Control Systems: Overview

Monday, Sep 10, 2018
Categories: engineering,
Series: AI methods for control systems,
Tags: matlab,

Overview of control theory

A control system is a mechanism that dictates the behaviour of another system. A controller relies on signals it can measure to estimate performance of the system. In response it sends out signals that influence the system towards the desired state.

A controller is designed by modelling a system as a set of differential equations in time. By analysing that model, a control strategy can be developed. There are two approaches to solving systems modelled as differential equatiosn:

Alternatively, the field of control systems can be divided by features or restrictions present in systems:

Yet another way to divide control theory is in how systems are modelled in time:

What is a system?

A system is any mechanism that takes an input and produces an output. How systems operate in inputs to produce an output or a response distinguishes them.

LTI Systems

A class of systems convenient to work with is called Linear Time Invariant (LTI) systems. Such systems satisfiy two properties:

Linearity

A linear system is one where the response scales or accumulates as the inputs scale or accumulate. There are two criteria for linearity:

$$ y(x_1 + x_2) = y(x_1) + y(x_2) $$

$$ y(k \cdot x) = k \cdot y(x) $$

Time-invariance

A time-invariant system does not change its response to the same output at different times. This means that the system output is purely a function of the input and not of time.

A time-invariant system:

$$ y(x(t)) = 3 \cdot x(t)^2 + x(t) $$

A time-variant system:

$$ y(x(t)) = t \cdot x(t) $$

System attributes

Stability

A stable system is one where in response to a bounded input, the output is also bounded. For example, a system susceptible to resonance frequencies will be unstable.

Initial time

The initial time of a system is before which there was no input.

Steady state response

The steady state response of a system is the output when the system has settled in response to inputs. It is defined as the output as $t \rightarrow \infty$. There are several parameters that describe the temporal behaviour of the system:

Order

The order of a system is a representation of its complexity. It can be denoted by various properties depending on the formulation of the system: