Digital Control

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ObjectiveControl System Terminology

Computer Based Control

Control Theory

Classical Approach to Analog Controller Design

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Control System Terminology

Control System Interconnection of components to provide a desired function Plant, Process - The portion of the system to be controlledController The portion of the system that does the controlling Digital Control System- Uses digital hardware (digital computer) Analog Control System- Electronic controller made of resistors , capacitors and operational amplifiers. Signals (i) Continuous time signals (defined for all time) (ii) Discrete time signals (defined at discrete instant of time ) *

*Advantage of Digital ControlReconfiguration, Flexibility (Controlled by changing software )

Wide selection of Control Algorithms

Integrated Control of Industrial System- (Production planning, scheduling, optimization, operations control)

Future Generation Control System (AI)

Radar Tracking*Footer Text*

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Servomechanism for Steering of Antenna

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*TachogeneratorVariable Speed DC Driveac

*Expansion slot of PCStep MotorDriver CircuitBridge and amplifier circuitPumpSumpV2V1Step MotorController CardA/D conversion cardLiquid Level Control System

Day 2

An Overview of classical approach to analog controller designBasic digital control schemePrinciple of signal conversion Basic discrete time signals

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*Computer Based Control1950 Digital Computer (main frame)1962 Digital Computer, Direct Digital Control ( no analog controller)1970 Small, faster, more reliable and cheap computer- Minicomputer1975 MicrocomputerDCCS Distributed Computer Control System for control of large and complex processSCADA- Supervisory Control And Data Acquisition Data Acquisition and Communication (ii) Event and alarm reporting (iii) Data processing (iv) Partial process control CIPS Computer Integrated Process System

Machine tool numerical control- hard- wired function was replaced by software- CNCCNC- Computerized Numerical ControlCIMS Computer Integrated Manufacturing SystemPLC Programmable Logic Controller*

Control Theory

1940 -1950 Classical control theory- Routh-Hurwitz , Root Locus, Nyquist, Bode, Nichols use transfer function in complex frequency (Laplace Variable s)domainLimited to SISO system and linear time invariant system1950- 1960 Modern Control System model in time domain- MIMO Lyapunov Stability criterion, pole placement by state feedback, state observers, optimal control Model based control Knowledge based control- Intelligent control using AI techniques ( Fuzzy Logic and Neural Network)

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An Overview of classical approach to analog controller design

Y(t) Controlled variable of the system m(t) Manipulated variable Yr(t) Desired value of controlled variable Gp(s) TF of controller systemH(s) TF of feedback element w(t) - Disturbance b(t) Feedback signal e(t) = yr y(t) System ErrorA(s) - TF of reference input elementr(t) Reference input compatible with b(t)e(t) Actuating error signalD(s) TF of controller u(t) Control signal( Has knowledge about the desired control action)GA(s) TF of actuator element (develop enough torque, pressure or heat )*

*--------- (1)The O/P equation Y(s) is given by

*----(2)----(3)

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*------(4)------(5)------(6)Sub (4) in (5) we obtain (6)From (6) we obtain the Reference Transfer Function M(s)------(7)

*----(8)----(9)Sub (8) in (9) and solving y(s)/W(s) will give the disturbance Transfer Function Mw(s) ----(10)

*The response to the simultaneous application of R(s) and W(s) is given by-----(11)From equation (7) & (10) M(s) and Mw(s) are closed loop Transfer FunctionsRoots of the characteristic equation are closed loop poles of the system

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Day 3

Time Domain Model for Discrete- Time System

State Variable Model

Difference Equation Model

Impulse Response Model

Transfer Function Model

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Time Domain Model for Discrete- Time SystemDiscrete time system is defined mathematically as a transformation, or an operator that maps an input sequence r(t) into an output sequence y(k).*State Variable ModelThe state equation and output equation of the system together give the state variable model of the system.

MATLAB Program

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*State Equation Output Equation

The state variable formulation as block diagram*

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The discrete time system of figure has one dynamic element, x(k) is the output of the dynamic element*0.050.04750.95Y(k)r(k)x(k)x(k+1)Study the response for the unit sep sequence and unit alternating sequence ( k) = 1 for k>=0 r(k)= (-1)k for k>=0 0 for k< 0 0 for k< 0++++

X(k+1) = 0.95x(k) + r(k) ; x(0) =0 -------(1)Y(k) = 0.0475 x(k) +0.05 r(k) --------(2)Solution for equation 1X(k) = (0.95)k-1 r(0)+ (0.95)k-2 r(1)+..r(k-1)Since X(k+1) = - x(k) + r(k) ; x(0) =0 X(0)=0; x(1)=r(0) ; x(2)= - r(0) +r(1); x(3)= -2 r(0) + r(1)+ r(2) x(k)= (-)k-1 r(0)+ (-)k-2 r(1)+..r(k-1) =

= since

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Difference Equation Model*

Impulse Response Model*------------ 1

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Transfer Function Model*Analytical study of a system is to set up mathematical equation to describe the system.Let us take a linear time invariant discrete time system that is initially relaxed at k=0

*Difference eq.

