Hello,

My answer has two parts: 1) the steady state part, and 2) the so-called transient part.

Part 1 - The steady state part.

In the case of a real plant and controller, say, an electrical plant and electrical controller, the components making up the plant and controller are not ideal components.  For example, real inductors exhibit inductance and parasitic resistance, due to the finite conductivity of the wire, and  parasitic capacitance due to the intra-winding capacitance between turns of insulated wire.  Real capacitors have parasitic inductance and parasitic leakage resistance, not to mention things like voltage coefficient (change in capacitance value due to the voltage difference applied across the capacitor plates).  There is also the precision problem, i.e., if I give you a 100 ohm carbon composite resistor, the resistor's true value may be 94.6 ohm due to manufacturing tolerances.  To the tolerance problem, we must add the fact that the values of real components drift slowly with time even under constant environmental conditions: aging, creep, etc.

All of these non-idealities mean that the expression you can write for GP(s) and GC(s) will never be correct.   In addition, almost all components are temperature dependent, which can lead to the plant and the controller drifting slowly with temperature because the components' values change with temperature, and temperature is always changing.  A capacitor with a pronounced voltage coefficient will drift with time also because the voltage difference across it is a function of time. 

As if all this was not enough, there is still the decomposition problem.  Can you, in fact, model the plant?  In many cases, it is almost impossible to accurately model the plant, and without an accurate plant model, how good is your controller going to be, even if you could build one out of perfect, i.e., ideal, components?  If the plant, itself, is drifting because its constituent components are drifting, how will you compensate for this in your controller?

In summary, every component making up your plant and controller is non-ideal and experiences drift in its values.  Even gauge blocks, used for checking micrometers, drift, i.e., the distance between their faces changes with time even under constant environmental conditions and careful handling.

Part 2 - the so-called transient part.

Implicit in Part 1 was the assumption that the system (plant + controller) was in steady state.  Control theory generally ignores the so-called transient solution in favor of the steady state solution.  In other words, when you solve for the differential equation (DE) of the system, y(s)=(GP(s)x(S))/(1-GP(s)GC(s)), the general solution is the sum of the complementary (a.k.a. transient) solution plus the particular (steady state) solution.  But ignoring the complementary solution has risks that are not obvious.  For example, suppose your plant has a  pole in the right half-plane, and you decide to remove it using pole zero cancellation.  In other words, if the numerator of the transfer function, H(s)=y(s)/x(s)=GP(s)/(1-GP(s)GC(s)), has an offending pole [called a zero, when it is in the numerator of H(s)], say, (s-a), then you might choose the controller such that the denominator of the transfer function has this same pole,and thus cancel the zero or pole in the numerator with the pole in the denominator.  With cancellation, the denominator of H(s) has no poles in the right half-plane and so the system is stable.  Unfortunately, this is a necessary but not sufficient condition for stability because the complementary solution still has the offending pole, i.e., the transient solution will blowup.  To see this clearly, realize that the complementary DE is (1-GP(s)GC(s))y(s)=0 and the particular DE is (1-GP(s)GC(s))y(s)=GP(s)x(s). 

Now, in real systems the variables cannot go to infinity, even for the complementary (transient) solution with a pole in the right half-plane, due to physical constraints.  But the transient behavior could look like a type of drift due to the presence of the offending pole.

I don't know if these rambling make any kind of sense, but it is the best I can do.

Regards,

Tom Cuff

It seems your drift of controlled variable means steady state error, I think it is due to inertia of the system. If we employ integral control it can be eliminated.

If your controlled variable is drifting away that means you have a steady state error that is function of time. Now this may arise due to many issues pertaining to the dynamic modelling and approximation of the system design. The basic thing to remember here is that our entire theory of control system is built on the premise of how precisely and accurately you could model your system. Even in case of linear systems, the components only behave in a linear manner while operating in a particular bandwidth of operation (For example take the characteristic of a BJT transistor, we can only have a linear model of operation of a BJT while operating within a particular bandwidth, otherwise we can no longer ignore the internal capacitance of the circuit.) Now BJT was just a basic example but also in case of electronic circuits, they act as the building blocks for more sophisticated components such as Op-Amps. Now Op-Amps are the essential part of designing modern day electronic control. So now you get the picture of how the game of approximation keeps playing in the background of linear control. This was just an example on electronic control, these same role of approximation can also be noticed in mechanical components such as springs and dampers (in case of mathematical modelling of dampers we only use the dependency of the damping forces on the velocity of the object while completely ignoring (most of the cases) the dependencies on higher order derivatives of velocity). So these approximations play a pivotal role in contributing time varying drifts to your system. You must understand that the nature we live in is highly non-linear and we only use the approximated linear models for the sake of reducing computational overloads on our controllers. But when left untreated for a long time, these unmodeled higher order dynamics may cause significant accumulated drift to your system. 

And this is the reason why in the last few decades people has started exploring newer ways to robustly (Robust Control) and adaptively (Adaptive Control) counter these non-linear control problems in highly precise systems.

I hope my answer has helped. 

Hello, 

Assuming that your control is well designed for the type of plant. Additional to the above mentioned causes of stationary state error, i.e., uncertainties due to unmodelled dynamics or change in parameters or external disturbances. I think that another important reason of stationary state error are saturations of actuators. On possible solution is to design a slow time closed loop response (less control gain).  

Hello,

I have added a proof that pole-zero cancellation of an unstable system does not render the system stable on my ResearchGate page, see the document titled "Pole-Zero Cancellation".

Regards,

Tom Cuff