lecture 1: introduction & mathematical modeling
TRANSCRIPT
Copyright ©1994-2012 by K. Pattipati
ECE 6095/4121
Digital Control of Mechatronic Systems
Lecture 1: Introduction &
Mathematical Modeling Prof. Krishna R. Pattipati
Prof. David L. Kleinman
Dept. of Electrical and Computer Engineering
University of Connecticut Contact: [email protected] (860) 486-2890
Copyright ©1994-2012 by K. Pattipati 2
Contact Information
– Room number: ITE 350
– Tel/Fax: (860) 486-2890/5585
– E-mail: [email protected]
Office Hours: Tuesday – Wednesday: 11:00-12:00 Noon
Very demanding course
– Homework every week and Design Projects (50%)
– Sensors and Actuators Presentation (10%)
– Class project (20%)
– Paper presentation (5%)
– Midterm – Take Home (15%)
Course Materials: http://huskyct.uconn.edu
Introduction
Copyright ©1994-2012 by K. Pattipati 3
Background Needed
Expected Background Knowledge (ECE 5101, ECE3111)
• Differential equations
• Continuous-time system modeling methods
– Transfer functions, state-space models (canonical forms, minimal
realizations)
– Controllability & Observability
– Transient response, especially for 2nd-order systems
• Stability theory for continuous-time systems
– Feedback, Routh-Hurwitz, Lyapunov Theory
• Graphical Tools
– Bode plot, Nyquist plot, Nichols Chart, and Root-locus
• Some basic knowledge of discrete-time systems
– Z-transforms, difference equations, and signal sampling
• Matrix theory and Linear Algebra
• Knowledge and use of MATLAB
Copyright ©1994-2012 by K. Pattipati 4
Mechatronic Systems Introduction & Overview
1. What is Mechatronics?
2. Elements of Mechatronics
3. Mechatronics Applications
4. Example of Mechatronics Systems
5. Mathematical Modeling of Mechatronic Systems
− Diesel Engine Driving a Pump
− Armature-controlled DC Motor
− Magnetic Levitation
− Inverted Pendulum
− Induction Motor Spray Painting in an Automotive Plant
6. Different Mathematical Representations of Systems
Copyright ©1994-2012 by K. Pattipati
What is Mechatronics? - 1
• The term “Mechatronics" was first coined by Tetsuro Mori, a senior engineer of
the Japanese company Yasakawa*, in 1969
− T. Mori, “Mechatronics,” Yasakawa Internal Trademark Application Memo, 21.131.01, July 12,
1969.
− R. Comerford, “Mecha … what?” IEEE Spectrum, 31(8), 46-49, 1994.
• Mechatronics refers to electro-mechanical systems and is centered on mechanics,
electronics, computing and control which, when combined, make possible the
generation of simpler, more economical, reliable and versatile systems
• Mechatronics is the synergistic integration of mechanical engineering, electronics
and intelligent computer control in design and manufacture of products and
processes
− F. Harshama, M. Tomizuka, and T. Fukuda, “Mechatronics-what is it, why, and how?-and
editorial,” IEEE/ASME Trans. on Mechatronics, 1(1), 1-4, 1996.
• Synergistic use of precision engineering, control theory, computer science, and
sensor and actuator technology to design improved products and processes.”
– S. Ashley, “Getting a hold on mechatronics,” Mechanical Engineering, 119(5), 1997.
* Makes servos, machine controllers, AC motor drives, switches and robots
5
Copyright ©1994-2012 by K. Pattipati
What is Mechatronics? - 2
• Field of study involving the analysis, design, synthesis, and selection of systems
that combine electronics and mechanical components with modern controls and
microprocessors.
− D. G. Alciatore and M. B. Histand, Introduction to Mechatronics and Measurement Systems, McGraw Hill, 1998.
− Good site for mechatronics: http://www.engr.colostate.edu/~dga/mechatronics/definitions.html
• Our working definition: Mechatronics is the synergistic integration of
sensors, actuators, signal conditioning, power electronics, decision and
control algorithms, and computer hardware and software to manage
complexity, uncertainty, and communication in engineered systems.
