Application of Classical Control

8/11/2019 Application of Classical Control

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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
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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



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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.


Philosophy



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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.


Philosophy



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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


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.

Documents

Application of Classical Control