LMIs in Control/pages/H inf Aircraft Optimization

Robust $$H_\infty $$ Aircraft Dynamics

The Optimization Problem
This Optimization problem involves the use of optimizing aircraft dynamics using the regulator framework, and optimizing the given aircraft parameters using the following set of aircraft dynamics. these aircraft dynamics are given non-dimensional characteristics defined by various parameters of the aircraft being tested. By making these characteristics non-dimensional, it allows for the problem to be scaled to larger porportions. For instance the longitudinal dynamics for an aircraft system are defined as suchː



\begin{align} A_p = \begin{bmatrix} X_u & X_w & 0 & - g_0 \cos(\vartheta_0) \\ \frac{Z_u}{1-Z_{\dot{w}}} & \frac{Z_w}{1-Z_{\dot{w}}} & \frac{u_0 + Z_q}{1-Z_{\dot{w}}} & \frac{-g_0 \sin(\vartheta_0)}{1 - Z_{dot{w}}}\\ M_u + \frac{M_{\dot{w}} Z_u}{1 - Z_{\dot{w}}} & M_w + \frac{M_{\dot{w}} Z_w}{1 - Z_{\dot{w}}} & M_w + \frac{(u_0 + Z_q) M_{\dot{w}}}{1 - Z_{dot{w}}} & \frac{-M_{\dot{w}} g_0 \sin(\vartheta_0)}{1 - Z_{dot{w}}}\\ 0 & 0 & 1 & 0\\	\end{bmatrix}\\ B_p = \begin{bmatrix} X_{\delta_e} & X_{\delta_T} \\ \frac{Z_{\delta_e}}{1-Z_{\dot{w}}} & \frac{Z_{\delta_T}}{1-Z_{\dot{w}}} \\ M_{\delta_e} + \frac{M_{\dot{w}} Z_{\delta_e}}{1 - Z_{\dot{w}}} & M_{\delta_T} + \frac{M_{\dot{w}} Z_{\delta_T}}{1 - Z_{\dot{w}}} \\ 0 & 0\\	\end{bmatrix}\\ C_p = I \\ D_p = 0 \\ \end{align}$$

This Optimization problem involves the same process used on the full-feedback control design; however, instead of optimizing the full output-feedback design of the Optimal output-feedback control design. This is done by defining the 9-matrix plant as such: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$. Using this type of optimization allows for stacking of optimization LMIs in order to achieve the controller synthesis for both a robust $$H_\infty $$ optimization.

The Data
The data is dependent on the type the state-space representation of the 9-matrix plant; therefore the following must be known for this LMI to be calculated: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$.

The LMI: Robust $$H_\infty$$ Optimal Aircraft Dynamics Control
There exists the scalar, $$ \gamma $$, along with the matrices $$ S > 0 $$, and $$ R $$ where:



\begin{align} \begin{bmatrix} N_R & 0 \\ 0 & I \\ \end{bmatrix}^{T} \begin{bmatrix} A R+R A^{T} & R C^T_1 & B_1 \\ C_1 R & -\gamma I & D_{11} \\ B^{T}_{1} & D^{T}_{11} & -\gamma I \\ \end{bmatrix} \begin{bmatrix} N_R & 0 \\ 0 & I \\ \end{bmatrix} < 0 \\

\begin{bmatrix} N_S & 0 \\ 0 & I \\ \end{bmatrix}^{T} \begin{bmatrix} A^{T} S+S A & S B_1 & C^{T}_1 \\ B^{T}_1 S & -\gamma I & D^{T}_{11} \\ C_1 & D_{11} & -\gamma I \\ \end{bmatrix} \begin{bmatrix} N_S & 0 \\ 0 & I \\ \end{bmatrix} < 0 \\

\begin{bmatrix} R & I \\ I & S \\ \end{bmatrix} \geq 0 \end{align}$$

Where the optimized controller matrices are defined as followsː



\begin{align} A_K = -N[A^T + S(A + B_2 F+L C_2)R+1/\gamma S(B_1 + L D_{21})B^{T}_1 + 1/\gamma C^{T}_1 (C_1 + D_{12}F)R]M^{T} \\ \\	B_K = N^{-1} S L \\ \\	C_K = F R M^{-T} \\ \\	D_K = 0; \end{align} $$

Where:



\begin{align} N = S \\ \\	M = S^{-1} - R \\ \\	F = -(D^{T}_{12} D_{12})^{-1} (\gamma B^{T}_2 R^{-1} + D^{T}_{12} C_1) \\ \\       L = -(\gamma S^{-1} C^{T}_2 + B_1 D^{T}_{21}) (D_{21} D^{T}_{21})^{-1} \\ \\ \end{align} $$

Conclusion:
The results from this LMI give a controller that is a robust $$ H_\infty $$, optimization which would be capable of stabilizing various aircraft Dynamics.

Implementation

 * Robust H_inf optimization example