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### Loop Shaping

Professor Walter W. Olson

Department of Mechanical, Industrial and Manufacturing Engineering

University of Toledo

Outline of Today’s Lecture

- Review
- PID Theory
- Integrator Windup
- Noise Improvement

- Static Error Constants (Review)
- Loop Shaping
- Loop Shaping with the Bode Plot
- Lead and Lag Compensators
- Lead design with Bode plot
- Lead design with root locus
- Lag design with Bode plot

PID: A Little Theory

- Consider a 1st order function where the 1st method of Ziegler Nichols applies
- The general transfer function for this system is
- The term is the transport lag and delays the action for t0 seconds. Therefore
- The term Ta is the time constant for the system. T measured on the graph is an estimate of this.

PID: A Little Theory

- The method 1 PI controller applied to the loop equation is

PID: A Little Theory

- In Method 2, the gain was increased until the system was nearly a perfect oscillatory system.
- Since the gain changes the oscillatory patterns, the lowest order system that this could represent would by a 3rd order system.
- For this system to oscillate, there must be a solution of the characteristic function for K real and positive where s=±wi

PID: A Little Theory

- Applying the PI Controller:

Integrator Windup

- We have tacitly assumed that the controlled devices could meet the demands of the controls that we designed.
- However real devices have limitations that may prevent the system from responding adequately to the control signal
- When this occurs with an integrating controller, the error which is used to amplify the control signal may build up and saturate the controller.
- We refer to this as “integrator windup”:
- the system can’t respond and the integrator signal is extremely large (often maxed out on a real controller)
- the result is an uncontrolled system that can not return to normal operating conditions until the controller is reset

Integrator Windup

- To avoid windup, a possible solution is to provide a correcting error from the actuator by adding another loop:(the actuator has to be extracted from the plant)

+

-1

+

+

+

+

+

+

+

+

-1

Derivative Noise Improvement

- A major problem with using the derivative part of the PID controller that the derivative has the effect of amplifying the high frequency components which, for most systems, is likely to be noise.

Without PID

With PID

Derivative Noise Improvement

- One way to improve the noise rejection at higher frequencies is to apply a second order filter that passes low frequency and rejects high frequency
- The natural frequency of the filter should be chosen aswith N chosen to give the controller the bandwidth necessary, usually in the range of 2 to 20
- The controller then has the design

Static Error Constants

- If the system is of type 0 at low frequencies will be level.
- A type 0 system, (that is, a system without a pole at the origin,)will have a static position error, Kp, equal to

- If the system is of type 1 (a single pole at the origin) it will have a slope of -20 dB/dec at low frequencies
- A type 1 system will have a static velocity error, Kv, equal to the value of the -20 dB/dec line where it crosses 1 radian per second

- If the system is of type 2 ( a double pole at the origin) it willhave a slope of -40 dB/dec at low frequencies
- A type 2 system has a static acceleration error,Ka, equal to the value of the -40 dB/dec line where it crosses 1 radian per second

Static Error Constants

Kv(dB)

Loop Shaping

Error

signal

E(s)

- We have seen that the open loop transfer function, has profound influences on the closed loop response
- The key concept in loop shaping designs is that there is some ideal open loop transfer (B(s)) that will provide the design specifications that we require of our closed loop system
- Loop shaping is a trial and error process:
- Everything is connected and nothing is independent
- What we gain in one area may (usually?) causes loss in other areas
- Often times, out best controller is a compromise between demands

- To perform loop shaping we can used either the root locus plots or the Bode plots depending on the type of response that we wish to achieve
- We have already considered an important form of loop shaping as the PID controller

Output

y(s)

Input

r(s)

Controller

C(s)

Plant

P(s)

Open Loop

Signal

B(s)

Sensor

-1

+

+

Loop Shaping with the Bode Plot

- The open loop Bode plot is the natural design tool when designing in the frequency domain.
- For the frequency domain, the common specifications are bandwidth, gain cross over frequency, gain margin, resonant frequency, resonant frequency gain, phase margin, static errors and high frequency roll off.

