EE 4237- State Space Control Laboratory University of MinnesotaDEPARTMENT OF ELECTRICAL ENGINEERING UNIVERSITY OF MINNESOTA EE4237 State Space Control Laboratory The first experiment is to introduce the use of Matlab and simulink in control system design. Students will learn the basics of Matlab software which is convenient and rapid solution of problems in numerical linear algebra. The second set of experiments will introduce DC motor control concepts using Quanser's QET DC Motor Control Trainer(DCMCT). The DC Motor Control Trainer consists of a motor instrumented with an encoder, and the motor is driven my a low-noise linear power amplifier. QET DCMCT provides a way to demonstrate the fundamentals of motor control, tuning and haptics using several integrated experiments. The third set of experiments will be based on a magnetic levitation apparatus manufactured by ECP. This consists of two electro-magnets used to levitate two magnets in the vertical space between them. The levitated magnets are constrained by a guiding rod which provides lateral stability while feedback via use of laser distance sensors and actuation through control of the electromagnets is used to stabilize and control their vertical motion. The fourth set of experiments will be based on torsional and inertial apparatus. This consists of a set of disks rigidly connected to a vertical flexible rod. The position and motion of the disks is controlled by feedback via optical encoders and torque applied by an electric motor. The fifth set has an introduction to another apparatus by ECP systems and that is the Gyroscope. Important concepts like nutation, precession will be studied. And the remaining set of experiments are related with inverted pendulum. Inverted pendulum accessory by ECP systems is attached to the torsion control system and control software will be required to invert the pendulum. Students will also try to balance the two-wheel robot made from LEGO mindstorms NXT.TABLE OF CONTENTS EXPERIMENT 0 Introduction to Matlab, Simulink, state feedback control and nonlinear state simulation (1 week) EXPERIMENT 1 DC motor control using DCMCT (2 weeks) EXPERIMENT 2 Magnetic Levitation System: Control of nonlinear compensated SISO system EXPERIMENT 3 Magnetic Levitation System: Multi input multi output control EXPERIMENT 4 Torsion Control System: Dynamic parameter identification EXPERIMENT 5 Torsion Control System: PD/PID control of rigid body dynamics EXPERIMENT 6 Torsion Control System: LQR control (1 week) EXPERIMENT 7 Control Moment Gyroscope: Nutation and Precession (1 week) EXPERIMENT 8 Control Moment Gyroscope: Full state LQR (1 week) EXPERIMENT 9 Inverted Pendulum using Torsion Control System (1 week) EXPERIMENT 10 Balancing robot using LEGO Mindstorms NXT (1 week) Notes: (3 weeks for experiments 2 and 3 and 3 weeks for experiments 4 and 5)DEPARTMENT OF ELECTRICAL ENGINEERING UNIVERSITY OF MINNESOTA EE 4237 EXPERIMENT 0: Introduction to Matlab, Simulink, state feedback control and nonlinear state simulation. Purpose: To introduce the use of Matlab in control system design. References: 1. PC-MATLAB User’s guide 2. Control System Toolbox User’s guide Equipment: IBM PC/AT with Printer Introduction: Matlab is a software system for doing numerical mathematics. The basic Matlab software is structured for convenient and rapid solution of problems in numerical linear algebra. In addition there are a number of toolboxes, which adapt Matlab to special tasks. We will study the basic Matlab package and the Control System Toolbox. The easiest way to get started on Matlab is to run demos 1 and 2 of the Matlab DEMO program, run the Matlab HELP programs and then start playing around with simple examples. Prelab: Look at the attached “help step” and “help simulink” printouts from Matlab. Procedure: 1. The basic Matlab system a. Throughout this and the other experiments the symbol <cr> means the ENTER or RETURN key on the keyboard. b. At the DOS prompt type matlab<cr> and then wait for the Matlab prompt, which is >>. At the Matlab prompt type demo<cr>. Follow the instructions to observe Demos 1) and 2). (Observe others if you want) c. At the Matlab prompt type help<cr> and note the available operators and functions. Next type help sin<cr> to see the type of help available for individual functions. Next try others. d. Enter a vector by typing x = [1 2 3]. Next type x’. Now create a matrix by typing A = [x’ x’ x’]. Find determinant of this matrix. Enter another matrix A that is not singular. Calculate A * x. Solve A * y = x for y. e. Generate the vector t = 0:10. Repeat for t = 0:0.1:1. f. Generate the logspaced vector w = logspace(-1,2) and note the entries. Read help logspace.2. Matlab Graphics a. Plot several cycles of a sinusoid. Put axis labels and a title on your plot. b. Define a random vector and plot it. c. Define two random vectors x and y and make a point plot of x vs y. d. Print out one of your plots on printer. 3. The control system toolbox a. Type ctrldemo<cr> and observe the demo. b. Type help rlocus<cr>, help bode<cr> and help step<cr> c. Plot the locus of the roots of (s3 + 6s2 + 5s) + K (s2 + 6s + 18) = 0 for 0 ≤ K≤ 200. On your plot indicate roots locations for K = 10. Re-scale your root locus plot showing only the second quadrant. (Hint: see help axis. Let (s2 + 6s + 18) as num and (s3 + 6s2 + 5s) as den. ) 4. Control system Example: A unity feedback control system has loop gain function L(s)= 200 (s + 5) /[s (s+1) (s+20)(s + 30)]. a. Plot a bode plot for L (jw). b. Plot a Nyquist plot for L (jw). c. Plot the closed loop poles and zeros on the sgrid (Type help sgrid<cr>.) d. Plot the closed loop step response. Put on axis labels and a title and make a hardcopy. Put your name and date on your plot using the Matlab function text (see help text <cr>.) e. Generate a discrete time transfer function L (z) using the Toolbox function c2d (do help c2d <cr>). Use a sampling period Ts = 0.01 sec. f. Plot the poles and zeros of the closed loop discrete time system on zgrid. 5. Simulink Example 1: a. Wire simulation for two-integrator feedback system shown in fig 0.1. b. Run simulation. c. Plot unit step response. d. Replace first integrator by the transfer function 2 / (s + 3) as shown in fig 0.2 and select the two feedback gains a and b to place closed loop poles at -1 + j2 and -1 - j2. e.