EE C128 / ME C134 { Feedback Control Systems

EE C128 / ME C134 – Feedback Control SystemsLecture – Chapter 9 – Design via Root Locus

Alexandre Bayen

Department of Electrical Engineering & Computer ScienceUniversity of California Berkeley

September 10, 2013

Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 1 / 41

Lecture abstract

Topics covered in this presentation

I Compensation to improve steady-state error

I Compensation to improve transient response

I Compensation to improve both

I Feedback compensation


Chapter outline

1 9 Design via root locus9.1 Introduction9.2 Improving steady-state error via cascade compensation9.3 Improving transient response via cascade compensation9.4 Improving steady-state error and transient response9.5 Feedback compensation9.6 Physical realization of compensation


9 Design via RL 9.1 Intro




Definitions, [1, p. 458]

Definition (compensator)

I A subsystem represented as a transfer function inserted into theforward or FB path for the purpose of improving the transientresponse or steady-state error.

I Changes the OL poles and zeros, thereby creating new RL that goesthrough the desired CL pole locations.

I Ideal / active – Use pure integration for improving steady-state erroror pure differentiation for improving transient response. Require theuse of active amplifiers and possible additional power sources.

I Passive – Implemented with passive elements such as resistors andcapacitors. Less expensive and do not require additional powersources for their operation. Their steady-state error is not driven tozero in cases where ideal compensators yield zero error.



System configurations, [1, p. 458]

I Cascade – The compensatingnetwork, G1(s), is placed atthe low-power end of theforward path in cascade withthe plant.

I FB – The compensator, H1(s),is placed in the FB path.

Figure: Compensation techniques: a.cascade; b. FB


9 Design via RL 9.2 Improving steady-state error via cascade compensation




2 methods, [1, p. 458]

I Ideal integral compensatorI PI controllerI Places an OL pole at the

origin (pure integrator) and azero close to the pole.

I Steady-state error goes tozero

I Active network

I Lag compensatorI Places a pole near the origin

(not pure integration) and azero close to the pole.

I Steady-state error does notgo to zero, but yields ameasurable reduction

I Passive network



Ideal integral compensation (PI), [1, p. 459]

Gc(s) = Ks+ a

s

I MethodI Original transient response

determined by location ofthe original OL poles

I Add a pole at originI RL no longer goes

through location ofprevious poles

I Add a zero close to the poleat the origin

I Zero location can betuned to cause RL to gothrough location ofprevious poles

Figure: Uncompensated




Gc(s) = Ks+ a

s







Figure: Compensator pole added




Gc(s) = Ks+ a

s







Figure: Compensator pole & zeroadded



PI controller, [1, p. 464]

Definition (PI controller)

Alternate name for an ideal integralcompensator that has bothproportional and integral control.

Gc(s) = KP +KI

s= KP

s+ KIKP

s)

where the value of the zero can beadjusted by varying KI

KP.

Figure: PI controller



Lag compensator, [1, p. 464]

I Pole-zero pair is moved left ofthe origin

I Steady-state errorI Static error constant

I Uncompensated

Kv0 = Kz1z2...

p1p2...

I Lag compensated

KvN = Kv0

zcpc

I Transient responseI Minimal effect if pole-zero

pair is placed near origin

Figure: a. Type 1 uncompensatedsystem; b. type 1 compensated system;c. compensator pole-zero plot



Lag compensator, [1, p. 464]

I Pole-zero pair is moved left ofthe origin

I Steady-state errorI Static error constant

I Uncompensated

Kv0 = Kz1z2...

p1p2...

I Lag compensated

KvN = Kv0

zcpc

I Transient responseI Minimal effect if pole-zero

pair is placed near origin

Figure: RL: a. before lagcompensation, b. after lagcompensation


9 Design via RL 9.3 Improving transient response via cascade compensation




2 methods, [1, p. 469]

I Ideal derivative compensatorI PD controllerI Add a zero (pure derivative)

to the forward path TFI Shorter response time

I Shorter Ts & Tp

I Same %OS

I Improvement in steady-stateerror not always guaranteed

I Warning: differentiation is anoisy process

I Level of noise is low, butthe frequency is highcompared to the signal

I Large, unwanted signalsI Saturation of components

I Active network

I Lead compensatorI Add a zero and a more

distant pole (not purederivative) to the forwardpath TF

I Pole farther from theimaginary axis than the zero

I The angular contributionof the compensator is stillpositive and thusapproximates anequivalent single zero

I Noise due to differentiationis reduced

I Passive network (cannotproduce a single zero)



Ideal derivative compensator (PD), [1, p. 470]

Gc(s) = s+ zc

I MethodI Uncompensated system

transient response isunacceptable

I Compensated systemtransient response variesbased on the zero location

I Same ζ ∝ %OSI Larger negative real part

∝ shorter Ts

I Larger imaginary part ∝shorter Tp

Figure: Uncompensated system




Gc(s) = s+ zc





∝ shorter Ts


Figure: Compensator zero at −2




Gc(s) = s+ zc





∝ shorter Ts






Gc(s) = s+ zc





∝ shorter Ts






Gc(s) = s+ zc





∝ shorter Ts


Figure: Uncompensated system andideal derivative compensation solutions



PD controller, [1, p. 476]

Definition (PD controller)

Alternate name for an idealderivative compensator that hasboth proportional and derivativecontrol.

Gc(s) = KP +KDs = KD(s+KP

KD)

where KPKD

is chosen to equal thenegative of the compensator zero,and KD is chosen to contribute tothe required loop-gain value.

