Optimal Control .Ppt
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What is Optimal Control Theory? Dynamic Systems: Evolving over time. Time: Discrete or continuous.
Optimal way to control a dynamic system.
Prerequisites: Calculus, Vectors and Matrices, ODE&PDE
Applications: Production, Finance/Economics, Marketing and others.
Basic Concepts and Definitions A dynamic system is described by state equation:
where x(t) is state variable, u(t) is control variable. The control aim is to maximize the objective function:
Usually the control variable u(t) will be constrained asfollows:
Sometimes, we consider the following constraints:
(1) Inequality constraint
(2) Constraints involving only state variables
(3) Terminal state
where X(T) is reachable set of the state variables at time T.
Formulations of Simple Control Models Example 1.1 A Production-Inventory Model.
We consider the production and inventory storage of a given good in order to meet an exogenous demand at minimum cost.
Table 1.1 The Production-Inventory Model of Example 1.1
Example 1.2 An Advertising Model. We consider a special case of the Nerlove-Arrow
advertising model.
Table 1.2 The Advertising Model of Example 1.2
Example 1.3 A Consumption Model. This model is summarized in Table 1.3:
Table 1.3 The Consumption Model of Example 1.3
History of Optimal Control Theory Calculus of Variations. Brachistochrone problem: path of least time Newton, Leibniz, Bernoulli brothers, Jacobi, Bolza. Pontryagin et al.(1958): Maximum Principle.
Figure 1.1 The Brachistochrone problem
Notation and Concepts Used = “is equal to” or “is defined to be equal to” or “is identically equal to.” := “is defined to be equal to.” “is identically equal to.” “is approximately equal to.” “implies.” “is a member of.”
Let y be an n-component column vector and z be an m-component row vector, i.e.,
when n = m, we can define the inner product
If
is an m x k matrix and B={bij} is a k x n matrix, C={cij}=AB,which is an m x n matrix with components:
Differentiating Vectors and Matrices with respect to Scalars
Let f : E1Ek be a k-dimensional function of a scalar variable t. If f is a row vector, then
If f is a column vector, then
Differentiating Scalars with respect to Vectors If F: En x Em E1, n 2, m 2, then the gradients Fy
and Fz are defined, respectively as;
Differentiating Vectors with respect to Vectors If F: En x Em Ek, is a k – dimensional vector function, f
either row or column, k 2; i,e;
where fi = fi (y,z), y En is column vector and z Em is row vector, n 2, m 2, then fz will denote the k x m matrix;
fy will denote the k x n matrix
Matrices fz and fy are known as Jacobian matrices.
Applying the rule (1.11) to Fy in (1.9) and the rule (1.12) to Fz in (1.10), respectively, we obtain Fyz=(Fy)z to be the n x m matrix
and Fzy = (Fz)y to be the m x n matrix
Product Rule for Differentiation
Let x En be a column vector and g(x) En be a row vector and f(x) En be a column vector, then
Vector Norm The norm of an m-component row or column vector z is
defined to be
Neighborhood Nzo of a point is
where > 0 is a small positive real number. A function F(z): Em E1 is said to be of the order o(z), if
The norm of an m-dimensional row or column vector function z(t), t [0, T], is defined to be
Some Special Notation left and right limits
xk: state variable at time k.
uk: control variable at time k.
k : adjoint variable at time k, k=0,1,2,…,T.
: difference operator. xk:= xk+1- xk.
xk*, uk*, and k, are quantities along an optimal path.
discrete time (employed in Chapters 8-9)
sat function
bang function
impulse control
If the impulse is applied at time t, then we calculate the objective function J as
Convex Set and Convex Hull A set D En is a convex set if y, z D,
py +(1-p)z D, for each p [0,1].
Given xi En, i=1,2,…,l, we define y En to be a convex combination of xi En, if pi 0 such that
The convex hull of a set D En is
Concave and Convex Function : D E1 defined on a convex set D En is concave
if for each pair y,z D and for all p [0,1],
If is changed to > for all y,z D with y z, and 0<p<1, then is called a strictly concave function.
If (x) is a differentiable function on the interval [a,b], then it is concave, if for each pair y,z [a,b],
(z) (y)+ x (y)(z-y).
If the function is twice differentiable, then it is concave if at each point in [a,b],
x x 0 . In case x is a vector, x x needs to be a negative definite
matrix.
If : D E1 defined on a convex set D En is a concave function, then - : D E1 is a convex function.
Figure 1.2 A Concave Function
Affine Function and Homogeneous Function of Degree One : En E1 is said to be affine, if (x) - (0) is linear. : En E1 is said to be homogeneous of degree one, if
(bx) = b (x), where b is a scalar constant.
Saddle Point : En x Em E1, a point En x Em is called a saddle
point of (x,y), if
Note also that
Figure 1.3 An Illustration of a Saddle Point
Linear Independence and Rank of a Matrix A set of vectors a1,a2,…,an En is said to be linearly
dependent if pi,, not all zero, such that
If the only set of pi for which (1.25) holds is p1= p2= ….= pn= 0, then the vectors are said to be linearly independent. The rank of an m x n matrix A is the maximum number of
linearly independent columns in A, written as rank (A). An m x n matrix is of full rank if rank (A) = n.