Fusion of Indirect Methods and Iterative Learning for Persistent Velocity Trajectory Optimization of a Sustainably Powered Autonomous Surface Vessel

I. INTRODUCTION

• The objective is to maximize distance traveled over a long mission, using distance as a proxy for information gathered by the ASV.The paper situates this need in the context of sparse or short-duration oceanographic measurements and persistent autonomous observations.Source: §I · p. 1
• Earlier MPC approximated the infinite-horizon problem with a finite-horizon optimization and terminal reward, but relied on solar forecasts and required substantial computation.The terminal reward was tuned heuristically, and forecast errors diminished performance.Source: §I · p. 1
• Pointwise SOC feasibility does not ensure future feasibility when day-to-day solar resources vary, leaving persistent feasibility unaddressed by prior finite-horizon cyclic approaches.The paper notes that SOC at its lower bound can fall below that bound during a future period with no solar resource.Source: §I · pp. 1–2
• The approach minimally tightens battery SOC constraints to guarantee persistent feasibility under a time-varying solar resource.The paper frames distance traveled as a proxy for information gathered while respecting battery capacity and available solar energy.Source: §I · pp. 1–2
• Indirect optimal control yields a switching law with constant optimal velocity during each SOC constraint-inactive interval, while iterative learning estimates that interval’s unknown value.The constant can differ across intervals.Source: §I · p. 2
• The ILC-based controller nearly matches MPC in simulation while requiring no solar forecast or its associated computational burden.The comparison uses a detailed SeaTrac SP-48 model and real solar data from the North Carolina coast.Source: §I · p. 2

II. MODEL & OPTIMIZATION PROBLEM

• The ASV velocity-control problem maximizes distance traveled as a fixed-final-state, free-final-time optimal control problem.The formulation is subject to dynamics and constraints.Source: A. Optimal Control Formulation · p. 2
• The formulation defines velocity u(t), state of charge b(t), incoming solar energy Pin(t), hotel load kh, and motor-power gain km.It also specifies minimum and maximum SOC and velocity bounds.Source: A. Optimal Control Formulation · p. 2
• The SeaTrac SP-48 is the reference vehicle model used in the simulations.Its parameters are listed in Table I.Source: A. Optimal Control Formulation · p. 2, Table I · p. 3

B. Solar Irradiance Model

• The study tests the proposed control strategies with both idealized and real solar-irradiance profiles.The idealized profile provides a relatively controlled environment, while the real profile is used to demonstrate performance in a realistic environment.Source: B. Solar Irradiance Model · p. 3
• In the idealized model, irradiance oscillates with period T, generally 24 hours, around a time-varying average D0(t) with amplitude D1(t).D0(t) and D1(t) depend on latitude and time of year; the simulations fix the boat's path along the latitude through Cape Hatteras, North Carolina.Source: B. Solar Irradiance Model · p. 3
• The idealized irradiance model omits reductions from cloud cover and other real-world phenomena.The study therefore also uses Cape Hatteras irradiance data for 2022 from the ERA5 global reanalysis.Source: B. Solar Irradiance Model · p. 3
• Fig. 3 shows the actual solar irradiance at Cape Hatteras in 2022.This is the real-data profile used alongside the idealized model.Source: Figure 3 · p. 3, B. Solar Irradiance Model · p. 3

III. OPTIMAL CONTROL FORMULATION: BARRIER FUNCTIONS & INDIRECT METHODS

• The barrier functions tighten the SOC constraints by the minimal amount needed to guarantee persistent feasibility.The optimal velocity is constant during each interval in which the tightened SOC constraints are inactive.Source: §III · p. 3

A. Barrier Functions for Persistent Feasibility

• Pointwise SOC feasibility at time t may not ensure future feasibility, so the constraints are reformulated as time-varying barriers for persistent feasibility.For example, at minimum SOC, a period without solar input lets the hotel load drive SOC below its minimum.Source: A. Barrier Functions for Persistent… · p. 3
• The energy deficit represents energy consumed under an assumed solar charging profile and hotel load when ASV velocity is zero throughout the mission.Fig. 4 presents the energy deficit curve.Source: A. Barrier Functions for Persistent… · p. 3, Figure 4 · p. 4
• The lower SOC barrier is computed from the energy deficit.Fig. 6 presents the SOC barriers as upper and lower limits that vary with time of day and guarantee persistent feasibility.Source: A. Barrier Functions for Persistent… · p. 4, Figure 6 · p. 4
• The energy surplus is computed to establish the upper SOC constraint.Fig. 5 shows the energy surplus curve.Source: A. Barrier Functions for Persistent… · p. 4, Figure 5 · p. 4
• The upper SOC barrier is computed from the energy surplus.Together, the upper and lower barriers establish time-varying SOC constraints.Source: A. Barrier Functions for Persistent… · p. 4

