Models in many disciplines are specified by algebraic equations. In filter engineering, they are always specified by differential equations (DEs), and in this chapter we develop the necessary background to enable us to use DEs as models. For each DE we will see that there is a unique transition matrix, and it is through the transition matrix that the DE is actually implemented. Our discussion is thus about DEs and their transition matrices, and how such matrices are derived.

Chapter Contents:

• 2.1 Linearity
• 2.1.1 Sets of linear equations
• 2.1.2 Linear independence
• 2.1.3 Linearity and differential equations
• 2.1.4 Constant coefficient linear DEs
• 2.1.5 Time-varying linear DEs
• 2.1.6 Nonlinear DEs
• 2.2 The two types of models
• 2.2.1 The external model
• 2.2.1.1 Fundamental assumption on which this book is based
• 2.2.2 The filter model
• 2.2.3 Our approach to DEs
• 2.3 Models based on polynomials
• 2.3.1 Notation
• 2.3.2 The transition matrix and the transition equation
• 2.3.3 The curve implied by a transition equation
• 2.3.4 The observed trajectory
• 2.3.5 Working in 3-dimensional space
• 2.3.6 Equally spaced observation instants
• 2.4 Models based on constant-coefficient linear DEs
• 2.4.1 One way to derive transition matrices
• 2.4.2 Every transition matrix is nonsingular
• 2.4.3 A general way to find transition matrices for constant-coefficient linear DEs
• 2.4.4 The DE governing a transition matrix
• 2.5 Models based on time-varying linear DEs
• 2.5.1 Comparison with constant-coefficient linear DEs
• 2.5.2 Obtaining the transition matrix Φ(tn + ζ, tn )
• 2.6 Models based on nonlinear DEs
• 2.6.1 The method of local linearization
• 2.6.2 Using the results from time-varying linear DEs
• 2.6.3 Summary
• 2.6.4 Examples of analytical solutions
• 2.7 Numerical partial differentiation
• Appendix 2.1 Linear independence of a set of vectors
• Appendix 2.2 The polynomial transition matrices
• Appendix 2.3 Derivation of the DE for the transition matrix Φ(tn + ζ, tn )
• Appendix 2.4 The method of local linearization
• Appendix 2.5 Proof of Theorem2.1: Every transition matrix Φ(ζ) is nonsingular
• Appendix 2.6 A general way to find transition matrices

Inspec keywords: differential equations; matrix algebra

Other keywords: algebraic equations; transition matrices; filter engineering; DE; differential equations

Subjects: Algebra, set theory, and graph theory; Function theory, analysis; Algebra; Algebra; Algebra