Set Invariance Research Articles

Safety-critical multi-tasks control for switched systems via several barrier functions

This paper investigates the problem of safety-critical multi-tasks control (SCMTC) for switched systems, where the multi-tasks control is achieved by ensuring the invariance and attraction of the task set. A new definition of safety is presented under the switched systems framework, which extends the existing safety definition for non-switched systems and reveals the essence of safety for switched systems. For two different types of task sets, a single barrier function (SBF) method and a multiple barrier functions method are proposed to solve the SCMTC problem, where the former requires each task set to be defined by a continuously differentiable function, while the latter requires that to be defined by multiple continuously differentiable functions. Both the two methods allow the safety and/or the invariance and attraction of the task sets to be unnecessary for individual subsystems. Also, two novel task-based switching signals are designed to guide the switching among subsystems that do not possess the invariance and attraction of task sets. Furthermore, a common barrier function method, as a special case of the SBF method, is proposed to solve the SCMTC problem for switched systems under arbitrary switchings. Finally, an illustrate example is given to verify the effectiveness of the proposed methods.

On the Viability and Invariance of Proper Sets under Continuity Inclusions in Wasserstein Spaces

On the Viability and Invariance of Proper Sets under Continuity Inclusions in Wasserstein Spaces

Tube-based model predictive control for linear systems with bounded disturbances and input delay

This paper proposes a tube-based model predictive control strategy for linear systems with bounded disturbances and input delay to ensure input-to-state stability. Firstly, the actual disturbed system is decomposed into a nominal system without disturbances and an error system. For the nominal system, solving an optimization problem, where the delayed control input is set as an optimization variable, yields a nominal control law that enables the nominal state signal to approach to zero. Then, for the error system, the Razumikhin approach is used to identify a robust control invariant set. Using the set invariance theorem, an ancillary control law is developed to confine the error state signal in the invariant set. Combining the two results, we obtain a control law that enables the state signal to remain within a robustly invariant tube. Finally, the effectiveness of the developed control strategy is validated by simulations.

Delay-dependent set-invariance for linear difference equations with multiple delays: a polyhedral approach

This paper revisits the set-invariance for linear delay-difference equations and proposes novel delay-dependent notions in contrast with the existing delay-independent constructions available in the literature. The main tool in this endeavour is the model transformation by means of a matrix parametrization, which opens the way to the elaboration of delay-dependent conditions for set invariance. For the class of polyhedral sets, it will be shown how these transformed models enable particularly structured invariance conditions. Aside from the mathematical developments, a discussion of the delay-dependent conditions with respect to the confinement of the state trajectories of the original system will be presented. The relationship will be established by means of constraints involving the initial states for the time-delay systems. The characterization of this set of admissible initial conditions can be analysed in terms of complexity and conservativeness with respect to the classical delay-independent set-invariance. An illustrative example is provided to underline that confinement of state trajectories in a set can be achieved even though this is not invariant according to the classical definition.

From Relaxed Constraint Satisfaction to $p-$Invariance of Sets

From Relaxed Constraint Satisfaction to $p-$Invariance of Sets

Robust Preview Path Tracking Control of Autonomous Vehicles Under Time-Varying System Delays and Saturation

Realistic uncertainties and disturbances cannot be ignored when striving towards high-performance autonomous driving. Meanwhile, advanced vehicular sensing technology enables more and more self-driving techniques to come out. Against this backdrop, this paper aims to holistically address the system uncertainties and disturbances in the path tracking control of autonomous vehicles (AVs) by the preview control theory to achieve superior control performance. First, unmeasureable multi-uncertainties and the external disturbance are considered simultaneously into the robust gain-scheduling framework with variable longitudinal velocity. Then, by the equivalent delay approach, ineluctable time-varying system delays are effectively handled. Next, the physical constraints of the steering system and direct yaw moment control (DYC) system are considered for practical implementation through robust set invariance theory. The parallel distributed compensation (PDC) technique and auxiliary feedback matrices approach are introduced for reducing design conservatism. After that, by incorporating the road curvature into the augmented system state, the path tracking preview control problem can be solved without online optimization. Thereafter, the closed-loop system stability and guaranteed performance have been proved through the Lyapunov- Krasovski approach. The proposed controller design is finally reformulated as the optimization problem based on linear matrix inequalities (LMIs) via specific convexification techniques. Finally, to verify the validity of the proposed controller, Software in Loop (SIL) tests with Carsim full-vehicle model under different dynamic conditions have been implemented.

