System Identification of MISO systems

Question

Currently I am trying to do system identification for a system with two inputs and one output.

For the identification i can only excite one input at a time because in order to excite the second input, the system already needs to be in closed loop control, to avoid damage. The second input basically is a disturbance.

So I excite the system with the first input u1 and measure the output y1. This gives me a state space system of the form $$ \dot{x_1} = A x_1 + B u_1, \quad\text{with}\quad y_1 = x_1 $$ During this measurement, the second input can be kept constant so that it has no effect on the system.

Then I can design the controller and excite the system with the second input u2

The expected system response would be $$ \dot{x}_2 = A_2 x_2 + B_2 u_2 \quad\text{with}\quad y_2 = x_2 $$

In the second measurement, I cant keep the first input constant since it has to be used to control the system. But since I already know the system response to the input u1 rom the first measurement i can simply calculate back, assuming that the actual system response is the sum of both subsystems: $$ y = y_1 +y_2 = x_1 + x_2$$

This means i can simply subtract the expected output of system 1 from the second measurement and then do the system identification for y2.

Two system equations have now emerged from the two measurements. My problem is that I cannot figure out how to combine the two equations into one.

Since the states of both systems describe the same physical quantities, I do not want to add any additional states.

The system equation of the two systems looks like this $$ \dot{x}_1 + \dot{x}_2 = A_1 x_1 + A_2 x_2 + B_1 u_1 + B_2 u_2 $$

So I need a mathematical way to combine the two system matrices A1 and A2 into a new matrix Anew.

$$ A_{new} (x_1+x_2) = A_1 x_1 + A_2 x_2 $$

Which reflects both systems

Can anyone point me to a solution for this problem?

Answer 1

It might be reasonable to treat the two separately identified systems as parallel, i.e., $y = (G_1 + G_2)u$, in which case the systems adds like

G1 + G2
A = 
 a1   0
  0  a2
B = 
 b1
 b2
C = 
 c1  c2
D = 
 d1 + d2

This gives you the desired output characteristics $y = y_1 + y_2$, but differs from the equations you have written down in that this creates a system with a number of states given by the sum of the number of states in the two separately identified systems.