*Z Transform Shifting theorem Z[y(k+n)]=zn Y(z)- zn Y(0)- zn-1Y(1)-. z2y(n-2)-zy(n-1)--------1In equation 1--------2--------3a--------3b

*Substituting 3a & 3b in 2 we get

*Transfer Function of Unit Delayer ( )

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Day 4Stability on the Z-Plane BIBO Stability Zero-input Stability Jury Stability Criterion **

*Stability on the Z-Plane & Jury Stability Criterion

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*BIBO Stability

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*Zero-input Stability

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*Jury Stability

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Practical aspects of the choice of Sampling RateLong sampling interval reduces computational load, need for rapid A/D conversion and hardware cost of the project.It result in degrading effectsLimiting Factor for Choice of Sampling RateInformation loss due to sampling (i) Real signals are not band limited (ii) Ideal lowpass filters are not physically realizable (iii) ZOH introduce reconstruction errors. Information loss due to Disturbance (i) High frequency noise appear as low frequency signals due to aliasing effect causing loss of information (ii) Cut off frequency of anti aliasing filter must be higher than system bandwidth*

Destabilizing Effects Due to conversion times and computation times digital algorithm contain dead- time .Algorithm- accuracy Effects Discretization process (ie) transformation of an algorithm from continuous time to discrete time form introduces error Word-length Effects

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Empirical rule for selection of sampling rate Practical experience and simulation results have produced useful rules for specification of minimum sampling rates.

(i) Table gives values from the experience of process industries(ii) Fast acting electromechanical system require shorter sampling intervals few milliseconds

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Type of Variable Sampling time(s) Flow 1 - 3 Level5 - 10 Pressure 1 - 5 Temperature10 - 20

The rule of thumb says, a sampling period needs to be selected much shorter than any of the time constants in the continuous time plant .For complex poles with imaginary part d the frequency of transient oscillation corresponding to the pole is d . Rule suggests sampling at the rate of 6 to 10 times per cycle.Sampling rates can be based on the bandwidth of the closed-loop system. Reasonable sampling rates are 10 to 30 times the bandwidth.For closed loop performance the sampling interval T should be equal to or less than, one-tenth of the desired settling time.*

Routh Stability criterion on the r - plane*

The outside of the unit circle in the z-plane is transformed to the right half of the new complex plane.

The boundary of the unit circle in the z-plane is transformed into the imaginary axis of the new complex plane.

The inside of the unit circle in the z-plane is transformed into the left half of the new complex plane.*

In the stability analysis using the bilinear transformation coupled with Routh stability criterion substitute (w+1)/(w-1) for z in the characteristic equation (z)=0 , where w=+j , to obtain the characteristic equation (w)=0

Problem: Consider the following Characteristic equation P(z) = z3-1.3z2-0.08z+0.24=0 . Determine whether or not any of the roots of the characteristic equation lie outside the unit circle in the z plane. Use the bilinear transformation and Routh stability criterion.*

-0.14w3+1.06w2+5.10w+1.98 = 0 w3-7.571w2-36.43w-14.14 = 0

w3 1 -36.43 w2 -7.571 -14.14 w1 -38.30 0 w0 -14.14

Since there is one sign change for the coefficient in the first column there is one root in the right half of the w plane. The system is unstable.*

1. s3+4.5s2+3.5s+1.5=0

s3 1 3.5S2 4.5 1.5S1 3.6= (4.5*3.5-1*1.5)/4.5 0S0 1.5=(3.16*1.5-4.5*0)/3.16 0

No sign change in the first column thus the system is stable

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2. Determine the stability of a system having following characteristic equation: s6+s5+5s4+3s3+2s2-4s-8=0

s6 1 5 2 -8S5 1 3 -4 0S4 2 6-8 0 Auxiliary equationS3 0000S2

A(s)= 2 S4+6S2-8dA(s)/ds= 8 S3+12S-0

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s6 1 5 2 -8S5 1 3 -4 0S4 2 6-8 0S3 812 0 0 coefficients of AuxiliaryS2 3 -8 0 0 equationS1 100/3 0 0 0S0 -8 0 0 0

There is one sign change in the first column thus the system is unstable*

3. F(s)=(s2+1)(s+1)(s+2)(s+3)The system has a pair of conjugate root on imaginary axis s=+-j1S5+6s4+6s3+12s2+5s+6+0S5 1 6 5S4 6 126S3 44S2 6 6 S1 0(12) 0S0 6A(s)= 6S2+6dA(s)/ds= 12sNo sign change in the first column thus the system is marginally stable

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Principles of Discreization*

Impulse Invariance*-------------1-------------1a-------------1b

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Example 1*

Example 1

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Step Invariance*----4

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Example 2*

Example 2

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*Example 3Find he response of the system shown in the figure to a unit impulse input.

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Implementation of digital controller*

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Pole- Placement Design and State ObserversCompensator Design by the Separation PrincipleConsider a linear completely controllable and completely observable systemState equation -------------(1)

State feedback control law ------------(2)Full order observer --------------(3)

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State feedback control law based on observer state ---------------(4)Sub (4) in (1) we get ) ----------(5)*) Fig: Combined state feedback control and state estimation

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Servo Design*

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*Control configuration of a servo systemw

Deadbeat control by state feedback and deadbeat observers*

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