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Computation of Limit Cycles forUncertain Nonlinear Fractional-order Systems using Interval Constraint
Propagation
P. S. V. Nataraj&
Rambabu Kalla
Systems & Control EngineeringIIT Bombay (India)
Outline
Problem Formulation Interval ArithmeticConstraint Satisfaction Problems Interval Constraint Satisfaction Problems Interval Constraint PropagationHC4 filter (with an example)Outline of the Algorithm (Flow Chart)Theoretical PropertiesApplication- Gas Turbine PlantComputation of Limit CyclesConclusions
Problem formulation for Limit Cycle Computation
Linear Fractional Order plant p(s,qg ) Nonlinear element with describing function N(a,ω,qn ) Example nonlinearity as relay with hysteresis type
(Nonlinearity with memory)
Nonlinear Feedback System
Problem formulation for Limit Cycle Computation contd.
},,{ qav
Variables
Specifications
Constraint
Specifications
Initial search domain
Specifications
Tools for problem solving
We use the following tools to solve the problem of computing the Limit Cycles for Uncertain Nonlinear fractional-order system
Tool of Interval arithmetic Tool of Interval Constraint Propagation
Brief review of above mentioned tools follows.
Another Arithmetic ParadigmInterval Arithmetic
Well known computing paradigms are Fixed & Floating point computer arithmetic paradigms.
Today, researchers are seriously considering another computer arithmetic paradigm—namely, interval arithmetic.
Interval arithmetic started with the work of Moore in 1966.
Interval arithmetic is a natural tool to deal with problems involving uncertainty such as Robust systems analysis & Control.
Interval Arithmetic
In interval arithmetic, if we add two intervals, we get, [a,b] + [c,d] = [a+c, b+d].
Then the lower bound of the result is rounded down to (a+c)- and the upper bound rounded up to (b+d)+.
In this way, the computed result C = [(a+c)-,(b+d)+]
is an interval that is known to contain the correct result.
Virtues of Interval Arithmetic
Methods based on interval mathematics, in particular interval-Newton methods, can enclose any and all solutions to a nonlinear equation system.
Also can find the global optimum of a nonlinear function.
These methods provide a mathematical and also
computational guarantee of reliability.
Constraint Satisfaction Problems
A set of variables ,For each variable with domain with
possible values for that variable, andA set of constraints, i.e., relations, that are
assumed to hold between the values of the variables.
The constraint satisfaction problem is to find, for each from to a value in for so that all constraints are satisfied.
Applications : AI & Operations Research
A Constraint Satisfaction Problem is characterized by :
},..,,{ 21 nvvv
iv iD
i 1 niD iv
Interval Constraint Satisfaction Problem
Set of variables
Set of Constraints
Initial search domain or box
Example :
},...,,{ n11 vvvv
},...,,{ 21 mcccc
},...,,{ n21 xxxD
]10,10[]10,10[ y x D
]1010[ ,y 10,10][x 122 yx
},{ yxv }1{ 22 yxc
Solving an ICSP
Constraint Propagation (Pruning)
Propagating through the constraint tree to reduce the search domain by throwing out the infeasible region
Constraint branching (Splitting)
Creating sub problems from the main problem and solving those sub problems.
HC4 Filter (For Pruning)
Forward Evaluation
Backward Propagation
Example :
]10,10[]10,10[ y x D
},{ yxv
}1{ 22 yxc
]1010[ ,y 10,10][x 122 yx
x
y
-1
-10
-10+10
-1
+10
+1
+1
=
+ 1
^ ^
X 2 Y 2[-10,10] [-10,10]
HC4 Filter (Tree Construction)
=
+ 1
^ ^
X 2 Y 2[-10,10] [-10,10]
HC4 Filter (Forward Evaluation)
[0,100] [0,100]
[0,200]
HC4 Filter (Backward Propagation)
X1 = [-99,1] ∩ [0,100]
=
+ 1
^ ^
X 2 Y 2[-10,10] [-10,10]
[0,100] [0,100]
[0,200] [1,1][1,1]
X1 = [0,1]
X1 + [0,100] = [1,1]
X1 = [-99,1]
X1 = [0,1]
[-1,+1]
Y1 = [0,1]
Y1 + [0,1] = [1,1]
Y1 = [0,1]
Y1 = [0,1] ∩ [0,100]
Y1 = [0,1]
[-1,+1]
x
y
-1
-10
-10+10
-1
+10
+1
+1
Algorithm
{}L
)(width x
x solsol LL
solL output
START
{}, solLL 0x
,0xc,v,
ENDHC4 usingbox thePrune
21
21
xx
xx x
LL
& toBisect
Y
N
L from box last upPick x
Y
N
Theoretical Properties
No need for approximation of FO system.
Reliable in the face of computational errors. We do not miss out any limit cycle points in the given search domain.
Accuracy guaranteed within user’s specification. The maximum error in the computed LC points cannot be more than the accuracy tolerance specified.
Computationally efficient (HC4 filter)
Errors resulting in the limit cycle locus due to approximate nature of describing function method cannot be avoided.
