Problems P.1: For the unity feedback system shown, where G(s): = 450(s+8)(s+12)(s +15) s(s+ 38) (s² + 2s +28) find the steady-state errors for the following test inputs: 25u(t), 37tu(t), 47t²u(t). R(s) + E(s) G(s) sR(s) 01+G(s) e(x) = lim C(s)
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- A stock-flow system models the level of water in a lake. Near a certain equilibrium point, there are three feedback loops: an amplifying feedback loop with strength of +0.55 per month, a stabilizing feedback loop with strength of -0.09 per month, and an amplifying feedback loop with strength of +0.79 per month. Calculate the strength of the overall feedback.Q.1 - The open loop transfer function for a unity - feedback systemis G(s)= XL‘ 7xs and r(t)=3t determine steady state error.If it is desired to reduce this existing error by 7% fined new value of gain of the system.1.block diagram physical meaning and the time response for different inputs
- 1. Give an example of open loop and closed loop system (one example each). Also state the input, control system, feedback and output parameter. Example. 1. Open Loop - Water Heater: Input - Water Temperature (Cold) System - Heating Element Output - Water Temperature (Hot) 2. Closed Loop - Air-conditioning System Input - Desired Room Temperature Control - Motor controller/Compressor/ACU Feedback - Temperature Sensing Output - Room TemperaturePLEASE GIVE COMPLETE AND DETAILED SOLUTION The oven shown in Figure 1 has a heating element, qi with capacitances C = 50 J/K. The corresponding temperature is T and ambient temperature is Ta. The thermal resistance of the oven wall is R = 2 K-s/J. The system has a steady-state error in the response, T(t) to a change in the reference input. The steady state error can be reduced if a proportional control action, Kp is included in the controller. Determine the range of values of Kp of the system so that the steady state error is lower than 0.4. (Hint: Kindly determine the steady state error at value 0.4)The engine, body, and tires of a racing vehicle affect the acceleration and speed attainable. The speed control of the car is represented by the model shown in Figure 2. (i). Determine the steady-state error of the system with unity gain and explain with suitable justification about the system type. A= 12 , B=5 , C=10 , D= 16 (ii). Determine the static error coefficient associated with the system. (iii). Assume the steady state error to be 10%. What will be the value of the controller gain, K?
- Determine the steady state time response x(t) of a viscously damped system subjected to harmonic base excitation for the following data : m = 15 kg, c = 25 N.s/m, k = 300 N/m, y(t) = 0.04 sin ( 5t ) m. %3DFor the Western Electric rule that if at least 2 of 3 consecutive points fall beyond 2σ to one side of the center line the process is out of control:a. FindtheType1errorrate.b. Find the Type 2 error rate if the process shifts from u to u + 2σ.a)is the aircraft stable about the equilibrium represented by the transfer function? b) Using proportional feedback,what is the range of acceptable gains for the closed loop systen to be stable? c) Design a feedback control system that allows the pilot to command a pitch angle with overshoot less than or equal to 4.15% and a natural frequency of greater than or equal to 0.99 rad/s d) Design a feedback control system that allows the pilot to command a pitch angle with the same overshoot and a natural frequency of one half the system in part c.
- For the calibration data of given below, (a) determine the static sensitivity of the system, (b) find the maximum nonlinearity, (c) find the value of the threshold. X (cm) 0.5 1.0 (cm) 0.4 1.0 5.0 10 15 20 25 30 35 40 45 50 6.9 15.8 24.7 36.4 49.9 60.3 75.1 87.0 98.9 110.1Exercise 3: Damped forced vibration. Find the solution ofu′′ +(1/8)u′ + u = 3 cos(ωt), u(0) = 2, u′(0) = 0 using sympy.dsolve for ω = 0.3 and ω = 1. Plot the solution, transient solution and steady-statesolution in a common plot for 0 < t < 10 for ω = 0.3 and ω = 1 respectively. Exercise 4: Undamped forced vibration. Plot the solution ofu′′ + u = 3 cos(ωt), u(0) = 0, u′(0) = 0 for 0 < t < 10 for ω = 0.3 and ω = 1 respectively. (You may find the solution either numerically with scipy.integrate.solve ivp, scipy.integrate.odeint or by finding and plotting the symbolic solution via sympy.dsolve)Remark: The vibration consists of beats; in the critical case ω = 1 solutions do not stay boundedName two effects of negative feedback.