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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.

Control barrier functions (CBFs) have been used as an effective tool for designing a family of controls that ensures the forward invariance of a set. When multiple CBFs are present, it is important that the set of controls satisfying all the barrier conditions is non-empty. In this paper, we investigate such a control-sharing property for multiple CBFs and provide sufficient and necessary conditions for the property to hold. Based on that, we study the tracking control design problem of an input–output linearizable system with multiple time-varying output constraints, where the output constraints are encoded as CBFs and the barrier conditions are expressed as hard constraints in a quadratic program (QP) whose feasibility is guaranteed by the control-sharing property of the CBFs. With the controller generated from the QP, the output constraints are always satisfied and the tracking objective is achieved when it is not conflicting with the constraints. The effectiveness of our control design method is illustrated by two examples.

Synthesizing interval observers for nonlinear systems is challenging partially because both stability and positivity need to be ensured simultaneously. This paper investigates the synthesis of interval observers for two classes of nonlinear systems: polytopic systems and conic systems. Conditions for positivity of the proposed interval observers are derived by expressing the nonlinear error dynamics as a linear differential inclusion. These positivity conditions are enforced along with stability conditions in convex programs which yield the interval observer gains and coupling terms. Two convex programs and one linear program are proposed for polytopic systems, and one convex program is proposed for conic systems.

This paper constructs observers for linear and Lipschitz nonlinear systems with self-triggered measurements by utilizing the interval observer framework, which guarantees that an upper bound on the error between the estimated state and the actual state are known. The self-triggered observers are designed based on periodically sampled interval observers which guarantees a positive uniform minimum inter-sampling time. Convex programs for constructing the observer gains and the triggering functions are proposed. Simulation results are given to show the effectiveness of the proposed algorithms.

Existing results on periodic event-triggered control mainly focus on linear systems, and their extension to nonlinear systems are largely open. This paper investigates the observer-based control design for incrementally conic nonlinear systems with periodic event-trigger mechanisms in both the input and output channels. The closed-loop system is modeled using an impulsive system approach, and sufficient conditions based on linear matrix inequalities are presented to guarantee the asymptotically practical stability of the closed-loop system. A single-link robot arm example is given to illustrate the theoretical results.

This paper investigates the observer-based output feedback control design problem for a class of nonlinear systems, where the nonlinear time-varying terms satisfy an incremental quadratic inequality that is parameterized by a set of multiplier matrices. The incremental quadratic constraints include a broad class of incrementally conic nonlinearities, such as the globally Lipschitz nonlinearity, the sector bounded nonlinearity, and the polytopic Jacobian nonlinearity, as special cases. We present LMI-based sufficient conditions for the quadratic stability of the closed-loop system where the block diagonal parameterization and the block anti-triangular parameterization of the multiplier matrices are discussed separately. A simulation example is given to illustrate the theoretical results.

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.

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., Comment: to be published in IEEE TAC

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.

Passivity indices are a pair of real numbers that quantitatively measure the shortage/excess of passivity of a system, which provide us with a useful tool for the analysis and design of compositional systems. For a system whose passivity indices are known, it can be passivated by adding the feedforward and/or feedback loop(s), such that its passivity indices both become non-negative. We investigate the achievable bounds of passivity indices of the passivated system and the corresponding bounds for the feedforward/feedback gains. Three cases depending on the sign of the original indices are discussed separately.

This paper addresses the synchronization problem of Boolean networks. Based on the matrix expression of logic, solvability conditions and design procedures of the synchronization of Boolean networks with outputs are given for both open-loop and feedback control. Necessary and sufficient conditions on open-loop control are proposed first with a constructive design procedure. Then sufficient condition for the feedback control case is obtained, and corresponding design procedure is proposed with the help of algorithms to solve logic matrix equations. Numerical examples are also provided to illustrate the proposed control design.

This paper investigates the control sharing property of two control barrier functions (CBFs), and its application to the control of strict feedback nonlinear systems with time-varying bounding output constraints. CBFs are used to design a family of controls that ensure the forward invariance of a set, which represents satisfaction of a safety specification. We call two CBFs have the control sharing property if a common control exists such that the two safety constraints can be satisfied simultaneously. A sufficient condition for the control sharing property is provided, and based on that, a control design method is given to the control of strict feedback nonlinear systems with time-varying output constraints. The effectiveness of the control design method is illustrated by an example.

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 (CBFs) 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 CBFs 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 CBFs 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. Note to Practitioners —Safety is of paramount importance for the control of automated vehicles. This paper is motivated by the problem of designing controllers that are provably correct for the simultaneous operation of two driver assistance modules, lane keeping and adaptive cruise control. This is a challenging problem partially, because the lateral and longitudinal dynamics of the vehicles are coupled, with few results known to exist that provide formal guarantees. In this paper, we employ an assume-guarantee formalism between these two subsystems, such that they can be considered individually; based on that, we use optimization to design safe sets that serves as “supervisors” for vehicle behavior, such that the trajectories of the closed-loop system are confined within the safe sets using predetermined bounds on wheel force and steering angle. The feedback controller is constructed by solving convex quadratic programs online, which can also be given in closed form, making the implementation much easier. One particular advantage of this control approach is that the safety set and the performance controller can be designed separately, which enables the integration of a legacy controller into a correct-by-construction solution.

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.

This paper presents a methodology for the development of control barrier functions (CBFs) through a backstepping inspired approach. Given a set defined as the superlevel set of a function, h, the main result is a constructive means for generating control barrier functions that guarantee forward invariance of this set. In particular, if the function defining the set has relative degree n, an iterative methodology utilizing higher order derivatives of h provably results in a control barrier function that can be explicitly derived. To demonstrate these formal results, they are applied in the context of bipedal robotic walking. Physical constraints, e.g., joint limits, are represented by control barrier functions and unified with control objectives expressed through control Lyapunov functions (CLFs) via quadratic program (QP) based controllers. The end result is the generation of stable walking satisfying physical realizability constraints for a model of the bipedal robot AMBER2.

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., Comment: This is an extended version of our IEEE CDC 2015 paper to appear in Japan