Stability vs Energy: The Hidden Trade-off

Problem Setup

A Stable Controller Can Still Drain the Mission

Start from the previous high-altitude flight problems. The aircraft is flying under wind disturbance. The controller must maintain altitude and pitch. The actuator has limits, and the controller updates at finite timing.

Now add the missing HAPS constraint: energy is limited.

Main idea: a controller can be stable but energetically expensive.

For high-altitude long-endurance aircraft, the best controller is not simply the one that reacts fastest. It is the controller that maintains acceptable stability while preserving energy.

Add Energy State

Battery Level Becomes a State of the System

To connect control behaviour with endurance, add an energy state:

$$ E(t) = \text{remaining battery energy} $$

Energy decreases due to baseline aircraft power and additional control effort:

$$ \dot{E} = -P_{base} - P_{control} $$

A simple way to model control power is:

$$ P_{control} = k_u u^2 $$

The squared term matters. Large control commands are disproportionately expensive. Small corrections are cheap, but repeated aggressive corrections can drain energy quickly over a long mission.

control command → control power → battery drain → endurance risk

Aircraft Model with Energy

Longitudinal Dynamics Plus Battery State

The same simplified HAPS longitudinal model is extended with battery energy.

$$ \dot{V} = -D(V) + a_u u + w_V(t) $$ $$ \dot{h} = V\sin(\theta) + w_h(t) $$ $$ \dot{\theta} = b_u u - c\theta $$ $$ \dot{E} = -P_{base} - k_u u^2 $$

The states are:

$V$ = velocity
$h$ = altitude
$\theta$ = pitch
$E$ = battery / energy remaining

This is the first page in the HAPS sequence where control is connected directly to endurance.

Two Controller Philosophies

Aggressive Stability vs Conservative Endurance

Controller A — Aggressive

The aggressive controller uses higher gain. It reacts strongly to altitude error, recovers quickly, and usually keeps the aircraft closer to the target altitude. But it uses more control effort and drains energy faster.

$$ u_A = K_A(h_{target} - h) - K_{\theta,A}\theta $$

Controller B — Conservative

The conservative controller uses lower gain. It reacts more gently, uses less control effort, and preserves energy. But it may allow larger altitude deviation under wind disturbance.

$$ u_C = K_C(h_{target} - h) - K_{\theta,C}\theta $$

Controller	Strength	Weakness
Aggressive	Fast recovery and smaller altitude error	High control effort and faster battery drain
Conservative	Lower energy usage and smoother control	Larger altitude deviation and possible stability risk

The Hidden Trade-off

Minimizing Error Is Not the Same as Minimizing Energy

A controller that minimizes altitude error may not minimize energy. A controller that minimizes energy may not maintain altitude tightly.

This trade-off can be expressed with a cost function:

$$ J = \int_0^T \left(q_h(h-h_{target})^2 + r_u u^2\right)dt $$

Here, $q_h$ penalizes altitude error and $r_u$ penalizes control effort. This connects directly to LQR and MPC thinking: the engineer must decide what is more expensive — error or effort.

Hidden trade-off: short simulations reward aggressive control, but long-duration missions punish unnecessary control effort.

Interactive Simulator

Compare Stability and Energy Usage

Use the sliders to compare an aggressive controller against a conservative controller. Watch altitude, pitch, control effort, battery level, power usage, and the stability-energy trade-off indicator.

Choose controller mode:

Aggressive controller Conservative controller Compare both

Show low-energy warning threshold

Wind gust strength: 6 m/s equivalent

Aggressive gain: 0.07

Conservative gain: 0.025

Energy capacity: 160 units

Baseline power: 0.25

Control power penalty: 18

Aircraft damping: 0.25

Simulation time: 120 s

Select a controller mode and run the simulation to generate results.

Stability vs Endurance

In Endurance Aircraft, Control Effort Is Not Free

Short simulations often reward aggressive control because the aircraft returns to target altitude quickly. But long-duration missions punish unnecessary control effort because every correction consumes energy.

A HAPS aircraft may need to remain airborne for days, weeks, or months. Even small inefficiencies accumulate over long durations.

Endurance lesson: a controller that looks best over 60 seconds may not be best over 60 hours.

This is why long-endurance control is not only about stability. It is also about energy management, disturbance rejection, and mission survival.

Connection to Existing Control Pages

How This Connects to LQR, MPC, and Saturation

This page links directly to your existing control progression. LQR formalizes the trade-off between state error and control effort. MPC extends that idea by planning ahead while respecting constraints. Saturation pages show that energy and actuator authority are both physical limits.

altitude error ↔ control effort ↔ battery usage ↔ mission endurance

Open LQR Control Page → Open MPC Page → Open Saturation Under Disturbance →

Engineering Interpretation

A Good Controller Must Stabilize Within Available Energy

A controller is not good just because it stabilizes the system. It must stabilize the system within available energy.

For HAPS, solar input, battery storage, actuator usage, and mission endurance are coupled. During daytime, solar power may recharge the system. At night, the aircraft depends on stored battery energy. Aggressive control at night may shorten mission endurance.

What the Engineer Should Check

Does the controller maintain altitude within acceptable limits?
How much average control effort is required?
How quickly does battery energy decrease?
Does the system remain stable near low battery?
Would a slightly slower controller extend endurance significantly?

Stability, power, and endurance must be evaluated together.

Python Implementation

Reproducing the Energy Trade-off in Python

The core idea is to integrate a battery state together with the aircraft dynamics.

import numpy as np

dt = 0.05
t_final = 120
t = np.arange(0, t_final, dt)

h_target = 1000
E_capacity = 160

V = np.zeros_like(t)
h = np.zeros_like(t)
theta = np.zeros_like(t)
E = np.zeros_like(t)
u = np.zeros_like(t)

V[0] = 25
h[0] = 950
theta[0] = 0
E[0] = E_capacity

K = 0.07              # aggressive gain
K_theta = 0.8
damping = 0.25
P_base = 0.25
k_u = 18

wind_strength = 6

for k in range(len(t)-1):

    wind = wind_strength * np.sin(2*np.pi*0.10*t[k])

    error = h_target - h[k]
    u[k] = K*error - K_theta*theta[k]

    P_control = k_u * u[k]**2
    P_total = P_base + P_control

    V_dot = -0.015*(V[k] - 25) + 0.04*u[k] + 0.03*wind
    h_dot = V[k]*np.sin(theta[k]) + 0.7*wind
    theta_dot = 1.2*u[k] - damping*theta[k]
    E_dot = -P_total

    V[k+1] = V[k] + V_dot*dt
    h[k+1] = h[k] + h_dot*dt
    theta[k+1] = theta[k] + theta_dot*dt
    E[k+1] = max(0, E[k] + E_dot*dt)

Takeaway

The Best Controller Is Not Necessarily the Fastest Controller

For long-endurance aircraft, the best controller is the one that balances stability, energy, and mission survival.

Aggressive control

Stable and fast, but energy-expensive.

Conservative control

Efficient, but may allow larger altitude error.

Mission-level control

Balances tracking, effort, and endurance.

Control design is constrained by energy, not just stability.