Nonlinear Nonlinear Control ofControl of

MechatronicMechatronicSystemsSystems

CLEMSONCLEMSONU N I V E R S I T Y

Darren DawsonDarren DawsonMcQueen Quattlebaum Professor

Electrical and Computer Engineering

• 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 PresentationOverview of Presentation

PART 1

PART 2

PART 3

Applications and Areas of InterestApplications 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 Nonlinear Control and Estimationand Estimation

The Mathematical ProblemThe 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

x f x y·= ( , )y g x y u

·= ( , , )u y x

Electrical Dynamics Mechanical Dynamics

x f x y·= ( , )y g x y u

·= ( , , )u y x

LinearController

fLinear

f

x

gLinear

g

y

u y xy x y u· =?( , , ) x x y

· =?( , )

x f x y·= ( , )?u y x

x f x y·= ( , )y g x y u

·= ( , , )u ? ?

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 ApproachThe Mathematical Solution or Approach

Mechatronics

Based Solution

AdvancedNonlinear Control

Design Techniques

RealtimeHardw are/Software+

NewControl

Solutions

u y x

NonlinearParameterEstimator

NonlinearController

y x y u· =?( , , ) x x y

· =?( , )

x f x y·= ( , )y g x y u

·= ( , , )u ? x

NonlinearObserver

NonlinearController

t

Transient Performance Envelopes

Control Design/Implementation CycleControl Design/Implementation Cycle

Testbed ConstructionSensors: 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 TechniquesSimulation Studies

Model-Based, Adaptive, Robust

Hamilton’s Hamilton’s Principle,Principle,Newton’s LawNewton’s Law

Control Control ObjectiveObjective

Problem FormulationTracking, Setpoint

Parametric UncertaintyBounded DisturbanceUnmeasurable Signals

ControlControlDesignDesign

Data Acquisition

MultiQ, ServoToGo I/O Board(encoders, D/A, A/D, digital I/O)

Real-Time OS,Real-Time OS,Driver Interface,Driver Interface,

Data HandlingData Handling

Signal Conditioning

Linear Power Amplifiers OPAMPS (gains, offsets)

Interface andInterface andSafety IssuesSafety Issues

ElectronicElectronicCompatibilityCompatibility

CodingCodingthe Controlthe ControlAlgorithmAlgorithm

Master ThesisStudents

PhDStudents

Motivating Example (Model Known)Motivating Example (Model Known)

• Dynamics:

Mass

bx3

asin(t) bx3

u(t)Nonlinear Damper

Disturbance Velocity

Control Input

a,b are constants

_x = ¡ bx3 ¡ asin(t)+u

• Tracking Control Objective: e= xd¡ x

• Open Loop Error System: _e= _xd ¡ _x = _xd +bx3 +asin(t) ¡ u

• Control Design:

• Closed Loop Error System: _e= ¡ K e

• Solution: e(t) = e(0)exp(¡ K t)

Feedforward Feedback

Assume a,b are known

Drive e(t) to zero

Exponential Stability

u = _xd + bx3 +asin(t) +K e

Motivating Example (Unknown Model)Motivating Example (Unknown Model)

• Open Loop Error System: _e= _xd ¡ _x = _xd +bx3 +asin(t) ¡ u

• Control Design:

a,b are unknownconstants

u = _xd +bb(t) x3 +ba(t) sin(t) + K x

Same controller as before, but and are functions of timeu = _x d + bb(t) x3 + ba (t) sin ( t) + K xu = _x d + bb(t) x3 + ba (t) sin ( t) + K x

How do we adjust and ?u = _x d + bb(t) x3 + ba (t) sin ( t) + K xu = _x d + bb(t) x3 + ba (t) sin ( t) + K x

Use the Lyapunov Stability Analysis to develop an adaptive control design tool for compensation of parametric uncertainty

• Closed Loop Error System: _e= ¡ K e+ ea(t) sin(t) +eb(t) x3ea(t) = a¡ ba(t)eb(t) = b¡ bb(t)

At this point, we have not fully developed the controller since and are yet to be determined.

