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U of M EE 4237 - Gyroscopic Torque Contro

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Procedure Successive LoopOuter Loop Pole PlacementFull State FeedbackDEPARTMENT OF ELECTRICAL ENGINEERING UNIVERSITY OF MINNESOTA EE 4237 State Space Control Laboratory Experiment 8: Gyroscopic Torque Control: Full State LQR Objective: 1. To study the LQR design and control implementation of Gyroscope system References: 1. ECP systems manual (Model 750) Apparatus: 1. Gyroscope Model 750 2. PC 3. Control Box Prelab Report: 1. What are the design steps in any LQR design? 2. Go through the given program and try to explain what is the functionality of the program. Postlab Report: • Answer the questions asked within or at the end of the procedure.ecpChapter 6. Experiments 1056.6 Gyroscopic Torque Control: Full State Feedback Linear Quadratic Regulator In this section, we employ a linear quadratic regulator (LQR) to the system using full state feedback1 for the control of the assembly about Axis 4. The block diagram of the scheme is given in Figure 6.6-1. As in the previous two sections, all tests in this section are performed with the apparatus in the configuration of Figure 6.6-2. kpfTime DomainLaplace DomainControlEffortu(t)k1k3k2q4Σr(t)-1Planty = Cxx==AAx ++BBuErrore(t)N2(s)D(s)k1+ k3sN4(s)N2(s)k2sΣkpfControlEffortu(t)Σr(t)-Errore(t)ω4ω21Σq4(s)q2(s) Figure 6.6-1. LQR Control Scheme 1 In a strict sense, the velocity states are observed rather than measured directly since they result from differentiation of the position feedback signals. In practice, however, the position signal resolution and sample rate are high enough that the rate signal behaves as a direct measurement.ecpChapter 6. Experiments 106EncoderBrakeCapstanPulleyGimbal Angles: q2o = 0, q3o = 0Axis 4 Brake: OFFAxis 3 Brake: ONAxis 2 Virtual Brake: OFF Figure 6.6-2. Configuration For Tests In This Section In order for the synthesis to converge to a unique solution and for there to be no unobservable or uncontrollable states, the plant model must be minimal. In the present case, the plant is 3rd order and therefore three states are selected. The equations of motion for this system involve time derivatives of positions about Axes 2 and 4. Because we are controlling motion about Axis 4, we select two states as q4 and ω4. Furthermore since the system is actuated via rate about Axis 2, we choose ω2 as the third state. The state and output vectors then become X = q4ω2ω4 , C = 100 (6.6-1) where regulation of the position about axis 4 is the control objective and hence is the only state included in C for control synthesis purposes. Thus the plant model is that of Eq.(5.4-13a) properly scaled for the control effort and encoder gains as was done in Section 6.1. (i.e. the model of Exercise E for the minimal realization of Special Case #2) 6.6.1 Design & Control Algorithm. 1. The following notation shall be used for LQ optimization: Feedback law: u= -Kx (6.6-2) where K=k1k2k3 (6.6-3)ecpChapter 6. Experiments 107 Perform LQR synthesis via the Riccati equation solution2 or numerical synthesis algorithms to find the controller K that minimizes the cost function (scalar control effort): J = x'Qx +u2rdt (6.6-4) In this synthesis choose Q=C'C so that the error at the intended output, q4, is minimized subject to the control effort cost. Perform synthesis for control effort weight values: r = 100, 10, 1.0, and 0.01. Calculate the closed loop poles for each case as the eigenvalues of [A–BK] 2. From this data, select a control effort weight to put the lowest pole frequency between 2.25 and 2.75 Hz. Use one of the above obtained K values if it meets this criteria, or interpolate between the appropriate r values and perform one last synthesis iteration. Do not use k1 values greater than 6, or k2 values greater than 0.08, or k3 values greater than 0.253 3. Calculate the scalar prefilter gain kpf by referring to Figure 6.6-1. As in the previous section, the goal is to have the output q4(s) scaled equal to the input r(s). 4. Write a suitable real-time algorithm that implements the control scheme of Figure 6.6-1 using your results from Steps 1, 2 and 3. Use Ts = 0.00884 s. Have your instructor or laboratory supervisor review and approve your routine before proceeding. 6.6.2 Control Implementation and Characterization Procedure 5. Setup the apparatus as per Figure 6.6-2 and initialize the rotor speed to 400 RPM. 6. Implement your algorithm from Step 3 making sure you set the sample period to 0.00884 sec. Safety check the system and verify that it is regulating about Axis 4. If not check your algorithm and repeat the above procedures. In all trials below, use a ruler or similar object to perturb the assembly about Axis 4 so that the rotor disk is oriented vertically before executing any of the maneuvers. 7. Setup to collect Commanded Position 1, Control Effort 2, Encoder 2 Position and Encoder 4 Position data. Setup Trajectory 1 as follows: Step input, 500 count Amplitude, 1000 ms. Dwell Time, 1 Repetition. Execute this trajectory and plot the Commanded Position 1 and Encoder 4 Position data on the left axis, and Control Effort 2 data on the right. Repeat this procedure for a 200 count step. Look closely at the shape of the curves in the first 250 ms. Is there a difference between the two cases? If so can you explain why? Save your plots. 8. Setup a Sine Sweep input with Amplitude = 50 counts, Start Frequency = 1.0 Hz, End Frequency = 10 Hz, Sweep Time = 30 sec, and Logarithmic Sweep checked. Execute 2See for example Kwakernaak and Sivan, "Linear Optimal Control Systems", Wiley & Sons, 1972. 3“k1“ scales control effort proportional to position error and, k2 and k4 scale control effort proportional to the respective velocities. Excessive values of k1 can lead to low stability margin and in the presence of time delays, instability. Large k2 or k3 cause excessive noise propagation and lead to "twitching" of the system.ecpChapter 6. Experiments 108this and plot the Encoder 4 data in Log(ω), dB format with Remove DC bias checked. Save your plot 9. Setup a Ramp input with Unidirectional Moves not checked, Distance = 6000 count, Velocity = 3000 counts/sec4, Dwell Time = 1000 ms, 2 Repetitions. Execute this trajectory and plot the Commanded Position 1 and Encoder 4 Position data. Execute this maneuver for your nominal design and plot the Commanded Position 1 and Encoder 4 Position data. You may also wish to


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