error: hinfsyn: (A, B2) must be stabilizable

[ovhnw]

New to H-infinity.

Got an error while trying to construct a controller.

Error persists with a simple example shown below. Model of a mass moving along one dimension. Control inputs and disturbance both effect plant in the same way (by applying a force).

pkg load control

% x = [x x_dot]'
A = [0 1;0 0];
B = [0 1]';
C = [1 0];
P21 = ss(A,B,C);
P22 = ss(A,B,C);
P11 = P21;
P12 = P22;

sys_for_h_inf = [   P11 P12;
                    P21 P22];
hinfsyn(sys_for_h_inf,1,1)

results in

error: hinfsyn: (A, B2) must be stabilizable
error: called from
    hinfsyn at line 231 column 5

(isctrb(A,B) is true/1)

I didn't know if the problem might not be well posed since there is no penalty for control effort, so I modified the problem to add control effort to the z output:

pkg load control

% x = [x x_dot]'
A = [0 1;0 0];
B = [0 1]';
C = [1 0];
P21 = ss(A,B,C);
P22 = ss(A,B,C);
P11 = ss(A,B,[C;0 0]);
P12 = ss(A,B,[C;0 0],[0;1]);

sys_for_h_inf = [   P11 P12;
                    P21 P22];
hinfsyn(sys_for_h_inf,1,1)

...same error
octave 9.4.0
archlinux

:) thanks

[ttl-octave]

I think your system P is not well defined. By stacking the same system in the way shown in your post, you are producing parallel dynamics which are identical. This results in non-controlability of the system part for which hinfsyn tries to find a stabilizing controller for.

Maybe the function augw might help here. However, a double integrator might generally cause problems

[ovhnw]

For this simple example, since the disturbances effect the plant the same way that the input does, and since the plant outputs are the very same variables that i want to limit, it seems like i need to have 4 identical transfer functions in the P matrix, yes?

But you are suggesting that replicating system dynamics is what is causing my error. So I need to create a system with identical transfer functions, without actually replicating the dynamics? Is this a limitation imposed by H-infinity, or by the Octave implementation?

The MATLAB documentation for hinfsyn, under section "input arguments" states that it converts P to a state space model. I don't know what is included with this conversion. This converted state space model only has one copy of the system dynamics, and is identical to the state space system expected by the SLICOT routines (SB10DD and SB10AD) referenced by the Octave function hinfsys. Does the MATLAB hinfsys perform a conversion (to eliminate redundant system dynamics) that the Octave hyinfsys does not?

The Wikipedia page for H-infinity first presents the H-infinity framework in terms of a system P broken into quadrants, and does not mention the form referenced by MATLAB and SLICOT, and does not mention any limitation on repeated system dynamics. However, when working with H-infinity, does one typically instead work with the form referenced by MATLAB and SLICOT?

I don't have MATLAB, so I can't compare, if this might be a compatibility issue, if MATLAB performs a conversion to eliminate redundant dynamics for you, but Octave does not.

If i'm understanding the limitations correctly, perhaps it would be useful to add a picture like below (copied from SLICOT docs) to the Octave docs to show the expected plant format, and pointing out that there must only be one copy of the plant dynamics, A?

                | A  | B1  B2  |   | A | B |
            P = |----|---------| = |---|---|
                | C1 | D11 D12 |   | C | D |
                | C2 | D21 D22 |

...unless this is common knowledge to those experienced with H-infinity :)

Thanks again :)

[ttl-octave]

Adding a more comprehensive documentation is definitely required. This will go on my todo list.

The limitation is a common one: the system has to be stabilizable from the last input(s), which are the outputs of the controller. In your case, there is only one input for four uncoupled subsystems with identical dynamics. This setup is not controllable and since you have a double integrator, not stabilizable. The structure shown in Wikipedia is the structure of the system's transfer matrix with entries from each input to each output.

I don't know what other programs are doing, but there is no conversion to a sort of minimal realization in Octave.

[ttl-octave]

I have extended the help for hinfsyn with commit ec2e297. Closing this issue.