u = _x d + bb(t) x3 + ba (t) sin ( t) + K xu = _x d + bb(t) x3 + ba (t) sin ( t) + K x

parameter error

u = _xd + bx3 +asin(t) +K e

( is UC)

Motivating Example (Unknown Model)Motivating Example (Unknown Model)

Fundamental Theorem

V (t) ¸ 0_V (t) · 0

ÄV (t)V (t) ¸ 0

V (t) ¸ 0

effects of conditions i) and ii)

i) If

ii) IfV (t) ¸ 0is bounded

iii) If is bounded

limt! 1

V (t) = 0_V (t) · 0limt! 1

V (t) = 0

satisfies condition i)

V (t) ¸ 0

finally becomes a constantV (t) ¸ 0

• Non-Negative Function: V =12

e2 +12

ea2 +12

eb2

• Time Derivative of V(t): _V = _ee¡ ea:ba ¡ eb

:bb

_e= ¡ K e+ ea(t) sin(t) +eb(t) x3

is bounded

examine condition ii)

design andu = _x d + bb(t) x3 + ba (t) sin ( t) + K xu = _x d + bb(t) x3 + ba (t) sin ( t) + K x

substitute the dynamics for

limt! 1

V (t) = constant

effects of condition iii)

_V (t) · 0

l imt! 1

e (t) = 0

Motivating Example (Unknown Model)Motivating Example (Unknown Model)

• Substitute Error System: _V = ¡ K e2 + ea³sin(t) e¡

:ba

´+eb

µx3e¡

:bb¶

How do we select and such that ?u = _x d + bb(t) x3 + ba (t) sin ( t) + K xu = _x d + bb(t) x3 + ba (t) sin ( t) + K x

• Update Law Design::ba= sin(t)e

:bb= x3e

• Substitute in Update Laws: _V = ¡ K e2 · 0 V (t) ¸ 0 _V (t ) · 0and

Fundamental Theorem is boundedV (t) ¸ 0 all signals are bounded

limt! 1

e(t) = 0limt! 1

V (t) = 0_V (t) · 0limt! 1

V (t) = 0Fundamental Theorem

u = _xd +µZ t

0x3 (¾)e(¾)d¾

¶x3 +

µZ t

0sin(¾)e(¾)d¾

¶sin(t) +K e

Feedforward Feedback

control structurederived fromstability analysis

control objective achieved

_V (t ) · 0

ÄV (t) is bounded

Boundary Control of a Boundary Control of a Flexible Rotor SystemFlexible Rotor System

Overview of Part II – Flexible Rotor Control ProblemOverview 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

Space-Based Systems that VibrateSpace-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

What is the Problem ?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 infinite dimensional nature of the PDE subsystem

– the nonlinearities associated with hybrid systems, and

– the coupling between the PDE and ODE subsystems

Problem

Model

Challenge

Hybrid System(PDE+ODE)

Based on aLinear/Discrete Model

DistributedControl

Linear ControlBoundaryControl

• 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)

AdvantagesDisadvantages

How are Flexible Systems Controlled ?How are Flexible Systems Controlled ?

Disadvantages

What is Boundary Control ?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

Rotor Displacement

Rotor Displacement

The Flexible Rotor ProblemThe Flexible Rotor Problem

Rotating Disk

Actuator Mass

f (t)1

f (t)2

(t)

Flexible Rotor

BoundaryControl Torque

Input

Boundary Control Force Inputs

Control Objective : Drive u(x,t) and v(x,t) to zero and force to track d(t)

f (t)1

x

u(x,t)

u(x,t) (t)

Cutaway

View

x u

v

(t)

x

v(x,t) f (t)2

v(x,t)

Comparison To Previous WorkComparison 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

Displacementconfined to 1-D

Rotation

1-D Problem1-D Problem

Neglects Nonlinearities& ODE/PDE Coupling

2-D Flexible Rotor Model2-D Flexible Rotor Model

• Field Equation (PDE Subsystem - Euler Bernoulli Model)

