, a slightly

different pole

location

Controller

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 compensation

Step response using the lead-lag controller

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 compensation

Cancellation

Root Locus

The lead-lag controller with notch compensation

w0 = 50

z = 0.01

Notch compensation

Too much overshoot !!

Also, exact cancellation might cause problems due to modelling errors

Thus, we need different parameters

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 compensation

The overshoot has been lowered to an acceptable level !

The oscillations have almost been removed !

So, we have reached a final design !

Notch compensation

Root Locus

The lead-lag controller with notch compensation

w0 = 60

z = 0.3

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
• Modifying the root locus method (direct application)
Time Delay

No delay

Exact

Root locus for some example system (heat exchanger)

(p,p) Approx. in Matlab :