DEPARTMENT OF ELECTRICAL ENGINEERING UNIVERSITY OF MINNESOTA EE 4237 State Space Control Laboratory Experiment 3: Multi-input, multi-output control Objective: 1. To study the MIMO control of Magnetic levitation system. References: 1. ECP systems manual (Model 730) Apparatus: 1. MagLev model 730 2. PC 3. Control Box 4. Magnets, Glass rod Comments: The plant model has two inputs and two outputs. Hence we have to apply MIMO control. The design requires LQR methodology. You can use Matlab to obtain values and explain steps of the design. A sample program is provided at the end. You will use it to experiment with and possibly modify. 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 104 6.8 MIMO Control In this experiment, full MIMO control is implemented on the two magnet system. The physical plant configuration is again that of the previous two sections except that now we apply inputs at both the upper and lower magnet locations. We shall implement a linear quadratic regulator using full state feedback where the states are defined to be the position and velocity of each magnet (i.e. those of Eq.(5.3-15)). The outputs are taken as y1 and y2; i.e. the output matrix is C = 10000010 (6.7-1) Here we employ the sensor and actuator nonlinearity compensation of the previous sections. The plant model was previously generated in Section 6.1. Recall that the underlying assumption in this model is that the sensor and actuator nonlinearity compensation functions are sufficiently near the inverse of the underlying plant nonlinearities that the new “plant” may be modeled as being linear for control design purposes. Recall also that there is no nonlinear compensation for the effective “spring” k12’. It is modeled as a linear spring whose value was derived by linearization about the operating point {y1ο,y2ο}. In the experiment that follows, we shall first implement SISO controllers (from previous sections) for the upper and lower magnets simultaneously and compare the resulting system performance with that of the MIMO design. Block diagram depictions of the full state feedback multivariable system is shown in Figure 6.8-1.ecpChapter 6. Experiments 105LQR Design: 1. Construct a state space model of the plant using the realization form of Eq (5.3-15) for the plant with nonlinear sensor and actuator compensation. Be sure to include the system gain, ksys, in your model. (You may have already generated this model as an exercise in Section 6.1) 2. The following notation shall be used for LQ optimization: Feedback law: u= -Kx (6.7-2) where K = k11k12k13k14k21k22k23k24 (6.7-3) Perform LQR synthesis via the Riccati equation solution1 or numerical synthesis algorithms to find the controller K which minimizes the cost function: J = x'Qx +u'Ru dt (6.7-4) In this synthesis choose Q=C'C so that the error at the intended outputs, y1 and y2, is minimized subject to the control effort cost R. Because of symmetry of the system choose R=rI where I is the 2x2 identity matrix and r is scalar. Perform synthesis for control effort weight values: r = 10, 1.0, 0.1, and 0.01. Calculate the closed loop poles for each case as the eigenvalues of [A–BK] 3) From this data, select a control effort weight to put the lowest pole frequency between 5.75 and 6.25 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..2 1See for example Kwakernaak and Sivan, "Linear Optimal Control Systems", Wiley & Sons, 1972. 2K1 ki1 and ki3K3 (i=1,2) scale control effort proportional to the respective position errors, ki2K2 and ki4K4 scale control effort proportional to the respective velocities. Excessive values of K1 or K3 can lead to low stability margin and in the presence of time delays, instability. Large K2 or K4 cause excessive noise propagation and lead to "twitching" of the system. - see Section 6.8 Generally, it is desirable to obtain the highest performance possible (e.g. best tracking, regulation, and disturbance rejection) through high gain subject to the above gain limitations. For the present system, gains that provide approximately 6 Hz closed loop bandwidth represent a reasonable trade between performance and noise and stability.ecpChapter 6. Experiments 1064) Generate a real-time routine that implements the above LQR design and includes the following. a) Nominal operating points y1ο and y2o at 1.0 and –2.0 cm respectively. b) Sample period Tσ=0.001768 s. c) For tracking, the prefilter gain kpf1 must be set equal to k11 +k13 and kpf2 set equal to k21 +k23.. Control Implementation Simultaneous SISO 5) Consider the simultaneous implementation of the two PD controllers with sensor and actuator nonlinearity compensation from Section 6.3. This identical configuration was used in Section 6.5 to control the upper magnet and thereby impart a disturbance to the lower one for the study of disturbance rejection. Modify this routine as follows: a) Set the nominal operating points to y1ο = 1.0 cm and y2o = –2.0 cm. b) Adjust the gravity offset parameters u1ο and u2o to account for the constant force due to k12 imposed on each magnet at the nominal separation distance y12o. c) Calculate and enter control gains such that ωn = 6 Hz and ζ=0.707 for each sensor/actuator/magnet system. d) Use cmd1_pos for the lower magnet reference input and cmd2_pos for the upper. e) Set the variables q10 =y1* and q12 = y2* for later data collection and plotting. Have your laboratory supervisor or instructor review your algorithm before proceeding. 6) Use the plastic safety clip so that the magnet rests at approximately –2.5 cm. Implement this routine and if the controller is stable and functioning properly (e.g. the magnets are levitating at roughly their proper heights) remove the plastic clip. Setup the following multivariable input: Trajectory 1: Step input, 15000 count amplitude, 1000 ms dwell, 4 repetitions, unidirectional; Trajectory 2: Step input, 15000 count amplitude, 1000 ms dwell, 4 repetitions, bidirectional (deselect unidirectional). Execute these two trajectories with a 500 ms delay of