Iterative Learning Control for Deterministic Systems

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Iterative Learning Control for Deterministic Systems (Advances in Industrial Control)

It begins with an introduction to the concept of learning control, including a comprehensive literature review. The text follows with a complete and unifying analysis of the learning control problem for linear LTI systems using a system-theoretic approach which offers insight into the nature of the solution of the learning control problem.

Additionally, several design methods are given for LTI learning control, incorporating a technique based on parameter estimation and a one-step learning control algorithm for finite-horizon problems. Further chapters focus upon learning control for deterministic nonlinear systems, and a time-varying learning controller is presented which can be applied to a class of nonlinear systems, including the models of typical robotic manipulators.

The book concludes with the application of artificial neural networks to the learning control problem. Three specific ways to neural nets for this purpose are discussed, including two methods which use backpropagation training and reinforcement learning.

The appendices in the book are particularly useful because they serve as a tutorial on artificial neural networks. Help Centre. My Wishlist Sign In Join. Be the first to write a review. Add to Wishlist.

It begins with an introduction to the concept of learning control, including a comprehensive literature review. The text follows with a complete and unifying analysis of the learning control problem for linear LTI systems using a system-theoretic approach which offers insight into the nature of the solution of the learning control problem.

Additionally, several design methods are given for LTI learning control, incorporating a technique based on parameter estimation and a one-step learning control algorithm for finite-horizon problems. Further chapters focus upon learning control for deterministic nonlinear systems, and a time-varying learning controller is presented which can be applied to a class of nonlinear systems, including the models of typical robotic manipulators.

The book concludes with the application of artificial neural networks to the learning control problem. Three specific ways to neural nets for this purpose are discussed, including two methods which use backpropagation training and reinforcement learning. The appendices in the book are particularly useful because they serve as a tutorial on artificial neural networks.

Benjamin Recht: Optimization Perspectives on Learning to Control (ICML 2018 tutorial)

We show the results of closed-loop tests of the controller with a nonlinear system model, which provide a partial validation of the controller and tool chain. The cost function of the underlying optimization problem consists of convex stage costs as well as Bregman distances centered around the a priori estimates.


The Bregman term allows to incorporate constraints into the cost function by means of relaxed barrier functions and to formulate the moving horizon estimator as an unconstrained optimization problem. The stability properties of the proposed estimator are investigated and the obtained results are illustrated by means of a numerical example. The low-level MPC makes use of an equivalent hydraulic model EHM that captures the dynamics of relevant internal electro-chemical states. This controller calculates the optimal charging current that satisfies the constraints associated with side reactions in anode and cathode.

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The high-level MPC solves a temperature feasibility problem using a battery thermal model by setting boundaries on the square of the current defined by the low-level control in the previous step. Information is exchanged between layers until the current proposed by low-level controller is feasible for the high-level controller requirements. Simulation results of the aforementioned scheme are compared with an alternative case in which the lower layer predictor uses an equivalent circuit model ECM.

We show that for the same thermal model and tuning parameters, similar performance regarding constraints satisfaction and convergence of the solution for the control action can be obtained for the EHM and ECM. Finally, the distributed approach is compared to a single MPC in which the temperature constraint is enforced by resorting to the thermal equilibrium current, which is calculated off-line.

The latter achieves similar results to the reference case at lower computational cost.

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Each of the secondary sensors is assigned to a sector of the operation area; a secondary sensor decides which target in its assigned sector to be identified and controls itself to identify the target. We formulate the decision-making process as an optimization problem to minimize the uncertainty of the targets' identities subject to the sensor dynamic constraints. The proposed algorithm is decentralized since the secondary sensors only communicate with the primary sensor for the target information, and need not to synchronize with each other.

By integrating the proposed algorithm with the existing multi-target tracking algorithms, we develop a closed-loop multi-target tracking and identity management algorithm. The effectiveness of the proposed algorithm is demonstrated with illustrative numerical examples. However, to ensure flight safety, guaranteed consistent uncertainty estimates are of equally crucial importance. Two easily implementable modifications to the Extended Set Membership Filter are shown to greatly reduce its conservatism.

The filter runs in fixed time and handles asynchronous observations. As another novelty, we take into account parametric uncertainty of the observation equation. The proposed filter is applied to a two unmanned aircraft systems UAS localization problem in simulation, with observation noise obtained from real sensors. We compare the new filter to a computationally very strong but much more expensive nonlinear set inversion algorithm and find that similar performances can be achieved.

