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-if QAUSI-STATIC then x=x_quasistatic, and x_dynamic and it’s time derivatives are zero. But dy/dt is not necessarily zero, nor is du/dt necessarily zero.

-if QAUSI-STATIC then x=x_quasistatic which can vary with time, but still x_dynamic and it’s time derivatives are zero. But dy/dt is not necessarily zero, nor is du/dt necessarily zero.

———— EDIT / PARTIAL ANSWER after comments with @aconcernedcitizen:

3. how do I obtain quasi-statics from dynamics formulation in state space?

I assert that the State Space Formulation uses dynamical information, so my system state variables: x, dx/dt, and d2x/dt2 become x_dynamic, dx/dt_dynamic, d2x/dt2_dynamic.

Written out in the first vector state space equations it would be:

[dx/dt_dynamic;d2x/dt2_dynamic]=A•[x_dynamic;dx/dt_dynamic]+B•[u;du/dt]

Inputs u, du/dt, and matrices A and B

B=[b11, b12; b21, b22] which is still satisfies the LTI condition, BUT is still able to switched OFF so in that sense is a function of which-process time, i.e. if dynamics are turned off then B=[0,0;0,0].

But that does NOT mean that u nor du/dt are turned off (zero) necessarily.

I assert that x=x_dynamic + x_static.

-If fully still-static (dead) then x=x_static, and x_dynamic ARE zero and that includes x_dynamic’s time derivatives. (Also probably for this case du/dt and dy/dt should be zero. The input/output would not be zero/zero 0/0 as because input=u,output=y=x_static)

-if QAUSI-STATIC then x=x_quasistatic, and x_dynamic and it’s time derivatives are zero. But dy/dt is not necessarily zero, nor is du/dt necessarily zero.

I assign the OUTPUT process variables y and dy/dt to be equal to the x_quasistatic, and dx/dt_quasistatic respectively.

So the second “state space formulation” for the OUTPUTS is:

[y;dy/dt]=C•[x_dynamic;dx/dt_dynamic]+D•[u;du/dt]

[y;dy/dt]->[x_quasistatic;dx/dt_quasistatic]

[x_dynamic;dx/dt_dynamic]->[0;0]

What that attempts to show is that the system is 1) it is at least a type of static system because d2x/dt2 is zero, 2) it is quasi-static because dx/dt is zero, but not x, 3) u and du/dt are not zero so the load is slowly applied which drives the output of the system, even those the states are zero.

[y; dy/dt] = C*[x;dx/dt] + D*[u; du/dt] [y; dy/dt] = C*[x; 0] + D*[u; du/dt]

Where A=[a11,a12;a21,a22] is a 2x2 state transition matrix Where B=[b11,b12;b21,b22] is a 2x2 matrix

What that attempts to show is that the system is 1) it is at least a type of static system because d2x/dt2 is zero, 2) it is quasi-static because dx/dt is zero, but not x, 3) u and du/dt are not zero so the load is slowly applied which drives the output of the system, even those the states are zero.

[y; dy/dt] = C*[x;dx/dt] + D*[u; du/dt] [y; dy/dt] = C*[x; 0] + D*[u; du/dt]

where C=[c11,c12;c21,c22] a 2x2 matrix where D=[d11,d12;d21,d22] a 2x2 matrix