I am interested in a LTI system
x_dot = A*x + B*u
y = C*x + D*u
(M*q_ddot + C*q_dot + K*q = f)
such that
f = vector of external forces
q = vector of displacements
u = [0 f^T 0]^T
x = [q q_dot]^T
y = [q^T q_dot^T q_ddot^T]^T
u is a vector of forces
A = [ 0 I
-M^-1*K -M^-1*C]
M is a mass matrix, C is damping matrix and K is stiffness matrix (all of them positive definite or semidefinite)
all of the eigenvalues of A has negative real part
I need to know whether exists a constant e such that
>= e * || y ||
If y = [q^T q_dot^T]^T and u = [0 f^T]^T (displacements and velocities only), I have shown that exists a constant e such that >= e * || y || (through Lyapunov equation), but I cannot find it when considering accelerations (D not equal to zero matrix). My conjecture is that the systems is still strictly output passive.
Without going into math, note that acceleration is not a "state" of your system. It is a derivative of the velocity (which is a state), so although for I:force and O:velocity the system is passive, this may not be the case for acceleration.
An example from linear random vibrations:
a white-noise force excited linear oscillator has a bounded displacement and velocity variances. But the acceleration variance is not defined... the integral is infinite!
I hope the above is a bit helpful!
Dear Agathoklis Giaralis
Your comments are highly appreciated. I thought acceleration could be bounded because it is a linear combination of states and inputs. Can you please provide me with a reference to fact you quoted about random vibrations? I think it could help me to find the counter example for my erroneous conjecture.
Do yoy know if boundedness of velocities H2 norm can be used to show or find boundedness of displacements in these oscilatory systems?
Thank you very much.
Greetings
Hernán
What I mention above can be found in any random vibrations book. See for example page 105 in "Roberts and Spanos, Random Vibrations and statistical linearization, Dover Publications, 2003".
Sorry, but the presentation leaves me confused. As things sometimes look, not always the question is "how?" but it may also need a "why?"
You write the state
x = [q q_dot]^T
and then the ouput
y = [q^T q_dot^T q_ddot^T]^T
In other words, your output not only contains the entire state, but you also seem to measure all accelerations. How? Why?
If D=0, then y=Cx and gives you what it gives. Same with y=Cs+Du, only more complex, because you add the direct input-output branch. How do you get that D adds perfect acceleration to your y?
Actually,, in both cases, you may have one input and one output and 1000 state variable, so the output is some combination of the 1000 variables.
It is known that, with proper velocity measuring (Mark Balas) or position-plus-velocity measuring (Barkana) and perfect sensors-actuators collocation, you can stabilize the structure and make the closed-loop system passive.
I hope it helps clarifying things and does not add confusion.
Cheers,
Itzhak
Thanks for your comments barkana.
With proper definition of D, y can get exact accelerations since q_ddot is the lower part of x_dot. So accelerations are a linear combination of displacements, velocities and exciting force.
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