DEPARTMENT OF MECHANICAL TECHNOLOGY

VI -SEMESTERAUTOMATIC CONTROL

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CHAPTER NO.6State space representation of Continuous

Time systems

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The Centre for Technology enabled Teaching & Learning , M G I, India DTEL(Department for Technology Enhanced Learning)

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DEPARTMENT OF MECHANICAL TECHNOLOGY VI -SEMESTER

CONTROL SYSTEMS ENGINEERING

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CHAPTER NO.6State space representation of Continuous Time

systems

CHAPTER 6:- SYLLABUS

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State space representation for Discrete time systems.

State equations, transfer function from state variable representation – solutions of the state equations.

Concepts of Controllability and Observability.

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4 Introduction to control system design lag lead compensation.

CHAPTER-6 SPECIFIC OBJECTIVE / COURSE OUTCOME

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Stability analysis using analytical and graphical techniques, 1

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The student will be able to:

To understand the concepts of time domain and frequency domain analysis of control system.

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LECTURE 1

5Linear Continuous-Time State Space Models

A continuous-time linear time-invariant state space model takes the for where x n is the state vector, u m is the control signal, y p is the output, x0 n is the state

vector at time t = t0 and A, B, C, and D are matrices of appropriate dimensions.

X(t) = AX(t) +B u(t) x(to) = xo

Y(t) =C x(t) +Du(t)

State equations

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Transfer Functions vs. State-Space Models

• Transfer functions provide only input and output behavior– No knowledge of the inner workings of the system– System is essentially a “black box” that performs some functions

• State-space models also represent the internal behavior of the system

H(s)X(s) Y(s)

State equations

Fig 6.1 Transfer Function

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LECTURE 2

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Linear State-Space Equations

tDvtCxty

tBvtAxtx

vector

vector

vectors

1

1

1,

Mty

Rtv

Ntxtx

RMDNMCRNBNNA

system matrix

input matrixoutput matrix

matrix representing directcoupling from system inputsto system outputs

If A, B, C, D are constant over time, then the system is also time invariant→ Linear Time Invariant (LTI) system.

State-Space Models

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LECTURE 2

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Transfer Functions vs. State-Space ModelsTransfer Functions

• Defined as G(s) = Y(s)/U(s)• Represents a normalized model of a process, i.e., can be used with any input.• Y(s) and U(s) are both written in deviation variable form.• The form of the transfer function indicates the dynamic behavior of the

process.

)()()()( 2 ds

Ccbss

Bas

AsY

dtptat eCteBeAty )sin()(

)()()(1)( 2 dscbssas

sG

Transfer Functions

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LECTURE

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LECTURE 3

Controllability

제 14 강

u x Ax B

rank[ ]c nP

2 1Controllability Matrix [ ]nc

P B AB A B A B

nn

nm Controllable

0c P

nn

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A system is completely controllable if there exists an unconstrained control u(t) that can transfer any initial state x(t0) to any other desired location x(t) in a finite time t0tT.

Controllability

Fig 6.2 Controllability

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LECTURE 4

제 14 강

uy= x Ax BCx

rank[ ]o nP

1Observability Matrix [ ]n To

P C CA CA

Controllable

0o P

1n

n1

nn

Observability

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Observability

A system is completely observable if and only if three exists a finite time T such that the national state x(0) can be demined from the observation history y(t) given the control u(t)

Fig 6.3 Observability

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LECTURE 5

1. Control system design and compensation

Design need to design the whole controller to satisfy the system

requirement.

Compensation only need to design part of the controller with known

structure.

2. Three elements for compensation

Original part of the system

Performance requirement

Compensation device

Stability criterion

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LECTURE 6

1. Frequency Response Based Method

Main idea ： By inserting the compensator, the Bode diagram of the original

system is altered to achieve performance requirements.

2. Root Locus Based Method Main idea: Inserting the compensator introduces new open-loop zeros and poles to change the closed-loop root locus to satisfy.

Method of compensator design

Original open-loop Bode diagram Bode diagram of compensator alteration of gain open-loop Bode diagram with compensation

Method of compensator design

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LECTURE 7

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13Transfer function:

Fig 6.4 Passive phase lag network

1R

2R

oEiEC

2

1 2

2

1 2

1

( ) 1( )

1 ( ) 1

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c

RCSG s

R RCS

R CsR R CsTs

Ts

Phase Lag Compensation

Phase Lag Compensation

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Fig 6.5 Maximum phase lag

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)(L

dec/dB20

lg20

lg10

)(

T/1 m T/1

m

1m T

1arcsin1m

The compensator has no filtering effect on the low frequency signal, but filters high frequency noise. The smaller β is ， the lower the noise frequency where the noise can pass.

Maximum phase lag

Stability criterion

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LECTURE 8

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Comments on phase lag compensation:1.Phase lag compensator is a low-pass filter. It changes the low-frequency part to reduce gain crossover frequency. The phase is of no consequence around the gain crossover frequency.2.Be able to amplify the magnitude of low-frequency part, and thus reduce the steady-sate error.3.The slope around gain crossover frequency is -20dB/ dec .Resonance peak is reduced, and the system is more stable.4.Reduce the gain crossover frequency, and then reduce the bandwidth. The rising time is increased. The system response slows down.

phase lag compensation

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THANK YOU

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References Books:1. Automatic control system by Farid Golnaraghi.2. Modern control System Engineering by

Katsuhiko Ogata3. Feedback Control System by R. A. Barapate4. Automatic control system by Benjamin

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