-
Feedback Control Systems. 11
FEEDBACK CONTROL
Control is a very common concept.e.g., Human-machine
interaction: Driving a car. MANUAL CONTROL.e.g., Independent
machine: Room temperature control. Furnace inwinter, air
conditioner in summer. Both controlled (turned on/off)by
thermostat. AUTOMATIC CONTROL.
DEFINITION: Control is the process of causing a system variable
toconform to some desired value, called a reference value.
(e.g.,variable=temperature)
DEFINITION: Feedback is the process of measuring the
controlledvariable (e.g., temperature) and using that information
to influence thevalue of the controlled variable.
Some applications: Airplane autopilots that can land a plane in
fog. Space telescopes with pointing accuracy of 106 degrees.
Disk-drive read heads with < 1 micron accuracy. Robots for
various applications. . . . (see next page for a synopsis).
-
FEEDBACK CONTROL 12
Typical Feedback ApplicationsCategories Specific
ApplicationsEcological Wildlife management and control; control of
plant chemical
wastes via monitoring lakes and rivers; air pollution
abatement;water control and distribution; flood control via dams
and re-sevoirs; forest growth management.
Medical Medical instrumentation for monitoring and control;
artificial limbs(prosthesis).
Homeappliances
Home heating, refrigeration, and airconditioning via
thermostaticcontrol; electronic sensing and control in clothes
dryers; humiditycontrollers; temperature control of ovens.
Power/energy Power system control and planning; feedback
instrumentation inoil recovery; optimal control of windmill blade
and solar panel sur-faces; optimal power distribution via power
factor control.
Transportation Control of roadway vehicle traffic flows using
sensors; automaticspeed control devices on automobiles; propulsion
control in railtransit systems; building elevators and
escalators.
Manufacturing Sensor-equipped robots for cutting, drilling die
casting, forging,welding, packaging, and assembling; chemical
process control;tension control windup processes in textile mills;
conveyor speedcontrol with optical pyrometer sensing in hot steel
rolling mills.
Aerospace andmilitary
Missile guidance and control; automatic piloting; spacecraft
con-trol; tracking systems; nuclear submarine navigation and
control;fire-control systems (artillery).
(reproduced from: J.R. Rowland, Linear Control Systems:
Modeling, Analysis, and Design, John Wiley & Sons, (New York:
1986), p. 6.)
-
FEEDBACK CONTROL 13
Rube Goldberg showed that a knowledge of feedback control
waseven useful to budding cartoonists!
Rube Goldberg walks in his sleep, strolls through a cactus field
in his bare feet, and screams out an idea for self-operatingnapkin:
As you raise spoon of soup (A) to your mouth it pulls string (B),
thereby jerking ladle (C) which throws cracker (D) pastparrot (E).
Parrot jumps after cracker and perch (F) tilts, upsetting seeds (G)
into pail (H). Extra weight in pail pulls cord (I),which opens and
lights automatic cigar lighter (J), setting off sky-rocket (K)
which causes sickle (L) to cut string (M) and allowpendulum with
attached napkin to swing back and forth thereby wiping off your
chin. After the meal, substitute a harmonica forthe napkin and
youll be able to entertain the guests with a little music.
Simple Feedback System Example
DesiredTemp.
RoomTemp.
Heat Loss (Qout)Qin
HouseFurnaceGasValveThermo-
stat
Block Diagram of furnace-controlled room temperature
controller.Identifies major components and omits details. Shows
information/energy flow.
Central component = PROCESS or PLANT, one of whose variableswe
want to control. e.g., Plant = ; Variable = .
-
FEEDBACK CONTROL 14
DISTURBANCE = some system input out of our direct control.e.g.,
Disturbance = . ACTUATOR = device that influences controlled
variable.
e.g., Actuator = . REFERENCE SENSOR measures desired system
output. OUTPUT SENSOR measures actual system output. * COMPENSATOR/
CONTROLLER = device that computes the
control effort to apply the the actuator, based on sensor
readings.e.g., combines the last three functions.
A More Abstract Block Diagram:
Ref.Value Output
Disturbance
PlantReferenceSensor
OutputSensor
Compen-sator Actuator
error
Negative Feedback
An Even More Abstract Block Diagram:
Ref.Value Output
Disturbance
PlantCompen-sator
-
FEEDBACK CONTROL 15
The Control Problem:
Reject disturbance. Acceptable steady state errors. Acceptable
transient response. Minimize sensitivity to parameter changes in
the plant.
Solution Reached By:
1. Choosing output sensors.
2. Choosing actuators.
*3. Developing plant, actuator, sensor equations (models).*4.
Designing compensator based on the models and design criteria.
*5. Evaluating design analytically, with simulation and
prototype.
*6. Iteration!!
A First Analysis: Auto Cruise Control
Want to control speed of automobile.1. Output sensor =
speedometer.2. Actuator = throttle and engine.
-
FEEDBACK CONTROL 16
Block diagram:
DesiredSpeed
ActualSpeed
Road Grade
Plant
Auto Body
OutputSensor
Speedometer
Sensor Noise
Compen-sator Actuator
EngineThrottle
ControlVariable
MeasuredSpeed
3. Model of system:
1. Operate system 55 MPH. Assume linear response near 55 MPH.2.
Measure: 1% change in throttle 10 MPH change in speed.3. Measure:
1% change in grade 5 MPH change in speed.4. Measure: Speedometer
accurate to a fraction of 1 MPH, so is
assumed exact.5. Functional block diagram:
Compen-satorr .t/ y.t/
w.t/
u.t/10
0:5
r .t/reference speed, MPH.u.t/throttle posn., %.y.t/actual
speed, MPH.w.t/road grade, %.
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FEEDBACK CONTROL 17
4. Design compensator/controller
First attempt = open loop controller.
Compensator
1=10r .t/ yol.t/
w.t/
u.t/10
0:5
yol.t/ D 10 .u.t/ 0:5w.t//D 10
r.t/
10 0:5w.t/
D r.t/ 5w.t/:
5. Evaluate design.
r.t/ D 55, w.t/ D 0% yol.t/ D 55. No Error. . . Good. r.t/ D 55,
w.t/ D 1% yol.t/ D 50. 10% Error. . . Not Good. r.t/ D 55, w.t/ D
2% yol.t/ D 45. 20% Error. . . Not Good. Suppose you load the trunk
with all your course materials and the
car becomes more sluggish.e.g., the gain 10 becomes 9
r.t/ D 55, w.t/ D 0% yol.t/ D 49:5. 10% Error. . . Not Good.
r.t/ D 55, w.t/ D 1% yol.t/ D : : :
6. Iterate design. Go to step #4.
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FEEDBACK CONTROL 18
4. Second attempt = Feedback controller
Compensatorz }| {100r .t/ ycl.t/
w.t/
u.t/10
0:5
Multiply your output error by 100; FEEDBACK GAIN D 100ycl.t/ D
10u.t/ 5w.t/u.t/ D 100.r.t/ ycl.t//
so; ycl.t/ D 1000r.t/ 1000ycl.t/ 5w.t/1001ycl.t/ D 1000r.t/
5w.t/
ycl.t/ D 0:999r.t/ 0:005w.t/5. Evaluate design.
r.t/ D 55, w.t/ D 0% ycl.t/ D 54:945 r.t/ D 55, w.t/ D 1% ycl.t/
D 54:94 r.t/ D 55, w.t/ D 2% ycl.t/ D 54:935 r.t/ D 55, w.t/ D 10%
ycl.t/ D 54:895 : : : 0:2% Error!
Feedback system rejects disturbances. Feedback system has steady
state error.
r.t/ D 55, w.t/ D 0%, plant=9, not 10 ycl.t/ D 54:939 Feedback
system less sensitive to system parameter
values.
NOTE! High feedback gain = good performance here. Not
alwaystrue! e.g., Public address amplifier.
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FEEDBACK CONTROL 19
4. Third attempt. Try to get rid of steady state error.
Compensator
r .t/ ycl.t/
w.t/
u.t/10
0:5
ycl.t/ D 10 .r.t/ ycl.t//
5w.t/D 10r.t/ 10ycl.t/ 5w.t/
ycl.t/ D 10r.t/ 5w.t/1C 10D
101C 10
| {z }set to 1: D1 110
r.t/
51C 10
w.t/
D r.t/ 510
w.t/:
Best results yet! (Try for yourself with D 100.)
A Brief History of Control
2nd century B.C.: Fluid level/Flowrate control. Still used to
flush toilets!
SupplyFloat
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FEEDBACK CONTROL 110
1624: Drebbels incubator Control temperatureusing
mechanicalfeedback.
WaterEggs
Riser
Float
Metal plate
GassesFlue
MercuryAlcoholFire
Damper
1787: Thomas Meads fly-ball governor. 1788: James Watts fly-ball
governor.
ButterflyTo engine
inlet
Pivot
valve
Ball
Rotation
Sleeve
Pulley from engine
Steam
Both control a rotating shaft.
1840: Airys telescope controller. and the machine (if I may
soexpress myself) became perfectly wild.
Discovery of instability in a feedback control system. Analysis
of system with differential equations. Beginnings of feedback
control theory.
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FEEDBACK CONTROL 111
1868: Maxwell found stability criteria for second and third
order(simple) systems. 1877: Routh found stability criteria for
more complex (general) linear
systems.
1893: Lyapunov: stability of nonlinear systems. 1932: Nyquists
graphical stability criteria (1945: Bodes simpler graphical method)
1936: PID Control. 1948: Evans root locus. 1960s: State variable
(modern) control design. 1980s: Post Modern H1 Robust control.
Adaptive control...?
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FEEDBACK CONTROL 112
1624
1728
1868
1877
1890
1932
1910
1927
1938
1942
1947
1948
1950
1956
1957
1960
1969
Chronological Historyof Feedback ControlDrebble, Incubator
Watt, Flyball governor
Maxwell, Flyball stability analysis
Routh, Stability
Liapunov, Nonlinear stability
Sperry, Gyroscope and autopilot
Black, Feedback electronic amplifier: Bush, Differential
analyzer
Nyquist, Nyquist stability criterion
Bode, Frequency response methods
Wiener, Optimal filter design: Ziegler-Nichols PID tuning
Hurewicz, Sampled data systems; Nichols, Nichols chart
Evans, Root locus
Kochenberger, Nonlinear analysis
Pontryagin, Maximum principle
Bellman, Dynamic programming
Draper, Inertial navigation; Kalman, Optimal estimation
Hoff, Microprocessor(reproduced from: G.F. Franklin, J.D.
Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, 3rd
edition,Addison Wesley, (Reading, Mass: 1994), flyleaf.)
munelCross-Out
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Feedback Control Systems. 21
SYSTEM MODELING IN THE TIME DOMAIN
Model = Set of equations used to represent a physical
system,relating output to input. Required to:
1. Understand system behavior (analysis).2. Design a controller
(synthesis). Developing the model 80%90% of the effort in designing
a
controller. Methods:
1. Analytic system modelingwe focus on these methods.2.
Empirical system identification. (In practice, there is always
an
empirical component to system modeling). No model is exact!
Inaccuracies due to:
1. Unknown parameter values.2. Unmodeled dynamics (to make
simpler model).
LTI Systems
This course teaches methods to control linear time invariant
(LTI)systems.
None exist! But, many are close enough. In the following,
suppose a system maps x.t/ 7! y.t/ as
y.t/ D T[x.t/]:
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SYSTEM MODELING IN THE TIME DOMAIN 22
LINEAR: A system is linear iffT
[ax1.t/C bx2.t/] D aT[x1.t/]C bT[x2.t/]:a.k.a.
superposition.
TIME INVARIANT: A system is time invariant iffy.t t0/ D T[x.t
t0/] 8 t0:
(Translation: If we input a specific signal x.t/ and record the
outputy.t/, then input a shifted version of the signal x.t/, the
output will be ashifted version of y.t/, with the same shift.)
KEY POINT: If a system is LTI, then it has an impulse response.
Thisentirely characterizes the systems dynamics. The Laplace
transformof the impulse response is the transfer function. Working
with thetransfer function eliminates the need to mess around with
trying tosolve complicated differential equations.
A Simple Model
Consider a resistor:i.t/
v.t/R
Ohms Law (model) says: v.t/ D i.t/ R. Is the resistor LTI? Is
the model of the resistor LTI?
EXAMPLE: Consider a 1 Ohm, 2 Watt resistor.
