Minimal invariant subspaces and reachability of 2D hybrid LTI systems (original) (raw)
Reachability, Observability and Minimality for a Class of 2D Continuous-Discrete Systems
2007
Reachability and observability criteria are obtained for 2D continuous-discrete time-variable Attasi type systems by using suitable 2D reachability and observability Gramians. Necessary and sufficient conditions of reachability and observability are derived for time-invariant systems. The duality between the two concepts is emphasized as well as their connection with the minimality of these systems.
Reachability computation for linear hybrid systems
1999
Linear hybrid systems are nite state machines with linear vector elds of the form _ x = Ax in each discrete location. Very recently, the reachability problem for classes of linear hybrid systems was shown to be decidable. In this paper, the decidability result is extended to capture classes of linear hybrid systems where in each location the dynamics are of the form _ x = Ax + Bu, f o r v arious types of inputs.
A Computatuional Analysis of the Reachability Problem for a Class of Hybrid Dynamical Systems
1996
Hybrid systems possess continuous dynamics de ned within regions of state spaces and discrete transitions among the regions. Many practical control veri cation and synthesis tasks can be reduced to reachability problems for these systems that decide if a particular state-space region is reachable from an initial operating region. In this paper, we present a computational analysis of the face reachability problem for a class of three-dimensional dynamical systems whose state spaces are de ned by piecewise constant vector elds and whose trajectories never return to a state-space region once they exit the region. These systems represent a restricted class of control systems whose dynamics results from a juxtaposition of piecewise parameterized vector elds. We had previously developed a computational algorithm for synthesizing the desired dynamics of a system in phase space by piecing together vector elds geometrically. We demonstrate in this paper that the reachability problem for this class of systems is decidable while the computation is provably intractable (i.e., PSPACE-hard). We prove the intractability via a reduction of satis ability of quanti ed boolean formulas to this reachability problem. This result sheds light on the computational complexity of phase-space based control synthesis methods and extends the work of Asarin, Maler, and Pnueli 2] that proves computational undecidability for three-dimensional constant-derivative systems.
Reachability Analysis of Linear Hybrid Systems via Block Decomposition
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2020
Reachability analysis aims at identifying states reachable by a system within a given time horizon. This task is known to be computationally expensive for linear hybrid systems. Reachability analysis works by iteratively applying continuous and discrete post operators to compute states reachable according to continuous and discrete dynamics, respectively. In this paper, we enhance both of these operators and make sure that most of the involved computations are performed in low-dimensional state space. In particular, we improve the continuous-post operator by performing computations in high-dimensional state space only for time intervals relevant for the subsequent application of the discrete-post operator. Furthermore, the new discrete-post operator performs low-dimensional computations by leveraging the structure of the guard and assignment of a considered transition. We illustrate the potential of our approach on a number of challenging benchmarks.
2010
This paper presents a method for complexity reduction in reachability analysis and safety-preserving controller synthesis via Schur-based decomposition. The decomposition results in either decoupled or weaklycoupled (lower dimensional) subsystems. Reachable sets, computed independently for each subsystem, are back-projected and intersected to yield an overapproximation of the actual reachable set. Moreover, applying this technique to a class of unstable LTI systems we show that when certain eigenvalue and state-constraint conditions are satised, further reduction of complexity is possible. Evaluating our method for a variety of examples we demonstrate that signicant reduction in the computational costs can be achieved. This technique has considerable potential utility for use in conjunction with computationally intensive reachability tools.
Observability and Geometric Approach of 2D Hybrid Systems
International Journal of Computers, 2021
A connection is emphasized between two branches of the Systems Theory, namely the Geometric Approach and 2D Systems, with a special regard to the concept of observability. An algorithm is provided which determines the maximal subspace which is invariant with respect to two commutative matrices and which is included in a given subspace. Observability criteria are obtained for a class of 2D systems by using a suitable 2D observability Gramian and some such criteria are derived for LTI 2D systems, as well as the geometric characterization of the subspace of unobservable states. The presented algorithm is applied to determine this subspace.
A Computational Analysis of the Reachability Problem for a Class of Hybrid Dynamical Systems
1997
Hybrid systems possess continuous dynamics defined within regions of state spaces and discrete transitions among the regions. Many practical control verification and synthesis tasks can be reduced to reach ability problems for these systems that decide if a particular state-space region is reachable from an initial operating region. In this paper, we present a computational analysis of the face reachability problem for a class of three-dimensional dynamical systems whose state spaces are defined by piecewise constant vector fields and whose trajectories never return to a state-space region once they exit the region. These systems represent a restricted class of control systems whose dynamics results from a juxtaposition of piecewise parameterized vector fields. We had previously developed a computational algorithm for synthesizing the desired dynamics of a system in phase space by piecing together vector fields geometrically. We demonstrate in this paper that the reachability problem for this class of systems is decidable while the computation is provably intractable (i.e., PSPACE-hard). We prove the intractability via a reduction of satisfiability of quantified boolean formulas to this reachability problem. This result sheds light on the computational complexity of phase-space based control synthesis methods and extends the work of Asarin, Maler, and Pnueli [2] that proves computational undecidability for three-dimensional constant-derivative systems.
Controllability and Gramians of 2D Continuous Time Linear Systems
2020
The controllability of a class of 2D linear time varying continuous time control systems is studied. The state space representation is provided and the formulas of the states and the input-output map of these systems are derived. The fundamental concepts of controllability and reachability are analysed and suitable controllability and reachability Gramians are constructed to characterize the controllable and the reachable time varying systems. In the case of time invariant 2D systems, some algorithms are developed to calculate different controllability Gramians as solutions of adequate Lyapunov type equations. Corresponding Matlab programs are implemented to solve these Lyapunov equations.
An improved reachability analysis method for strongly linear hybrid systems (extended abstract)
Springer eBooks, 1997
This paper addresses the exact computation of the set of reachable states of a strongly linear hybrid system. It proposes an approach that is an extension of classical state-space exploration. This approach uses a new operation, based on a cycle analysis in the control graph of the system, for generating sets of reachable states, as well as a powerful representation system for sets of values. The method broadens the range of hybrid systems for which a finite and exact representation of the set of reachable states can be computed. In particular, the state-space exploration may be performed even if the set of variable values reachable at a given control location cannot be expressed as a finite union of convex regions. The technique is illustrated on a very simple example.
Reachability analysis of complex planar hybrid systems
Science of Computer Programming, 2013
Hybrid systems are systems that exhibit both discrete and continuous behavior. Reachability, the question of whether a system in one state can reach some other state, is undecidable for hybrid systems in general. The Generalized Polygonal Hybrid System (GSPDI) is a restricted form of hybrid automaton where reachability is decidable. It is limited to two continuous variables that uniquely determine which location the automaton is in, and restricted in that the discrete transitions does not allow changes in the state, only the location, of the automaton. One application of GSPDIs is for approximating control systems and verifying the safety of such systems. In this paper we present the following two contributions: i) An optimized algorithm that answers reachability questions for GSPDIs, where all cycles in the reachability graph are accelerated. ii) An algorithm by which more complex planar hybrid systems are over-approximated by GSPDIs subject to two measures of precision. We prove soundness, completeness, and termination of both algorithms, and discuss their implementation.