Confidence-Aware Safe and Stable Control of Control-Affine Systems

Designing control inputs that satisfy safety requirements is crucial in safety-critical nonlinear control, and this task becomes particularly challenging when full-state measurements are unavailable. This work addresses the problem of synthesizing safe and stable control for control-affine systems via output feedback (using an observer) while reducing the estimation error of the observer.

The gist

This work presents an optimization-based control approach that addresses the design of safe and stabilizing controls for control-affine nonlinear systems using output feedback, specifically focusing on enhancing state confidence.

Observer Dynamics and Confidence Metrics

The paper introduces an EKF (Extended Kalman Filter) based nonlinear observer to estimate the system states from partial measurements. The observer dynamics are given by equation (2), where the estimated state is updated using a time-varying gain matrix K(t) defined by a Riccati equation (5). Crucially, the paper defines two metrics related to the observer's performance:

1. The uncertainty of the estimated states, denoted by P(t).

2. The confidence of the observer, denoted by S(t) = P−1(t).

The dynamics of this confidence metric S(t) are derived from equation (7):

S˙ = −κS − A⊤S − SA + C⊤R−1C − SQS.

Assumption 1 establishes a bound on the eigenvalues of P(t), such that pI ≤ P(t) ≤ p¯I, which is used to define the confidence metric S(t).

Stability and Safety Frameworks

The authors adapt Control Lyapunov Functions (CLFs) and Control Barrier Functions (CBFs) to the output feedback setting. The CLF definition is extended to include the estimated state domain, requiring a class C2 positive definite function V: X ∪ X → R+. A function V is defined as an observer-based exponentially stabilizing CLF if it satisfies conditions related to its Lie derivatives and a lower bound on its value (Definition 1).

For safety, the paper introduces an observer-based CBF (Definition 3). This requires finding a class C2 function h: X ∪ X → R that satisfies:

sup u∈U Lfh(x) + Lgh(x)u − r−1pK¯ hK2qM(0) ≥ −αh(x), where α ≤ θ. The resulting set of safe control inputs is defined by the constraint in equation (24):

Kcbf(t, x, z ˆ) =

Confidence-Aware Optimization Problems

The core contribution involves formulating two confidence-aware optimization problems to synthesize the controller. These problems are designed to optimize a metric of the observer's confidence:

1. The first problem (P1) seeks to minimize a weighted combination of tracking error and the minimum eigenvalue of the future confidence matrix:

π(t, x, z ˆ) = arg min u∈Rnu u⊤u − c1λmin(S(t + ∆t)) s.t. LfV (ˆx) + LgV (ˆx)u + γV (ˆx) ≤ δ, Lfh(ˆx) + Lgh(ˆx)u + αh(ˆx) + ∇h(ˆx)⊤S−1(t)C⊤R−1[z − q(ˆx)] ≥ 0.

2. A second problem (P2), used when a nominal controller πn is available, minimizes the tracking error relative to the nominal controller while penalizing the loss of confidence:

π(t, x, z ˆ) = arg min u∈Rnu∥u − πn(ˆx)∥2 − c1λmin(S(t + ∆t)) s.t. Lfh(ˆx) + Lgh(ˆx)u + αh(ˆx) + ∇h(ˆx)⊤S−1(t)C⊤R−1[z − q(ˆx)] ≥ 0.

Feasibility and Lipschitz Continuity

The paper proves the feasibility of these optimization problems by showing that the constraints can be written in a form T[u, δ]⊤ ≤ [b1, b2]⊤ with linearly independent rows, guaranteeing a unique minimizer due to strong convexity of the objective function. Furthermore, it demonstrates the Lipschitz continuity of the resulting controller π with respect to its arguments (xˆ and z) by analyzing the Hessian of the objective function and using regularity conditions from related literature.

Validation through Simulation

The effectiveness of this approach is validated through simulation studies on two illustrative examples:

1. A second-order nonlinear system stabilization problem, where confidence optimization (c1 = 1000) leads to different control inputs but ensures safety and stability, and reduces the largest eigenvalue of P(t).

Here are the specific improvements that can be made to AI systems by leveraging the concepts from this scientific paper, along with a description of what these improved systems could achieve:

The core contribution of this work is developing a framework for synthesizing safe and stable control inputs for nonlinear systems when full state measurements are unavailable, by intelligently combining an observer (EKF) with safety constraints (CBF) and performance objectives (CLF/Optimization).

Here are the specific improvements and their resulting capabilities:

1. Dominance of the Confidence-Aware Control Synthesis Framework:

2. Integration of Optimization for State Uncertainty Reduction:

3. Adaptive Safety Enforcement via Observer-Based CBFs:

4. Implementation of Tracking Objectives alongside Safety Constraints (Hybrid Control):

The improved AI systems, utilizing these techniques, can achieve the following specific capabilities:

1. A system that can maintain stable operation and guarantee safety for a nonlinear physical agent (like a robot or aircraft) even when its internal state is only partially observable (e.g., due to sensor occlusion or noise).

2. The ability to proactively design control inputs that not only keep the system within predefined safe operational boundaries but also actively adjust the control strategy to minimize the uncertainty of its own state estimation in real-time, thereby knowing more about itself.

3. A robust navigation system for autonomous vehicles (like unicycles or mobile robots) that can successfully navigate complex environments (e.g., avoiding obstacles) while simultaneously optimizing its sensor usage to improve the accuracy of its position and orientation estimates, leading to smoother control inputs and lower actuator wear.

4. A sophisticated trajectory generation system for perception-based tasks where the AI can optimize a path not just for reaching a goal, but also for maximizing the information it gathers from available sensors (e.g., LiDAR or cameras), ensuring that the chosen path is both safe and maximally informative regarding the environment's structure.

5. A control architecture capable of seamlessly blending nominal performance tracking (e.g., following a desired path) with hard safety constraints, where the system can dynamically adjust its tracking error based on its current confidence in its state estimate, leading to high-performance control that is inherently safer than traditional methods because it accounts for estimation risk.

Designing control inputs that satisfy safety requirements is crucial in safety-critical nonlinear control, and this task becomes particularly challenging when full-state measurements are unavailable. In this work, we address the problem of synthesizing safe and stable control for control-affine systems via output feedback (using an observer) while reducing the estimation error of the observer. To achieve this, we adapt control Lyapunov function (CLF) and control barrier function (CBF) techniques to the output feedback setting. Building upon the existing CLF-CBF-QP (Quadratic Program) and CBF-QP frameworks, we formulate two confidence-aware optimization problems and establish the Lipschitz continuity of the obtained solutions. To validate our approach, we conduct simulation studies on two illustrative examples. The simulation studies indicate both improvements in the observer's estimation accuracy and the fulfillment of safety and control requirements.