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Loop-Shaping Controller Design

One of the most powerful yet simple controller synthesis tools is loopsyn. Given an LTI plant, you specify the shape of the open-loop systems frequency response plot that you want. Then loopsyn computes a stabilizing controller that best approximates your specified loop shape.

For example, consider the 2-by-2 NASA HiMAT aircraft model (Safonov, Laub, and Hartmann [1]), depicted in the following diagram.

The control variables are elevon and canard actuators ( $\delta_e$ and $\delta_c$). The output variables are angle of attack ( $\alpha$) and attitude angle ( $\theta$). The model has six states, given by:

$$x = \left[ {\matrix{
   {{x_1}}  \cr
  {{x_2}}  \cr
  {{x_3}}  \cr
  {{x_4}}  \cr
  {{x_5}}  \cr
  {{x_6}}  \cr
} } \right] = \left[ {\matrix{
  {\dot \alpha }  \cr
  \alpha   \cr
  {\dot \theta }  \cr
  \theta   \cr
  {{x_e}}  \cr
  {{x_\delta }}  \cr
} } \right],$$

where $x_e$ and $x_{\delta}$ are the elevator and canard actuator states, respectively.

Create HiMAT Model

The following commands create a state-space model G of the aircraft.

ag =[ -2.2567e-02  -3.6617e+01  -1.8897e+01  -3.2090e+01   3.2509e+00  -7.6257e-01;
       9.2572e-05  -1.8997e+00   9.8312e-01  -7.2562e-04  -1.7080e-01  -4.9652e-03;
       1.2338e-02   1.1720e+01  -2.6316e+00   8.7582e-04  -3.1604e+01   2.2396e+01;
       0            0   1.0000e+00            0            0            0;
       0            0            0            0  -3.0000e+01            0;
       0            0            0            0            0  -3.0000e+01];
bg = [ 0     0;
       0     0;
       0     0;
       0     0;
       30     0;
       0    30];
cg = [ 0     1     0     0     0     0;
       0     0     0     1     0     0];
dg = [ 0     0;
       0     0];
G = ss(ag,bg,cg,dg);

Design Controller

To design a controller to shape the frequency response (singular-value) plot so that the system has approximately a bandwidth of 10 rad/s, specify your target desired loop shape $G_{d}(s) = 10/s$. Then use loopsyn to find a loop-shaping controller for G that optimally matches the desired loop shape Gd.

s = zpk('s');
w0 = 10;
Gd = w0/(s+.001);
[K,CL,GAM] = loopsyn(G,Gd);

Examine the open-loop frequency response with the resulting controller, K.

sigma(G*K,'r',Gd,'k-.',Gd/GAM,'k:',Gd*GAM,'k:',{.1,30})
legend('Achieved Loop Shape','Target Loop Shape','Gd/GAM','Gd*GAM')

Examine the closed-loop response as well.

T = feedback(G*K,eye(2));
sigma(T,ss(GAM),'r*',{.1,30});
legend('Closed loop','GAM')
grid

The returned value GAM is an indicator of the accuracy to which the optimal loop shape matches your desired loop shape. GAM is an upper bound on the resonant peak magnitude of the closed-loop transfer function T = feedback(G*K,eye(2)). In this case, GAM = 1.6024 = 4 dB, as the singular value plots show. The plots also show that the achieved loop shape matches the desired target Gd to within about GAM dB.

References

[1] Safonov, M.G., Laub, A.J., and Hartmann, G., "Feedback Properties of Multivariable Systems: The Role and Use of Return Difference Matrix," IEEE Trans. of Automat. Contr., 1981, AC-26(1), pp. 47-65.

See Also

Related Topics

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