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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
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• 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
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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
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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 ? ?
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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
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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
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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
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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
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( 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
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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
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Boundary Control of a Boundary Control of a Flexible Rotor SystemFlexible Rotor System
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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• 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 ]
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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
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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
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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
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Free-End Snapshot of RotorFree-End Snapshot of Rotor
Flexible Rotor
Magnetic Bearing
2-Axis Shear Sensor
Actuator Mass
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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]
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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