Gas Turbine Engine
A critical system in aircraft is the gas turbine engine.
A gas turbine engine provides thrust under all conditions enveloping flight spectrum of altitude And speed.
It is a is a complex machine consisting of a number of rotating and stationary components having aerodynamic and thermodynamic properties.
Schematic of Gas Turbine
Afterburner flow DistributorNozzle
Digital ElectronicControl Unit
VG
MainFuel
Reheat
Compressor Variable Geometry ( VG) Main Burners
Nozzle actuators
Fuel in
Manual Fuel Control Linkage
PLA
Gearbox
Hydromec-hanical
Systems
A Single Spool Turbojet Power Plant consists of an intake, a compressor, a combustion chamber, a turbine and a propelling nozzle.
A Twin Spool turbofan power plant consists of an
intake, a low pressure compressor, high pressure compressor, a combustion chamber, a high pressure turbine, a low pressure turbine, a mixer, and a propelling nozzle.
Various Gas Turbine Configurations
23
NOZZLE
Combustor HPT LPT
Fuel FlowVariable Geometry LPC: Low Pressure Compressor
HPC: High Pressure CompressorLPT: Low Pressure TurbineHPT: High Pressure Turbine
HPCFLOW IN FLOW
OUT
LPC
Schematic of a Twin Spool Gas Turbine Engine
Application to a Twin Spool Gas Turbine Plant
Operating Regime : 90% to 93% high pressure spool speed demand.
Input : Fuel rate to the gas turbine
Output : High pressure spool speed
Steady state values for i/p & o/p are 0.2442 Kg/sec & 20,620 rpm for 90% HP spool speed.
Identification at 90% HP spool speed
Input : PRBS signal
Sampling Time : 0.01 sec
Method : Output-Error Identification
Model Orders : OE221 to OE999
Identification- fractional and integer order models are obtained using output-error identification technique.
FO Model Identification
1)(
2121
1
sasa
bsp
FO model Structure used
OE identification technique with GL approximation and N = 25.
Gas Turbine Plant Cont. (Input Perturbation, 90% HP spool speed )
Gas Turbine Plant Cont. (Output, 90% HP spool speed)
FO Model Identified (90% HP spool speed)
FO Model Identified is
11356.000734.0
9705.103)(
8421.06807.1
sssp
Model Validation (Input, 90% HP spool speed)
Model Validation (90% HP spool speed)
Respective MSEs for FO & IO are 2.34e-04 & 2.57e-03
FO Model Identified (93% HP spool speed)
Similarly at another operating regime of 93% HP spool speed, the FO Model identified is
11818.00130.0
9238.110)(
7089.06062.1
sssp
Combined FO Model for 90-93% HP spool speed
Combined model for 90-93% HP spool speed obtained by identification is
],8421.0,7089.0[
],6807.1,6062.1[],1818.0,1356.0[
],0130.0,00734.0[],9238.110,9705.103[
,1
)(
2
12
11
21
1
21
a
ab
sasa
bsp
Computation of Limit Cycles for combined FO Model for GT
Gas turbine model
Nonlinear Element (Relay D with hysteresis H)
Specifications
Search domain and accuracy
Initial search domain
Limit Cycles are computed to an accuracy of
)( 000 qYx
Limit Cycle Locus with variation in b1
Limit Cycle Locus with variation in a1
Limit Cycle Locus with variation in a2
Limit Cycle Locus with variation in a3
Limit Cycle Locus with variation in α1
Limit Cycle Locus with variation in α2
Limit Cycle Locus with variation in D
Limit Cycle Locus with variation in H
The limit cycle frequency decreases with a1, α1, D but increases with b1, a2, α2 and H.
The limit cycle amplitude decreases with a2, α2 but increases with b1, a1, α1, H, D.
Both the limit cycle frequency and amplitude increase with b1, H.
Note that since a3 parameter has been fixed at unity and so does not vary, there is no variation in the limit cycle frequency and amplitude for this parameter.
Analysis of Limit Cycles for the combined FO model of the Gas Turbine
Analysis of Limit Cycles for the combined FO model of the Gas Turbine
Example-2
Model
Specifications
)( 000 qYx
Limit Cycle Locus with variation in q1
Limit Cycle Locus with variation in q2
Limit Cycle Locus with variation in q3
Limit Cycle Locus with variation in q4
Limit Cycle Locus with variation in q5
Limit Cycle Locus with variation in q6
Limit Cycle Locus with variation in q7
Limit Cycle Locus with variation in H
The limit cycle frequency decreases with q2 and q6 but increases with q3, q7 and H.
There is a very slight increase of limit cycle amplitude with q1, q4 and q5.
The limit cycle amplitude increases with q2, q3 and H but decreases with q6 and q7.
Both limit cycle frequency and amplitude increase with q3 and H and both decrease with q6.
Analysis of Limit Cycles of Ex-2
Analysis of Limit Cycles of Ex-2
Conclusions
Algorithms for computation of Limit Cycles for Nonlinear uncertain FOS.
Problem is formulated as ICSP and HC4 filter is used for pruning the search domain.
Demonstrated on practical application of a Gas Turbine plant.
