1. Nonlinear Control of MechatronicSystemsCLEMSON U N I V E R S I T Y Darren Dawson McQueen Quattlebaum Professor Electrical and Computer Engineering

2. Research Overview Applications and Areas of Interest Key Elements of the Research Program A Motivating Example The Flexible Rotor Problem Introduction and Problem Formulation Motivation for Control Design Control Structure Experimental Results Administrative Plans Academic Qualifications Departmental Goals Attaining the Goals Overview of Presentation PART 1 PART 2 PART 3

3. Applications and Areas of Interest Mobile Platforms UUV, UAV, and UGV Satellites & Aircraft Automotive Systems Steer-By-Wire Thermal Management Hydraulic Actuators Spark Ignition CVT Mechanical Systems Textile and Paper Handling Overhead Cranes Flexible Beams and Cables MEMS Gyros Robotics Position/Force Control Redundant and Dual Robots Path Planning Fault Detection Teleoperation and Haptics Electrical/Computer Systems Electric Motors Magnetic Bearings Visual Servoing Structure from Motion Nonlinear Control and Estimation

4. The Mathematical Problem Typical Electromechanical System Model Classical Control Solution Obstacles to Increased Performance –System Model often contains Hard Nonlinearities –Parameters in the Model are usually Unknown –Actuator Dynamics cannot be Neglected –System States are Difficult or Costly to Measure

5. Nonlinear Lyapunov-Based Techniques Provide –Controllers Designed for the Full-Order Nonlinear Models –Adaptive Update Laws for On-line Estimation of Unknown Parameters –Observers or Filters for State Measurement Replacement –Analysis that Predicts System Performance by Providing Envelopes for the Transient Response The Mathematical Solution or Approach Mechatronics Based Solution Transient Performance Envelopes

6. Control Design/Implementation Cycle Testbed Construction Sensors: Encoders, Force Sensor, Camera Actuators: Motors, Electromagnets, Speakers Software Development QMotor 3.0 (QNX, C++) RTLT 1.0 (RT-Linux, Simulink ) Mathematical Model PDE-ODE model (flexible systems) ODE model (rigid systems) Stability Analysis Lyapunov Techniques Simulation Studies Model-Based, Adaptive, Robust Hamilton’sPrinciple, Newton’s Law ControlObjective Problem Formulation Tracking, Setpoint Parametric Uncertainty Bounded Disturbance Unmeasurable Signals ControlDesign Data Acquisition MultiQ, ServoToGo I/O Board (encoders, D/A, A/D, digital I/O) Real-Time OS, Driver Interface, Data Handling Signal Conditioning Linear Power Amplifiers OPAMPS (gains, offsets) Interface and Safety Issues ElectronicCompatibility Coding the Control Algorithm Master Thesis Students PhD Students

7. Motivating Example (Model Known) Dynamics: Mass Nonlinear Damper DisturbanceVelocity Control Input a, b are constants Tracking Control Objective: Open Loop Error System: Control Design: Closed Loop Error System: Solution: Feedforward Feedback Assume a, b are known Drive e(t) to zero Exponential Stability

8. Motivating Example (Unknown Model) Open Loop Error System: Control Design: a, b are unknown constants Same controller as before, but and are functions of time How do we adjust and ? Use the Lyapunov Stability Analysis to develop an adaptive control design tool for compensation of parametric uncertainty Closed Loop Error System: At this point, we have not fully developed the controller since and are yet to be determined. parameter error

9. ( is UC) Motivating Example (Unknown Model) Fundamental Theorem effects of conditions i) and ii) i) If ii) If is bounded iii) If is bounded satisfies condition i) finally becomes a constant Non-Negative Function: Time Derivative of V(t): is bounded examine condition ii) design and substitute the dynamics for constant effects of condition iii)

10. Motivating Example (Unknown Model) Substitute Error System: How do we select and such that ? Update Law Design: Substitute in Update Laws: and Fundamental Theorem is bounded all signals are bounded Fundamental Theorem Feedforward Feedback control structure derived from stability analysis control objective achieved is bounded

