Damped response

From ASCEND

In classical control theory, a second-order system has a transfer function like

We can model a system like this fairly easily with ASCEND. First we need to reduce the model to first order differential equations:

s²Y + 2sY + 2Y = AX

Now, let sY = Z so that we have a pair of equations

Taking the inverse Laplace transform and rearranging,

This can be coded into ASCEND in the usual syntax (see below).

Here is the output from the above system, with A = 2, x(t) = 1, plotted using OpenOffice. Different values of the coefficients in the equation give different levels of damping, resonant frequency, etc.

This plot shows that the above transfer function has roots at as expected:

Here is the model that was used to produce these solutions:

REQUIRE "ivpsystem.a4l";
 REQUIRE "atoms.a4l";
 IMPORT "johnpye/extpy/extpy";
 IMPORT "johnpye/roots";

 (*
    'Test problem'
    We create a set of models with artificially-selected roots
    and then check that the roots plot is as expected.
 *)
 MODEL simpleroots;

    range IS_A set OF integer_constant;
    range :== [7..8];
    x[range], dx_dt[range] IS_A solver_var;

    (* roots at -1 +/i j *)
    dx_dt[7] = 2.0 * 1.0 - 2.0 * x[7] - 2.0 * x[8];
    dx_dt[8] = x[7];

    t IS_A time;
 METHODS
    METHOD values;
        x[7] := 0;
        x[8] := 0;
    END values;

    METHOD ode_init;
        FOR i IN [range] DO
            x[i].ode_type := 1;
            dx_dt[i].ode_type := 2;
            x[i].ode_id := i;
            dx_dt[i].ode_id := i;
            x[i].obs_id := i;
        END FOR;
        t.ode_type := -1;
        t := 0 {s};
    END ode_init;

    METHOD on_load;
        RUN default_self;
        RUN reset; RUN values;
        RUN ode_init;
        RUN roots;
    END on_load;

    METHOD roots;
        EXTERNAL roots(SELF);
    END roots;
 END simpleroots;

See also IDA