Also, simulation results illustrate the effective reduction of filter conservatism by a small number of iterative updates. The design is based on a decomposition into locally observable and unobservable substates and on properties of homogeneous systems. Each observer node can reconstruct in finite-time its locally observable substate with its measurements only. Then exploiting the coupling, a finite-time converging observer is constructed for the remaining states by adding the consensus terms.

A numerical example illustrates the result. In doing so, the system can be split into smaller subsystems that can be estimated and controlled more easily. While current state-of-the-art fusion methods for distributed estimation assume the fusion of estimates referring to the full dimension of the state, little effort has been made to account for the fusion of unequal state vectors referring to smaller subsystems of the full system. In this paper, a novel method to fuse overlapping state vectors using a deterministic sample-based fusion method is proposed.

These deterministic samples can be used to account for the correlated and uncorrelated noise terms and are therefore able to reconstruct the joint covariance matrix in a distributed fashion. The performance of the proposed fusion method is compared to other state-of-the-art methods. Moving horizon estimation involves solving a quadratic program to minimize the estimation error relative to a model over a fixed window of past input-output observations.

By exploiting the spatial structure of a chain, two algorithms for solving this quadratic program are considered. Both algorithms can be distributed in the sense that the computations associated with each sub-system component of the state depend only on information associated with the immediate neighbours. The algorithms differ in the way that the linear Karush-Kuhn-Tucker conditions for optimality are solved. Computational and information dependency overheads are analyzed. Numerical results are presented for a 1-D mass-spring-damper chain. Inspired by the alternate attitude and position distributed optimization framework discussed in [1], we propose an estimation scheme that exploits the unit dual quaternion algebra to describe the sensors pose.

This representation is beneficial in the formulation of the optimization scheme in that it allows to solve the localization problem without designing two interlaced position and orientation estimators, thus improving the estimation error distribution over the two pose components and the overall localization performance. Furthermore, the numerical experimentation asserts the robustness of the proposed algorithm w.


Standard feedforward methods to achieve these goals are inversion-based feedforward control and input shaping. The former achieves perfect tracking but lacks robustness to uncertainties in the system dynamics while the latter may account for robustness at the expense of reduced tracking performance. In this work, a continuous time finite-impulse-response FIR prefilter design for linear systems is proposed which allows to trade off tracking performance against robustness. The approach is based on a finite time linear quadratic LQ optimal control formulation of the filtering task.

Robustness is taken into account by penalizing large magnitudes of the trajectory sensitivity. Tracking delay can be reduced by adjusting the filter length which is a free design parameter. With the proposed prefiltering scheme fast and robust tracking can be achieved.

An important application is simultaneous tracking and vibration suppression for motion systems with uncertain elasticity parameters which is used as a simulation example. The special case where the unknown inputs have direct feedthrough to the outputs is considered. Necessary and sufficient conditions are given under which an unbiased pseudo state and input estimation is possible. The prediction step of the fractional Kalman filter uses an approximation as it can not be calculated exactly in practice.

Therefore, the estimation procedure is suboptimal. Nevertheless, the method yields accurate estimates of pseudo states and unknown inputs which is illustrated by means of an academic example. By sending a sample only when some triggering condition is fulfilled, we can ensure that the transmitted samples actually carry innovation. However, in an event-based system, the state estimation problem becomes complicated, as the information of not receiving a measurement must be taken into consideration.

Recent research has examined the feasibility of using particle filters for solving the event-based state estimation problem. To the best of our knowledge, only the simple bootstrap particle filter has so far been considered in this setting. We argue that, as this filter does not fully utilize the current measurement, it is not well suited for state estimation in event-based systems. We propose an extension to the auxiliary particle filter for systems with event-based measurements, in which certain existing techniques for finding an approximation of the fully adapted filter can easily be utilized.

In a simulation study, we demonstrate that at new measurement events, the benefits of using the auxiliary particle filter increases when fewer measurements are being sent. The algorithm uses numerically stable square-root formulas, can handle simultaneous independent measurements and non-equally spaced abscissas, and can compute state estimates at points between the data abscissas.

The state space model's parameters, including driving noise intensity, measurement variance, and initial state, are determined from the given data sequence using maximum likelihood estimation computed using an expectation maximisation iteration. In tests with synthetic biomechanics data, the algorithm is found to be more accurate compared to a widely used open source automatic numerical differentiation algorithm, especially for acceleration estimation.

