An algebraic approach to supervisory control

Supervisory control problems are formulated in terms of a process model where the mechanism of control is expressed in terms of an algebraic operator with the plant and supervision processes as its arguments. The solution subspaces for supervisory processes restrict the observation and the control capability of supervision. The main result corresponds to decentralized marked supervision under partial observations, and specific cases are derived from this result in a unified, algebraic way. The result and its derivation demonstrate the relative simplicity of the algebraic process formulation.     

An algebraic approach to iterative learning control

In this paper discrete-time iterative learning control (ILC) systems are analysed from an algebraic point of view. The algebraic analysis shows that a linear-time invariant single-input–single-output model can always represented equivalently as a static multivariable plant due to the finiteness of the time-axis. Furthermore, in this framework the ILC synthesis problem becomes a tracking problem of a multi-channel step-function. The internal model principle states that for asymptotic tracking (i.e. convergent learning) it is required that an ILC algorithm has to contain an integrator along the iteration axis, but at the same time the resulting closed-loop system should be stable. The question of stability can then be answered by analysing the closed-loop poles along the iteration axis using standard results from multivariable polynomial systems theory. This convergence theory suggests that time-varying ILC control laws should be typically used instead of time-invariant control laws in order to guarantee good transient tracking behaviour. Based on this suggestion a new adaptive ILC algorithm is derived, which results in monotonic convergence for an arbitrary linear discrete-time plant. This adaptive algorithm also has important implications in terms of future research work—as a concrete example it demonstrates that ILC algorithms containing adaptive and time-varying components can result in enhanced convergence properties when compared to fixed parameter ILC algorithms. Hence it can be expected that further research on adaptive learning mechanisms will provide a new useful source of high-performance ILC algorithms.     

Alternative algebraic approach to stabilization for linear parabolic boundary control systems

In this paper, we develop an alternative algebraic approach to feedback stabilization problems of linear parabolic boundary control systems. The control system is general in the sense that no Riesz basis associated with the elliptic operator is expected. The control system contains a finite dimensional dynamic compensator, the dimension of which has been so far determined only by the actuators on the boundary. One of the purposes of the paper is to show a new alternative scheme that the dimension of the compensator could be determined by the sensors on the boundary: it leads to a realization of stabilizing compensators of lower dimension by comparison. This is achieved by introducing a new operator equation connecting two states of the controlled plant and the compensator. The other purpose is a generalization of the condition posed on the sensors. When the elliptic operator admits generalized eigenfunctions, the sharpest criterion is proposed on the minimum choice of the necessary number of the sensors. In fact, only one sensor is enough for the stabilization in the best case (the algebraic multiplicities of the corresponding eigenvalues may, of course, be greater than 1).     

ADMiRe: an algebraic data mining approach to system performance analysis

Performance analysis of computing systems is an increasingly difficult task due to growing system complexity. Traditional tools rely on ad hoc procedures. With these, determining which of the manifold system and workload parameters to examine is often a lengthy and highly speculative process. The analysis is often incomplete and, therefore, prone to revealing faulty conclusions and not uncovering useful tuning knowledge. We address this problem by introducing a data mining approach called ADMiRe (analyzer for data mining results). In this scheme, regression analysis is first applied to performance data to discover correlations between various system and workload parameters. The results of this analysis are summarized in sets of regression rules. The user can then formulate intuitive algebraic expressions to manipulate these sets of rules to capture critical information. To demonstrate this approach, we use ADMiRe to analyze an Oracle database system running the TPC-C (Transaction Processing Performance Council) benchmark. The results generated by ADMiRe were confirmed by Oracle experts. We also show that by applying ADMiRe to Microsoft Internet Information Server performance data, we can improve system performance by 20 percent.     

Mechanisms for naming An algebraic approach with an application to Java

The present paper investigates the hypothesis that a variety of mechanisms for naming can be understood as algebraic concepts. These concepts are developed and then they are applied to aspects of Java to see whether indeed they lead to compact characterizations of the language's mechanisms for naming. Focus is on object oriented themes: inheritance, polymorphism and encapsulation.     

