Time-constrained temporal logic control of multi-affine systems

In this paper, we consider the problem of controlling a dynamical system such that its trajectories satisfy a temporal logic property in a given amount of time. We focus on multi-affine systems and specifications given as syntactically co-safe linear temporal logic formulas over rectangular regions in the state space. The proposed algorithm is based on estimating the time bounds for facet reachability problems and solving a time optimal reachability problem on the product between a weighted transition system and an automaton that enforces the satisfaction of the specification. A random optimization algorithm is used to iteratively improve the solution.     

Temporal logic control for stochastic linear systems using abstraction refinement of probabilistic games

We consider the problem of computing the set of initial states of a dynamical system such that there exists a control strategy to ensure that the trajectories satisfy a temporal logic specification with probability 1 (almost-surely). We focus on discrete-time, stochastic linear dynamics and specifications given as formulas of the Generalized Reactivity(1) fragment of Linear Temporal Logic over linear predicates in the states of the system. We propose a solution based on iterative abstraction-refinement, and turn-based 2-player probabilistic games. While the theoretical guarantee of our algorithm after any finite number of iterations is only a partial solution, we show that if our algorithm terminates, then the result is the set of all satisfying initial states. Moreover, for any (partial) solution our algorithm synthesizes witness control strategies to ensure almost-sure satisfaction of the temporal logic specification. While the proposed algorithm guarantees progress and soundness in every iteration, it is computationally demanding. We offer an alternative, more efficient solution for the reachability properties that decomposes the problem into a series of smaller problems of the same type. All algorithms are demonstrated on an illustrative case study.     

On witnessed models in fuzzy logic

Witnessed models of fuzzy predicate logic are models in which each quantified formula is witnessed, i.e. the truth value of a universally quantified formula is the minimum of the values of its instances and similarly for existential quantification (maximum). Systematic theory of known fuzzy logics endowed with this semantics is developed with special attention paid to problems of arithmetical complexity of sets of tautologies and of satisfiable formulas. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Past–future separation and normal forms in temporal predicate logic specifications

Temporal specifications are often used when phenomena are modelled where dynamics play a main role. If simulation is one of the aims of modelling, usually a restricted, executable modelling language format is used, based on some form of past to future implications. In this paper a detailed transformation procedure is described that takes any temporal predicate logic specification and generates a specification in a past-implies-future normal format. The procedure works for temporal specifications in which the atoms either express time ordering relations or are state-related, i.e., include only one time variable.     

Tracking control of high-performance robots via stabilizing controllers for uncertain systems

The development of flexible manufacturing systems calls for industrial robots characterized by robustness of performance with regard to the variations of the loads and real time specification of the trajectory in the work space. In this paper, the design of a feedback controller guaranteeing such performance is considered. At first, the manipulator dynamics are embedded into a larger class of uncertain dynamical systems and a class of feedback controls is proposed that guarantees uniform ultimate boundedness of the tracking error. Successively, the methodology is specialized for the case of robotic manipulators to track trajectories described in task-oriented coordinates; the proposed control algorithm operates without requiring any explicit coordinate transformation.     

Double-Pullback Transitions and Coalgebraic Loose Semantics for Graph Transformation Systems

The classical algebraic approach to graph transformation is a mathematical theory based on categorical techniques with several interesting applications in computer science. In this paper, a new semantics of graph transformation systems (in the algebraic, double-pushout (DPO) approach) is proposed in order to make them suitable for the specification of concurrent and reactive systems. Classically, a graph transformation system comes with a fixed behavioral interpretation. Firstly, all transformation steps are intended to be completely specified by the rules of the system, that is, there is an implicit frame condition: it is assumed that there is a complete control about the evolution of the system. Hence, the interaction between the system and its (possibly unknown) environment, which is essential in a reactive system, cannot be modeled explicitly. Secondly, each sequence of transformation steps represents a legal computation of the system, and this makes it difficult to model systems with control. The first issue is addressed by providing graph transformation rules with a loose semantics, allowing for unspecified effects which are interpreted as activities of the environment. This is formalized by the notion of double-pullback transitions, which replace (and generalize) the well-known double-pushout diagrams by allowing for spontaneous changes in the context of a rule application. Two characterizations of double-pullback transitions are provided: the first one describes them in terms of extended direct DPO derivations, and the second one as incomplete views of parallel or amalgamated derivations. The issue of constraining the behavior of a system to transformation sequences satisfying certain properties is addressed instead by introducing a general notion of logic of behavioral constraints, which includes instances like start graphs, application and consistency conditions, and temporal logic constraints. The loose semantics of a system with restricted behavior is defined as a category of coalgebras over a suitable functor. Such category has a final object which includes all finite and infinite transition sequences satisfying the constraints.     

