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Modern Control Systems

Matthew M. Peet Illinois Institute of Technology

Lecture 1: Modern Control Systems

MMAE 543: Modern Control Systems

This course is on Modern Control Systems

• (^) Techniques Developed in the Last 50 years
• Computational Methods

I (^) No Root Locus

I (^) No Bode Plots

I (^) No pretty pictures

Classical Control Systems is a different Course

• (^) MMAE 443 - Systems Analysis and Control

We focus on State-Space Methods

• (^) In the time-domain
• We use large state-space matrices

d

dt

x 1 (t)

x 2 (t)

x 3 (t)

x 4 (t)

x 1 (t)

x 2 (t)

x 3 (t)

x 4 (t)

[

u 1 (t)

u 2 (t)

]

• We require Matlab

I (^) Need robust control toolbox (Available in MMAE computer lab)

MMAE 543: Modern Control Systems

A Computational Class

• Divides modern from classical
• Solutions by hand are too complicated
• A new definition of solution

I (^) Design an algorithm

• We have classes of solutions

I (^) Convex Optimization

I (^) Linear Matrix Inequalities

A Mathematical Class

• (^) We will use mathematical shorthand

I (^) ∀ is “for all”

I (^) X ∩ Y means “intersection of X and Y ”

I (^) X ∪ Y means “union of X and Y ”

I (^) x ∈ Y means x is an element of Y

• (^) We will use proofs.

I (^) Based on definitions

I (^) Wording is important

Proof Example

Begin with a theorem statement

Theorem 1.

lim x→∞

1 + x

To prove such a things, we need to understand what it means.

Definition 2.

The z is limit limx→∞ f (x) if for any δ > 0 , there exists an  > 0 such that for

any x > , ‖z − f (x)‖ ≤ δ

Example: The Shower

Goal: Not too hot, not too cold

• (^) Adjust water temperature to within acceptable tolerance

Human-in-the-Loop Control: How to do it?

• (^) Too cold - rotate knob clockwise
• (^) Too hot - rotate knob counterclockwise
• (^) Much too cold - rotate faster
• (^) Much too hot - rotate faster

If we go too fast, we overshoot

• Abstract the system using block diagrams

Example: Heating System

Air conditioning is an abstract version of the shower problem

Sensor: Thermometer

• (^) When sensed temperature is below desired, the thermostat opens the gas

valve.

• (^) Feedback is Desired Temperature - Sensor Temperature = Error in

Temperature

• The thermostat is desired to drive the error to zero

Examples of Systems

College

Definition 4.

The System to be controlled is called the Plant.

Control Systems

Definition 5.

A Control System is a system which modifies the inputs to the plant to

produce a desired output.

The Basic Types of Control

The first basic type of control is Open Loop.

Definition 7.

An Open Loop Controller has actuation, but no measurement.

The Two Basic types of Control

The second basic type of control is Closed Loop.

Definition 8.

The Sensor is the mechanism by which the controller detects the outputs of the

plant.

Definition 9.

A Closed Loop Controller uses Sensors in addition to Actuators.

Control System

Plant

Sensor

Lets go through some detailed examples.

History of Water Clocks

Time passed is amount of water in pot.

Problem: Water flow varies by amount of water in the top pot.

Solution: Maintain a constant water level in top pot.

History of Water Clocks

Problem: Manually refilling the top pot is labor intensive and inaccurate.

Solution: Design a control System (Inputs, Outputs?).

History of Water Clocks

Heron (Hero) of Alexandria c. 10 AD

As any good student, Hero used Ctesibius’ water clock

to perform party tricks.

The self-replenishing wine bowl. (Inputs, Outputs?)

History of Water Clocks

The Pipe Organ

Ctesibius himself applied the principle of pneumatic control to create a pipe

organ.