Control theory
- This article is about an engineering theory called control theory. There is also a sociological theory of deviant behavior that is called control theory.
Table of contents |
2 Classical control theory 3 Stability 4 State space representation 5 Simple example of state space representation 6 Controllability and observability 7 See also |
An example
As an example, consider cruise control. In this case, the system is a car. The goal of the cruise control is to keep it at a constant speed. So, the output variable of the system is the speed of the car. The primary means to control the speed of the car is the amount of gas being fed into the engine.
A simple way to implement cruise control is to lock the position of the gas pedal the moment the driver engages cruise control. This is fine if the car is driving on perfectly flat terrain. On hilly terrain, the car will accelerate when going downhill and slow down when going uphill; something its driver may find highly undesirable.
This type of controller is called an open-loop controller because there is no direct connection between the output of the system and its input. One of the main disadvantages of this type of controller is the sensitivity to the dynamics of the system under control.
Classical control theory
To avoid the problems of the open-loop controller, control theory introduces feedback. The output of the system y is fed back to the reference value r. The controller C then takes the difference between the reference and the output, the error e, to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.
A simple feedback control loop
If we assume the controller C and the plant P are linear, time-invariant and all single input, single output, we can analyse the system above by using the Laplace transform on the variables. This gives us the following relations:
Stability
Stability in control theory means that for any bounded input over any amount of time, the output will also be bounded. Mathematically, that means for a system to be stable all the poles of its transfer function must lie in the left half of the complex plane. Or simpler put, the real part of every complex number that makes the transfer function become infinite, has to be negative for the whole system to be stable.
State space representation
To get a coherent model for systems with multiple inputs and multiple outputs, we need a way to record every relation between any input variable and any output variable. With n inputs and m outputs, we have to write down mn Laplace transforms to encode all the information about a system. A more compact representation of a system is its state space representation using p internal states:
- .
The internal states are stored in the x(t) vector
- is the input vector
- is the output vector
Simple example of state space representation
Let's take the simplest possible example of the state space representation. Here we have a single input variable and a single output variable. Also the internal state of the system is only represented by a single variable.
- y(t) is the output variable
- u(t) is the input variable
- x(t) is the internal state variable
Controllability and observability
Controllability is a measure for the ability to use a system's external inputs to manipulate its internal state. Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals.