Reachable Set Estimation Research Articles

Estimation and synthesis of reachable set for delta operator systems

This paper considers the problems of reachable set estimation and synthesis of delta operator systems under bounded peak disturbances. A basic reachable set estimation tool for delta operator systems is developed, which can be reduced to the z-domain or s-domain forms when sampling period is 1 or tends to 0. Through the obtained generalized criterion of reachable set, the ellipsoidal reachable set estimation condition of delta operator systems is proposed. Moreover, the state-feedback controller design problem for delta operator systems is investigated. The intention of the controller is to bound the closed-loop system to a given ellipsoid or to make the reachable set of system trajectory as small as possible. Finally, the effectiveness of the obtained results are illustrated by several numerical examples.

Regional Analysis of Slope-Restricted Lurie Systems

This paper considers the stability analysis of nonlinear Lurie type systems where the nonlinearity is both (locally) sector and slope restricted. Convex conditions for verifying stability, computing outer estimates of reachable sets, and upper bounds on the induced $\mathcal {L}_2$ gain in a local or global domain are proposed. The conditions use a Lyapunov function that is quadratic on both the states and the nonlinearity and has an integral term on the nonlinearity. Numerical examples outline the benefits of the proposed approach.

Semidefinite Approximations of Reachable Sets for Discrete-time Polynomial Systems

We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints. Assuming ...

Reachable set estimation for switched positive systems with mixed time‐varying delays and bounded disturbances

In this study, the problem of reachable set estimation of switched positive systems with discrete and distributed delays subject to bounded disturbances is studied. The authors first investigate the reachable set of unswitched positive systems. Then, applying a new approach which is different from the Lyapunov functional method, they give a sufficient condition for the existence of a ball such that all the solutions of the switched positive system converge exponentially within it under certain switching rules. In addition, a precise estimation of the decay rate is also presented. Finally, the effectiveness of the theoretical result is illustrated by a numerical example.

Reachable set estimation for Markovian jump neural networks with time-varying delay

This paper is concerned with the reachable set estimation for Markovian jump neural networks with time-varying delay and bounded peak inputs. The objective is to find a description of a reachable set that is containing all reachable states starting from the origin. In the framework of Lyapunov–Krasovskii functional method, an appropriate Lyapunov–Krasovskii functional is constructed firstly. Then by using the Wirtinger-based integral inequality and the extended reciprocally convex matrix inequality, an ellipsoidal description of the reachable set for the considered neural networks is derived. Finally, a numerical example with simulation results is provided to verify the effectiveness of our results.

Reachable set estimation and dissipativity for discrete-time T–S fuzzy singular systems with time-varying delays

This paper studies the problems of reachable set estimation and dissipativity analysis for a class of nonlinear singular system with time-varying delays and bounded input disturbances. The nonlinear singular system is modeled by Takagi–Sugeno (T–S) fuzzy singular delay system. Firstly, based on new Lyapunov–Krasovskii functional, in which the difference of triple-summable terms and both the upper and the lower bound of time-varying delays are considered, and by using the free-weighting matrix technique, reciprocally convex combination approach, a new delay-dependent reachable set estimation and dissipativity condition is obtained. Furthermore, the derived condition can be converted into linear matrix inequalities (LMIs), and it can guarantee the reachable set to be bounded by the intersection of ellipsoids and the T–S fuzzy singular delay system is strictly (G1,G2,G3)−δ̃−dissipative. Finally, a numerical example is provided to demonstrate the effectiveness of the obtained results in this paper.

Reachable set estimation for switched positive systems

ABSTRACTThis paper focuses on the problem of reachable set estimation for discrete-time switched positive systems under two possible classes of exogenous disturbance. The multiple linear copositive Lyapunov function approach is applied to determine the bounding hyper-pyramids for the reachable set. Based on some Lyapunov-based inequalities and the linear version of the S-procedure technique, the bounding hyper-pyramids for the reachable set can be determined by solving a set of inequalities. Two optimisation methods are adopted to make the bounding hyper-pyramids as small as possible. Genetic Algorithm (GA) is utilised to search for the optimal value of the decision variables in the obtained inequalities. Finally, numerical examples are given to illustrate the effectiveness of the theoretical findings.