-
SYSTEM MODELING IN THE TIME DOMAIN 23
Apply 1 Volt;1 Amp of current is predicted to flow.Power
dissipated D V 2=R D 1 Watt. Now apply 10 Volts.
10 Amps of current is predicted to flow.Power dissipated D V 2=R
D 100 Watts! Model will no longer be accurate. True behavior
depends on input signal levelnonlinear. Model is accurate in
certain range of input-signal values.
Dynamics of Mechanical Systems I (translational) Newtons
Law:
XF D ma. Vector sum of forces = mass of object
times inertial acceleration.
Free-body diagrams are a tool to apply this law.EXAMPLE: Cruise
control model.
Write the equations of motion for the speed and forward motion
of acar assuming that the engine imparts a forward force of
u.t/.
1. Assume rotational inertia of wheels is negligible.2. Assume
that friction is proportional to cars speed (viscous friction).
XF D ma
u.t/ b .x.t/ D m ..x.t/or;
..
x.t/C bm
.
x.t/ D u.t/m
b.
x.t/ u.t/
x.t/
m
-
SYSTEM MODELING IN THE TIME DOMAIN 24
If the variable of interest is speed (v.t/ D .x.t/), not
position,.
v.t/C bmv.t/ D u.t/
m Notice that the differential equation has output variables on
the left
of D, and input variables on the right.IMPORTANT POINT: All of
our models of dynamical systems will be
differential equations involving the input (e.g., u.t/) and its
derivativesand the output (e.g., y.t/) and its derivatives. No
other signals(intermediate variables) are allowed in our
solutions.
EXAMPLE: Car suspension.Each wheel in a car suspension system
has a tire, shock absorberand spring. Write the one-dimensional
(vertical) equations ofmotion for the car body and wheel.
Quarter-car model
Road SurfaceInertial Reference
kw
bks
m1
m2
x.t/
y.t/
r .t/
Free-body diagram:ks.y.t/ x.t// b.
.
y.t/ .x.t//
kw.x.t/ r .t// ks.y.t/ x.t// b. .y.t/ .x.t//
m1 m2x.t/ y.t/
-
SYSTEM MODELING IN THE TIME DOMAIN 25
The force from the spring is proportional to its stretch. The
force from theshock absorber is proportional to the rate-of-change
of its stretch.X
F D mab
.
y.t/ .x.t/C ks .y.t/ x.t// kw .x.t/ r.t// D m1 ..x.t/
ks .y.t/ x.t// b
.
y.t/ .x.t/D m2
..
y.t/
Re-arrange:..
x.t/C bm1.
.
x.t/ .y.t//C ksm1.x.t/ y.t//C kw
m1x.t/ D kw
m1r.t/
..
y.t/C bm2.
.
y.t/ .x.t//C ksm2.y.t/ x.t// D 0
Important components for mechanical-translational systems:
x1.t/
x1.t/
x2.t/
x2.t/f .t/ D k.x1.t/ x2.t//
f .t/ D b. .x1.t/ .x2.t//
k
b
m1. Mass
2. Spring
3. Damper
Dynamics of Mechanical Systems II (rotational) Newtons Law is
modified to be:
XM D J (or I)
Vector sum of moments = moment of inertia times
angularacceleration. (momentDtorque).
EXAMPLE: Satellite control.Satellites require attitude control
so that sensors, antennas, etc., areproperly pointed. Lets consider
one axis of rotation.
-
SYSTEM MODELING IN THE TIME DOMAIN 26
.t/
d
Gas jetFc.t/
moment D Fc.t/ d; so;Fc.t/d D J
..
.t/..
.t/ D Fc.t/dJNote: Output of system .t/ integrates
torques twicedouble-integrator plant.EXAMPLE: Torsional
pendulum.A torsional pendulum is used, for example, in clocks
enclosed in glassdomes. A similar device is the read-write head on
a hard-disk drive.
k
bJ
;
k: Springiness of suspension wire.
b: Viscous friction.
XM D J ...t/
J..
.t/ D .t/ b ..t/ k.t/..
.t/C bJ.
.t/C kJ .t/ D.t/
J
Important components for mechanical rotational systems:
1.t/
1.t/
2.t/
2.t/ .t/ D k.1.t/ 2.t//
.t/ D b. .1.t/.
2.t//
k
b
J1. Inertia
2. Spring
3. Damper
-
SYSTEM MODELING IN THE TIME DOMAIN 27
EXAMPLE: NONLINEAR Rotational Pendulum.
l
.t/; .t/
mg
Moment of intertia: J D ml2:XM D J ...t/
J..
.t/ D .t/ mgl sin..t//..
.t/C gl sin..t//| {z }Nonlinear!
D .t/ml2
If motion is small, sin ..t// .t/...
.t/C gl .t/ D.t/
ml2Linear.
This is a preview of linearization.
Summary of Developing Models for Rigid Bodies:1. Assign
variables such as x.t/ and .t/ that are both necessary and
sufficient to describe and arbitrary position of the object.2.
Draw a free-body diagram of each component, and indicate all
forces
acting on each body and the accelerations of the center of mass
withrespect to an inertial reference.
3. Apply Newtons laws:X
F D ma;X
M D J.4. Combine the equations to eliminate internal forces.5.
The final form must be in terms of ONLY the input to the system
and
its derivatives, and the output of the system and its
derivatives.
Dynamics of Electrical Circuits
Kirchhoffs Laws: Current Law (KCL): The algebraic sum of
currents entering a node
equals the algebraic sum of currents leaving the node. Voltage
Law (KVL): The algebraic sum of all voltages taken around
a closed path in a circuit is zero.
-
SYSTEM MODELING IN THE TIME DOMAIN 28
Node analysis is a tool to apply these laws. (i.e., select one
node asreference (e.g., ground) and assume all other voltages are
unknown.Write equations for the unknowns using KCL. KVL must be
used forvoltage sources.)
EXAMPLE: Bridged-Tee circuit.
R1 R2
C1
C2
vi.t/
Select reference = .
KVL at : v.t/ D vi.t/. KCL at : v.t/ v.t/
R1 v.t/ v.t/
R2 C1 .v.t/ D 0.
KCL at : v.t/ v.t/R2
C C2. .v.t/ .v.t// D 0.
v.t/ v.t/C R2C2. .v.t/ .v.t// D 0v.t/C R2C2. .v.t/ .v.t// D
v.t/
v.t/v.t/C R2C2. .v.t/ .v.t//
R1
v.t/C R2C2. .v.t/ .v.t//
v.t/R2
C1
.
v.t/C R2C2. ..v.t/ ..v.t// D 0
R2(v.t/
v.t/C R2C2. .v.t/ .v.t//
R1(v.t/C R2C2. .v.t/ .v.t//
v.t/
-
SYSTEM MODELING IN THE TIME DOMAIN 29
R1 R2C1
.
v.t/C R2C2. ..v.t/ ..v.t// D 0:(
R1 R22C1C2
..
v.t/C(R22C2 C R1 R2C2 C R1 R2C1
.
v.t/C .R2/ v.t/D .R1 R22C1C2/
..
v.t/C .R22C2 C R1 R2C2/.
v.t/C .R2/v.t/
Important components for electrical systems:
1. Resistor
2. Capacitor
3. Inductor
4. Voltage source
5. Current source
6. OperationalAmplifier
v.t/
v.t/
v.t/
v.t/
i.t/
i.t/
i.t/
i.t/
vs
is
vd.t/i.t/
iC.t/ vo.t/
v.t/ D Ri.t/
i.t/ D C d v.t/dt
v.t/ D L d i.t/dt
v.t/ D vs
i.t/ D is
vd.t/ D 0i.t/ D iC.t/ D 0vo.t/ D Ao.vC.t/ v.t//
as Ao !1
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SYSTEM MODELING IN THE TIME DOMAIN 210
EXAMPLE: Op-amp circuit.
R1
R2 C
vi.t/ vo.t/
i.t/ D vi.t/R1
vo.t/ D R2i.t/ vc.t/:
dvo.t/dtD R2di.t/dt
dvc.t/dt
D R2R1
dvi.t/dt i.t/
C
D R2R1
dvi.t/dt 1
R1Cvi.t/
R1C.
vo.t/ D R2C .vi.t/ vi.t/(as C !1, we get an inverting
amplifier.)
Dynamics of Electro-Mechanical Systems
These are systems that convert energy from electrical to
mechanical,or vice versa.
EXAMPLE: DC Generator.
Assume generator is driven at constant speed. Generator has
field windings (input), and rotor/armature windings
(output).R f Ra Lai f .t/ ia.t/
e f .t/ L f eg.t/ Zlea.t/
| {z }Fieldcircuit
| {z }Rotorcircuit
| {z }Loadcircuit
-
SYSTEM MODELING IN THE TIME DOMAIN 211
e f .t/ D R f i f .t/C L f di f .t/dt e f .t/ is input, i f .t/
is output.
eg.t/ D Kd.t/dt K depends on generator structure.D Kgi f .t/.
d.t/=dt = angular velocity = cst.
= flux, proportional to i f .t/.
eg.t/ D Raia.t/C La dia.t/dt C ea.t/. eg.t/ is input, ia.t/ is
output. ea.t/ D Zlia.t/. ia.t/ is input, ea.t/ is output.
e f .t/ ea.t/i f .t/ eg.t/ ia.t/Field
circuit KgRotorcircuit Zl
This is a preview of a block diagram used to simplify our
understandingof the system dynamics.EXAMPLE: DC Motor
(servo-motor). Directly generates rotational motion. Indirectly
generates translational motion.
ea.t/ eb.t/
ia.t/ Ra La b
J.t/; .t/
| {z }Armature
| {z }Load
Mechanical resistance of load is translated into an
electricalresistance called the back e.m.f.
eb.t/ D Ke d.t/dt Ke D K, as with generator.
ea.t/ D Raia.t/C La dia.t/dt C eb.t/ .t/ D K ia.t/ K D K1
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SYSTEM MODELING IN THE TIME DOMAIN 212
Combining these equations of motion, recall Newton:XM D J
J..
.t/ D .t/ b ..t/D K ia.t/ b
.
.t/
Assume (FOR NOW ONLY) electrical response is faster
thanmechanical. La 0.
J..
.t/ D K
ea.t/ eb.t/Ra
b ..t/
J..
.t/C
b C KKeRa
| {z }
back emf indistinguishable from friction!
.
.t/ D KRa ea.t/
Dynamics of Heat Flow/ Dynamics of Fluid Flow
These two subjects will not be covered here. Refer to texts
onthermodynamics or fluid-dynamics.
Transformers and Gears
Ideally, both of these devices simply scale their input
value.
Transformer : N1N2D e1
e2D i2
i1
Gears : r1r2D 21D 12
r1 r2
System Identification (SYS ID) When we generate models of system
dynamics, we are performing
system identifications.
-
SYSTEM MODELING IN THE TIME DOMAIN 213
When we use known properties from physics and knowledge of
thesystems structure (as we have done here) we are performing
whitebox system ID.
If the system is very complex, or if the physics are not
wellunderstood, we need to use input/output data to generate a
systemmodel: black-box system ID.
A topic for the whole course!
Linearization
We will study how to control linear systems. Linear systems are
rare. We can linearize a non-linear systemthe controller designed
for
linearized model will work on the true nonlinear system (but not
aswell as a controller designed directly for the non-linear
system.)
KEY POINT: We can convert any differential equation into a
first-ordervector differential equation:
.
Ex D f .Ex; u/ I Ex D vector; u D input:If the system is linear,
this will be of the form:
.
Ex D AEx C BuI A and B are constant matrices:EXAMPLE: Torsional
pendulum (pg. 226)
..
.t/C bJ.
.t/C kJ .t/ D.t/
J
let"
x1.t/
x2.t/
#D".t/.
.t/
#
-
SYSTEM MODELING IN THE TIME DOMAIN 214
.
x2.t/C bJ x2.t/CkJ
x1.t/ D .t/J".
x1.t/.
x2.t/
#D"
0 1k=J b=J
#| {z }
A
"x1.t/
x2.t/
#C"
01=J
#| {z }
B
.t/:
So, our model of the torsional pendulum is linear.EXAMPLE:
Rotational pendulum (pg. 227)
..
.t/C gl sin..t// D.t/
ml2
let"
x1.t/
x2.t/
#D".t/.