The matrix partial realization problem: an algebraic approach

This paper introduces a new approach to the multivariable partial realization problem. It formulates the problem in algebraic terminology, which sheds new light on the nature of the problem and our existing knowledge of it. The structure of the state space is analysed in terms of polynomial models. The main idea is analysis of the algebraic structure of the kernels of truncated Hankel maps. The elements of such kernels are directly related to the partial realization problem, in the sense that the columns of the denominator matrix of a partial realization are elements of such kernels. The numerator matrix is determined by an appropriate denominator matrix     

The exact model-matching problem for linear time-varying systems: an algebraic approach

The exact model-matching problem is formulated and solved for linear time-varying systems. The condition for the existence of a proper solution, which is well known in the time-invariant case, is proven here to still be valid in the time-varying case. The properness is characterized using the Smith-MacMillan form at infinity, recently defined by the authors for the transfer matrices with time-varying coefficients.     

Towards an algebraic theory of information integration

Information integration systems provide uniform interfaces to varieties of heterogeneous information sources. For query answering in such systems, the current generation of query answering algorithms in local-as-view (source-centric) information integration systems all produce what has been thought of as “the best obtainable” answer, given the circumstances that the source-centric approach introduces incomplete information into the virtual global relations. However, this “best obtainable” answer does not include all information that can be extracted from the sources because it does not allow partial information. Neither does the “best obtainable” answer allow for composition of queries, meaning that querying a result of a previous query will not be equivalentto the composition of the two queries. In this paper, we provide a foundation for information integration, based on the algebraic theory of incomplete information. Our framework allows us to define the semantics of partial facts and introduce the notion of the exact answer—that is the answer that includes partial facts. We show that querying under the exact answer semantics is compositional. We also present two methods for actually computing the exact answer. The first method is tableau-based, and it is a generalization of the “inverse-rules” approach. The second, much more efficient method, is a generalization of the rewriting approach, and it is based on partial containment mappings introduced in the paper.     

Design of LPV observers for LPV time-delay systems: an algebraic approach

The design of reduced order observer for linear parameter varying (LPV) time-delay systems is addressed. Necessary conditions guaranteeing critical structural properties for the observation error dynamics are first provided through nonlinear algebraic matrix equalities. An explicit parametrisation of the family of observers fulfilling these necessary conditions is then derived. Finally, an approach based on linear matrix inequalities is provided and used to select a suitable observer within this family, according to some criterion; e.g. maximisation of the delay margin or guaranteed suboptimal ?₂-gain. Examples from the literature illustrate the efficiency of the approach.     

Attractor dynamics approach to formation control: theory and application

In this paper we show how non-linear attractor dynamics can be used as a framework to control teams of autonomous mobile robots that should navigate according to a predefined geometric formation. The environment does not need to be known a priori and may change over time. Implicit to the control architecture are some important features such as establishing and moving the formation, split and join of formations (when necessary to avoid obstacles). Formations are defined by a formation matrix. By manipulating this formation matrix it is also possible to switch formations at run time. Examples of simulation results and implementations with real robots (teams of Khepera robots and medium size mobile robots), demonstrate formation switch, static and dynamic obstacle avoidance and split and join formations without the need for any explicit coordination scheme. Robustness against environmental perturbations is intrinsically achieved because the behaviour of each robot is generated as a time series of asymptotically stable states, which contribute to the asymptotic stability of the overall control system.     

On a quantitative theory of robust adaptive control: an intervalplant approach

This paper considers a robust direct-model reference adaptive control scheme, where the components of the estimated controller parameter vector are constrained to vary in certain prespecified intervals. This interval constraint facilitates the quantitative analysis of the robustness of the closed-loop adaptive control system. Extending existing results from the area of robust parametric stability, a tractable procedure is developed for verifying a condition which guarantees the boundedness of all the closed-loop signals. The robustness verification using this procedure involves the calculation of worst-case “shifted” H_∞ norms over certain extremal one-parameter families. This makes the condition easily verifiable, whereas otherwise one is faced with the formidable task of determining the worst-case norms over higher-dimensional compact convex sets in the plant and controller parameter spaces. A simple example is used in the paper to illustrate the theoretical development     

Game theory approach to optimal control problem with multi-channel control

In this paper, we mainly study the optimal control problem with multi-channel control. The main contribution is to treat the optimal control problem as a leader-follower problem with multi-hierarchy decision makers in game theory. According to the sequence of the decision makers, the optimal controller is derived by the maximum principle. It is worth mentioning that the optimal solution is in the feedback form related to two symmetric Riccati equations with same dimension as the original state. Furthermore, the obtained solution is sufficient and necessary.     

Towards an algebraic theory of typed mobile processes

The impact of types on the algebraic theory of the π-calculus is studied. The type system has capability types. They allow one to distinguish between the ability to read from a channel, to write to a channel, and both to read and to write. They also give rise to a natural and powerful subtyping relation.

