International Workshop on Hybrid Systems: Modeling, Simulation and Optimization

 

Contributed Papers

Title:         Use of Hybrid System Modeling Technique for Robot-Assisted Rehabilitation Systems

Authors:  Duygun Erol

                  Department of Electrical &Electronics Engineering, Yeditepe University, Kayisdagi, Istanbul, Turkey

Abstract:

Recent research in rehabilitation indicates that tasks that focus on activities of daily living (ADL) is likely to show significant increase in motor recovery after stroke. The availability of providing a rehabilitation therapy that requires performing ADL tasks, however, is limited by the amount of costly therapist's time they involve and the ability of the therapist to provide controlled, quantifiable and repeatable assistance to movement. Consequently, robot-assisted rehabilitation that can quantitatively monitor and adapt to patient progress, and ensure consistency during rehabilitation has become an active research area to provide a solution to these problems. Even though existing arm and hand rehabilitation systems have shown promise of clinical utility, they are limited by their inability to simultaneously assist both arm and hand movements. This limitation is critical because the stroke therapy literature supports the idea that the ADL-focused tasks, which engage patients to perform the tasks in enriched environments, have shown significant increase in the motor recovery after stroke.

It is possible to integrate an arm assistive device and a hand assistive device to provide the necessary motion for ADL-focused therapy. However, none of the existing controllers used for robot-assisted rehabilitation can be directly used for this purpose because they are not suited for controlling multiple systems in a coordinated manner. In this work, we address the intelligent controller design issue of a robot-assisted rehabilitation system that can simultaneously coordinate both arm and hand motion to perform ADL tasks. Initially, we have designed a robot-assisted rehabilitation system that consists of an arm assistive device and a hand assistive device. Later, we address the controller design issue of this robot-assisted rehabilitation system that can simultaneously coordinate both arm and hand motion to perform ADL tasks using an intelligent control architecture.

In this architecture, low-level arm assistive controller and low-level hand assistive controller is used to provide assistance to the subject’s arm and hand movement, respectively. A high-level controller (HLC) is used to allocate task responsibility between the low-level assistive controllers (LLACs) based on the task requirements and specific events that may arise during the task performance. HLC plays the role of a human supervisor (therapist) who would otherwise monitor the task, assess whether the task needs to be updated and determine the activation of the assistive devices. However, in general, the HLC and the LLACs may not communicate directly because each may operate in different domains. While the LLACs may operate in a continuous way, the HLC may need to make intermittent decisions in a discrete manner. Hybrid system theory provides mathematical tools that can accommodate both continuous and discrete systems in a unified manner. Thus, we take advantage of using a hybrid system model to design the proposed intelligent control architecture.

A hybrid system based control mechanism could be useful in rehabilitation context in terms of coordinating decision making and assisting, monitoring safety, and managing and modifying code for automation. To our knowledge, such a mechanism has not been explored in rehabilitation robotics.

Title:         Dynamic Modeling and Optimization of Circadian Clock

Authors:  Ugur Kaplan, I.Halil Kavakli, Metin Turkay
                   College of Engineering, Koc University, Istanbul, Turkey

Abstract:

TBA

Title:         A Model of Angiogenesis by Hybrid Systems with Delay on the Piecewise Constant part

Authors:  Bulent Karasozen, Hakan Oktem, Mustafa Kahraman
              
    Department of Mathematics & Institute of Applied Mathematics, METU, Ankara, Turkey

Abstract:

An important approach in understanding the cancer dynamics is the modeling of angiogenesis process. There are several attempts in modeling this process. An important factor to consider in modeling the angiogenesis is the time delays caused by the physical distance between the tumor and the vessel. Some studies has suggested that those delays can cause oscillatory behavior in the angiogenesis process. In this work we employed piecewise linear hybrid systems with delay on the piecewise constant part. Our approach is based on piecewise linearizing the system and involving the delay between threshold crossing and the state transition. The use of piecewise linear systems with a single threshold for each variable suggest advances in understanding the qualitative dynamical behavior of dynamical systems and various uses in modeling of genetic interactions has already been demonstrated. Our approximation allow tractable approximation of angiogenesis process incorporating more variables and effect of some possible external inputs can also be involved in the simulation.