.t/
#"
.
x1.t/.
x2.t/
#D24 x2.t/g
lsin.x1.t//
35C
24 01
ml2
35 .t/:
Not linear because we cannot make a constant A matrix.
Small Signal Linearization
Uses a Taylor-series expansion of the differential equation
aroundsome operating condition. (Equilibrium value where.
x0 D 0 D f .x0; u0/).
let x D x0 C x x0 D operating stateu D u0 C u u0 D nominal
control value:.
x D f .x; u/: Taylor-series expansion:
.
x D .x0 C .x f .x0; u0/C Ax C Bu plus higher-order terms
-
SYSTEM MODELING IN THE TIME DOMAIN 215
Subtract out equillibrium (nominal) solution;
.
x D Ax C Bu;which is linear. This is exactly how we linearized
the rotationalpendulum before, with 0 D 0I 0 D 0:
sin./ D 3
3!C
5
5!
:Feedback Linearization (computed torque) For rotational
pendulum, ml2 ...t/C mgl sin..t// D .t/.
COMPUTE: .t/ D mgl sin..t//C u.t/. THEN: ml2
..
.t/ D u.t/, no matter how large .t/ becomes! Sometimes used in
robotics and airplane flight control, but very
computationally intensive.
Analogous Systems
The linearized differential equations of many very different
physicalsystems appear identical.
One would suppose they behave in similar ways (dynamic
response)and can be controlled with similar controllers.
-
SYSTEM MODELING IN THE TIME DOMAIN 216
Mechanical Translational m ..x.t/C b .x.t/C kx.t/ D
u.t/Mechanical Rotational J
..
.t/C b ..t/C k.t/ D .t/Satellite J
..
.t/ D f .t/ dDC Motor (for La D 0) J
..
.t/C
b C kkeRa
.
.t/ D kRa ea.t/Generator .La L f /
..
ea.t/C .L f .Ra C Rl/C La R f / .ea.t/CR f .Ra C Rl/ D .kg Rl/e
f .t/
These are all of the forma2
..
x.t/C a1 .x.t/C a0x.t/ D b2 ..u.t/C b1 .u.t/C b0u.t/which is
called a second-order form.
Therefore, we have seen very specific examples of a very
generalclass of system. If we learn how to control the general
class, we canapply this knowledge to specific systems.
-
Feedback Control Systems. 31
DYNAMIC RESPONSE
We can now model dynamic systems with differential equations.
What do these equations mean? How does this system respond to
certain inputs? If we add dynamics (a controller) how will the
system respond? How SHOULD the system respond? (specifications)
KEY, ESSENTIAL, VITAL, TOOL: Laplace Transform
Some Important Input Signals
Several signals recur throughout this course. The unit step
function:
1.t/ D(
1; t 0I0; otherwise:
t
1.t/
The unit ramp function:
r.t/ D(
t; t 0I0; otherwise:
t
r .t/
The unit parabola function:
p.t/ D8 0. h.t/ D ekt1.t/. Response of this system to general
input:
y.t/ DZ 11
h. /u.t / d
DZ 11
ek1. /u.t / d
DZ 1
0eku.t / d:
Transfer Function
Response to impulse = impulse response h.t/. Response to general
input = messy convolution: h.t/ u.t/.
Response to Sinusoid? Cosinusoid?
A cos.!t/ D A2(e j!t C e j!t
Break it Down: Response to Exponential?
Let u.t/ D est; where s is complex.y.t/ D
Z 11
h. /u.t / d
DZ 11
h. /es.t/ d
DZ 11
h. /estes d
-
DYNAMIC RESPONSE 35
D estZ 11
h. /es d| {z }Transfer function; H.s/
D est H.s/: An input of the form est decouples the convolution
into two
independent parts: a part depending on est and a part depending
onh.t/. [Complex exponentials are eigenfunctions of all LTI
systems.]
EXAMPLE:.
y.t/C ky.t/ D u.t/ D est :but ; y.t/ D H.s/est; .y.t/ D s
H.s/est;s H.s/est C k H.s/est D est
H.s/ D 1s C k .I never integrated!/
y.t/ D est
s C k :Response to Cosinusoid (revisited)Let sD j! u.t/De j!t
y.t/DH. j!/e j!t
sD j! u.t/De j!t y.t/DH. j!/e j!tu.t/DA cos.!t/ y.t/DA
2H. j!/e j!t C H. j!/e j!t
Now; H. j!/ 4D Me jH. j!/ D Me j .can be shown for h.t/
real/
y.t/ D AM2e j .!tC/ C e j .!tC/
D AM cos.!t C /: The response of an LTI system to a sinusoid is
a sinusoid! (of the
same frequency).
-
DYNAMIC RESPONSE 36
EXAMPLE: Frequency response of our first order system:
H.s/ D 1s C k
H. j!/ D 1j! C kM D jH. j!/j D 1p
!2 C k2 D 6 H. j!/ D tan1
!k
y.t/ D Ap
!2 C k2 cos!t tan1
!k
:
Can we use these results to simplify convolution and get an
easierway to understand dynamic response?
The Laplace L Transform
We have seen that if a system has an impulse response h.t/, we
cancompute a transfer function H.s/,
H.s/ DZ 11
h.t/est dt:
Since we deal with causal systems (possibly with an impulse att
D 0), we can integrate from 0 instead of negative infinity.
H.s/ DZ 1
0h.t/est dt:
This is called the one-sided (uni-lateral) Laplace transform of
h.t/.
-
DYNAMIC RESPONSE 37
Laplace Transforms of Common SignalsName Time function, f .t/
Laplace tx., F.s/Unit impulse .t/ 1
Unit step 1.t/ 1s
Unit ramp t 1.t/ 1s2
nth order ramp tn 1.t/ n!snC1
Sine sin.bt/1.t/ bs2 C b2
Cosine cos.bt/1.t/ ss2 C b2
Damped sine eat sin.bt/1.t/ b.s C a/2 C b2
Damped cosine eat cos.bt/1.t/ s C a.s C a/2 C b2
Diverging sine t sin.bt/1.t/ 2bs.s2 C b2/2
Diverging cosine t cos.bt/1.t/ s2 b2
.s2 C b2/2
Properties of the Laplace Transform
Superposition: L fa f1.t/C b f2.t/g D aF1.s/C bF2.s/. Time
delay: L f f .t /g D esF.s/. Time Scaling: L f f .at/g D 1jajF
sa
.
(useful if original equations are expressed poorly in time
scale. e.g.,measuring disk-drive seek speed in hours).
Differentiation:
L
n.
f .t/oD s F.s/ f .0/
L
n..
f .t/oD s2F.s/ s f .0/ .f .0/
L
f .m/.t/} D sm F.s/ sm1 f .0/ : : : f .m1/.0/:
-
DYNAMIC RESPONSE 38
Integration: LZ t
0f . / d
D 1
sF.s/.
Convolution: Recall that y.t/ D h.t/ u.t/Y .s/ D L fy.t/g D L
fh.t/ u.t/gD L
Z tD0
h. /u.t / d
DZ 1
tD0
Z tD0
h. /u.t / d est dt
DZ 1D0
Z 1tD
h. /u.t / est dt d:
t
D t
Region ofintegration
Multiply by eses
Y .s/ DZ 1D0
h. /esZ 1
tDu.t /es.t/ dt d:
Let t 0 D t :Y .s/ D
Z 1D0
h. /es dZ 1
t 0D0u.t 0/est
0 dt 0
Y .s/ D H.s/U .s/: The Laplace transform unwraps convolution for
general input
signals. Makes system easy to analyze.
The Inverse Laplace Transform
The inverse Laplace Transform converts F.s/! f .t/. Once we get
an intuitive feel for F.s/, we wont need to do this often.
-
DYNAMIC RESPONSE 39
The main tool for ILT is partial-fraction-expansion.Assume :
F.s/ D b0s
m C b1sm1 C C bmsn C aasn1 C C an
D kQm
iD1.s zi/QniD1.s pi/
.zeros/
.poles/
D c1s p1 C
c2
s p2 C Ccn
s pn if fpigdistinct:
so; .s p1/F.s/ D c1 C c2.s p1/s p2 C C
cn.s p1/s pn
let s D p1 : c1 D .s p1/F.s/jsDp1ci D .s pi/F.s/jsDpi
f .t/ DnX
iD1cie
pi t1.t/ since Lekt1.t/
D 1s k :
EXAMPLE: F.s/ D 5s2 C 3s C 2 D
5.s C 1/.s C 2/.
c1 D .s C 1/F.s/sD1D 5
s C 2sD1D 5
c2 D .s C 2/F.s/sD2D 5
s C 1sD2D 5
f .t/ D .5et 5e2t/1.t/: If F.s/ has repeated roots, we must
modify the procedure. e.g.,
repeated three times:
F.s/ D k.s p1/3.s p2/
D c1;1s p1 C
c1;2
.s p1/2 Cc1;3
.s p1/3 Cc2
s p2 C
c1;3 D .s p1/3F.s/sDp1
-
DYNAMIC RESPONSE 310
c1;2 D
dds(.s p1/3F.s/
sDp1
c1;1 D 12
d2
ds2(.s p1/3F.s/
sDp1
cx;ki D 1i!
di
dsi(.s pi/k F.s/
sDpi
:
EXAMPLE: Find ILT of s C 3.s C 1/.s C 2/2 .
ans: f .t/ D .2et 2e2t te2t|{z}from repeated root.
/1.t/:
TEDIOUS. Use Matlab. e.g., F.s/ D 5
s2 C 3s C 2.
Example 1. Example 2.
>> Fnum = [0 0 5]; >> Fnum = [0 0 1 3];
>> Fden = [1 3 2]; >> Fden = conv([1 1],conv([1
2],[1 2]));
[r,p,k] = residue(Fnum,Fden); [r,p,k] = residue(Fnum,Fden);
r = -5 r = -2
5 -1
p = -2 2
-1 p = -2
k = [] -2
-1
k = []
When you use residue and get repeated roots, BE SURE to typehelp
residue to correctly interpret the result.
-
DYNAMIC RESPONSE 311Using the Laplace Transform to Solve
Problems
We can use the Laplace transform to solve both homogeneous
andforced differential equations.
EXAMPLE:..
y.t/C y.t/ D 0; y.0/ D ; .y.0/ D :s2Y .s/ s C Y .s/ D 0
Y .s/.s2 C 1/ D s C Y .s/ D s C
s2 C 1D s
s2 C 1 C
s2 C 1:From tables, y.t/ D [ cos.t/C sin.t/]1.t/. If initial
conditions are zero, things are very simple.
EXAMPLE:..
y.t/C5 .y.t/C4y.t/ D u.t/; y.0/ D 0; .y.0/ D 0; u.t/ D
2e2t1.t/:s2Y .s/C 5sY .s/C 4Y .s/ D 2
s C 2Y .s/ D 2
.s C 2/.s C 1/.s C 4/D 1
s C 2 C2=3
s C 1 C1=3
s C 4:
From tables, y.t/ De2t C 2
3et C 1
3e4t
1.t/.
Time Response vs. Pole Locations: 1st-Order Pole(s) (stable)
Pole = root of denominator of H.s/ D b.s/=a.s/. Poles qualitatively
determine the behavior of the system.
-
DYNAMIC RESPONSE 312
Zeros quantify this relationship.EXAMPLE: H.s/ D 1
s C h.t/ D e t1.t/:
If > 0, pole is at s < 0, STABLE i.e., impulse response
decays, andany bounded input produces bounded output.
If < 0, pole is at s > 0, UNSTABLE. is time constant
factor: D 1= .
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1impulse([0 1],[1 1]);
Time (sec )t D
1e
e t
h.t/
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1step([0 1],[1 1]);
Time (sec )t D
y.t/
KK .1 et= /
System response. K D dc gain
Response to initial condition! 0.
Time Response vs. Pole Locations: 2nd-Order Poles (stable)
H.s/ D b0s2 C a1s C a2 D
K!2ns2 C 2!ns C !2n
.standard form/:
D damping ratio:!n D natural frequency or undamped
frequency:
h.t/ D !np1 2e
t .sin.!dt// 1.t/;
where; D !n;!d D !n
p1 2 D damped frequency:
-
DYNAMIC RESPONSE 313
D sin1. /=.s/
-
DYNAMIC RESPONSE 314
=.s/
-
DYNAMIC RESPONSE 315
tr = Rise time = time to reach vicinity of new set point. ts =
Settling time = time for transients to decay (to 5%, 2%, 1%). Mp =
Percent overshoot. tp = Time to peak.
