Linear (LTI) Models

What Is a Plant?

Typically, control engineers begin by developing a mathematical description of the dynamic system that they want to control. The system to be controlled is called a plant. As an example of a plant, this section uses the DC motor. This section develops the differential equations that describe the electromechanical properties of a DC motor with an inertial load. It then shows you how to use the Control System Toolbox™ functions to build linear models based on these equations.

Linear Model Representations

You can use Control System Toolbox functions to create the following model representations:

SISO Example: The DC Motor

A simple model of a DC motor driving an inertial load shows the angular rate of the load, ω(t), as the output and applied voltage, υapp(t), as the input. The ultimate goal of this example is to control the angular rate by varying the applied voltage. This figure shows a simple model of the DC motor.

In this model, the dynamics of the motor itself are idealized; for instance, the magnetic field is assumed to be constant. The resistance of the circuit is denoted by R and the self-inductance of the armature by L. If you are unfamiliar with the basics of DC motor modeling, consult any basic text on physical modeling. With this simple model and basic laws of physics, it is possible to develop differential equations that describe the behavior of this electromechanical system. In this example, the relationships between electric potential and mechanical force are Faraday's law of induction and Ampère's law for the force on a conductor moving through a magnetic field.

τ(t)=Kmi(t)

υemf(t)=Kbω(t)

J (dw/dt)=∑τi=−Kfω(t)+Kmi(t)

υapp(t)−υemf(t)=L (di/dt)+ R i(t)

υapp(t)=L (di/dt)+ R i(t)+Kbω(t)

di/dt=−(R/L) i(t)−(Kb /L)ω(t)+(1/L)υapp(t)

dω/dt=−(1/J) Kfω(t)+(1/J)Kmi(t)

d/dt[i; ω]= [ -R/L    -Kb /L

                   -Km /J    -Kf /j ]    [i ; w] + [ i/L  0] *υapp(t)

y(t)=[0 1]⋅[i  ω]+[0]⋅υapp(t)

Building SISO Models

After you develop a set of differential equations that describe your plant, you can construct SISO models using simple commands. The following sections discuss

R= 2.0 % Ohms
L= 0.5 % Henrys
Km = .015 % torque constant
Kb = .015 % emf constant
Kf = 0.2 % Nms
J= 0.02 % kg.m^2

A = [-R/L -Kb/L; Km/J -Kf/J]
B = [1/L; 0];
C = [0 1];
D = [0];
sys_dc = ss(A,B,C,D)

a = 
                        x1           x2
           x1           -4        -0.03
           x2         0.75          -10
 
 
b = 
                        u1
           x1            2
           x2            0
 
 
c = 
                        x1           x2
           y1            0            1
 
 
d = 
                        u1
           y1            0

sys_tf = tf(sys_dc)

Transfer function:
       1.5
------------------
s^2 + 14 s + 40.02

sys_zpk = zpk (sys_dc)
 
Zero/pole/gain:
        1.5
-------------------
(s+4.004) (s+9.996)

sys = tf(num,den)               % Transfer function
sys = zpk(z,p,k)                % Zero/pole/gain
sys = ss(a,b,c,d)               % State-space
sys = frd(response,frequencies) % Frequency response data

sys_tf = tf(1.5,[1 14 40.02])
 
Transfer function:
       1.5
------------------
s^2 + 14 s + 40.02

s = tf('s');
sys_tf = 1.5/(s^2+14*s+40.02)

Transfer function:
        1.5
--------------------
s^2 + 14 s + 40.02

sys_zpk = zpk ([], [- 9.996 -4.004], 1.5)

Zero/pole/gain:
        1.5
-------------------
(s+9.996) (s+4.004)

Constructing Discrete Time Systems

The Control System Toolbox software provides full support for discrete-time systems. You can create discrete systems in the same way that you create analog systems; the only difference is that you must specify a sample time period for any model you build. For example,

sys_disc = tf(1, [1 1], .01);

creates a SISO model in the transfer function format.

Transfer function:
  1
-----
with +1
 
Sample time: 0.01

sys_delay = tf(1, [1 1], 0.01,'ioDelay',5)

Transfer function:
           1
z ^ (- 5) * -----
         with +1
 
Sample time: 0.01

Adding Delays to Linear Models

You can add time delays to linear models by specifying an input delay, output delay, or I/O delay when building a model. For example, to add an I/O delay to the DC motor, use this code.

sys_tfdelay = tf(1.5,[1 14 40.02],'ioDelay',0.05)

This command constructs the DC motor transfer function, but adds a 0.05 second delay.

Transfer function:
                      1.5
exp(-0.05*s) * ------------------
               s^2 + 14 s + 40.02

For more information about adding time delays to models, see Time Delays in Linear Systems.

LTI Objects

For convenience, the Control System Toolbox software uses custom data structures called LTI objects to store model-related data. For example, the variable sys_dc created for the DC motor example is called an SS object. There are also TF, ZPK, and FRD objects for transfer function, zero/pole/gain, and frequency data response models respectively. The four LTI objects encapsulate the model data and enable you to manipulate linear systems as single entities rather than as collections of vectors or matrices.

To see what LTI objects contain, use the get command. This code describes the contents of sys_dc from the DC motor example.

get(sys_dc)
                A: [2×2 double]
                B: [2×1 double]
                C: [0 1]
                D: 0
                E: []
           Scaled: 0
        StateName: {2×1 cell}
        StateUnit: {2×1 cell}
    InternalDelay: [0×1 double]
       InputDelay: 0
      OutputDelay: 0
               Ts: 0
         TimeUnit: 'seconds'
        InputName: {''}
        InputUnit: {''}
       InputGroup: [1×1 struct]
       OutputName: {''}
       OutputUnit: {''}
      OutputGroup: [1×1 struct]
            Notes: [0×1 string]
         UserData: []
             Name: ''
     SamplingGrid: [1×1 struct]

You can manipulate the data contained in LTI objects using the set command; see the Control System Toolbox online reference pages for descriptions of set and get.

Another convenient way to set or retrieve LTI model properties is to access them directly using dot notation. For example, if you want to access the value of the A matrix, instead of using get, you can type

sys_dc.A

at the MATLAB^® prompt. This notation returns the A matrix.

years =

   -4.0000   -0.0300
    0.7500  -10.0000

Similarly, if you want to change the values of the A matrix, you can do so directly, as this code shows.

A_new = [-4.5 -0.05; 0.8 -12.0];
sys_dc.A = A_new;