Modelling, control and supervision for a class of hybrid systems

The aim of this thesis is to narrow the gap between two different control techniques: the continuous control and the discrete event control techniques DES. This gap can be reduced by the study of Hybrid systems, and by interpreting as Hybrid systems the majority of large-scale systems. In particular, when looking deeply into a process, it is often possible to identify interaction between discrete and continuous signals. Hybrid systems are systems that have both continuous, and discrete signals. Continuous signals are generally supposed continuous and differentiable in time, since discrete signals are neither continuous nor differentiable in time due to their abrupt changes in time. Continuous signals often represent the measure of natural physical magnitudes such as temperature, pressure etc. The discrete signals are normally artificial signals, operated by human artefacts as current, voltage, light etc. Typical processes modelled as Hybrid systems are production systems, chemical process, or continuos production when time and continuous measures interacts with the transport, and stock inventory system. Complex systems as manufacturing lines are hybrid in a global sense. They can be decomposed into several subsystems, and their links. Another motivation for the study of Hybrid systems is the tools developed by other research domains. These tools benefit from the use of temporal logic for the analysis of several properties of Hybrid systems model, and use it to design systems and controllers, which satisfies physical or imposed restrictions. This thesis is focused in particular types of systems with discrete and continuous signals in interaction. That can be modelled hard non-linealities, such as hysteresis, jumps in the state, limit cycles, etc. and their possible non-deterministic future behaviour expressed by an interpretable model description. The Hybrid systems treated in this work are systems with several discrete states, always less than thirty states (it can arrive to NP hard problem), and continuous dynamics evolving with expression: with Ki ¡ Rn constant vectors or matrices for X components vector. In several states the continuous evolution can be several of them Ki = 0. In this formulation, the mathematics can express Time invariant linear system. By the use of this expression for a local part, the combination of several local linear models is possible to represent non-linear systems. And with the interaction with discrete events of the system the model can compose non-linear Hybrid systems. Especially multistage processes with high continuous dynamics are well represented by the proposed methodology. Sate vectors with more than two components, as third order models or higher is well approximated by the proposed approximation. Flexible belt transmission, chemical reactions with initial start-up and mobile robots with important friction are several physical systems, which profits from the benefits of proposed methodology (accuracy). The motivation of this thesis is to obtain a solution that can control and drive the Hybrid systems from the origin or starting point to the goal. How to obtain this solution, and which is the best solution in terms of one cost function subject to the physical restrictions and control actions is analysed. Hybrid systems that have several possible states, different ways to drive the system to the goal and different continuous control signals are problems that motivate this research. The requirements of the system on which we work is: a model that can represent the behaviour of the non-linear systems, and that possibilities the prediction of possible future behaviour for the model, in order to apply an supervisor which decides the optimal and secure action to drive the system toward the goal. Specific problems can be determined by the use of this kind of hybrid models are: - The unity of order. - Control the system along a reachable path. - Control the system in a safe path. - Optimise the cost function. - Modularity of control The proposed model solves the specified problems in the switching models problem, the initial condition calculus and the unity of the order models. Continuous and discrete phenomena are represented in Linear hybrid models, defined with defined eighth-tuple parameters to model different types of hybrid phenomena. Applying a transformation over the state vector : for LTI system we obtain from a two-dimensional SS a single parameter, alpha, which still maintains the dynamical information. Combining this parameter with the system output, a complete description of the system is obtained in a form of a graph in polar representation. Using Tagaki-Sugeno type III is a fuzzy model which include linear time invariant LTI models for each local model, the fuzzyfication of different LTI local model gives as a result a non-linear time invariant model. In our case the output and the alpha measure govern the membership function. Hybrid systems control is a huge task, the processes need to be guided from the Starting point to the desired End point, passing a through of different specific states and points in the trajectory. The system can be structured in different levels of abstraction and the control in three layers for the Hybrid systems from planning the process to produce the actions, these are the planning, the process and control layer. In this case the algorithms will be applied to robotics ¡V a domain where improvements are well accepted ¡V it is expected to find a simple repetitive processes for which the extra effort in complexity can be compensated by some cost reductions. It may be also interesting to implement some control optimisation to processes such as fuel injection, DC-DC converters etc. In order to apply the RW theory of discrete event systems on a Hybrid system, we must abstract the continuous signals and to project the events generated for these signals, to obtain new sets of observable and controllable events. Ramadge & Wonham¡¦s theory along with the TCT software give a Controllable Sublanguage of the legal language generated for a Discrete Event System (DES). Continuous abstraction transforms predicates over continuous variables into controllable or uncontrollable events, and modifies the set of uncontrollable, controllable observable and unobservable events. Continuous signals produce into the system virtual events, when this crosses the bound limits. If this event is deterministic, they can be projected. It is necessary to determine the controllability of this event, in order to assign this to the corresponding set, , controllable, uncontrollable, observable and unobservable set of events. Find optimal trajectories in order to minimise some cost function is the goal of the modelling procedure. Mathematical model for the system allows the user to apply mathematical techniques over this expression. These possibilities are, to minimise a specific cost function, to obtain optimal controllers and to approximate a specific trajectory. The combination of the Dynamic Programming with Bellman Principle of optimality, give us the procedure to solve the minimum time trajectory for Hybrid systems. The problem is greater when there exists interaction between adjacent states. In Hybrid systems the problem is to determine the partial set points to be applied at the local models. Optimal controller can be implemented in each local model in order to assure the minimisation of the local costs. The solution of this problem needs to give us the trajectory to follow the system. Trajectory marked by a set of set points to force the system to passing over them. Several ways are possible to drive the system from the Starting point Xi to the End point Xf. Different ways are interesting in: dynamic sense, minimum states, approximation at set points, etc. These ways need to be safe and viable and RchW. And only one of them must to be applied, normally the best, which minimises the proposed cost function. A Reachable Way, this means the controllable way and safe, will be evaluated in order to obtain which one minimises the cost function. Contribution of this work is a complete framework to work with the majority Hybrid systems, the procedures to model, control and supervise are defined and explained and its use is demonstrated. Also explained is the procedure to model the systems to be analysed for automatic verification. Great improvements were obtained by using this methodology in comparison to using other piecewise linear approximations. It is demonstrated in particular cases this methodology can provide best approximation. The most important contribution of this work, is the Alpha approximation for non-linear systems with high dynamics While this kind of process is not typical, but in this case the Alpha approximation is the best linear approximation to use, and give a compact representation.

