EE392m - Spring 2005Gorinevsky
Control Engineering 3-1
Lecture 3 – Basic Feedback
Simple control design and analysis • Linear model with feedback control • Use simple model design control validate • Simple P loop with an integrator • Velocity estimation • Time scale • Cascaded control loops
EE392m - Spring 2005Gorinevsky
Control Engineering 3-2
Feedback Stability – State Space
• Closed-loop dynamics
• Stability is described by the closed-loop poles
Cxy
BuAxdtdx
=
+=)( dyyKu −−=
BKCAAK −=
Simple feedback control
dKK yBxAdtdx +=
)eig(}{ Kj A=λ
BKBK =
EE392m - Spring 2005Gorinevsky
Control Engineering 3-3
Closed-loop eigenvalues
• Roots = poles = eigenvalues
• Can be plotted for different gains K
• Root locus plot
F16 Longitudinal Model Example
[ ]03.5700
18.00
1015.217.0
08.1082.01095.2100091.0002.11054.248.02.3282.81093.1
3
12
4
2
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
⋅−=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅
−⋅−−−⋅−
=−
−
−
−
C
BA
)(eig BKCA −
% Take A from % the F16 example>> eig(A)
ans =-1.9125 -0.1519 + 0.1143i-0.1519 - 0.1143i0.0970
% Closed-loop poles >> K = -0.2;>> eig(A-B*K*C)
ans =-1.4419 -0.0185 -0.3294 + 1.1694i-0.3294 - 1.1694i
EE392m - Spring 2005Gorinevsky
Control Engineering 3-4
Closed-loop poles
• Transfer function poles tells you everything about stability• Model-based analysis for a simple feedback example:
)()(
dyyKuusHy−−=
=dd ysLy
KsHKsHy )()(1
)( =+
=
• If H(s) is a rational transfer function • Then L(s) also is a rational transfer function • Stability is determined by the poles of L(s) • Same results as for the state space analysis
EE392m - Spring 2005Gorinevsky
Control Engineering 3-5
Control of a 1st order system
• Simplest dynamics, 1st order system • Simple feedback works just fine
– Static output feedback is sometimes called P control– P = ‘proportional’– The name ‘P’ is used in process industries and in servosystems,
less in flight control
• Closed-loop dynamics are very well understood • Can be used as a design template for more complex
systems, cascade loops
EE392m - Spring 2005Gorinevsky
Control Engineering 3-6
Control of a 1st order system
• First order system, integrator dynamics
• P Control
• Closed loop dynamics
• Eigenvalue=pole
xy
buxdtdx
⋅=
+⋅=
1
0
1
0
===
CbB
A
)( dyykbdtdx −−=
kb−=λ
)( dyyku −−=
EE392m - Spring 2005Gorinevsky
Control Engineering 3-7
Example: Utilization control in a video server
P control - example• Integrator plant:
wbuy +=&
)( dP yyku −−=
• P controller:
admission rate
CPU
u(t)
completion rate
server utilization-u(t)
y(t)
d(t)
y(t)
yd(t)
-d(t)
Video stream i– processing time c[i], period p[i]– CPU utilization: U[i]=c[i]/p[i]
tUnew
∆∆=
tUdone
∆∆=
EE392m - Spring 2005Gorinevsky
Control Engineering 3-8
P control• Closed-loop dynamics
wbks
ybks
bkyP
dP
P
++
+= 1
wybkybky dPP +=+&
• Steady-state (s = 0)• Step response:
• Frequency response (bandwidth=2π/T)
SSP
dSS wbk
yy 1+=
( )TtSS
Pd
Tt ewbk
yeyty // 11)0()( −− −⋅⎟⎟⎠
⎞⎜⎜⎝
⎛++=
1)/(
)/()(ˆ)(ˆ)(ˆ
22 +
+=
P
Pd
bk
bkiwiyiy
ωωω
ω0.01 0.1 1 10-20
-15-10-5 0
0 2 400.20.40.60.8
1
y(t)
|y(iω)|ti
tidd
eidtd
eiytyω
ω
ω
ω
)(ˆ)(
)(ˆ)(
=
=
)/(1 PbkT =
EE392m - Spring 2005Gorinevsky
Control Engineering 3-9
Control and Error Peaking• Fast poles are not necessarily good• This might mean large peak resonse• Example: P control of an integrator
( )tbkbkth PP −= exp)(
0
slow response
fast response
• Engineering design is a series of tradeoffs
dyhy *= - closed-loop impulse response
dP yhku *= - control impulse response
EE392m - Spring 2005Gorinevsky
Control Engineering 3-10
Example:• flow through a valve
• Valves:– Mechanical: fluid or gas– Electrical: power– Computing: tasks– Comm: packets
I control
• Introduce integrator into control
• Closed-loop dynamics
)(,
dI yykvvu
−−==&
,wudy +⋅=
wdkssy
dksdky
Id
I
I
++
+=
y(t)
u(t)in-flow
out-flow
valve
• 0th order (feedthrough) system
EE392m - Spring 2005Gorinevsky
