Key Commands:

A. Overview of the MatLab Control System Toolbox

MatLab offers a special toolbox for analyzing control systems called (surprise!) the Control System Toolbox. To see the various commands provided by this toolbox, type the command:

A simple demonstration of some of the features of this toolbox can be seen by running the command:

Run this command and you will be rewarded with a number of plots that may not mean much to you yet. That's ok, we'll get there soon.

In version 5 of MatLab, many of the functions in the control toolbox require a special data object called a system, or SYS. A SYS object is an internal MatLab representation of an LTI (linear time invariant) dynamical system. A SYS object may be created in several ways. In this studio we will focus on creating system objects from transfer function polynomials using the tf command.

(NOTE: COMMANDS CAMPATIBLE WITH MATLAB VERSION 4 ARE DISCUSSED FOR THOSE WHO WANT TO USE PRE-v5 AT HOME. IN GENERAL, COMMANDS WHICH REQUIRE A SYS DATA OBJECT IN v5 CAN BE USED IN MATLAB v4 BY PASSING (num, den) INSTEAD OF (sys) TO THOSE FUNCTIONS.)

B. Model Creation via tf()

C. Poles and zeros

A transfer function can be described by its poles (roots of the denominator) and zeros (roots of the numerator). The roots() function can be used to find the roots of a polynomial. To find the zeros and poles of our example system, type:

Alternately, the zeros and poles may be found simultaneously using the tf2zp() function, where tf2zp stands for "transfer function to zeros & poles". To do this, type:

You should find that zeros and poles are the same as zeros2 and poles2 . In both cases, you should find there are no finite zeros, and the poles are located at s = -2.3874 and s = -0.2792.

Another useful MatLab function, called zp2tf() , can be used to define a transfer function from a set of poles and zeros. In this case, the arguments of the function are the poles and zeros placed in row vector form, and the result of the function is 2 vectors containing the numerator and denominator polynomials. For example, the poles of our example are located at s = -2.3874 and s = -0.2792, and there are no finite zeros. To find the transfer function given this information, type:

Which should return the following:

The third argument of the zp2tf() function is the scalar gain of the transfer function. Why is this needed? Given a set of poles, there are an infinite number of polynomials which have those poles as their roots; if we find one such polynomial, then any scalar multiple of that polynomial will possess the same roots. When you give MatLab a set of zeros and poles, it chooses polynomials for the numerator and denominator which are scaled so that the highest order coefficient of each polynomial is equal to one. In order to produce the desired transfer function, you need to tell MatLab what the scalar gain should be based on this fact. For our example system, a scalar gain of 1/3 is used since, by observation of the original transfer function, we would need to divide both the numerator and denominator through by 3 to produce numerator and denominator polynomials with the highest order coefficient of each polynomial equal to one.

A final way to go between a transfer function representation of a system and a pole/zero representation is to use the poly() function. This is the inverse of the roots() function, but now we provide the roots of a polynomial as arguments to the function (in vector form), and the result of the function is a polynomial which possesses those roots. Just as with zp2tf() , poly( ) returns a polynomial with a highest-order coefficient of one. Using our previously defined variables, type:

These commands will produce the following output:

To produce the original transfer function, a scalar value of 1/3 must be multiplied on the numerator. This is analogous to including the scalar gain of 1/3 to the zp2tf() function.

Another way of finding poles of a system, along with the zeros, is by using Matlab's pole(), zero(), and pzmap() functions. Each of these functions works on a SYS data type. Use the help command for more info on these functions. You will use them in the assignment at the end of this lab.

 

D. Partial Fraction Expansion and Residues

MatLab can also be used to perform partial fraction expansion of transfer function expressions. This can be very handy for examining the response of individual components of a larger transfer function. First, verify that you still have the original numerator and denominator polynomials, num and den , properly defined. To perform a partial fraction expansion on this transfer function, type:

R is a vector containing the residues, P is a vector containing the component poles, and K is sometimes called the direct term .

The residue() function breaks the transfer function defined by num and den into its n component residues, where n is the order of the denominator:

num(s)       R(1)       R(2)             R(n)
------  =  -------- + -------- + ... + -------- + K(s) 
den(s)     s - P(1)   s - P(2)         s - P(n)

E. Impulse response

Now that you know how to create SYS data objects and represent systems in terms of transfer functions and poles/zeros, we can look at the time response of linear systems. In this studio we'll consider the response of a linear system to two types of inputs: impulse and step functions.

The impulse response of an LTI system can be plotted using the impulse() function. To try this with our example system, type:

recall that in Matlab v4, you need to replace the 'sys' object with 'num,den', i.e.:

The output should look like:

F. Step response

Step response can be found using MatLab's step() function, which operates the same way as impulse(). In this case, the forcing function is a unit step applied at time t=0s. To plot the step response of our system, type:

or

After typing the plot command, you should see the following:

 

H. Time response of 2nd order systems

Let us briefly consider the response of second order systems. You are given the following second order (open-loop) system whose transfer function is:

             1           G(s)
      R(s)= ---    _______________
             s     |     b        |   C(s)
      -------------| ------------ |--------->  
                   |s^2 + a s + b |    
                   ---------------

Depending on the pole locations, the system is either overdamped, critically damped, underdamped, or undamped. The following examples illustrate the different cases:

(1) Overdamped

                 4
      G(s) = ------------                (1)
             s^2 + 5 s +4

Create the system in Matlab and apply the step() function. The step response of system (1) is

(2) Critically damped

                 4
      G(s) = ------------                (4)
             s^2 +4 s + 4

Create the system in Matlab and apply the step() function. The step response of system (4) is

(3) Underdamped

                 4
      G(s) = ------------                (2)
             s^2 + s +4

Create the system in Matlab and apply the step() function. The step response of system (2) is

(4) Undamped

                 4
      G(s) = ------------                (3)
              s^2  + 4

The step response of system (3) is

 

We will shortly be working with 2nd order systems in lecture. Understanding the relationship between pole locations and dynamic behavior of 2nd order systems is important for many control problems.

Assignments

1. For the above four cases, do the following:

2. Derive the transfer function of the following electrical network by hand. The input is the voltage vi(t) and the output is the voltage vo(t) .

3. Write a MatLab m-file which will do the following (turn in m-file and output file including all plots)