Let us consider the system described in the following Figure.

This system can be represented by the transfer functions:

or the state space representation:

Its dynamics is characterized by 2 poles or eigenvalues:

- : the "slow" one,
- : the "fast" one.

G=ss([0 1;-10 -11],[0;10],eye(2),zeros(2,1));

damp(G)

Its step response:

figure

step(G)

Clearly the responses (it is more particularly true for ) are dominated by the slow eigenvalue . Indeed, the long-term response in the time-domain corresponds to the low-frequency response in the frequency-domain, i.e. when tends to 0. On the transfer functions, that means the terms in can be neglected w.r.t to terms in s or :

Thus, it is possible to find a first order system quite representative of the long-term response of the 2nd order system . Such an operation is called a model reduction.

Gslow=tf({10;[10 0]},[11 10]);

hold on

step(Gslow)

legend('G(s)','Gslow(s)')

The short-term step response ():

figure

step(G,0.2)

Clearly the response is dominated by the fast eigenvalue . Indeed, the short-term response in the time-domain corresponds to the high-frequency response in the frequency-domain, i.e. when tends to âˆž. On the transfer functions, that means the terms in can be neglected w.r.t to terms in s or :

Thus, it is possible to find a first order transfer quite representative of the short-term response of . For , it is not possible to find a first order transfer representative of its short-term behavior. We will choose 0 and thus: . Such an operation is also called a model reduction but it is completely different from the previous one.

Gfast=tf({0;10},[1 11]);

hold on

step(Gfast)

legend('G(s)','Gfast(s)')

The previous operations on transfer functions become tricky when we have have to cope with high order systems, with several inputs and outputs (ex: the longitudinal model of an aircraft). It is thus highly recommended to eliminate state variables directly in the state-space representation (see: doc modred). The function modred proposes 2 methods to eleminate state space variables:

- 'matchdc' (i.e.: match DC gain, the defauft method ): to be used to eliminate the "fast dynamics" in order to have a reduced model representative of the long-term behavior.

In our example is fast w.r.t. , i.e.: reaches its steady state almost instaneoulsly w.r.t. to the settling time of . So we will assume ( does not evolve any more). Then, in the second row of the state equation (1), we get: . By replacing in the first row of the state equation and in the output equation (2), the first order reduced model reads:

Gslow=modred(G,2);

tf(Gslow)

- 'truncate': to be used to eliminate the "slow dynamics" in order to have a reduced model representative of the short-term behavior.

In our example is slow w.r.t to , i.e.: within the time constant of , has not time enough to change from its equilibrium value. So we will assume . Then considering the second row of the state equation and the output equation (2), the first order reduced model reads:

Gfast=modred(G,1,'truncate');

tf(Gfast)

This reduced model is quite obvious considering the structure of the model depicted in the block-diagram.