L02 - Linear time-invariant (LTI) dynamical systems ; state space and transfer function representations

System

A composition, a whole of members.

NO CIRCUITS!

We don't care about circuits in this course but we will use them sometimes as examples

Continuous-Time ()

That's just what we call it. We always asssume there's an initial time. If none is specified, assume

Differential Equations are Involved

Quantities in a system are related by differential equations

We will always know the input.

What we look for is the output.

Goal is solution of a system

Computing the evolution in time of given the evolution in time of and the mathematical model of the system

Typical Inputs

- Dirac Delta, function that is zero everywhere but at x = 0, where it's infinite. Physically it doesn't make sense but it can be used mathematically to make sense of things.

- Unitary Amplitude Step (Step Function!)
Zero at any point before 0. Shift applies to this. Constant if we start with

- Linear ramp, 0 before 0, linear after 0, basically,

- Exponential function

- Harmonics IMPORTANT!

Static System

Evolves in time but the input-output relationship is static, in other words, the output depends on only the input at that instant.
WE DO NOT NEED THIS! WE WILL ONLY BE STUDYING DYNAMICAL SYSTEMS.

Dynamic System

Input-output relationship is dynamic, depends not only on input but also on history of the input and initial conditions of the system.

Proper definition of a LTI system

The system is Linear. Time-Invariant does not refer to inputs and outputs, it refers to the system parameters.
In a Linear Time-Invariant (LTI) system, inputs and outputs do change. The system parameters are the ones that don't change.

Matrix Representation and Proper definition of LTI systems

The output equation is algebraic, linear.

is the state matrix
is the input matrix
is the output matrix
is the input-output matrix

If a model is too complex, it can't be modelled mathematically. If it's too simple, it's unrealistic.

We will want to solve the state-space equation. After that, the output equation is trivial.

Matlab - Control Systems Toolbox

ss - Matlab command, defines LTI system given matrices

Lagrange Equation WILL NOT BE NEEDED, but could be used in some cases

- Zero-input response, state response when input is zero,
- Zero-state response. Does not refer to state being zero. It's the state response for

These are also known as the natural and force responses.

The same way, we have these for the output responses:
- Zero-input output response
- Zero-state output response