First order system(Sample Question 1)Example question 2L3 and L4Analysis of Vehicle Speed ModelDraw a block diagram of an integral controller applied to the 1st orderBy making a number of assumptions concerning friction etc.,themovement of a motor vehicle can be represented by the followingTransfer Function stated above.Add the following labels:vehicle speed,standard 1st order Transfer Function,control input,control model and set point.Use the rules of blockGdiagram analysis to find the closed-loop Transfer Function.Find thesteady state gain of the closed-loop Transfer Function and comment onwhere X is the vehicle speed (mph)and U is the engine force.In thismodel,the engine force is represented as a scaled value between D and1.0,where 1.0 implies maximum force,0.5 half of this and so on.set point.Using the Transfer Function model above,and T=5.2 seconds andG=87,sketch the response of vehicle speed to a step in engine.Block Diagram (Lecture 6)[10 Marks)force from zero to 0.75 units.Label your sketch to define the steadystote speed and time constant (stating all the relevant numericalvalues and units).Suggest two limitations of this model.Answer 1-Marks Allocation.Convert TF into standard first order TF.Steady state gain K =G=87Close Loop Transfer Function [3 Marks]After 5.2 s (the time constant),the speed 0.63x65.25 =41.11 mphY=s Ts+1/Ts2 +s)KGGD=Cont'dKGTs2+s+KGModel limitations:no engine wear;no change in engine temperatureor efficiency;assume friction is proportional to vehicle speed;assumeno change in frictional coefficient due to change of road surface,winddirection;assume mass of the vehicle remains constant;assume nochange in gradient of the road surface;etc.Anything at all toCont'drecognise that the mathematical equation above is not the real world.KG.the steady vehicle speed is.where d is a constant set point.x will track the set point.The steadystate error between the vehicle speed and the set point will be zero,i.e.the actual vehicle speed will track the desired vehicle speed.Maris]5.2Answer-4 (Marks allocation)time (seconds)Standard objectives (stability,satisfactory transient,etc.)The system should be stable.[3 Marks)The output should track the desired output response in the steady state,i.e.there should be no steady state error.[3 Marks]The transient behaviour of the system should be satisfactory [3 Marks]The control system should reject disturbances.[3 Marks]You have to relate your answers with a given case study.Case StudyStabilityThe engineer asks for the optimum temperature of the incubator,thisis the desired output (set point).Obviously,the heating/coolingsystem should be stable.The system should be stable and hence,thetemperature would not decrease/increase to infinity (colder thanabsolute zero or hotter than the sun).More importantly thetemperature should be maintained within a range(question 1 andquestion 2)to ensure the baby is neither too hot nor too cold.TrackingThe engineer's second question ascertains the allowable steady stateerror(SSE).The actual(measured)temperature should track thedesired(optimum)temperature at steady state with zero SSE.Forexample,if the optimum temperature is 20.5C,the actualtemperature should be 20.5C plus or minus a clinically acceptedtolerance and not any other value outside of this!Transient behaviourThe engineer investigates about the response of the control system intheir fourth question.At what rate should the temperature adjust tomaintain optimum conditions?Rapid fluctuations may not benecessary or detrimental if the allowable range is more than a coupleof degrees.Equally,temperature optimisation may be futile if tooslow.The actual temperature should reach the set temperature,butover how long?This relates to transient behaviour and the dynamicresponse;i.e.how long should it take "before"steady state isreached.Furthermore,oscillations in the temperature are undesirableand an overshoot of the desired temperature plus/minus theuncertainty must not occur otherwise the baby will overheat or gettoo cold.DisturbancesThe engineer asks for details on the ambient temperature of the ward and how muchthis varies throughout the day.Here they are determining the extent of externalinfluences over the measured temperature inside the incubator.This is a reference todisturbances(the fluctuations in the ambient temperature is a disturbance).The controlsystem designed should 'reject'any influences over the intemal temperature of theincubator.For example,the sun shining through the window of the ward,the doors andports of the incubator being left open,excess heat generated by the baby.Anything thatwill cause the temperature inside the incubator to increase or decrease-the controllershould adjust the control input to bring the temperature back to the desired level.