# Find system behavior: P or I type?

In control theory, how do I find out of which type a system is? P-type or I-type (P = proportional and I = integral)

Consider the following nonlinear system with x as output:

$$\dot{x} = A(x) + Bu$$

$$y = x$$

Is this system of I-type because of 1/s (integrator)

$$x = \int\dot{x}$$

Or is it not that simple to claim of which type the system is? How is it done?

EDIT

I am using a PI controller. And the complete state space representation using A(x) is (a, b and g are constants > 0)

$$\dot{x} = -a\sqrt{2gx}+bu$$

• What's A(x) ? A as function of x? – Eugene Sh. Jun 8 '15 at 21:55
• yes exactly, this should describe the nonlinear behaviour – fjp Jun 8 '15 at 21:57
• P = proportional, and I = integral? I had never heard of P-type and I-type before (I have taken controls before...), and Google didn't show anything. – Mewa Jun 8 '15 at 21:59
• correct, I couldn't find anything on this topic too. Therefore I asked here. Though I think I heard that a control system can behave in different ways (P or I). And depending on this behaviour you can choose a suitable controller (P or I but mostly PI). However, I really don't know how to find out of which type it is. – fjp Jun 8 '15 at 22:02
• I think you should search for instead PID (proportional-integral-derivative) control. P and I are commonly used in control schemes, and the D sometimes, in very small amounts. – rdtsc Jun 8 '15 at 22:11

For single input , single output linear systems we define 'type' according to the number of integrators (number of free 's' in the denominator) in the open loop transfer function. A type zero system would have zero integrators, a type one, one integrator, and so forth.

The significance of classifying system types gives you an idea of how the system will behave to a specific type of input signal if you apply simple, proportional feedback to the open loop system.

For a type 0 system with step input you would see a finite steady state error, proportional to the loop gain.

For a type 1 system with step input you would see a zero steady state error.

This page provides a more complete summary (table) for type systems (denoted by N) and the expected output.

For nonlinear systems you cannot generally apply the same principles so a 'type' is not defined.

For type you might be confusing the 'type' of control compensator (filter): P, I, PI or PID types which more often refer to the control compensator that's applied in the control loop to control a system. As an open loop transfer function yo determine type (0, 1,2 , etc.) you can consider the compensator together with the plant as the open loop transfer function - to define the type you would get after closing the loop.

• thanks for your help! open loop transfer function tf means without the feedback. So I would have to consider the controller and the plant: tf(PI)*tf(plant) to say of which type my system is? Considering the plant would be linear. Is the compensator used to "linearize" the nonlinear plant? – fjp Jun 9 '15 at 13:23
• @evolved yes - in terms of transfer functions the open loop system is the product of the controller and plant. – docscience Jun 9 '15 at 13:26
• @evolved You can always chose what to call it. Either separately linearization and compensator, or all lumped together as compensator. If the linearization 'undoes' the nonlinearity of your system then the system effectively becomes linear, and you can apply the principles of system type. – docscience Jun 9 '15 at 13:33
• I am confused with the control compensator. Is this the controller or a linearization and controller – fjp Jun 9 '15 at 13:35
• @evolved Again you can chose to define compensator to include both of those components, but traditionally 'compensator' is used as a linear filter in the control loop used to stabilize the closed loops system and meet certain performance criteria. – docscience Jun 9 '15 at 20:10

Several things are wrong here:

(i) if A(x) defines a non-linear system you cannot apply state space analysis without linearising around an operating point via the Jacobian

(ii) x=xdot/s doesn't really make any sense because you're mixing time domain and Laplace domain, but even so, that expression is true for every dynamic system under the sun, and just means that the x is the integral of the derivative of x

(iii) p-type and I-type are meaningless terms when applied to a complete system - they may refer to the controller within the system, as subsets of 'PID' or 'three-term' control.

If you can define A(x) and B and we may be able to advise on the appropriate analysis method.

• thanks again for your effort and sorry for not upvoting. I haven't enough reputation. I chose docscience answer because of the hint that the principles of system 'type' can only be applied to linear systems. – fjp Jun 9 '15 at 13:42