• An Embedded system, a component of mechatronics system, is a combination
of hardware and software designed to run on its own without human intervention,
and may be required to respond to events in real-time
• When these systems are networked, they are called Cyber-physical systems
− Numerous applications: Zero-accident highways, smart grid, smart buildings, tele-
operation rooms, aerospace and transportation, robotics and intelligent machines,…..
6
Copyright ©1994-2012 by K. Pattipati
A Venn Diagram View of Mechatronics
Communication
Networks
Protocols,
Architectures
Cyber-physical
Systems (convergence of
Computing,
Communication
and Control)
Embedded
Systems
Computer
Systems Micro
Control Simulation
Analog
Circuits System
Models
Digital
Control
Electro-
Mechanical
Systems
Actuators
and
Sensors
Digital
Circuits
Electrical
Systems
Mechanical
Systems
7
Copyright ©1994-2012 by K. Pattipati
Mechatronics Applications
• Smart consumer products: home automation and security, microwave oven, toaster, dish washer, laundry washer-dryer, climate control,..
• Medical: implant-devices, assisted surgery, body area networks (BANs),..
• Defense: unmanned air, ground and undersea vehicles, smart munitions, jet engines,…
• Manufacturing: robotics, machines, processes, etc.
• Automotive: climate control, generation II antilock brake systems, active suspension, cruise control, air bags, engine management, safety, navigation, tele-operation, tele-diagnosis, backup collision sensing, rain sensing, etc.
• Cyber-physical Networked Systems: distributed robotics, tele-robotics, intelligent highways, smart grid, smart buildings, etc.
8
Copyright ©1994-2012 by K. Pattipati
Examples of Mechatronic Systems -1
Computer disk drive Asimo Humanoid
(Honda)
Mars Curiosity Rover
Robocup Team Qurio Humanoid Office Copier
9
Copyright ©1994-2012 by K. Pattipati
Examples of Mechatronic Systems -2
Aviation Underwater Robot Flying UAV
Big Dog Robot HEXAPOD Robot Micro Robot
10
Copyright ©1994-2012 by K. Pattipati
Control History Perspective
• First generation: Analog Control (prior to 1960)
– Technology: Feedback amplifiers and Proportional-Integral-Derivative (PID) controllers
– Theory: Frequency domain analysis
– Tools: Bode Plot, Root Locus, Nyquist Plot, Nichols Chart
• Second generation: Digital Control
– Around 1960-1970
– Technology: Digital computers and embedded microprocessors
– Theory: State-space design
– Real-Time Scheduling
• Third generation: Networked Control (Cyberphysical Systems .. phrase coined around 2006)
– Convergence of computing, communication and control throughout 1990s and beyond
– Theory: Hybrid systems, Formal methods, …
– Technology: Wireless and wired networks for remote sensing and actuation
– Numerous applications: Zero-accident highways, smart grid, tele-operation rooms,
aerospace and transportation, robotics and intelligent machines, health,…..
11
Copyright ©1994-2012 by K. Pattipati
Feedback Control System
12
Adapted from
Astrom & Murray, 2011
• Process = actuators + dynamic system + sensors
• Noise and external disturbances perturb the dynamics of the process
• Controller = pre-filter + A/D + computer + D/A+ system clock
• Operator input = reference signal (set point, desired input)
Copyright © 1994-2012 by K. Pattipati 13
Control System Design Process
A) Mathematical Model of System (Process) to be Controlled
MATH MODEL OF SYSTEM
PERFORMANCE MEASURES
CONTROL
DESIGN
EVALUATION &
SIMULATION OK?