11. Boundary Control of a Flexible Rotor System

12. Overview of Part II – Flexible Rotor Control Problem Examples of Flexible Systems Background on Flexible Systems Research Flexible Rotor Problem Formulation Comparison to Previous Work Flexible Rotor System Model Control Objectives Heuristic Design of Control Model-Based Boundary Controller Adaptive Control Redesign Experimental Results Concluding Remarks

13. Space-Based Systems that Vibrate Long-Reach Robot Manipulators often Exhibit Vibration Aircraft Wings may Exhibit Vibration Other Light-Weight Components on Space Probes may Vibrate Cassini : Mission to Saturn

14. What is the Problem ? Mechanical systems containing flexible parts are subject to undesirable vibrations under motion or disturbances. Mathematically, these hybrid systems are composed of rigid and flexible subsystems that are described by –a ordinary differential equation (ODE) subsystem, –a partial differential equation (PDE) subsystem, and –a set of boundary conditions (static or dynamic) Control design for hybrid systems is complicated due to –the i nfinite dimensional nature of the PDE subsystem –the nonlinearities associated with hybrid systems, and –the coupling between the PDE and ODE subsystems Problem Model Challenge

15. Hybrid System (PDE+ODE) Based on a Linear/Discrete Model Distributed Control Linear Control Boundary Control Requires large number of sensors and actuators or smart structures Difficult and costly to implement Uses infinite dimensional system model (no spillover) Simple control structure Requires very few actuators/sensors Can excite unmodeled high-order vibration modes (spillover) Yields a controller that might require a high order observer (robustness problems) Advantages Disadvantages How are Flexible Systems Controlled ? Disadvantages

16. What is Boundary Control ? Heuristically, boundary control involves the design/use of virtual dampers to reduce the vibration associated with flexible components Virtual damping can be applied to the end of the rotor via a magnetic bearing The nonlinearities and the coupling between the rigid/flexible subsystems mandate the design of a nonlinear damper-like scheme Flexible Rotor Virtual Dampers Applied Torque Virtual Dampers suck the energy out of the system Rotor at rest A Lyapunov-type analysis is used to derive the structure of the nonlinear damper-like control scheme

17. Rotor Displacement Rotor Displacement The Flexible Rotor Problem Rotating Disk Actuator Mass   (t) Flexible Rotor Boundary Control Torque Input Boundary Control Force Inputs Control Objective : Drive u(x,t) and v(x,t) to zero and force to track  d (t) x u(x,t)  (t) CutawayViewCutawayView x u v  x v(x,t)

18. Comparison To Previous Work Morgul (1994), Laousy (1996) - [1-D Problem] –Exponentially stabilized the system with a free-end boundary control force –Desired angular velocity setpoint had to be sufficiently small –Neglected the disk and free-end dynamics (Morgul) –Neglected the free-end dynamics (Laousy) Proposed Control - [2-D Problem] –Exponentially stabilizes the system with a free-end boundary control force –No magnitude restrictions on the desired angular velocity –Includes both the disk and free-end dynamics (Includes Nonlinearities & Coupling) –Controller provides for angular velocity tracking –Redesigned adaptive controller compensates for parametric uncertainty Displacement confined to 1-D Rotation 1-D Problem Neglects Nonlinearities & ODE/PDE Coupling

19. 2-D Flexible Rotor Model Field Equation (PDE Subsystem - Euler Bernoulli Model) Boundary Conditions where Disk Dynamics (ODE Subsystem: J - Disk Inertia) EI -bending stiffness &  mass per unit length Free-End Dynamics (ODE Subsystem: m - actuator mass ) Beam is clamped at the disk No applied Torque at the Free End Composite Rotor Displacement

20. Control Objectives Angular velocity tracking error regulation Auxiliary tracking signal regulation where is the desired angular velocity trajectory 0 Rotor displacement regulation 0 0 Application Based Laws of Nature Analysis Generated Free-End Velocity Angular Velocity Free-End Displacement Free-End Shear Reasons

