Understand Forces and Moments: Identify all forces and moments acting on the AUV, including gravitational, buoyant, hydrodynamic, and propulsive forces.
Apply Newton-Euler Equations: Use Newton-Euler equations to relate linear/angular accelerations to the sum of external forces/moments:
Hydrodynamic Modeling: Include hydrodynamic effects like added mass, damping, and restoring forces in your equations of motion.
Equations of Motion: Formulate the equations of motion, which describe how the AUV’s state evolves over time based on the applied forces and moments.
Choose Operating Point: Identify a nominal operating point (e.g., cruising at constant depth and speed) around which to linearize the system.
Taylor Series Expansion: Use Taylor series expansion to linearize the nonlinear equations of motion about the operating point. This involves computing the Jacobian matrices (partial derivatives) of the nonlinear functions with respect to the states and inputs.
Linear Approximation: Obtain a linear approximation of the system dynamics in the form: $\dot{x} = f(x, u) ≈ f(x_0, u_0) + A(x - x_0) + B(u - u_0)$ Where A and B are Jacobian matrices evaluated at the operating point $(x_0, u_0)$
Extract Matrices: Identify the A and B matrices from the linear approximation of the system dynamics.
State Space Representation: Formulate the state-space representation using the A and B matrices: $\dot{x} = Ax + Bu$
Check Controllability: Ensure that the system is controllable by checking the rank of the controllability matrix $(C = [B, A,B ,A^2,B, ..., A^{n-1}B])$. The system is controllable if $rank(C) = n$, where n is the number of states.
Q and R Matrices: Choose the Q (state cost) and R (control input cost) matrices, considering the importance of each state and control input in the cost function.
Cost Function: Define the cost function to be minimized: $J = \int_{0}^{\infty} (x^TQx + u^TRu) dt$