Two variants of typed bisimilarity are considered, both in their late and in their early version. For both of them, proof systems that are sound and complete on the closed finite terms are given. For one of the two variants, a complete axiomatisation for the open finite terms is also presented.     

An algebraic approach to shape-from-image problems

This paper presents a new method for recovering three-dimensional shapes of polyhedral objects from their single-view images. The problem of recovery is formulated in a constrained optimization problem, in which the constraints reflect the assumption that the scene is composed of polyhedral objects, and the objective function to be minimized is a weighted sum of quadric errors of surface information such as shading and texture. For practical purpose it is decomposed into the two more tractable problems: a linear programming problem and an unconstrained optimization problem. In the present method the global constraints placed by the polyhedron assumption are represented in terms of linear algebra, whereas similar constraints have usually been represented in terms of a gradient space. Moreover, superstrictness of the constraints can be circumvented by a new concept ‘position-free incidence structure’. For this reason the present method has several advantages: it can recover the polyhedral shape even if image data are incorrect due to vertex-position errors, it can deal with perspective projection as well as orthographic projection, the number of variables in the optimization problem is very small (three or a little greater than three), and any kinds of surface information can be incorporated in a unifying manner.     

Optimal control using an algebraic method for control-affine non-linear systems

A deterministic optimal control problem is solved for a control-affine non-linear system with a non-quadratic cost function. We algebraically solve the Hamilton–Jacobi equation for the gradient of the value function. This eliminates the need to explicitly solve the solution of a Hamilton–Jacobi partial differential equation. We interpret the value function in terms of the control Lyapunov function. Then we provide the stabilizing controller and the stability margins. Furthermore, we derive an optimal controller for a control-affine non-linear system using the state dependent Riccati equation (SDRE) method; this method gives a similar optimal controller as the controller from the algebraic method. We also find the optimal controller when the cost function is the exponential-of-integral case, which is known as risk-sensitive (RS) control. Finally, we show that SDRE and RS methods give equivalent optimal controllers for non-linear deterministic systems. Examples demonstrate the proposed methods.     

An algebraic approach to feature interactions

The various approaches proposed to provide communication between CAD systems and process planning systems share the major problem that, due to geometric interactions among features, there may be several equally valid sets of manufacturable features describing the same part, and different sets of features may differ in their manufacturability. Thus, to produce a good process plan-or, in some cases, any plan at ll-it may be necessary to interpret the part as a different set of features than the one initially obtained from the CAD model. This is addressed using an algebra of features. Given a set of features describing a machinable part, other equally valid interpretations of the part can be produced by performing operations in the algebra. This will enable automated process planning systems to examine these interpretations in order to see which one is most appropriate for use in manufacturing. The feature algebra has been implemented for a restricted domain and integrated with the Protosolid solid modeling system and the EFHA process planning system     

An algebraic approach to continuous collision detection for ellipsoids

We present algebraic expressions for characterizing three configurations formed by two ellipsoids in R³ that are relevant to collision detection: separation, external touching and overlapping. These conditions are given in terms of explicit formulae expressed by the subresultant sequence of the characteristic polynomial of the two ellipsoids and its derivative. For any two ellipsoids, the signs of these formulae can easily be evaluated to classify their configuration. Furthermore, based on these algebraic conditions, an efficient method is developed for continuous collision detection of two moving ellipsoids under arbitrary motions.     

Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems

Algebraic multilevel preconditioners for algebraic problems arising from the discretization of a class of systems of coupled elliptic partial differential equations (PDEs) are presented. These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary matrix corresponding to a single PDE. The preconditioners are implemented in a parallel computing framework and are tested on two representative PDE systems. The results of the numerical experiments show the effectiveness and the scalability of the proposed methods. A convergence theory for the twolevel case is presented.     

Two-mode control: an oculomotor-based approach to tracking systems

The paper aims to use the knowledge about how the visual system organizes the components of oculomotor system to propose a new tracking paradigm. The tracking system is assumed to be described by linear time-invariant discrete-time state-space equations. The approach described, motivated by the behavior of the visual system, is to switch off the smooth controller whenever a violation occurs and design a time-optimal control action, i.e., a “saccade”, to drive the control system so that the constraint is satisfied after the shortest possible time interval. After that, the smooth controller is switched back into the loop. The way this switching is performed is critical for obtaining “good behavior”. A method is proposed which is based on a careful definition of the target set for the saccade. The tracking system proposed in this paper is closely related to recent results in linear optimal and robust control theory