application of classical control
TRANSCRIPT
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Control Guidance & Simulation Entity
Classical Control Theory:
- Application to Launch Vehicles
Presented By:
Kapil Kumar Sharma, (Sci./Eng.SD)
Control Design Division, CGSE
M. V. Dhekane, Deputy Director
Control Guidance & Simulation Entity, VSSC
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The concept of the feedback loop
http://en.wikipedia.org/wiki/File:Feedback_loop_with_descriptions.svghttp://en.wikipedia.org/wiki/File:Feedback_loop_with_descriptions.svghttp://en.wikipedia.org/wiki/File:Feedback_loop_with_descriptions.svghttp://en.wikipedia.org/wiki/File:Feedback_loop_with_descriptions.svghttp://en.wikipedia.org/wiki/File:Feedback_loop_with_descriptions.svghttp://en.wikipedia.org/wiki/File:Feedback_loop_with_descriptions.svg -
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Control System Design Objectives
Primary Objectives:
1. Dynamic stability2. Accuracy
3. Speed of response
Addition Considerations:
4. Robustness (insensitivity to parameter variation)5. Cost of control
6. System reliability
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Control System Design Steps
Define the control system objectives.
Identify the system boundaries.
define the input, output and disturbancevariables
Determine a mathematical modelfor the
components and subsystems.
Combine the subsystems to form a model forthe whole system.
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Control System Design Steps
Apply analysis and design techniques to
determine the control system structure andparameter values of the control components,to meet the design objectives.
Testthe control design on a computer
simulation of the system. Implement and testthe design on the actual
process or plant.
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Digital Auto-Pilot (DAP)
Launch Vehicle Autopilot is an inner loop of the Navigation,
Guidance and Control (NGC) subsystem.
Controls the Attitude of the vehicle in Pitch, Yaw and Roll
channels from lift-off till end of flight.
It ensures the stabilisation of the attitude in the presence of
disturbing forces and moments caused by various sources.
Steers the vehicle along a desired trajectory, maintaining the
structural integrity.
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SpecificationOverall length : 49.128 m
Lift off Mass : 401.8 t
GTO payload : 1600 kg
Definition
(4L40+S125)+L37.5+C12.5
GSLV-MK2
Sensors : RESINS at EB ,
RGP at M
Actuators: SITVC, EGC(L40)
EGC(L37.5), CSCE,
OSS
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GuidanceLaw
ControlLaw
PlantDynamics
Attitude Sensing
+ +
Pos/Velocity Sensing
Target
Attitude
Commands Actuator
Commands
Navigation Guidance and Control Loop
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System
Input (d
Disturbance (d)
Controller
Output (q
c
command
Design Problem
Behavior of the output in presence of the disturbance maynot be satisfactory
Controller has to ensure satisfactory response of the system
rejecting the disturbance
System will be modeled by using differential equations
The controller will process the input signal to achievesatisfactory output.
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i). Mathematical model of vehicle dynamics
ii). Design to meet specifications
iii). On-board implementation of AutopilotAlgorithm
iv). Validation
Design and development of Autopilot
consists following important elements.
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Control Configuration
Control Power PlantPhysical Location / Alignment
Control Force / Moment
Dynamics
SensorsPlacement
Dynamics
BafflesPlacement
Characteristics
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Model of the Systemto be controlledPlant :Rigid body, Flexibility, Slosh Dynamics
Actuators :SITVC, EGS, FNC, RCS
Sensors :RGP,RESINS, LAP
Disturbances
Thrust misalignment, CG offset, DifferentialThrust, Winds
Tracking CommandsGenerated by Guidance Law
Specifications
Inputs for the Design
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PL NT MODEL
RIGID BODY DYN MICS
LAUNCH VEHICLE MODELLING
CONTROL
POWER
PLANT
SENSOR
DYNAMICS
LIQUID SLOSH
STRUCTURES
PROPULSION
AERO DYNAMICS
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Input Data:
Plant Model :Mass, Length, Diameter, CG, MI of vehicle
Control Power Plant :Actuator model (transfer function),
Nozzle mass, length, Inertia
Sensor Dynamics :Attitude & body rate transfer functions
Liquid Slosh :Pendulum Mass, Length, Distance of
pendulum hinge point from CG, Un-damped natural frequency, Damping ratio
Structures :Bending mode Frequency, Generalized
mass, Mode shape, Mode slope,
Damping ratio
Launch Vehicle Modelling
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Input Data:
Propulsion :Thrust
Aerodynamic :Aerodynamic (Lateral/Side) force coefficients,
Center of Pressure
Trajectory :Altitude, Mach Number, Inertial velocity
Atmosphere :Density
Dispersion level :Specified Uncertainty bounds
(example: Aero coefficients 3%,
Bending mode frequency 10% etc.)