 

Title:         Simulation of Mating Behavior in Flies

Authors:  Mehmet Kayým, Aykut Kence
                   Department of Biology, Middle East Technical University, Ankara, Turkey

Abstract:

Mating behavior plays a key role in determining the fitness of individuals within and among populations and in maintaining the integrity of gene pools. We are able to construct simulation program which describes matings between individuals in a population in flies. This hybrid mating system contains both a continuous and a discrete dynamic behavior. The individuals formed in the mating pool, where preferential (assortative) mating takes place, are assigned to random vigor values according to a Gaussian distribution (continuous). The success of mating and duration of mating time between a male and a female are determined by differences in their vigor. Assigning individuals their genotypes according to expected frequencies and observed frequencies for the next generation constitutes the discrete part of the model. The model can simulate the mating time and selection due to mating behavior rather well. It is possible to investigate the behavioral isolation during speciation.

 

Title:         Multi-Dimensional Particle Swarm Optimization

Authors:  Serkan Kiranyaz*, Turker Ince**, Alper Yildirim***, Moncef Gabbouj*

                   *Tampere University of Technology, Tampere, Finland

                   **Izmir Economy University, Izmir, Turkey

                   ***TUBITAK, Ankara, Turkey

Abstract:

The behavior of a single organism in a swarm is often insignificant but their collective and social behavior is of paramount importance. The particle swarm optimization (PSO) was introduced by Kennedy and Eberhart in 1995 as a population based stochastic search and optimization process. It is originated from the computer simulation of the individuals (particles or living organisms) in a bird flock or fish school, which basically show a natural behavior when they search for some target (e.g. food). In the basic PSO algorithm, the particles are initially distributed randomly over the search space with a random velocity and the goal is to converge to the global optimum of a function or a system. Each particle keeps track of its position in the search space and its best solution so far achieved. This is the personal best value (the so-called pbest) and the PSO process also keeps track of the global best solution so far achieved by the swarm with its particle index (the so called gbest). So during their journey with discrete time iterations, the velocity of each agent in the next iteration is computed by the best position of the swarm (position of the particle gbest as the social component), the best personal position of the particle (pbest as the cognitive component), and its current velocity (the memory term). Both social and cognitive components contribute randomly to the position of the agent in the next iteration. In principle, PSO follows the same path of the other evolutionary algorithms (EAs) such as Genetic Algorithm (GA), Genetic Programming (GP), Evolutionary Strategies (ES) and Evolutionary Programming (EP). The common point of all is that EAs are in population based nature and thus they can avoid being trapped in a local optimum. Thus they can find the optimum solutions; however, this is never guaranteed. In this study, we propose a novel optimization technique, the so-called Multi-Dimensional Particle Swarm Optimization (MD PSO), which re-forms the native structure of swarm particles in such a way that they can make inter-dimensional passes with a dedicated dimensional PSO process. Therefore, in a multidimensional search space where the optimum dimension is unknown, swarm particles can seek for both positional and dimensional optima. This eventually negates the necessity of setting a fixed dimension a priori, which is a common drawback for the family of swarm optimizers. Therefore, instead of operating at a fixed dimension N, the MD PSO algorithm is designed to seek both positional and dimensional optima within a dimension range, ( min max D ≤N≤ D ). In order to accomplish this, each particle has two sets of components, each of which has been subjected to two independent and consecutive processes. The first one is a regular positional PSO, i.e. the traditional velocity updates and due positional shifts in N dimensional search (solution) space. The second one is a dimensional PSO, which allows the particle to navigate through dimensions. Accordingly, each particle keeps track of its last position, velocity and personal best position (pbest) in a particular dimension so that when it revisits that the same dimension at a later time, it can perform its regular “positional” fly using this information. The dimensional PSO process of each particle may then move the particle to another dimension where it will remember its positional status and keep “flying” within the positional PSO process in this dimension, and so on. The swarm, on the other hand, keeps track of the gbest particles in all dimensions, each of which respectively indicates the best (global) position so far achieved and can thus be used in the regular velocity update equation for that dimension. Similarly the dimensional PSO process of each particle uses its personal best dimension in which the personal best fitness score has so far been achieved. Finally, the swarm keeps track of the global best dimension, dbest, among all the personal best dimensions. The gbest particle in dbest dimension represents the optimum solution and dimension, respectively. We investigated the application of the proposed method over two well-known domains, nonlinear function minimization and data clustering. An extensive set of experiments show that in both application domains, MD PSO can converge to the global optimum at the true (optimum) dimension as long as the basic PSO can on a fixed dimension.

