Control Systems: Overview

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:

• Classical methods use various transforms (Laplace, Fourier) to convert the set of equations from the time domain into the frequency domain. This can bybass many complications of handling derivatives and integrals.

• Modern methods simply a higher order differential equation into a set of lower order differential equations which can be manipulated more easily. These sets of equations are called State equations.

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

• Robust control attempts to acount for inherent inaccuracies an uncertainties in measurement of signals in the systemm.

• Adaptive control tries to evolve its control strategy as the characteristics of the system change over time which renders a static system model useless.

• Optimal control uses some measure to evaluate the state of a system. It tries to maximize that value over time.

• Non-linear control wrestles with systems that cannot be modelled as linear differential equations and so do not lend themselves to a host of methods applicable to solving linear equations.

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

• Continuous systems vary smoothly as time increases. This is the default assumption when modelling systems in control theory as differential equations and transforms are indended for continuous variables.

• Discrete systems vary in time steps. This means there are discontinuities in the system model (infinite gradients etc.). For these class of systems, the z-transform can be used to represent them in a more continuous-friendly form.

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:

• Additivity: The sum of outputs of two inputs is the same as the output of the sum of two inputs. Mathematically, if $y$ is the system function, and $x_1$ and $x_2$ are inputs, then:

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

• Homogeneity: If the input scales by a factor $k$, then the output scales by the same factor:

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

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:

• Step response is the output of the system in response to a constant $1$ input starting at initial time.

• Target value is the output desired from a given input.

• Rise time is the time taken to get within a margin of the target value for the first time.

• Percent overshoot is how much the response overcompensated relative to the target value.

• Steady state error is the final, constant error of the response as $t \rightarrow \infty$.

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:

• The highest exponent in the Laplace domain,
• Number of variables in state space equations (see Modern control methods),
• Number of energy storage elements in the model (e.g capacitors, inductors, not resistors).