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Design Methods of Linear Control Systems
Visiting ProfessorMehmet Dal
Department of Electrical and ComputerEngineering at TUM, Munchen, Germany
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2015
Contents
Introduction to Automatic Control System
Control System Design Methods
Simulation and Implementation
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To analysis and controls design for a linear time-invariant (LTI) system, it can be represented in several ways, devided into twogroups:
1) Time domanin (t)• Differential equation• Difference equation for discrete-time domain• State variable form
2) Frequency Domain• Transfer function• Block diagram or flow graph• Impulse response
Each description can be converted to others, and provides differentapproach to analysis and controls design
System modelling and analysis
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Order of the systems and their properties• The order of a system reflects its number of energy
strorage elements.• A serial RC circuit can be cosidered a simple example of
First order system (low pass filter) and a serial RLC circuitis of a Second order.
• For both systems, the input voltage and the output voltageare selected as the input and the output, respectively.
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First order system (series RC circuit)
dt
tduti
Cdt
tdiR
dt
tduCtituti
CtRi c
)()(
1)(
)()(),()(
1)(
1) differential equation of the circuit
)()()(
)(1
)(1)(
tButAxdt
tdx
tuRC
tuRCdt
tdu
uB
x
c
Ax
c
2) state space equation
1
11
1
)(
)(
)()()1
(
)()(1
)(
RCSCs
R
Cssu
su
susICs
R
ssusIC
sRsI
c
3) transfer function
From the KVL
4) block diagram
block simplificationfomula
Responce of first order system
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• A first order system corresponds to delay element, like a Low Pass Filter (LPF)
• The time constant of the circuit
1) Time response
0,0
,
),1(
)(0
1
t
tV
tteV
tu s
tRC
s
c
0
0
,
,0)(
ttV
tttu
s
Normalized transfer function of the systemgives its characteristic equation and poles
RCas
assChas
asG
/1
0)(
)(
2) Frequance response
f(Hz)
Gai
n
• Bandwidth (BW) for low pass system is defined as frequency, where the magnitudeof voltage gain dropes by a factor of .
• If Bode diagram is ploted for a=2, it shows that bandwidth is equal to polemagnatude,
)3(707.02/1 dB
RC
/1 aBW
dBV
V
u
u
s
sc 3)707.0
(20log)(20log
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Unite step response
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t(s)
uc(t
)
1)(for),1(1)(
transformLaplasoftablethefrom
find becan responsedomen timeits
)(
1)()()(
is)(output then the
/1)(and)(
tuetu
ass
a
as
a
ssGsusu
su
ssuas
asG
atc
c
c
A good notes from thetext book
The system transfer fynction andunite step input defined as follows
69.02.2 dr TandT
SECOND ORDER SYSTEMSMany useful systems are of second order, for high order systems oftenly thedominant (low frequency) pole pairs are analiyzed to approximate the system witha second order transfer function. Therefore, the transfer function of second ordersystem is very important
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the general form of second order system transfer function
the poles of the system from equating H(s) to 0 are
22
2
22
2
22)(
)()(
nn
n
n
n
sssssR
sCsG
22 2)( nnssH
The denominator is the Characteristic polynomial
22,1 1 nn js
)( 22
2,1
nj
js
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Step respose of second order system
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The rise time tr, which is the time required for the step response to risefrom 0.1 to 0.9 of its steady-state value. The settling time ts is the time required for the signal to effectively reach its steady-state value. For the pure exponential response
input step unite,/1)(,2
1)(
22
2
ssRsss
sCnn
n
5or4,2.2 sr tt unit step response:
2
1
1(
10012
n
p
p
T
eM
21100
ePOS
Effects of Damping ratio
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2
1for 21 2 nr
2
22,1
1and
1
nn
nn js
*Underdamped case if 0 < ξ <1The resonant frequency is given for ξ=0.7 by
*Undamped case: if ξ = 0 and α= 0, β=ω then the poles are at s = ± jβ on the imaginary axis, osilations exits
*Overdamped case: if ξ =1 then β=0, α=ω the poles are on the real axis, both at s = - α
* The maximum value of overshot of the Bode plot at resonance is given by
