Solve State Space Model HW: Find A, B, C, q

[kev08106]

Hi I have a problem I need to solve for a class soon.
The prof gave a us a homework problem where he gives us a state space representation for one system and connect it in cascade to another system and need the equivalent state space parameters:
Heres the question:
x(t)----> |system1 | --> y(t) --> |system2 |----> z(t)

system 1 is defined by:
dq1(t)/dt = A1(t)q1(t) + B1(t)x(t)
y(t) = C1(t)q1(t)

system 2 is defined by:
dq2(t)/dt = A2(t)q2(t) + B2(2)y(t)
z(t) = C2(t)q2(t)

the combination of the system is defined:
dq(t)/dt = A(t)q(t) + B(t)x(t)
z(t) = C(t)q(t)

It says find q(t), A(t), B(t), and C(t) in similar quantities for the separate systems.

 

[Valkarie]

I was thinking that q(t) = [q1(t) q2(t)] and then A(t) = [A1(t) 0; B1(t) A2(t)], B(t) = [B1(t) 0; 0 B2(t)], and C(t) = [C1(t) 0; 0 C2(t)] but I am not sure if this is correct. Any help would be appreciated. Thank you.

 

[piercas]

To solve this problem, we first need to understand the concept of a state space model. A state space model is a mathematical representation of a physical system that describes the relationship between the input, output, and internal state of the system. In this case, the input is x(t), the output is z(t), and the internal state is represented by q(t).

To find the state space parameters for the combined system, we need to combine the equations for system 1 and system 2. This can be done by substituting the equation for y(t) from system 1 into the equation for dq2(t)/dt in system 2, giving us:

dq2(t)/dt = A2(t)q2(t) + B2(2)C1(t)q1(t)

Next, we can substitute the equation for q1(t) from system 1 into the equation for dq2(t)/dt, giving us:

dq2(t)/dt = A2(t)q2(t) + B2(2)C1(t)(A1(t)q1(t) + B1(t)x(t))

We can then simplify this equation by factoring out q1(t) and x(t), giving us:

dq2(t)/dt = (A2(t) + B2(2)C1(t)A1(t))q1(t) + B2(2)C1(t)B1(t)x(t)

This equation now has the same form as the equations for system 1 and system 2, with A(t) = A2(t) + B2(2)C1(t)A1(t), B(t) = B2(2)C1(t)B1(t), and C(t) = C2(t). Therefore, to find the state space parameters for the combined system, we can simply substitute these values into the equations for system 1 and system 2.

To find q(t), we can use the equation for q1(t) from system 1, which is:

q1(t) = e^(∫A1(t)dt)q1(0) + ∫e^(∫A1(t)dt)B1(t)x(t)dt

We can then substitute this equation into the equation for q2(t) from system 2, giving us:

q2(t) = e^(∫A2(t)dt)q2(0