Switched systems approach to state bounding for time delay systems

This paper investigates the problems of reachable set estimation for dynamic systems with time-varying delay and random delay. Different from the traditional Lyapunov-Krasovskii functional method, a switched systems approach is adopted to study the reachable set estimation problem for time-varying delay systems. Through augmentation, time-varying delay systems can be transformed into switched delay-free systems. The reachable set of a time-varying delay system is estimated based on its corresponding augmented switched system. The number of decision variables in the obtained reachable set estimation condition is small compared with the Lyapunov-Krasovskii functional method. In addition, the problem of mean square estimation of the system state for random delay systems is also studied. The effectiveness of the theoretical findings is verified through several simulation examples.

External Estimates for Reachable Sets of a Control System with Uncertainty and Combined Nonlinearity

The problem of estimating the trajectory tubes of a nonlinear control dynamic system with uncertainty in the initial data is studied. It is assumed that the dynamic system has a special structure in which the nonlinear terms are defined by quadratic forms in the state coordinates and the values of uncertain initial states and admissible controls are subject to ellipsoidal constraints. The matrix of the linear terms in the velocities of the system is not known exactly; it belongs to a given compact set in the corresponding space. Thus, the dynamics of the system is complicated by the presence of bilinear components in the righthand sides of the differential equations of the system. We consider a complicated case and generalize the author’s earlier results. More exactly, we assume the simultaneous presence in the dynamics of the system of bilinear functions and quadratic forms (without the assumption of their positive definiteness) and we also take into account the uncertainty in the initial data and the impact of the control actions, which may also be treated here as undefined additive disturbances. The presence of all these factors greatly complicates the study of the problem and requires an adequate analysis, which constitutes the main purpose of this study. The paper presents algorithms for estimating the reachable sets of a nonlinear control system of this type. The results are illustrated by examples.

State estimation for bilinear impulsive control systems under uncertainties

The state estimation problem for uncertain impulsive control systems with a special structure is considered. The initial states are taken to be unknown but bounded with given bounds. We assume here that the coefficients of the matrix included in the differential equations are not exactly known, but belong to the given compact set in the corresponding space. We present here algorithms that allow to find the external ellipsoidal estimates of reachable sets for such bilinear impulsive uncertain systems.

Reachable set estimation for discrete-time bilinear systems with time-varying delays

The problem of reachable set estimation is studied for discrete-time bilinear system in this paper. Time-varying delays and bounded input disturbances are both considered in bilinear system. The aim is to find reachable set that converges from all the states of system with initial conditions. By constructing Lyapunov–Krasovskii functional, sufficient delay-dependent less conservative stable conditions of reachable set estimation are obtained for bilinear delay system via the reciprocally convex combination and delay partition approaches. The derived theorem can guarantee that all the states of system with initial conditions from some domain are bounded in an ellipsoid and all the states from other domain are converged exponentially within a ball. One simulation example is presented to illustrate the correctness of the derived result in this paper.

Reachable set estimation for Takagi-Sugeno fuzzy systems against unknown output delays with application to tracking control of AUVs

In this paper, we address the problem of reachable set estimation for continuous-time Takagi-Sugeno (T-S) fuzzy systems subject to unknown output delays. Based on the reachable set concept, a new controller design method is also discussed for such systems. An effective method is developed to attenuate the negative impact from the unknown output delays, which likely degrade the performance/stability of systems. First, an augmented fuzzy observer is proposed to capacitate a synchronous estimation for the system state and the disturbance term owing to the unknown output delays, which ensures that the reachable set of the estimation error is limited via the intersection operation of ellipsoids. Then, a compensation technique is employed to eliminate the influence on the system performance stemmed from the unknown output delays. Finally, the effectiveness and correctness of the obtained theories are verified by the tracking control of autonomous underwater vehicles.

Output Reachable Set Estimation and Verification for Multilayer Neural Networks.

In this brief, the output reachable estimation and safety verification problems for multilayer perceptron (MLP) neural networks are addressed. First, a conception called maximum sensitivity is introduced, and for a class of MLPs whose activation functions are monotonic functions, the maximum sensitivity can be computed via solving convex optimization problems. Then, using a simulation-based method, the output reachable set estimation problem for neural networks is formulated into a chain of optimization problems. Finally, an automated safety verification is developed based on the output reachable set estimation result. An application to the safety verification for a robotic arm model with two joints is presented to show the effectiveness of the proposed approaches.

