EE C128 / ME C134 – Feedback Control Systems
Lecture – Chapter 9 – Design via Root Locus
Alexandre Bayen
Department of Electrical Engineering & Computer Science
University 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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 2 / 41
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 3 / 41
9 Design via RL 9.1 Intro
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 4 / 41
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.
IIdeal / active – Use pure integration for improving steady-state erroror pure di↵erentiation for improving transient response. Require theuse of active amplifiers and possible additional power sources.
IPassive – 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.
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 5 / 41
9 Design via RL 9.1 Intro
System configurations, [1, p. 458]
ICascade – The compensatingnetwork, G1(s), is placed atthe low-power end of theforward path in cascade withthe plant.
IFB – The compensator, H1(s),is placed in the FB path.
Figure: Compensation techniques: a.cascade; b. FB
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 6 / 41
9 Design via RL 9.2 Improving steady-state error via cascade compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 7 / 41
9 Design via RL 9.2 Improving steady-state error via cascade compensation
2 methods, [1, p. 458]
IIdeal integral compensator
I 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
ILag compensator
I 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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 8 / 41
9 Design via RL 9.2 Improving steady-state error via cascade compensation
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 of
previous poles
I Add a zero close to the poleat the origin
IZero location can be
tuned to cause RL to go
through location of
previous poles
Figure: Uncompensated
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 9 / 41
9 Design via RL 9.2 Improving steady-state error via cascade compensation
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 of
previous poles
I Add a zero close to the poleat the origin
IZero location can be
tuned to cause RL to go
through location of
previous poles
Figure: Compensator pole added
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 10 / 41
9 Design via RL 9.2 Improving steady-state error via cascade compensation
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 of
previous poles
I Add a zero close to the poleat the origin
IZero location can be
tuned to cause RL to go
through location of
previous poles
Figure: Compensator pole & zeroadded
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 11 / 41
9 Design via RL 9.2 Improving steady-state error via cascade compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 12 / 41
9 Design via RL 9.2 Improving steady-state error via cascade compensation
Lag compensator, [1, p. 464]
I Pole-zero pair is moved left ofthe origin
I Steady-state errorI Static error constant
IUncompensated
Kv0 = Kz1z2...p1p2...
ILag compensated
KvN = Kv0
zcpc
I Transient responseI Minimal e↵ect if pole-zero
pair is placed near origin
Figure: a. Type 1 uncompensatedsystem; b. type 1 compensated system;c. compensator pole-zero plot
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9 Design via RL 9.2 Improving steady-state error via cascade compensation
Lag compensator, [1, p. 464]
I Pole-zero pair is moved left ofthe origin
I Steady-state errorI Static error constant
IUncompensated
Kv0 = Kz1z2...p1p2...
ILag compensated
KvN = Kv0
zcpc
I Transient responseI Minimal e↵ect if pole-zero
pair is placed near origin
Figure: RL: a. before lagcompensation, b. after lagcompensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 14 / 41
9 Design via RL 9.3 Improving transient response via cascade compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 15 / 41
9 Design via RL 9.3 Improving transient response via cascade compensation
2 methods, [1, p. 469]
IIdeal derivative compensator
I PD controllerI Add a zero (pure derivative)
to the forward path TFI Shorter response time
IShorter Ts & Tp
ISame %OS
I Improvement in steady-stateerror not always guaranteed
I Warning: di↵erentiation is anoisy process
ILevel of noise is low, but
the frequency is high
compared to the signal
ILarge, unwanted signals
ISaturation of components
I Active network
ILead compensator
I Add a zero and a moredistant pole (not purederivative) to the forwardpath TF
I Pole farther from theimaginary axis than the zero
IThe angular contribution
of the compensator is still
positive and thus
approximates an
equivalent single zero
I Noise due to di↵erentiationis reduced
I Passive network (cannotproduce a single zero)
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 16 / 41
9 Design via RL 9.3 Improving transient response via cascade compensation
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
ISame ⇣ / %OS
ILarger negative real part
/ shorter Ts
ILarger imaginary part /shorter Tp
Figure: Uncompensated system
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 17 / 41
9 Design via RL 9.3 Improving transient response via cascade compensation
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
ISame ⇣ / %OS
ILarger negative real part
