Hybrid Automata

A secondary objective of this approach is to permit more formal investigation of mind-body inter- actions. A common modern theory of mind views the mind as a kind of computer; from the perspective of hybrid systems, one might view the mind as a controller (or set of controllers) and the body as the continuous portion acted upon by the controllers. In effect, the mind’s state determines the actions of the body; actions alter the observed world, in turn creating changes in the mind, which produce new actions, and so on. As with hybrid automata, however, the languages used for mind and body differ wildly; one can characterize physical actions in mechanical or physics terminology, which is generally disjoint from the terminology of beliefs, desires, and intentions that is sometimes used to describe mental states. By characterizing hybrid automata via a single set of interactions, I hope to provide a common mechanics- based description which might be used to characterize mind-body interactions. It bears emphasizing that this work does not, itself, attempt to characterize minds in this common language; it is purely my objec- tive here to construct a unified description for hybrid automaton, which might permit such mind-body descriptions in the future.

In this paper, we applied ideas from formal methods in computer science to the problem of verification of floodgate management strategy. We demonstrated its effectiveness in a linear arrangement of reservoirs. Several directions of future research remain open. Many of these are essentially relaxations in the assumptions and generalizations of the problem statement that we addressed in this paper, yielding hybrid automata that are more faithful models to different systems:

In this section, we declare how to encode the implementation of epistemic hybrid automata for the previous scenario in CLP, the reason of choosing CLP that it is enriched with a lot of domain solver as interval domain (IC) that is used to represent the continuous evolution of automata and symbolic domain to represent fired events. The code is implemented using eclipse prolog [18]. Our model follows the definition of EPH, where each automaton is defined by the automaton location, the real variables, time and the agent knowledge.

3. OPTIMAL SOLUTION FOR RECTANGULAR HYBRID AUTOMATA This section deals with the optimal control prob- lem for rectangular hybrid automata. Before pro- ceeding with the main theorem, we discuss the re- lationship between the open and closed loop con- trol. In each discrete state q, there are two types of controllers that generate the control input, namely the open loop controllers and the closed loop (feedback) controllers. Let u q ∈ [U q , U q ]

Rectangular hybrid automata. Puri and Varaiya [PV94] show that for a re- stricted subclass of rectangular hybrid automata the reachability problem is decid- able. The restrictions are twofold: 1) the jump predicate is specified using boolean combinations of inequalities in which a value of the variable is compared against an integer constant; 2) if a discrete transition can occur at a non-integer time-point, the jump predicate guarantees reinitialisation of every variable whose flow changes. Henzinger et al. [HKPP98] consider a different subclass of rectangular hybrid automata, referred to as the initialised rectangular hybrid automata , and show that the reachability problem for this class is PSPACE-complete. PSPACE-completeness follows from a polynomial-time reduction from an initialised rectangular hybrid au- tomaton with n variables to a timed automaton with 2 n +1 variables. The reduction consists of an intermediate reduction to a multi-rate linear automaton with 2n + 1 variables. Every variable of the original initialised hybrid automaton is encoded using two variables of the multi-rate linear hybrid automaton. Intuitively, the two variables “keep track” of the upper and lower bounds on the value of the variable in the original initialised rectangular hybrid automaton. The two restrictions nec- essary for the result to hold are as follows: 1) values of two variables with different flows are never compared; 2) whenever the flow of a variable changes the value of that variable is reinitialised (reset). The authors also prove that if any of the two restrictions are removed then the reachability-problem becomes undecidable.

Considering the high computational complex- ity of the MPC on-line algorithm presented in (Bemporad et al., 1999), we formulate a semi- explicit(sub-optimal) method that reduces the computational burden. For this, we remove the off-line choices for the switching part, by selecting the shortest discrete path. It is then shown that the shortest path can be used to derive a semi- explicit algorithm for hybrid automata, instead of solving the mixed integer programming problem (NP hard) for each discrete-time instant.

Abstract: Hybrid systems are dynamical systems consisting of interacting discrete event and continuous state subsystems. A controlled hybrid automaton is a hybrid automaton whose continuous-state dynamics are described by inhomogeneous differential equations. This paper presents a sufficient condition for the existence of global non-terminating solutions in controlled hybrid automata. The condition is based on a recursive algorithm that can always terminate after a finite number of iterations to a limit set of states, i.e. the fixed point of the recursion. If the fixed point is non-empty, then there exists a measurable control under which the hybrid automaton generates a global non-terminating solution. The more important is that this result can also be used to infer the existence of global solutions to compositions of controlled hybrid automata, thereby providing a foundation for the analysis of large scale hybrid systems. The controlled hybrid automata model can be used for robotics system modeling and control. By solving the global non-terminating solution to controlled hybrid automata, the biped robots can be guaranteed to keep the walking gait without falling down.

the shortest path can be used to derive a semi- explicit algorithm for hybrid automata. The de- sign proposed in this paper is computationally ef- ficient due to the fact that the global optimization problem has been decomposed into several consec- utive local optimization problems. However, the price to be paid for the computational saving is that the result is sub-optimal.

