## 3 Tracking Control Laws  
### Assumptions and Definitions  
This paper considers the case where the vehicle is not equipped with planning rejection controllers. The planing force is large and occurs at dominating frequencies of about 10-50 Hz. Due to this high frequency, very fast and expensive actuators are required to actively stabilize and control the vehicle under planing. Since our control laws avoid planing completely, we can reduce the drag associated to planing force and do not require expensive actuators.  
Due to fuel consumption, center of gravity, mass, and mass moments of inertia matrix change as time goes on. Let _vehicle parameters_ denote these changing values for convenience. Let \(\rm pm(t)\) denote the _vehicle parameters_ at time step t. Let T denote the time step when the fuel tank becomes empty.  
We assume that the vehicle can check its fuel consumption in real time. We further assume that the vehicle can access a table showing the relationship between fuel consumption and the vehicle parameters. Using this table, the vehicle can check its changing parameters.  
We further assume that our vehicle is equipped with Inertial Measurement Units as well as depth sensors. Thus, the vehicle can measure its depth and attitude (roll, pitch, and yaw) in real time. Using these measurements, the vehicle generates its control inputs.  
As we control the depth of the vehicle, we do not control the lateral motion of the vehicle. The cavitator, fins 1 and 3 are called _elevators_, providing some component of steady lift to support the body. We control the cavitator, fins 1 and 3 to change the depth of the vehicle. Fins 1 and 3 are located on both sides of the vehicle. See Figure 1.  
Fins 2 and 4 are called _rudders_ that stabilize the vehicle in roll. In ideal environments without disturbances, the roll of the vehicle does not change as time goes on. Thus, we do not have to control the rudders (fins 2 and 4). However, due to disturbances in real environments, the roll of the vehicle changes as time goes on. This situation occurred as we add zero-mean Gaussian noise to control inputs of the vehicle to simulate disturbance in inputs. To stabilize the roll to zero in noisy environments, we control rudders, as presented in Section 3.6.  
Let _target depth_ denote the desired depth of the vehicle. We say that the pitch is _aligned_ in the following two cases:1. The target depth is deeper than the depth of the vehicle and the pitch of the vehicle is negative.
2. The target depth is shallower than the depth of the vehicle and the pitch of the vehicle is positive.  
Once the pitch is aligned, then the vehicle will reach the target depth as long as the vehicle moves straight while not rotating about the CG.  
Note that the control inputs, the deflection angles of fins or cavitator, cannot change abruptly due to the actuator limit. Considering this constraint, let input command denote the desired input of the vehicle. Once an input command is set, then the control input changes to reduce the difference between the command and the current input.  
For instance, let \(\delta_{\rm c}\) denote an input command for the cavitator pitch angle. Also, let \(\delta_{\rm k}\) denote the cavitator angle at time step \(\rm k\). Let \(A_{\rm f}\) denote the maximum angular rate of the cavitator pitch. In the case where \(\delta_{\rm c}>\delta_{\rm k}\), we set \(\delta_{\rm k+1}\) as \(\delta_{\rm k}+\min(\rm dt*A_{\rm f},\delta_{\rm c}-\delta_{\rm k})\). In the case where \(\delta_{\rm c}\leq\delta_{\rm k}\), we set \(\delta_{\rm k+1}\) as \(\delta_{\rm k}-\min(\rm dt\times A_{\rm f},\delta_{\rm k}-\delta_{\rm c})\). Here, \(\rm dt\) is the sampling interval. Also, \(\min(\beta_{1},\beta_{2})\) denotes the smaller value between \(\beta_{1}\) and \(\beta_{2}\).  
Let elevator command denote an input angle command associated to the cavitator, fins 1 and 3. Also, let rudder command denote an input angle command associated to fins 2 and 4.