**Linear Time Invariant**

LTI concept is very important to understand linear control theory. Two basic characteristics of LTI is linear and time invariant.

**Equation 1 - Linearity**

Let:

y=f(x), then ay=f(ax).

Intepretation:

A function taking x and returning y, if multiplied with a constant, a in x, will give a product of a and y.

**Equation 2 - Time**

Let:

y(t) = g(x,t), then y(t-b) = g(x,t-b)

Intepretation:

A function y in time, t, of g in x and time, if the time is delayed by b, the output will give the same.

**DC Motor Nonlinearities**

A direct current motor is subject to at least two types of nonlininearities, namely the saturation and dead zone. Another type of nonlinearity is the performance due to the decay of the motor mechanism and carbon brush. This is a type variant system but usually was not taken into account due to slow decay.

**Dead Zone**

Let's say V = 10 volts applied to the motor. After steady state, the motor will turn with an angular velocity of 100 rpm. If V applied is 1 volts, the motor will turn with 100/10 = 10 rpm. But given V = 0.1 volts, the motor will not turn due to the dead zone. Example of cause of dead zone is Coulomb friction. Coulomb friction is hard to model and it changes with many variables.

**Saturation**

Let's put it again V = 10 volts yield 100 rpm, giving 20 volts will yield 200 rpm, but giving 30 volts might only produce 240 rpm. This might not be the nonlinearity in the motor but in the motor driver. It depends on the voltage supplied to the motor.

**Decay**

Let's say again, V = 10 volts yields 100 rpm, but after one year, V = 10 volts might yield only 95 rpm.

**Concluding Remark**

Nonlinearity is a pain in control system. This is why simulation and real application is so different.

**Check out the full series**

Part I

Part II

Part III

Part II, III - Interlude

Part IV

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