However, sometimes the dynamical system evolves in such a way that online learning is preferable. This paper addresses the online joint state estimation and learning problem for nonlinear dynamical systems. We leverage a recently developed reduced-rank formulation of Gaussian-process state-space models GP-SSMs , and develop a recursive formulation for updating the sufficient statistics associated with the GP-SSM by exploiting marginalization and conjugate priors. The results indicate that our method efficiently learns the system jointly with estimating the state, and that the approach for certain scenarios gives similar performance as more computation-heavy offline approaches.

To confirm the correctness of our results, we have proved that the newly constructed UD based array computational schemes are algebraically equivalent to the "straight" conventional information filter. Although all these information-type algorithms are theoretically equivalent, their computational properties are different. The newly proposed algorithms are numerically robust to machine round-off errors due to the numerically stable orthogonal transformations applied on each iteration. Additionally, algorithm eUD-IF has the extended array form, i.

So, our results extend the existing class of numerically efficient KF implementation methods and can be used in practical applications. For the aim, the relationship between the Stokes-Dirac structure and the topological geometry of the manifolds is clarified in terms of harmonic differential forms. The performance enhancement is achieved by simultaneous design of the power flow and controllers.

The power flow design problem is formulated as parameter design in the parameter-dependent power system model. We propose a solution method to the parameter design, which is based on iteratively solving linear matrix inequalities. In order to further improve the damping performance of the power system, controller design is combined with the parameter design. Different from the previous works, we account for the dynamics of RL power lines and consider generic interconnections of loads and DGUs.

The stability of the ImG hinges on the passivity of all components with specific input-output pairs and the properties of the electrical interconnections. For the DGUs, we provide necessary and sufficient conditions for the existence of control gains guaranteeing passivity. In addition, we show that controllers decoupling the direct- and quadrature axis components of the electric variables can never render the DGU passive.

Theoretical results are backed by simulation studies. Coupling the wind farm with advanced energy storage systems represents, in principle, a good solution for these problems. To date, several researches have been conducted on storage technology, but the problem of finding the best ESS solution is still open. Indeed, every storage technology has its own constraints and limitations in terms of capital cost, response time, operational, maintenance and degradation issues.

This highlights the importance of advanced control algorithms for energy storage management systems to mitigate the problems outlined. Via a MPC based controller and mixed-integer linear constraints and dynamics, we address the problem of satisfying a forecasted power demand. In particular, we consider a control problem of power consumers. The behavior of the control system including consumers is inevitably affected by their decisions.

We aim to construct a stable control system that is consistent with consumers' preferences. To this end, the preferences are described as cost functions to formulate the problem of the control system design. In general, the assumption that the cost functions of all consumers are available for the design is too severe for practical systems. In this paper, we find a special class of the cost functions and propose a preference-independent design of the system.

Finally, we confirm the usefulness of the proposed system through numerical simulation. First, a detailed simulation model of the complete power train of a generic railway vehicle, supplied by an electric energy source, is described. The presented operating strategy comprises the power management between the two power sources of the vehicle, namely the catenary as well as the ESS. To derive the energy optimal parametrisation for the proposed power management approach, a Grey Wolf Optimiser GWO is applied, which is inspired by the hunting behavior of grey wolves in nature.

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Worcester Polytechnic Inst Kontopyrgos, Marios Worcester Polytechnic Institute Keywords: Distributed parameter systems Abstract: This work considers a level-set based method for path planning of human evacuation in a hazardous indoor environment. The accumulated inhalation of a hazardous substance is calculated via the line integral of the substance concentration over the selected path. The candidate paths are generated by calculating the constant angle paths towards the exit if the critical level-set of the gas concentration has not been reached.

Once the path overlaps with the selected level set, a new set of angles is calculated so that the path follows the level-set. Simulations of both spatially and spatiotemporally varying fields of the hazardous gas are examined. Worcester Polytechnic Inst Keywords: Observers for linear systems , Distributed parameter systems , Computational methods Abstract: This paper provides the necessary system-theoretic conditions for the establishment of the well-posedness and convergence of filters for of parabolic PDEs in 2D rectangular domains.

By decomposing the spatial domain into an inner subdomain that includes the sensing device s and an outer subdomain that does not include the sensor s , the resulting state estimators can employ different numerical grids to compute the associated filter gains. The ultimate goal is to have variable spatial resolution of the filters that is dependent on the sensor location. Different from multi-grid methods, the proposed DD filters provide additional flexibility on the numerical implementation of the proposed filters and the eventual code parallelization.

Two asymptotic gains are studied: the gain in the L2 spatial norm and the gain in the spatial sup norm. It is shown that the asymptotic gain property holds in the L2 norm of the displacement without any assumption for the damping coefficients. The derivation of the upper bounds for the asymptotic gains is performed by either employing an eigenfunction expansion methodology or by means of a small-gain argument, whereas a novel frequency analysis methodology is employed for the derivation of the lower bounds for the asymptotic gains.

