Continuous time LTI (Linear time invariant) systems

Introduction to continuous time LTI systems

Continuous LTI system stands for Linear Time invariant system. LTI systems theory plays a key role in designing most of dynamic system. Most of the practical systems of interest can be modeled as linear time in variant systems or at least approximations of them around nominal operating point because of its(theory of LTI systems) simplicity in dealing complex real systems. The analysis of LTI systems is equipped with well developed theory rich in its content. 

Definition of continuous time LTI systems

A continuous time LTI system is one which deals with continuous time signals and satisfies both the principles of linearity and time invariance. The principles of linearity and time invariance can be stated as follows

 

Principle of Linearity

Consider a linear system S characterized by the transformation operator T[]. Let x1, x2 are the inputs applied to it and y1, y2 are the outputs. For a system to be linear it has to satisfy both the principles of homogeneity and superposition. The following equations hold for a linear system

     y= T[x1], y2 = T[x2]        

Principle of homogeneity: T [a*x1] = a*y1, T [b*x2] = =b*y2

Principle of superposition: T [x1] + T [x2] = a*y1+b*y2

Linearity: T [a*x1] + T [b*x2] = a*y1+b*y2

Where a, b are constants.

Principle of Time invariance

The principle of time variance states that the behavior of the system should not change with time. In a time invariant system if input is delayed by time t0 the output will also gets delayed by t0. Mathematically it is specified as follows

y(t-t0) = T[x(t-t0)]

Examples of Linear time invariant systems

Consider a system characterized by following input output relationship y(t) = A*x(t)(signal amplifier output in linear range). Now we will check whether this system is linear time invariant or not

linearity:

Homogeneity

If an input of a*x(t) is applied to this system the output = A*a*x(t) which is equal to a*A*x(t)= a*y(t). Hence this system satisfies homogeneity principle.

Superposition

let x1(t) , x2(t) be two distinct inputs applied to this system then

         y1(t)=A*x1(t),y2(t)=A*x2(t)

if we apply a input to equal to sum of individual inputs a*x1(t), b*x2(t) 

                                                                 x'(t) = a*x1(t)+b*x2(t) then

                                                          output y'(t) = A*a*x1(t)+A*b*x2(t)

                                                                        y'(t) =a*y1(t)+b*y2(t)
hence the system satisfies superposition principle.

Since the system satisfies both linearity and superposition the system is a linear system.

Time invariance:

If input is delayed by to i.e. input = x(t-to) the

        output = A*x(t-to) (equation 1)

If output is delayed by to i.e. y(t-to)(replace t by to in the RHS and LHS of the equation

y(t-to) =A* x(t-to) (equation 2)

From equation 1 and 2 it is clear that this system is time invariant.

Summing up the properties of linearity and time invariance the system characterized by output y(t)=A*x(t) is a linear time invariant system.

Whether this system is LTI system or not

Y(t) = sin(x(t))

linearity:

Homogeneity

If an input of a*x(t) is applied to this system the output = A*sin(a*x(t)) which is not equal to a*y(t) = a*sin(x(t)). Hence this system does not satisfy homogeneity principle.

Superposition

let x1(t) , x2(t) be two distinct inputs applied to this system then

y1(t)=sin(x1(t)),y2(t)=sin(x2(t))

if we apply an input to equal to sum of individual inputs a*x1(t), b*x2(t) 

                                        x'(t) = a*x1(t)+b*x2(t) then

                           output y'(t) =sin(a*x1(t))+sin(b*x2(t)) ≠ a*y1(t)+b*y2(t)      

hence the system does not satisfy superposition principle.

Since the system does not satisfy linearity and superposition the system is a Non-linear system.

Time invariance:

If input is delayed by to i.e. input = x(t-to) the

                                                             output =sin(x(t-to)) (equation 1)

If output is delayed by to i.e. y(t-to)(replace t by to in the RHS and LHS of the equation

                                       y(t-to) = sin(x(t-to)) (equation 2)

From equation 1 and 2 it is clear that this system is time invariant.

Summing up the properties of non linearity and time invariance the system characterized by output y(t)=sin(x(t)) is a not a linear time invariant system. It is a non linear time invariant system.

A Linear time invariant system in time domain can be described by differential equations of the form

                                     \sum_{p=0}^{N}a_{k}*d^{k}y(t)/dt^{k} = \sum_{p=0}^{N}b_{k}*d^{k}x(t)/dt^{k}

Where x(t) is input to the system, y(t) is output of the system, ak and bk are constant coefficients independent of time.

Example: A RC circuit shown in the figure is characterized by differential equation

                                    dy(t)/dt+R*C*y(t)=x(t)/(R*C) 

Where y(t) is output voltage, x(t) is input voltage.

The solution of the above equation is given as    

\dpi{120} y(t)*e^{-t/(R*C)}= \int_{-\infty }^{t} x(\tau )*e^{-\tau/(R*C)}*d\tau+constant

if the constant is set to zero you can easily verify that the system is linear time invariant system.

Note: If ak and bk are functions of time then the system is linear time variant system.

S.NO.        System input output relationship             LTI system/Not a LTI system
1.                      y(t) = dx/dt                                  Linear Time Invariant system
2.                       y(t) = t*x(t)                                  linear time variant system
3.                        y(t) = t2*x2(t)                           non linear time variant system
4.                     dy/dt +t*y(t) = x(t)                          linear time variant system
5.                      y(t) = ln(x(t))                               Non linear time invariant system
6.                      y(t) = x(t)+constant                     Non linear time invariant system

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