 

Title:         Time-optimal Control of Automobile Test Drives with Gear Shifts

Authors:  Christian Kirches, Sebastian Sager, Hans Georg Bock

                   Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Germany

Abstract:

Mixed–integer optimal control problems (MIOCPs) in ordinary differential equations (ODEs) have gained increasing interest over the last years. This is probably due to the fact that the underlying processes have a high potential for optimization.

As a typical example, we discuss the choice of gears in automobile driving [3, 5]. We treat a benchmark problem from automobile test-driving presented in [1], where a prescribed test track has to be completed in minimum time.

Our computations make use of a direct method in which infinite–dimensional control functions are discretized by basis functions and corresponding finite–dimensional parameters that enter into the optimization problem. The gear choice decisions are reformulated using the technique of outer convexification with respect to the binary controls, as developed in [4, 5]. For time-optimal control problems like the discussed benchmark problem, this reformulation yields optimal solutions of the relaxed problem that exhibit a bang–bang structure, and are thus already integer feasible.

For the benchmark problem at hand, we reproduce the computational results that have been presented in [1] while reducing the computation time required by more than three orders of magnitude. When compared to an alternative approach from [2] we still achieve a reduction in computation time of about one order of magnitude.

In a second step we extend the benchmark problem to compute periodic solutions on elliptical tracks. Here, more complex gear shift structures appear, which necessitate the inclusion of an additional nonlinear constraint limiting the engine’s revolutionary speed. For the elliptical tracks we obtain and present nonintuitive time-optimal periodic solutions that would be considerably more difficult to compute using the approach of [2].

Finally we mention recent developments towards more realistic race tracks, as well as towards mixed-integer optimal control on moving horizons.

References

[1] M. Gerdts. Solving mixed-integer optimal control problems by branch&bound: A case study from automobile test-driving with gear shift. Optimal Control Applications and Methods, 26:1–18, 2005.

[2] M. Gerdts. A variable time transformation method for mixed-integer optimal control problems. Optimal Control Applications and Methods, 27(3):169–182, 2006.

[3] C. Kirches, S. Sager, H.G. Bock, and J.P. Schloder. Time-optimal control of automobile test drives with gear shifts. Optimal Control Applications and Methods, 2009. (submitted).

[4] S. Sager. Numerical methods for mixed–integer optimal control problems. Der andere Verlag, T¨onning, Lubeck, Marburg, 2005. ISBN 3-89959-416-9. Available at http://sager1.de/sebastian/downloads/Sager2005.pdf.

[5] S. Sager, G. Reinelt, and H.G. Bock. Direct methods with maximal lower bound for mixed-integer optimal control problems. Mathematical Programming, published online at http://dx.doi.org/10.1007/s10107-007-0185-6 on 14 August 2007, 2008.

Title:         Modeling Hybrid Systems using Constraint Logic Programming

Authors:  Ammar Mohammed

                  Computer Science Department, Koblenz-Landau University, Universitaetsstrasse 1, 56070 Koblenz, Germany

Abstract:

Hybrid systems are the result of merging the two most commonly used models of dynamical systems, namely continuous dynamical systems defined by differential equations, and discrete event systems defined by automata. One can view hybrid systems as constrained systems. The constraints are used to describe the possible flows, invariants and transitions on one hand, and to mark certain parts of the state space (e.g. the set of initial states, or the set of unsafe state) on the other hand. Therefore, it is natural to use Constraint Logic Programming (CLP) as an approach to model hybrid systems. One advantage from this approach is that the problem of analyzing hybrid systems can be solved by execution of the corresponding CLP programs. This paper uses CLP to model hybrid concurrent systems. This is possible by controlling the execution of the concurrent hybrid systems using time constraints of events. We will demonstrate the approach by using Eclipse-CLP interval arithmetic.

