Zero Input & Zero State Responses, Time Domain

Contents

Introduction

This document discusses zero input and zero state responses using only step inputs and only time domain analysis.  If you understand Laplace Transforms, there are easier ways to implement the zero input / zero state method.  The document starts with a brief review of the method of homogeneous and particular solutions (also sometimes called the natural and forced responses).  The zero input and zero state solutions are then introduced.  At this point using the zero input and zero state solutions will be more laborious than using the homogeneous and particular solutions, but we will develop other methods that naturally lead to zero input and zero state solutions and it is useful to see how these relate to homogeneous and particular solutions.

Homogeneous and particular

We will demonstrate the solution of a problem via homogeneous and particular responses with an example.

Example: Solution via homogeneous and particular responses

Consider the system shown.  In this diagram C=1F, R=2Ω.  The input, ein(t) is a 2 volt step (i.e., ein(t)=0 for t<0, and ein(t)=2 for t≥0.  We write this as ein(t)=2·γ(t);  At t=0- (i.e., immediately before t=0) there is initially -1.5 volts on the capacitor (eout(0-)=-1.5).

The differential equation representing this system is

To find eout(t) we can split the problem up into a homogeneous part (eout,h(t)) and a particular solution (eout,p(t)).

Homogeneous solution.

To find the homogeneous solution, we set the input to zero, and assume that the solution is of the form A·est.

so

Particular Solution

To find the particular solution, we assume the particular response of the output is a constant (since the input is a constant) and solve.

so

Complete Solution

To calculate the complete response we add the homogeneous and particular solutions

To find the unknown coefficient, A, we use the initial condition at t=0+ (i.e., immediately after t=0).  Since the voltage across a capacitor can't change instantaneously eout(0-)=eout(0+)=-1.5.  So

and

We can show the solutions with Matlab

t=linspace(0,10);
ein=ones(size(t))*2;
eh = -3.5*exp(-t/2);
ep = 2*ones(size(t));
ec = eh + ep;
plot(t,ein,t,eh,t,ep,'--',t,ec);
legend('e_{in}','e_{out,h}','e_{out,p}','e_{out,c}');
axis([0, 10, -4 4])
xlabel('Time');  ylabel('Voltage');
title('Homogeneous, Particular and Complete Solution');
grid


Finding the homogeneous and particular solutions is a general technique for solving differential equations of the sort that we will encounter (different inputs require different forms of the particular response, but we will only consider step inputs (i.e., the input is constant for t>0) for now.  However, contemplation of the technique begs the question: What, physically, do the homogeneous and particular response represent.  The particular response represents the response of the system after any initial transients have died out, but the the homogeneous response doesn't really represent anything physical.  The reason we use it is that it is mathematically correct and yields the right answer.

Zero Input/Zero State

Another method for solving differential equations of the type we will encounter is called the zero input/zero state method.  One advantage of this method is that the zero input and zero state solutions have precise, physical, interpretations.  It also turns out that techniques we will develop lend themselves to this solution method.  Unfortunately, with the time domain methods we have available, the technique will be harder than the homogeneous/particular technique.  We introduce it now so that you can make a connection between the two techniques, and can understand the (subtle) differences.

As expected from the name, the zero input solution is that part of the solution found by setting the input to zero.  We then find the solution due to initial conditions (contrast this to the homogeneous solution whose form we can find, but whose unknown coefficients we can't find until after we know the particular response).  The zero state response is the part of the solution that is due to the input with all initial conditions set to zero.

To see how this technique is used, we will repeat the previous problem.

Example 2a: Zero input / Zero state solution

Solve the differential equation (note: this problem was solved above using homogeneous and particular solutions).

Zero Input

To solve the zero input problem, we set the input to zero and change eout to eout,zi to indicate that it is now the zero input solution, and as before we assume a form of the solution

However, with the zero input/zero state technique, we can immediately solve for the unknown coefficient from the initial conditions

To reiterate, this is the response due to the initial conditions alone (the input does not affect this response)

Zero State

To solve the zero state problem, we set the initial conditions to zero and change eout to eout,zs and proceed as before.

Here we have a quandary - how do we fine eout,zs(t)?  At this point, the only way to do this is by solving in terms of homogeneous and particular solutions.  However, there are other techniques (i.e., convolution and the Laplace Transform) that we will soon learn about to find the zero state response directly.

The homogeneous solution is found as before:

The particular solution is also found as before

The zero state response is simply the sum of the two

and we get the unknown coefficient from initial conditions (recall eout,zs(0-)=0, and since eout is accross a capacitor eout,zs(0+)=eout,zs(0-).

Complete Solution

The complete solution is simply the sum of the zero input and zero state response - all three are plotted below.

Note: as expected this is the same result we obtained previously.

t=linspace(0,10);
ein=ones(size(t))*2;
ezi = -1.5*exp(-t/2);
ezs = 2-2*exp(-t/2);
ec = ezi + ezs;
plot(t,ein,t,ezi,t,ezs,t,ec);
legend('e_{in}','e_{out,zi}','e_{out,zs}','e_{out,c}');
axis([0, 10, -4 4])
xlabel('Time');  ylabel('Voltage');
title('Zero input, zero state and Complete Solution');
grid

The following examples show how the zero input / zero state solution can simplify the solution of differential equations as the input and/or initial conditions change.

Example 2b: Step Input with x(0-)=2

Solve the differential equation

with

Zero input solution

Since this is the same problem we were solving before (Example 2a), if the input is multiplied by a constant, so is the output.

Zero State Solution

We have solved this previously (Example 2a) and needn't do any more work

Complete Solution

The complete solutions is simply the sum of the zero state and zero input solution

Example 2c: Double the size of the input

Solve the differential equation

Zero input solution

Zero State Solution

The input is the same as in Example 2a and 2b, but scaled by a factor of 2, sow we scale the output by the same factor

Complete Solution

The complete solutions is simply the sum of the zero state and zero input solution

Example 2d: Change both input and initial conditions

Solve the differential equation

Zero input solution

The initial condition is the same as in Example 2b, so we don't need to solve it again.

Zero State Solution

Complete Solution

The complete solutions is simply the sum of the zero state and zero input solution

The computational efficiency of the zero input / zero state solution is shown by the last example in which no extra work needed to be done to get the solution.

Key Concept: Zero Input, Zero State and Complete Response

The transient response of a system can be found by splitting a problem into two parts.

1. The zero input part of the response is the response due to initial conditions alone (with the input set to zero).
2. The zero state part of the response is the response due to the system input alone (with initial conditions set to zero).

The complete response is simply the sum of the zero input and zero state solutions.

ZI/ZS vs Homogeneous/Particular

The zero input / zero state solution is advantageous relative to the homogeneous and particular solution because the two components have specific physical interpretations.  The disadvantage (so far) is that we need to calculate the zero state solution using homogeneous and particular solutions.  However, we will use another method, called convolution, to directly calculate the zero state solution.

References