# SRV02-Series Rotary Experiment # 1. Position Control. Student Handout

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1 SRV02-Series Rotary Experiment # 1 Position Control Student Handout

2 SRV02-Series Rotary Experiment # 1 Position Control Student Handout 1. Objectives The objective in this experiment is to introduce the student to the fundamentals of control using the PID family of compensators. At the end of this session, you should know the following: How to mathematically model the servo plant from first principles. An understanding of the different tuning parameters in the controller. To design and simulate a PV controller to meet the required specifications. To implement your controller and evaluate its performance. 2. System Requirements To complete this lab, the following hardware is required: [1] Quanser UPM 2405/1503 Power Module or equivalent. [1] Quanser MultiQ PCI / MQ3 or equivalent. [1] Quanser SRVO2-(E) servo plant. [1] PC equipped with the required software as stated in the WinCon user manual. The suggested configuration for this experiment is the SRV02-E(T) in the High-Gear configuration with a UPM 2405 power module and a gain cable of 5. It is assumed that the student has successfully completed Experiment #0 of the SRVO2 and is familiar in using WinCon to control the plant through Simulink. It is also assumed that all the sensors and actuators are connected as per dictated in the Experiment #0 as well as the SRVO2 User s Manual. Page # 2 Revision: 01

3 3. Mathematical Model This section of the lab should be read over and completely understood before attending the lab. It is encouraged for the student to work through the derivations as well as to get a thorough understanding of the underlying mechanics. For a complete listing of the symbols used in this derivation as well as the model, refer to Appendix A - SRV02 Nomenclature at the back of this handout. We shall begin by examining the electrical component of the motor first. In Figure 1, you see the electrical schematic of the armature circuit. Figure 1 - Armature circuit in the time-domain Using Kirchhoff s voltage law, we obtain the following equation: [3.1] Since L m << R m, we can disregard the motor inductance leaving us with: [3.2] We know that the back emf created by the motor is proportional to the motor shaft velocity such that: [3.3] We now shift over to the mechanical aspect of the motor and begin by applying Newton s 2 nd law of motion to the motor shaft: [3.4] Page # 3 Revision: 01

4 Where is the load torque seen thru the gears. And is the efficiency of the gearbox. We now apply the 2 nd law of motion at the load of the motor: [3.5] Where B eq is the viscous damping coefficient as seen at the output. Substituting [3.4] into [ 3.5], we are left with: [3.6] We know that and (where is the motor efficiency), we can re-write [3.6] as: [3.7] Finally, we can combine the electrical and mechanical equations by substituting [3.3] into [3.7], yielding our desired transfer function: [3.8] Where: This can be interpreted as the being the equivalent moment of inertia of the motor system as seen at the output. Page # 4 Revision: 01

5 3.1 Pre-Lab Assignment This lab involves designing a PV controller for the servo plant. We have omitted the integral gain (PID), as the main purpose of the integral gain is to reduce the steady state error by introducing a pole located at s = 0 in the open loop. Looking back at the derived equation [3.8], we can clearly see that the plant already has a pole at s = 0. For this reason, and for the sake of simplicity, the required specifications will be met using only a proportional (P) and velocity/derivative (V) controller. In the classical sense, a PD controller would have the form: C(s) =K p + K d s. Placing this controller into the forward path would result in introducing an unwanted zero in the closed loop transfer function. As a result of the zero, the closed loop transfer function no longer fits equation [3.10] and it becomes increasingly difficult to design the controller to meet time specifications. With this limitation in mind, we can now make the transition to use a state-feedback (PV) approach that can meet our requirements and results in a closed loop transfer function of the form seen in equation [3.10]. The PV controller that will be implemented in this lab can be seen in Figure 2 below and has the form: [3.9] Figure 2 - PV Controller for the SRV02 Plant Page # 5 Revision: 01

6 The purpose of this lab is to design a controller using the 2 nd order transfer function representation of the form: [3.10] with characteristic equation: [3.11] *Coincidentally, the characteristic equations of the PV and PD controller closed loop transfer functions are equal. A PV controller in essence is a PD controller without the unwanted zero, allowing the designer to meet the required specifications using only the characteristic equation. 1) Obtain the transfer function of the closed loop model in Figure 2. Extract the characteristic equation and fit it to the form seen in equation [3.11]. Obtain 2 equations expressing and as functions of K p and K v as these are the only 2 variables in your system. Using your newly obtained formulas and referring to your in-class notes, what changes to your response would you expect to see by varying the values of K p and K v? (What happens to when you increase/decrease K p?) (What happens to when you increase/decrease K v and/or K p?) Keep your answers simple. (i.e. will and increase or decrease?) 2) For the in-lab portion of this experiment, you are required to design a PV controller that will yield the following time requirements: The Overshoot should be less than 5% ( 0.707). The time to first peak should be 100ms (T p = 0.100). Using the formulas from Question 1, choose values for K p and K v to meet the requirements. *Hint: [3.12] Page # 6 Revision: 01

9 Figure 3 - Step Response to a Command of 30 degrees We can see by looking at Figure 3, that the position response has a 100ms time to peak and an overshoot of less than 5%. The system requirements have been met and implemented using a PV controller. Page # 9 Revision: 01

11 Appendix - A: SRV02 Nomenclature Symbol Description MATLAB Variable Nominal Value SI Units V m I m Armature circuit input voltage Armature circuit current R m Armature resistance Rm 2.6 L m E emf Armature inductance Motor back-emf voltage Motor shaft position Motor shaft angular velocity Load shaft position Load shaft angular velocity T m T l Torque generated by the motor Torque applied at the load K m Back-emf constant Km K t Motor-torque constant Kt J m Motor moment of inertia Jmotor 3.87 e-7 J eq Equivalent moment of inertia at the load Jeq 2.0 e-3 B eq Equivalent viscous damping coefficient Beq 4.0 e-3 K g SRV02 system gear ratio (motor->load) Kg 70 (14x5) Gearbox efficiency Eff_G 0.9 Motor efficiency Eff_M 0.69 Undamped natural frequency Damping ratio Wn zeta K p Proportional gain Kp K v Velocity gain Kv T p Time to peak Tp Page # 11 Revision: 01

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