tum control design - Technische Universität München · PDF fileTo analysis and controls design for a linear time-invariant (LTI) system, ... First order system(seriesRC circuit)

18.12.2015

1

Design Methods of Linear Control Systems

Visiting ProfessorMehmet Dal

Department of Electrical and ComputerEngineering at TUM, Munchen, Germany

TUM, Germany, 2015 by m.dal 1

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


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


• 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


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


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


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


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)


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


* 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

TUM, Germany, 2015 by m.dal

14

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


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


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


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


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


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

TUM, Germany, 2015, m.dal 23

)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


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()(.

)(

)()(


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


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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14


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)


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 )


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


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)]


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


• 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


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


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


Building A DC motor model


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


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.


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

tum control design - Technische Universität München · PDF fileTo analysis and controls design for a linear time-invariant (LTI) system, ... First order system(seriesRC circuit)

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