🚁 Lyapunov-based Deep Neural Network

📚 Description

🎯 Problem Statement

Consider a quadrotor UAV operating under intermittent feedback conditions where state measurements are periodically unavailable. The challenge is to design an adaptive controller that maintains stability and tracking performance during both feedback-available and feedback-lost periods while learning unknown system dynamics.

Key Innovation: Lyapunov-based Deep Neural Network (LbDNN) adaptation with mathematically rigorous stability guarantees under intermittent feedback loss scenarios.

🚁 Nonlinear System Dynamics

Second-Order Nonlinear System:

                        $$\ddot{x} = f(x,\dot{x}) + g(x,\dot{x})u$$
                    
State Definitions:

                        • $x, \dot{x} \in \mathbb{R}^{n}$: States with available measurements

                        • $\ddot{x} \in \mathbb{R}^{n}$: Unknown state-derivative

                        • $f: \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$: Unknown continuously differentiable drift function

                        • $g: \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n \times m}$: Known locally Lipschitz control effectiveness matrix

                        • $u \in \mathbb{R}^{m}$: Control input

For Quadrotor Application: The state vector $x = [x, y, z, \phi, \theta, \psi]^T \in \mathbb{R}^6$ represents position and orientation, with control inputs $u = [F_x, F_y, F_z, \tau_\phi, \tau_\theta, \tau_\psi]^T \in \mathbb{R}^6$ representing forces and torques applied to achieve desired trajectory tracking.

� Tracking Error Formulation

Tracking Error Definition:

                        $$e \triangleq x - x_d(t)$$
                    
Auxiliary Error (Sliding Variable):

                        $$r \triangleq \dot{e} + \alpha_1 e$$
                    
Error Dynamics:

                        $$\dot{r} = f(x,\dot{x}) + g(x,\dot{x})u - \ddot{x}_d(t) + \alpha_1(r - \alpha_1 e)$$

Design Parameter: $\alpha_1 \in \mathbb{R}_{>0}$ is a constant control gain that determines the convergence rate of the tracking error dynamics and affects the sliding surface slope.

⚙️ Control Law Design

Control Input Design:

                        $$u = g^+(x,\dot{x})\left(\ddot{x}_d(t) - (\alpha_1 + k_r)r + (\alpha_1^2 - 1)e - \Phi(X,\hat{\theta})\right)$$
                    
Control Parameters:

                        • $g^+(x,\dot{x})$: Moore-Penrose pseudoinverse of control effectiveness matrix

                        • $k_r \in \mathbb{R}_{>0}$: Sliding mode gain

                        • $\Phi(X,\hat{\theta})$: DNN approximation of unknown dynamics

                        • $X = [x^T, \dot{x}^T]^T \in \mathbb{R}^{2n}$: Augmented state vector

Control Strategy: The control law combines feedforward compensation (desired acceleration), feedback stabilization (sliding mode terms), and adaptive uncertainty compensation (neural network approximation).

🔄 Dynamic State-Derivative Observer

Observer Dynamics:

                        $$\dot{\hat{r}} = g(x,\dot{x})u - \ddot{x}_d(t) + \alpha_1(r - \alpha_1 e) + \hat{f} + \alpha_2\tilde{r}$$
                    
                        $$\dot{\hat{f}} = k_f(\dot{\tilde{r}} + \alpha_2\tilde{r}) + \tilde{r}$$
                    
Observer Error Dynamics:

                        $$\dot{\tilde{r}} = \tilde{f} - \alpha_2\tilde{r}, \qquad \dot{\tilde{f}} = \dot{f} - k_f\tilde{f} - \tilde{r}$$
                    
Observer Variables:

                        • $\hat{r}, \hat{f} \in \mathbb{R}^n$: Observer estimates of $r$ and $f$

                        • $\tilde{r} \triangleq r - \hat{r}$, $\tilde{f} \triangleq f(x,\dot{x}) - \hat{f}$: Observer errors

                        • $\alpha_2, k_f \in \mathbb{R}_{>0}$: Constant observer gains

                        • $\dot{f} \triangleq \frac{\partial f}{\partial x}\dot{x} + \frac{\partial f}{\partial \dot{x}}\ddot{x}$: Time derivative of drift function

Implementation Note: Since $\dot{\tilde{r}}$ is unknown, equation (2) is implemented by integration: $$\hat{f}(t) = \hat{f}(t_0) + k_f\tilde{r}(t) - k_f\tilde{r}(t_0) + \int_{t_0}^t (k_f\alpha_2 + 1)\tilde{r}(\tau)d\tau$$

