Maximal Controlled Invariant Set Research Articles

On proving that an unsafe controller is not proven safe

Cyber-physical systems are often safety-critical and their correctness is crucial, such as in the case of automated driving. Using formal mathematical methods is one way to guarantee correctness and improve safety. Although these methods have shown their usefulness, care must be taken because modelling errors might result in proving a faulty controller safe, which is potentially catastrophic in practice. This paper deals with two such modelling errors in differential dynamic logic, a formal specification and verification language for hybrid systems, which are mathematical models of cyber-physical systems. The main contributions are to provide conditions under which these two modelling errors cannot cause a faulty controller to be proven safe, and to show how these conditions can be proven with help of the interactive theorem prover KeYmaera X. The problems are illustrated with a real world example of a safety controller for automated driving, and it is shown that the formulated conditions have the intended effect both for a faulty and a correct controller. It is also shown how the formulated conditions aid in finding a loop invariant candidate to prove properties of hybrid systems with feedback loops. Furthermore, the relation between such a loop invariant and the characterisation of the maximal control invariant set is discussed.

Output Trackability of Boolean Control Networks via Ledley Antecedence Solution

The Ledley solution method is applied to analysis and control design of output trackability of Boolean control networks (BCNs) in this brief. By constructing the maximal control invariant set with respect to a defined state set, the output tracking problem is converted into a set stabilization problem and a corresponding condition for verifying solvability is obtained. Then an effective algorithm is provided to design the optimal state feedback controllers. Finally, we give an illustrated example.

Systems synchronisation in Max-Plus algebra: a controlled invariance perspective In memoriam Édouard Wagneur

We study the concept of systems synchronisation in Max-Plus algebra. We show that the problem of synchronising Max-Plus linear systems is twofold. It can be stated in terms of controlled invariance, and of coreachability. The controlled invariance is the property of a set for which a suitable control exists, that maintains the trajectory in the set. We recall some properties of controlled invariance, in particular the existence of a maximal controlled invariant set included into any given set. Beyond the determination of the admissible controlled invariant set, we define and study a problem of coreachability. The controlled invariant set is the target and we study the existence of a control law that brings the trajectory to the target from the initial state within a specific number of steps. We illustrate the study with an example from the literature.

Model-Predictive Control With Generalized Zone Tracking

In this paper, we propose a new framework for model-predictive control (MPC) with generalized zone tracking. The proposed zone MPC tracks a generalized target set of system state and input which is not necessarily control-invariant. In this context, the classical MPC theory no longer applies because the target zone may not be stable in the sense of Lyapunov. We extend LaSalle's invariance principle and develop new theories for stability analysis of zone MPC. It is proved that under the zone MPC design, the system converges to the maximal control invariant set in the target zone. Sufficient conditions for asymptotic stability of the maximal control-invariant set are also discussed. By tracking the generalized target zone, the proposed zone MPC is able to: (i) yield smaller zone tracking errors than all existing methods which essentially track some steady-state subset of the target zone, and (ii) allow more admissible operations and release more degrees of freedom to achieve other economic objectives. Further discussions are made on extending the prediction horizon of the zone MPC based on an auxiliary control law as well as handling a secondary economic objective via a second-step economic optimization.

Safety Verification Methods for Human-Driven Vehicles at Traffic Intersections: Optimal Driver-Adaptive Supervisory Control

We design an optimal, driver-adaptive supervisor for collision avoidance at an intersection. The algorithm is able to identify optimal corrections to the human-decided inputs and to keep the system collision free. To determine the set of safe control actions, we exploit the notion of maximal controlled invariant set. We leverage results from scheduling theory to verify the safety of a given control input, and propose an efficient optimization algorithm providing optimal solutions with respect to the drivers’ intent. We also present an approximate supervisor algorithm that can be solved in polynomial time and has guaranteed error bounds. Finally, we validate our approach with simulation results, as well as on naturalistic data.

Event-triggered MPC design for distributed systems with network communications

This paper deals with the communication problem in the distributed system, considering the limited battery power in the wireless network and redundant transmission among nodes. We design an event-triggered model predictive control U+0028 ET-MPC U+0029 strategy to reduce the unnecessary communication while promising the system performance. On one hand, for a linear discrete time-invariant system, a triggering condition is derived based on the Lyapunov stability. Here, in order to further reduce the communication rate, we enforce a triggering condition only when the Lyapunov function will exceed its value at the last triggered time, but an average decrease is guaranteed. On the other hand, the feasibility is ensured by minimizing and optimizing the terminal constrained set between the maximal control invariant set and the target terminal set. Finally, we provide a simulation to verify the theoretical results. It U+02BC s shown that the proposed strategy achieves a good trade-off between the closed-loop system performance and communication rate.

