Qualitative Action Systems

This study presents the implementation of a hybrid control strategy that is applied to a brushless AC (BLAC) motor drive. Hybrid control is a general approach for control of switching-based hybrid systems (HS). This class of HS includes a continuous process, controlled by a discrete controller with a finite number of states. The overall stability of the system is shown with the use of Lyapunov technique. The Lyapunov functions contain a term that penalises incremental energy of control error, torque and stator current, which enhances the stability. The closed-loop system, with the proposed control law, provides good transient response and good regulation of the BLAC motor control. A new logical field programmable gate array current (torque) controller is developed, based on the Lyapunov theory. The reference tracking performance of speed and torque (current) is demonstrated in terms of transient characteristics through simulation and experimental results.

The article gives the analysis of optimizing the digital controller of the automatic control system (ACS) of a continuous object. The procedure for replacing a continuous controller with a discrete controller has been analyzed. The original continuous optimized ACS has been illustrated. It has been stated that the continuous ACS and its elements (all or particular) can be modeled by discrete models described by the difference equations, and not the differential ones, as is done in the case of continuous models. The specification of the sampling period of the PI controller model equal to 0.001 s has been given, the coefficients of the Z-transfer function of the digital PI controller, which is equivalent to a continuous controller, are considered. Of practical importance in the control theory is the representation of a discrete model of a controller, which helps obtain a digital transformation algorithm carried out by a controller over a control error, organize a discrete-digital control of a continuous object using a digital processor. The model with a continuous and discrete controller is illustrated. A formula for calculating the values of the lattice function of the output signal of a discrete controller is obtained. It has been found that sampling and quantization of the error signal, as well as restoring the control signal, should be carried out with the proper accuracy, which should provide the required control accuracy. There are considered the models of discrete-digital control with the correct selection of the parameters of discrete PI controllers, a model with a finite time of the transient process and a structure of its contour, given a fifth-order discrete function with a five-step transition to the steady state. It has been inferred that reducing the sampling period and quantization step significantly increases the processor speed requirements.

Optimization problems that involve discrete variables are exposed to the conflict between being a powerful modeling tool and often being hard to solve. Infinite-dimensional processes, as e.g. described by differential equations, underlying the optimization may lead to the need to solve for distributed discrete control variables. This work analyzes approximation arguments that replace the need for solving the optimization problem by the need for first solving a relaxation and second computing appropriate roundings to regain discrete controls. We provide sufficient conditions on rounding algorithms and their grid refinement strategies that allow to prove approximation of the relaxed controls by the discrete controls in weaker topologies, a feature due to the infinite-dimensional vantage point. If the control-to-state mapping of the underlying process exhibits suitable compactness properties, state vector approximation follows in the norm topology as well as, under additional assumptions, optimality principles of the computed discrete controls. The conditions are verified for representatives of the family of Sum-Up Rounding algorithms. We apply the arguments on different classes of mixed-integer optimization problems that are constrained by partial differential equations. Specifically, we consider discrete control inputs, which are distributed in the time domain, for evolution equations that are governed by a differential operator that generates a strongly continuous semigroup, discrete control inputs, which are distributed in multi-dimensional spatial domains, for elliptic boundary value problems and discrete control inputs, which are distributed in space-time cylinders, for evolution equations that are governed by differential operators such that the corresponding Cauchy problem satisfies maximal parabolic regularity. Furthermore, we apply the arguments outside the scope of partial differential equations to a signal reconstruction problem. Computational results illustrate the findings.

A third-order nonlinear differential equation with two sets of boundary conditions is considered. These boundary value problems arise as boundary layer problems from a model of large scale ocean circulation. Using geometrical techniques from qualitative differential equations, such as Wazewski’s theorem, invariant manifolds, and Lyapunov functions, the existence of solutions for each boundary value problem is given in a uniform way for all positive values of a parameter of the differential equation.