🧠 Composite Adaptation Law

Prediction Error Design:

                        $$E \triangleq \hat{f} - \Phi(X,\hat{\theta})$$
                    
Composite Least Squares Adaptation:

                        $$\dot{\hat{\theta}} = \text{proj}\left(-k_{\hat{\theta}}\Gamma(t)\hat{\theta} + \Gamma(t)\Phi'^T(X,\hat{\theta})(r + \alpha_3 E)\right)$$
                    
Time-Varying Adaptation Gain:

                        $$\frac{d}{dt}\Gamma^{-1} = \begin{cases}
                        -\beta(t)\Gamma^{-1} + \Phi'^T(X,\hat{\theta})\Phi'(X,\hat{\theta}), & \text{if } \lambda_{\Gamma,\min} < \lambda_{\min}(\Gamma) \text{ and } \lambda_{\max}(\Gamma) < \lambda_{\Gamma,\max} \\
                        0_{p \times p}, & \text{otherwise}
                        \end{cases}$$
                    

Key Innovation: The prediction error $E$ uses the dynamic state-derivative estimator $\hat{f}$ instead of traditional approaches, enabling composite adaptation for non-linearly parameterized (NIP) neural networks.

Adaptation Parameters:

                        • $\alpha_3, k_{\hat{\theta}} \in \mathbb{R}_{>0}$: Constant adaptation gains

                        • $\Phi'(X,\hat{\theta}) \triangleq \frac{\partial \Phi(X,\hat{\theta})}{\partial \hat{\theta}} \in \mathbb{R}^{n \times p}$: Jacobian matrix

                        • $\text{proj}(\cdot)$: Continuous projection operator ensuring $\hat{\theta}(t) \in \mathcal{B}_{\bar{\theta}}$

                        • $\beta(t) \triangleq \beta_0\left(1 - \frac{\|\Gamma(t)\|}{\varkappa_0}\right)$: Bounded forgetting factor

�️ Intermittent Feedback Framework

Time Partition:

                        • Feedback Available: $t \in [t_{2i}, t_{2i+1})$, $i \in \mathbb{N}$

                        • Feedback Lost: $t \in [t_{2i+1}, t_{2i+2})$, $i \in \mathbb{N}$

                        • Total feedback loss duration: $\sum_i (t_{2i+2} - t_{2i+1}) \leq T_{\max}$

Control During Feedback Loss:

                        • Neural network weights frozen: $\dot{\hat{\theta}} = 0$

                        • Observer adaptation suspended: $\dot{\hat{r}} = 0$, $\dot{\hat{f}} = 0$

                        • Open-loop control using last known states and frozen DNN

During feedback-available periods, the controller has access to state measurements $x(t)$, $\dot{x}(t)$ and performs active adaptation. During feedback-lost periods, the controller operates open-loop using the last known state information and frozen learned dynamics.

🌪️ Turbulent Nonlinearity Modes

The simulation incorporates nine distinct turbulent nonlinearity modes that represent real-world aerodynamic and mechanical phenomena that significantly challenge quadrotor control systems. Each mode introduces specific state-dependent disturbances that test the adaptive controller's robustness.

🌬️ 1. Aerodynamic Stall & Rotor Interference

Mathematical Model:

                            $\text{AoA}_{\text{eff}} = \arctan\left(\frac{\dot{z}}{\sqrt{\dot{x}^2 + \dot{y}^2 + 0.001}}\right)$ 

                            $S_{\text{stall}} = 1 - 0.8 \exp\left(-\frac{(\text{AoA}_{\text{eff}} - 0.3)^2}{0.15^2}\right)$ 

                            $F_{\text{interference}} = 0.1 \sin(2\phi) \cos(3\theta) |\mathbf{v}|$

Physical Origin: When the effective angle of attack exceeds critical values (~17°), airflow separates from rotor blades causing dramatic lift loss. Rotor wake interference creates asymmetric forces that depend on attitude angles and velocity magnitude. This nonlinearity is particularly dangerous during aggressive maneuvers.

