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  1. 1. INTRODUCTION :Automatic control is the research area and theoretical base for mechanizationand automation, employing methods from mathematics and engineering. Acentral concept is that of the system which is to be controlled, such as a radder,propeller or ballistic missile. The systems studied within automatic control aremostly the linear systems. Automatic control is also a methodology orphilosophyof analyzing and designing a system that can self-regulate a plant (suchas a machine or a process) operating condition or parameters by the controllerwith minimal human intervention. A regulator such as a thermostat is an exampleof a device studied in automatic control.An automatic control system is a preset closed-loop control system that requires no operatoraction. This assumes the process remains in the normal range for the control system. Anautomatic control system has twoprocess variables associated with it: a controlled variable anda manipulatedvariable.A controlled variable is the process variable that is maintained at aspecified value or within aspecified range..A manipulated variable is the processvariable that is acted on by the control system to maintainthe controlled variable at the specified value or within the specified range.Functions of Automatic Control-In any automatic control system, the four basicfunctions that occur are:MeasurementComparisonComputationCorrection Elements of Automatic Control-The three functional elements needed to performthe functions of an automatic control systemare:A measurement elementan errordetection elementA final control element.Components : Sensor(s), which measure some physical state such as temperature or liquid level.
  2. 2. Responder (s), which may be simple electrical or mechanical systems or complex special purpose digital controllers or general purpose computers. Actuator (s), which effect a response to the sensor(s) under the command of the responder, for example, by controlling a gas flow to a burner in a heating system or electricity to a motor in a refrigerator or pump.Feedback Control System :A feedback or closed loop control system is one where the input has control overthe output(controlled variable). In this system the controlled variable is measuredand fed back to the controller through a path (or loop). Some or all of the systemoutputs are measured and used by the controller. The controller then compares adesired plant value with the actual measured output value and acts to reduce thedifference between the two to zero value.Types of feedback:When feedback acts in response to an event/phenomenon, it can influence theinput signal in one of two ways:An in-phase feedback signal when feedback acts in response to anevent/phenomenon, it can influence the input signal in, where a positive-goingwave on the input leads to a positive-going change on the output, will amplify theinput signal, leading to more modification. This is known as positive feedback.A feedback signal which is inverted, where a positive-going change on the inputleads to a negative-going change on the output, will dampen the effect of theinput signal, leading to less modification. This is known as negative .
  3. 3. I/PI Comparator Control Unit Signal Final Control Process O/P Unit Processing Unit Plant/P Detecting Unit Measuring Unit Signal Conditioner Transmitting Unit This is the basic block diagram of feedback control system. The different types of system functions and their order will be discussed below. A transfer function (also known as the system function or network function) is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function (hence a function of spatial frequency) i.e. the intensity distribution caused by a point object in the field of view. The term is often used exclusively to refer to linear, time-invariant systems (LTI) Most real systems have non-linear input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") have behavior that is close enough to
  4. 4. linear that LTI system theory is an acceptable representation of the input/outputbehavior.In its simplest form for continuous-time input signal x(t) and output y(t), thetransfer function is the linear mapping of the Lap lace transform of the input, X(s),to the output Y(s):Type 0 System:G(s) is said to be type 0 if the number of poles of G(s) at the origin is equal tozero.Given a transfer function G(s),G(s) is type 0 when n = 0.When we compute the steady-state error, G(s) must correspond to the transferfunction of the forward loop.
