0.1 Lecture 2: introduction to systems

Lecture #2:

INTRODUCTION TO SYSTEMS

Motivation: To describe man-made and natural systems, in forms of mathematical expressions in time and in frequency domains. Many physical systems are described by linear differential equations or by linear difference equations.

Outline:

• Man-made systems — modular, hierarchic design. Natural systems — modular, hierarchic analysis. Dynamic analogies

• Classification of systems

• Reducing differential equations to algebraic equations by using complex notations. Role of complex exponential time functions in LTI systems. Sources of linear differential equations. Homogeneous solution — exponential solution, natural frequencies. Particular solution — system function, poles&zeros. Total solution — initial conditions, steady-state.

• Reducing difference equations to algebraic equations. Role of complex geometric (exponential) time functions in DT LTI systems. Linear difference equations arise as system. Homogeneous solution — geometric (exponential) solution, natural frequencies. Particular solution — system function, poles&zeros. Total solution — initial conditions, steady-state.

Signals and systems

This subject deals with mathematical methods used to describe signals and to analyze and synthesize systems.

• Signals are variables that carry information

• Systems process input signals to produce output signals.

Last time — SIGNALS; Today — SYSTEMS.

I. MAN-MADE SYSTEMS — MODULAR, HIERARCHIC DESIGN

Robot car

1/ Robot car block diagram

Hierarchic design — top (1st) level includes: wheel position controller, digital camera, and image processing software.

2/ Wheel position controller block diagram

Hierarchic design — 2nd level is a block diagram of the wheel controller which includes: amplifier, motor, and shaft decoder.

3/ Motor dynamics

Hierarchic design—3rd level includes a more detailed description of the motor. Important quantities are:

• Current in motor windings i(t),

• Motor shaft angular displacement θ(t),

• Motor parameters — viscous damping constant B, moment of inertia J, electromechanical constant k.

The torque balance equation is:

$\text{ki}(t)-B\frac{\mathrm{d\theta }(t)}{\text{dt}}=J\frac{{\mathrm{d\theta }}^2(t)}{{\text{dt}}^2}$

4/ Observation

• Man-made complex systems are designed in a modular, hierarchical fashion often expressed in nested block diagrams.

• Block input/output relations provide a communication mechanism for team projects.

• Optimization of system performance requires excellent tools to characterize signal transformations at each level of the hierarchy.

II. NATURALLY OCCURRING SYSTEMS — MODULAR, HIERARCHIC ANALYSIS

Human speech production system — anatomy

1/ Human speech production system — block diagram

2/ Observation

• Naturally occurring systems are not designed in a modular fashion — they have evolved.

• To understand these systems, we impose a hierarchy and parse the system into modules whose function can be characterized.