🌍 2. Ground Effect

Mathematical Model:

                            $\text{GE}_{\text{factor}} = \left(\frac{D_{\text{rotor}}}{h_{\text{ground}}}\right)^2 \exp\left(-\frac{h_{\text{ground}}}{D_{\text{rotor}}}\right)$ 

                            $F_{\text{lift,aug}} = \text{GE}_{\text{factor}} \cdot 0.15 \tanh(5\dot{z})$ 

                            $\tau_{\text{coupling}} = 0.05 \cdot \text{GE}_{\text{factor}} [\sin(\phi), \sin(\theta), 0]^T$

Physical Origin: When operating near the ground (typically $h < 2D_{\text{rotor}}$), the downwash reflects off the surface creating a "cushioning" effect that increases effective lift. This exponentially decaying phenomenon also induces attitude-dependent moments due to asymmetric pressure distribution.

🌀 3. Vortex Ring State

Mathematical Model:

                            $\text{VRS}_{\text{condition}} = (\dot{z} < -2.0) \land (|\mathbf{v}_{\text{horizontal}}| < 1.5)$ 

                            $I_{\text{VRS}} = \exp\left(-\frac{(|\mathbf{v}_{\text{horizontal}}| - 0.5)^2}{0.3^2}\right)$ 

                            $F_{\text{VRS}} = -I_{\text{VRS}} \cdot 8.0 \sin(0.5t) \hat{\mathbf{z}}$

Physical Origin: During rapid descent with low forward speed, the rotor enters its own wake, creating a toroidal vortex that severely disrupts lift generation. This dangerous flight condition causes oscillatory forces and loss of control authority, requiring immediate recovery maneuvers.

🏗️ 4. Structural Flexibility

Mathematical Model:

                            $F_{\text{flex},x} = 0.1 \sin(\phi) (\dot{x}^2 + \dot{y}^2)$ 

                            $F_{\text{flex},y} = 0.08 \cos(\theta) \dot{z} \dot{\phi}$ 

                            $\tau_{\text{flex},\psi} = 0.12 (\phi^2 + \theta^2) \text{sign}(\dot{\psi})$

Physical Origin: Real quadrotor frames exhibit elastic deformation under aerodynamic loads and inertial forces. This creates complex coupling between translational and rotational dynamics, where structural bending produces position-dependent attitude disturbances and vice versa.

🔋 5. Actuator Nonlinearity & Battery Degradation

Mathematical Model:

                            $\beta_{\text{battery}}(t) = 1 - \alpha_{\text{deg}} \tanh(t/30)$ 

                            $\xi_{\text{motor}} = 0.1(1 - \beta_{\text{battery}})$ 

                            $F_{\text{nonlin}} = \xi_{\text{motor}} [\sin(2t)\dot{x}, \cos(2.3t)\dot{y}, \sin(1.7t)\dot{z}]^T$

Physical Origin: Electric motors exhibit nonlinear torque-speed characteristics, especially under varying voltage conditions. Battery degradation reduces available power, causing time-varying actuator effectiveness and introducing harmonic disturbances in the control response.

🌊 6. Atmospheric Chaos & Multi-Scale Turbulence

Mathematical Model:

                            $W_{\text{large}} = I_{\text{turb}} \sin(0.05t) \cos(0.3x + 0.2y)$ 

                            $W_{\text{small}} = 0.5 I_{\text{turb}} \sin(2.1t + \sigma_{\text{state}}) \cos(5.7t)$ 

                            $W_{\text{gust}} = I_{\text{turb}} \exp\left(-\frac{(t \bmod 20 - 10)^2}{9}\right)$

Physical Origin: Atmospheric turbulence exhibits complex multi-scale behavior with large eddies (building-scale), small eddies (vehicle-scale), and intermittent gust fronts. This creates spatially and temporally correlated disturbances that challenge predictive control algorithms.

⚡ 7. Reynolds Number-Dependent Drag

Mathematical Model:

                            $\text{Re} = \frac{|\mathbf{v}| L_{\text{char}}}{\nu_{\text{air}}}$ 

                            $C_D(\text{Re}) = 0.1 + \frac{0.4}{1 + \exp(-(Re - 1000)/200)}$ 

                            $\mathbf{F}_{\text{drag}} = -C_D \cdot |\mathbf{v}| \mathbf{v} \odot \text{sign}(\mathbf{v})$

Physical Origin: The aerodynamic drag coefficient varies significantly with Reynolds number, transitioning from laminar to turbulent flow regimes. This velocity-dependent nonlinearity affects both the magnitude and direction of aerodynamic forces acting on the vehicle.

