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G(s) C(s) R(s) 1+ G(s)H(s)* (1) The characteristic equation of the closed-loop system is obtained by setting the denominator poly- nomial of the transfer function to zero. Thus, the roots of the characteristic equation must satisfy 1+G(s)H(s) = 0. (2) Now we are ready to investigate the conditions under which equation (2) is satisfied. Let us express G(s)H(s) as G(s)H(s) = KG₁(s)H₁(s). (3) 1 where G₁(s) H₁ (s) does not contain the variable parameter K, but retains the same dynamic prop- erties as G(s)H(s). Since equation (2) gives G(s)H(s) = -1, equation (3) can be written as G₁(s) H₁(s) = = (4) K To satisfy equation (4), the following conditions must be satisfied simultaneously: Conditions on Magnitude: |G₁(s) H₁(s) = -∞ <K <∞. 3 K Conditions on Angles: ZG₁(s) H₁(s) = (2k + 1), K≥ 0. ZG₁(s) H₁(s) = 2km, K≤ 0. (5) (6) (7)
be: 1 Gnom(s) = OL(S) Tm(s) 1.12725s² + 1.2s (8) However, in some cases (e.g. when evaluating the effect of shaft resonance), a more accurate model of the drive system which includes the effect of non-ideal motor-load coupling, is required. Again using mechanical principles, a more accurate model (Model 1) of the servomechanism was determined to be: 0.12s + 372 G₁(s) = OL(S) Tm(s) (9) 0.298s4 +0.9804s³ +419.481s² + 446.4s
T, = π n 1 - 2 ان لیا
cos %OS = exp Since percent overshoot is given by = SWn Wn -சர் الا =S. = x 100%,
Experiment 1: 2nd order model, no filter Consider a feedback control system with proportional control. Using the Nominal Model given in equation (8) for the above described servomechanism, determine the proportional gain Kp which will result in a response with 9% (0.1%) overshoot. To achieve this, complete the following steps: a) Write instructions to create the transfer functions representing the filter Gf(s), model G(s) = Gnom(s) (equation 8), and feedback transfer function (H(s) = 1). b) Write instructions to find the equivalent forward transfer function Geg(s), and the open-loop transfer function Geg(s).H(s). c) Plot the root locus of the system. d) Plot a close-up of the root locus with axes going from 1 to 0 on the real axis, and -2 to 2 on the imaginary axis. Do this in separate figure window. e) Overlay the 9% overshoot line on the close-up root locus. f) Allow interactive selection of the point where the root locus crosses the overshoot line. Retrieve and record the gain at that point, as well as all of the closed-loop poles for that gain. g) Generate the step response for the system with your determined gain for an 9% overshoot. 6 h) Retrieve and tabulate the characteristics of the actual response. Ensure the overshoot is within 0.1%. If the overshoot is not sufficiently accurate, improve the resolution of the root locus and recollect results. i) Use the root locus to determine the stability region of the system.
Experiment 2: 2nd order model, with filter Often a filter is used for noise suppression and improving frequency response characteristics. Con- sider the system described by Figure 4 with a first order filter with transfer function: Gf(s) = 1 s+8 (10) The Nominal model of equation (8) is still used for G(s). Redesign the system for 9% (±0.2%) Controller Filter Servomechanism OR(S) Oz (s) C(s) Gf(s) G(s) Figure 4: Block diagram of control system using proportional control. overshoot. To achieve this, complete the following steps: a) Make a copy of your code from experiment 1. Modify your code to create the transfer functions representing the filter Gf(s) as given by equation (10). In this experiment, continue to use the model G(s) = Gnom(s) (equation 8), and feedback transfer function (H(s) = 1). b) Plot the root locus of the system. c) Plot a close-up of the root locus with suitable axes limits. Overlay the 9% overshoot line on the close-up root locus, as before. d) Use interactive selection of the point where the root locus crosses the overshoot line, to retrieve the gain at that point, as well as all of the closed-loop poles for that gain. e) Generate the step response for the system with your determined gain for an 9% overshoot. f) Retrieve and tabulate the characteristics of the actual response. Ensure the overshoot is within 0.2%. If the overshoot is not sufficiently accurate, improve the resolution of the root locus and recollect results. 7 g) Use the root locus to determine the stability region of the system.
Experiment 3: Higher order model, with filter Consider a feedback control system shown in Figure 4 with proportional control, and first order filter given by equation (10). Using Model 1 given in equation (9) for the servomechanism, determine the proportional gain K, which will result in a response with 9% (±0.2%) overshoot. To achieve this, complete the following steps: a) Make a copy of your code from experiment 2. Modify your code to create the transfer function representing model 1 G₁(s) as given by equation (9). In this experiment, continue to use the first order filter G, (s) given by equation (10), and feedback transfer function (H(s) = 1). b) Plot the root locus of the system. c) Plot a close-up of the root locus with suitable axes limits. Overlay the 9% overshoot line on the close-up root locus, as before. d) Use interactive selection of the point where the root locus crosses the overshoot line, to retrieve the gain at that point, as well as all of the closed-loop poles for that gain. e) Generate the step response for the system with your determined gain for an 9% overshoot. f) Retrieve and tabulate the characteristics of the actual response. Ensure the overshoot is within 0.2%. If the overshoot is not sufficiently accurate, improve the resolution of the root locus and recollect results. g) Use the root locus to determine the stability region of the system.