By definition, all linear hybrid automata are DLHAs. Our system dynamically changes its structure by sending and receiving messages. However, the messages stati- cally determine the structure, and the system is a linear hybrid automaton with a set of queues. It is basically equivalent to the reachability analysis of a linear hybrid automa- ton. Therefore, the reachability problem of DRSs is undecidable, and this algorithm might not terminate [16].

Uncertain input/output timing adversely affects the performance of digital control loops but is virtually unavoidable in complex real-time systems. To still prove safety we are required to prove worst-case behavior, such as the maximum position error of a quadcopter, in the presence of timing uncertainty. To address this challenge, we present a series of benchmark problems for the verification of hybrid automata, which are a formalism that captures both the discrete- time and continuous-time aspects of real-time control systems. However, first experiments with set-based reachability tools suggest that verification is only possible for simple examples.

Abstract— This paper presents a hybrid control model, which is based on the Real-Time Unified Modeling Language (UML) and hybrid automata in order to conveniently analyze, design and implement industrial Hybrid Dynamic Systems (HDS). This model also creates a communication pattern, which can permit the designed components to be customized and reused in new control applications for various HDS types. The paper shows out step-by-step the specification of an industrial HDS modeled by the specialization of hybrid automata, the analysis and design of HDS controllers by using the Real-Time UML, which allow us to quickly find out the main control capsules, their ports and communication protocols in order to precisely model and tightly allocate control structures and dynamic behaviors for the implementation of industrial HDS. Following this proposed model, the application to KC05.21/16-20 project entitled “Research, design and manufacture geothermal cooling systems for Base Transceiver Station” is finally presented.

As the increasing complexity of AGV systems in commercial and industrial scenarios, control and coordination of the AGVs become much more difficult under uncertain dynamic environment. Along with higher requirements of complex task, the increased ability and responsiveness for AGV con- trol systems have been received much consideration in the last decade. The multiagent-based or behaviors-based approaches have been the focus of the researchers. For examples, Choudhary et al. [1] proposed a multiagent-based framework representing zone controlled AGV environment incorporating various behaviors like path generation, collision and deadlock avoidance, etc. Christopher et al. [2] implemented behaviors- based intelligent distributed fuzzy logic control systems integrating the presentation of human knowledge to a meca- num wheeled AGV, the navigational and collision avoidance behaviors of AGV were controlled by using IF-THEN rules. The above mentioned approaches could achieve improved reliability and reduce complexity of AGV control systems to some extent. However, the AGV is a complex nonlinear system with nonholonomic constraint. It can not be controlled by smooth linear time invariant controls laws [3], which most of the assumptions made in the controls literature are not satisfied. It is especially hard to provide provable guarantees on safety and performance to its behavior control systems (BCS) which control the AGV to perform various tasks in uncertain dynamic environment. One successful approach is to decompose it into hybrid systems that intermix discrete modes and continuous dynamics. Recently years, a rapid growth of interest in hybrid systems has developed efficient tools for synthesis and analysis of such complex systems. This hybrid, hierarchical approach to the design and control of BCS for AGV has proven to be very successful. For examples, Kress-Gazit et al. [4] applied linear-temporal logic specifica- tions for generating robot behaviors. Hua et al. [5] applied motion description languages to solve the pose stabilization problem of nonholonomic wheeled mobile robots. Gayan et al. [3] explained a hybrid control strategy developed to coor- dinate multiple autonomous mobile robots with nonholo- nomic constraints in an obstacles populated environment. Wang et al. [6] adopted a hybrid input/output automata to decompose the behavior control systems of mobile robots and different languages were used to define the behavior speci- fications. Hua et al. [7] addressed a 3-layered hierarchical hybrid control structure to pose stabilization controller for wheeled mobile robots. Pu et al. [8] resolved the parking problem for wheeled mobile robot by using hybrid automata- based approach.

Our increasing reliance on complex embedded and cyber-physical systems calls for the develop- ment of methods for the verification of hybrid systems, which are systems in which behaviour is described as an interplay between discrete and continuous components. In this paper, we con- sider a well-known formalism for the description of hybrid systems, namely hybrid automata [ACH + 95], which comprise a finite-state graph, to represent the discrete part of the system, and a finite set of real-valued variables, to represent the continuous part of the system. Interaction be- tween the discrete and continuous parts of the system is represented by labelling the graph with conditions on the variables and their first derivatives. Hence we can express that, as time passes and the system resides in a particular node of the graph, the rate of change of the variables is de- scribed in a certain way; we can also describe how the system moves from node to node when the

Decidability issues associated to hybrid automata have been studied in [9], [10] where the reader is referred for more details. In [9] initialised rectangular automata are stated as the boundary between decidability and undecidability of the reachability problem for hybrid automata. In [11] the problem of synthesis of controllers for generalised rectangular automata, named linear hybrid systems, has been addressed. Reachability was shown to be semidecidable for this class of systems and studies for zeno behaviour, time diverging and partial observability have been considered. The synthesis of controllers for hybrid systems, in general, and for safety and eventuality specifications, using optimal control and games theory , have been studied in [15] and [14] respectively.

