arXiv subjects

Xiangru Xu

Publications and source records attributed to Xiangru Xu.

34 records · Page 2Linked to original sources

arXiv2022

Small-Gain Theorem for Safety Verification under High-Relative-Degree Constraints

This paper develops a small-gain technique for the safety analysis and verification of interconnected systems with high-relative-degree safety constraints. In this technique, input-to-state safety (ISSf) is used to characterize how the safety of a subsystem is influenced by the external input, and ISSf-barrier functions (ISSf-BFs) with high relative degree are employed to capture the safety of subsystems. With a coordination transform, the relationship between ISSf-BFs and the existing high-relative-degree (or high-order) barrier functions is established in order to simplify the ISSf analysis. With the help of high-relative-degree ISSf-BFs, a small-gain theorem is proposed for safety verification. It is shown that, under the small-gain condition, i) the interconnection of ISSf subsystems is still ISSf; and ii) the overall interconnected system is input-to-state stable (ISS) with respect to the compositional safe set. The effectiveness of the proposed small-gain theorem is illustrated on the output-constrained decentralized control of two inverted pendulums connected by a spring mounted on two carts.

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arXiv2021

Flatness-based Quadcopter Trajectory Planning and Tracking with Continuous-time Safety Guarantees

This work presents a convex optimization framework for the planning and tracking of quadcopter trajectories with continuous-time safety guarantees. Using B-spline basis functions and the differential flatness property of quadcopters, a second-order cone program is formulated to generate optimal trajectories that respect safe state and input constraints in the continuous-time sense. A quadratic program (QP) based on control barrier functions is proposed to guarantee bounded trajectory tracking in continuous time by filtering a nominal controller, where the QP is shown to be always feasible. Furthermore, conditions that ensure the safe tracking controller respects thrust, roll angle, and pitch angle constraints are also proposed. The effectiveness of the proposed framework is demonstrated by real-world experiments using a Crazyflie2.1 nano quadcopter.

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arXiv2021

Observer-based Control Barrier Functions for Safety Critical Systems

This paper considers the safety-critical control design problem with output measurements. An observer-based safety control framework that integrates the estimation error quantified observer and the control barrier function (CBF) approach is proposed. The function approximation technique is employed to approximate the uncertainties introduced by the state estimation error, and an adaptive CBF approach is proposed to design the safe controller which is obtained by solving a convex quadratic program (QP). Theoretical results for CBFs with a relative degree 1 and a higher relative degree are given individually. The effectiveness of the proposed control approach is demonstrated by two numerical examples.

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arXiv2021

Control Barrier Function Meets Interval Analysis: Safety-Critical Control with Measurement and Actuation Uncertainties

This paper presents a framework for designing provably safe feedback controllers for sampled-data control affine systems with measurement and actuation uncertainties. Based on the interval Taylor model of nonlinear functions, a sampled-data control barrier function (CBF) condition is proposed which ensures the forward invariance of a safe set for sampled-data systems. Reachable set overapproximation and Lasserre's hierarchy of polynomial optimization are used for finding the margin term in the sampled-data CBF condition. Sufficient conditions for a safe controller in the presence of measurement and actuation uncertainties are proposed, for CBFs with relative degree 1 and higher relative degree individually. The effectiveness of the proposed method is illustrated by two numerical examples and an experimental example that implements the proposed controller on the Crazyflie quadcopter in real-time.

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arXiv2021

Finding Matrix Sequences with a High Asymptotic Growth Rate for Linear Constrained Switching Systems

Linear constrained switching systems are linear switched systems whose switching sequences are constrained by a deterministic finite automaton. This work investigates how to generate a sequence of matrices with an asymptotic growth rate close to the constrained joint spectral radius (CJSR) for constrained switching systems, based on our previous result that reveals the equivalence of a constrained switching system and a lifted arbitrary switching system. By using the dual solution of a sum-of-squares optimization program, an algorithm is designed for the lifted arbitrary switching system to produce a sequence of matrices with an asymptotic growth rate that is close to the CJSR of the original constrained switching system. It is also shown that a type of existing algorithms designed for arbitrary switching systems can be applied to the lifted system such that the desired sequence of matrices can be generated for the constrained switching system. Several numerical examples are provided to illustrate the better performance of the proposed algorithms compared with existing ones.

