The traditional approach to controlling systems with many degrees of freedom (e.g., aircraft, submarines, etc) is to assume that each degree of freedom is more or less independent and then use a standard single-input single-output such as PID.  While there are cases where this is generally true– an airplane, for example, almost always has a pitch less than 20 degrees– there are some cases where one genuinely needs a more complex approach.  Enter the so-called Linear Quadratic Regulator, or “LQR” controller.   An LQR-based controller uses matrices to represent each of the system’s states.  A ship, for example, might represent its state as a heading and a speed, while an aircraft might have heading, pitch, roll, altitude, airspeed, and so on.  One can also describe how the system will respond to no input– that is, no thrust from an engine or whatever– with a matrix (remember all those linear differential equations from Physics 112?).  Similarly, a matrix is used to describe the effect of all the inputs.  By using non-diagonal matrices, this model can even account for various interactions and cross-coupling between axes.   Thus, the next state can be written as x’ = Ax + Bu where x is the state, u is the “input” to the system, and A and B are the matricies that describe the dynamics of the system.  If one knows x (from sensor values) and x’ (from how the pilot sets the autopilot) and has previously-calculated values for A and B (discovered through weeks of careful testing by the control engineers), it is possible– but not trivial– to solve for the optimal value of u.  Fortunately, there are a few tricks to avoid computing the entire problem every time step– often or the order of 1-10 times a second– but that’s still a lot of matrix math.      http://documents.wolfram.com/applications/control/OptimalControlSystemsDesign/10.1.html          http://en.wikipedia.org/wiki/Linear-quadratic_regulator