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Case 1: Open Loop (OL) Control la) Calculate PD (OL) Case 2: Closed-Loop (CL) Control Consider now a closed-loop depicted in Figure 1b: r(t) + e(t) u(t) y(t) C P S Figure lb: CL Control of Room Temperature

Case 1: Open Loop (OL) Control la) Calculate PD (OL) Case 2: Closed-Loop (CL) Control Consider now a closed-loop depicted in Figure 1b: r(t) + e(t) u(t) y(t) C P S Figure lb: CL Control of Room Temperature

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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please help solve the problem shown in photo. Thanks!

PROBLEM 1 We wish to control the temperature y(t) of a room with an electrical heater driven by an input voltage u(t). This heater provides continuous control action, as opposed to the on-off heaters that most of us have at home. Let us define the set electric heater-room (see Figure 1a) as “the plant". Plant of Gain P u(t) y(t) Heater Room Figure la: Plant consisting of Electric Heater + Room In the fall season, a voltage u(t) =10 [V] produces a room temperature yall = 70 ['F], and *F therefore, the plant gain is P= 7 V In wintertime, with the outside low temperature as a perturbation, the same u(t) = 10 [V] 'F produces a room temperature of ywinter = 60 ['F], so the plant gain becomes P=6 V For CASE 1 and CASE 2 below, let us measure the robustness of the control system, by calculating the following percent difference (PD): PD=fall -Ywinter -x100 y winter The smaller PD, the more insensitive to perturbations is our control system. In other words, the smaller PD, the more robust the control system is. Case 1: Open Loop (OL) Control la) Calculate PD (OL) Case 2: Closed-Loop (CL) Control Consider now a closed-loop depicted in Figure 1b: r(t) + e(t) u(t) y(t) C P Figure lb: CL Control of Room Temperature The following gains are used: [V] Sensor gain: S = 0.1 ['F] [V] Controller gain: C=100 [V] lb) What should be the value of the reference signal r(t)to obtain a temperature y(t) in the vicinity of y(t) = 70 ['F](Hint: Consider the desired output, the sensor gain and the fact that the resulting error will be very small: e(t) =[r(t)- y,(t)]→0) Using P=7 *F (fall time), calculate the resulting temperature yfall and the lc) corresponding control voltage uall ld) Using P= 6 *F |(wintertime), calculate the resulting temperature ywinter and the corresponding control voltage uwinter le) Calculate PD (CL). Compare with PD (OL). Explain qualitatively the reason for the increased robustness in closed loop (Hint, look at the control signal uinte, in OL and in CL)
Transcribed Image Text:PROBLEM 1 We wish to control the temperature y(t) of a room with an electrical heater driven by an input voltage u(t). This heater provides continuous control action, as opposed to the on-off heaters that most of us have at home. Let us define the set electric heater-room (see Figure 1a) as “the plant". Plant of Gain P u(t) y(t) Heater Room Figure la: Plant consisting of Electric Heater + Room In the fall season, a voltage u(t) =10 [V] produces a room temperature yall = 70 ['F], and *F therefore, the plant gain is P= 7 V In wintertime, with the outside low temperature as a perturbation, the same u(t) = 10 [V] 'F produces a room temperature of ywinter = 60 ['F], so the plant gain becomes P=6 V For CASE 1 and CASE 2 below, let us measure the robustness of the control system, by calculating the following percent difference (PD): PD=fall -Ywinter -x100 y winter The smaller PD, the more insensitive to perturbations is our control system. In other words, the smaller PD, the more robust the control system is. Case 1: Open Loop (OL) Control la) Calculate PD (OL) Case 2: Closed-Loop (CL) Control Consider now a closed-loop depicted in Figure 1b: r(t) + e(t) u(t) y(t) C P Figure lb: CL Control of Room Temperature The following gains are used: [V] Sensor gain: S = 0.1 ['F] [V] Controller gain: C=100 [V] lb) What should be the value of the reference signal r(t)to obtain a temperature y(t) in the vicinity of y(t) = 70 ['F](Hint: Consider the desired output, the sensor gain and the fact that the resulting error will be very small: e(t) =[r(t)- y,(t)]→0) Using P=7 *F (fall time), calculate the resulting temperature yfall and the lc) corresponding control voltage uall ld) Using P= 6 *F |(wintertime), calculate the resulting temperature ywinter and the corresponding control voltage uwinter le) Calculate PD (CL). Compare with PD (OL). Explain qualitatively the reason for the increased robustness in closed loop (Hint, look at the control signal uinte, in OL and in CL)
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