trying to use the command hinfsyn on a rotary inverted pendulum for stabilization analysis

Hello

I am trying to follow this paper

http://ietresearch.onlinelibrary.wiley.com/doi/epdf/10.1049/tje2.12078

I have made following code for making the system looking like this

The code is given below

A=[0 0 1 0;
   0 0 0 1;
   0 39.32 -14.52 0;
   0 81.78 -13.98 0;];
B=[0;0;25.54;24.59];
C=[1 0 0 0;
    0 1 0 0];
D=[0;0;];
sys=ss(A,B,C,D);
sys.InputName={'u'};
sys.OutputName={'theta','alpha'};


s=tf('s');
Wact=0.002*(s+0.01)/(s+10);
Wact.u='u'; Wact.y='e1';
Wn1=ss(0.05);
Wn1.u='d1'; Wn1.y='wn1';
Wn2=ss(0.0275);
Wn2.u='d2'; Wn2.y='wn2';
Wx1=151.5/(s+50.5);
Wx1.u='theta'; Wx1.y='e2';
Wx2=202/(s+50.5);
Wx2.u='alpha'; Wx2.y='e3';


sdmeas=sumblk('y1 = theta+wn1');
abmeas=sumblk('y2 = alpha+wn2');
ICinputs={'d1','d2','u'};
ICoutputs={'e1','e2','e3','y1','y2'};
qsys=connect(sys,Wact,Wn1,Wn2,Wx1,Wx2,sdmeas,abmeas,ICinputs,ICoutputs);

When I am running the command hinfsyn I get following gamma

[~,~,gamma]=hinfsyn(qsys,2,1)

gamma = 0.3644

while in the paper they have

gamma=0.6951 in section 3.1 H-infinty controll. Where is my mistake or any solution to get the same answer as in the paper ?

Thanks for all answers

Tor Erik Haavik

[Paul]

Hi CyberRobotics,

Some observations ....

The C-matrix in the code above does not match eqn (28) in the linked paper. Though eqn (28) in the linked paper doesn't match the surrounding text in the paper (C(2,4) = 1?). So either they used a wrong C-matrix or eqn (28) has a typo.

Also, Figure 5 in the paper (and shown above) appears to be missing the reference command input for theta that would typically be included in the model to design a command following system (as appears to be the goal here) where the feedback to the controller (K(s)) would be the theta error signal.

Then Figure 6 says the step response is from Vm, but Vm is the control input and not an input to the closed-loop system.

Later in the paper the authors use acker() for pole placement, but MathWorks has recommended not using acker for a long time. place is recommended (though in this case acker did produce the correct result). As far as I can tell, acker is not included in the formal doc for the Control Systems Toolbox.

The authors state: "Obviously, there is no way we can have robustness analyzed for uncertainties with this [state feedback] structure." That statement is incorrect.

Figure 5 doesn't match the text. The text states that Wn1 is applied to the disturbance on theta and Wn2 to the disturbance on alpha. But Figure 5 shows the exact opposite. If you reverse the wn inputs to sdmeas and abmeas you'll get a gamma of 0.61, which is closer to the reported result.

[CyberRobotics]

So do think that a gamma of 0.61 and if I have understand correct my code works correctly just a bad article explaining the stuff?

[Paul]

Based on the article, it's hard to say what is "correct" because the results from the article are not able to be replicated, at least for me. Every time I look at a different section I have more concerns. I'm really curious about how the authors developed Figure 6.