What is Order and Type of a System in Control Theory , First Order Vs Second Order Vs...


  • System is a generalized term for something that process an input to produce output or response. System may be physical like electrical or mechanical or it may be based on algorithm like information system. We will focus on physical system.

A system from inside comprises of various components. Because of combination of those components, a particular system have particular property which tells how a system process input to produce an output.
System property or behavior can be approximated by mathematical equations.
If the system behavior resembles first order differential equation or equations, it is called First order system. 
i.e the mathematical equation used to model the relation between the I/P and O/P of the system is a first order differential equation.

If resembles second order differential equation or equations it is second order system.
For example: RC circuit is an electrical system. It is composed of a resistor and a capacitor. Input to the system is either current or voltage. Output may be current or voltage. So, the mathematical model can be formed using KVL or KCL. You will see that the input-output equation is a first order differential equation. You can say that, the system is a first order system.
Similarly RLC circuit is a second order system.

https://www.quora.com/What-are-first-order-and-second-order-systems

http://engineering.ju.edu.jo/Laboratories/01-Introduction-to-Control-Systems.pdf

http://engineering.ju.edu.jo/Laboratories/05-%20Mathematical%20Modeling%20of%20DC%20Motor.pdf

http://engineering.ju.edu.jo/Laboratories/02-Mathematical-Modeling-Using-Simulink.pdf

  • First order systems are, by definition, systems whose input-output relationship is a first order differential equation. A first order differential equation contains a first order derivative but no derivative higher than first order – the order of a differential equation is the order of the highest order derivative present in the equation. 
  • First order systems contain a single energy storage element. In general, the order of the inputoutput differential equation will be the same as the number of independent energy storage elements in the system. Independent energy storage cannot be combined with other energy storage elements to form a single equivalent energy storage element. For example, we previously learned that two capacitors in parallel can be modeled as a single equivalent capacitor – therefore, a parallel combination of two capacitors forms a single independent energy storage element. 
  • There is another concept in control systems called type of the system, which is not to be confused with order of the system. The type of the system gives the number of integrators([1][/s]) the system has. The type of the system tells us what possible inputs the system can follow/track with zero steady state error. A type-1 system can follow a step input with zero steady state error, type-2 can track a ramp input, type-3 a parabolic input and so on.
  • The order of a linear system is the number of derivatives (or integrals) needed to model the dynamics of the system. When taking the Laplace transform of the model’s differential equations, those derivatives will be replaced by the “s” variable and you can build the transfer function between the input and the output. The largest exponent of “s” in the denominator (or the number of poles) will also match the order of the system. 
hence by definition, we also say that the order of a system is the value of the highest exponent that appears in the denominator of the transfer function.
So, the first difference between first order and second order system is the number of poles.

The order of a system is equal to the number of poles.


That being said, a first order system having one pole, that pole must be real (any complex or imaginary pole must also have its conjugate in order to have a real temporal response). The second order system can have complex poles, which makes the temporal response possibly exhibit oscillations. If both poles of a second order system are real, it will react as 2 first order system put together (no oscillation). Note: I used transfer function as an example, the same can be said if you use a State-Space model. Such a model divides the dynamic equations into several equations having only one derivative per equation (state equation). A first order system would only have one state equation and a second order system would have 2. Those state equations can be represented in matrix form as x˙=Ax+Bu where x is the vector of states, u is the input, and A and B are matrices (assuming your equations are linear). The eigenvalues of A would be the same as the poles of the transfer function (one for a first order system, 2 for a second order system).

https://www.edaboard.com/showthread.php?96790-Definition-1st-order-system-and-2nd-order-system

  • 1st order system contains 1 energy storing element and 1 element which dissipates energy. e.g mass - damper system. system represented by (s+2)/(s+5) is 1st order system as second-order system has no zeros in the transfer function.
  • Is it possible that the pole of a first-order system is a complex number?

Not if the transfer function represents a physical system, such as a control system or filter.
Poles of a transfer function representing a physical system are either real, or occur in complex conjugate pairs only. Of course, a pole can be complex, but since it does not represent a physical system, it would be of interest only to pure mathematicians, not engineers.
  • Assume that there is a 1st-order physical system with complex pole, what will be step response of such system? Can we solve it 'mathematically'?
A 1st order system with a complex pole can be solved, but the solution would be meaningless in a physical sense. For example, consider the following transfer function, which has a single complex pole:
F1(s) = 1/(s + (a+jb)) = [1/(a+jb)] / [1 +(1 + (1/a+jb)s].
Calculate the step response of the system represented by this transfer function:
F2(s) = [1/s]F1(s).
The solution is:
f(t) = [1/(a+jb)][1-exp(-t/)a+jb)).
This is a perfectly good mathematical solution, but it has no meaning in a physical sense, because of the 1/(a+jb) factor.

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