On reachability analysis for nonlinear control systems with state constraints (original) (raw)
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On Reachable Sets for a Class of Nonlinear Systems with Constraints
Journal of Mathematical Analysis and Applications, 1999
The determination of the reachable set for a class of nonlinear systems with control and state trajectory constraints is investigated. The main result links this problem with the determination of the set of admissible controls, for which procedures already exist. The paper also gives a procedure to generate an admissible control which steers the system to a reachable state.
Reachability Analysis of Nonlinear Systems Using Conservative Approximation
2003
In this paper we present an approach to approximate reachability computation for nonlinear continuous systems. Rather than studying a complex nonlinear system x = g(x), we study an approximating system x = f(x) which is easier to handle. The class of approximating systems we consider in this paper is piecewise linear, obtained by interpolating g over a mesh. In order to be conservative, we add a bounded input in the approximating system to account for the interpolation error. We thus develop a reachability method for systems with input, based on the relation between such systems and the corresponding autonomous systems in terms of reachable sets. This method is then extended to the approximate piecewise linear systems arising in our construction. The final result is a reachability algorithm for nonlinear continuous systems which allows to compute conservative approximations with as great degree of accuracy as desired, and more importantly, it has good convergence rate. If g is a C 2 function, our method is of order 2. Furthermore, the method can be straightforwardly extended to hybrid systems.
Reachability of Nonlinear Systems With Unknown Dynamics
IEEE Transactions on Automatic Control
Determining the reachable set for a given nonlinear control system is crucial for system control and planning. However, computing such a set is impossible if the system's dynamics are not fully known. This paper is motivated by a scenario where a system suffers an adverse event mid-operation, resulting in a substantial change to the system's dynamics, rendering them largely unknown. Our objective is to conservatively approximate the system's reachable set solely from its local dynamics at a single point and the bounds on the rate of change of its dynamics. We translate this knowledge about the system dynamics into an ordinary differential inclusion. We then derive a conservative approximation of the velocities available to the system at every system state. An inclusion using this approximation can be interpreted as a control system; the trajectories of the derived control system are guaranteed to be trajectories of the unknown system. To illustrate the practical implementation and consequences of our work, we apply our algorithm to a simplified model of an unmanned aerial vehicle. Notice of Previous Publication. This manuscript substantially improves the work of [1]. Theory has been generalized to include a class of non-invertible matrices and improved to provide a larger set of reachable states. All lemmas, corollaries, and Theorems 1, 2, and 4 are entirely novel. Theorem 3 has been slightly modified from existing theorems in [1] given our new results.
Nonlinear Systems: Approximating Reach Sets
2004
We describe techniques to generate useful reachability information for nonlinear dynamical systems. These techniques can be automated for polynomial systems using algorithms from computational algebraic geometry. The generated information can be incorporated into other approaches for doing reachability computation. It can also be used when abstracting hybrid systems that contain modes with nonlinear dynamics. These techniques are most naturally embedded in the hybrid qualitative abstraction approach proposed by the authors previously. They also show that the formal qualitative abstraction approach is well suited for dealing with nonlinear systems.
SIAM Journal on Control and Optimization, 2010
We consider a target problem for a nonlinear system under state constraints. We give a new continuous level-set approach for characterizing the optimal times and the backward-reachability sets. This approach leads to a characterization via a Hamilton-Jacobi equation, without assuming any controllability assumption. We also treat the case of time-dependent state constraints, as well as a target problem for a two-player game with state constraints. Our method gives a good framework for numerical approximations, and some numerical illustrations are included in the paper.