2
1for
12
12
Mp and
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Second order system (Series RLC Circuits)
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differential equation (2nd order) for this circuit
the transfer function for the current:
the transfer function for the voltage across the capacitor
the poles of the system
22
2
2
2
1
42
nL
Rs
LCL
R
L
Rs
0)()()()(
)2(and)1(from
)2()(
)(currenttheforand
)1()()()(
)(
from
2
tutudt
tduRC
dt
tduLC
dt
tducti
tutudt
tdiLtRi
KVL
ccc
c
c
Or it can be rewritten for the current
1)(
)(2
RCsLCs
Cs
su
si
dttiC
tudt
tdu
c
i
dt
tidL
dt
tdiR c )(
1)(,
)()()(2
22
1
1
)(
)()(
2
RCsLCssu
susG c
1)( 2 RCsLCssH
from characteristic equation of
Other terms of a second order system
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* If the angle θ is zero, then the two polesare coincident. This condition correspondsto critical damping corresponds to thecondition
C
LR
LCL
Rn
n
2
1
2
022
22 frequency osilasyon
cos ratiodamping
1 frequency resonant natural
21conatant time
2:termdecay
n
n
nLC
R
LL
R
pole positions
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For the case where: α > ωnwe have two real poles generated One the these poles will move towards the lefton the real axis and the other to the right. The system response is now very slow,and it is said to be overdamped. There are no oscillations.
Definition of several other terms cont.
Another important property of a series RLC circuit is the impedance transfer function.
If we let s = jω (i.e. the resonant frequency), and substitute this into
Clearly the magnitude of this expression has a minimum value when the imaginaryterm is zero. Therefore:
nLC
LC 1
012
Cs
RCsLCs
si
susZ
1
)(
)()(
2
Quality Factor
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Quality Factor: Another important measure of resonant second order circuits
• the total energy is:
• the energy dissipated over a period To is
If Q = 0.5 then
which is the same expression for the resistance when the circuit is criticallydamped
LCC
L
RQ
R
Lf
RLIf
LIQ
n
m
m
1 using
1
2
2
12
1
2 02
0
2
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Test of second order system analysis
1
1
)(
)()(
2
RCsLCssu
susG c
Build a transfer function block in Matlab/simulink, find paramters andsimulate the model for step input to explore the three different cases:1) undamped, 2) underdaped and 3) overdamped showing the systemoutput, uc(s), on the scope screen
Feedback control system design objectives
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Properties of control systems:• StabilityFor the bounded input signal, the output must be bounded and if input is zero then output must be zero then such a control system is said to be stable system
• Performance-Dynamic stability-Accuracy
Dynamic overshootingOscillation durationSteady-state error
-Speed of (Transient) response
Additional considerations:• Robustness (insensitivity to
parametervariation)to models (uncertainties andnonlinearities)to disturbances, and tonoises
• Cost of control• System reliability
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Requirement of Good Control System
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Bandwidth: An operating frequency range decides the bandwidth ofcontrol system. Bandwidth should be large as possible for frequencyresponse of good control system.
Speed: It is the time taken by control system to achieve its stableoutput. A good control system possesses high speed. The transientperiod for such system is very small.
Oscillation: A small numbers of oscillation or constant oscillation ofoutput tend to system to be stable.
Brief view of control techniques• Classical control: Proportional-integral-derivative (PID) control, developed in1940s and used for control of industrial processes. Examples: chemical plants,commercial aeroplanes.
• Optimal control: Linear quadratic Gaussian control (LQG), Kalman filter, H2control, developed in 1960s to optimize a certain ‘cost index’ and boomed byNASA Apollo Project.
• Adaptive control: Uses online identification of the process parameters,thereby obtaining strong robustness properties. Adaptive control was appliedfor the first time in the aerospace industry in the 1950s.
• Robust control: H∞ control, developed in 1980s & 90s to achieve robust performance and/or stability in the presence of small modeling errors. Example: military systems.