Computing Robust Controlled Invariant Sets of Linear Systems

We consider controllable linear discrete-time systems with bounded perturbations and present two methods to compute robust controlled invariant sets. The first method tolerates an arbitrarily small constraint violation to compute an arbitrarily precise outer approximation of the maximal robust controlled invariant set, while the second method provides an inner approximation. The outer approximation scheme is $\delta$-complete, given that the constraint sets are formulated as finite unions of polytopes.

On the Computation of $\lambda$- Contractive Sets for Linear Constrained Systems

We present two theoretical results on the computation of $\lambda$ - contractive sets for linear systems with state and input constraints. First, we show that it is possible to a priori compute a number of iterations that is sufficient to approximate the maximal $\lambda$ - contractive set with a given precision using 1-step sets. Second, based on the former result, we provide a procedure for choosing $\lambda$ so that the associated maximal $\lambda$ - contractive set is guaranteed to approximate the maximal controlled invariant set with a given accuracy.

Set‐theoretic analysis of the isolated ramp metering problem

SummaryAnalysis of ramp metering, as an effective freeway traffic control solution, is focused in the paper. The studied traffic control problem is discussed in a set‐theoretic context in order to quantitatively characterize its effectivenes. The notions of maximal robust controlled invariant set, as well as t‐step robust controllable set are defined and used for analyzing the ramp metering problem independently of the control policy applied. Algorithms are developed to compute these sets, with special attention to their practical interpretations. Numerical examples with graphical representation of the proposed methodology are given to examine local ramp metering and conclude implications to control strategies. Copyright © 2015 John Wiley & Sons, Ltd.

Least Restrictive Supervisors for Intersection Collision Avoidance: A Scheduling Approach

We consider a cooperative conflict resolution problem that finds application, for example, in vehicle intersection crossing. We seek to determine minimally restrictive supervisors, which allow agents to choose all possible control actions that keep the system safe, that is, conflict free. This is achieved by determining the maximal controlled invariant set, and then by determining control actions that keep the system state inside this set. By exploiting the natural monotonicity of the agents dynamics along their paths, we translate this problem into an equivalent scheduling problem. This allows us to leverage existing results in the scheduling literature to obtain both exact and approximate solutions. The approximate algorithms have polynomial complexity and can handle large problems with guaranteed approximation bounds. We illustrate the application of the proposed algorithms through simulations in which vehicles crossing an intersection are overridden by the supervisor only when necessary to maintain safety.

Safety preserving control synthesis for sampled data systems

In sampled data systems the controller receives periodically sampled state feedback about the evolution of a continuous time plant, and must choose a constant control signal to apply between these updates; however, unlike purely discrete time models the evolution of the plant between updates is important. In this paper we describe an abstract algorithm for approximating the discriminating kernel (also known as the maximal robust control invariant set) for a sampled data system with continuous state space, and then use this operator to construct a switched, set-valued feedback control policy which ensures safety. We show that the approximation is conservative for sampled data systems. We then demonstrate that the key operations–the tensor products of two sets, invariance kernels, and a pair of projections–can be implemented in two formulations: one based on the Hamilton–Jacobi partial differential equation which can handle nonlinear dynamics but which scales poorly with state space dimension, and one based on ellipsoids which scales well with state space dimension but which is restricted to linear dynamics. Each version of the algorithm is demonstrated numerically on a simple example.

A new class of Lyapunov functions for the constrained stabilization of linear systems

The constrained stabilization of linear uncertain systems is investigated via the set-theoretic framework of control Lyapunov R-functions. A novel composition rule allows the design of a composite control Lyapunov function with external level set that exactly shapes the maximal controlled invariant set and inner sublevel sets arbitrarily close to any choice of smooth ones, generalizing both polyhedral and truncated ellipsoidal control Lyapunov functions. The feasibility test of the proposed smooth control Lyapunov functions can be cast into matrix inequalities conditions. The constrained linear quadratic control is addressed as an application.

Robust vehicle lateral stabilisation via set-based methods for uncertain piecewise affine systems

The paper presents the design of a lateral stability controller for ground vehicles based on front steering and four wheels independent braking. The control objective is to track yaw rate and lateral velocity reference signals while avoiding front and rear wheel traction force saturation. Control design is based on an approximate piecewise-affine nonlinear dynamical model of the vehicle. Vehicle longitudinal velocity and drivers steering input are modelled as measured disturbances taking values in a compact set. A time-optimal control strategy which ensures convergence into a maximal robust control invariant (RCI) set is proposed. This paper presents the uncertain model, the RCI computation, and the control algorithm. Experimental tests at high-speed on ice with aggressive driver manoeuvres show the effectiveness of the proposed scheme.

Decentralized Robust Control Invariance for a Network of Storage Devices

Robust control of networked storage devices is considered. Each storage device is modeled as a single-state, discrete-time integrator with bounded control input, and subject to additive disturbance. The values of the disturbance are unknown but are assumed to belong to bounded sets. Nodes exchange stored resource through links of limited capacity. Characterization of the maximal robust control invariant (RCI) set is provided and decentralization of the robust feasibility problem is considered. A notion of decentralized RCI set is introduced and a parametrization of a family of decentralized RCI sets is proposed. It is shown that the set of parameters for which the decentralized RCI sets are non-empty is a polyhedral set. The result offers a possibility to use convex optimization for selection of bounds on the admissible flow through the network which ensure feasibility of the proposed decentralized design.