Figure: PD controller



Lead compensator, [1, p. 477]

I MethodI Select a dominant 2nd-order

pole on the s-planeI The sum of the angles from

the uncompensated system’spoles and zeros to the designpoint can be found

I The difference between 180◦

and the sum of the anglesmust be the angularcontribution required by thecompensator

I An ∞ number of leadcompensators could be usedto meet the transientresponse requirement

Figure: Geometry of lead compensation



Lead compensator, [1, p. 477]

I MethodI Select a dominant 2nd-order

pole on the s-planeI The sum of the angles from

the uncompensated system’spoles and zeros to the designpoint can be found

I The difference between 180◦

and the sum of the anglesmust be the angularcontribution required by thecompensator

I An ∞ number of leadcompensators could be usedto meet the transientresponse requirement

Figure: 3 of the ∞ possible leadcompensator solutions


9 Design via RL 9.4 Improving steady-state error & transient response




2 methods, [1, p. 482]

Steady-state error or transient response? Which should we improve 1st?Either way, the 1st improvement is deteriorated. We will follow thetextbook: 1st design for transient response and 2nd design for steady-stateerror.

I PID controllerI Active networkI PD controller followed by a

PI controller

I Lag-lead compensatorI Passive networkI Lead compensator followed

by a lag compensator



PID controller, [1, p. 482]

Definition (PID controller)

Alternate name for an active PDcontroller followed by an active PIcontroller.

Gc(s) = KP +KI

s+KDs

=KD

(s2 + KP

KDs+ KI

KD

)s

which has 2 zeros and 1 pole at theorigin.

Figure: PID controller



PID controller design technique steps, [1, p. 482]

1. Evaluate the performance of the uncompensated system to determinehow much improvement in transient response is required.

2. Design the PD controller to meet the transient responsespecifications. The design includes the zero location and the loopgain.

3. Simulate the system to be sure all requirements have been met.

4. Redesign if the simulation shows that requirements have not beenmet.

5. Design the PI controller to yield the required steady-state error.

6. Determine the gains, KP , KI , & KD.


8. Redesign if simulation shows that requirements have not been met.



Lag-lead compensator design technique steps, [1, p. 487]

1. Evaluate the performance of the uncompensated system to determinehow much improvement in transient response is required.

2. Design the lead compensator to meet the transient responsespecifications. The design includes the zero location, pole location,and the loop gain.


4. Redesign if the simulation shows that requirements have not beenmet.

5. Evaluate the steady-state error performance for the lead-compensatedsystem to determine how much more improvement in steady-stateerror is required.

6. Design the lag compensator to yield the required steady-state error.


8. Redesign if simulation shows that requirements have not been met.



Notch filter motivation, [1, p. 492]

I High frequency vibrationmodes

I Desired CL response may bedifficult to obtain

I Modeled as part of theplant’s TF by pairs ofcomplex poles near theimaginary axis

I In a CL configuration, thesepoles can move closer to oreven cross the imaginary axis

I Result in instability orhigh-frequency oscillations Figure: a. RL before cascading notch

filter; b. typical CL step responsebefore cascading notch filter



Notch filter design, [1, p. 492]

I MethodI Place 2 zeros close to the

low-damping-ratio poles ofthe plant as well as 2 realpoles

Figure: a. RL after cascading notchfilter; b. CL step response aftercascading notch filter


9 Design via RL 9.5 FB compensation




FB vs. cascade compensation, [1, p. 495]

I Methods of reshaping the RLto intersect CL s-plane polesthat yield a desired transientresponse

I Cascade compensatorI FB compensator

I Approach 1 – Similar tocascade compensation,but poles and zeros areadded via H(s)

I Approach 2 – Designspecified performance forthe minor loop then themajor loop.

Figure: Generic control system with FBcompensation



FB vs. cascade compensation, [1, p. 495]

I Can yield faster responses

I Can be used in cases wherenoise problems preclude the useof cascade compensation

I May not require additionalamplification

I Typically the design consists offinding the gains, K, K1, andKf after establishing adynamic form of Hc(s)

Figure: Generic control system with FBcompensation



Approach 1, [1, p. 496]

I MethodI Reduce the generic control system with

FB compensation to the equivalent blockdiagram

I Loop gain

G(s)H(s) = K1G1(s)[KfHc(s)+KG2(s)]

I Loop gain without FB

G(s)H(s) = K1G1(s)G2(s)

I Adding FB replaces the poles and zeros ofG2(s) with those of [KfHc(s) +KG2(s)]

Figure: Equivalent blockdiagram



Approach 2, [1, p. 500]

I MethodI Minor loop is a forward-path

TF whose poles can beadjusted with the minor-loopgain with a pure derivativerather than with additionalpoles and zeros, as incascade compensation

I Minor-loop poles thenbecome the OL poles for theentire control system

I The CL poles are set by themajor loop gain, as incascade compensation

Figure: Equivalent block diagram


9 Design via RL 9.6 Physical realization of compensation




Active-circuit realization, [1, p. 504]

I Inverting operational amplifier

Figure: Operational amplifierconfigured for TF realization Table: Active realization of controllers

and compensators, using an operationalamplifier



Passive-circuit realization, [1, p. 504]

I Remember: 2 networks mustbe isolated to ensure that onenetwork does not load theother

Table: Active realization of controllersand compensators, using an operationalamplifier



Lag-lead compensator realization, [1, p. 505]

I Active circuit

Figure: Lag-lead compensatorimplemented with operationalamplifiers

I Passive circuit

Figure: Lag-lead compensatorimplemented with cascaded lag andlead networks with isolation



Bibliography

Norman S. Nise. Control Systems Engineering, 2011.


EE C128 / ME C134 { Feedback Control Systems

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