• Nonlinear control: Currently hot research topics, developed to handle nonlinear systems with high performances. Examples: military systems such as aircraft, missiles.
• Intelligent control: Predictive control, neural networks, fuzzy logic, machine learning, evolutionary computation and genetic algorithms, researched heavily in 1990s, developed to handle systems with unknown models. Examples: economic systems, social systems, human systems.
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Methods of Analysis and Design in Linear Control System Cosidered
• Mathematical Models of Systems– Laplace transforms and transfer functions– State-space model• Feedback Characteristics and Performance– Time-domain performance specifications– Stability, transient and steady-state responses– Ziegler–Nichols algorithm• Model based analytical design– Full state-feedbck (pole placement)• Complex-domain method– Root locus method for analysis and design of control systems• Frequency-domain method– Frequency-domain performance specifications– Bode plot diagrams for analysis and design of control systems• Design of control system
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Three terms control, Prortional Integral Derivative (PID)
A PID control algorithm involves three separate parameters namely; Kp, Ki, Kdconstant gain values. • The proportional value determines the reaction to the current error, • The integral value determines the reaction based on the sum of recent
errors, and • The derivative value determines the reaction based on the rate at which the
error has been changing, By tuning the three constants in the PID controller algorithm, the controller can provide control action designed for specific process requirements.
The response of the controller can be described in terms• Steady state error, e• Overshoots, Mp• The degree of system oscillation, which corresponds to settling time ts[Ogata-2002, Dorf and Bishop-2005].
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Characteristics of P, I, and D Controllers
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Determining the values for Ki, Kp and Kd with the correlations in the tablemay be used. But changing one of these variables can change the effectof the other two, therefore these values are not exactly accurate, becauseKi, Kp and Kd are dependent of each other. For this reason, the tableshould only be used as a reference when you determine parameters Ki,Kp and Kd with the use of the trial-error method is used.
Model of PMDC motor with transfer function block
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L
t
t tikdt
dJ
e
omittedisif B
)2(
)1(
L
t
t
e
e
tikBωdt
dJ
kuRidt
diL
e
b
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;
State space model of PM DC motor
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)2(
11
)1(11
L
t
t
e
e
tJ
ikJ
ωJ
b
dt
d
kL
uLL
Ri
dt
di
e
b
uL
iL
k
L
k
J
k
J
b
idt
d
BxA
x
0
1
xCy
i
01
Simulation model for state variables i and ω
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Cascade controlled system (DC motor drives)• An electric motor drive system can have three cascade controller• Design should be started from the festest control loop, in this
case, it is the most inner loop, (current control loop). The current ismore faster then machanical variables, speed and position.
0,0and7
loopcontrolcurrentfor204
speed,for %5Mp
s
eesB
JT
msTt
im
cs
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PI controller model based design
pic
pi
pi
icio
ioic K
LT
K
Ls
sK
LsF
sF
sF
sI
sIsF
,
1
1)(
)(1
)(
)(
)()(
*
spics t
LKTt 44
msR
LTT
Ls
K
sTRsT
sTKsFGG
sE
sIsF ei
pi
ei
ipiioscio 1.1for
)1(
1)1()(.
)(
)()(
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If the back-emf is assumed as disturbance of the system, then it can be neglected, eb=0, the open loop system transfer function
1) Designing current loop control
Closed loop transfer functions
Speed loop control
* For an optimal second-order control system in set-point control is given by
* Open loop gain Kpω is set for critical damping 2/1
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2) Designing speed loop control
J
kKssT
J
kK
JkK
JkK
kK
sTJs
sTJs
kKsTJs
kK
sFtp
i
tp
tp
tp
tp
i
i
tp
i
tp
mcl
2/
/
1)1(
1
)1(1
)1()(
JBTTB
ksTs
B
J
K
sT
Bk
sTsT
sTKsF mt
i
p
mt
i
pmc /,
1
1
1
)1(
/1
1
1)1()(
)1()(
sTJs
kKsF
i
tpmo
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t
pip
tpipn
pi
in
Lk
JKK
JL
kKK
L
K
T2
24
12
pinni
n
pitp
i
i
tp
tpi
tp
cl KssT
JL
KkK
Tss
JT
kK
J
kKssT
J
kK
sF ,21
)(22
2
22
pins K
Lt
84
21100
ePOS
Speed loop control
2
1
1(
10012
n
p
p
T
eM
Ziegler–Nichols Methods
• Most useful when a mathematical model of the plant is not available.
