Introduction

Signal

a function of time

System

a process that transforms one signal (input) into another signal (output); a bounded collection of interacting eleemets that gve rise to some collective bahavior of interest

Bounded - System Definition

means that it is separable from its environment (phsyically or abstractly) and thus the behavior takes form as some exchange (material/energy) across the boundary

Elements - System Definition

the system is made up of identifiable parts (sub system) and there can be many levels (system hierarchy)

Interacting - System Definition

behavior of one point influences the behavior of another part (ex. series, parallel, and feedback connections)

Collection - System Definition

all elements/interactions are taken together to give the overall behavior; complexity can emerge from considering everything simultaneously

Behavior - System Definition

time evolution of some obserable output of the system in response to some input (cam have more than 1 input/output)

Dynamic System

a system in which the present value of an outpur depends on past values of an input and the initial conditions; has memory; includes state variables; variables are extensive or intensive ex. biological systems

Static System

a system in which the present value of an output depends only on the present value of an input; no memory

State Variables

the internal variables, that together with any input, are sufficient to determine any output; ex. RLC circuit (i_L(t) and v_c(t))

Extensive Variable

a variable where the total must be accounted for by addition in a dynamic system; ex. mass, volume, and charge

Intensive Variable

a variable that may not be added to get a total in a dynamic system; ex. pressure or density

Modeling

how dynamic systems concepts are emplyed in engineering; the representation of systems by mathematical expressions involving input, state, and output (putting math to a dynamic system)

Partial Differential Equations (PDEs)

derivatives with respect to (w.r.t) space and time; more accruate

Ordinary Differential Equations (ODEs)

derivatives with respect to (w.r.t) time only; more tractable

Reasons for Modeling

1. develop deeper understanding of physiological/biological systems

2. expand knowledge through hypothesis generation

3. use as a substitute to experimentation

4. predict system behavior

5. use for system identification

6. employ simulations for demos/education

Reasons for Modeling #1

develop a deeper understanding of physiological/biological systems; possible when can describe the system in sufficient detail, predict system behavior that has not been previously observed, and explain mechanisms that give rise to the system behavior

Reasons for Modeling #2

expand knowledge through hypothesis generation; lead to enhanced appreciation of limits of understanding so can help generate new hypotheses to be tested

Reasons for Modeling #3

use as a substitute to experimentation; bc with a good model, it’s efficient to conduct virtual experiments first to guide subsquent actual experimentation

Reasons for Modeling #4

predict system behavior; this is useful bc can help forecast a future event (ex. wll a patient be susceptible to a heart attack in this certain time period?), reveal signals that aren’t accessible by measurement (ex/ blood flow), and determine what will happen if some change is made to the system

Reasons for Modeling #5

use for system identification; can adjust the parameters of the model to fit measurements from the actual system (ex. a patient’s mods); can be helpful in diagnostics and monitoring

Reasons for Modeling #6

emplot simulations fro demos/education

Steps in Modeling

1. build model

2. solve model eqns.

3. validate model eqns.

4. apply validated model for design and discovery

Steps in Modeling #1 - Build the model

a. need to establish a level of sophistication

b. define the system (establish boundaries (system/environment), identify inputs/outputs)

c. construct the model eqns. (make assumptions, apply phsyical laws and constitutive relations, and manipulate model eqns. for classification)

Physical Laws

any law that follows conservations laws; ex. conservation of mass, energy, or momentum; always true

Constitutive Relations

used for intensive variables; ex. Hooke’s law (F=k(L-L0)), Ohm’s law (V = IR); empirical

Classifications of Model Eqns.

order, linear vs non-linear, time-invariant vs time-varying, and autonomous (homogenous) vs non-autonomous

Steps in Modeling #2 - Solve Model Eqns.

use analytical for LTI models (low order, lineaer, and time-invariant) and numerical otherwise

Steps in Modeling #3 - Validate Model Eqns.

3 types - descriptive, predictive, and explanative

Descriptive Validation

model fits the measurements; weakest bc this is essentially the bare minimum

Predictive Validation

model forecasts previosly unobserved measurements; strongest

Explanative Validation

model explains mechanisms