Control Engineering 3-11
P and I control• P control of an integrator
• I control of a 0th order system. Basically, the same feedback loop
wbuy +=&)( dP yyku −−=
d
kI ∫ -yd
wy)( dI yyku −−=&
,wudy +⋅=
kp
b∫-yd
w
y
EE392m - Spring 2005Gorinevsky
Control Engineering 3-12
First order estimation - differentiator
• Differentiating filter• Velocity estimation: y v
• Observer
• Velocity estimation filter
xy
vdtdx
=
=
xy
yyLdtxd
ˆˆ
)ˆ(ˆ
=
−= )ˆ(ˆˆ yyL
dtxdv −==
xsL
sv 11ˆ −+
=L
1∫ -y
y
v
Model:
EE392m - Spring 2005Gorinevsky
Control Engineering 3-13
First order estimation – example
InputSignal
Feedback gainL = 2
OutputSignal
ysL
sv 11ˆ −+
=
0 1 2 3 4 5 6 7 8 9 10-1
0
1
2
3
4
5INPUT SIGNAL y
0 1 2 3 4 5 6 7 8 9 10
-1
-0.5
0
0.5
1
TIME
ESTIMATED DERIVATIVE v
EE392m - Spring 2005Gorinevsky
Control Engineering 3-14 Val
idat
ion
an
d ve
rifi
cati
onD
esig
n a
nd
anal
ysis
Simplified design and analysis
• Simple design model– Often 0th or 1st or 2nd order model – Will consider typical examples– Use cascade loops
• Approximate model, robustness
• Analyze using a more detailed linear model
• Validate through simulation,
Control
Simple Design Model
Control
Detailed Simulation
Model
EE392m - Spring 2005Gorinevsky
Control Engineering 3-15
Example
e
e-
.q..qq
.q..V
δαθ
δαα
180081 820
101529100210
3
−−==
⋅−+−==
&
&
&
[ ] xy
uxdtdx
C
BA
⋅=
⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
⋅−+⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−=
−
4434421
4434421444 3444 21
03.570
18.00
1015.2
08.1082.010091.0002.1 3
>> eig(A)ans =
0-0.1880-1.9120
z
x
δe
α θV
eu δ=
θ3.57=y
• F16 longitudinal model, constant velocity• Assume that the velocity is maintained by regulating thrust
EE392m - Spring 2005Gorinevsky
Control Engineering 3-16
F16 Attitude Control• Simulated step response• At slow time scale, the integrator dynamics are dominant• Approximate by a simple integrator model
0 2 4 6 8 10 12
-200
-150
-100
-50
0
TIME (s)
STEP RESPONSE
edtd δθ 02−=
EE392m - Spring 2005Gorinevsky
Control Engineering 3-17
Simple model
‘Detailed’ model
F16 Attitude Control• P control design for an integrator
• Time responses for the simple model and for the ‘detailed’ model
)( dPe k θθδ −=
edtd δθ 02−=
dθ
005.0=Pk
)( dPe k θθδ −=
CxBAxx e
=+=
θδ&
dθ
10)20/(1 == PkT
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
1.2
1.4
TIME (s)
STEP RESPONSE
EE392m - Spring 2005Gorinevsky
Control Engineering 3-18
Time scale • The same plot at different scales • Bandwidth 1/Timescale
• Simple 2nd order model example: H(s) = 1/(1+s+s2)
0 0.5 10
0.1
0.2
0.3
0.4
0 5 100
0.5
1
1.5
0 50 1000
0.5
1
1.5H ≈ 1/s2 H = 1/(1+s+s2) H ≈ 1Fast Intermediate Slow Time scales:
EE392m - Spring 2005Gorinevsky
Control Engineering 3-19
Time Scale and Frequency Response
Frequency response for the example:
H(s) = 1/(1+s+s2)
Bandwidth=1/Timescale
• The bandwidth is limited by model uncertainty: Lectures 9-10
-80
-60
-40
-20
0
20
Mag
nitu
de (
dB)
10-2
10-1
100
101
102
-180
-135
-90
-45
0
Pha
se (d
eg)
Bode Diagram
Frequency (rad/sec)
0 0.5 10
0.1
0.2
0.3
0.4
0 5 100
0.5
1
1.5
0 50 1000
0.5
1
1.5
Fast
Slow
Time scales:
Intermediate
EE392m - Spring 2005Gorinevsky
Control Engineering 3-20
Feedback loop time scale
• Slow feedback loop– I control– Plant as a feedthrough
• Fast feedback loop– P control, plant as an integrator– PD control, plant as a double integrator
EE392m - Spring 2005Gorinevsky
Control Engineering 3-21
Cascaded loop design• Inner loop has faster time scale than outer loop • In the outer loop time scale, consider the inner loop as
a 0th or 1st order system that follows its setpoint input
inner loop
-inner loop
setpoint
output
Plant
Inner Loop Control
Outer Loop Control
-
outer loop
setpoint
EE392m - Spring 2005Gorinevsky
Control Engineering 3-22
Servomotor Speed Control Example
• The control goal is to track a velocity setpoint
• Mechanical time constant TJ is dominant.
