Linear Systems and Signals Notes
Notes for Linear Systems and Signals Fall 2016
Professor: Sophocles Orfanidis
Chapter 2 Time Domain Analysis of Continuous-Time Systems
A total system response is:
Total Response = zero-input response + zero-state response
- zero-input response:
- zero-state response: is the system’s response to any external input, say, when the system is in a zero-state (absence of any other energies)
Given an Nth order differential equation have the following
Zero-State Response of Differential Equation
Zero-Input Response of Differential Equation
From these two equation we can obtain the general solution to a differential equation as
Solving for the Zero-Input Response
For a quick review of our differential equations we should remember that all terms within the zero-input response should be of the form:
Recall that the zero-input term is equal to the following:
Note the term . This is a polynomial which denotes the differentials in the equation and can help us evaluate which terms will be present in our zero-input response.
Given a polynomial we can solve for to find the roots of the polynomial to help find the zero-input response.
We call the characteristic polynomial
The characteristic equation is
Non-Repeating Real Roots Polynomial
In the case that the roots of polynomial are real roots we can use the following process to solve for the zero-input response.
Assume the roots of polynomial with number of roots to have the roots
- Once we’ve found our roots , then we can find the characteristic terms to be equal to
- This makes the zero-input equation
- From this we need for solve for our constants . To do this we need to:
- Find the the 1 to nth derivative of the zero-input equation
- Find/Get the the zero-input response at for the 0 to nth derivative equations.
- Solve the system of equations to get
Repeating Real Roots Polynomial
Given the steps found above for Non-Repeating Real Roots Polynomial we can follow a similar procedure except that given we have, say, repeating roots of in the polynomial, then the roots which correspond for are represented in the following way:
This makes the zero-input response equal to:
Using that information you can solve for the contants of the zero-input response by the same method as above for real-rooted polynomials.
Complex Roots Polynomial
Polynomials with complex () roots are slightly more complex.
It is imperative to remember Euler’s Formula
Given a polynomial with complex roots, due to the nature of the quadratic equation, , any complex roots must appear in pairs, such that they are complex conjugates, say, and .
If we treat these roots similar to the real-rooted polynomials then we would end up with an equation similar to the following:
For a real system, the response of should be real as well. That is only the case such if and are conjugates.
We should then set and where we introduce a new parameter,
This will then result in our zero-input response being
This allows us to separate the roots into real and imaginary components.
Then doing some algebra..:
Which finally gives us a nice simplified zero-input equation where we can solve for the constants and :
The Unit Impulse Response
Given our differential equation before
We want to define a new quantity which we call the unit impulse response, .
This quantity allows us to impart initial conditions on a system given the correct system parameters.
Let’s redefine and by
Where , the unit-impulse function. We then define the unit-impulse to be
We should also note that if the order of is less than that of then
The Zero-State Response of a Systems
To determine the zero state response of a system, excluding the systems initial conditions we can calculate such a response by the following, given:
- , the input function
- , the output response
- , the impulse response
Then we can say that
This is called the convolution of two functions. For more information, refer to the convolution section below.
Classical Solutions to Differential Equations - Coefficient Matching
Recall that the total response of a system can be characterized by either (We just reviewed the zero state+input method above)
- Zero-State + Zero-Input
- Homogeneous(Natural) Response + Forced Response
In this section we’ll review how to solve a system of equations based on the Homogeneous and forced response solutions using the method of coefficient matching. This method is typically the same method taught in introductory differential equations courses.
In this method we’re given a differential equation of the form .
The basic methodology for solving via the coefficient matching is as follows:
- Solve for the characteristic polynomial and use that to find the characteristic modes. From construct the natural response equation
- After obtaining the form of the natural response with arbitrary coefficients use the following table to determine the the forced response equation
3. Use the differentials from the characteristic polynomial and differentiate the forced solution the required number of times to obtain a new set of equations with unkown coefficients which are added together on the left side of the equation. 4. Use the differentials for the input to create a new equation. This equation should have terms which match up in degree with or function with terms from the forced response. This goes on the right side of the original equation. 5. Using all of the different degree terms from the polynomials, create a system of equations to solve for the unkown coefficients. 6. Finally after solving for the coefficients, we can add the new forced response to our natural response with our unkown coefficients . Use the given intial conditions to find the unknown coefficients with the natural+forced response system.
The resulting coefficient values will give the final solution which includes the natual + forced responses of the system which should be the same as the zero-state + zero-input response.
Chapter 2: Discretization Schemes and Convolution
Given the simple differential equation:
The solution involves:
So how can we solve this for any ?
One possible way to to refer to the definition of an integral - the area under a curve.
With this in mind we can apply some approximation techniques that we may have learned back in our calculus courses to estimate the value of
There are three methods which involve approximations:
- Forward Euler
- Backward Euler
- Trapezoidal (Binilear Approximation)
In order to shorten our notation we’ll use the following symbols:
Working with linear systems we’re able to add parts of systems together to find the solution to a whole system
If we set
Given the system
This can modified to
This can then be expanded via our previous definitions
If we then play around some more we an obtain:
Using this it’s now possible to solve this system because it’s defined recursively (see how we define as being equal to an expression with )
Another Discretization Scheme
Matlab uses this method as the default. It’s called Zero-Order Hold
It can be invoked with
Given an exponential differential equation:
If we want to write all of our ’s in terms of we can simple multiply through by . This gives:
A linear time-invariant systems (LTI) satisfies the property that order in which an operation is performed (e.g time delay or transfer function) commute (the final value from the system is the same no matter the order).
The same linear combination of inputs should result in the same output.
A convolution is defined as:
Convolutions are special in that they allow us to calculate the zero-state responses of a system. It is not always the simplest way, but it does provide one method of calculation.
Convolution has a few key properties to keep in mind:
- Defined such that if then
Chapter 4: Solving Differential Equations - Laplace method
There is yet another way to solve differential equations by using something called the Lapalce Transform
The laplace is defined as