Rise Time
All step responses rise in roughly the same amount of time
(seepg. 3313.) Take D 0:5 to be average. time from 0:1 to 0:9 is
approx !ntr D 1:8:
tr 1:8!n:
We could make this more accurate, but note: Only valid for
2nd-order systems with no zeros. Use this as approximate design
rule of thumb and iterate design
until spec. is met.
Peak Time and Overshoot
Step response can be found from ILT of H.s/=s:y.t/ D 1 e t
cos.!dt/C
!dsin.!dt/
;
!d D !np
1 2; D !n: Peak occurs when .y.t/ D 0
.
y.t/ D e t
cos.!dt/C !d
sin.!dt/ e t .!d sin.!dt/C cos.!dt//
D e t 2
!dsin.!dt/C !d sin.!dt/
D 0:
-
DYNAMIC RESPONSE 316
So,!dtp D ;
tp D !dD !np
1 2 :
Mp D e=p
1 2 100. (common values: Mp D 16% for D 0:5; Mp D 5% for D 0:7).
0 0.2 0.4 0.6 0.8 1.00
102030405060708090
100
Mp,
%
Settling Time
Determined mostly by decaying exponentiale!n ts D : : : D 0:01;
0:02; or 0:05
EXAMPLE:
D 0:01e!n ts D 0:01!n ts D 4:6
ts D 4:6!nD 4:6
ts0:01 ts D 4:6=0:02 ts D 3:9=0:05 ts D 3:0=
Design Synthesis
Specifications on tr , ts, Mp determine pole locations. !n
1:8=tr . fn.Mp/: (read off of versus Mp graph on page 3316) 4:6=ts:
(for examplesettling to 1%)
-
DYNAMIC RESPONSE 317
=.s/
-
DYNAMIC RESPONSE 318
Mp D 0; D 1: From transfer fn: 2 D 2!n !n D 1: !2n D 12 D 0:5Ka;
Ka D 2:0 Note: ts D 4:6 seconds. We will need a better controller
than this for a
pen plotter!
Time Response vs. Pole Locations: Higher Order Systems
We have looked at first-order and second-order systems
withoutzeros, and with unity gain.
Non-unity gain
If we multiply by K , the dc gain is K . tr , ts, Mp, tp are not
affected.Add a zero to a second-order system
H1.s/ =2
.s C 1/.s C 2/ H2.s/ =2.s C 1:1/
1:1.s C 1/.s C 2/=
2s C 1
2s C 2 =
21:1
0:1
s C 1 C0:9
s C 2
=
0:18s C 1 C
1:64s C 2
Same dc gain (at s D 0). Coefficient of .s C 1/ pole GREATLY
reduced. General conclusion: a zero near a pole tends to cancel the
effect of
that pole. How about transient response?
H.s/ D .s=!n/C 1.s=!n/2 C 2 s=!n C 1:
-
DYNAMIC RESPONSE 319
Zero at s D : Poles at
-
DYNAMIC RESPONSE 320
0 2 4 6 8 100.5
0
0.5
1
1.5
2
H .s/
Ho.s/
Hd.s/
Time (sec)
y.t/
2nd-order min-phase step resp.
0 2 4 6 8 101.5
1
0.5
0
0.5
1
1.5
Ho.s/
H .s/
Hd.s/
Time (sec)
y.t/
2nd-order nonmin-phase step resp.
Add a pole to a second order system
H.s/ D 1.s=!n C 1/[.s=!n/2 C 2 s=!n C 1]:
Original poles at
-
DYNAMIC RESPONSE 321
Extra zero in RHP depresses overshoot, and may cause
stepresponse to start in wrong direction. DELAY .
Extra pole in LHP increases rise-time if extra pole is within a
factor of 4 from the real part of complex poles.
UnstableregionDominantInsignificant
=.s/
-
DYNAMIC RESPONSE 322
Important tools: block diagram manipulation and Masons rule.
Block Diagram Manipulation
We have already seen block diagrams (see pg. 113). Shows
information/energy flow in a system. When used with Laplace
transforms, can simplify complex system
dynamics.
Four BASIC configurations:H .s/ Y .s/U.s/ Y .s/ D H .s/U.s/
H1.s/ H2.s/ Y .s/U.s/ Y .s/ D [H1.s/H2.s/] U.s/
H1.s/
H2.s/
Y .s/U.s/ Y .s/ D [H1.s/C H2.s/] U.s/
H1.s/
H2.s/
Y .s/R.s/U1.s/
Y2.s/ U2.s/
U1.s/ D R.s/ Y2.s/Y2.s/ D H2.s/H1.s/U1.s/
so; U1.s/ D R.s/ H2.s/H1.s/U1.s/D R.s/
1C H2.s/H1.s/Y .s/ D H1.s/U1.s/
D H1.s/1C H2.s/H1.s/R.s/
Alternate representation
-
DYNAMIC RESPONSE 323H .s/
H1.s/
H1.s/
H1.s/ H2.s/
H2.s/
H2.s/Y .s/
Y .s/
Y .s/
Y .s/
U.s/
U.s/
U.s/
R.s/
EXAMPLE: Recall dc generator dynamics from page 2210R f Ra Lai f
.t/ ia.t/
e f .t/ L f eg.t/ Zlea.t/
| {z }Fieldcircuit
| {z }Rotorcircuit
| {z }Loadcircuit
e f .t/ ea.t/i f .t/ eg.t/ ia.t/Field
circuit KgRotorcircuit Zl
Compute the transfer functions of the four blocks.
e f D R f i f C L f ddt i fE f .s/ D R f I f .s/C L f s I f .s/I
f .s/E f .s/
D 1R f C L f s :
eg.t/ D Kgi f .t/Eg.s/ D Kg I f .s/Eg.s/I f .s/
D Kg:
-
DYNAMIC RESPONSE 324
ea.t/ D ia.t/ZlEa.s/ D Zl Ia.s/Ea.s/Ia.s/
D Zl:
eg.t/ D Raia.t/C La ddt ia.t/C ea.t/Eg.s/ D Ra Ia.s/C Las Ia.s/C
Ea.s/
D .Ra C Las C Zl/ Ia.s/Ia.s/Eg.s/
D 1Las C Ra C Zl :
Put everything together.Ea.s/E f .s/
D Ea.s/Ia.s/
Ia.s/Eg.s/
Eg.s/I f .s/
I f .s/E f .s/
D Kg Zl(L f s C R f
.Las C Ra C Zl/:
Block Diagram Algebra
()
()
()
H .s/
H .s/
H .s/H .s/
H .s/
1H .s/
H1.s/H1.s/ H2.s/
H2.s/
1H2.s/
Y .s/Y .s/
Y .s/Y .s/
Y1.s/Y1.s/
Y2.s/Y2.s/
U.s/U.s/
U1.s/U1.s/
U2.s/U2.s/
R.s/R.s/
Unity Feedback
-
DYNAMIC RESPONSE 325
EXAMPLE: Simplify:
H1.s/ H2.s/
H3.s/
H4.s/
H5.s/
H6.s/
Y .s/R.s/
H1.s/1 H1.s/H3.s/ H2.s/
H4.s/
H5.s/
H6.s/
Y .s/R.s/
H1.s/1 H1.s/H3.s/ H2.s/
H4.s/
H5.s/
H6.s/H2.s/
Y .s/R.s/
| {z }H1.s/H2.s/
1H1.s/H3.s/
1CH1.s/H2.s/H4.s/1H1.s/H3.s/
| {z }
H5.s/CH6.s/H2.s/
H1.s/H2.s/H5.s/C H1.s/H6.s/1 H1.s/H3.s/C H1.s/H2.s/H4.s/R.s/ Y
.s/
-
DYNAMIC RESPONSE 326Masons Rule
If you dont care to simplify a block diagram using block
diagrammanipulation, you can use Masons Rule.
Node = Common input to several blocks, or output of
summingjunction.
H1.s/
H2.s/
Y .s/U.s/
Path = Sequence of connected blocks, from one variable to
another,in the direction of signal flow, without including any
variable more thanonce.
Forward Path = Path from input to output. Loop = Path from node
back to itself. Path Gain = Product of all transfer functions in a
path. Loop Gain = Product of all transfer functions in a loop.
Nontouching = Two loops are nontouching if they have no nodes
in
common.
G1 G2 G3 G4 G5
G6
H1 H2YR
2 Loops: , Nontouching. 2 Forward paths: Touches both loops.
Does not touch first loop, but touches second loop.
-
DYNAMIC RESPONSE 327
Masons RuleT D 1
1
pXkD1
Mk1k D 11
(M111 C M212 C C Mp1p
1 D 1 .sum of ALL .touching or not/ individual loop gains/C.sum
of the products of loop gains of all possiblecombinations of
nontouching loops taken two at a time/.sum of the products of loop
gains of all possiblecombinations of nontouching loops taken three
at a time/C
Mk D Path gain of kth forward path1k D Value of 1 for that part
of the flow graph not touching the kth
forward path:
Key to using rule = organization! Above example:
Find the loop- and the forward path-gains. Loops: L1 : ;
Gain=H1G2. L2 : ; Gain=H2G4.
Forward Paths: M1 : ; Gain=G1G2G3G4G5 M2 : ; Gain=G6G4G5.
1 D 1 .L1 C L2/C L1L2 D 1C H1G2 C H2G4 C H1 H2G2G4. 1k is
easiest determined by redrawing flowgraph without path k.
-
DYNAMIC RESPONSE 328
G1 G2 G3
G6
H1
H1
H2
H2
Path 1 removed. No loops.11 D 1 0
Path 2 removed. One loop withgain G2 H1. 12 D 1 .G2 H1/
T D M111 C M2121
D G1G2G3G4G5 C G6G4G5.1C H1G2/1C H1G2 C H2G4 C H1 H2G2G4 :
Masons rule may be shorter that block-diagram manipulation. Use
the method with which you are most comfortable.
-
Feedback Control Systems. 41
BASIC PROPERTIES OF FEEDBACK
Two basic types of control systems:OPEN LOOP:
Ctrlrr .t/ y.t/
Disturbance
Plant
CLOSED LOOP:
Ctrlrr .t/ y.t/
Disturbance
Plant
Sensor
We will compare these systems in a number of ways:
disturbancerejection, sensitivity, dynamic tracking, steady state
error and stability.
DC Motor Speed Control
Recall equations of motion for a dc motor but add
loadtorque.
ea.t/ eb.t/
ia.t/ Ra La b
J.t/; .t/
| {z }Armature
| {z }Load
l , load
Assume that we are trying to control motor speed:
-
BASIC PROPERTIES OF FEEDBACK 42
J..
C b .ke
.
C La diadt C RaiaDD
k ia C lea
9=; let output y D
.
;
disturbance w 4D l:
J.
y C bykey C La diadt C Raia
DD
k ia C wea
9=; s JY .s/C bY .s/ D k Ia.s/CW .s/keY .s/C sLa Ia.s/C Ra Ia.s/
D Ea.s/
.J Lass C bLas C J Ras C bRa C kke/Y .s/ D kEa.s/C .Ra C Las/W
.s/ This can be re-written as (magic happens)
.1s C 1/.2s C 1/Y .s/ D AEa.s/C BW .s/1 Ra J=.kke/ D mechanical
time constant2 La=Ra D electrical time constantA D k=.bRa C kke/B D
Ra=.bRa C kke/
So,Y .s/ D A
.1s C 1/.2s C 1/Ea.s/CB
.1s C 1/.2s C 1/W .s/: If ea.t/ D ea 1.t/ (constant) and w.t/ D
w 1.t/ (constant), What is
steady state output?
Recall Laplace-transform final value theorem:
If a signal has a constant final value, it may be found asyss D
lim
s!0sY .s/:
So,Ea.s/ D ea
s; W .s/ D w
s; yss D Aea C Bw:
This is the response of the open-loop system (without a
controller).
-
BASIC PROPERTIES OF FEEDBACK 43
Ctrlrr .t/ y.t/
Dist. w.t/
A.1s C 1/.2s C 1/
BA
ea.t/
Motor
Lets make a simple controller for the open-loop system.ea.t/ D
Kolr.t/; .gain of Kol/
Choose Kol so that there is no steady-state error when w D 0.yss
D AKolrss C Bwss) Kol D 1=A:
Is closed-loop any better?