• Boundary Conditions q (0; t) = qx (0; t ) = qx x ( L ; t) = 0

½³

qtt (x; t ) + 2S qt (x ; t) _µ ( t ) + S q (x ; t) ĵ (t ) ¡ q ( x; t) _µ2

( t)´

+ E I qx x xx ( x ; t) = 0

½³

qtt (x; t ) + 2S qt (x ; t) _µ ( t ) + S q (x ; t) ĵ (t ) ¡ q ( x; t) _µ2

( t)´

+ E I qx x xx ( x ; t) = 0

q (x; t) =£

u (x ; t ) v ( x; t)¤T

where

F (t) =£

f 1 ( t) f 2 ( t)¤T

where

J ĵ (t ) = ¿ ( t )• Disk Dynamics (ODE Subsystem: J - Disk Inertia)

S =·

0 ¡ 11 0

¸;

EI -bending stiffness & mass per unit length

• Free-End Dynamics (ODE Subsystem: m - actuator mass )

m·qt t (L ; t ) + 2S qt ( L ; t) _µ ( t) + S q ( L ; t) ĵ ( t) ¡ q (L ; t) _µ

2(t)

¸¡ E I qx x x (L ; t) = F (t )

Beam is clamped at the disk No applied Torque at the Free End

Composite Rotor Displacement

Control ObjectivesControl Objectives

• Angular velocity tracking error regulation

• Auxiliary tracking signal regulation

where is the desired angular velocity trajectorye ( t) = _µ ( t) ¡ ! d

e ( t) = _µ ( t) ¡ ! d 0

• Rotor displacement regulation

q (x; t) =£

u (x ; t ) v ( x; t)¤T 0

´ ( t) = qt ( L ; t) + _µ ( t) S q ( L ; t) ¡ qx xx ( L ; t ) 0

ApplicationBased

Laws ofNature

AnalysisGenerated

Free-EndVelocity

AngularVelocity

Free-EndDisplacement

Free-EndShear

ReasonsReasons

e ( t) = _µ ( t) ¡ ! d

e ( t) = _µ ( t) ¡ ! d

Heuristic Control Design - Flexible Rotor SubsystemsHeuristic Control Design - Flexible Rotor Subsystems

Flexible Rotor Dynamics

Rotating Disk Dynamics

Free-EndDynamics

Input Force

Clamped Boundary

FreeBoundary

Input Torque

RotorRotorDisplacementDisplacement

Angular Velocity

Free EndMotion

Heuristic Control Design - Dynamic CouplingHeuristic Control Design - Dynamic Coupling

Flexible Rotor Dynamics

Rotating Disk Dynamics

Free-EndDynamics

Input Force

Clamped Boundary

FreeBoundary

Input Torque

PDE/ODECoupling

PDE/ODECoupling

ODE/ODECoupling

RotorRotorDisplacementDisplacement

Angular Velocity

Free EndMotion

Heuristic Control Design - Control ObjectivesHeuristic Control Design - Control Objectives

Flexible Rotor Dynamics

Rotating Disk Dynamics

Free-EndDynamics

AuxiliaryTracking Signal

Input Force

Clamped Boundary

FreeBoundary

RotorRotorDisplacementDisplacement

Angular VelocityTracking Error

Input Torque

q(x,t) 0

td(t)(L,t) 0

PDE/ODECoupling

PDE/ODECoupling

ODE/ODECoupling

ControlControlObjectivesObjectives

{

Design Boundary Control

{

Design Boundary Control

Model-Based Boundary Control LawModel-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