NO B
C
YES
A
B) Performance Measures and Concerns
Mathematical criteria that are driven by customer's qualitative/quantittaive specifications
for behavior of the closed-loop (feedback) system
(1) Stability of the closed-loop system
(2) Steady-state accuracy, Integral absolute error (IAE), integral square error (ISE)
(3) Speed of response/transient
(4) Sensitivity/robustness
C) Evaluation and Simulation
− Computer-aided Engineering (CAE) software tools such as MATLAB/Simulink,LabVIEW
− Hardware-in-the-loop simulation
Focus of the course
Copyright ©2012 by K.R. Pattipati 14
Relationships among LTI Modeling Techniques
Transfer Functions
G(s)
Ordinary Differential
Equations (ODE)
State Variables
Signal Flow Graphs
sdt
d
• Physical Variables
• Standard Controllable form (SCF)
• Standard Observable form (SOF)
• Modal (Diagonal) form
DuxCy
BuxAx
DBAsICsG 1)()(
SCF, SOF
Modal
LTI: Linear Time-invariant
Useful for
Nonlinear
Systems
as well
(SFG as Simulink
Diagram. Mason’s
Rule not applicable)
Copyright ©2012 by K.R. Pattipati 15
Math Modeling involves Diverse Disciplines - 1
• Classic Book
− R. H. Cannon. Dynamics of Physical Systems. Dover, 2003 or McGraw-Hill, 1967.
• Mechanics
− V. I. Arnold. Mathematical Methods in Classical Mechanics. Springer, 1978.
− H. Goldstein. Classical Mechanics. Addison-Wesley, Cambridge, MA, 1953.
• Thermal
− H. S. Carslaw and J. C. Jaeger. Conduction of Heat in Solids. Clarendon Press,1959.
• Fluids
− J. F. Blackburn, G. Reethof, and J. L. Shearer. Fluid Power Control. MIT Press,1960.
• Vehicles
− M. A. Abkowitz. Stability and Motion Control of Ocean Vehicles. MIT Press, 1969.
− J. H. Blakelock. Automatic Control of Aircraft and Missiles. Addison-Wesley, 1991.
− J. R. Ellis. Vehicle Handling Dynamics. Mechanical Engineering Pubs., London,1994.
− U. Kiencke and L. Nielsen. Automotive Control Systems: For Engine, Driveline, and
Vehicle. Springer, Berlin, 2000.
Copyright ©2012 by K.R. Pattipati 16
Math Modeling involves Diverse Disciplines - 2
• Robotics
− M. W. Spong & M. Vidyasagar, Dynamics and Control of Robot Manipulators. Wiley, ‘89.
− R. M. Murray, Z. Li, and S. S. Sastry. A Mathematical Introduction to Robotic
Manipulation. CRC Press, 1994.
• Power Systems
− P. Kundur. Power System Stability and Control. McGraw-Hill, New York, 1993.
• Acoustics
− L. L. Beranek. Acoustics. McGraw-Hill, New York, 1954.
• Micromechanical Systems
− S. D. Senturia. Microsystem Design. Kluwer, Boston, MA, 2001.
• Biological Systems
− J. D. Murray. Mathematical Biology, Vols. I and II. Springer-Verlag, 2004.
− H. R. Wilson. Spikes, Decisions, and Actions: The Dynamical Foundations of
Neuroscience. Oxford University Press, Oxford, UK, 1999.
Copyright © 2012 by K.R. Pattipati 17
Force-Voltage Analogy for Translational Systems
dt
diLV
dt
dvMMaf
RiVdt
dxBBvf
dticc
qVdtvktkxf
1)(
Key Mechanical Elements:
• spring
• viscous damper
• mass
K
)(tx
)(tf
B
)(tx
)(tf
x
fM 1
capacitor spring
resistor damper
inductor mass
current velocity
chargeposition
voltage force
CK
RB
LM
iv
qx
Vf
Analogy
spring
damper
mass
Copyright © 2012 by K.R. Pattipati 18
Torque-Voltage Analogy for Rotational Systems
J
K
B
u(t)
input torque
,
dt
diLVJJT
RiVBBT
dtiCC
qVdtKKT
1
Analogy
C/1Capacitor Spring
ResistorDamper
Inductor inertia ofMoment
Current Velocity
Chargent Displaceme
VoltageTorque
K
RB
LJ
i
q
VT
Damper
Torsion Spring
Copyright © 2012 by K.R. Pattipati 19
Diesel Engine Driving a Pump
K
1
Electrical Analog
1
1
1
1B
2BJT System Equations
KTB
KKT
haveweTB
ceAlso
KT
TJJ
B
BJKBT
pumplocity of Angular ve
Tthe springTorque in
States
1
1
1
1
2
2111
,1
sin,
)(
1
)()(
,
,
:
Ty
KTB
KK
JJ
B
T
01
0
1
1
2
Input:
Output:
Draw Signal Flow Graph and Compute Transfer Function,
Steady state gain? System type?