Ctrlrr .t/ y.t/
Dist. w.t/
A.1s C 1/.2s C 1/
BA
ea.t/
Tachometer
1
Lets make a similar controller for the closed-loop system (with
thepossibility of a different value of K .)
ea.t/ D Kcl.r.t/ y.t// The transfer function for the closed-loop
system is:
Y .s/ D AKcl.1s C 1/.2s C 1/ .R.s/ Y .s//C
B.1s C 1/.2s C 1/W .s/
D AKcl.1s C 1/.2s C 1/C AKcl R.s/C
B.1s C 1/.2s C 1/C AKcl W .s/
yss D AKcl1C AKcl rss; assuming w D 0:
-
BASIC PROPERTIES OF FEEDBACK 44
If AKcl 1; yss rss. Open-loop with load:
yss D AKolrss C Bwss D rss C Bwssy D Bwss:
Closed-loop with load:yss D AKcl1C AKcl rss C
B1C AKclwss
y B1C AKclwss:
which is much better than open-loop since AKcl 1.ADVANTAGE OF
FEEDBACK: Better disturbance rejection (by factor of1C AKcl).
Sensitivity
The steady-state gain of the open-loop system is: 1.0 How does
this change if the motor constant A changes?
A! A C AGol C Gol D Kol.A C A/
D 1A.A C A/
D 1C AA|{z}
gain error
:
In relative terms: GolGolD A
AD 1:0|{z}sensitivity
AA:
Therefore, a 10% change in A 10% change in gain.
Sensitivity=1.0.
-
BASIC PROPERTIES OF FEEDBACK 45
Steady-state gain of closed-loop system is: AKcl1C AKcl :
Gcl C Gcl D .A C A/Kcl1C .A C A/Kcl : From calculus (law of
total differential)
Gcl D dGcld A Aor
GclGclD
A
GcldGcld A
| {z }
sensitivity SGclA
AA
SGclA DA
AKcl=.1C AKcl/.1C AKcl/Kcl Kcl.AKcl/
.1C AKcl/2
D 11C AKcl :
ADVANTAGE OF FEEDBACK: Lower sensitivity to modeling error (by
afactor of 1C AKcl)
Dynamic Tracking
Steady-state response of closed-loop better than open-loop:
Betterdisturbance rejection, better (lower) sensitivity. What about
transient response? Open-loop system: Poles at roots of .1s C 1/.2s
C 1/
s D 1=1; s D 1=2: Closed-loop system: Poles at roots of .1s C
1/.2s C 1/C AKcl:
s D .1 C 2/p.1 C 2/2 412.1C AKcl/
212:
-
BASIC PROPERTIES OF FEEDBACK 46
FEEDBACK MOVES POLES System may have faster/slower response
System may be more/less damped System may become unstable!!! Often
a high gain Kcl results in instability. We need design tools to
help us design the dynamic response of the closed-loop system.
For this dc motor example, we can get step responses of the
following form:
0 2 4 6 8 100
0.5
1
1.5
High Kcl
Low Kcl
Time (sec)
y.t/
PID Control
General control setup:
D.s/R.s/ Y .s/
Disturbance
G.s/
Need to design controller D.s/. One option is PID (Proportional
Integral Derivative) control design.
-
BASIC PROPERTIES OF FEEDBACK 47
Extremely popular. 90C% of all controllers are PID. Doesnt mean
that they are great, just popular.
We just saw proportional control where u.t/ D K e.t/, or D.s/ D
K . Proportional control allows non-zero steady state error.
Increases speed of response. Larger transient overshoot.
Integral control eliminates steady state error. D.s/ D K Is:
Dynamic response gets worse.
Derivative control damps dynamic response. D.s/ D K
Ds.Proportional Control
u.t/ D K .r.t/ y.t// D K e.t/ : : : D.s/ D K : May have steady
state error, may not be able to completely reject a
constant disturbance.
EXAMPLE: Determine behavior of closed loop poles for the dc
motor.
Y .s/R.s/
D AK.1s C 1/.2s C 1/C AK :
Poles are roots of .1s C 1/.2s C 1/C AK : Without feedback, K !
0.
s1 D 1=1; s2 D 1=2: With feedback,
s1; s2 D .1 C 2/p.1 C 2/2 412.1C AK /
212:
-
BASIC PROPERTIES OF FEEDBACK 48
K D 0K D 0 11
12
K D .1 2/2
412 A
1 C 2212
-
BASIC PROPERTIES OF FEEDBACK 49
12..
y.t/C .1 C 2/.
y.t/C y.t/ D A
KTI
Z t0.r. / y. // d
C Bw.t/:
Differentiate,
12...
y.t/C .1 C 2/..
y.t/C .y.t/ D AKTI.r.t/ y.t//C B .w.t/
12...
y.t/C .1 C 2/..
y.t/C .y.t/C AKTI
y.t/ D AKTI
r.t/C B .w.t/:
If r.t/ D cst, w.t/ D cst, .w.t/ D 0,AKTI
yss D AKTI rss no error: Steady-state improves Dynamic response
degrades.
Very oscillatory. Possibly unstable.
Can be improved by adding propor-tional term to integral
term.
K D 0K D 0 11
12
-
BASIC PROPERTIES OF FEEDBACK 410
TD = Derivative time. PURE DERIVATIVE CONTROL IMPRACTICAL SINCE
DERIVATIVE
MAGNIFIES SENSOR NOISE! Practical version = Lead control, which
we will study later. Stabilizes system. Does nothing to reduce
constant error! If .e.t/ D 0, then u.t/ D 0. Motor Control: Poles
at roots of 12s2 C .1 C 2 C AK TD/s C 1 D 0.
TD enters term, can make damping better.
PD = Proportional plus derivative controlD.s/ D K .1C TDs/
Root locus for dc motor, PD control.
11
12
1TD
-
BASIC PROPERTIES OF FEEDBACK 411
For speed control problem,u.t/ D K
.r.t/ y.t//C 1
TI
Z t0.r. / y. // d C TD. .r.t/
.
y.t//:
(math happens). Solve for poles12TI s3 C TI ..1 C 2/C AK TD/s2 C
TI .1C AK /s C AK D 0
s3 C1 C 2 C AK TD
12
s2 C
1C AK12
s C AK
12TID 0:
Three coefficients, three parameters. We can put poles
anywhere!Complete control of dynamics in this case. Entire transfer
functions are:
Y .s/W .s/
D TI BsTI12s3 C TI .1 C 2/s2 C TI .1C AK /s C AK :
Y .s/R.s/
D AK .TI s C 1/TI12s3 C TI .1 C 2/s2 C TI .1C AK /s C AK :
We can plot responses in MATLAB:num = [TI*B 0];den =
[TI*TAU1*TAU2 TI*(TAU1+TAU2) TI*(1+A*K) A*K];step(num, den)
0 0.01 0.02 0.03 0.04 0.050
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time (sec)
y.t/
(rad/
sec) P
PI
PID
Step Reference Response
0 0.01 0.02 0.03 0.04 0.050
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time (sec)
y.t/
(rad/
sec) P
PIPID
Step Disturbance Response
Ziegler - Nichols Tuning of PID Controllers
Rules of Thumb for selecting K , TI , TD.
-
BASIC PROPERTIES OF FEEDBACK 412
Not optimal in any sensejust provide good performance.METHOD I:
If system has step response like this,
Slope, A=A
d
Y .s/U .s/
D Aeds
s C 1;(first-order system plus delay)
We can easily identify A, d, from this step response. Dont need
complex model! Tuning criteria: Ripple in impulse response decays
to 25% of its value
in one period of ripple
Period1
0:25
RESULTING TUNING RULES:P PI PID
K D Ad
K D 0:9Ad
TI D d0:3
K D 1:2Ad
TI D 2dTD D 0:5d
METHOD II: Configure system as
Kur .t/ y.t/Plant
Turn up gain Ku until system produces oscillations (on
stabilityboundary) Ku = ultimate gain.
-
BASIC PROPERTIES OF FEEDBACK 413
Period, Pu1 RESULTING TUNING RULES:
P PI PID
K D 0:5KuK D 0:45KuTI D 11:2 Pu
K D 0:6KuTI D 0:5PuTD D Pu8
Practical Problem: Integrator Overload
Integrator in PI or PID control can cause problems. For example,
suppose there is saturation in the actuator.
Error will not decrease. Integrator will integrate a constant
error and its value will blow up.
Solution = integrator anti-windup. Turn off integration when
actuatorsaturates.
K
KaK
TI s
e.t/ u.t/ umin
umax
Doing this is NECESSARY in any practical implementation.
Omission leads to bad response, instability.
-
BASIC PROPERTIES OF FEEDBACK 414
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time (sec)
Step Response
With antiwindup
Without antiwindup
y.t/
0 2 4 6 8 100.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time (sec)
Control Effort
With antiwindup
Without antiwindup
u.t/
Steady-State Error
System error is any difference between r.t/ and y.t/. Two
sources:
1. Imprecise tracking of r.t/.2. Disturbance affecting the
system output.
Steady-State Error (w.r.t. Reference Input) We have already seen
examples of CL systems that have some
tracking error (proportional ctrl) or not (integral ctrl) to a
step input.We will formalize this concept here.
Start with very general control structure:T .s/r .t/ y.t/
The closed-loop transfer function for the whole system,T .s/ D Y
.s/
R.s/:
The error isE.s/ D R.s/ Y .s/
-
BASIC PROPERTIES OF FEEDBACK 415
D R.s/ T .s/R.s/D [1 T .s/] R.s/:
Assume conditions of final value theorem are satisfied (i.e., T
.s/ isstable)
ess D limt!1 e.t/ D lims!0 s[1 T .s/]R.s/:
We use test inputs of the type: r.t/ D tk
k!1.t/; R.s/ D 1
skC1.
t
1.t/
t
t 1.t/
t
t2
2 1.t/
As k increases, tracking is progressively harder.
ess D lims!0
1 T .s/sk
D
8>:
0; type > K Iconstant; type D K I1; type < K :
If system type = 0, constant steady-state error for step input,
infinites.s. error for ramp or parabolic input.
If system type = 1, no steady-state error for step input,
constant s.s.error for ramp input, infinite s.s. error for
parabolic input.
If system type = 2, no steady-state error for step or ramp
inputs,constant s.s. error for parabolic inputs.
And so forth, for higher-order system types. Ramp responses of
different system types:
-
BASIC PROPERTIES OF FEEDBACK 416
t
tType 0 System
t
tType 1 System
t
tType 2 System
DANGER: Higher order sounds better but they are harder to
stabilizeand design. Transient response may be poor.
UNITY FEEDBACK: SPECIAL CASE
NOTE: The following method is a special case of the above
generalmethod. Be careful to use the apropriate method for the
problem athand!
Unity-feedback is when the control system looks like:Gol.s/r .t/
y.t/
There are some important simplifications:T .s/ D Gol.s/
1C Gol.s/1 T .s/ D
1C Gol.s/1C Gol.s/
Gol.s/
1C Gol.s/D 1
1C Gol.s/:So,
E.s/ D 11C Gol.s/R.s/:
-
BASIC PROPERTIES OF FEEDBACK 417
For the test inputs R.s/ D 1skC1
ess D lims!0
s E.s/
D lims!0
1[1C Gol.s/]sk
For type 0,ess D lim
s!01
1C Gol.s/ D1
1C K p ; K p D lims!0 Gol.s/:
For type 1,ess D lim
s!01
1C Gol.s/1sD lim
s!01
sGol.s/D 1
Kv; Kv D lim
s!0sGol.s/:
For type 2,ess D lim
s!01
1C Gol.s/1s2D lim
s!01
s2Gol.s/D 1
Ka; Ka D lim
s!0s2Gol.s/:
These formulas only meaningful for unity-feedback!K p D lim
s!0Gol.s/: position error constant
Kv D lims!0
sGol.s/: velocity error constantKa D lim
s!0s2Gol.s/: acceleration error constant
Steady-state tracking errors ess for unity-feedback case
ONLY.Sys. Type Step Input Ramp Input Parabola Input
Type 0 11C K p 1 1
Type 1 0 1Kv
1Type 2 0 0 1
Ka
-
BASIC PROPERTIES OF FEEDBACK 418EXAMPLES:
(1) Consider Gol.s/ D s C 1.s C 2/.s C 3/; System type?