-3 db

Roll off Rate dB/dec

Resonant peak gain, dB

Bandwidth rps

Resonant peak frequency rps

Gain cross over

frequency rps

Loop Shaping with the Bode Plot

Increase of gain

also increases

bandwidth and

resonant gain

Break frequency

corresponds to the

component pole or zero

Poles bend the magnitude and phase down

Zeros bend the magnitude and the phase up

Lead and Lag Compensators

- The compensator with a transfer functionis called a lead compensator if a<b and a lag compensator if b>a
- The lead and the lag compensator can be used together
- Note: the compensator does add a steady state gain ofthat needs to be accounted for in the final design
- There are analytical methods for designing these compensators (See Ogata or Franklin and Powell)

Lead Compensator

- The lead compensator is used to improve stability and to improve transient characteristics.
- The lead compensator can be designed using either frequency response or root locus methods
- Usually, the transient characteristics are better addressed using the root locus methods
- Addressing excessive phase lag is better addressed using the frequency methods

- The pole of the system is usually limited by physical limitations of the components use to implement the compensator
- In the lead compensator, the zero and pole are usually separated in frequency from about .4 decades to 1.5 decades depending on the design

Lead Compensator (Frequency Design)

Mechanical

Lead

Compensator

Note:

the lead compensator opensup the high frequency regionwhich could cause noise problems

The Lead compensator addsphase

b1

xi

b2

x0

k

y

f

wm

Example

- An aircraft has a pitch rate control as shown. Design a lead compensator for this system for a static velocity error of 4/sec, and a phase margin of 40 degrees.

Aircraft Pitch

Rate Dynamics

Compensator

+

+

Y

R

C(s)

-1

Example

Current System:

When design a lead

compensator first adjust

the gain to meet the static

error condition

In this case the gain needs

the be increase by 180 or

20Log10180= 45.1 dB

added

-33dB

Example

- Gain Adjusted System
- Then noting where the
- phase currently is, that
- is the desired location for
- the peak of the lead phase
- = spec – Pm=40-0.153
- + a small safety = 55 deg
- Finally adjust them as
- necessary

12 dB

Example

Initial design: Phase good but Kvnot

Gain needs to be increased by about 20 dB

Final design:

Example

- An aircraft has a pitch rate control as shown. The response of the pitch control is under damped, sluggish has an objectionable transient vibration mode. Design a lead compensator that provides a damping ratio of between 0.45 to 0.50 and 5% settling time of 150 seconds which reduces the vibration mode as much as possible.

+

+

Y

R

C(s)

-1

Example

The root locus from the complex poles has very little damping and causes the vibration seen in the response. There is a pole at -0.01 on the real axis that is dominant and causes the sluggish behavior.

Strategy: Use a lead compensator to bend the curves to the left and into the 0.6 damping region. The zero of the compensator should counteract the vibrational mode

Example

The initial design with the pole at -5 and the zero at -1 had the desired effectof bending the root locus to the left and removing most of the vibration. However

the pole is still too close to the origin such that 0.6 damping can not be achieved.

Example

We achieved the specifications once the pole of the compensator was moved

out to -9 and we adjusted the gain for the 0.6 damping.

Lag Compensator

- Lag compensators are used to improve steady state characteristics where the transient characteristics are adequate and to attenuate high frequency noise
- In order to not change the transient characteristics, the zero and pole are located near the origin on the root locus plot
- The starting point for the design on a root locus is to start with a pole location at about s = -0.001 and then locate the pole as needed for the desired effect

- In order to not give up too much phase, the zero and pole are located away from the phase margin frequency

Lag Compensator

Mechanical

Lag

Compensator

xi

b2

k

a

b

b1

Note that the lag compensator causes

a drop in the magnitude and phase

This could be useful in reducing bandwidth, and improving gain margin; however it might reduce phase margin

x0

Example

- A linear motor has an open loop rate transfer function ofIt is desired that the system have a static velocity error constant greater than 20/sec, a phase margin of 45 degrees plus or minus 5 degrees and gain crossover frequency of 1 radian/sec. Design a lag compensator for this system.

Linear Motor

Rate Dynamics

Compensator

+

+

Y

R

C(s)

-1

Example

Current System:

Phase margin is

low and the static

velocity error constant

must be improved.

Start by correcting the

static velocity error constant

4.9dB

Example

Gain Adjusted:

Gain and Phase

Margin problems

try placing a pole

at -0.01 rps and

adjust the zero

Need to shape

the curve like this

Need to move PM to here

Example

Final Design

Summary

- Static Error Constants (Review)
- Loop Shaping
- Loop Shaping with the Bode Plot
- Lead and Lag Compensators
- Lead design with Bode plot
- Lead design with root locus
- Lag design with Bode plot

Next: Sensitivity Analysis

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