PHAVerLite PHAVerLite is a variant of the stand-alone verification tool PHAVer, sharing the same capabilities and formal soundness guarantees. It is worth stressing that PHAVerLite, being a stand-alone tool, differs from the PHAVer-lite SpaceEx plugin that participated in the friendly competition in 2018. For instance, while PHAVer-lite was able to accept input specified using the SpaceEx syntax for hybrid automata, at present PHAVerLite can only accept input specified using the PHAVer syntax. The main difference with respect to PHAVer is the adoption of the new polyhedra library PPLite [8]: thanks to a novel representation and conversion algorithm [7] for NNC (Not Necessarily Closed) polyhedra, PPLite is able to obtain significant efficiency improvements with respect to the classical polyhedra implementation used in PHAVer (which is based on the Parma Polyhedra Library [5]). The development of PHAVerLite was motivated by the desire to go beyond the main change above and also revisit many of the key design and implementation choices of the original PHAVer: this allowed to experiment with novel algorithms or design tradeoffs, also exploiting some of the more recent advances in the implementation of operators on the polyhedral domains. At present, PHAVerLite has only been used to analyze systems characterized by piecewise constant dynamics; also note that a few of the PHAVer functionalities (e.g., the computation of simulation relations) have been deliberately removed.

XSpeed The tool XSpeed implements algorithms for reachability analysis of continuous and hybrid systems with linear dynamics. The focus of the tool is to exploit the modern multicore architectures to enhance the performance through parallel computations. The algorithms in XSpeed are based on symbolic states represented using support functions. The tool can analyze hybrid automata models in the SpaceEx input format. It allows to compute the reachability in bounded depth as well as reachability till fixed point. XSpeed realizes two algorithms to enhance the performance of reachability analysis of purely continuous systems. The first is the parallel support function sampling algorithm and the second is the time-slicing algorithm [10, 11]. The performance of hybrid systems reachability analysis is enhanced using an adaptation of the G.J. Holzmann’s parallel BFS algorithm in the SPIN model checker, called the AGJH algorithm [5]. In addition, a task parallel and an asynchronous variant of AGJH are also implemented in the tool. The details of the algorithms in XSpeed can be found in [6]. The tool is available at http://xspeed.nitmeghalaya.in/

BACH BACH [12, 11] is a bounded reachability checker for Linear Hybrid Automata (LHA) model, Hybrid Systems with Piecewise Constant Dynamics (HPWC) in the term of ARCH competition. The tool provides GUI for LHA modeling and also bounded reachability checkers for both single automaton and automata network. Be different from classical bounded checkers of LHA, which encodes the “complete” bounded state space of the system into a huge SMT

XSpeed The tool XSpeed implements algorithms for reachability analysis of continuous and hybrid systems with linear dynamics. The focus of the tool is to exploit the modern multicore architectures to enhance the performance through parallel computations. The algorithms in XSpeed are based on symbolic states represented using support functions. The tool can analyze hybrid automata models in the SpaceEx input format. It allows to compute the reachability in bounded depth as well as reachability till fixed point. XSpeed realizes two algorithms to enhance the performance of reachability analysis of purely continuous systems. The first is the parallel support function sampling algorithm and the second is the time-slicing algorithm [10, 11]. The performance of hybrid systems reachability analysis is enhanced using an adaptation of the G.J. Holzmann’s parallel BFS algorithm in the SPIN model checker, called the AGJH algorithm [5]. In addition, a task parallel and an asynchronous variant of AGJH are also implemented in the tool. The details of the algorithms in XSpeed can be found in [6]. The tool is available at http://xspeed.nitmeghalaya.in/

On the other hand, many works have been proposed in the model checking and control communities to han- dle hybrid systems. Some examples include (Cimatti et al. 2015; Cavada et al. 2014; Cimatti, Mover, and Tonetta 2012; Tabuada, Pappas, and Lima 2002; Maly et al. 2013), sampling-based planners (Karaman et al. 2011; Lahijanian, Kavraki, and Vardi 2014). Another related direction is falsi- fication of hybrid systems (i.e., guiding the search towards the error states, that can be easily cast as a planning problem) (Plaku, Kavraki, and Vardi 2013). However, while all these works aim to address a similar problem, that cannot be used to handle PDDL+ models. Some recent works (Bogomolov et al. 2014; 2015) are trying to define a formal translation be- tween PDDL+ and standard hybrid automata, but so far only an over-approximation has been defined, that allows the use of those tools only for proving plan non-existence.

Automata theory has proved to be a useful tool in theoretical computer science since last couple of decades. The algebraic automaton is an advanced form of auto- mata which has very interesting properties and structures from algebraic theory of mathematics. The applications of algebraic theory of automata are not limited to com- puters but are being seen in many other disciplines of science and engineering, for example, representing cha- racteristics of natural phenomena in biology [25] and modeling of chemical systems using cellular automata [26]. Another interesting application, in which a system is described for synthesizing physics-based animation programs based on hybrid automata, is presented in [27]. Because of having special algebraic characteristic, an automaton can easily be extended to develop algebraic automata.