Gol.0/ D 12 3 D16:
Therefore, typeD 0, ess to unit stepD 11C 1=6 D67:
(2) Consider Gol.s/ D .s C 1/.s C 10/.s 5/.s2 C 3s/.s4C s2 C 1/
; System type?
Gol.0/ D 1 10 .5/0 1 D 1 Type > 0:
sGol.s/ D .s C 1/.s C 10/.s 5/.s C 3/.s4 C s2 C 1/
sGol.s/jsD0 D 1 10 .5/3 1 D50
3:
Therefore, typeD 1, ess to unit rampD 350 :
(3) Consider Gol.s/ D s2 C 2s C 1
s4 C 3s3 C 2s2 ; System type?.
Gol.0/ D 10 D 1 Type > 0
sGol.s/ D s2 C 2s C 1
s3 C 3s2 C 2ssGol.s/jsD0 D 10 D 1 Type > 1
s2Gol.s/ D s2 C 2s C 1
s2 C 3s C 2s2Gol.s/
sD0 D
12
Type D 2:
-
BASIC PROPERTIES OF FEEDBACK 419
Therefore, typeD 2, ess to unit parabolaD 2.KEY POINT: Open-loop
G.s/ D Gol.s/ tells us about closed-loop s.s.
response.
EXAMPLE: DC-motor example with Proportional control.
Ctrlrr .t/ y.t/A
.1s C 1/.2s C 1/
Controller: D.s/ D K .D.s/G.s/ D K A
.1s C 1/.2s C 1/; lims!0 D.s/G.s/ D K A:
So system is type 0, with s.s. error to step input of 11C K A :
(This
agrees with prior results.)EXAMPLE: DC-motor example with PI
control
Controller: D.s/ D K
1C 1TI s
:
D.s/G.s/ D K A CK ATI s
.1s C 1/.2s C 1/:lims!0
D.s/G.s/ D 1
lims!0
s D.s/G.s/ D K ATI:
System is type 1, with s.s. error to ramp input of TIK A
.
EXAMPLE: DC-motor with another integrator
-
BASIC PROPERTIES OF FEEDBACK 420
Controller: D.s/ D K
1C 1TI sC 1
TI s2
:
D.s/G.s/ D K A CK ATI sC K ATI s2
.1s C 1/.2s C 1/lims!0
s2D.s/G.s/ D K ATI:
System is type 2, with s.s. error to parabolic input of TIK
A
.
Steady State Error (w.r.t. Disturbance) Recall, system error is
any difference between r.t/ and y.t/. One source of system error is
disturbance. Can find system type with respect to disturbance.
Y .s/W .s/
D Tw.s/such that
Y .s/ D T .s/R.s/C Tw.s/W .s/: We do not wish the output to have
ANY disturbance term in it, so the
output error due to the disturbance is equal to the output due
to thedisturbance.
ess D yss D lims!0
sTw.s/W .s/:
Type 0 system (w.r.t. disturbance) has constant lims!0
Tw.s/:
Type 1 system (w.r.t. disturbance) has constant lims!0
Tw.s/s
:
Type 2 system (w.r.t. disturbance) has constant lims!0
Tw.s/s2
:
-
Feedback Control Systems. 51
STABILITY ANALYSIS
Many classifications of stability in system analysis. For LTI
systems, all are basically the same BIBO.
Stable Neutral Unstable
If input is bounded: ju.t/j < K1, then output is also
boundedjy.t/j < K2. In the time domain,
y.t/ DZ 11
h. /u.t / d
jy.t/j DZ 11
h. /u.t / d
-
STABILITY ANALYSIS 52
Z 11jh. /j ju.t /j d
K1Z 11jh. /j d:
So if a system is BIBO stable,Z 11jh. /j d < 1.
y.t/u.t/ C Stable? Note: .h.t/ D 1.t//.
In the Laplace domain,H.s/ D Y .s/
R.s/D K
Qm1 .s zi/Qn
1.s pi/m n:
(assume poles unique)h.t/ D
nXiD1
kiepi t
Z 11jh. /j d
-
STABILITY ANALYSIS 53
EXAMPLE: T .s/ D 2s2 C 3s C 2. Is T .s/ stable?
T .s/ D 2.s C 1/.s C 2/;
Roots at s D 1; s D 2: Stable!EXAMPLE: T .s/ D 10s C 24
s3 C 2s2 11s 12 . Is T .s/ stable?
T .s/ D 10.s C 2:4/.s C 1/.s 3/.s C 4/;
Roots at s D 1, s D C3, s D 4. Unstable!EXAMPLE: T .s/ D s
s2 C 1. Is T .s/ stable?
T .s/ D s.s j/.s C j/;
Roots at s D j . MARGINALLY stable. Use input = sin.t/.
Y .s/ D ss2 C 1
1s2 C 1 D
s
.s2 C 1/2y.t/ D t sin.t/ : : : unbounded:
MARGINALLY stable = unstable (bounded impulse response,
butunbounded output for some inputs.)
Routh Hurwitz Stability Criterion
Factoring high-degree polynomials to find roots is tedious
andnumerically not well conditioned.
Want other stability tests, and also MARGINS of stability.
-
STABILITY ANALYSIS 54
.1/ Routh test
.2/ Root locus
.3/ Nyquist test
.4/ Bode stability margins
9>>>>=>>>>;
Can you tell this isan important topic?
In 1868, Maxwell found conditions on the coefficients of a
transferfunction polynomial of 2nd and 3rd order to guarantee
stability.
It became the subject of the 1877 Adams Prize to
determineconditions for stability for higher-order polynomials.
Routh won this prize, and the method is still useful.Case 0
Consider the denominator a.s/.
2nd order:a.s/ D s2 C a1s C a0 D .s p1/.s p2/ D s2 .p1 C p2/s C
p1 p2:3rd order:
a.s/ D s3 C a2s2 C a1s C a0D .s p1/.s p2/.s p3/D s2 .p1 C p2/s C
p1 p2 s p3D s3 .p1 C p2 C p3/s2 C .p1 p2 C p1 p3 C p2 p3/s p1 p2
p3:
TREND: Stability none of the coefficients ai can be 0.Does this
trend continue?
-
STABILITY ANALYSIS 55
an1 D .1/ sum of all roots.an2 D sum of products of roots taken
two at a time.an3 D .1/ sum of products of roots taken three at a
time.
:::
a0 D .1/n product of all roots. Conclusions:
1. If any coefficient ai D 0, not all roots in LHP.2. If any
coefficient ai < 0, at least one root in RHP.
TEST: If any coefficient ai 0, system is unstable.EXAMPLE: a.s/
D s2 C 0s C 1. From conclusion (1), not all roots in LHP. Roots at
s D j . Marginally stable.
EXAMPLE: a.s/ D s3 C 2s2 11s 12. From conclusion (2), at least
one root is in RHP. Roots at s D 1, s D 4, s D C3. Unstable.
EXAMPLE: a.s/ D s3 C s2 C 2s C 8: We dont know yet if this
system is stable or not.
Roots at s D 2, s D 12 jp
152
. Unstable, but how do we find outwithout factoring?
-
STABILITY ANALYSIS 56
Case 1 Once we have determined that ai 0 8 i , we need to run
the Routh
test.
Very mechanical, not intuitive. Proof difficult. a.s/ D sn C
an1sn1 C C a1s C a0: Form Routh Array.
sn an an2 an4 sn1 an1 an3 an5 sn2 b1 b2 sn3 c1 c2 :::
s1 j1s0 k1
b1 D 1an1
an an2an1 an3
b2 D 1an1
an an4an1 an5
c1 D 1b1
an1 an3b1 b2
c2 D 1b1
an1 an5b1 b3
TEST: Number of unstable roots = number of sign changes in left
column.
EXAMPLE: a.s/ D s3 C s2 C 2s C 8.
s3 1 2s2 1 8s1 6s0 8
b1 D 11
1 21 8 D .8 2/ D 6
c1 D 16
1 86 0 D 16.0C 68/ D 8
Two sign changes (1!6; 6! 8) Two roots in RHP.
-
STABILITY ANALYSIS 57
EXAMPLE: a.s/ D s2 C a1s C a0.
s2 1 a0s1 a1
s0 a0
b1 D 1a1
1 a0a1 0 D 1a1 .0 a0a1/ D a0
Stable iff a1 > 0, a0 > 0.EXAMPLE: a.s/ D s3 C a2s2 C a1s
C a0.
s3 1 a1s2 a2 a0
s1 a1 a0a2
s0 a0
b1 D 1a2
1 a1a2 a0 D 1a2 .a0 a1a2/D a1 a0
a2
c1 D 1b1
a2 a0b1 0 D 1b1 .a0b1/ D a0:
Stable iff a2 > 0, a0 > 0, a1 > a0a2
.
Case 2 Sometimes find an element in first column = 0 0
0(!)
Replace with as ! 0.
-
STABILITY ANALYSIS 58
EXAMPLE: a.s/ D s5 C 2s4 C 2s3 C 4s2 C 11s C 10.
s5 1 2 11s4 2 4 10s3 0 6
new s3 6
s212
10
s1 6s0 10
If > 0, two sign changes.If < 0, two sign changes.
Two poles in RHP.
b1 D 12
1 22 4 D 12 .4 4/ D 0
b2 D 12
1 112 10 D 12 .10 22/ D 6
c1 D 1
2 4 6 D 1 .12 4/ 12
c2 D 1
2 10 0 D 1 .0 10/ D 10
d1 D 12
612
10
D
12.10 C 72
/ 6
e1 D 16
12
10
6 0
D16.0 60/ D 10
Case 3 Sometimes an entire row in Routh array = 0. This means
that polynomial factors such that one factor has
conjugate-MIRROR-roots.=.s/
-
STABILITY ANALYSIS 59
Complete Routh array by making polynomial a1.s/ from last
non-zerorow in array. Poly has even orders of s only!!
a1.s/ is a factor of a.s/. It is missing some orders of s, so IS
NOTSTABLE (by case 0.) We want to see if it has roots in the RHP or
only on the j!-axis. Replace zero row with da1.s/
dsand continue.
EXAMPLE: a.s/ D s4 C s3 C 3s2 C 2s C 2.
s4 1 3 2s3 1 2 0s2 1 2s1 0
new s1 2s0 2
No sign changes in Routharray. So, no RHP roots.Roots of a1.s/
at s D j
p2.
Marginally stable.
b1 D 11
1 31 2 D 1
b2 D 11
1 21 0 D 2
c1 D 11
1 21 2 D 0
a1.s/ D s2 C 2da1.s/
dsD 2s
d1 D 12
1 22 0 D 2
-
STABILITY ANALYSIS 510
EXAMPLE: a.s/ D s4 C 4 (Hard).
s4 1 0 4s3 0 0
new s3 4 0s2 0 4
new s2 4
s116
s0 4
Two sign changes. Therefore2 RHP poles. Other twopoles are
mirrors in LHP.
a1.s/ D s4 C 4I da1.s/ds D 4s3:
b1 D 14
1 04 0 D 0
b2 D 14
1 44 0 D 4
c1 D 1
4 0 4 D 16
d1 D 16
416
0
D 4
Design Tool
Consider the system:
r .t/ y.t/Ks C 1
s.s 1/.s C 6/
T .s/ D Y .s/R.s/
D KsC1
s.s1/.sC6/1C K sC1
s.s1/.sC6/
D K .s C 1/s.s 1/.s C 6/C K .s C 1/
a.s/ D s.s 1/.s C 6/C K .s C 1/
-
STABILITY ANALYSIS 511
D s3 C 5s2 C .K 6/s C K :
s3 1 K 6s2 5 K
s14K 30
5s0 K
For stability of theclosed-loop system, K > 0,and K >
30=4.
b1 D 15
1 K 65 K D 15 .K 5.K 6//D .4K 30/=5:
c1 D 1b1
5 Kb1 0 D 1b1 .b1K / D K
Step response for different values of K .
0 2 4 6 8 10 120.5
0
0.5
1
1.5
2
2.5
K =7:5K =13K =25
Ampl
itude
Time (sec.)EXAMPLE:
r .t/ y.t/K
1C 1TI s
1
.s C 1/.s C 2/
-
STABILITY ANALYSIS 512
T .s/ D Y .s/R.s/
DK TI sCK
s.sC1/.sC2/1C K TI sCK
s.sC1/.sC2/
D K TI s C Ks.s C 1/.s C 2/C K TI s C K :
a.s/ D s3 C 3s2 C .2C K TI /s C K .
s3 1 2C K TIs2 3 K
s16C 3K TI K
3s0 K
b1 D 13
1 2C K TI3 K D 13 .K 3.2C K TI //
c1 D 1b1
3 Kb1 0 D 1b1 .b1K / D K
For stability of the closed-loop system, 0 < K < 61 3TI
for TI 0 for TI >13
.