The ℒ1 controller synthesis for piecewise continuous nonlinear systems via set invariance principles

AbstractThis article provides a synthesis method of the controller for piecewise continuous nonlinear systems in terms of set invariance principles. Following the arguments of Filippov's differential equations, solutions of piecewise continuous differential equations are first characterized by set‐valued functions. Based on such characteristics of solutions equipped with set‐valued functions, the performance of piecewise continuous nonlinear systems, that is, the gain from the associated disturbance to regulated output, is formulated. To clarify the existence of a controller such that the resulting closed‐loop system satisfies the performance in the presence of piecewise continuity, the so‐called generalized controlled invariance domain is proposed. Taking this advanced argument allows us to establish a synthesis procedure for the controller by showing a lower‐semicontinuity of the set‐valued functions involved in the generalized controlled invariance domain. Finally, some numerical examples are provided to evaluate the validity of the existence condition for the controller.

Data‐driven safe gain‐scheduling control

AbstractData‐based safe gain‐scheduling controllers are presented for discrete‐time linear parameter‐varying systems (LPV) with polytopic models. First, ‐contractivity conditions are provided under which the safety and stability of the LPV systems are unified through Minkowski functions of the safe sets. Then, a data‐based representation of the closed‐loop LPV system is provided, which requires less restrictive data richness conditions than identifying the system dynamics. This sample‐efficient closed‐loop data‐based representation is leveraged to design data‐driven gain‐scheduling controllers that guarantee ‐contractivity and, thus, invariance of the safe sets. It is also shown that the problem of designing a data‐driven gain‐scheduling controller for a polyhedral (ellipsoidal) safe set amounts to a linear program (a semi‐definite program). The motivation behind direct learning of a safe controller is that identifying an LPV system requires satisfying the persistence of the excitation (PE) condition. It is shown in this paper, however, that directly learning a safe controller and bypassing the system identification can be achieved without satisfying the PE condition. This data‐richness reduction is of vital importance, especially for LPV systems that are open‐loop unstable, and collecting rich samples to satisfy the PE condition can jeopardize their safety. A simulation example is provided to show the effectiveness of the presented approach.

Characterization of Positive Invariance of Quadratic Convex Sets for Discrete-Time Systems Using Optimization Approaches

A positively invariant set is an important concept in dynamical systems. The study of positively invariant set conditions for discrete-time systems is one interesting topic in both theoretical studies and practical applications research. Different methods for characterizing the invariance of different types of sets have been established. For example, the ellipsoidal and the Lorenz cone, which are quadratic convex sets, have different properties from a polyhedral set. This paper presents an optimization method and a dual optimization method to characterize the positive invariance of the ellipsoidal and the Lorenz cone. The proposed methods are applicable to both linear and nonlinear discrete-time systems. Using nonlinear programming and an induced norm, the positive invariance condition problems are transformed into optimization problems, and the dual optimization method is also used to give equivalent dual forms. Fewer results on the positive invariance condition of Lorenz cones can be found than for the other type of set; this paper fulfills the results of this problem. In addition, the proposed methods in this paper provide more options for checking the positive invariance of quadratic convex sets from the perspective of optimization and dual optimization. The effectiveness of this method is demonstrated by numerical examples.

Control barrier functionals: Safety‐critical control for time delay systems

AbstarctThis work presents a theoretical framework for the safety‐critical control of time delay systems. The theory of control barrier functions, that provides formal safety guarantees for delay‐free systems, is extended to systems with state delay. The notion of control barrier functionals is introduced, to attain formal safety guarantees by enforcing the forward invariance of safe sets defined in the infinite dimensional state space. The proposed framework is able to handle multiple delays and distributed delays both in the dynamics and in the safety condition, and provides an affine constraint on the control input that yields provable safety. This constraint can be incorporated into optimization problems to synthesize pointwise optimal and provable safe controllers. The applicability of the proposed method is demonstrated by numerical simulation examples.

Invariance of sets under mutational inclusions on metric spaces

Invariance of sets under mutational inclusions on metric spaces

Dual optimization approach to set invariance conditions for discrete-time dynamic systems

Dual optimization approach to set invariance conditions for discrete-time dynamic systems

Computation of parameter dependent robust invariant sets for LPV models with guaranteed performance

This paper presents an iterative algorithm to compute a Robust Control Invariant (RCI) set, along with an invariance-inducing control law, for Linear Parameter-Varying (LPV) systems. As real-time measurements of the scheduling parameters are typically available, we allow the RCI set description and the invariance-inducing controller to be scheduling parameter dependent. Thus, the considered formulation leads to parameter-dependent conditions for the set invariance, which are replaced by sufficient Linear Matrix Inequalities (LMIs) via Polya’s relaxation. These LMI conditions are then combined with a novel volume maximization approach in a Semidefinite Programming (SDP) problem, which aims at computing the desirably large RCI set. Besides ensuring invariance, it is also possible to guarantee performance within the RCI set by imposing a chosen quadratic performance level as an additional constraint in the SDP problem. Using numerical examples, we show that the presented iterative algorithm can generate RCI sets for large parameter variations where commonly used robust approaches fail.

