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8/11/2019 2.3 Calculus of Variations Fixed Ends With Constraints
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FIXED-END TIME AND FIXED-ENDPROBLEM: WITH CONSTRAINTS
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Approach 1: Direct methodAs a necessary condition for extrema, we have
Without constraint case: optimum is where
.For with constraint case,
and
are not arbitrary, but
related as:
Arbitrarily choose one of two variables, say , as anindependent variable. Then becomes dependent variable.Then provided .
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Approach 1: Direct methodTherefore,
As is arbitrary and can not be zero, .Or,
Alternatively,
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Approach 1: Direct methodThe following two equations solved simultaneously to find a
solution of the problem:
Some Facts:
First equation is also know as Jacobian of and . Tedious to solve for higher order problems.
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Approach 2: Lagrange Multiplier methodAn augmented Lagrangian function is formed: Where, is Lagrange multiplier, a parameter to be determined.With , .Therefore, necessary condition for extrema is that
Since and are functions of both and :
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Approach 2: Lagrange Multiplier method
Since both and are not independent, let isindependent. Then is dependent differential.Further, let is so chosen that one of the coefficient becomeszero. Let the coefficient of
is made zero by choosing a value
of as , that is Therefore, .Since is independent variable:
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Approach 2: Lagrange Multiplier method
Further,
Combining the three equations:
To be solved simultaneously
By eliminating from first two equations: The same can be extended for multiple constraints, i.e.
,
,
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Extrema of Functionals with Constraints
subject to the constraint (plant or system equations)
with fixed end-point conditions:
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Six Steps for Solution
Step 1: Lagrangian
Step 2: Variations and Increment
Step 3: First Variation
Step 4: Fundamental Theorem
Step 5: Fundamental Lemma
Step 6: Euler-Lagrange Equation
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Step 1 : Lagrangian
where
is the Lagrange multiplier and
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Step 2 : Variations and Increment
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Step 3 : First Variations (contd)
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Step 4 : Fundamental Theorem
vanishes, i.e.
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Step 5 : Fundamental Lemma
Then the first variation
becomes zero.
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ExampleMinimize the performance index
with boundary conditions and and subject tothe condition (plant equation)
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Solution by Direct method
; ; ; Replacing
from the condition in
, one gets
Now, one can minimize in straight forward way.
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Solution by Direct method
; ; ; Invoke Euler-Lagrange equation:
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Solution by Direct method
; ; ; Solution of Euler-Lagrange equation:
Simplifying,
Solution of the above is:
The constants and can be determined using boundaryconditions as:
and
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Solution by Lagrange Multiplier Method
; ; ; The condition can be written as:
Now, we form an augmented functional:
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Solution by Lagrange Multiplier Method
; ; ;
Invoke the optimality conditions:
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Solution by Lagrange Multiplier Method
; ; ;
Invoke the optimality conditions:
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Solution by Lagrange Multiplier Method
; ; ;
;
;
From last two equations:
Replacing above in the first equation:
Then the same solution prevails as the direct method.
Optimal Control: Calculus of Variations 62