LMIs in Control/Click here to continue/Applications of Linear systems/Damping Improvement of a F-16 Aircraft through Linear Matrix Inequalities

Damping Improvement of a F-16 Aircraft through Linear Matrix Inequalities

Introduction
The dynamic response characteristics of an aircraft are highly nonlinear. Generally flight control systems have been designed using mathematical models of an aircraft linearized around several operating points, the controller parameters are programed in accordance with the flight conditions. Among all the techniques presented in control, the linear matrix inequality becomes a possible tool to solve various optimization problems of the F-16 aircraft. The main advantage of this technique is to allow the simultaneous treatment of various performance and robustness requirements. This is because of the emergence of interior point algorithms for the solution of convex optimization problems, which made it possible to solve the linear matrix inequalities much faster and more efficiently.

The System
$$ \begin{align} \dot{x} &= Ax + Bu\\ \end{align} $$

$$ \begin{align} y = Cx \end{align} $$

where x is the state vector, y is the output vector, and u is the input vector and these are given by.

$$ \begin{align} \dot{x} = \begin{bmatrix} \dot{\Phi}\\ \dot{\beta}\\ \dot{P}\\ \dot{R}\\ \dot{\delta_a}\\ \dot{\delta_r}\\ \end{bmatrix} , y = \begin{bmatrix} \Phi\\ \Beta\\ P\\ R\\ \end{bmatrix} , u = \begin{bmatrix} \delta_a\\ \delta_r\\ \end{bmatrix} \end{align} $$

Here Phi, Beta, P, R, delta_a, delta_r represent the aircraft's roll angle, the angle of side slip, roll rate, pitch rate, aileron and rudder deflections respectively. The structure of the controller to be used to control the F-16 aircraft is pre-defined, which is an important feature, considering the practical application of control systems. the restricted structure is given by the following transfer function.

$$ \begin{align} K_{y_k \rightarrow u_l} = \frac{a_{y_k \rightarrow u_l} \cdot s^2 + b_{y_k \rightarrow u_l} \cdot s + c_{y_k \rightarrow u_l}}{s^2 + (p_1+p_2) \cdot s + p_1 \cdot p_2} \end{align} $$

In this work, we work with predefined poles and we determine the gains and the zeroes given by the values of ayk_ul, byk_ul and cyk_ul of the controller constrained to feasible values. The control method comprises of applying an output feedback control to the F-16 system. The controller can be written in state space form as.

$$ \begin{align} \dot{x_c} &= A_c \cdot X_c + B_c\cdot y\\ \end{align} $$

$$ \begin{align} u = C_c \cdot X_c + D_c\cdot y\\ \end{align} $$

Where A_c and B_c are given by

$$ \begin{align} A_c = \begin{bmatrix} 0 & 1 & \cdot & \cdot & \cdot & 0 & 0\\ k & j & \cdot & \cdot & \cdot & 0 & 0\\ \cdot & \cdot & \cdot &    \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot &  \cdot &   \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot &    \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & \cdot & \cdot & \cdot & 0 & 1\\ 0 & 0 & \cdot & \cdot & \cdot & k & j\\ \end{bmatrix} \end{align} $$

$$ \begin{align} B_c = \begin{bmatrix} 0 & 1 & \cdot & \cdot & \cdot & 0 & 0\\ \cdot & \cdot & \cdot &    \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot &  \cdot &   \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot &    \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & \cdot & \cdot & \cdot & 0 & 1\\ \end{bmatrix} \end{align} $$

Where k = -p1*p2 and j = p1+p2, p1 and p2 are the poles of the system. Applying the controller to the model, we obtain the following description of the system in closed loop:

$$ \begin{align} \begin{bmatrix} \dot{x}\\ \dot{x_c} \end{bmatrix}= \begin{bmatrix} A + B\cdot D_c\cdot C & B\cdot C_c \\ B_c\cdot C & A_c \end{bmatrix} \cdot \begin{bmatrix} x\\ x_c \end{bmatrix} \end{align} $$

using the design of dynamic controllers the above problem can be rewritten as a static state feedback output problem as:

$$ \begin{align} \dot{x_m} &= A_m \cdot X_m + B_m\cdot y_m\\ \end{align} $$

$$ \begin{align} y = C_m \cdot u_m\\ \end{align} $$

where Am, Bm Cm and the static controller output Kc is given as follows:

$$ \begin{align} A_m = \begin{bmatrix} A & B\cdot C_c\\ 0 & A_c \end{bmatrix} , B_m = \begin{bmatrix} B & 0\\ 0 & I \end{bmatrix} , C_m = \begin{bmatrix} C & 0 \end{bmatrix} , K_c = \begin{bmatrix} D_c\\ B_c \end{bmatrix} \end{align} $$

Output Feedback Control
Applying the control law Um = Kc.y, we can change the position of the system poles in the closed loop as

$$ \begin{align} \dot{x_m} &= (A_m + B_m\cdot K_c\cdot C_m)\cdot x_m = A_cl \cdot x_m\\ \end{align} $$

$$ \begin{align} A_cl\cdot Q &= A_m\cdot Q + B_m\cdot K_c\cdot C_m\cdot Q\\ \end{align} $$

The above system equation is linearized by considering the following variable transformation

$$ \begin{align} K_c\cdot c_m\cdot Q &= N\cdot C_m\\ \end{align} $$

$$ \begin{align} A_cl\cdot Q &= A_m\cdot Q + B_m\cdot N\cdot C_m\\ \end{align} $$

Linear Matrix Inequality
$$ \begin{align} \begin{bmatrix} f\cdot A_cl\cdot Q + f\cdot Q\cdot A_cl^T & g\cdot A_cl\cdot Q + g\cdot Q\cdot A_cl^T\\ g\cdot A_cl\cdot Q + g\cdot Q\cdot A_cl^T & f\cdot A_cl\cdot Q + f\cdot Q\cdot A_cl^T \end{bmatrix} < 0 \end{align} $$

where,

$$ \begin{align} A_cl\cdot Q &= A_m\cdot Q + B_m\cdot N\cdot C_m\\ \end{align} $$

$$ \begin{align} Q\cdot A_cl^T &= Q_m\cdot A_m^T + C_m^T\cdot N^T\cdot B_m^T\\ \end{align} $$

$$ \begin{align} f &= \sin\theta\\ \end{align} $$

$$ \begin{align} g &= \cos\theta\\ \end{align} $$

$$ \begin{align} 2\cdot \alpha\cdot Q + A_m\cdot Q + B_m\cdot N\cdot C_m + Q\cdot A_m^T + C_m^T\dot N^T\cdot B_m^T < 0\\ \end{align} $$

Conclusions
The above methodology can be used for pole placement of linearized systems around various operating points of the complex plane, defined by the region of the intersection of the linear matrix ineequality. The controller used in the above methodology has a fixed structure. When applied to the pole placement through the linear matrix inequality to the lateral F-16 aircraft system, there was an improvement in all the properties considered for specific performance conditions.