Xiangru Xu

Aaron D. Ames, Xiangru Xu, Jessy W. Grizzle et al.

Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions -- expressed as control barrier functions -- to be unified with performance objectives -- expressed as control Lyapunov functions -- in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds.

Xiangru Xu, Paulo Tabuada, Jessy W. Grizzle et al.

Barrier functions (also called certificates) have been an important tool for the verification of hybrid systems, and have also played important roles in optimization and multi-objective control. The extension of a barrier function to a controlled system results in a control barrier function. This can be thought of as being analogous to how Sontag extended Lyapunov functions to control Lyapunov functions in order to enable controller synthesis for stabilization tasks. A control barrier function enables controller synthesis for safety requirements specified by forward invariance of a set using a Lyapunov-like condition. This paper develops several important extensions to the notion of a control barrier function. The first involves robustness under perturbations to the vector field defining the system. Input-to-State stability conditions are given that provide for forward invariance, when disturbances are present, of a "relaxation" of set rendered invariant without disturbances. A control barrier function can be combined with a control Lyapunov function in a quadratic program to achieve a control objective subject to safety guarantees. The second result of the paper gives conditions for the control law obtained by solving the quadratic program to be Lipschitz continuous and therefore to gives rise to well-defined solutions of the resulting closed-loop system.

Xiangru Xu, Jessy W. Grizzle, Paulo Tabuada et al.

This paper develops a control approach with correctness guarantees for the simultaneous operation of lane keeping and adaptive cruise control. The safety specifications for these driver assistance modules are expressed in terms of set invariance. Control barrier functions are used to design a family of control solutions that guarantee the forward invariance of a set, which implies satisfaction of the safety specifications. The control barrier functions are synthesized through a combination of sum-of-squares program and physics-based modeling and optimization. A real-time quadratic program is posed to combine the control barrier functions with the performance-based controllers, which can be either expressed as control Lyapunov function conditions or as black-box legacy controllers. In both cases, the resulting feedback control guarantees the safety of the composed driver assistance modules in a formally correct manner. Importantly, the quadratic program admits a closed-form solution that can be easily implemented. The effectiveness of the control approach is demonstrated by simulations in the industry-standard vehicle simulator Carsim.

Xiangru Xu, Behcet Acikmese

This paper studies the constrained switching (linear) system which is a discrete-time switched linear system whose switching sequences are constrained by a deterministic finite automaton. The stability of a constrained switching system is characterized by its constrained joint spectral radius that is known to be difficult to compute or approximate. Using the semi-tensor product of matrices, the matrix-form expression of a constrained switching system is shown to be equivalent to that of a lifted arbitrary switching system. Then the constrained joint/generalized spectral radius of a constrained switching system is proved to be equal to the joint/generalized spectral radius of its lifted arbitrary switching system which can be approximated by off-the-shelf algorithms.

Xiangru Xu, Necmiye Ozay, Vijay Gupta

This paper studies the performance of a continuous controller when implemented on digital devices via sampling and quantization, by leveraging passivity analysis. Degradation of passivity indices from a continuous-time control system to its sampled, input and output quantized model is studied using a notion of quasi-passivity. Based on that, the passivity property of a feedback-connected system where the continuous controller is replaced by its sampled and quantized model is studied, and conditions that ensure the state boundedness of the interconnected system are provided. Additionally, the approximate bisimulation-based control implementation where the controller is replaced by its approximate bisimilar symbolic model whose states are also quantized is analyzed. Several examples are provided to illustrate the theoretical results.

Xiangru Xu, Adam M. Tahir, Behcet Acikmese

Periodic event-triggered control (PETC) evaluates the triggering rule periodically and is well-suited for implementation on digital platforms. This paper investigates PETC design for nonlinear systems affected by external disturbances under the impulsive system formulation. Sufficient conditions are provided to ensure the input-to-state stability of the resulting closed-loop system for the state feedback and the observer-based output feedback configurations separately. For each configuration, the sampling period and the triggering functions are provided explicitly. Sufficient conditions in the form of linear matrix inequalities are provided for the PETC design of incrementally quadratic nonlinear systems. Two examples are given to illustrate the effectiveness of the proposed method.

Yuhao Zhang, Xiangru Xu

Recurrent neural networks (RNNs) are widely employed to model complex dynamical systems due to their hidden-state structure, which inherently captures temporal dependencies. This work presents a hybrid zonotope-based approach for computing exact forward and backward reachable sets of closed-loop RNN systems with ReLU activation functions. The method formulates state-pair sets to compute reachable sets as hybrid zonotopes without requiring unrolling. To improve scalability, a tunable relaxation scheme is proposed that ranks unstable ReLU units across all layers using a triangle-area score and selectively applies convex relaxations within a fixed binary limit in the hybrid zonotopes. This scheme enables an explicit tradeoff between computational complexity and approximation accuracy, with exact reachability as a special case. In addition, a sufficient condition is derived to certify the safety of closed-loop RNN systems. Numerical examples demonstrate the effectiveness of the proposed approach.

Yuhao Zhang, Xiangru Xu

Feedforward neural networks are widely used in autonomous systems, particularly for control and perception tasks within the system loop. However, their vulnerability to adversarial attacks necessitates formal verification before deployment in safety-critical applications. Existing set propagation-based reachability analysis methods for feedforward neural networks often struggle to achieve both scalability and accuracy. This work presents a novel set-based approach for computing the reachable sets of convolutional neural networks. The proposed method leverages a hybrid zonotope representation and an efficient neural network reduction technique, providing a flexible trade-off between computational complexity and approximation accuracy. Numerical examples are presented to demonstrate the effectiveness of the proposed approach.

Xiangru Xu, Necmiye Ozay, Vijay Gupta

In this paper, we present some preliminary results for compositional analysis of heterogeneous systems containing both discrete state models and continuous systems using consistent notions of dissipativity and passivity. We study the following problem: given a physical plant model and a continuous feedback controller designed using traditional control techniques, how is the closed-loop passivity affected when the continuous controller is replaced by a discrete (i.e., symbolic) implementation within this framework? Specifically, we give quantitative results on performance degradation when the discrete control implementation is approximately bisimilar to the continuous controller, and based on them, we provide conditions that guarantee the boundedness property of the closed-loop system.