1
2
1
Alternate States:
( , ) ( , )x
BK x
J
Kx x
B
Copyright © 2012 by K.R. Pattipati 20
Analysis of Engine Driving a Pump
2
1
2
1
0 0.25 0.5 0
20 4 20
10 101 0 ( ) 3.32; 0.64;
4.25 11 11n
B
J J
KT T K T TK
B
y G s dc gainT s s
2 2
1
2
20 . / ;
2 . / ;
5 . .sec/ ;
0.5 . .sec/
K N m rad
J kg m rad
B N m rad
B N m rad
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Step Response
Time (seconds)
Am
plit
ude
-60
-50
-40
-30
-20
-10
0
Magnitu
de (
dB
)
10-1
100
101
102
-180
-135
-90
-45
0
Phase (
deg)
Bode Diagram
Frequency (rad/s)
Copyright © 2012 by K.R. Pattipati
DC Motors
DC Motors convert DC energy into mechanical energy used in disk drives, robots, tape
transport mechanisms
Nice animation at:
− http://www.wisc-online.com/objects/ViewObject.aspx?ID=IAU13208
− http://www.wisc-online.com/objects/ViewObject.aspx?ID=IAU11508
− http://www.wisc-online.com/objects/ViewObject.aspx?ID=IAU13708
Torque of a DC motor:
− Field controlled: iA constant; iF controlled;
− Armature controlled: iF constant; iA controlled;
Three types of DC motors: series wound, shunt and compound
currentArmatureicurrentFieldiiKiT AFAFm ;;
FFm iKT
ATm iKT
21
Copyright © 2012 by K.R. Pattipati 22
Armature Controlled DC Motors (Fixed Field )
Fixed field is created by permanent magnets surrounding the armature
Applying an input voltage to the armature circuit causes a current in the coils of the
armature
This generates a magnetic field which is repelled by the permanent field causing
the motor to spin
The torque generated is proportional to the armature current
Modeling a DC Motor
Vi
RA LA
iA
)(tKv
Reverse voltage; opposes
input voltage Vi
)(tKv
Mechanical Action Armature
JM JL
BM
T =KTiA
motor load
Electro-mechanical system
Copyright © 2012 by K.R. Pattipati 23
TF of an Armature Controlled DC Motor
Modeling a DC Motor
Vi
RA LA
iA
)(tKv
Mechanical Action Armature
JM JL
BM
T =KTiA
motor load
Computing (s)/Vi(s)
)1()()()()(
)(
ssKsIsLsIRsV
tKdt
diLiRv
vAAAAi
vA
AAAi
Armature Equation:
)2()()(
)()()(
)()(
2
2
sK
Bss
K
JJsI
ssBsIKssJJ
tBiKtJJ
T
M
T
LMA
MATLM
MATLM
Mechanical Dynamics Equation:
Substitute (2) into (1) and solve for (s)/Vi(s)
VTMAMALMALMA
T
i KKBRsBLJJRsJJLs
K
sV
s
))(()()(
)(2
Copyright © 2012 by K.R. Pattipati 24
SS and SFG of an Armature Controlled DC Motor
A
v
A
AA
Ai
A
vA
AAAi
L
Kt
L
Ri
Lv
dt
di
tKdt
diLiRv
)(1
)(
Motor Equations:
LM
M
LM
TA
MATLM
JJ
Bt
JJ
Kit
tBiKtJJ
)()(
)()(
Declare states as 1 2 3, ,Ax i x x
State Space:
3
2
1
3
2
1
3
2
1
100
0
0
/1
010
0)/()/(
0//
x
x
x
y
v
L
x
x
x
JJBJJK
LKLR
x
x
x
i
A
LMMLMT
AvAA
vi y= 1/s 1/s 1/s
KT/(JM+JL)
-BM/(JM+JL)
-RA/LA
-KV/LA
1/LA 1
x1 x2 x3
SFG:
2
1 ; 0.001 ; 5 / ;
5 sec/ ; 20 / / sec;
( ) 1 . .sec /
A A T
v M
M L
R L H K N m A
K volt rad B kg m
J J N m rad
Copyright © 2012 by K.R. Pattipati 25
Steady-state Characteristics
m
f
A
f
i
fvT
AfATm
v
AAi
TKi
R
Ki
v
KiKK
iKiiKT
K
iRv
2
Steady-state Equations:
• Armature resistance, RA
• Field current (field flux), iF
• Armature voltage, vi
Speed Control Techniques:
10
0
0
0
S
m
A
if
S
f
im
T
T
R
vKiTtorquestalling
Ki
vspeedloadNoT
Torque
Speed
Maximum
Torque Ra increasing
Torque
Speed
Max.
