Dynamic Optimization

Abstract

Optimization of linear systems of the type (7.1) usually relies upon minimization of the quadratic criterion

$$I = \mathop \smallint \limits_0^\infty ({x^T}Q + {u^T}Ru)dt,$$

((9.1))

where Q is a positive semi-definite symmetric matrix, and R is a positive definite symmetric matrix. If the controlled variable is of the form y = Dx where y∈ℝ^k, k<n, and D is constant kxn matrix, then taking

$$Q = {D^T}D\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0$$

((9.2))

one obtains that the first term in the criterion (9.1) is yTy and characterizes the degree of deviation of the controlled variable from the zero state. The second term in the criterion defines the penalty for controll “expenses”. The relation between the weight matrices Q and R defines the tradeoff between the two contradictory desires such as to have a rapidly decaying control process and to reduce power consumption for its realization.