Qualitative differential equations (QDE) are a popular formalism for modelling dynamic systems qualitatively. This paper briefly introduces the basic concepts of this approach, including types of constraints used in QDE and the QSIM simulation algorithm. It also discusses the relation of QSIM to other approaches to qualitative modelling, such as confluences and Qualitative Physics Theory, and the advantages and limitations of QSIM. Selected annotated bibliography is also given.KeywordsQualitative BehaviourInductive LogicQualitative ModelInductive Logic ProgrammingQualitative ReasoningThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Empirical support for the laws of software evolution cannot be rigorously tested as long as they are stated in natural language [1, 2]. Let’s consider the laws that directly relate to measurable phenomena (law 1 to 7) the eighth law “feedback system” is for the moment excluded of this discussion. Let’s assume that sufficient empirical data is available. One approach that has been used to assess the empirical support for the first seven laws uses quantitative models that are consistent with the laws. One fits such models to the empirical data and assesses the magnitude of the residuals. If these are small, the law will be supported by the data. If not, the law will remain unsupported. To illustrate this approach, let’s consider the 6th law “continuing growth”, which states that “the functional capability of E-type systems must be continually increased to maintain user satisfaction over the system lifetime”. Possible formal models of the law include the following: S(i) = a.i+b Eq. 1 S(i) = S(i-1)+e/[S(i-1)] Eq. 2 S(t) = t Eq. 3 where S represents the size of a system measured, for example, in number of modules, t is the age of the system, and i is the release sequence number. a, b, g, and e are parameters suitably chosen to minimise the fitting error. Eqs. 1-3 provide models of system growth. In fact, Eq. 1 replicates the linear growth pattern of OS 360 (rsn 1-19). Eqs. 2 and 3 replicate within segments the growth of systems studied in FEAST [3, 4]. However, Eqs. 1-3 imply additional detail than that in the statement of the 6 law. Each model prescribes a particular type of growth trajectory. The law prescribes none. For this reason none of the models in isolation will be sufficient if used with the sole purpose of testing the 6th law. Data suggesting a trajectory, different than that of Eqs. 1-3, as for example in [5], is still supportive of the law. Models such as those of Eqs. 1-3 in isolation are not proper tests. A workaround to this problem uses a formalism that considers the phenomenon at a higher level of abstraction than that implied by conventional quantitative modelling, such as qualitative differential equations (QDE) [6]. For example, the 6th law can be expressed in QDE form as: dS/dt = M(t) Eq. 4 M represents any monotonically increasing function. A simple test based on Eq. 4 evaluates S(t) – S(t-1) over the whole data set and checks whether they are greater than zero. If so, the data supports the law. Since a superimposed ripple have been observed in growth patterns [1], data smoothing techniques may have to be applied before the test. However, the fuller potential of the qualitative simulation techniques becomes apparent if one considers that the laws are interrelated. The combined action of the forces implied by the laws may lead to more complex behaviours than monotonic increase or decrease. In order to address this, one can write several QDEs formalising several “measurable” laws. The resulting system of QDEs can be simulated [6, 7] and the outputs compared with that implied by empirical observations. This qualitative simulation approach accounts in a natural way for the fact that the laws are interrelated. Eq. 2 and 3 illustrate interrelationship to some extent. They imply positive but decreasing growth rate and in doing so are consistent with both the 2nd law “increasing complexity” and the 6th law. In applying the conventional quantitative approach, one requires the ingenuity of the model builders for identification of models that acknowledge more than one law. This is not simple. The use of qualitative simulation techniques, a topic for further research [7], are likely to facilitate assessment of joint support in a systematic way and at a higher level of abstraction. The eighth law has been justified based on several observations [1, 2]. Unfortunately, a direct rigorous test of its empirical support is difficult to achieve since the behaviours of even simple feedback systems have not been fully codified [8]. The first seven laws can be rationalised in terms of feedback loop structures in the total E-type evolution process [2]. Hence, positive support to any of the first seven laws provides a degree of support to the eighth law.