The method of uniform monotonous approximation of the reachable set border for a controllable system
Journal of Global Optimization, 2015
A numerical method of a two-dimensional non-linear controllable system reachable set boundary approximation is considered. In order to approximate the boundary right piecewise linear closed contours are used: a set of broken lines on a plane. As an application of the proposed technique a method of finding linear functional global extremum is described, including its use for systems with arbitrary dimensionality. Keywords Reachable set • Optimal control • Non-linear dynamic system 1 Introduction Reachable sets (RS) are among classical objects of investigation in the optimal control theory. The possibility to successfully operate with RS of controllable systems can greatly simplify the solutions of a number of traditional extreme problems, such as local and global search for a functional extremum, parametric identification, optimal control synthesis, system phase state estimation, control normalization, trajectories pencil control, and others. The problem of finding the RS is closely related to another classical problem of constructing integral funnels of differential inclusions [4
A Comprehensive Method for Reachability Analysis of Uncertain Nonlinear Hybrid Systems
IEEE Transactions on Automatic Control, 2016
Reachability analysis of nonlinear uncertain hybrid systems, i.e. continuous-discrete dynamical systems whose continuous dynamics, guard sets and reset functions are defined by nonlinear functions, can be decomposed in three algorithmic steps: computing the reachable set when the system is in a given operation mode, computing the discrete transitions, i.e. detecting and localizing when (and where) the continuous flowpipe intersects the guard sets, and aggregating the multiple trajectories that result from an uncertain transition once the whole flowpipe has transitioned so that the algorithm can resume. This paper proposes a comprehensive method that provides a nicely integrated solution to the hybrid reachability problem. At the core of the method is the concept of MSPB, i.e. geometrical object obtained as the Minkowski sum of a parallelotope and an axes aligned box. MSPB are a way to control the over-approximation of the Taylor's interval integration method. As they happen to be a specific type of zonotope, they articulate perfectly with the zonotope bounding method that we propose to enclose in an optimal way the set of flowpipe trajectories generated by the transition process. The method is evaluated both theoretically by analysing its complexity and empirically by applying it to well-chosen hybrid nonlinear examples.
arXiv (Cornell University), 2022
This work presents a method of efficiently computing inner and outer approximations of forward reachable sets for nonlinear control systems with changed dynamics and diminished control authority, given an a priori computed reachable set for the nominal system. The method functions by shrinking or inflating a precomputed reachable set based on prior knowledge of the system's trajectory deviation growth dynamics, depending on whether an inner approximation or outer approximation is desired. These dynamics determine an upper bound on the minimal deviation between two trajectories emanating from the same point that are generated on the nominal system using nominal control inputs, and by the impaired system based on the diminished set of control inputs, respectively. The dynamics depend on the given Hausdorff distance bound between the nominal set of admissible controls and the possibly unknown impaired space of admissible controls, as well as a bound on the rate change between the nominal and off-nominal dynamics. Because of its computational efficiency compared to direct computation of the off-nominal reachable set, this procedure can be applied to on-board fault-tolerant path planning and failure recovery. In addition, the proposed algorithm does not require convexity of the reachable sets unlike our previous work, thereby making it suitable for general use. We raise a number of implementational considerations for our algorithm, and we present three illustrative examples, namely an application to the heading dynamics of a ship, a lower triangular dynamical system, and a system of coupled linear subsystems.
Linearized control systems and small-time reachable sets
IMA Journal of Mathematical Control and Information, 1995
Using linear approximations of nonlinear systems has long been a practice to design control laws. In this paper, an analysis is given involving linear approximation of the nonlinear control system and small-time reachable sets in IR 2. A useful concept, the swing-out, which is a measure of nonlinearity, is de ned. This is used t o examine the relationship between the small-time reachable sets of the nonlinear control system and its linear approximation. Behaviour of the nonlinear system under a control law is examined within this context. More facts are given about the swing-out for some special cases.
Reachability properties of constrained linear systems
Journal of Optimization Theory and Applications, 1992
This paper presents reachability results for a linear control system and arbitrary terminal points p in R" with controls constrained within a compact set U containing the origin. The well-known results for p = 0 are then a special case of our results. Geometric properties of the reachable set are presented and include: general containment properties which describe conditions that guarantee the inclusion of a reachable set in another reachable set; classification of the set of all points p that ensure the equivalence of two (different) reachable sets and the properties of this set. The topological properties of the reachable set to a point p depends on the control set U, the final point p, and the location of the spectrum of the system in the complex plane. We characterize the geometric properties of the reachable set when the spectrum lies in the closed right-half plane, the open left-half plane, or a combination of both.