KfUPM Optimal Control of Bilinear System Worksheet

Static Optimization In this chapter we discuss optimization when time is not a parameter.
 The discussion is preparatory to dealing with time-varying systems in
subsequent chapters.
 1.1 OPTIMIZATION WITHOUT CONSTRAINTS
 A scalar performance index L(u) is given which is a function of a control
or decision vector u ∈ Rm.
 It is desired to determine the value of u that results in a minimum value of
L(u), (How?)
Static Optimization
 Write the Taylor series expansion for an increment in L as
 where O(3) represents terms of order three.
 The gradient of L with respect to u is the column vector
and the Hessian (curvature) matrix is
Static Optimization
 A critical or stationary point is characterized by the value of u such that
=0
 Suppose that we are at a critical point, so
Lu=0
 For the critical point u to be a local minimum, it is required that
=>0 (Luu is positive definte)
 If Luu 0
 It is a maximum if Luu < 0.  If | Luu | < 0 some times and | Luu | >0,
then u* is a saddle point.
 If |Q| = 0, then u* is a singular point and
in this case Luu does not provide
sufficient information for characterizing
the nature of the critical point.
 Let
1 1 
Q
, S  0 1

1 2
 2  1 0  1 
*
1 T
u  Q S   
 



 1 1  1  1
L*  1 / 2 (Check it ?)
Static Optimization




Example 1 .1-2 . Optimization by Scalar Manipulations
The previous example discussed the optimization in terms of vectors and the gradient.
As an alternative approach, we could deal entirely in terms of scalar quantities.
To demonstrate, let
 where u1 and u2 are scalars.
 Take the derivatives of L with respect to all arguments and equate them to zero:
Solving this system of equations yield
Optimization with Equality Constraints
Define the Hamiltonian function
The necessary conditions for a minimum
point of L(x, u) that satisfies the constraint
f(x, u) = 0 are:
Optimization with Equality Constraints
Example 1.2-1. Quadratic Surface with Linear Constraint
 Suppose the performance index is as given in Example 1.1-1:
and
 Then the Hamiltonian is
 What are the conditions for a stationary point?
Optimization with Equality Constraints
 What are the conditions for a stationary point?
 The conditions are:
 So, the stationary point is
 The Hessian matrix Luu=2 (How?)
Numerical Solution Methods
 Many software routines are available for unconstrained optimization.
 The numerical solution of the constrained optimization problem of
minimizing L(x, u) subject to f(x, u) = 0 can be obtained using the
MATLAB function constr.m which is available under the Optimization
Toolbox.
 This function takes in the user-defined subroutine funct.m, which computes
the value of the function, the constraints, and the initial conditions.
Homework 3
 Solve the following problems from Chapter 1 or Frank Lewis Book
 1.1.1
 1.2.6 (note that in the x value replace = by + after the first term)
Static Optimization
 In this chapter we discuss optimization when time is not a parameter.
 The discussion is preparatory to dealing with time-varying systems in
subsequent chapters.
 1.1 OPTIMIZATION WITHOUT CONSTRAINTS
 A scalar performance index L(u) is given which is a function of a control
or decision vector u ∈ Rm.
 It is desired to determine the value of u that results in a minimum value of
L(u), (How?)
Static Optimization
 Write the Taylor series expansion for an increment in L as
 where O(3) represents terms of order three.
 The gradient of L with respect to u is the column vector
and the Hessian (curvature) matrix is
Static Optimization
 A critical or stationary point is characterized by the value of u such that
=0
 Suppose that we are at a critical point, so
Lu=0
 For the critical point u to be a local minimum, it is required that
=>0 (Luu is positive definte)
 If Luu 0
 It is a maximum if Luu < 0.  If | Luu | < 0 some times and | Luu | >0,
then u* is a saddle point.
 If |Q| = 0, then u* is a singular point and
in this case Luu does not provide
sufficient information for characterizing
the nature of the critical point.
 Let
1 1 
Q
, S  0 1

1 2
 2  1 0  1 
*
1 T
u  Q S   
 



 1 1  1  1
L*  1 / 2 (Check it ?)
Static Optimization




Example 1 .1-2 . Optimization by Scalar Manipulations
The previous example discussed the optimization in terms of vectors and the gradient.
As an alternative approach, we could deal entirely in terms of scalar quantities.
To demonstrate, let
 where u1 and u2 are scalars.
 Take the derivatives of L with respect to all arguments and equate them to zero:
Solving this system of equations yield
Optimization with Equality Constraints
Define the Hamiltonian function
The necessary conditions for a minimum
point of L(x, u) that satisfies the constraint
f(x, u) = 0 are:
Optimization with Equality Constraints
Example 1.2-1. Quadratic Surface with Linear Constraint
 Suppose the performance index is as given in Example 1.1-1:
and
 Then the Hamiltonian is
 What are the conditions for a stationary point?
Optimization with Equality Constraints
 What are the conditions for a stationary point?
 The conditions are:
 So, the stationary point is
 The Hessian matrix Luu=2 (How?)
Numerical Solution Methods
 Many software routines are available for unconstrained optimization.
 The numerical solution of the constrained optimization problem of
minimizing L(x, u) subject to f(x, u) = 0 can be obtained using the
MATLAB function constr.m which is available under the Optimization
Toolbox.
 This function takes in the user-defined subroutine funct.m, which computes
the value of the function, the constraints, and the initial conditions.
Homework 3
 Solve the following problems from Chapter 1 or Frank Lewis Book
 1.1.1
 1.2.6 (note that in the x value replace = by + after the first term)