ENGR 315Mathematical Models of SystemsConsider the problem of controlling an inverted pendulum on a moving base. See figure below. The design objective is balance the pendulum in the presence of disturbance inputs.ReportReport due next laboratory period.Bode PlotsGain and Phase MarginBandwidth FrequencyClosed-loop performanceENGR 315MATLAB / SIMULINK - Laboratory # 8Mathematical Models of SystemsObjectives: To learn about frequency response methods for designing analyzing and control systems.Equipment:Computer Lab PCResources:1 - Modern Control Systems, Dorf and Bishop2 – MATLAB, Simulink, Control System Toolbox, Class NotesExperiment:Consider the problem of controlling an inverted pendulum on a moving base. See figure below. The design objective is balance the pendulum in the presence of disturbance inputs.Let Ms = 10kg, Mb = 100kg, L=1m, g=9.81m/s^2, a=5, and b=10. The design specifications, based on a unit step disturbance, are.1. settling time (2% criterion) less than 10 seconds2. percent overshoot less than 40%, and 3. steady-state tracking error less than 0.1degree in the presence of the disturbance.Develop a set of interactive Matlab scripts to aid in the control system design. The first script should accomplish at least the following:1. compute the closed-loop transfer function from the disturbance to the output with K as an adjustable parameter2. Draw the Bode plot of the closed-loop system3. Automatically compute and output Mpw and wr.As an intermediate manual step, use Mpw and wr, and Fig. 8.11 in section 8.2 to estimate zeta and wn. The section script should perform at least the following function: estimate the settling time and percent overshoot using zeta and wn, as input variables. If the performance specifications are not satisfied, change K and iterate on the design using the first two script. After completion of the first two steps, the final step is to test the design by simulation. The function of the third script is to1. plot the response of theta (t), to a unit step disturbance with K as an adjustable parameter, and2. label the plot appropriately.Utilizing the interactive scripts, design the controller to meet the specifications using frequency response Bode methods. To start the design process, use analytic methods to compute the minimum value of K to meet the steady-state tracking error specification. Use the minimum K as the first guess in the design iteration. Finally calculate the gain and phase margin for the minimum and final K.ReportSummarize your observations and attach relevant MATLAB scripts, Simulink diagrams and plots.Report due next laboratory period.Bode Plots Bode plot is the representation of the magnitude and phase of G(j*w) (where the frequency vector w contains only positive frequencies). To see the Bode plot of a transfer function, you can use the Matlab bode command. For example, bode(50,[1 9 30 40])displays the Bode plots for the transfer function: 50 / (s^3 + 9 s^2 + 30 s + 40)Gain and Phase Margin Let's say that we have the following system: where K is a variable (constant) gain and G(s) is the plant under consideration. The gain margin is defined as the change in open loop gain required to make the system unstable.Systems with greater gain margins can withstand greater changes in system parameters before becoming unstable in closed loop. Keep in mind that unity gain in magnitude is equal to a gain of zero in dB.The phase margin is defined as the change in open loop phase shift required to make a closed loop system unstable. The phase margin is the difference in phase between the phase curve and -180 deg at the point corresponding to the frequency that gives us a gain of 0dB (the gain cross over frequency, Wgc). Likewise, the gain margin is the difference between the magnitude curve and 0dB at the point corresponding to the frequency that gives us a phase of -180 deg (the phase cross over frequency,Wpc).We can find the gain and phase margins for a system directly, by using Matlab. Just enter the margin command. This command returns the gain and phase margins, the gain and phase cross over frequencies, and a graphical representation of these on the Bode plot. Let's check it out: margin(50,[1 9 30 40])Bandwidth FrequencyThe bandwidth frequency is defined as the frequency at which the closed-loop magnitude response is equal to -3 dB. However, when we design via frequency response, we are interested in predicting the closed-loop behavior from the open-loop response. Therefore, we will use a second-order system approximation and say that the bandwidth frequency equals the frequency at which the open-loop magnitude response is between -6 and - 7.5dB, assuming the open loop phase response is between -135 deg and -225 deg. For a complete derivation of this approximation, consult your textbook. If you would like to see how the bandwidth of a system can be found mathematically from the closed-loop damping ratio and natural frequency, the relevant equations as well as some plots and Matlab code are given on our Bandwidth Frequency page. In order to illustrate the importance of the bandwidth frequency, we will show how the output changes with different input frequencies. We will find that sinusoidal inputs with frequency less than Wbw (the bandwidth frequency) are tracked "reasonably well" by the system. Sinusoidal inputs with frequency greater than Wbw are attenuated (in magnitude) by a factor of 0.707 or greater (and are also shifted in phase). Let's say that we have the following closed-loop transfer function representing a system: 1---------------s^2 + 0.5 s + 1First of all, let's find the bandwidth frequency by looking at the Bode plot: bode (1, [1 0.5 1 ])Since this is the closed-loop transfer function, our bandwidth frequency will be the frequency corresponding to a gain of -3 dB. looking at the plot, we find that it is approximately 1.4 rad/s. We can also read off the plot that for an input frequency of 0.3 radians, the output sinusoid should have a magnitude about one and the phase should be shifted by perhaps a few degrees (behind the input). For an input frequency of 3 rad/sec, the output magnitude should be about -20dB (or 1/10 as large as the input) and the phase should be nearly -180 (almost exactly out-of-phase). We can use the lsim command to simulate the response of the system to sinusoidal inputs. First, consider a sinusoidal input with a frequency lower than Wbw. We must also keep in mind that we want to
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