Torque Field Current Decreasing
Trated Torque
Speed
Max.
Torque
Trated
Vi increasing
• good speed regulation
• maintains max. torque capability
• Slow transient response
• Does not maintain
max. torque capability
• Power loss in Ra
• Does not maintain
max. torque capability
• Poor speed regulation
4
2.
)1(
:
0max
0
0
s
sm
TP
atMax
TTP
powerMax
Copyright © 2012 by K.R. Pattipati
Magnetic Levitation - 1
Nonlinear System Equations
2
0 05
tan
0 0001
tan 0 01
Re tan 1
M Mass of Ball . kg
h Dis ce of Ball from Magnet
K Coefficient in N - m / Amp .
L Induc ce in H . H
R si ce in Ω Ω
V
i
1B
RL
M
g
h
2
2
iMh Mg K
h
diV L Ri
dt
1 2 3
2
31 2 2
1
3 3
; ;
;
1
x h x h x i
xKx x x g
M x
Rx x V
L L
1
1 0 3 0 0
2 2
03 0
0 0
Linearize around equilibrium
point:
e
e e
e
x h x i V
Mgx Mghx i
K K
V Ri
:IR sensor measurement
y ah b
Adapted from
Murray, 2004
See Kuo’s Book
Copyright © 2012 by K.R. Pattipati
Magnetic Levitation - 2
Linearized System Equations:
1
1 1 1 0 1 2 2 3 3 3 0 3
1 2
2
3 32 1 33 2
1
3 3
1
; ;
22
1
e
e e
e e
e
x x x h x x x x x x i x
x x
x KxKx x x
M x Mx
Rx x V
L L
y a x
1
1 0 3 0
2 2
03 0
0 0 3
Linearize around equilibrium
point: 0.01
0.7004
0.7004
e
e e
e
e
x h m x i
Mgx Mghx i A
K K
V Ri Rx V
2
0 05
tan
0 0001
tan 0 01
Re tan 1
M Mass of Ball . kg
h Dis ce of Ball from Magnet
K Coefficient in N - m / Amp .
L Induc ce in H . H
R si ce in Ω Ω
1 1
2 2
3 3
1
2
3
0 1 0 0
1962 0 28.1 0
0 0 100 100
0 0
x x
x x V
x x
x
y a x
x
3 2
2
2
( ) 2810( )
( ) 100 1962 196,200
2810 =
( 100)( 1962)
Neglecting motor dynamics
28.1( )
( 1962)
Y s aG s
V s s s s
a
s s
aG s
s
Open loop
unstable
Copyright © 2012 by K.R. Pattipati
Inverted Pendulum
System Equations (prove in HW): states
2
2
1
2
cos sin .
0 cos ( ) sin
:
F M m ml p cp ml
ml J ml mgl
y poutputs
y
Mass of base; mass of system to be balanced
Moment of intertia of system to be balanced
Distance from the base to the center of mass to be balanced
, Coefficients of viscous friction
M m
J
l
c
g
Acceleration due to gravity
[ ]p p
Astrom & Murray, 2011
Copyright © 2012 by K.R. Pattipati
Inverted Pendulum
System Equations (prove in HW): states [ ]p p
2 21
2 2
2
2 2
2 2
sin . ( / )sin cos ( / ) cos .; :
( / ) cos
sin cos . sin cos ( / ) cos
( / ) cos
t t
t t
t t
t t
p
p
yml mg ml J cp J ml Fdoutputs
M m ml J ypdt
ml M gl cl p M m l F
J M m ml
p
2;t tLet M M m J J ml
2 2
2 2
1
2
0 0 1 0 0
0 0 0 1 0;
0 / / / /
0 / / / /
1 0 0 0:
0 1 0 0
t t
t t
t t
p p
dF M J m l
m l g cJ mlp p Jdt
M mgl cml M ml
p
youtputs
y p
Linearized Equations: 20 sin & cos 1and small is negligible
Copyright © 2012 by K.R. Pattipati 30
Spray Painting in an Automotive Plant -1
System Equations
• Inputs: Motor Torque, Tm is a function of line frequency, 0; Source voltage, v and
Motor speed, m (which is a state variable) There is inherent feedback already!