F (t) = ¡h

k s ´ ( t) + E I qx xx (L ; t ) + m³

_µ2

( t) q (L ; t) ¡ _µ (t ) S qt (L ; t) ¡ qx x x t (L ; t)´ i

F (t) = ¡h

k s ´ ( t) + E I qx xx (L ; t ) + m³

_µ2

( t) q (L ; t) ¡ _µ (t ) S qt (L ; t) ¡ qx x x t (L ; t)´ i

F (t) = ¡h

k s ´ ( t) + E I qx xx (L ; t ) + m³

_µ2

( t) q (L ; t) ¡ _µ (t ) S qt (L ; t) ¡ qx x x t (L ; t)´ i

F (t) = ¡h

k s ´ ( t) + E I qx xx (L ; t ) + m³

_µ2

( t) q (L ; t) ¡ _µ (t ) S qt (L ; t) ¡ qx x x t (L ; t)´ i

F (t) = ¡h

k s ´ ( t) + E I qx xx (L ; t ) + m³

_µ2

( t) q (L ; t) ¡ _µ (t ) S qt (L ; t) ¡ qx x x t (L ; t)´ i

where is the free-end displacement, is the

free-end velocity, and is the free-end shear

´ ( t) = qt ( L ; t) + _µ ( t ) S q ( L ; t) ¡ qx xx ( L ; t ) ´ ( t) = qt ( L ; t) + _µ ( t) S q ( L ; t) ¡ qx xx ( L ; t )´ ( t) = qt ( L ; t) + _µ ( t ) S q ( L ; t) ¡ qx xx ( L ; t )

¿ (t) = ¡ kr e(t) +J _! d (t)

Only Boundary Terms

• The boundary control force and torque are designed to yield

m_́(t) = ¡ ks´ (t) and J _e(t) = ¡ kre(t) Exponentially Stable Closed-Loop Error Systems

Auxiliary Tracking Signal Angular Velocity

Standard Tracking Control

• If the control gain is selected to satisfy the following sufficient condition,

Stability ResultStability Result

ks >E I2

then the angular velocity tracking error and the rotor displacement are globally exponentially regulated as given by

ks >E I2

RotorEnergy

AngularVelocity TrackingError

E I2L3

kq(x;t)k2 · kE R (t)k; je(t)j · · 0 exp(¡ · 1t)

RotorDisplacement

By Means ofan IntegralInequality

Directly from previous inequalities ( )

_V · ¡ · V

l i mt! 1

kq (x ; t )k ; j e ( t )j = 0 8x 2 [0; L ]

Adaptive ControlAdaptive ControlRobustness - Parametric UncertaintyRobustness - 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 m ( t) = _µ2

( t) q ( L ; t) ¡ _µ (t ) S qt (L ; t) ¡ qxx xt (L ; t) :

, and are positive adaptive update gains

F ( t) = ¡hks ´ ( t) + dE I ( t) qxx x (L ; t) + cm ( t) m ( t)

i

:

:bJ (t) = ¡ ° j _! d (t) e(t)

¿ (t) = ¡ kre(t)+ bJ (t) _! d (t)

:bm (t) = °m T

m (t) ´ (t)

°e °m °j

:dEI (t) = ° eq

Txxx (L;t) ´ (t)

AnalysisGenerated

Block Diagram Overview of the Adaptive Boundary ControllerBlock Diagram Overview of the Adaptive Boundary Controller

Flexible RotorSystem

Disk Torque Control

Free-End Force Control

Parameter UpdateLaw Disk Position,

Free-End Shear,Free-End Displacement

Sensor Measurements:

Rotor VibrationRegulation

Disk VelocityTracking

TechronLinear Power

Amplifiers

Multi Q I/O Board

Camera Decoder Board

Pentium166 MHzHost PC System

Hall EffectCurrentSensors

Shear Sensor

Amplifier

BDC Motor

InstrumentationAmplifiers

boundary controltorque applied via belt-pulley transmission

via slip ringassembly

Encoder

A/D

D/AMagnetic Bearing AppliesBoundary ControlForce Linear

CCD Cameras

Rotating Disk

Two-AxisShear Sensor

Flexible Rotor

LED

Actuator Mass

Experimental SetupExperimental Setup

x uv

Free-End Snapshot of RotorFree-End Snapshot of Rotor

Flexible Rotor

Magnetic Bearing

2-Axis Shear Sensor

Actuator Mass

Free-End Displacement RegulationFree-End Displacement Regulation(Velocity Setpoint Regulation Objective)(Velocity Setpoint Regulation Objective)

0 10 255 2015Time [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 issimilar

[m]

d = 380 [rpm]

Technical ConclusionsTechnical 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