A two-dimensional metric temporal logic

We introduce a two-dimensional metric (interval) temporal logic whose internal and external time flows are dense linear orderings. We provide a suitable semantics and a sequent calculus with axioms for equality and extralogical axioms. Then we prove completeness and a semantic partial cut elimination theorem down to formulas of a certain type.     

Infinitary S5-Epistemic Logic

It is known that a theory in S5-epistemic logic with several agents may have numerous models. This is because each such model specifies also what an agent knows about infinite intersections of events, while the expressive power of the logic is limited to finite conjunctions of formulas. We show that this asymmetry between syntax and semantics persists also when infinite conjunctions (up to some given cardinality) are permitted in the language. We develop a strengthened S5-axiomatic system for such infinitary logics, and prove a strong completeness theorem for them. Then we show that in every such logic there is always a theory with more than one model.     

Tableau method for residuated logic

Residuated logic is a generalization of intuitionistic logic, which does not assume the idempotence of the conjunction operator. Such generalized conjunction operators have proved important in expert systems (in the area of Approximate Reasoning) and in some areas of Theoretical Computer Science. Here we generalize the intuitionistic tableau procedure and prove that this generalized tableau method is sound for the semantics (the class of residuated algebras) of residuated propositional calculus (RPC). Since the axioms of RPC are complete for the semantics we may conclude that whenever a formula 0 is tableau provable, it is deducible in RPC. We present two different approaches for constructing residuated algebras which give us countermodels for some formulas φ which are not tableau provable. The first uses the fact that the theory of residuated algebras is equational, to construct quotients of free algebras. The second uses finite algebras. We end by discussing a number of open questions.     

On State Feedback Stabilization of Singular Systems with Random Abrupt Changes

This paper deals with the class of continuous-time singular linear systems with random abrupt changes. The state feedback stabilization and its robustness for this class of systems with norm-bounded uncertainties are tackled. Sufficient conditions for designing either a stabilizing controller or a robust stabilizing controller are developed in the LMI setting. The developed sufficient conditions are used to synthesize the state feedback controller that guarantees that either the nominal system or the uncertain system is piecewise regular, impulse free and stochastically stable or robust stochastically stable. The research of this author was supported by NSERC, Grant RGPIN36444-02.     

Randomized methods based on new Monte Carlo schemes for control and optimization

We address randomized methods for control and optimization based on generating points uniformly distributed in a set. For control systems this sets are either stability domain in the space of feedback controllers, or quadratic stability domain, or robust stability domain, or level set for a performance specification. By generating random points in the prescribed set one can optimize some additional performance index. To implement such approach we exploit two modern Monte Carlo schemes for generating points which are approximately uniformly distributed in a given convex set. Both methods use boundary oracle to find an intersection of a ray and the set. The first method is Hit-and-Run, the second is sometimes called Shake-and-Bake. We estimate the rate of convergence for such methods and demonstrate the link with the center of gravity method. Numerical simulation results look very promising.     

不确定非线性系统的鲁棒自适应控制器

在ｂａｃｋｓｔｅｐｐｉｎｇ程序中，把非线性自适应控制和鲁棒控制连接起来，为参数化的严格反馈系统在不确定性存在的情况下，建立了一种鲁棒自适应控制方案．非线性自适应控制被用来处理系统的线性参数化部分，而鲁棒控制通过引进非线性阻尼项被用来处理不确定性部分．与现有的方案不同，作者给出了非线性阻尼项的无限种选择，而不是仅仅一种选择．通过使用一种合适的选择，能够设计一个鲁棒自适应控制器．它不仅能够保证对不确定性的鲁棒性，而且能够使输出误差任意小，以及用较小的控制努力取得较好的性能．     

Deterministic and stochastic approaches to supervisory control design for networked systems with time-varying communication delays

This paper proposes a supervisory control structure for networked systems with time-varying delays. The control structure, in which a supervisor triggers the most appropriate controller from a multi-controller unit, aims at improving the closed-loop performance relative to what can be obtained using a single robust controller. Our analysis considers average dwell-time switching and is based on a novel multiple Lyapunov–Krasovskii functional. We develop stability conditions that can be verified by semi-definite programming, and show that the associated state feedback synthesis problem also can be solved using convex optimization tools. Extensions of the analysis and synthesis procedures to the case when the evolution of the delay mode is described by a Markov chain are also developed. Simulations on small and large-scale networked control systems are used to illustrate the effectiveness of our approach.     