• Output: Pump speed, p
• Is the system linear? No, because Tm is a nonlinear function of {0, v and m}
• How to get state equations? Gears are not perfect; They have efficiency, < 1.
• How to get induction motor torque, Tm?
r : Gear Speed Ratio
Copyright © 2012 by K.R. Pattipati 31
Spray Painting in an Automotive Plant -1
System Equations
r : Gear Speed Ratio
r < 1 if p < m
Torque scales by 1/r
Electrical Analog
Electrical Analog
p
pb
mT
rTp
mmb
mJ
pT
pk
1
mr pJ
pmr
p
p
p
p
p
pppppp
ppmppmpp
m
m
p
m
m
m
mm
mpmmmm
ppm
TJJ
bbJT
krkrkT
TJ
TJ
r
J
b
TTr
bJ
TStates
1)3(
)()2(
1
)1(
,,:
Tm = Nonlinear f(0, v, m)
p= r m in steady state
supply radian frequency
supply voltage
Copyright © 2012 by K.R. Pattipati 32
Magnetic Torque in an Induction Motor -1
polesofNumberP
Hzfrequencyplyf
PP
fn
rpmspeedsSynchronou
sync
)(sup
60120
:)(
0
00
• Very good animation and introduction to induction motors at
− http://www.wisc-online.com/objects/ViewObject.aspx?ID=IAU10108
− http://www.wisc-online.com/objects/ViewObject.aspx?ID=IAU13508
Slip, S = 0 Slip, S = 1
0
0 0
0
0
( ) ( )
, 1
0 ; 1 0
.
m
sync m m m
sync
m m
sync
Slip
Rotor Mechanical speed rpm n
n nslip S
n
S S
slip speed sn
slip radian frequency s
radian freq of voltage induced in rotor
T0
Copyright © 2012 by K.R. Pattipati 33
Magnetic Torque in an Induction Motor - 2
• Equivalent circuit of an induction motor. Assume 3 phases − Similar to a transformer except that the secondary windings rotate
− Frequency of voltage induced in the rotor is s0
− Voltage at s=0 is zero (no torque), Maximum at s=1
− ER0 = E1 NR / NS = E1 /aeff; ER = s ER0
stator
air gap
rotor
Equivalent circuit
Rotor Equivalent circuit
0
0
0
0
RR
R
RR
RR
jXs
R
E
jsXR
sEI
Rotor Equivalent circuit
2
2 0
2
2
2
1 0
eff R
eff R
R
eff
eff R
Seff
R
X a X
R a R
II
a
E a E
Na
N
Copyright © 2012 by K.R. Pattipati 34
Magnetic Torque in an Induction Motor - 3
• Final Equivalent circuit of an induction motor. Assume 3 phases
Actual rotor resistance
Resistance equivalent to
mechanical load
2
0
2
22
0
000
2
2
2
20
2
2
10
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
02
2
22
20
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2)(3
13)1(3
,
)1(3,
mmm
m
mmSm
q
m
m
mmm
m
thatsuchSsmallforq
qTT
XR
RVTTwhere
SX
R
qST
SX
R
X
RS
TTXSR
SRVPT
XS
RS
SRVP
XS
R
VIimpedancestatorNeglecting
S
SRIPPowerMechanical
So, induction motors can be controlled by
varying 20, RorV
Copyright © 2012 by K.R. Pattipati 35
Steady-state Operating Point & Linearization
• Steady-state operating point
• State equations in matrix form
)(
)(;;
00
22
0
2
0
22
0
000
2
m
mmp
m
m
mm
p
mmmpppp
q
qbr
b
T
q
qT
brbTrbT
p
p
m
m
m
p
p
m
ppp
pp
mmm
p
p
m
Ty
T
J
T
JbJ
KrK
JrJb
T
100
0
0
/1
//10
0
0//
Nonlinearity enters only through Tm !
