8.1 Feedback control systems for underwater vehicles

Underwater vehicles need precise control to navigate the depths. Feedback control systems are key, using sensors, controllers, and actuators to keep vehicles on course. These systems continuously measure the vehicle's state and make adjustments to reach desired positions and orientations.

PID controllers are widely used in underwater robotics. They calculate control inputs based on error signals, helping vehicles maintain depth, heading, and speed. Tuning PID gains is crucial for optimal performance, balancing response time and stability in challenging underwater environments.

Feedback Control for Underwater Vehicles

Principles and Components of Feedback Control

• Feedback control systems continuously measure the output of a system and compare it to a desired reference value or setpoint
  • The difference between the measured output and the setpoint is the error signal
  • Feedback control enables underwater vehicles to operate autonomously by automatically correcting for disturbances and uncertainties in the environment or vehicle dynamics
• The three main components of a feedback control loop for underwater vehicles:
  • Sensors to measure the controlled variable (pressure sensor for depth, compass or gyro for heading, DVL for velocity)
  • Controller to compute the required actuator inputs based on the error signal (PID, state-space, adaptive control)
  • Actuators to generate forces and moments to correct the vehicle's state (thrusters, control surfaces, variable buoyancy)

Types of Control Systems

• Open-loop control applies predetermined inputs without regard to the system's actual output
  • Example: A thruster commanded to run at a fixed rpm regardless of the vehicle's resulting speed
• Closed-loop control continuously adjusts the inputs based on measured output
  • Example: A depth controller that adjusts thruster rpm based on the difference between the desired and actual depth from a pressure sensor
• Underwater vehicles primarily rely on closed-loop feedback control to maintain depth, heading, velocity and other parameters in the presence of disturbances like currents, waves and changing payloads

PID Controller Design for AUVs

PID Control Algorithm

• Proportional-Integral-Derivative (PID) control is a widely used feedback control method that calculates actuator inputs as a weighted sum of the error signal, its integral, and derivative

  • The proportional term ($K_p$) provides a restoring force proportional to the current error
    • Example: A large depth error generates a large thruster command to quickly correct the depth
  • The integral term ($K_i$) accumulates error over time to eliminate steady-state offset
    • Example: A small persistent depth error is integrated to generate an additional thruster command that corrects for buoyancy changes
  • The derivative term ($K_d$) responds to the rate of change of error to improve transient response and stability
    • Example: A rapidly changing depth error produces a counteracting thruster command to prevent overshoot

• The PID control law in the time domain is: $u(t) = K_p e(t) + K_i \int_{0}^{t} e(\tau) d\tau + K_d \frac{de}{dt}$

  where $u$ is the control input and $e$ is the error signal

PID Tuning and Implementation

• PID gains ($K_p$, $K_i$, $K_d$) must be carefully tuned to achieve the desired control performance in terms of rise time, overshoot, settling time, and steady-state error
  • Tuning methods include manual iteration, Ziegler-Nichols, and optimization algorithms
  • Example: Increasing $K_p$ speeds up the response but may lead to overshoot, while increasing $K_d$ dampens overshoot but slows the response
• Implementing PID control for underwater vehicles requires:
  • Defining the control variables and setpoints (depth, heading, surge velocity)
  • Selecting appropriate sensors and actuators with sufficient bandwidth and authority
  • Developing the control software to execute the PID algorithm at an appropriate sample rate
  • Integrator anti-windup logic to handle actuator saturation
    • Example: Preventing the integral term from accumulating when thrusters are at maximum rpm
• Separate decoupled PID loops are commonly used for each degree of freedom, but coupling between axes (pitch affects depth) may require MIMO control methods
• Nonlinear PID variants such as gain scheduling can provide improved performance over a wide operating envelope
  • Example: Reducing gains at high speeds where drag is higher to prevent oscillations

