MIMO Models

State-Space Model of Jet Transport Aircraft

This example shows how to build a MIMO model of a jet transport. Because the development of a physical model for a jet aircraft is lengthy, only the state-space equations are presented here. See any standard text in aviation for a more complete discussion of the physics behind aircraft flight.

The jet model during cruise flight at MACH = 0.8 and H = 40,000 ft. is

A = [-0.0558   -0.9968    0.0802    0.0415
      0.5980   -0.1150   -0.0318         0
     -3.0500    0.3880   -0.4650         0
           0    0.0805    1.0000         0];

B = [ 0.0073         0
     -0.4750    0.0077
      0.1530    0.1430
           0         0];

C = [0     1     0     0
     0     0     0     1];

D = [0     0
     0     0];

Use the following commands to specify this state-space model as an LTI object and attach names to the states, inputs, and outputs.

states = {'beta' 'yaw' 'roll' 'phi'};
inputs = {'rudder' 'aileron'};
outputs = {'yaw rate' 'bank angle'};

sys_mimo = ss(A,B,C,D,'statename',states,...
'inputname',inputs,...
'outputname',outputs);

You can display the LTI model by typing sys_mimo.

sys_mimo
 
a = 
                      beta          yaw         roll          phi
         beta      -0.0558      -0.9968       0.0802       0.0415
          yaw        0.598       -0.115      -0.0318            0
         roll        -3.05        0.388       -0.465            0
          phi            0       0.0805            1            0
 
 
b = 
                    rudder      aileron
         beta       0.0073            0
          yaw       -0.475       0.0077
         roll        0.153        0.143
          phi            0            0
 
 
c = 
                      beta          yaw         roll          phi
     yaw rate            0            1            0            0
   bank angle            0            0            0            1
 
 
d = 
                    rudder      aileron
     yaw rate            0            0
   bank angle            0            0
 
Continuous-time model.

The model has two inputs and two outputs. The units are radians for beta (sideslip angle) and phi (bank angle) and radians/sec for yaw (yaw rate) and roll (roll rate). The rudder and aileron deflections are in degrees.

As in the SISO case, use tf to derive the transfer function representation.

tf(sys_mimo)
 
Transfer function from input "rudder" to output...
               -0.475 s^3 - 0.2479 s^2 - 0.1187 s - 0.05633
 yaw rate:  ---------------------------------------------------
            s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674
 
                         0.1148 s^2 - 0.2004 s - 1.373
 bank angle:  ---------------------------------------------------
              s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674
 
Transfer function from input "aileron" to output...
            0.0077 s^3 - 0.0005372 s^2 + 0.008688 s + 0.004523
 yaw rate:  ---------------------------------------------------
            s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674
 
                        0.1436 s^2 + 0.02737 s + 0.1104
 bank angle:  ---------------------------------------------------
              s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674

Constructing MIMO Transfer Functions

MIMO transfer functions are two-dimensional arrays of elementary SISO transfer functions. There are two ways to specify MIMO transfer function models:

Concatenation of SISO transfer function models
Using tf with cell array arguments

Concatenation of SISO Models

Consider the following single-input, two-output transfer function.

$H (s) = [\begin{matrix} \frac{s - 1}{s + 1} \\ \frac{s + 2}{s^{2} + 4 s + 5} \end{matrix}]$

You can specify H(s) by concatenation of its SISO entries. For instance,

h11 = tf([1 -1],[1 1]);		
h21 = tf([1 2],[1 4 5]);

or, equivalently,

s = tf('s')
h11 = (s-1)/(s+1);
h21 = (s+2)/(s^2+4*s+5);

can be concatenated to form H(s).

H = [h11; h21]

This syntax mimics standard matrix concatenation and tends to be easier and more readable for MIMO systems with many inputs and/or outputs.

Using the `tf` Function with Cell Arrays

Alternatively, to define MIMO transfer functions using tf, you need two cell arrays (say, N and D) to represent the sets of numerator and denominator polynomials, respectively. See Cell Arrays for more details on cell arrays.

For example, for the rational transfer matrix H(s), the two cell arrays N and D should contain the row-vector representations of the polynomial entries of

$N (s) = [\frac{s - 1}{s + 2}] D (s) = [\frac{s + 1}{s^{2} + 4 s + 5}]$

You can specify this MIMO transfer matrix H(s) by typing

N = {[1 -1];[1 2]};   % Cell array for N(s)
D = {[1 1];[1 4 5]}; % Cell array for D(s)
H = tf(N,D)

These commands return the following result:

Transfer function from input to output...
      s - 1
 #1:  -----
      s + 1
 
          s + 2
 #2:  -------------
      s^2 + 4 s + 5

Notice that both N and D have the same dimensions as H. For a general MIMO transfer matrix H(s), the cell array entries N{i,j} and D{i,j} should be row-vector representations of the numerator and denominator of H_ij(s), the ijth entry of the transfer matrix H(s).

Accessing I/O Pairs in MIMO Systems

After you define a MIMO system, you can access and manipulate I/O pairs by specifying input and output pairs of the system. For instance, if sys_mimo is a MIMO system with two inputs and three outputs,

sys_mimo(3,1)

extracts the subsystem, mapping the first input to the third output. Row indices select the outputs and column indices select the inputs. Similarly,

sys_mimo(3,1) = tf(1,[1 0])