Constrained Optimal Controller Design of Aerial Robotics Based on Invariant Sets

Constrained optimal control problem of unmanned aerial vehicle (UAV) which is also called aerial robotics is studied in this paper. The nonlinear system of small unmanned helicopter with bounded disturbance is abstracted and modeled by PWA hybrid systems model. As the complexity on-line computation for a class of large hybrid systems, an explicit optimal controller for hybrid systems based on multi-parametric quadratic programming (mp-QP) is proposed. The feasible domain which is the maximal controlled invariant sets for hybrid systems is partitioned in backward dynamic programming by mp-QP method. At each step, one step reachable sets are computed, optimal control laws are constructed to the corresponding regions, and the explicit optimal controller is obtained. Finally simulation results verify the effectiveness of the proposed control method.

Robust Electronic Stability Control via Set-Based Methods for Uncertain Piecewise Affine Systems

AbstractThe paper presents the design of a lateral stability controller for ground vehicles based on front steering and four wheels independent braking. The control objective is to track yaw rate and lateral velocity reference signals while avoiding front and rear wheel traction force saturation. Control design is based on an approximate piecewise–affine nonlinear dynamical model of the vehicle. Vehicle longitudinal velocity and drivers steering input are modeled as measured disturbances taking values in a compact set. A time–optimal control strategy which ensures convergence into a maximal robust control invariant (RCI) set is proposed. This paper presents the uncertain model, the RCI computation, and the control algorithm. Simulations results on a high–fidelity vehicle model show the effectiveness of the propose scheme.

On efficient computation of low-complexity controlled invariant sets for uncertain linear systems

A method is presented for determining invariant low-complexity polytopic sets and associated linear feedback laws for linear systems with polytopic uncertainty. Conditions based on the relationship between 2- and ∞-norms are used to define an initial invariant low-complexity polytope as the solution of a semi-definite program. The problem of computing a maximal controlled invariant low-complexity polytopic set is then formulated as a bilinearly constrained problem, and a relaxation of this problem is derived as an iterative sequence of convex programs. The proposed method scales linearly with the state dimension, which allows the possibility of determining low-complexity robust controlled invariant sets for high-order systems.

STRONG FEASIBILITY IN INPUT-MOVE-BLOCKING MODEL PREDICTIVE CONTROL

Time-invariant input-move-blocking regimes are used in many practical online model predictive control systems in order to reduce the computational complexity of the associated finite-horizon optimal control problem, and have been shown to be beneficial for offline model predictive control methods also. However, until now there exists no method to ensure strong feasibility. In this paper a least-restrictive method to enforce strong feasibility in time-invariant input-move-blocking model predictive control problems is proposed, where the state of the first prediction step is constrained to a novel type of controlled invariant set, called here a controlled invariant feasible set. An algorithm to determine maximal controlled invariant feasible sets is proposed. This algorithm is shown to be semi-decidable for the case of linear, time-invariant plants with time-invariant, polytopic state and control input constraint sets.

Aircraft Autolander Safety Analysis Through Optimal Control-Based Reach Set Computation

and safety analysis of autoland systems. It is shown to be applicable to specific phases of landing: descent, flare, and touchdown. The method is based on optimal control and level set methods; it simultaneously computes a maximal controlled invariant set and a set-valued control law guaranteed to keep the aircraft within a safe set of states under autopilot mode switching. The method is applied to the sequenced flap and slat deflections of a simplified model of a DC9-30. The paper concludes with a demonstration of the method on higher dimensional aircraft models.

Computation of Maximal Safe Sets for Switching Systems

The problem of determining maximal safe sets and hybrid controllers is computationally intractable because of the mathematical generality of hybrid system models. Given the practical and theoretical relevance of the problem, finding implementable procedures that could at least approximate the maximal safe set is important. To this end, we begin by restricting our attention to a special class of hybrid systems: switching systems. We exploit the structural properties of the graph describing the discrete part of a switching system to develop an efficient procedure for the computation of the safe set. This procedure requires the computation of a maximal controlled invariant set. We then restrict our attention to linear discrete-time systems for which there is a wealth of results available in the literature for the determination of maximal controlled invariant sets. However, even for this class of systems, the computation may not converge in a finite number of steps. We then propose to compute inner approximations that are controlled invariant and for which a procedure that terminates in a finite number of steps can be obtained. A tight bound on the error can be given by comparing the inner approximation with the classical outer approximation of the maximal controlled invariant set. Our procedure is applied to the idle-speed regulation problem in engine control to demonstrate its efficiency.