🚁 8. Propwash-Induced Coupling

Mathematical Model:

                            $v_{\text{wash}} = 5.0 + 2.0|\dot{z}|$ 

                            $F_{\text{wash},x} = 0.05 v_{\text{wash}} \sin(2\phi) \cos(\psi)$ 

                            $F_{\text{wash},y} = 0.05 v_{\text{wash}} \sin(2\theta) \sin(\psi)$ 

                            $F_{\text{wash},z} = -0.1 v_{\text{wash}} (1 + 0.3\cos\phi \cos\theta)$

Physical Origin: The high-velocity downwash from rotors creates attitude-dependent forces that couple translational and rotational dynamics. These propeller-induced effects vary with rotor disk loading and attitude angles, creating cross-axis disturbances during maneuvering.

🧲 9. Magnetic Interference & Electronic Coupling

Mathematical Model:

                            $B_{\text{field}}(t) = 0.02(1 + 0.5\sin(0.1t))$ 

                            $|\boldsymbol{\omega}| = \sqrt{\dot{\phi}^2 + \dot{\theta}^2 + \dot{\psi}^2}$ 

                            $F_{\text{mag}} = B_{\text{field}} [\sin\phi (x^2+y^2)/10, \cos\theta \cdot z \cdot \text{sign}(\dot{\psi}), 0]^T$ 

                            $\tau_{\text{mag}} = B_{\text{field}} \tanh(|\boldsymbol{\omega}|) \cdot 0.1$

Physical Origin: Electric motors and power electronics generate time-varying magnetic fields that interact with the Earth's magnetic field and metallic structures. These electromagnetic effects couple with vehicle dynamics, especially during high angular rate maneuvers, causing subtle but persistent disturbances.

💡 Collective Impact on Adaptive Control

These nine turbulent nonlinearity modes can be selectively activated to test different aspects of the adaptive controller's robustness. When combined, they create a highly challenging flight environment that requires sophisticated adaptation mechanisms.

State-Dependent Coupling: Most effects depend on current position, velocity, and attitude states
Multi-Time-Scale Dynamics: Effects range from fast oscillations to slow drift phenomena
Cross-Axis Coupling: Disturbances in one axis affect control authority in others
Unknown Parameters: Physical constants vary with environmental conditions and aging

🚁 Lyapunov-based deep neural network - Intermittent Feedback

Controller Type

Initial Conditions

Intermittent Feedback

Neural Network

Control Gains

Adaptation Gains

Position PID Gains

Orientation PID Gains

Environment

🌪️ Turbulent Nonlinearity

📚 Description

🎯 Problem Statement

🚁 Nonlinear System Dynamics

� Tracking Error Formulation

⚙️ Control Law Design

🔄 Dynamic State-Derivative Observer

🧠 Composite Adaptation Law

�️ Intermittent Feedback Framework

🌪️ Turbulent Nonlinearity Modes

🌬️ 1. Aerodynamic Stall & Rotor Interference

🌍 2. Ground Effect

🌀 3. Vortex Ring State

🏗️ 4. Structural Flexibility

🔋 5. Actuator Nonlinearity & Battery Degradation

🌊 6. Atmospheric Chaos & Multi-Scale Turbulence

⚡ 7. Reynolds Number-Dependent Drag

🚁 8. Propwash-Induced Coupling

🧲 9. Magnetic Interference & Electronic Coupling

💡 Collective Impact on Adaptive Control

🚁 Lyapunov-based deep neural network - Intermittent Feedback

Controller Type

Initial Conditions

Intermittent Feedback

Neural Network

Control Gains

Adaptation Gains

Position PID Gains

Orientation PID Gains

Environment

🌪️ Turbulent Nonlinearity

🎯 Select Initial Condition Scenario

Hovering Start

Large Displacement

Moving Start

Inverted Start

High Angular Rate

Random Conditions

📚 Description

🎯 Problem Statement

🚁 Nonlinear System Dynamics

� Tracking Error Formulation

⚙️ Control Law Design

🔄 Dynamic State-Derivative Observer

🧠 Composite Adaptation Law

�️ Intermittent Feedback Framework

🌪️ Turbulent Nonlinearity Modes

🌬️ 1. Aerodynamic Stall & Rotor Interference

🌍 2. Ground Effect

🌀 3. Vortex Ring State

🏗️ 4. Structural Flexibility

🔋 5. Actuator Nonlinearity & Battery Degradation

🌊 6. Atmospheric Chaos & Multi-Scale Turbulence

⚡ 7. Reynolds Number-Dependent Drag

🚁 8. Propwash-Induced Coupling

🧲 9. Magnetic Interference & Electronic Coupling

💡 Collective Impact on Adaptive Control