B. Indirect Methods for Deriving the Optimal Velocity Profile

• The optimal control law is specified after reformulating the original pointwise SOC constraints using the barrier functions.The formulation retains the other constraints from the original problem.Source: B. Indirect Methods for Deriving the… · p. 4
• When the tightened SOC constraints are inactive, the optimal velocity is constant.This is the section's main result for the unconstrained intervals.Source: B. Indirect Methods for Deriving the… · p. 4
• The proof introduces an augmented state for violations of the tightened SOC constraint and derives a Hamiltonian and co-state equations.The proof uses co-states p1(t) and p2(t), the Heaviside step function, and the unit impulse function.Source: B. Indirect Methods for Deriving the… · pp. 4–5
• When tightened SOC constraints are inactive, the relevant Heaviside and impulse terms vanish, making p1 constant and yielding constant unconstrained velocity.The indirect method does not straightforwardly solve for p1, which may differ across consecutive inactive intervals.Source: B. Indirect Methods for Deriving the… · p. 5

IV. REAL-TIME REALIZATION OF THE OPTIMAL VELOCITY PROFILE THROUGH ITERATIVE LEARNING

• The optimal velocity is constant during each constraint-inactive interval, with switching between constrained and unconstrained control.The theory establishes the form of the strategy but not the optimal constant value for each interval.Source: §IV · p. 5
• Sampled-time implementation and sensor noise make the switching controller prone to significant chatter that must be mitigated for practical use.Source: §IV · p. 5
• The derived theory does not establish the optimal value of the constant velocity in each constraint-inactive interval.Source: §IV · p. 5

A. Discretization

• A buffer is added to the control solution to address switching chatter as SOC barriers change between active and inactive.Discretization and noisy SOC measurements can complicate this chatter in experimental implementations.Source: A. Discretization · p. 5
• Fig. 7 depicts the iterative approach for learning the optimal constant speed.Source: Figure 7 · p. 5
• The buffer width δ is tunable and measured in watt-hours, while u* denotes the optimal unconstrained velocity.Source: A. Discretization · p. 5

B. Iterative Learning

• Each iteration represents a 24-hour solar cycle, and an initial p1 estimate is updated to compute the commanded velocity.The approach uses two update laws to estimate the optimal costate, which is constant but unknown within each constraint-inactive interval.Source: B. Iterative Learning · p. 5
• The once-per-iteration update uses the proportional gain kp on the difference between end-of-day SOC and desired terminal SOC.This update is applied at the end of each day.Source: B. Iterative Learning · p. 5
• The continuous rate-based update uses kd on the difference between current SOC and SOC at the same time on the previous day.It updates the velocity from the previous iteration and acts as a damping term on control updates.Source: B. Iterative Learning · p. 5
• Under 15 iterations, the costate converges to p1 = -0.0012, corresponding to u = 1.83 m · s^-1, with kp = 5×10^-5 and kd = 1×10^-5.The simulation repeats environmental conditions across iterations; the costate and velocity trajectories are shown in Figs. 8 and 9.Source: B. Iterative Learning · pp. 5–6, Figure 8 · p. 6, Figure 9 · p. 6

A. Comparison Strategies

• The constant-velocity reference balances energy expended over a one-year simulation with available solar energy, then runs without SOC constraints as a performance reference.It is not a tight upper bound because the ASV is subject to SOC constraints in reality.Source: A. Comparison Strategies · p. 6
• A second comparison uses the same constant-velocity strategy only when tightened SOC constraints are inactive, otherwise setting velocity to a limit to satisfy the constraint at equality.Source: A. Comparison Strategies · p. 6
• The proposed indirect-methods-plus-ILC strategy and the two comparison strategies are also compared with the MPC strategy implemented in.Source: A. Comparison Strategies · p. 6

V. RESULTS

• The evaluation compares strategies for maximizing ASV distance traveled over a one-year mission in two solar scenarios.The scenarios use an idealized irradiance model and ERA5 data for Cape Hatteras.Source: A. Comparison Strategies · p. 6

• The ILC strategy nearly matches MPC performance under both idealized and real-world solar profiles.Under the idealized profile it also outperforms constant-velocity benchmarks.Source: B. Simulation Results · p. 6, Figure 10 · p. 7
• ILC velocity and SOC are more volatile as learning responds to solar variation, but the idealized-profile volatility is not associated with appreciable performance reduction.The real-world velocity estimate takes time to adjust to day-to-day solar changes.Source: B. Simulation Results · p. 6, Figure 11 · p. 7
• ILC requires no solar forecast and is simpler to implement, with two gains to tune rather than MPC's detailed model and terminal-reward tuning.The reported real-world sensitivity reflects time needed for estimated velocity to adjust to day-to-day irradiance variation, while MPC accesses a perfect forecast.Source: B. Simulation Results · pp. 6–7
• 20 seconds versus 21 hours and 36 minutes: ILC and MPC simulation times, respectively, for the same year-long simulation at a 6-minute time step.The simulations ran on the specified Intel Xeon W-2125 computer.Source: B. Simulation Results · p. 7

VI. CONCLUSIONS AND FUTURE WORK

• Barrier functions tighten SOC constraints by the minimum amount required for persistent feasibility, and indirect methods yield a provably optimal switching law with ILC estimates of constant velocity in constraint-inactive intervals.Source: §VI · pp. 7–8
• Using real solar data and a SeaTrac SP-48 model, the approach outperforms simple benchmarks and nearly matches more complex MPC that requires a solar forecast.Source: §VI · p. 8
• Future work will replace distance maximization with information maximization, combine velocity optimization with path planning, and experimentally validate the algorithms on the SP-48.Source: §VI · p. 8