  5. 5. Type 1 System:G(s) is said to be type 1 if the number of poles of G(s) at the origin is equal to one.Given a transfer function G(s),G(s) is type 1 when n = 1.When we compute the stedy-state error,G(s) must correspond to the transferfunction of the forward loop.Type 2 System:G(s) is said to be type 2 if the number of poles of G(s) at the origin is equal to two.Given a transfer function G(s),
  6. 6. G(s) is type 1 when n = 2.When we compute the stedy-state error,G(s) must correspond to the transferfunction of the forward loop.Time response of a first order control system subjected to unit stepinput function :Output for the system-- C(s)=R(s)*[1/(sT+1)] , R(s)=(1/s)Therefore, C(s)=(1/s)-(1/(s+(1/T)))Taking inverse laplace transform, we get:c(t)=1-e^(-t/T)The error is given as:e(t)= e^(-t/T)The steady state error:ess= Lim e^(-t/T)=0t∞
  7. 7. Time response of a second order control system subjected to unit stepinput function :Output of the system—C(s)=R(s)*[(w^2) / (s^2+2*z*w*s+w^2)]Where, z=zeta—damping ratio w=natural frequency of oscillationAfter taking the inverse laplace, we get:c(t)=1- [{(e^(-zwt))/{(1-z^2)^(1/2)}} * x] ,where,x=sin[{w*((1-z^2)*t))^(1/2)} + tan^(-1){((1-z^2)^(1/2)) / z}]The error is given as:e(t)= [{(e^(-zwt))/{(1-z^2)^(1/2)}} * x] ,where, x=sin[{w*((1-z^2)*t))^(1/2)} + tan^(-1){((1-z^2)^(1/2)) / z}]The steady state error is given as:ess= Lim [{(e^(-zwt))/{(1-z^2)^(1/2)}} * x]t∞The steady-state error depends on the type of inputs and the type ofthe forward-loop. It can be computed easily using error constants :Static position error coefficient Kp :
  8. 8. Kp= lim G(s)s0For a type 0 system,Kp=lim *K(Tas+1)(Tbs+1)…+/*(T1s+1)(T2s+1)…+ = Ks0For a type 1 or higher system,Kp=lim *K(Tas+1)(Tbs+1)…+/*(s^N) (T1s+1)(T2s+1)…+ = ∞ , for N>=1s0Static velocity error coefficient Kv :Kv= limsG(s)s0For a type 0 system,Kv=lim *s*K(Tas+1)(Tbs+1)…+/*(T1s+1)(T2s+1)…+ = 0s0For a type 1 system,Kv=lim *s*K(Tas+1)(Tbs+1)…+/*s* (T1s+1)(T2s+1)…+ = Ks0For a type 2 or higher system,Kv=lim [s*K(Tas+1)(Tbs+1)…+/*(s^N) (T1s+1)(T2s+1)…+ =∞ , for N>=2s0
  9. 9. Static acceleration error coefficient Ka :Ka= lim (s^2)G(s)s0For a type 0 system,Ka=lim *(s^2)*K(Tas+1)(Tbs+1)…+/*(T1s+1)(T2s+1)…+ = 0s0For a type 1 system,Ka=lim *(s^2)*K(Tas+1)(Tbs+1)…+/*s* (T1s+1)(T2s+1)…+ = 0s0For a type 2 or higher system,Ka=lim *(s^2)*K(Tas+1)(Tbs+1)…+/*(s^2) (T1s+1)(T2s+1)…+ =Ks0MATLAB :MATLAB, which stands for MATrix LABoratory, is a technical computingenvironmentfor high-performance numeric computation and visualization.SIMULINK is a part of MATLAB that can be used to simulate dynamic systems. Tofacilitate model definition, SIMULINK adds a new class of windows called blockdiagramwindows. In these windows, models are created and edited primarily bymousedrivencommands. Part of mastering SIMULINK is to become familiarwithmanipulating model components within these windows.
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  17. 17. Type 0 System :G(s)=1/(s+1) Scope 1 res0 s+1 Step Transfer Fcn To Workspace Type 0 Step Response 0.5 0.45 0.4 0.35 0.3 Amplitude 0.25 0.2 0.15 0.1 0.05 0 0 2 4 6 8 10 12 14 16 18 20 Time (s)Fig : Type 0 Response
  18. 18. Type 1 Response :G(s)=1/s(s+1) Scope 1 res1 s2 +s Step Transfer Fcn To Workspace Type 1 Step Response 1.4 1.2 1 0.8 Amplitude 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Time (s)Fig : Type 1 Response