• Use simple model
model
usTsT
GyIJ )1)(1( ++
=
Transfer function
sec02.0sec,1.0 == IJ TT
MotorPower amp sensor
speed vcontrolvoltage u
setpoint vd
controller -
uTG
sy
J
⋅= 1
vy =duRIILcIbvvJ
=+=+
&
&
1=G
EE392m - Spring 2005Gorinevsky
Control Engineering 3-23
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.2
0.4
0.6
0.8
1
OPEN LOOP STEP RESPONSE y
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.2
0.4
0.6
0.8
1
CLOSED LOOP STEP RESPONSE
Servomotor Example, cont’d
• Design P control for the simple model uTG
sv
J
⋅= 1)( dV vvku −−=
‘Detailed’ model
Simplified model
01.0/
110
=⋅
=
=
JPloop
V
TGkT
k
EE392m - Spring 2005Gorinevsky
Control Engineering 3-24
Servomotor Position Control
• Cascaded with the speed control loop • The control goal is to track the
position setpoint• Speed loop integrator yields the
dominant dynamics dv
dtdx =
• Use simple model of the plant (inner loop)
MotorPower amp sensor
position xcontrolvoltage u setpoint vd
speedcontrol -
us
y ⋅= 1
speed v
setpoint xd
position control -
EE392m - Spring 2005Gorinevsky
Control Engineering 3-25
Servomotor Position, cont’d
• Design P control for the simple model dvs
y ⋅= 1)( dPd yykv −−=
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.1
0.2
0.3
0.4
0.5STEP RESPONSE OF POSTION y WITH OUTER LOOP OPEN
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.2
0.4
0.6
0.8
1
CLOSED LOOP STEP RESPONSE
‘Detailed’ model
Simplified model
05.01
120
=⋅
=
=
Ploop
P
kT
k
EE392m - Spring 2005Gorinevsky
Control Engineering 3-26
Electric Motor Servo
• Broadly available products• 2-4 cascaded loops depending on the sensor hardware, motor
hardware, the application, and the required control functions
Lecture 6
EE392m - Spring 2005Gorinevsky
Control Engineering 3-27
Aircraft Cascaded Loops
• In practice, multivarable control design might be done for attitude control
• Otherwise, aircraft is represented as a chain of several integrators
EE392m - Spring 2005Gorinevsky
Control Engineering 3-28
Flight Control
Guidance and Autopilot
FMS/MMS
Basic cascaded loops in aircraft
Actuator servos
Angular rate
Angular position; Attitude
Translational velocity
Translational position
WaypointAltitude, coordinates
Commanded airspeed
Angular rate command
Elevator positionFlight ActuatorsActuators
Attitude command
x
α θ V
EE392m - Spring 2005Gorinevsky
Control Engineering 3-29
Aircraft Cascaded LoopsEmbeded servo avionics• Actuators, 100 hz bandwidth
Flight Control box• Angular rate, 2Hz bandwidth• Angular position, 0.5Hz bandwidth
Autopilot/Guidance • Translational velocity, 5 sec• Translational position, 30 sec
FMS - Flight Management System• Waypoint, 100-1000 sec
EE392m - Spring 2005Gorinevsky
Control Engineering 3-30
• Descent/Abort Guidance
• Dale Enns, Honeywell, 1989/1997
X-38 - Space Station Lifeboat
Cascaded Loop Example