Reachable set estimation for switched linear systems with dwell-time switching

In this work we address the problem of (outer) estimation of reachable sets in switched linear systems subject to dwell-time switching. After giving some conditions that exploit the well-known properties of exponential decrease/bounded increase of the Lyapunov function (i.e. exponential decrease in between switching times and bounded increase at switching times), we overcome the need for such properties. This is done by introducing a new notion of τ-reachable set, i.e. the set that can be reached by the trajectories defined at time τ after the switch. Such extended notion of reachable set can be used to parametrize the estimate of the reachable set as a function of the distance in time from the switch. Two approaches are provided to implement such parametrization: the first approach exploits the evolution of the system in between switches via the matrix exponential of the state subsystem matrix; the second approach exploits a time-scheduled Lyapunov function. A numerical example is provided to show the effectiveness of the proposed methods and computational cost is addressed.

Reachable set estimation for discrete‐time switched system with application to time‐delay system

SummaryThis paper addresses the problem of reachable set estimation and synthesis for a class of discrete‐time switched linear systems with time delay and bounded peak disturbance. Combined with the feature of mode‐dependent average dwell time switching, a new algorithm is developed to estimate the reachable set of switched system, which is both quasi‐time‐dependent and mode‐dependent. Then, the proposed method is applied to time‐delay system and a sufficient condition is presented to guarantee the asymptotic stability and estimate the bounding ellipsoid. Furthermore, the quasi‐time‐dependent controller is designed to stabilize the system and restrict the closed‐loop system states to an ellipsoidal bound. Examples are presented to illustrate the effectiveness and advantages of the obtained theorems.

The HJB Approach and State Estimation for Control Systems with Uncertainty

This paper deals with the comparison principle for the first-order ODEs of the Hamilton - Jacobi - Bellman (HJB) type which describe solutions of the dynamical control system with a special structure, with combined nonlinearity of quadratic and bilinear kinds and with uncertainty in initial states and in system parameters. The uncertainty studied here is of set-membership type when only the bounding set for unknown items is given. The ellipsoidal estimates of reachable sets are derived using the special structure of studied control system. The techniques of generalized solutions of Hamilton - Jacobi - Bellman inequalities together with previously established results of ellipsoidal calculus are applied to find the external set-valued estimates of reachable sets as the level sets of a related cost functional.

Reachable Set Estimation for a Class of Memristor-Based Neural Networks With Time-Varying Delays

This paper investigates the reachable set estimation problem for a class of memristor-based neural networks with time-varying delays and bounded disturbances. By constructing a Lyapunov–Krasovskii functional, a sufficient condition for the solvability of the addressed problem is established based on linear matrix inequality. This condition ensuring the existence of an ellipsoid that contains all the states under initial conditions. A stability criterion of memristor-based neural networks with time-varying delays is also given. Two numerical examples are provided to show the effectiveness of the proposed methods.

Estimation Techniques for Bilinear Control Systems

Abstract The problem of estimating reachable sets of impulsive control systems with uncertainties in initial states and in the matrix of the system is studied. We assume that the initial states to be unknown but belong to a given star-shaped symmetric nondegenerate polytope. The matrix included in the differential equations of the system dynamics is uncertain and only bounds on admissible values of this matrix coefficients are known. We present here the approach that allow to find ellipsoidal estimates of reachable sets which uses the special structure of the bilinear control system. The algorithm of constructing such ellipsoidal set-valued estimates and numerical simulation results are given.

Differential Equations for Ellipsoidal Estimates of Reachable Sets for a Class of Control Systems with Nonlinearity and Uncertainty

The problem of estimating reachable sets of nonlinear dynamical control systems with combined nonlinearity of quadratic and bilinear types and with uncertainty in initial states and in system parameters is studied. We assume that the uncertainty is of set-membership kind when only the bounding sets for unknown parameters and functions are given. We find the ellipsoidal estimates of reachable sets using the special bilinear-quadratic structure of studied control system. The main result consists in deriving the related differential equations which describe the dynamics of the ellipsoidal estimates of reachable sets of the control system under study, related numerical simulation results are also given.

State Estimates of Bilinear Discrete-Time Systems with Integral Constraints through Polyhedral Techniques

Abstract We consider bilinear discrete-time systems for the case when interval bounds on the coefficients of the system are imposed, additive input terms are restricted by integral constraints, and initial states are restricted by given sets. Several ways for constructing external parallelepiped-valued estimates of reachable sets of such systems are proposed. One of the above techniques is based on obtaining recurrence relations for reachable sets in the ”extended” phase space and constructing corresponding estimates in the form of polytopes of some special shape.