Launch Vehicle Modelling
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Assumptions:
Time Slice Approach:Time varying mass and inertia properties are frozen over a short period of time
(Short Period Dynamics)
Small Angle Approximation:Deviations from Reference trajectories are small so that trigonometric non-
linearity, and higher order terms contributions are neglected.
Decoupling of Attitude Dynamics:Due to axis symmetry of launch vehicles, Pitch/Yaw/Roll motions are assumed
to be decoupled.
NOTE: There is significant amount of coupling in Yaw/Roll motion for aircraft.
Linear Time Invariant :Non-linearity of actuator/sensors (Dead-zone, slew rate etc.) are neglected at
design phase.
(All above assumptions lead to LTI systems properties.)
Launch Vehicle Modeling
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Design Specifications
Primary Specification-
Tracing Error < 1 Degree
Robustness Specifications
Rigid body
Aero margin > 6 dB
Phase margin > 30 Degree
Gain margin > 6 dB
Bending modes
Phase Margin > 40 Deg. : Phase Stabil isat ion
Attend. margin > 6 dB : Gain Stabil isat ion
Slosh modes
Phase margin > 30 Degree
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Design Methodology
Classical Design TechniqueTracking Error Specification -
Bandwidth / Damping
Gain Design- PID
Frequency Domain Design
Roll off, Notch, Lag-Lead filter
Analysis using
Root Locus, Bode plot, Nyquist plot
i i i i
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Rigid Body Rotational Dynamics
2
Assumptions: & sin( ) =
( )
( )
sin( )
where,
,
C C
C C
C CC
C
I L l T l
I L l T l
T l L
s
s s
l
I I
q d
q q d
q d d
q
d
: Attitude lC: Control moment arm
: Control variable I : Moment of inertia
: Angle of attack l: Aerodynamic moment arm
L : (QSCN ) Q: Dynamic Pressure, S Reference areaCN : Aerodynamic Normal Force Coefficient, TC: Controlling Thrust
q
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Root locus analysis
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N i t Pl t @I + T
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No. of open loop poles in RHP : P = 1
# counter-clockwise encirclement : N =1
Nyquist Plot @Ign. + T sec
Z = PN = 0
(STABLE)
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G i D i M th d 2 (R t l it i )
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Gain Design Method-2 (Root-locus criteria):
)
)
2
2 2 2
2
2
2
2
P
2
R
1 1 02
1 12
Characteristic eq.:
For s = - 1
Compute K from angle criteria
( K does not contribute in angle)
n
a C
P R
a a a
a C
P R
a
n n
a
n
a
K K ss s s
j
K K ss s s
)
)
2
0
2 2 2
2
2 2 2
P
Apply magnitude criteria,
Calculate K
1 1802
1 12
.
a C
P R
a a a
a C
P R
a a a
K K ss s s
K K ss s s
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SLOSH
Lateral oscillations of liquids in tank is slosh
Modelled by replacing liquid mass with rigid
mass and a harmonic oscillator such as spring or
pendulum
Pendulum parameters are function of tank shape
and liquid level
M d li Li id Sl h
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Modeling Liquid Slosh
Ref: NASA SP-106
The Force & Moments produced by sloshing of the liquid fuel is
analyzed by an equivalent Mass and PendulumOr an equivalent
Spring MassAnalogy.
Ref: Elements of Control Theory by A. L. Greensite
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Slosh Dynamics
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Slosh Dynamics
2 2
0
th
th
th
th
( 2 ) ( ) ( )
: of i pendulum
: distance of i pendulum hinge point from body cg
: damping of i Pendulum: Undamped natural Frequenc
where
y i
,
of P
pi pi pi pi i pi pi
pi
pi
pi
pi
L s s w U L
L Length
q q
0
endulum
: Pendulum angle
: Lateral acceleration
: Forward inertial velocity
: attitude rate
i
w
U
q
NOTE: Complex Pole-zero pair is introduced in Rigid body dynamics by each
Slosh mode.
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N i t l t t i l h iti it i t t
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Nyquist plot at maximum slosh sensitivity instant :
- Rigid Body with Slosh, Actuator, Sensor& delay
- Slosh margin 5.5 degree& 8.9 degree
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Lag-lead compensator
(phase margin improvement)
Lag-Lead Compensator Bode plot
Modeling Flexibility Dynamics
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Modeling Flexibility Dynamics
Ref: Elements of Control Theory by A. L. Greensite
( ) ( )( ) ( )
i i
i
u q t Bending Mode displacement:( )
(
t
t)
h
h
:Generalized Mode Shape of i Bending mode
:Generalized Co-ordinate of i
(
Bending mode
)
( )
i
i
q t
Where,
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Flexibility Dynamics
1. Generalized mode shape is a function of vehicle length and mass
distribution etc. Since mass of vehicle is rapidly decreasing, Mode shape
is changing w.r.t. time. (Predicted with uncertainty bounds)
2. Generalized Co-ordinate dynamics is represented by second order
differential equation.