0 2 4 6 8 10 12
0
0.2
0.4
0.6
0.8
1
1.2
K D 3; TI D 0:5
K D 1; TI D 0:25
K D 1; TI D 1Ampl
itude
Time (sec.)
-
Feedback Control Systems. 61
ROOT-LOCUS ANALYSIS
Recall step response: we have seen that pole locations in the
systemtransfer function determine performance characteristics; such
as risetime, overshoot, settling time.
We have also seen that feedback can change pole locations in
thesystem transfer function and therefore performance is
changed.
Suppose that we have one variable parameter in our control
system.Then, we make a parametric plot of pole locations as that
parameterchanges. The poles are the roots of the denominator of the
transferfunction (a.k.a. the characteristic equation.) This plot is
a plot of thelocus of the roots or the ROOT LOCUS PLOT (First
suggested byEvans.)
VERY IMPORTANT NOTE: Root locus is a parametric plot (vs. K ) of
theroots of an equation
1C K b.s/a.s/D 0:
A common control configuration is
r .t/ y.t/K G.s/
Unity feedback, proportional gain. We will generalize control
configuration later.
-
ROOT-LOCUS ANALYSIS 62
Closed-loop transfer function
T .s/ D K G.s/1C K G.s/:
Poles = roots of 1C K G.s/ D 0. Assume plant transfer function
G.s/ is rational polynomial:
G.s/ D b.s/a.s/
; such that
b.s/ D .s z1/.s z2/ .s zm/ .b.s/ is monic:/a.s/ D .s p1/.s p2/
.s pn/ n m .a.s/ is monic:/
[a.s/ may be assumed monic without loss of generality. If b.s/
is notmonic, then its gain is just absorbed as part of K in 1C K
G.s/ D 0] zi are zeros of G.s/, the OPEN-LOOP transfer function. pi
are poles of G.s/, the OPEN-LOOP transfer function. CLOSED-LOOP
poles are roots of equation
1C K G.s/ D 0a.s/C K b.s/ D 0;
which clearly move as a function of K .
Zeros are unaffected by feedback.EXAMPLE: G.s/ D 1
s.s C 2/: Find the root locus. a.s/ D s.s C 2/; b.s/ D 1. Locus
of roots: (aside, stable for all K > 0 .)
s.s C 2/C K D 0s2 C 2s C K D 0:
-
ROOT-LOCUS ANALYSIS 63
For this simple system we can easily find the roots.s1;2 D 2
p4 4K
2D 1p1 K :
Roots are real and negative for 0 < K < 1. Roots are
complex conjugates for K > 1.
s1; K D 0s2; K D 0K D 1
=.s/
=>; K D 2
or we can locate the point on the root locus wherej
-
ROOT-LOCUS ANALYSIS 64
[A value of s D s1 is on the (0) complementary root locus iff1C
K G.s1/ D 0 for some REAL value of K , 1 < K < 0.] So, K D
1
G.s/ Note, G.s/ is complex, so this is really two equations!
jK j D 1G.s/
(6.1)6 G.s/ D 6
1K
: (6.2)
Since K is real and positive, 6 K D 0.Therefore, 6 G.s/ D 180
l360; l D 0; 1; 2; : : : So once we know a point on the root locus,
we can use the
magnitude equation 6.1 to find the gain K that produced it. We
will use the angle equation 6.2 to plot the locus. i.e., the locus
of
the roots = all points on s-plane where 6 G.s/ D 180
l360:NOTES:
1. We will derive techniques so that we dont need to test every
point onthe s-plane!
2. The angle criteria explains why the root locus is sometimes
called the180 root locus.
3. Similarly, the complementary root locus is also called the 0
rootlocus.
KEY TOOL: For any point on the s-plane6 G.s/ D
X6 .due to zeros/
X6 .due to poles/:
EXAMPLE: G.s/ D .s z1/.s p1/.s p2/:
-
ROOT-LOCUS ANALYSIS 65
p1p2z1
Test points1
=.s/
-
ROOT-LOCUS ANALYSIS 66
If the test point s1 is to the left of 1 pole and 1 zero: 6
G.s1/ D 180 .180/ D 360 D 0. 2 poles: 6 G.s1/ D 180 180 D 360 D 0.
2 zeros: 6 G.s1/ D 180 C 180 D 360 D 0.
NOT ON THE LOCUS.
GENERAL RULE #1All points on the real axis to the left of an odd
number of poles and zerosare part of the root locus.
EXAMPLE: G.s/ D 1s.s C 4C 4 j/.s C 4 4 j/:
=.s/
-
ROOT-LOCUS ANALYSIS 67Locus Not on the Real Axis
OBSERVATION: Because we assume that system models
arerational-polynomial with real coefficients, all poles must be
either realor complex conjugate pairs. Therefore, THE ROOT LOCUS
ISSYMMETRICAL WITH RESPECT TO THE REAL AXIS. What happens at K D
0,1?
1C K G.s/ D 1C K .s z1/.s z2/ .s zm/.s p1/.s p2/ .s pn/ D 0
.s p1/.s p2/ .s pn/C K .s z1/.s z2/ .s zm/ D 0: At K D 0 the
closed loop poles equal the open-loop poles. As K approaches1
.s p1/.s p2/ .s pn/K
C .s z1/.s z2/ .s zm/ D 0Ithe n poles approach the zeros of the
open-loop transfer function,INCLUDING THE n m ZEROS AT C1. Plug s D
1 into G.s/ and notice that it equals zero if m < n. The idea
of1 in the complex plane is a number with infinite
magnitude and some angle. To find where the n m remaining poles
go as K !1, consider that
the m finite zeros have canceled m of the poles. Looking back at
theremaining n m poles (standing at C1), we have approximately
1C K 1.s /nm ;
or n m poles clustered/centered at . We need to determine , the
center of the locus, and the directions
that the poles take.
-
ROOT-LOCUS ANALYSIS 68
Assume s1 D Re j is on the locus, R large and fixed, variable.
Weuse geometry to see what must be for s1 to be on the locus.
Since all of the open-loop poles are at approximately the same
place, the angle of 1C K G.s1/ is 180 if the n m angles from to s1
sumto 180.
.n m/l D 180 C l360; l D 0; 1; : : : ; n m 1or
l D 180 C 360.l 1/
n m ; l D 1; 2; : : : ; n m: So, if
n m D 1I D C180:
There is one pole going to C1along the negative real axis:
n m D 2I D 90:
There are two poles going toC1 vertically:
n m D 3I D 60; 180:
One goes left and the other twogo at plus and minus 60:
(etc)
To find the center, , reconsider our approximation b.s/a.s/ 1.s
/nm
as jsj ! 1. Divide (by long division) b.s/
a.s/and 1
.s /nm and match the toptwo coefficients of s, the dominant
ones.
An easier way...
-
ROOT-LOCUS ANALYSIS 69
Note that roots of denominator of G.s/ satisfy:sn C a1sn1 C
a2sn2 C C an D .s p1/.s p2/ .s pn/
a1 D X
pi :
Note that the roots of the denominator of T .s/ aresn C a1sn1 C
a2sn2 C C an C K .sm C b1sm1 C C bm/ D 0:
If n m > 1 then the .n 1/st coefficient of the closed loop
systemis such that a1 D
Xri where ri are the closed-loop poles.
We know that m poles go to the zeros of G.s/, and assume
theother n m are clustered at 1
.s /nm . Therefore, the asymptoticsum of roots is .n m/
Xzi :
Putting this all together,Xri D .n m/ C
Xzi D
Xpi
or
DP
pi P zi.n m/ :
GENERAL RULE #2 All poles go from their open-loop locations at K
D 0 to:
The zeros of G.s/, or To C1.
Those going to C1 go along asymptotesl D 180
C 360.l 1/n m
centered at D
Ppi P zin m :
-
ROOT-LOCUS ANALYSIS 610EXAMPLES:
Additional Techniques
The two general rules given, plus some experience are enough
tosketch root loci. Some additional rules help when there is
ambiguity.(As in examples , )
-
ROOT-LOCUS ANALYSIS 611Departure Angles, Arrival Angles
We know asymptotically where poles go, but need to know how
theystart, and how they end up there.
Importance: One of the following systems is stable for all K
> 0, theother is not. Which one?
=.s/
-
ROOT-LOCUS ANALYSIS 612
whereN1 D Angle from Np1 to s0 90:1 D Angle from p1 to s0:2 D
Angle from p2 to s0 135:
1 D 45: Can now draw departure of poles on locus. Single-pole
departure rule:
dep DX
6 .zeros/X
6 .remaining poles/ 180 360l: Multiple-pole departure rule:
(multiplicity q 1)
qdep DX
6 .zeros/X
6 .remaining poles/ 180 360l: Multiple-zero arrival rule:
(multiplicity q 1)
q arr DX
6 .poles/X
6 .remaining zeros/C 180 360l: Note: The idea of adding 360l is
to add enough angle to get the
result within 180. Also, if there is multiplicity, then l counts
off thedifferent angles.
Imaginary Axis Crossings
Routh stability test can be run to find value for K D K 0 that
causesmarginal stability.
Substitute K 0 and find roots of a.s/C K 0b.s/ D 0:
Alternatively, substitute K 0, let s D j!0, solve for
a. j!0/C K 0b. j!0/ D 0: (Real and Imaginary parts)
-
ROOT-LOCUS ANALYSIS 613Points with Multiple Roots
Sometimes, branches of the locus intersect. (see , on pg.6610).
Computing points of intersection can clarify ambiguous loci.
Consider two poles approaching each other on the real axis:
=.s/
-
ROOT-LOCUS ANALYSIS 614
Some similar loci for which finding saddle points helps
clarifyambiguity ...
=.s/
-
ROOT-LOCUS ANALYSIS 615
The magnitude equation may be used to find the value of K to get
aspecific set of closed-loop poles.
That is, K D 1G.s0/
is the gain to put a pole at s0, if s0 is on the locus.Summary:
Root-Locus Drawing Rules180 Locus The steps in drawing a 180 root
locus follow from the basic phase definition. This is the locus
of
1C K b.s/a.s/D 0; K 0
phase of b.s/
a.s/D 180
:
They are STEP 1: On the s-plane, mark poles (roots of a.s/) by
an and zeros (roots of a.s/) with an .
There will be a branch of the locus departing from every pole
and a branch arriving at every zero. STEP 2: Draw the locus on the
real axis to the left of an odd number of real poles plus zeros.
STEP 3: Draw the asymptotes, centered at and leaving at angles ,
where
n m D number of asymptotes:n D order of a.s/m D order of
b.s/
DP
pi P zin m D
a1 C b1n m
l D 180 C .l 1/360
n m ; l D 1; 2; : : : n m:
For n m > 0, there will be a branch of the locus approaching
each asymptote and departing toinfinity. For n m < 0, there will
be a branch of the locus arriving from infinity along
eachasymptote.
STEP 4: Compare locus departure angles from the poles and
arrival angles at the zeros whereqdep D
X i
Xi 180 l360
q arr DX
i X
i C 180 l360;where q is the order of the pole or zero and l
takes on q integer values so that the angles arebetween 180 . i is
the angle of the line going from the i th zero to the pole or zero
whoseangle of departure or arrival is being computed. Similarly, i
is the angle of the line from the i thpole.
STEP 5: If further refinement is required at the stability
boundary, assume s0 D j!0 andcompute the point(s) where the locus
crosses the imaginary axis for positive K .
STEP 6: For the case of multiple roots, two loci come together
at 180 and break away at 90.Three loci segments approach each other
at angles of 120 and depart at angles rotated by 60.
-
ROOT-LOCUS ANALYSIS 616 STEP 7 Complete the locus, using the
facts developed in the previous steps and making
reference to the illustrative loci for guidance. The loci
branches start at poles and end at zeros orinfinity.