/ shorter Ts
ILarger imaginary part /shorter Tp
Figure: Compensator zero at �2
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 18 / 41
9 Design via RL 9.3 Improving transient response via cascade compensation
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
ISame ⇣ / %OS
ILarger negative real part
/ shorter Ts
ILarger imaginary part /shorter Tp
Figure: Compensator zero at �3
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 19 / 41
9 Design via RL 9.3 Improving transient response via cascade compensation
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
ISame ⇣ / %OS
ILarger negative real part
/ shorter Ts
ILarger imaginary part /shorter Tp
Figure: Compensator zero at �4
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 20 / 41
9 Design via RL 9.3 Improving transient response via cascade compensation
Ideal derivative compensator (PD), [1, p. 472]
Gc(s) = s+ zc
I MethodI Uncompensated system
transient response isunacceptable
I Compensated systemtransient response variesbased on the zero location
ISame ⇣ / %OS
ILarger negative real part
/ shorter Ts
ILarger imaginary part /shorter Tp
Figure: Uncompensated system andideal derivative compensation solutions
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 21 / 41
9 Design via RL 9.3 Improving transient response via cascade compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 22 / 41
9 Design via RL 9.3 Improving transient response via cascade 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 di↵erence between 180�
and the sum of the anglesmust be the angularcontribution required by thecompensator
I An 1 number of leadcompensators could be usedto meet the transientresponse requirement
Figure: Geometry of lead compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 23 / 41
9 Design via RL 9.3 Improving transient response via cascade 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 di↵erence between 180�
and the sum of the anglesmust be the angularcontribution required by thecompensator
I An 1 number of leadcompensators could be usedto meet the transientresponse requirement
Figure: 3 of the 1 possible leadcompensator solutions
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 24 / 41
9 Design via RL 9.4 Improving steady-state error & transient response
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 25 / 41
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.
IPID controller
I Active networkI PD controller followed by a
PI controller
ILag-lead compensator
I Passive networkI Lead compensator followed
by a lag compensator
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 26 / 41
9 Design via RL 9.4 Improving steady-state error & transient response
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 27 / 41
9 Design via RL 9.4 Improving steady-state error & transient response
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.
7. Simulate the system to be sure all requirements have been met.
8. Redesign if simulation shows that requirements have not been met.
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 28 / 41
9 Design via RL 9.4 Improving steady-state error & transient response
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.
3. Simulate the system to be sure all requirements have been met.
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.
7. Simulate the system to be sure all requirements have been met.
8. Redesign if simulation shows that requirements have not been met.
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 29 / 41
9 Design via RL 9.4 Improving steady-state error & transient response
Notch filter motivation, [1, p. 492]
I High frequency vibrationmodes
I Desired CL response may bedi�cult 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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 30 / 41
9 Design via RL 9.4 Improving steady-state error & transient response
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 31 / 41
9 Design via RL 9.5 FB compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 32 / 41
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
IApproach 1 – Similar to
cascade compensation,
but poles and zeros are
added via H(s)I
Approach 2 – Design
specified performance for
the minor loop then the
major loop.
Figure: Generic control system with FBcompensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 33 / 41
9 Design via RL 9.5 FB compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 34 / 41
9 Design via RL 9.5 FB compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 35 / 41
9 Design via RL 9.5 FB compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 36 / 41
9 Design via RL 9.6 Physical realization of compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 37 / 41
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 38 / 41
9 Design via RL 9.6 Physical realization of compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 39 / 41
9 Design via RL 9.6 Physical realization of compensation
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
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 40 / 41
9 Design via RL 9.6 Physical realization of compensation
Bibliography
Norman S. Nise. Control Systems Engineering, 2011.
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 41 / 41