• Proportional‐integral‐derivative (PID) control framework is a method to control uncertain systems
• Many different PID tuning rules available
• Transfer function of a PID controller
• The three term control signal
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The S-shaped step responseZiegler–Nichols Tuning Formula (first method)
• The S-shaped curve may becharacterized by two parameters:delay time L and time constant T
• The transfer function of such a plantmay be approximated by a first-order system with a transport delay
Table 1.
Ziegler‐Nichols PID Tuning (Second Method)(Use the proportional controller to force sustained oscillations)
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In this method, the closed-loop systembehavior is observed. A P-controller is usedto tune the system towards oscillationboundary. The gain is increased until thesystem is on the oscillation boundary. Thenthe output of the system oscillates withconstant amplitude and frequency.The parameters of the controller arecalculated according to table 2.
Table 2.
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State-Feedback Control(pole-placement control )
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The method of feed-backing all the state variables to the input of thesystem through a suitable feedback matrix in the control strategy isknown as the full-state variable feedback control technique.
In this approach, the poles or eigenvalues of the closed loop system canbe placed arbitrarily at the specified location.
Placing the poles or eigenvalues of the closed-loop system at specifiedlocations arbitrarily if and only if the system is controllable. Poleplacement is easier if the system is given in controllable form.
Thus, the aim is to design a feedback controller that will move some orall of the open-loop poles of the measured system to the desired closed-loop pole location as specified. Hence, this approach is also known asthe pole-placement control (or Pole Assignment, Pole Allocation)design.
State‐space representation
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Cx
BAxx
y
RyRuRxudt
d n ,,,
m
m
n
n
R
u
u
u
R
x
x
x
dt
d
2
1
2
1
0)0( uxxxBuAxx
State feedback control law
mxn
mnm
n
R
kk
kk
1
111
, KKxu
K is the controller gain matrixRequires measurement of all state variables measurable
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Control Design using Pole Placement• Objectives
– Choose eigenvalues of closed-loop system– Decrease response time of open-loop stable system– Stabilize open-loop unstable system
The state feedback controller is designed using pole placement technique viaAckermann’s formula
x*(t) = (A − BK)x(t)y=Cx
the characteristic polynomial forthis closed-loop system is thedeterminant of [sI - (A-BK)]
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Introducing the Reference Input
• Since the matrices A and BK are both 2x2 matrices, there should be 2 poles for the system.
• By designing a full-state feedback controller, we can move these two poles anywhere we want them.
• first try to place them at -5+j and -5-j (note that this corresponds to a ξ = 0.98 which gives 0.1% overshoot and a sigma = 5 which leads to a 1 sec settling time).
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Drawback: One of the major disadvantage of state feedback controllerdesign by using only the pole-placement is the introduction of a largesteady-state error. In order to compensate this problem, an integralcontrol is added where it will eliminate the steady-state error inresponding to a step input.
In brief the pole assignment technique is somewhat similar to the rootlocus method in that a closed loop poles are placed at desired locations.The basic difference is that in root locus design only the dominant closedloop poles are placed at the desired locations, while in the poleassignment technique all the closed loop poles are placed at the desiredlocations. [Ogata-1998, Ogata-2002, Dorf and Bishop-2005].