21. Heuristic Control Design - Flexible Rotor Subsystems Flexible Rotor Dynamics Rotating Disk Dynamics Free-End Dynamics Input Force Clamped Boundary Free Boundary Input Torque RotorDisplacement Angular Velocity Free End Motion

22. Heuristic Control Design - Dynamic Coupling Flexible Rotor Dynamics Rotating Disk Dynamics Free-End Dynamics Input Force Clamped Boundary Free Boundary Input Torque PDE/ODE Coupling PDE/ODE Coupling ODE/ODE Coupling RotorDisplacement Angular Velocity Free End Motion

23. Heuristic Control Design - Control Objectives Flexible Rotor Dynamics Rotating Disk Dynamics Free-End Dynamics Auxiliary Tracking Signal Input Force Clamped Boundary Free Boundary RotorDisplacement Angular Velocity Tracking Error Input Torque q(x,t) 0  t  d (t)  (L,t) 0 PDE/ODE Coupling PDE/ODE Coupling ODE/ODE Coupling ControlObjectives { Design Boundary Control {

24. Model-Based Boundary Control Law Based on the stability analysis, the boundary control force applied to the free end of the rotor is given by The boundary control torque applied to the disk is given by where is the free-end displacement, is the free-end velocity, and is the free-end shear Only Boundary Terms The boundary control force and torque are designed to yield and Exponentially Stable Closed-Loop Error Systems Auxiliary Tracking SignalAngular Velocity Standard Tracking Control

25. If the control gain is selected to satisfy the following sufficient condition, Stability Result then the angular velocity tracking error and the rotor displacement are globally exponentially regulated as given by Rotor Energy Angular Velocity Tracking Error Rotor Displacement By Means of an Integral Inequality Directly from previous inequalities ( )

26. Adaptive Control Robustness - Parametric Uncertainty The boundary control force and torque are redesigned as a certainty equivalence adaptive controller as follows The adaptive update laws for the bending stiffness, the free-end mass and the inertia of the disk are shown below where, and are positive adaptive update gains Analysis Generated

27. Block Diagram Overview of the Adaptive Boundary Controller Flexible Rotor System Disk Torque Control Free-End Force Control Parameter Update Law Disk Position, Free-End Shear, Free-End Displacement Sensor Measurements: Rotor Vibration Regulation Disk Velocity Tracking

28. Techron Linear Power Amplifiers Multi Q I/O Board Camera Decoder Board Pentium 166 MHz Host PC System Hall Effect Current Sensors Shear Sensor Amplifier BDC Motor Instrumentation Amplifiers boundary control torque applied via belt-pulley transmission via slip ring assembly Encoder A/D D/A Magnetic Bearing Applies Boundary Control Force Linear CCD Cameras Rotating Disk Two-Axis Shear Sensor Flexible Rotor LED Actuator Mass Experimental Setup x u v

29. Free-End Snapshot of Rotor Flexible Rotor Magnetic Bearing 2-Axis Shear Sensor Actuator Mass

30. Free-End Displacement Regulation (Velocity Setpoint Regulation Objective) 0102552015 Time [s] 0.02 0 -0.02 Open Loop Damper Peak Model-Based Controller Displacement = 4.7% (approx.) x Peak Open Loop Displacement = 26% (approx.) x Peak Damper Displacement Model Based One direction & other direction is similar [m]  d = 380 [rpm]

31. Technical Conclusions Developed a model-based boundary control strategy for the hybrid model of a 2-D flexible rotor – Exponentially regulated the rotor displacement and the angular velocity tracking error – Uses measurements of the link’s free-end displacement, free-end shear, angular velocity, and the time derivatives of some of these quantities Developed an adaptive boundary controller for the flexible rotor –Asymptotically regulated the rotor displacement and the angular velocity tracking error –Compensated for parametric uncertainties in the system Both controllers were implemented on a flexible rotor test-stand The controllers account for the disk inertia and free-end dynamics No restriction on the magnitude of the desired angular velocity; moreover, a solution for the angular velocity tracking problem was proposed