Stability and Performance of Underwater Robot Control

Stability Analysis

• The stability of a closed-loop system refers to its ability to converge to an equilibrium point from different initial conditions and in response to disturbances
  • An unstable system will have diverging oscillations or unbounded outputs
  • Example: An improperly tuned depth controller causing the AUV to repeatedly overshoot and undershoot the target depth
• For linear time-invariant (LTI) systems, stability can be analyzed using the locations of the closed-loop poles in the s-plane
  • A system is stable if all poles are in the left-half plane
  • Example: A system with poles at $s=-1$ and $s=-2$ is stable, while a system with poles at $s=1$ and $s=-1$ is unstable
• Stability margins quantify how close the system is to instability under variations in gain or phase
  • Gain margin is the factor by which the gain can be increased before instability
  • Phase margin is the additional phase lag at the gain crossover frequency required for instability
  • Example: A gain margin of 6 dB means the controller gain can be doubled before the system goes unstable

Performance Metrics

• Time-domain performance specifications for control systems include:
  • Rise time: time to reach a specified fraction of the steady-state value
  • Overshoot: how much the response exceeds the final value
  • Settling time: time for the response to converge within a percentage of the final value
  • Steady-state error: difference between the setpoint and actual value after transients die out
  • Example: A depth controller with a 5 second rise time, 10% overshoot, 20 second settling time, and 0.1 m steady-state error
• Frequency-domain analysis using Bode plots shows the magnitude and phase response of the system over a range of input frequencies
  • Bandwidth is the frequency at which the output magnitude drops by 3 dB
  • Higher bandwidth improves disturbance rejection and tracking
  • Example: A heading controller with a 1 Hz bandwidth can effectively reject wave disturbances but not high frequency sensor noise
• Underwater vehicle dynamics are inherently nonlinear, so local linearization or describing functions must be used to apply linear stability and performance metrics
  • Global stability of nonlinear systems is difficult to guarantee

Challenges in Underwater Vehicle Control

Nonlinear Vehicle Dynamics

• Hydrodynamic drag forces on underwater vehicles are nonlinear functions of velocity, usually modeled as a quadratic drag law
  • Linear approximations are only valid for small deviations from the operating point
  • Example: Drag force $F_D = -\frac{1}{2} \rho C_D A v |v|$, where $\rho$ is water density, $C_D$ is the drag coefficient, $A$ is the frontal area, and $v$ is velocity
• Underwater vehicles experience added mass forces which are proportional to acceleration
  • This increases the effective inertia of the vehicle and cannot be neglected in the equations of motion
  • Example: Added mass force $F_A = -m_a \dot{v}$, where $m_a$ is the added mass coefficient and $\dot{v}$ is acceleration
• Control fin and thruster actuators often have position and rate limits, introducing saturation nonlinearities
  • Anti-windup logic or control allocation schemes are needed to manage saturations
  • Example: A thruster with a maximum rpm of 1000 cannot generate the 1200 rpm commanded by the PID controller, so the integral term must be limited

Environmental Disturbances

• Ocean currents act as a disturbance to the vehicle's velocity relative to the water
  • Current profiles can vary with depth and time
  • Feedforward control or estimation of current velocities can mitigate the effects
  • Example: A 0.5 m/s tidal current causes an AUV to deviate from its desired track if not accounted for in the control system
• Waves generate oscillatory disturbances to the vehicle's position and orientation
  • Wave filtering or adaptive control techniques may be needed for shallow water operations
  • Example: Heave and pitch motions from surface waves can disrupt an AUV trying to hold a constant depth
• The vehicle's buoyancy, weight, and center of gravity can change during a mission due to varying payloads or water density
  • Trim and ballast systems or adaptive control can compensate for these changes
  • Example: An AUV deploying a 10 kg payload becomes more positively buoyant, requiring the depth controller to adjust the thruster forces

Sensor Limitations

• Sensor noise, bias, and delays degrade the accuracy of feedback measurements
  • State estimators (Kalman filters) can fuse data from multiple sensors to improve feedback signal quality
  • Example: Compass measurements corrupted by electric motor noise can be filtered and combined with gyro data for a cleaner heading feedback signal