 

Title:         An Introduction of Hybrid Systems with Memory

Authors:  Hakan Oktem, A. Hayfavi, N. Caliskan, N. Gokgoz

                   Institute of Applied Mathematics, METU, Ankara, Turkey

Abstract:

For a wide range of switching systems in nature and technology the system’s behaviour and response to external inputs are determined not only by the initial value but the whole history. Especially, for systems requiring history memorization capabilities like many biological systems, this is a requirement. Various types of functional differential equations (FDE) were suggested for systems with function memory. Thus, abstraction of the initial function is an important requirement for modeling concrete functional systems. Furthermore, combining the functional features and switching behaviour require tools for studying such systems. In this work we define Hybrid Systems with memory as a collection of: A finite number of states, an invariant set and a vector field for each state where the state’s ODE is valid within its invariant set, a set of edges representing the possible state transitions, a set of guard conditions and a reset map for each edge and a memory which is updated at each state transition. Unlike the conventional Hybrid systems, the ODE of each state is determined by both the state and the memory. We discuss different schemes and illustrate with simple examples.

 

Title:         Reformulations and Numerical Methods for Control Problems with Integer-Valued Functions

Authors:  Sebastian Sager, Christian Kirches, Hans Georg Bock

                   Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Germany

Abstract:

Mixed–integer optimal control problems (MIOCPs) in ordinary differential equations (ODEs) have gained increasing interest over the last years. This is probably due to the fact that the underlying processes have a high potential for optimization. Typical examples are the choice of gears in transport,[5, 2], or processes in chemical engineering involving on-off valves, [4, 1].

Although the first MIOCPs were already solved in the early eighties, the so–called indirect methods used there do not seem appropriate for generic large–scale optimal control problems with underlying nonlinear differential algebraic equation systems. Instead direct methods, in particular all–at–once approaches have become the methods of choice for most practical problems.

In direct methods infinite–dimensional control functions are discretized by basis functions and corresponding finite–dimensional parameters that enter into the optimization problem. The drawback of direct methods with binary control functions obviously is that they lead to high–dimensional vectors of binary variables. For many practical applications a fine control discretization is required, however. Therefore, techniques from mixed–integer nonlinear programming like Branch&Bound or Outer Approximation will work only on limited and small time horizons because of the exponentially growing complexity of the problem.

We propose to use an outer convexification with respect to the binary controls. The reformulated control problem has two main advantages compared to standard relaxations. First, especially for time-optimal control problems, the optimal solution of the relaxed problem often exhibits a bang–bang structure, and is thus already integer feasible. Second, theoretical results have recently been found, [3, 5] that show that even for path-constrained and sensitivity-seeking arcs the optimal solution of the relaxed problem yields the exact lower bound on the minimum of the integer problem. This allows to calculate the loss of performance, if a coarser control discretization grid, a simplified switching structure for the optimization of switching times, or heuristics are used.

The strength of this approach was shown by solving several challenging applications in areas as diverse as systems biology, chemical engineering and transport either for the first time or with computational costs reduced by several orders of magnitude.

References

[1] Y. Kawajiri and L.T. Biegler. Large-scale optimization strategies for zone configuration of simulated moving beds. In 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering, pages 131–136. Elsevier, 2006.

[2] C. Kirches, S. Sager, H.G. Bock, and J.P. Schloder. Time-optimal control of automobile test drives with gear shifts. Optimal Control Applications and Methods, 2009. (submitted).

[3] S. Sager. Numerical methods for mixed–integer optimal control problems. Der andere Verlag, T¨onning, Lubeck, Marburg, 2005. ISBN 3-89959-416-9. Available at http://sager1.de/sebastian/downloads/Sager2005.pdf.

[4] S. Sager, M. Diehl, G. Singh, A. K¨upper, and S. Engell. Determining SMB superstructures by mixedinteger control. In K.-H. Waldmann and U.M. Stocker, editors, Proceedings OR2006, pages 37–44, Karlsruhe, 2007. Springer.

[5] S. Sager, G. Reinelt, and H.G. Bock. Direct methods with maximal lower bound for mixed-integer optimal control problems. Mathematical Programming, published online at http://dx.doi.org/10.1007/s10107-007-0185-6 on 14 August 2007, 2008.

