Linear differential equations with complex exponential Inputs Motivation: Many physical systems are described by linear differential equations Many physical systems are described by linear differential equations Reducing differential equations to algebraic equations Reducing differential equations to algebraic equations Link between structural and functional descriptions of systems Link between structural and functional descriptions of systems Role of complex exponential time functions in LTI systems Role of complex exponential time functions in LTI systems

Linear, ordinary differential equations arise for a variety of system descriptions Electric network

Mechanical system The simplest possible model of a muscle (the linearized Hill model) consists of the mechanical network shown below which relates the rate of change of the length of the muscle v(t) to the external force on the muscle f e (t). f c is the internal force generated by the muscle, K is its stiffness and B is its damping. The muscle velocity can be expressed as v(t) = v c (t)+v s (t).

Mechanical system, contd Combining the muscle velocity equation with the constitutive laws for the elements yields:

General differential equation For many (lumped-parameter or compartmental) systems, the input x(t) and the output y(t) are related by an nth order differential equation of the form: We seek a solution of this system for an input that is of the form x(t) = Xe st u(t) where X and s are in general complex quantities and u(t) is the unit step function. We will also assume that the system is at rest for t < 0, so that y(t) = 0 for t < 0.

Homogeneous solution Exponential solution Let the homogeneous solution be y h (t), then the homogeneous equation is: To solve this equation we assume a solution of the form: y h (t) = Ae λt. SinceWe have therefore:

Example natural frequencies of a network The differential equation relating v(t) to i(t) is: The characteristic polynomial is:

Edison4-- simulation

Simulated Results

Electrical Parallel System

System Response to Step Input

Series System

Simulation of Series to Step Input

Example natural frequencies of a network, contd The roots of this quadratic characteristic polynomial, the natural frequencies, are: The locus of natural frequencies is plotted for L = C = 1 with R increasing in the directions of the arrows. Note the natural frequencies are in the left-half plane for all values of R > 0. As R, the natural frequencies approach the imaginary axis. What is the physical significance of this behavior?

Example reconstruction of differential equation from H(s) Suppose: what is the differential equation that relates y(t) to x(t)? Cross multiply the equation and multiply both sides by e st to obtain: (s+1)Y e st = sXe st which yields sY e st +Y e st = sXe st, from which we can obtain the differential equation

Problem 3-1 Given the system function: Determine the natural frequencies of the system. Determine a differential equation that relates x(t) and y(t).

Solution ….