Dominant mode: illustration
1. A second order system:
Let us consider the system 
described in the following Figure.
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)
2. Long-term dominant mode:
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 
:
) 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.
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)')
3. Short-term dominant mode:
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 ∞. On the transfer functions, that means the terms in can be neglected w.r.t to terms in s or :
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 ∞. 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.
. 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)')
4. Model simplification by state elimination.
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:
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:
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.