 

Title:         A Mode-Based Hybrid Controller Design for Agile Maneuvering Unmanned F-16 Aircraft

Authors:  Nazim Kemal Ure, Gokhan Inalhan
              
    Faculty of Aeronautics and Astronautics, Istanb
ul Technical University, Istanbul, Turkey

Abstract:

The ability of a combat unmanned air vehicles to autonomously design and execute agile maneuvers in complex and dynamic environments, is a low-level enabling technology for future air scenarios driven by performance and safety goals. In this work, we present a mode-based hybrid controller design for a full-flight envelope maneuvering autonomous F-16 aircraft. The controller design is structured around a finite state automaton that spans the full-flight-envelope maneuvers of a generic aircraft model, and we design a nonlinear sliding manifold control system that tracks the outputs of this automaton.  

In general it is a very challenging task to describe general motion of an unmanned air vehicle and design a single control law that handles the reference flight trajectory tracking commands. From the inspection of the well known smooth aerobatic maneuvers and more complex combat maneuvers, we observe that this task can be quantized by decomposing general maneuvers to maneuver modes in which both the system dynamics and also the control task is simplified. The trajectory generation is then simplified to the task of time and duration sequencing of the maneuver modes and the selection of specific maneuver parameters referred as modal inputs. Two constraints arise for the building motion alphabet for such a system. First constraint arises from maneuver sequencing. Due to physical considerations, maneuver execution cannot necessarily be arbitrary. For this, we provide a set of rules reflected on a mode transition chart that describes which maneuver mode can be executed after another. Second constraint is associated with modal inputs. Due to aerodynamic, structural and actuator limitations, modal inputs must lie inside the flight envelope during the execution. We describe a coordinated finite state automaton that combines the translational, rolling and loop plane specifications of the maneuver. Transition logic, domains and trajectory acceptance conditions of the automata are formed by the constraints arising from the above dynamic transition limitations. Overall, this framework reduces the complexity of motion planning problem into lower dimensional modal input search and mode sequencing problem, which generates feasible reference maneuvers. 

In second part of this work, we focus on the design of a nonlinear sliding manifold control system which tracks the maneuvers generated by the modes within the automaton. For each mode of the system, a sliding manifold is defined in the state space of the aircraft. The functional feedback form that is considered can robustly track both minimum and non-minimum phase outputs on these manifolds. In addition, the use of higher mode sliding mode controllers diminishes the control signal chattering problem that is seen in high-performance maneuver executions. 

The overall system is applied on a realistic unmanned version of a F-16 combat aircraft model. The model includes full-flight-envelope nonlinear aerodynamic database with stall, actuator saturation and

rate limits. The nonlinear control system shows the ability to track the planned aggressive maneuver sequences that are not trackable by the tested classical control designs.

 

Title:         A Switching Control Approach to Stabilization of Parameter Varying Time Delay Systems

Authors:  Peng Yan*, Hitay Ozbay**
                   *Enterprise Design Engineering, Seagate Technology, 1280 Disc Drive, Shakopee, MN 55379, USA

                   **Department of Electrical & Electronics Engineering, Bilkent University, Ankara, 06800, Turkey

Abstract:

Many time varying time delay systems can be described as parameter varying systems where the system parameters are scheduled along a measurable parameter trajectory. A good example of parameter varying time delay systems is the data congestion control model for TCP networks, where all the parameters of the dynamical model, including the time delay RTT(round trip time), are dependent on instantaneous queue length at the bottleneck network node.

The analysis and control of LPV (Linear Parameter Varying) delay free systems have been discussed widely, among which two important methods are (1) gain scheduling method and (2) switching control method. There are, however, very limited results on LPV systems with time delays, due to the general difficulty of infinite dimensionality. This paper proposes a switching control method for robust stabilization of parameter varying time delay systems, where each candidate state feedback controller guarantees robust stability at the selected operating interval and the switching rules are developed to cover the whole operating range. This paper provides dwell time based stability condition for switched time varying time delay systems, which is an extension of [Yan, P., Ozbay, H. (2008). Stability Analysis of Switched Time Delay Systems. SIAM Journal on Control and Optimization, 47(2):936-949]. Based on the parameter trajectory, the switching logic with hysteresis determined by dwell time is discussed to guarantee asymptotical stability of the time varying time delay systems in the whole operating range. The proposed method offers a new look into the synthesis of time varying time delay systems.