STEP 8 Select the desired point on the locus that meets the
specifications (s0), then use themagnitude condition to find that
the value of K associated with that point is
K D 1jb .s0/ =a .s0/j :
EXAMPLE:
r .t/ y.t/K.s C 1/.s C 2/
s.s C 4/
=.s/
-
ROOT-LOCUS ANALYSIS 617
Compute T .s/ to determine the characteristic equation as a
functionof K .
T .s/ D KsC2
s101C K .sC2/.sC3/
.sC1/.s10/
D K .s C 2/.s C 1/.s C 1/.s 10/C K .s C 2/.s C 3/
Poles of T .s/ at.s C 1/.s 10/C K .s C 2/.s C 3/ D 0
or
1C K .s C 2/.s C 3/.s C 1/.s 10/ D 0:
=.s/
-
ROOT-LOCUS ANALYSIS 618
Routh Arrays2 K C 1 6K 10 K > 1s1 5K 9 K > 9
5D 1:8 stability criterion
s0 6K 10 K > 106D 1:66
When K D 9=5, the s1 row of the Routh array is zerotop
rowbecomes a factor of the characteristic equation. Imag.-axis
crossingswhere
95C 1
s2 C
69
5 10
D 0
14s2 C 4 D 0 : : : s D jr
27:
=.s/
-
ROOT-LOCUS ANALYSIS 619
D .2C 2/s3 C .18 5C 10 9/s2 C.20C 45C 12 45/s C .50 54/ D 0
D 14s2 C 32s 4 D 0
s D 32p
322 4.14/.4/28
D 32p
124828
roots at f0:118; 2:40g. Now we can complete the locus:=.s/
-
ROOT-LOCUS ANALYSIS 620
3) Asymptotes:n m D 2
DP
pi P zi2
D 0C 0C .a/ .1/2
D 1 a2
D f0:5; 24:5; 4gl D 180
C 360.l 1/2
D 90:4) Departure angles for two poles at s D 0.
2dep DX
6 .zeros/X
6 .remaining poles/ 180 360lD 0 0 180 360l
dep D 90 360
2l D 90:
5) Will always be stable for K > 0.6) Breakaway points:
K .s/ D s2.s C a/s C 1
dds
K .s/ D .s C 1/.3s2 C 2as/ s2.s C a/.1/
.s C 1/2D 0
D s (2s2 C .a C 3/s C 2a D 0breakaway at
(0;a C 3p.a C 3/2 4.2/.2a/
2.2/
)
saDf2;50;9g D1:25 jp7=16| {z }
not on locus; .2:04 and 24:46/; 3} and 0
-
ROOT-LOCUS ANALYSIS 621
=.s/
-
ROOT-LOCUS ANALYSIS 622
6 5 4 3 2 1 0 1 2 36
4
2
0
2
4
6
Real Axis
Imag
Axis TAKE ME TO YOUR
LEADER!
Summary: Root-Locus Drawing Rules0 Locus
We have assumed that 0 K
-
ROOT-LOCUS ANALYSIS 623 STEP 3: Draw the asymptotes, centered at
and leaving at angles , where
n m D number of asymptotes:n D order of a.s/m D order of
b.s/
DP
pi P zin m D
a1 C b1n m
l D .l 1/360
n m ; l D 1; 2; : : : n m:
For n m > 0, there will be a branch of the locus approaching
each asymptote and departing toinfinity. For n m < 0, there will
be a branch of the locus arriving from infinity along
eachasymptote.
STEP 4: Compare locus departure angles from the poles and
arrival angles at the zeros where qdep D
X i
Xi l360
q arr DX
i X
i l360;where q is the order of the pole or zero and l takes on q
integer values so that the angles arebetween 180 . i is the angle
of the line going from the i th zero to the pole or zero whoseangle
of departure or arrival is being computed. Similarly, i is the
angle of the line from the i thpole.
STEP 5: If further refinement is required at the stability
boundary, assume s0 D j!0 andcompute the point(s) where the locus
crosses the imaginary axis for positive K .
STEP 6: For the case of multiple roots, two loci come together
at 180 and break away at 90.Three loci segments approach each other
at angles of 120 and depart at angles rotated by 60.
STEP 7: Complete the locus, using the facts developed in the
previous steps and makingreference to the illustrative loci for
guidance. The loci branches start at poles and end at zeros
orinfinity.
STEP 8: Select the desired point on the locus that meets the
specifications (s0), then use themagnitude condition to find that
the value of K associated with that point is
K D 1jb .s0/ =a .s0/j :
Extensions to Root Locus MethodTime Delay
A system with a time delay has the formG.s/ D edsG 0.s/
where d is the delay and G 0.s/ is the non-delayed system.
-
ROOT-LOCUS ANALYSIS 624
This is not in rational-polynomial form. We cannot use root
locustechniques directly.
METHOD 1:
Approximate eds by
b0s C b1a0s C 1
; a polynomial.
Pad approximation.eds 1 .ds=2/
1C .ds=2/ First-order approximation
1 ds=2C .ds/2=12
1C ds=2C .ds/2=12 Second-order approximation.
11C ds Very crude.
Extremely important for digital control!!!METHOD 2:
Directly plot locus using phase condition. i.e., 6 G.s/ D .d!/C
6 G 0.s/ if s D C j!. i.e., look for places where 6 G 0.s/ D 180 C
d! C 360l: Fix !, search horizontally for locus.
-
Feedback Control Systems. 71
ROOT-LOCUS DESIGN
We have seen how to draw a root locus for a given plant
dynamics. We include a variable gain K proportional control. What
if desired pole locations are not on this locus? We need to modify
the locus itself by adding extra
dynamicscompensator:
r .t/ y.t/K G.s/D.s/
We redraw the locus and pick K in order to put the poles where
wewant them. HOW?
T .s/ D K D.s/G.s/1C K D.s/G.s/ let G
0.s/ D D.s/G.s/
D K G0.s/
1C K G 0.s/ We know how to draw this locus! Adding a compensator
effectively adds dynamics to the plant. What types of (1)
compensation should we use, and (2) how do we
figure out where to put the additional dynamics?
Types of Compensator Dynamics
There are 3 classical types of controllers with some
importantvariations.
-
ROOT-LOCUS DESIGN 72
1) Proportional FeedbackD.s/ D 1: u.t/ D K e.t/T .s/ D K
G.s/
1C K G.s/: Same as what we have already looked at. Controller
only consists of
a gain knob. We have to take the locus as given since we have
noextra dynamics to modify it. Usually a very limited approach, but
a good place to start.
2) Integral FeedbackD.s/ D 1
TI su.t/ D K
TI
Z t0
e. / d
T .s/ DKTI
G.s/s
1C KTI G.s/s:
Usually used to reduce/eliminate steady-state error. i.e., if
e.t/constant, u.t/ will become very large and hopefully correct the
error. Penalty: Steady-state response improves, but often
transient
response degrades.
Steady-State Response We apply a step (ramp, parabola) input.
How well does our (closed
loop) system track this? Ultimately, we would like zero error.
e.t/ D 0. (Maybe 1%2% in
reality) Recall: For a unity-feedback control system as drawn
above, the
steady-state error to a unit-step input is:
ess D 11C K D.0/G.0/:
-
ROOT-LOCUS DESIGN 73
If we make D.s/ D 1TI s
, then as s ! 0, D.s/!1
ess! 11C1 D 0: So by adding the integrator into the compensator,
the error has been
reduced from 11C K p to zero for systems that do not have any
free
integrators.
EXAMPLE: G.s/ D 1.s C a/.s C b/; a > b > 0:
Proportional feedback, D.s/ D 1, G.0/ D 1ab
, ess D 11C Kab:
a b
=.s/
-
ROOT-LOCUS DESIGN 74
Proportional-Integral (PI) Control Now, D.s/ D 1C 1
TI sD
s C .1=TI /s
. Both a pole and a zero.
a b
=.s/
jp0j, the phase added to the open-loop transferfunction is
negative: : : phase lag
Pole often placed very close to zero. e.g., p0 0:01. Zero is
placed near pole. e.g., z0 0:1.
-
ROOT-LOCUS DESIGN 75
a b
=.s/
-
ROOT-LOCUS DESIGN 76
a b
=.s/
-
ROOT-LOCUS DESIGN 77
a1 and b1 are chosen to make locus go through s D s1,a1s1 C
a0b1s1 C 1
G.s1/ D 1
for that point to be on the root locus.
Magnitudea1s1 C a0b1s1 C 1
jG.s1/j D 1 Phase 6
a1s1 C a0b1s1 C 1
C 6 G.s1/ D 180:
(math happens)a1 D sin./C a0jG.s1/j sin. /js1jjG.s1/j sin. /b1 D
sin. C /C a0jG.s1/j sin./js1j sin. /
9>>=>>;
s1 D js1je jG.s1/ D jG.s1/je j :
EXAMPLE: G.s/ D 1s2: s1 D 2
p2e j135 D 2C 2 j .
The point s1 is chosen to achieve D 0:707 and D 0:5 sec. a0: We
cannot compute a0 since 1
s2
sD0!1. So, arbitrarily choose
a0 D 2: a1: Note, D 135; D 270 because
G.s1/ D 1s2
sD2p2e j135
D 18
e j270:
a1 D sin.135/C 2.1=8/ sin.45/
.2p
2/.1=8/ sin.270/ D.1=p
2/.1C 1=4/p2=4
D 52:
b1:b1 D sin.135
/C 2.1=8/ sin.135/.2p2/ sin.270/ D
.1=p2/.1 1=4/2p2 D
316:
-
ROOT-LOCUS DESIGN 78
So, the compensator is:D.s/ D .5=2/s C 2
.3=16/s C 1:
6 5 4 3 2 1 0 14
3
2
1
0
1
2
3
4
Real Axis
Imag
Axis
Example locus passing through (-2,2)
Proportional-Integral-Derivative (PID) Control Most common type
of control, even though often approximated by
Lead-Lag control.
D.s/ D K
1C 1TI sC TDs
:
K D sin. C /TI js1jjG.s1/j sin./[TI js1j C 2 cos./] TD D sin.
/js1jjG.s1/j sin./ C
KTI js1j2 , where s1 D js1je
j and
G.s1/ D jG.s1/je j for both cases. TI chosen to match some
design criteria. e.g., Steady-state error.
Compensator Implementation
Analog compensators commonly use op-amp circuits. See the
following pages: : :
-
ROOT-LOCUS DESIGN 79
-
ROOT-LOCUS DESIGN 710
-
ROOT-LOCUS DESIGN 711
z1
K D p1z1
1
V1 V2
1p1
p1 z1p1
LEAD
z1
-
Feedback Control Systems. 81
FREQUENCY-RESPONSE ANALYSIS
Advantages and disadvantages to root-locus design
approach:ADVANTAGES:
Good indicator of transient response. Explicitly shows location
of closed-loop poles. Tradeoffs are
clear.
DISADVANTAGES:
Requires transfer function of plant be known. Difficult to infer
all performance values. Hard to extract steady-state response
(sinusoidal inputs).
Frequency-response methods can be used to supplement root locus:
Can infer performance and stability from same plot. Can use
measured data when no model is available. Design process is
independent of system order (# poles). Time delays handled
correctly (es ). Graphical techniques (analysis/synthesis) are
quite simple.
Frequency Response
We want to know how a linear system responds to sinusoidal
input, insteady state.
-
FREQUENCY-RESPONSE ANALYSIS 82
Consider systemY .s/U .s/
D G.s/: Consider the input:
u.t/ D u0 sin.!t/U .s/ D u0!
s2 C !2 : With zero initial conditions,
Y .s/ D G.s/ u0!s2C !2 :
Do a partial-fraction expansion (assume distinct roots)Y .s/ D
1
s a1 C2
s a2 C Cn
s an C0
s C j! C0
s j!y.t/ D 1ea1t C 2ea2t C C neant| {z }
If stable; these decay to zero:C2j0j sin.!t C /
yss.t/ D 2j0j sin.!t C /; D tan1=.0/
-
FREQUENCY-RESPONSE ANALYSIS 83
D tan1=.G. j!//
-
FREQUENCY-RESPONSE ANALYSIS 84
Phase = phase of numerator phase of denominatortan1
=.num/
-
FREQUENCY-RESPONSE ANALYSIS 85Plot Method #2: Magnitude and
Phase Plots
We can replot the data by separating the plots for magnitude
andphase making two plots versus frequency.
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
Frequency, (rads/sec.)
jG.
j!/j
0 1 2 3 4 5 69