Copyright © 2012 by K.R. Pattipati 36
Steady-state Operating Point & Linearization
• Linearized state equations:
• The required derivatives are evaluated at steady-state operating points
yyyTTTTTT mmmppppppmmm ;;;;
0 2
0
2
/ / /
/ / / 0
0 0 0 0
0 1/ / 0 0 0
0 0 1
m m m
m m m m mm m m m m
p pp p
p p pp p
m
p
p
T V T T R
J J JT b J r J V
T TK r K
J b J R
Ty
max2
max
2
max
2
max20
2
2
max2
max
2
max
2
max
2
max
2
max22
0
2
0
2
max20
max22
max
2
max
20
2
2
2
2
max2
00000
0/1
/1
/1
3
;0)/1
/21(
/1
)1(/1
/1
3
;0/1
6;0
/1
/13
;;1
;1;1
T
T
T
T
m
T
T
T
T
T
T
m
T
mT
T
T
m
m
T
m
m
mm
SSforSS
SS
SSR
SV
R
T
SSforSS
SSS
SS
SSS
SSR
VT
SSR
SV
V
TSSfor
SS
SSRVT
X
RSatTorqueMaximum
SSSS
Copyright © 2012 by K.R. Pattipati 37
• We consider n state (x), m input (u), p output (y) systems (vectors are columns)
• The system dynamics are continuous, not discrete.
( , ); 0 initial stat
(1.1
e
, )(
x t f x t u t x
y t g x t u t
1 1 1
; ;
n m p
x t u t y t
x t u t y t
x t u t y t
(a) Linearized state space model
System matrix
Control matrix
Output matrix
I/O coupling matrix
A n n
B n m
C p n
D p m
• Multivariable LTI: linear and time-invariant Superposition and constant parameter
systems
;
(1.2)
[ ] [ ]; [ ] [ ]
; [ ] [ ]; [ ] [ ]
i iij ij
j j
i iij ij
j j
f fx t A x t B u t A a B b
x u
g gy t C x t D u t C c D d
x u
Math. Representation of (Sub)system Dynamics - 1
Copyright © 2012 by K.R. Pattipati 38
Math. Representation of (Sub)system Dynamics - 2
(b) Transfer function representation
1 1
1
0;
Laplace transform variable
( ) ( ) ( ) ( ) [ ( ) ] ( ) ( ) ( )
( ) ( ) ( ); 1,2,..,
( ) [ ( )] ( )
( )( ) |
(
(1.3)
) i
m
k kj j
j
kj
kkj u i
j
s
x s sI A B u s y s C sI A B D u s G s u s
y s g s u s k p
G s g s pby m transfer function matrix TFM
y sg s
u s
j
1
1; 1
1; 1
1; 1
m p SISO
m p SIMO
m p MISO
m p MIMO
(c) Impulse response representation
( )
0 0 0
( ) [Im Re ] [ ( )]
0; 0( )
( );
(
0
:
( ) ( ) ( ) ( ) ( 1.( 4) ) ( ) )
At
t t tA t
G s L pulse sponse L G t
tG t
Ce B D t t
Convolution Integral Formula
y t G t u d G u t d C e B u d D u t
Copyright © 2012 by K.R. Pattipati
(d) Variation 1: Disturbances, d and Measurement noise, v
(
( )
( ) ( ) = ( ) ( ) ( ) ( ) ( ) 1.5)
d
d d
x t Ax t Bu t B d t
y t Cx t Du t D d t v t y s G s u s G s d s v s
(e) Variation 2: Descriptor representation
1 (1 = ( ) ( ) [ ( ) ] ( ) .6)
Ex t Ax t Bu t
y t Cx t Du t
y s G s u s C sE A B D u s
Math. Representation of (Sub)system Dynamics - 3
39