• To put it simply: we figure out how the process behaves naturally, we determine how we want it to behave, and we insert a controller to make it do what we want.

• CONTROL: Using artificial means to manipulate the world: with a particular goal.

• Continuous: the values to be controlled change smoothly. e.g. the speed of a car

• Discrete: The value to be controlled are easily described as on-off. e.g. the car motor is on-off (like basic pneumatics). NOTE: all systems are continuous (Except for Heisenberg’s electrons) but they can be treated as discrete for simplicity.

• Linear: Can be described with a simple differential equation (not a very accurate explanation).

e.g. a car can be driving around a track and can pass same the same spot at a constant velocity. But, the longer the car runs, the mass decreases, and it travels faster, but requires less gas, .......... etc. Basically, the math gets tougher, and the problem becomes non-linear.

This is the preferred starting point for simplicity, and a common approximation for real world problems.

• Non-Linear: Not Linear. This is how the world works, but is very complicated. Especially when trying to do mathematical approximations. (Note: if the coefficients in a differential equation change it is non-linear)

• Temporal/sequential: the controller must not just keep track of things that changes, but it must know the time, or how long since something happened.

non-linear: as rocket approaches sun, gravity increases, so control must change.

linear: We are driving the perfect car with no friction, with no drag, and can predict how it will work perfectly.

discrete: “When I do this, that always happens!” For example, when the power is turned on, the press closes!

How to tell the difference? An elevator is the perfect example.

1. The elevator must move towards a floor when a button is pushed.

2. The elevator must open a door when it is at a floor.

3. It must have the door closed before it moves.

Linear (our approximation commonly used in industry) NOTE: each floor will have a certain motor position, and we know what that position is.

1. If the desired position changes to a new value, accelerate quickly towards the new position.

2. As the elevator approaches the correct position, slow down.

Non-linear (not yet common in industry) e.g. Fuzzy logic, Neural Networks, Adaptive Control

2. Decelerate as you approach the final position.

3. Allow faster motion while moving.

4. Compensate for cable stretch, and changing spring constant, etc.

• Control systems are present in most systems we see. Example: elevator height.

• Some systems are naturally unstable and tend to self destruct. Example: balanced broom.

• Other systems are self regulating, and tend to some stable state: these are good candidates for open loop control. Example: city water tank

• Other systems need some sort of regulation mechanisms added, these are called closed loop systems. Example: car cruise control

• If a system is simple we will say that it is linear. Example: perfect elevator

• A system can be subject to complicating factors that make it difficult/impossible to model mathematically. Example: elevator in tall building with stretchy cable

• If a system has switched states it can be described as discrete. Example: buttons to call elevator, open/close doors

• Temporal/sequential systems can change over time, this requires sequential control. Example: washing machine

Problem 1.1 Consider a heater in your house, how can different forms of control be applied to this problem?

Problem 1.2 Why is discrete control so popular when continuous control allows more precision?