LABORATORY GUIDE 3 DOF Gyro Gyrosco scope pe Experi Experimen mentt for MATLA MATLAB B /Simu Simulink link Use Users rs Developed by: Jacob Apkarian, Ph.D., Quanser Amirpasha Javid, B. Eng., Quanser
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PREFACE Preparing laboratory experiments can be time-consuming. Quanser understands time constraints of teaching and research professors. That’s why Quanser’s control laboratory solution s come with proven practical exercises. The courseware is designed to save you time, give students a solid understanding of various control concepts and provide maximum value for your investment. Quanser 3 DOF Gyroscope courseware materials are supplied in a format of the Laboratory Guide. The Lab Guide contains lab assignments for students.
This courseware sample is prepared for users of The MathWorks’ MATLAB /Simulink software in conjunction with Quanser’s QUARC real-time control software. A version of the curriculum for National Instruments LabVIEW™ users is also available.
The following material provides an abbreviated example of in-lab procedures for the 3 DOF Gyroscope experiment. Please note that the examples are not complete as they are intended to give you a brief overview of the structure and content of the courseware materials you will receive with the plant.
TABLE OF CONTENTS
PREFACE ......................................................................................................................
PAGE 1
INTRODUCTION TO QUANSER 3 DOF GYROSCOPE COURSEWARE SAMPLE ............. PAGE 3 LABORATORY GUIDE TABLE OF CONTENTS ................................................................ PAGE 3 BACKGROUND SECTION – SAMPLE ............................................................................ PAGE 4 LAB EXPERIMENTS SECTION – SAMPLE ...................................................................... PAGE 5
1. INTRODUCTION TO Q UANSER 3 DOF GYROSCOPE COURSEWARE SAMPLE Quanser courseware materials provide step-by-step pedagogy for a wide range of control challenges. Starting with the basic principles, students can progress to more advanced applications and cultivate a deep understanding of control theories. Quanser 3 DOF Gyroscope courseware covers topics, such as: Obtain a state-space representation of the o pen-loop system Design a state-feedback gain for the closed-loop system using the Linear-Quadratic Regulator (LQR) optimization Simulate the system and ensure it is stabilized using the designed state-feedback control Implement your state-feedback controller on the 3D GYRO system and evaluate its actual performance
2. LABORATORY GUIDE TABLE OF CONTENTS The full Table of Contents of the Quanser 3 DOF Gyroscope Laboratory Guide is shown here:
1. INTRODUCTION 2. BACKGROUND 2.1. MODELING 2.1.1. MODEL CONVENTION 2.1.2. EQUATIONS OF MOTION 2.1.3. LINEAR STATE -SPACE MODEL 2.2. CONTROL 2.2.1. STABILITY 2.2.2. CONTROLLABILITY 2.2.3. LINEAR Q UADRATIC REGULAR (LQR) 2.2.4. FEEDBACK CONTROL 3. LAB EXPERIMENTS 3.1. SIMULATION 3.1.1. PROCEDURE 3.2. IMPLEMENTATION 3.2.1. PROCEDURE 3.2.2. ANALYSIS 4. SYSTEM REQUIREMENTS 4.1. OVERVIEW OF FILES 4.2. SETUP FOR SIMULATION 4.3. SETUP FOR RUNNING ON 3 DOF GYROSCOPE REFERENCES
3. BACKGROUND SECTION - SAMPLE Equations of Motion
The equations of motion representing the angular rate of the red gimbal, Ψ, and the outer blue gimbal, , are ([1]):
where
(2.1)
2
Jy = 0.0036 kgm 2 Jz = 0.0226 kgm 2 h = 0.44 kgm /s
The moment of inertia about the y -axis angle, , is Jy and the moment of inertia about the z-axis angle (red gimbal), Ψ , is denoted as Jz. The constant h is calculated based on the moment of inertia of the gyroscope rotor about its own axis and its velocity. Be cause the outer gray rectangular fr ame is fixed, the only actuated axis is the y-axis. The control input in the single-input single-output (SISO) system is the torque applied in the y-axis, τy .
Linear State-Space Model The linear state-space equations are
and
where x is the state, u is the control input, A, B, C , and D are state-space matrices.
(2.2) (2.3)
For this system, the state and output are defined
and
Solving for the acceleration terms in the linear equations of motion given in Equation 2.1 we get
Substituting the state in , we obtain the following state-space matrices:
and
(2.4)
4. LAB EXPERIMENTS SECTION - SAMPLE Simulation In this section we will use the Simulink diagram shown in Figure 3.1 to simulate the c losed-loop control of the 3 DOF Gyroscope system. The system is simulated using the model summarized in Sect ion 2.1. The Simulink model uses the state-feedback control described in Se ction 2.2.4. The feedback gain K is found using the Matlab LQR command (LQR is described briefly in Section 2.2.3).
Table 3.1: Simulink model used to simulate the 3 DOF Gyroscope.
IMPORTANT: Before you can conduct these experiments, you need to make sure that the lab files are configured according to your setup. If they have not been configured already, t hen you need to go to Section 4 to configure the lab files first. Procedure Follow these steps to simulate the 3 DOF Gyroscope: 1. Make sure the LQR weighting matrices in setup_3dgyro.m are set to
and R=5 2. Run the script to generate the gain
Remark: When tuning the LQR, Q(2 ; 2) effects the red gimbal proportional gain while Q(1 ; 1) effects the red gimbal derivative gain (which reduces the overshoot). Q(4 ; 4) affects the red gimbal integral gain which is used to minimize the steady state e rror. 3. Open the red gimbal position scope, Psi (deg), the torque scope, Torque (N.m). 4. Start the simulation. The scopes should be displaying responses similar to Figure 3.2.
Figure 3.2: Simulated closed-loop response.
Implementation The q_3dgyro_lqr Simulink diagram shown in Figure 3.3 is used to perform the red gimbal position control on the 3 DOF Gyroscope. The HIL Interfacing subsystem contains drivers that interface with the DC motor and sensors of the 3 DOF Gyroscope system.
Figure 3.3: Simulink model used to run controller on the 3 DOF Gyroscope.
IMPORTANT: Before you can conduct these experiments, you need to make sure that the lab files are configured according to your setup. If they have not been configured already, then you ne ed to go to Section 4 to configure the lab files first.
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