Practical stability of nonlinear stochastic hybrid parabolic systems of Itô-type: Vector Lyapunov functions approach

The Arduino Due is a microcontroller board based on the Atmel SAM3X8E ARM Cortex-M3 CPU and we can use it to make the speed control and simulation of a PMSM through the Bulirsch-Stoer implementation. The power system that we want to research on contains three parts: an inverter, an electrical machine and a controller. Firstly, we use the classical Electric Machine Models of PMSM, then we present it as a set of Ordinary Differential Equation. Secondly, we solve the se equations with the Bulirsch-Stoer algorithm and compare it with the results of Matlab. Finally, we design a suitable control algorithm which contains speed PI controller and current PI controller for the system. The simulation has to deal with a hybrid system, described on the one hand by continuous differential equations solving the model of the electrical machine, and on the other hand by a discrete controller. The continuous part of the system is described by state-space equations that can be solved by the Bulirsch-Stoer algorithm.

Hybrid system is a dynamic system that involves continuous, discrete behaviors, and the interactions between continuous physical components and discrete controllers. In this paper a hybrid modeling language (called HML) for hybrid systems is extended with templates to achieve code reuse. For the formal analysis of the corresponding hybrid system models in this modeling language, these models are translated into SMT (satisfiability modulo theories) formulas as the input to an SMT solver dReal which retains the capability of bounded reachability analysis for non-linear hybrid systems. Moreover, dReal can produce data for potential traces of hybrid systems, thus it can be employed to simulate on hybrid systems. In this paper the simulation and reachability analysis are integrated in a prototype tool (open source). We present a case study for an inverted pendulum with PID (Proportional-Integral-Derivative) controllers and a rod reactor system for temperature control, both are verified to demonstrate the efficiency of the prototype tool. We conclude that, this modeling language is capable of modeling and verification of hybrid systems based on simulation and bounded reachability analysis.

Hybrid systems are characterized by a composition of discrete and continuous dynamics.In particular, the system has a continuous evolution and occasional jumps.The jumps are caused either by controllable, uncontrollable external events or by its continuous evolution.The continuous evolution and these jumps in control loops are the origins from the most stringent real-time demands.With the necessity to launch more satellites, Brazilian National Institute for Space Research (INPE) has been carrying out research on modeling and verifying hybrid systems, of which its main focus is to obtain a better balance between dependability, schedule, and cost.We are attempting to use Object Management Group (OMG) specifications to model discrete events.We are focusing mainly in Modelica (with some degree of Scicoslab) to model continuous dynamics.Another concern addressed by this, INPE, research is to be independent from commercial tools, establishing itself on open source software.This paper presents an approach to implement Model-Driven Architecture in hybrid systems based on vendor neutral specifications.It shows how the models are defined, traced and used, as well as a set of tools for this.SysML (Systems Modeling Language), and MARTE (Modeling and Analysis of Real-Time Embedded Systems) allowed us to define a Computation Independent Model focused mainly on high-level structure and behavior (state oriented).At the end, a case study is presented (inverted pendulum).From this case study, we have concluded that the proposed approach can complement uncovered topics in current research applied to hybrid systems development and maintenance.

Kuipers' QSIM algorithm for tracking the monotonicity properties of solutions to differential equations has been revisited by Dordan by placing it in a rigorous mathematical framework. The Dordan QSIM algorithm provides the transition laws from one qualitative cell to the others. We take up this idea and revisit it at the light of recent advances in the field of "hybrid systems" and, more generally, "impulse differential equations and inclusions". Let us consider a family of "qualitative cells Q(α)" indexed by a parameter a ϵ A: We introduce a dynamical system on the discrete set of qualitative states prescribing an order of visit of the qualitative cells and an evolutionary system govening the "continuous" evolution of a system, such as a control system. The question arises to study and characterize the set of any pairs of qualitative and quantitative initial states from which start at least one order of visit of the qualitative cells and an continuous evolution visiting the qualitative cells in the prescribed order. This paper is devoted to the issues regarding this question using tools of set-valued analysis and viability theory.

In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between DNA, RNA, proteins, and small molecules. As most genetic regulatory networks of interest involve many components connected through interlocking positive and negative feedback loops, an intuitive understanding of their dynamics is hard to obtain. As a consequence, formal methods and computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This paper reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, Boolean networks and their generalizations, ordinary and partial differential equations, qualitative differential equations, stochastic equations, and rule-based formalisms. In addition, the paper discusses how these formalisms have been used in the simulation of the behavior of actual regulatory systems.

