LMIs in Control/pages/Maximum Natural Frequency in State Feedback

The System:
The number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix $$B$$. This can be achieved by adding an Apkarian filter in the input of the system.

The Optimization Problem:
Apkarian Filter 

Let consider our TS-LIA model. This can be re written in linear form as:



\dot{x} = A(z(t)) x + B(z(t)) u

$$

The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown, where $$ A_F$$ = −100$$ I_4$$, $$B_F$$ = 100$$I_4$$ and $$I_4$$ ∈ $$R^{4\times4}$$ is the identity matrix.


 * $$ \dot{x_F} = A_F x_F + B_F u_F; y_f = x_f$$.

When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. $$u$$ = $$y_F$$ ). Then, the extended model is:



\dot{x_c}= \begin{bmatrix} A(z(t))&B(z(t))\\ 0&A_F\\ \end{bmatrix} x_c + \begin{bmatrix}0\\B_F\end{bmatrix} u_F = A_e(z(t)) x_e + B_e u_f;

x_e = \begin{bmatrix}x\\x_F \end{bmatrix} $$

This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.

State Feedback Controller Design

Let consider the state feedback control law for the extended TS-LIA model:$$ \dot{x_e} = \sum_{i=1}^{32} h_i(z(t)) [ A_{ei} x_e + B_{ei} u_F ]$$, where the state feedback control laws are :$$ u_F= \sum_{i=1}^{32} h_i(z(t)) K_i x(t) $$, we get the closed loop system  :$$ \dot{x_e} = \sum_{i=1}^{32} \sum_{j=1}^{32} h_j(z(t)) [ A_{ei} x_e + B_{ei} K_j ]x_e$$

The LMI:
The design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:


 * $$L \otimes P + M \otimes P (A_{ei} + B_e K_i) + MT \otimes (A_ei + B_e K_i)^T P < 0$$ ∀i = 1, . . ., 32.

A pair of conjugate complex poles s of the closed loop system can be written as $$s$$ = − $${\xi} \omega_n \pm j\omega_d $$where $$\xi$$ is the damping ratio,$$\omega_n$$ is the undamped natural frequency and $$\omega_d $$ is the frequency response defined as $$ \omega_d = \omega_n$$ $$ \sqrt{1-\xi^2}$$.Three different LMI regions have been considered, each one related with a performance specification regarding $$ \alpha = \xi \omega_n,\omega_n$$ and $$\xi$$:

Maximizing Natural Frequency:

Natural frequency is related with the maximum frequency response in the undamped case ($$ \xi $$ = 0). If we want to set a maximum $$ \omega_n $$ condition, the LMI region associated is $$S_\rho$$ = [s = x + jy | |x + jy| < $$\rho$$], ::$$ L_\rho = \begin{bmatrix}-{\rho}&0\\0&\rho \end{bmatrix}, M_\rho = \begin{bmatrix}0&1\\0&0 \end{bmatrix} $$. Resulting LMI condition is:



\begin{bmatrix} - \rho P&P(A_{ei}+B_{ei} K_i)\\ (A_{ei}+B_{ei} K_i)^T P& - \rho P \end{bmatrix} < 0 $$ ∀i = 1,. . ., 32.

Conclusion:
The LMI is feasible.

Related LMIs

 * Apkarian Filter and State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Apkarian_Filter-and_State_Feedback
 * Minimum Decay Rate in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Minimum_Decay_Rate_in_State_Feedback#The_LMI%3A