Toward Minimal Data Rate Enforcing Regular Safety Properties: An Invariance Entropy Approach

The study of minimal data rate for control using some notions of entropy has been so far limited to classical control tasks such as set invariance, state-estimation, or stabilization. In this work, for the first time, we present a study on sufficient data rates to enforce regular safety properties over uncertain systems with dynamics described by set valued maps. Every regular safety property has a set of bad prefixes which can be modelled by a deterministic finite automaton (DFA). The main idea is to construct a hybrid system by taking the product of the deterministic finite automata with the given system and studying the invariance feedback entropy (IFE) of controlled invariant sets of the hybrid system. If there exists a nonempty controlled invariant set for the hybrid system satisfying a certain property then there exists a coder-controller with a data rate not less than the IFE that can enforce the regular safety property over the original control system. We demonstrate the effectiveness of our results by designing a coder-controller enforcing a regular safety property over a linear control system.

A Control Barrier Function Approach for Observer-based Visually Safe Pursuit Control with Spherical Obstacles*

Pursuing a target in the presence of obstacles requires that an autonomous mobile robot keeps sight of the target in a way robust to the target's unknown behavior. This paper presents visually safe pursuit control, which keeps a target with the unknown motion inside the camera's field of view while preventing occlusion caused by spherical obstacles. Framed as forward invariance of sets in the SE(3) state space, visual safety is ensured by the Control Barrier Functions (CBFs) approach. Concretely, by showing the Input-to-State stability of the vision-based observer that estimates the target's motion, we design safety certificates for visual safety that accommodate uncertainties in the target's motion. This enables us to synthesize a safe controller as a Quadratic Programming problem. Finally, the theoretical results are verified via a simulation of a visual pursuit scenario.

Systematic Design of Discrete-Time Control Barrier Functions Using Maximal Output Admissible Sets

This paper introduces a novel discrete-time control barrier function (DCBF) that stems directly from discrete-time set invariance theory. The proposed DCBF provides necessary and sufficient conditions for certifying control invariance and can be used to synthesize a constrained control policy. Moreover, the DCBF can be constructed for arbitrary sets of state and input constraints by taking advantage of maximal output admissible set theory. The resulting DCBF-based controller is proven to be safe and recursively feasible. Numerical examples showcase the effectiveness of the scheme by comparing it to existing constrained control approaches.

Safety Assist Control for Nonautonomous Control-affine Systems via Time-varying Control Barrier Function

The recent research on nonlinear control systems tends to prioritize safety beyond stability, and control barrier functions (CBFs) have been gaining popularity as a CBF-based controller can theoretically ensure safety of control systems with the notion of set invariance. In this paper, we define a time-varying CBF (Tv-CBF) for a nonautonomous control-affine system with a human operator input to address a time-dependent safety constraint problem. Then, we propose a human assist control law using a Tv-CBF that satisfies time-dependent safety constraints. We also prove that solutions to an ordinary differential equation denoting control systems are forward complete by employing the proposed human assist controller. Lastly, the effectiveness of the proposed method is confirmed through computer simulation.

Safety Verification of Neural-Network-Based Controllers: A Set Invariance Approach

Safety Verification of Neural-Network-Based Controllers: A Set Invariance Approach

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support functions.

Safety Compliant Control for Robotic Manipulator With Task and Input Constraints

Increasingly, robots are working in close proximity with humans, and in dynamically changing environments. We present a new control architecture that guarantees safety under such conditions, while simultaneously ensuring that task goals are satisfied. Our approach combines Control Barrier Functions (to provide safety guarantees) with Rapidly Exponentially Stabilizing Control Lyapunov functions (to satisfy task constraints and ensure exponential convergence). We formulate the problem completely in the operational space, using super ellipsoids to define safe sets which are dynamically adapted to restrict the robot’s operational space in response to moving obstacles (including humans). Control barrier functions, which can be implemented in real time using quadratic programming, ensure the forward invariance of these safe sets, while minimally perturbing prescribed control trajectories. We demonstrate our approach on a seven degree-of-freedom Kuka LBR iiwa robotic manipulator, both in a physics engine-based simulation and with real hardware experiments.