Benefits of State-Space Method for Modern Control Systems

In general, control systems aim to realise control or balancing tasks in a shorter period time from the human reflex time.

After deciding on a certain control element, the type of signal to be received from or sent to the control system must be decided.

When discussing a control system, first issue to consider is the element that will realise control in the system.

If there is any conversion in question, it must be decided whether to add a converted to the system.

Then, the system must be holistically considered to create a mathematical model and algorithm.

After creating of the algorithm and modelling, control systems are classified under two groups.

The first is comprised of a single input and output.

This method is used in fields which do not necessitate high accuracy.

This a simpler model able to cover a larger number of systems with a single equation, where certain assumptions are made e.g. for demonstration purposes at bachelor level university education in relevant engineering departments.

Such models are equations solved with the "Classical Method."

To explain most common general assumptions in an example;

Considering any mass-coil limb installed on the device, complexity at movement can be analysed with two equations;

- fmy (lxcosΘ) – fnx (lxsinΘ) = IaΘ’’ ... (1)

- fnx = m(d2/dx2) (xlsinΘ) ... (2)

Applying the conversions;

sinΘ = Θ,

cosΘ=1,

Θ, Θ’=0

on these equations, we derive;

- fmy (lx) – fnx (lxΘ) = IaΘ’’ (1*)

- fnx = m(x’’+IΘ’’) (2*)

and linearise the equations.

Or we can decrease its grade and thus derive the desired result by the classical method.

The second method is based on directly solving all equations without any assumptions and completely performing all relevant analyses.

This method is preferred in cases where human life is directly affected or where high accuracy is needed in control actions.

After all analyses are concluded general equation application is commenced.

In application stage, the method which do not present any problem and operate in the desired tolerance range is called as the "State Space."

The method provides a more comfortable solution proposition against complex equations and provides a notation method suggesting a more accurate control theory.

Complexity of a system is not only related to outputs, but can also be related to the inputs.

A system including multiple inputs and outputs turns into a non-linear system, but such a system can still be solved with this method.

However, despite all these advantages, state-space notation might not provide us with what is needed or the equation can become stuck.

Thus, the classical system is saved from being mothballed due to these continued complaints and the inability to find a solution to this issue as yet.

It can be seen that classical method can be of help at a certain point or at the beginning in certain cases.

A correlation must be established between the two methods to realise the synergy mentioned above.

Solution of the equation can be facilitated by use of both methods thanks to a correlation between the transfer function and state-space notation.

Adil Can Kavcar, 07 April 2023, Izmir-Turkey