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