A new approach for the solution of the optimal set-point tracking problem for nonlinear systems

In this paper, a discrete time dynamic integrated system optimisation and parameter estimation algorithm is applied to the solution of the nonlinear tracking optimal control problem. A version of the algorithm with a linear-quadratic model-based problem is developed and implemented in software. The algorithm implemented is tested with simulation examples.

Model updating of the non-linear vibrating structures through Volterra series and proper orthogonal decomposition

Classical procedures for model updating in non-linear mechanical systems based on vibration data can fail because the common linear metrics are not sensitive for non-linear behavior caused by gaps, backlash, bolts, joints, materials, etc. Several strategies were proposed in the literature in order to allow a correct representative model of non-linear structures. The present paper evaluates the performance of two approaches based on different objective functions. The first one is a time domain methodology based on the proper orthogonal decomposition constructed from the output time histories. The second approach uses objective functions with multiples convolutions described by the first and second order discrete-time Volterra kernels. In order to discuss the results, a benchmark of a clamped-clamped beam with an pre-applied static load is simulated and updated using proper orthogonal decomposition and Volterra Series. The comparisons and discussions of the results show the practical applicability and drawbacks of both approaches.

Setting Inventory Levels of CONWIP Flow Lines via Linear Programming

This paper treats the problem of setting the inventory level and optimizing the buffer allocation of closed-loop flow lines operating under the constant-work-in-process (CONWIP) protocol. We solve a very large but simple linear program that models an entire simulation run of a closed-loop flow line in discrete time to determine a production rate estimate of the system. This approach introduced in Helber, Schimmelpfeng, Stolletz, and Lagershausen (2011) for open flow lines with limited buffer capacities is extended to closed-loop CONWIP flow lines. Via this method, both the CONWIP level and the buffer allocation can be optimized simultaneously. The first part of a numerical study deals with the accuracy of the method. In the second part, we focus on the relationship between the CONWIP inventory level and the short-term profit. The accuracy of the method turns out to be best for such configurations that maximize production rate and/or short-term profit.

Stable Model Predictive Control for Integrating Systems with Optimizing Targets

This paper concern the development of a stable model predictive controller (MPC) to be integrated with real time optimization (RTO) in the control structure of a process system with stable and integrating outputs. The real time process optimizer produces Optimal targets for the system inputs and for Outputs that Should be dynamically implemented by the MPC controller. This paper is based oil a previous work (Comput. Chem. Eng. 2005, 29, 1089) where a nominally stable MPC was proposed for systems with the conventional control approach where only the outputs have set points. This work is also based oil the work of Gonzalez et at. (J. Process Control 2009, 19, 110) where the zone control of stable systems is studied. The new control for is obtained by defining ail extended control objective that includes input targets and zone controller the outputs. Additional decision variables are also defined to increase the set of feasible solutions to the control problem. The hard constraints resulting from the cancellation of the integrating modes Lit the end of the control horizon are softened,, and the resulting control problem is made feasible to a large class of unknown disturbances and changes of the optimizing targets. The methods are illustrated with the simulated application of the proposed,approaches to a distillation column of the oil refining industry.

Designing real-time systems based on mono-master Profibus-DP networks

Profibus networks are widely used as the communication infrastructure for supporting distributed computer-controlled applications. Most of the times, these applications impose strict real-time requirements. Profibus-DP has gradually become the preferred Profibus application profile. It is usually implemented as a mono-master Profibus network, and is optimised for speed and efficiency. The aim of this paper is to analyse the real-time behaviour of this class of Profibus networks. Importantly, we develop a new methodology for evaluating the worst-case message response time in systems where high-priority and cyclic low-priority Profibus traffic coexist. The proposed analysis constitutes a powerful tool to guarantee prior to runtime the real-time behaviour of a distributed computer-controlled system based on a Profibus network, where the realtime traffic is supported either by high-priority or by cyclic poll Profibus messages.