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arXiv2020

Small-Gain Theorem for Safety Verification of Interconnected Systems

A small-gain theorem in the formulation of barrier function is developed in this work for safety verification of interconnected systems. This result is helpful to verify input-to-state safety (ISSf) of the overall system from the safety information encoded in the subsystem's ISSf-barrier function. Also, it can be used to obtain a safety set in a higher dimensional space from the safety sets in two lower dimensional spaces.

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arXiv2020

Approximation of The Constrained Joint Spectral Radius via Algebraic Lifting

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.

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arXiv2020

Periodic Event-triggered Control for Incrementally Quadratic Nonlinear Systems

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.

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arXiv2020

Observer-Based Controllers for Incrementally Quadratic Nonlinear Systems with Disturbances

Robust global stabilization of nonlinear systems by observer-based feedback controllers is a challenging task. This article investigates the problem of designing observer-based stabilizing controllers for incrementally quadratic nonlinear systems with external disturbances. The nonlinearities considered in the system model satisfy the incremental quadratic constraints, which are characterized by incremental multiplier matrices and encompass many common nonlinearities. The simultaneous search for the observer and the controller gain matrices is formulated as a feasibility problem of linear matrix inequalities, for two parameterizations (i.e., the block diagonal parameterization and the block anti-triangular parameterization) of the incremental multiplier matrices, respectively. The closed-loop system implementing the observer-based feedback controller is proven to be input-to-state stable with respect to external disturbances. Using the proposed continuous-time observer-based controllers, event-triggered controllers with time regularization are constructed for globally Lipschitz systems, such that the closed-loop system is Zeno-free and input-to-state practically stable.

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arXiv2019

Passivity-Based Analysis of Sampled and Quantized Control Implementations

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.

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arXiv2019

Successive Convexification: A Superlinearly Convergent Algorithm for Non-convex Optimal Control Problems

This paper presents the SCvx algorithm, a successive convexification algorithm designed to solve non-convex constrained optimal control problems with global convergence and superlinear convergence-rate guarantees. The proposed algorithm can handle nonlinear dynamics and non-convex state and control constraints. It solves the original problem to optimality by successively linearizing non-convex dynamics and constraints about the solution of the previous iteration. The resulting convex subproblems are numerically tractable, and can be computed quickly and reliably using convex optimization solvers, making the SCvx algorithm well suited for real-time applications. Analysis is presented to show that the algorithm converges both globally and superlinearly, guaranteeing i) local optimality recovery: if the converged solution is feasible with respect to the original problem, then it is also a local optimum; ii) strong convergence: if the Kurdyka-Lojasiewicz (KL) inequality holds at the converged solution, then the solution is unique. The superlinear rate of convergence is obtained by exploiting the structure of optimal control problems, showcasing that faster rate of convergence can be achieved by leveraging specific problem properties when compared to generic nonlinear programming methods. Numerical simulations are performed for a non-convex quad-rotor motion planning problem, and corresponding results obtained using Sequential Quadratic Programming (SQP) and general purpose Interior Point Method (IPM) solvers are provided for comparison. The results show that the convergence rate of the SCvx algorithm is indeed superlinear, and that SCvx outperforms the other two methods by converging in less number of iterations.

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arXiv2017

Correctness Guarantees for the Composition of Lane Keeping and Adaptive Cruise Control

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.

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arXiv2016

Robustness of Control Barrier Functions for Safety Critical Control

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.

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arXiv2016

Control Barrier Function Based Quadratic Programs for Safety Critical Systems

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.

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arXiv2015

Passivity Degradation In Discrete Control Implementations: An Approximate Bisimulation Approach

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.

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arXiv2014

Leader-following Consensus of Multi-agent Systems over Finite Fields

The leader-following consensus problem of multi-agent systems over finite fields ${\mathbb F}_p$ is considered in this paper. Dynamics of each agent is governed by a linear equation over ${\mathbb F}_p$, where a distributed control protocol is utilized by the followers.Sufficient and/or necessary conditions on system matrices and graph weights in ${\mathbb F}_p$ are provided for the followers to track the leader.

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