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Imaginary Axis (seconds) присов) -15- The root locus for a plant G(s) with gain K is given in Figure 1. Root Locus 15 10-045 10 15 20 RealAxis (seconds) Figure 1 Find an open loop transfer function for which this is a valid root locus, if the open loop steady state response for a unit step is 2. Find the gain K such that the closed-loop response's 2% settling time is 4 seconds using two methods: 1) by hand AND 2) using MATLAB. Add a PD controller of s+2 (i.c., add a zero of -2 to the open loop transfer function) and plot the root locus in MATLAB. Explain the effect of the PD controller on the stability of the system. Hint: For Q6 (b), the 2% settling time is 4 seconds, that the 7, is 1, therefore the real part of the poles should be -1. There are two ways to solve it from here. (Method I without Matlab) Step 1: From the open loop transfer function in part (a), we can write the characteristic equation with an unknown gain K. Step 2: Because (w, is 1, it means the complex poles will be -1± jwa. We can write the second (ideal) characteristic equation. Step 3: From comparing the coefficients of the corresponding s terms of those two characteristic equations, we can get the unknown K. (Method 2 using Matlab) The value of K can be found from the intersection of a straight vertical line at -1 with the root locus once you plot the root locus with MATLAB. 20 15 bo

Imaginary Axis (seconds) присов) -15- The root locus for a plant G(s) with gain K is given in Figure 1. Root Locus 15 10-045 10 15 20 RealAxis (seconds) Figure 1 Find an open loop transfer function for which this is a valid root locus, if the open loop steady state response for a unit step is 2. Find the gain K such that the closed-loop response's 2% settling time is 4 seconds using two methods: 1) by hand AND 2) using MATLAB. Add a PD controller of s+2 (i.c., add a zero of -2 to the open loop transfer function) and plot the root locus in MATLAB. Explain the effect of the PD controller on the stability of the system. Hint: For Q6 (b), the 2% settling time is 4 seconds, that the 7, is 1, therefore the real part of the poles should be -1. There are two ways to solve it from here. (Method I without Matlab) Step 1: From the open loop transfer function in part (a), we can write the characteristic equation with an unknown gain K. Step 2: Because (w, is 1, it means the complex poles will be -1± jwa. We can write the second (ideal) characteristic equation. Step 3: From comparing the coefficients of the corresponding s terms of those two characteristic equations, we can get the unknown K. (Method 2 using Matlab) The value of K can be found from the intersection of a straight vertical line at -1 with the root locus once you plot the root locus with MATLAB. 20 15 bo

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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Imaginary Axis (seconds) присов) -15- The root locus for a plant G(s) with gain K is given in Figure 1. Root Locus 15 10-045 10 15 20 RealAxis (seconds) Figure 1 Find an open loop transfer function for which this is a valid root locus, if the open loop steady state response for a unit step is 2. Find the gain K such that the closed-loop response's 2% settling time is 4 seconds using two methods: 1) by hand AND 2) using MATLAB. Add a PD controller of s+2 (i.c., add a zero of -2 to the open loop transfer function) and plot the root locus in MATLAB. Explain the effect of the PD controller on the stability of the system. Hint: For Q6 (b), the 2% settling time is 4 seconds, that the 7, is 1, therefore the real part of the poles should be -1. There are two ways to solve it from here. (Method I without Matlab) Step 1: From the open loop transfer function in part (a), we can write the characteristic equation with an unknown gain K. Step 2: Because (w, is 1, it means the complex poles will be -1± jwa. We can write the second (ideal) characteristic equation. Step 3: From comparing the coefficients of the corresponding s terms of those two characteristic equations, we can get the unknown K. (Method 2 using Matlab) The value of K can be found from the intersection of a straight vertical line at -1 with the root locus once you plot the root locus with MATLAB. 20 15 bo
Transcribed Image Text:Imaginary Axis (seconds) присов) -15- The root locus for a plant G(s) with gain K is given in Figure 1. Root Locus 15 10-045 10 15 20 RealAxis (seconds) Figure 1 Find an open loop transfer function for which this is a valid root locus, if the open loop steady state response for a unit step is 2. Find the gain K such that the closed-loop response's 2% settling time is 4 seconds using two methods: 1) by hand AND 2) using MATLAB. Add a PD controller of s+2 (i.c., add a zero of -2 to the open loop transfer function) and plot the root locus in MATLAB. Explain the effect of the PD controller on the stability of the system. Hint: For Q6 (b), the 2% settling time is 4 seconds, that the 7, is 1, therefore the real part of the poles should be -1. There are two ways to solve it from here. (Method I without Matlab) Step 1: From the open loop transfer function in part (a), we can write the characteristic equation with an unknown gain K. Step 2: Because (w, is 1, it means the complex poles will be -1± jwa. We can write the second (ideal) characteristic equation. Step 3: From comparing the coefficients of the corresponding s terms of those two characteristic equations, we can get the unknown K. (Method 2 using Matlab) The value of K can be found from the intersection of a straight vertical line at -1 with the root locus once you plot the root locus with MATLAB. 20 15 bo
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