The basic difference with root locus design
State feedback Control design
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• Closed‐loop system
• Design objective– Choose K such that sA‐BK) are placed at the
desired locations– Closed‐loop characteristic equation
– Desired closed‐loop characteristic equation
– Equate powers to determine K
nxnRdt
d BKAxBKABKxAxBuAx
x)(
0)( 011
1 asasass n
nn BKA
0011
1 sss n
nn
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Controllability (Reachability)
• Eigenvalues can be placed arbitrarily if and only if system is controllable
• Single input (m = 1)– Controllability matrix
– System is controllable if Co is nonsingular
• Multiple inputs (m > 1)– Controllability matrix
– System is controllable if rank(Co) = n
nxnn RCo BAABB 1
nxmRCo
Illustrative Example
• Linear model
• Open‐loop stability– si(A) = ‐0.438, ‐4.56
– Origin is a stable steady state
• Controllability
uudt
dBAxx
x
0
1
42
11
20
112
1
0
1
42
11
ABB
AB
Co 02)( nCorank
Co is non-sigular so thatthe system is completlycontrollable
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Illustrative Example contin.
• Characteristic equation
• Desired characteristic equation
• Controller gains
224)5()2)(1()4)(1(
42
11)()(
42
11
0
1
42
11
2112
21
21
2121
kksksksks
s
kksss
kkkk
BKAIBKA
BKA
12.07.0)4.0)(3.0( 2 ssss
66.712.0224
30.47.05
221
11
kkk
kk
State‐space model of DC motor
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uL
iL
k
L
k
J
k
J
b
idt
d
BxA
x
0
1
xCyi
;01
)2(
11
)1(11
b
e
e
e
L
t
t
kL
uLL
Ri
dt
di
tJ
ikJ
ωJ
B
dt
d
kkk te
For a control system defined in state-space form and depending on the pole assignment the control signal will be
Kxu Where the control signal u is determined bythe instantaneous states x1=i and x2=ωThe rest of the design presuderes follows thepreious illustrative example
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Hardware Test Setup
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PM DC motor and its Parameters
Department of EnergyTechnology, AAU 2012 by m.dal
42
Vdc (V) 12 Terminal nominal voltageN (rpm) 5800 No_load speedTe (Nm) 94e-3 Stall torqueTmax (Nm) 28.4e-3 Max continious torqueImax (A) 1.5 Max continious currentRa (Ω ) 2.5 Terminal resistanceLa (H) 300e-6 Rotor inductanceLd (H) 3e-3 inductivity of added coilJ kg m2 17.6e-7 Rotor inertiaB (Nms/rad) 1.41e-6 Viscous damping conctantTc (Nm) 0 Coloumb frictionKe Vs/rad 19.5e-3 back-EMF constantKm 19.5e-3 torque constantRd (Ω ) 0.69 resistance of added coilJb1 kg m2 8.11e-6 inertia of swing-wheel
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Content of DC motor Interface Block
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Building A DC motor model
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Lab Procedures for Model Building• Build a dynamic model for a separately PM DC motor. • A suggested system block diagram is shown in given block diagram• Refer to your lecture notes for details.• Print the model including block diagrams of all subsystems.The DC motor has the nameplate data and parameters given the table in slide 49. The load torque TL is assumed to be zero.
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Group study
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Grup1 Grup2 Grup3 Grup4
Simulation methods=========
tasks
Separate Simulink blocks
Func, mux, integrator, sum blocks
Sys func.Block(m‐file)
Embedded Matlab func.
blocks
building and simulation of IM
designingcontroller and
PWM
Integration and simu. of drive
scheme.
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By Prof. Bill Messner, Carnegie Mellon University
and Prof. Dawn Tilbury, University of Michigan.
http://www.engin.umich.edu/class/ctms/matlab42/index.htm
Control design tutorial
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Disturbance Observer
M. Nakao, K. Ohnishi, and K. Miyachi, “A robust decentralized joint control based on interference estimation,” InProc. IEEE Int. Conf Robotics and Automat., vol. 1, pp. 326-331, 1987.
Compensation for the disturbance torque on the rotor shaft makes a drive robust against load changes and unmodeled torques.
Bωdt
dJikt
Bωdt
dJtt
tL
Le
te, m
tl
sωJikgs
gt ntnL
ˆ ik
gs
gJ
gs
ggJ tnnn
2
Note: The ideal derivative is not realizable in digital implementation so that the differentiation task is performed by a Low Pass Filter (LPF).
Rotor