Applications of Category Theory to the Area of Algebraic Specification in Computer Science

The theory of algebraic specifications – one of the most important mathematical approaches to the specification of abstract data types and software systems – is reviewed from a mathematical and a computer science point of view. The important role of category theory in this area is discussed and it is shown how the following selected problems are treated using category theory: First, a unified framework for specification logics, second compositional semantics, third partial algebras and their specification, and fourth specifications and models for concurrent systems. For the solution of two of the problems classifying categories are used. They allow to present categories of algebras as functor categories and to derive a number of important properties from well known results for functor categories.     

Probabilization of Logics: Completeness and Decidability

The probabilization of a logic system consists of enriching the language (the formulas) and the semantics (the models) with probabilistic features. Such an operation is said to be exogenous if the enrichment is done on top, without internal changes to the structure, and is called endogenous otherwise. These two different enrichments can be applied simultaneously to the language and semantics of a same logic. We address the problem of studying the transference of metaproperties, such as completeness and decidability, to the exogenous probabilization of an abstract logic system. First, we setup the necessary framework to handle the probabilization of a satisfaction system by proving transference results within a more general context. In this setup, we define a combination mechanism of logics through morphisms and prove sufficient condition to guarantee completeness and decidability. Then, we demonstrate that probabilization is a special case of this exogenous combination method, and that it fulfills the general conditions to obtain transference of completeness and decidability. Finally, we motivate the applicability of our technique by analyzing the probabilization of the linear temporal logic over Markov chains, which constitutes an endogenous probabilization. The results are obtained first by studying the exogenous semantics, and then by establishing an equivalence with the original probabilization given by Markov chains.     

Decentralized robust control for a class of uncertain large-scale interconnected nonlinear dynamical systems

The problem of the decentralized robust control for a class of large-scale interconnected nonlinear dynamical systems with input interconnection and external interconnection perturbations is considered. Based on the stabilizability of each nominal isolated subsystem (i.e., the isolated subsystem in the absence of interconnection perturbations), a class of decentralized local state feedback controllers is proposed, and some sufficient conditions are derived by making use of the Lyapunov stability criterion such that uncertain large-scale interconnected systems can be stabilized asymptotically by these decentralized state feedback controllers. For large-scale systems with only input interconnection perturbations, such decentralized controllers become a class of decentralized stabilizing state feedback controllers. That is, the decentralized stability of such large-scale systems can be guaranteed always by using the decentralized state feedback controllers proposed in the paper. Finally, a numerical example is given to demonstrate the validity of the results.     

Adaptive testing for specification coverage and refinement in CPS models

Design of correct cyber–physical systems (CPS) is of uttermost importance for safety-critical applications. This crucial yet extremely challenging property is often addressed in practice by simulation-based methods. The simulation activity can be made more systematic and rigorous by using formal specifications to express requirements and guide the testing of the system.In this paper, we develop a procedure for generating tests from formal specifications given in Signal Temporal Logic (STL), a declarative language used to express CPS requirements. The proposed test generation method is adaptive with the aim at achieving specification coverage. We devise to this goal cooperative reachability games, which we enhance with numerical optimization to facilitate exercising various parts of specifications. The resulting approach is effective in finding specification violations, but also in increasing confidence (via coverage) that the specification is satisfied. In the latter case, we also propose a method for automatically refining the specification into its part that is actually implemented, thus gaining additional insight into the system-under-test.     

Model checking MASL specification of distributed real-time systems

We present model checking algorithms for MASL specification of distributed real-time systems. The proposed algorithms use symbolic model checking approach by analogy with model checking algorithms for branching-time temporal logic CTL and alternating-time temporal logic ATL. For the fixed environment case, the algorithm is polynomial-time in the specification length and sizes of the sets of system states and actions. For the dynamic environment case, the algorithm is polynomial-time in the model size, but it is exponential-time in the structure of environment specification.     

Unifying computers and dynamical systems using the theory of synchronous concurrent algorithms

A synchronous concurrent algorithm (SCA) is a parallel deterministic algorithm based on a network of modules and channels, computing and communicating data in parallel, and synchronised by a global clock with discrete time. Many types of algorithms, computer architectures, and mathematical models of physical and biological systems are examples of SCAs. For example, conventional digital hardware is made from components that are SCAs and many computational models possess the essential features of SCAs, including systolic arrays, neural networks, cellular automata and coupled map lattices.In this paper we formalise the general concept of an SCA equipped with a global clock in order to analyse precisely (i) specifications of their spatio-temporal behaviour; and (ii) the senses in which the algorithms are correct. We start the mathematical study of SCA computation, specification and correctness using methods based on computation on many-sorted topological algebras and equational logic. We show that specifications can be given equationally and, hence, that the correctness of SCAs can be reduced to the validity of equations in certain computable algebras. Since the idea of an SCA is general, our methods and results apply to each of the particular classes of algorithms and dynamical systems above.