Optimal controllers with complex order derivatives

This paper studies the optimization of complex-order algorithms for the discrete-time control of linear and nonlinear systems. The fundamentals of fractional systems and genetic algorithms are introduced. Based on these concepts, complexorder control schemes and their implementation are evaluated in the perspective of evolutionary optimization. The results demonstrate not only that complex-order derivatives constitute a valuable alternative for deriving control algorithms, but also the feasibility of the adopted optimization strategy.

Evaluating Android OS for embedded real-time systems

Since its official public release, Android has captured the interest from companies, developers and the general audience. From that time up to now, this software platform has been constantly improved either in terms of features or supported hardware and, at the same time, extended to new types of devices different from the originally intended mobile ones. However, there is a feature that has not been explored yet - its real-time capabilities. This paper intends to explore this gap and provide a basis for discussion on the suitability of Android in order to be used in Open Real-Time environments. By analysing the software platform, with the main focus on the virtual machine and its underlying operating system environments, we are able to point out its current limitations and, therefore, provide a hint on different perspectives of directions in order to make Android suitable for these environments. It is our position that Android may provide a suitable architecture for real-time embedded systems, but the real-time community should address its limitations in a joint effort at all of the platform layers.

Handling QoS dependencies in distributed cooperative real-time systems

Due to the growing complexity and adaptability requirements of real-time embedded systems, which often exhibit unrestricted inter-dependencies among supported services and user-imposed quality constraints, it is increasingly difficult to optimise the level of service of a dynamic task set within an useful and bounded time. This is even more difficult when intending to benefit from the full potential of an open distributed cooperating environment, where service characteristics are not known beforehand. This paper proposes an iterative refinement approach for a service’s QoS configuration taking into account services’ inter-dependencies and quality constraints, and trading off the achieved solution’s quality for the cost of computation. Extensive simulations demonstrate that the proposed anytime algorithm is able to quickly find a good initial solution and effectively optimises the rate at which the quality of the current solution improves as the algorithm is given more time to run. The added benefits of the proposed approach clearly surpass its reducedoverhead.

Optimal tuning of fractional controllers using genetic algorithms

This study addresses the optimization of fractional algorithms for the discrete-time control of linear and non-linear systems. The paper starts by analyzing the fundamentals of fractional control systems and genetic algorithms. In a second phase the paper evaluates the problem in an optimization perspective. The results demonstrate the feasibility of the evolutionary strategy and the adaptability to distinct types of systems.

Periodic paths on nonautonomous graphs

We define nonautonomous graphs as a class of dynamic graphs in discrete time whose time-dependence consists in connecting or disconnecting edges. We study periodic paths in these graphs, and the associated zeta functions. Based on the analytic properties of these zeta functions we obtain explicit formulae for the number of n-periodic paths, as the sum of the nth powers of some specific algebraic numbers.

A fractional linear system view of the fractional brownian motion

Nonlinear Dynamics, Vol. 38

Performance of fractional PID algorithms controlling nonlinear systems with saturation and backlash phenomena

The development of fractional-order controllers is currently one of the most promising fields of research. However, most of the work in this area addresses the case of linear systems. This paper reports on the analysis of fractional-order control of nonlinear systems. The performance of discrete fractional-order PID controllers in the presence of several nonlinearities is discussed. Some results are provided that indicate the superior robustness of such algorithms.

Complementary filter design with three frequency bands: robot attitude estimation

This paper extents the by now classic sensor fusion complementary filter (CF) design, involving two sensors, to the case where three sensors that provide measurements in different bands are available. This paper shows that the use of classical CF techniques to tackle a generic three sensors fusion problem, based solely on their frequency domain characteristics, leads to a minimal realization, stable, sub-optimal solution, denoted as Complementary Filters3 (CF3). Then, a new approach for the estimation problem at hand is used, based on optimal linear Kalman filtering techniques. Moreover, the solution is shown to preserve the complementary property, i.e. the sum of the three transfer functions of the respective sensors add up to one, both in continuous and discrete time domains. This new class of filters are denoted as Complementary Kalman Filters3 (CKF3). The attitude estimation of a mobile robot is addressed, based on data from a rate gyroscope, a digital compass, and odometry. The experimental results obtained are reported.

System Modeling and Control Through Fractional-Order Algorithms

The theory of fractional calculus goes back to the beginning of thr throry of differential calculus but its inherent complexity postponed the applications of the associated concepts. In the last decade the progress in the areas of chaos and fractals revealed subtle relationships with the fractional calculus leading to an increasing interest in the development of the new paradigm. In the area of automaticcontrol preliminary work has already been carried out but the proposed algorithms are restricted to the frequency domain. The paper discusses the design of fractional-order discrete-time controllers. The algorithms studied adopt the time domein, which makes them suited for z-transform analusis and discrete-time implementation. The performance of discrete-time fractional-order controllers with linear and non-linear systems is also investigated.