 

Title:         Fuzzy L1 Norm Strategy for Model Based Control

Authors:  Zehra ZEYBEK

                   Ankara University, Department of Chemical Engineering, 06100 Tandoðan, Ankara, Turkey

Abstract:

The formulation of Model Based control (MBC) depends heavily on the quality of the model chosen. It is therefore of greatest importance  to select a model structure and a set of model parameters where there are a great deal of uncertainty as well as vague phenomena  to obtain a model with sufficient predictive precision. So, this paper considers the parameter estimation problem of MBC model formulated new technique, based on fuzzy linear programming which utilizes  fuzzy parameters consisting of an ordered pair  which describes the center and width of the fuzzy parameters respectively and the application of this model to the control of  stable and unstable chemical process. Minimization of L1 criterion is used to find the fuzzy model parameters, which leads to computationally favorable linear programming problems and allows the possibility to include a priori information in the form of linear constraints without making the computations more complex. And also the control strategy proposed in this work relies on solving the continuous L1 regulation and tracking problem.

The Auto-Regression model with eXogenous input or ARX model for identification has been considered.. This model was built using fuzzy linear regression techniques and non-fuzzy output data. Also fuzzy linear regression can be applied to any model that uses conventional non-fuzzy linear regression by fuzzyfying  the linear regression model using a fuzzy linear function together with the fuzzy parameters of the triangular membership functions.  In this correspondence  we adhere to the “primal problem” of solution  of the standard form of the simplex method and find an on-line adaptation for it.

The estimation of ARX model requires the model orders  to be specified. If the orders are chosen too large, many parameters have to be estimated with a corresponding loss of efficiency. On the other hand, if the orders are too small then the estimates become inconsistent. The identification problems of determining the black box model structure is partially solved by computing sample cross-correlation and auto- correlation functions from the input ,output data.

In our approach we develop a parameterization of the plant that is affine in the unknown parameters. The “computed variable” is given by the model.

                                                                                  (1)

Where  is the “computed or estimated value of the variable y, are known functions of states and are the p number of unknown parameters of the model to be determined. In this study , was assumed to be a symmetric triangular fuzzy number with center  and half-width, , .Fig.3 shows triangular fuzzy regression coefficient . Assuming all the measurements of pair have same precision. The estimation problem is basically a problem of finding estimates for the parameters  are determined so that the function  is as close to the data as possible, e.g., one takes estimate which minimizes an L1 norm.

                                                                                                  (2)

Then the fuzzy linear regression model can be rewritten as follows:

      (3)

The following linear  programming (LP) formulation was employed to estimate

 

This inclusion of subjective information in forecasting is considered in fuzzy linear regression because one cannot appropriately fit the real world into a classical  mathematical model.

For example The center and width of the constant of (A0) are both zero which represents the condition that these regression models pass the primary point or are not decided by the centers and widths of the parameters (A1-Ap). The small widths of the parameters indicate lower levels of fuzziness. The reliability of the fuzzy parameters is also high.  If values of the parameters are between positive and negative values, they represent that the variable may be diminished in some situations.

 

As the first step of  control design, an error function  is defined as follows:                                                                                                                      (4)

where c is a constant parameter vector , found by fuzzy L1 norm programming. is the state variable A cost function J is built as follows:

                                                                                                            (5)

Control motion is said to be asymptotically generated in the system (12) if the state variable  fullfills  .                                                                                               (6)

 

The following two examples were illustrated application of the fuzzy L1 norm control on the processes. One example  used is a linearized Fuzzy CARIMA model of  temperature of well mixed reactor with cooling Jacket. The  second example is the level model of the sulfur tank on the unstable condition.

 

Title:         Modeling and Optimal Control for Freeway Onramp Metering Based on the Piecewise Affine System

Authors:  Liguo Zhang

                  School of Electronic and Control Engineering, Beijing University of Technology, Beijing, China

Abstract:

The freeway onramp metering control problem is developed using a hybrid piecewise affine model (PWA), which is a modified version of Daganzo’s cell transmission model (CTM). The modeling formulation captures both free flow and congested flow modes, and includes the constraint conditions on the onramp metering rates, queue lengths and mainline jam density. Receding horizon control for freeway onramp metering is solved using the well-known multi-parametric programming approach. The technique is tested and compared with the popular Percent-occupancy and Alinea schemes based on the PARAMICS simulation environment.

 

               

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