2 ( ) ( ) ( ) 2 ( )( )( = 1, 2, 3...2 [ ] ) ,i i i i c iTs s q
Mi d
3. Flexibility deflection is picked-up by attitude sensor & Flexible-
deflection rate is picked-up by rate sensor.( )
( )
( )( )where,
( )
( )
i
R RG
i
i
R PG
i
ii
q i
q i
q q
q q
4. Complex Pole-zero pair is introduced in Rigid body dynamics by each
Generalized Co-ordinate.
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Bending Mode Stabilization
Gain Stabilization:
Attenuate control loop gain at desired frequency, to ensurestability regardless of control loop phase uncertainty.
(Second/Higher BM are usually Gain stabilized)
Phase Stabilization:
Provide proper phase characteristics at desired frequency toobtain a close loop damping, that is greater than the passive
damping.
(First/Second BM are usually Gain stabilized)
Gain-Phase Stabilization:A Rigid body/Flexible mode is said to be gain-phase stabilize if it
is close loop stable with finite gain and phase margin.
Rigid Body + Flexibility
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Rigid Body + Flexibility
Rigid Body poles : Two poles at origin [K/(s2- a)]
Actuator : Second order system [w2/s2+2*z*w*s+w2]
Slosh : 3 bending mode (complex pole zeros pair per bending mode)
Uncompensated Bode plot Uncompensated Nyquist plot
Lag Compensator in Rate pathCompensated Nyquist Plot
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Lag-lead & Roll off filter in forward path
BM1, BM2are Phase stabilized
BM3Gain stabilized
Compensated Nyquist Plot
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Time Response Analysis
Step response analysis to check the timedomain specifications ( like percentageovershoot, rise time, settling time )are met.
In the time response all known nonlinearitieslike saturation of actuator , slew rate limit ofactuator etc. has been modeled properly.
All related states are monitored
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Integrator limit Rate Control Limit
Dead-zone
Non-Linear Control
Block Diagram : ON-OFF Control with Dead-zone
d qpK
Cq
KRs
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Control Law- Error computation
Gain Schedule
Filters Integrator
Nonlinear control Logics
Design parameters
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Finite Word length Machine
Memory and Execution Time
Fixed point representation Accuracy and Scaling
Overflow problems
Transportation and computationaldelays
Implementation Aspects
C t l D i & V lid ti
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Nominal
Plant (Vehicle + disturbances)
Non-failure Cases Failure Cases
Off
Nominal(3
)
Stress(3
- 6
) Over Stressed
unrealistic Cases
Act. failure Propulsion fail
Control Design & Validation
Philosophy
C t l D i & V lid ti
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Nominal:Control design to meet performance
specifications.
Ensures Stability.
Design for linear zone. Small perturbations.
Off nominal:
Validation to ensure stability.
Slight performance deviations which areacceptable.
Design tested with nonlinearities touched.
Non-linear dynamics exercised.
Control Design & Validation
Philosophy
C t l D i & V lid ti
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Stress:Stability to be ensured.
Large deviations in performance is expected and
accepted.
To be used mainly for software reliability
Over stress:
Even stability is not ensured.
To be used for Limits of Performanceassessment.
To define specification on other systems.
Control Design & Validation
Philosophy
C t l D i & V lid ti
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Instability due to Inadequate control
Dynamics. (Control lagging disturbance)
High frequency instability
Coupling P/Y/R
Why systems fail where statically controlis available ?
Solution: Redesign of Autopilot accounting for
Nonlinear dynamics of Actuator
Saturation
Disturbance Time constant
Disturbance value
Control Design & Validation
Philosophy
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Simulated Input Profile (SIP)
OBC In Loop Simulation (OILS)
Hardware In Loop Simulation (HLS)
Actuator In Loop Simulation (ALS)
Flight Test
Post flight analysis Disturbance calculation-
model matching Model Update / Design Update
Validation
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Complexity of the model
Robust Design Requirement
Fault tolerance
SISO to MIMO
Unified Design Approach
Design Automation Code Automation
Challenges in Design
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Acknowledgement
We sincerely acknowledge the contributions of all our seniors and
colleagues in CGDG,VSSC towards the development of Launch Vehicle
control design philosophy.