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Preventing State Duplication in System Interconnections

This example shows guidelines for building minimum-order models of LTI system interconnections.

Model Interconnections

You can connect LTI models using the operators +, *, [,], [;] and the commands series, parallel, feedback, and lft. To prevent duplication of some of the dynamics and ensure that the resulting model has minimal order, it is important that you follow some simple rules:

  • Convert all models to the state-space representation before connecting them

  • Respect the block diagram structure

  • Avoid closed-form expressions and transfer function algebra.

As an illustration, this example compares two ways to compute a state-space model for the following block diagram

where

G = [1 , tf(1,[1 0]) , 5];
Fa = tf([1 1] , [1 2 5]);
Fb = tf([1 2] , [1 3 7]);

Recommended Method

The best way to connect these three blocks is to convert them to state space and treat the block diagram as a series connection of G with [Fa;Fb]:

H1 = [ss(Fa) ; Fb] * G;

To find the order of H1, type

order(H1)
ans =

     5

The order 5 is minimal. Note that because SS has higher precedence than TF, it is enough to convert one of the blocks to state-space (the remaining conversions take place automatically).

Order-Inflating Method

Observe that the overall transfer function is

$$ H(s) = \left( \begin{array}{c} F_a (s) G(s) \\ F_b (s) G(s)
\end{array} \right) $$

Therefore, you can also connect the three blocks and compute H by typing

H2 = ss([Fa * G ; Fb * G]);

Verify that the frequency responses of H1 and H2 match:

bode(H1,'b',H2,'r--')

While H2 is a valid model, its order is 14, almost three times higher than that of H1:

order(H2)
ans =

    14

H2 has higher order because:

  • G appears twice in this expression

  • The dynamics of Fa and Fb get replicated three time when evaluating Fa*G and Fb*G

  • The state-space conversion is performed on a 2x3 MIMO transfer matrix with four entries of order 2 and two entries of order 3, yielding a total order of 14.

Using a closed-form expression for the overall transfer function is a bad idea in general as it will typically inflate the order and introduce lots of cancelling pole/zero dynamics.

Conclusion

When connecting LTI models, avoid introducing duplicate dynamics by staying away from closed-form expressions, working with the state-space representation, and breaking block diagrams down to elementary series, parallel, and feedback connections. When in doubt, use the function connect which automatically converts all models to state space and is guaranteed to produce minimal realizations of block diagrams.

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