Modern control systems are hybrid – continuous or mixed objects are controlled by discrete controllers. Colored Petri nets are theexcellent mean of simulation of control algorithms. Continuous objects can be described by differential equations. These equations canbe represented as a structural scheme. An interface between those to parts of a model is very important as well as evaluation of the time,because it can be one of the factors of control. The simulation system realized in the program package CENTAURUS-C is presented.Interface between both parts of a model is realized via specialized transition, in Petri net, named – process. Three default real variables– $Out, $In and $Prev are used in the interface. A process can carry the value of $Out into a structure of an object as a step signal andreceive signal from any output of blocks in structure as a value of the variable $In. Value of variable $Prev is a value of $Inremembered on previous step of simulation. Variable $Out can be used in an expression of input chord of a process and init the process.Variables $In and $Prev can be used in guard expression of a process to set the end of a process – firing of the specialized transition.Time in the presented system is an attribute of processes and transitions – delay. The delays of transitions are represented by specialexpressions and can be changed dynamically. Ill. 7, bibl. 11 (in Lithuanian; summaries in Lithuanian, English, Russian).Modern control systems are hybrid – continuous or mixed objects are controlled by discrete controllers. Colored Petri nets are theexcellent mean of simulation of control algorithms. Continuous objects can be described by differential equations. These equations canbe represented as a structural scheme. An interface between those to parts of a model is very important as well as evaluation of the time,because it can be one of the factors of control. The simulation system realized in the program package CENTAURUS-C is presented.Interface between both parts of a model is realized via specialized transition, in Petri net, named – process. Three default real variables– $Out, $In and $Prev are used in the interface. A process can carry the value of $Out into a structure of an object as a step signal andreceive signal from any output of blocks in structure as a value of the variable $In. Value of variable $Prev is a value of $Inremembered on previous step of simulation. Variable $Out can be used in an expression of input chord of a process and init the process.Variables $In and $Prev can be used in guard expression of a process to set the end of a process – firing of the specialized transition.Time in the presented system is an attribute of processes and transitions – delay. The delays of transitions are represented by specialexpressions and can be changed dynamically. Ill. 7, bibl. 11 (in Lithuanian; summaries in Lithuanian, English, Russian).

We introduce a class of MPDs which greatly simplify Reinforcement Learning. They have discrete state spaces and continuous control spaces. The controls have the effect of rescaling the transition probabilities of an underlying Markov chain. A control cost penalizing KL divergence between controlled and uncontrolled transition probabilities makes the minimization problem convex, and allows analytical computation of the optimal controls given the optimal value function. An exponential transformation of the optimal value function makes the minimized Bellman equation linear. Apart from their theoretical significance, the new MDPs enable efficient approximations to traditional MDPs. Shortest path problems are approximated to arbitrary precision with largest eigenvalue problems, yielding an O (n) algorithm. Accurate approximations to generic MDPs are obtained via continuous embedding reminiscent of LP relaxation in integer programming. Off-policy learning of the optimal value function is possible without need for state-action values; the new algorithm (Z-learning) outperforms Q-learning.

In this paper, we consider a class of stochastic resource allocation problems where resources assigned to a task may fail probabilistically to complete assigned tasks. Failures to complete a task are observed before new resource allocations are selected. The resulting temporal resource allocation problem is a stochastic control problem, with a discrete state space and control space that grow in cardinality exponentially with the number of tasks. We modify this optimal control problem by expanding the admissible control space, and show that the resulting control problem can be solved exactly by efficient algorithms in time that grows nearly linear with the number of tasks. The approximate control problem also provides a bound on the achievable performance for the original control problem. The approximation is used as part of a model predictive control (MPC) algorithm to generate resource allocations over time in response to information on task completion status. We show in computational experiments that, for single resource class problems, the resulting MPC algorithm achieves nearly the same performance as the optimal dynamic programming algorithm while reducing computation time by over four orders of magnitude. In multiple resource class experiments involving 1000 tasks, the model predictive control performance is within 4% of the performance bound obtained by the solution of the expanded control space problem.