� Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Laboratory Session 5: Elimination of Steady-State Error Using Integral Control Action1 Laboratory Objectives: (i) To investigate the elimination of steady-state error through the use of integral (I), and proportional plus integral (PI) control. (ii) To compare your experimental results with a Simulink digital simulation. Introduction: In the previous laboratory experiments you have noted that there was a steady-state error to a constant angular velocity command, and that the error magnitude depended on the degree of viscous damping present. In many control problems it is desirable to eliminate the steady-state error, and the most common way of doing this is through the use of integral control action and proportional plus integral (PI) control. A PI controller has a transfer function 1 Gc(s) = Kp + Ki s with a block-diagram K-+P I C o n t r o l l e rs e t p o i n tisr ( t )e ( t )Kp++v ( t )cy ( t )and a time domain response t vc(t) = Kpe(t) + Ki e(t)dt 0 where vc(t) is the controller output. A description of how the integral component acts to eliminate steady-state error is given in Appendix A. Please take a few minutes to read through and understand the Appendix. 1October 22, 2007 1The Experimental Setup: The set-up is the same as in Lab. 4, using the 2.004 PID Controller as shown below: Use the tachometer low-pass filter as you did in Lab 4. In this lab, in addition to proportional control you will be using integral control, adjusted by the knob labelled Int. Gain (Ki) on the front panel. In digital control systems such as this, real-time integration is done through an approximate numerical algorithm, such as rectangular integration, where the integral is represented as a sum sn sn = sn−1 + enΔT where en is the error at the nth iteration, and ΔT is the time step, or trapezoidal integration sn = sn−1 + (en−1 + en) ΔT/2 Experiment #1: Verification of Integrator Performance Verify that the integrator is functioning correctly using the following steps: (a) Connect the computer-based controller, but keep the power amp turned off for all parts of this experiment. (b) Set the function generator to produce a step (square) function of amplitude 1 volt, at a frequency of 1 Hz. (c) Open the controller, and select a sampling rate of 100 samples/sec. (Maintain this value for all parts of the lab.) (d) Set Kp = 0 and Ki = 1 on the front panel. Start the controller and observe the error trace. (If the tachometer is noisy, you might want to disconnect it and ground the input). Visually confirm that the error trace is the integral of the input. Either save and plot the output, or make a sketch of it. (e) Add a 0.5 volt offset to the square wave and repeat part (d). (f) Now set a 1 Hz. triangular wave (no offset) on to the function generator and repeat the experiment. Experiment #2: Proportional Control Obtain a “baseline” step response with pro-portional control. Basically repeat the Lab. 4 step response measurement to demonstrate the existence of the steady-state error: 2� � (a) Set Kp = 3, and Ki = 0, with a sampling rate of 100 samples/second. Install one damping magnet. (b) Set the function generator to produce a DC signal of 1 volt magnitude. (c) Record and plot the closed-loop step response, and measure the steady-state error. Experiment #3: Pure Integral Control (a) Now investigate pure integral control by repeating Expt. #2 with Kp = 0, and Ki = 3 so that 3 Gc(s) = . s When using integral control, make sure that the power amp is turned on before starting the controller. This avoids the problem of “integrator wind-up.” Has integral control helped with the steady-state error? Can you tell? What has happened to the transient response? Plot your results. (b) Remove the damping magnet and repeat part (a). Is the response “better” or “worse”. Discuss your results with your lab instructor. Look at the closed-loop characteristic equation from Appendix A, and discuss how the closed-loop roots are affected by the values of B and Ki. In particular, think about what happens to the closed-loop if the viscous damping B = 0. Experiment #4: PI Control: In this experiment, use PI Control, that is with 1 Kps + Ki s + Ki/KpGc(s) = Kp + Ki = = Kps s s (a) Start with Kp = 3, Ki = 1, and a single magnet for damping. Use the same function generator settings, and record and save the step response. (Note – use the pan and zoom tools to select a complete positive step section of the response before saving it to MATLAB.) Is the response more satisfactory? (b) Repeat (a) with Ki = 5 and 10. In each case save the response to MATLAB, and make a plot of the positive step response. (c) Qualitatively examine the effect of integral control by using a finger to add a constant disturbance torque to the flywheel. Observe the controller output (blue/grey trace). Make a note of what happens. Compare your three plots. Briefly describe how the value of Ki has affected 1) any “over-shoot” in the step response, 2) the time to the peak response, and the time to reach the steady-state response. Experiment #5: Compare your results with a Simulink Simulation: Simulink is one of the most widely used computer tools for control system analysis and design. It is an integral part of MATLAB, and is a drag-and-drop block-diagram time-domain simulation 3language. Simulink provides a graphical work-space where you can create very complex sys-tem models without writing a single line of code. Later in this course we will introduce you to programming in Simulink, but for now we provide you with a Simulink model of the lab setup and ask you to run it and compare your experimental step-responses with the Simulink simulation. The figure above shows the ”pre-wired” Simulink simulation for this lab. You can change the values of Kp and Ki by double-clicking on the appropriate block and entering the new value. You can display the ’scope by double-clicking on the icon, and then resizing the window. The input block at the far left is a Simulink step function, so that the simulation will display the closed-loop step-response. Many other functions may be found in the “sources” library. Three signals are “multiplexed” on to the scope (input,
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