The graphical illustration of the upper and lower bounds for the gains shows that the asymptotic gain in the L2 norm is estimated much more accurately than the asymptotic gain in the sup norm. Sevilla Keywords: Distributed parameter systems , Distributed control , Linear systems Abstract: We study the problem of stabilizing reaction-diffusion equations within a prescribed time. We employ the backstepping method and select a target equation whose state and its spatial derivative converge to zero within the prescribed time.

This stability is related back to the plant by characterizing the growth-in-time of the kernel associated to the backstepping transformation. We extract from the backstepping transformation a time-varying boundary feedback controller which stabilizes the plant in the prescribed time. Using the same technique, we then propose an observer that incorporates flux measurements at the boundary which are scaled by a time-varying observer gain; we show that the state estimate error and its spatial derivative converge to zero within the prescribed time. We combine the prescribed-time stabilizing controller and observer results to develop output feedback, for which we verify the separation principle.

Finally, a simulation is provided where output feedback is utilized to stabilize an unstable reaction-diffusion equation within a prescribed time. The autonomous equation is shown to be exponentially stable with respect to the square integral norm. On the basis of this result, the task of reference tracking using a distributed control input is investigated and, in particular, the optimal control problem associated to the minimization of a power functional is addressed.

In relation to these systems, we study an object we call the semistability Gramian, which serves as a generalization of the ordinary controllability Gramian valid for semistable systems. This Gramian is then given geometric as well as algebraic characterization via a Lyapunov equation. We then proceed to show that under a commutativity assumption relating the original and reduced systems, and as long as the semistability is preserved, we may derive a priori error formulas in H2-norm in terms of the trace of this Gramian.

At each step the algorithm alternates between solving a quasi-definite linear system with a constant coefficient matrix and a projection onto convex sets. The solver is able to exploit chordal sparsity in the problem data and to detect infeasible problems. The low per-iteration computational cost makes the method particularly efficient for large problems, e. Our Julia implementation is open-source, extensible, integrated into the Julia optimisation ecosystem and performs well on a variety of large convex problem classes.

The calibration tool applied to create the control maps, named "Off-line parameterization tool", was designed based on the Design of Experiments method.

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The tool was designed to work both fully automatically and semi-automatically. Many reports on engine calibration have taken the Design of Experiments DoE approach, but their implementations in choosing experimental design types and optimization processes are different compared to this paper. The unique aspect of this research lies in the significant properties of the Off-line parameterization tool. First, this tool is flexible: it is able to work with multiple inputs and multiple outputs.

Second, it can reduce the calibration time as the engine running time is kept as short as possible and all the data processing work is accomplished automatically. We design an online algorithmic framework based on prediction-correction, which employs splitting methods to solve the sampled instances of the time-varying problem. We describe the prediction-correction scheme and two splitting methods, the forward-backward and the Douglas-Rachford. Then by using a result for generalized equations, we prove convergence of the generated sequence of approximate optimizers to a neighborhood of the optimal solution trajectory.

Simulation results for a leader following formation in robotics assess the performance of the proposed algorithm. However, it can exhibit undesirable periodic oscillatory behaviour in some applications that slows its convergence. Restart schemes seek to improve the convergence of FGM algorithms by suppressing the oscillatory behaviour.

Recently, a restart scheme for FGM has been proposed that provides linear convergence for non strongly convex optimization problems that satisfy a quadratic functional growth condition. However, the proposed algorithm requires prior knowledge of the optimal value of the objective function or of the quadratic functional growth parameter.

In this paper we present a restart scheme for FISTA algorithm, with global linear convergence, for non strongly convex optimization problems that satisfy the quadratic growth condition without requiring the aforementioned values. We present some numerical simulations that suggest that the proposed approach outperforms other restart FISTA schemes. We show that its Lebesgue volume vol K can be approximated as closely as desired by solving a sequence of generalized eigenvalue problems with respect to a pair of Hankel matrices of increasing size, whose entries are obtained in closed form.

An extension to the non-homogeneous case is also briefly described. The set of block factor-width-two matrices is a proper cone and we compute a closed-form expression for its dual cone. We use these cones to build hierarchies of inner and outer approximations of the cone of positive semidefinite matrices.

The main feature of these cones is that they enable a decomposition of a large semidefinite constraint into a number of smaller semidefinite constraints. As the main application of these classes of matrices, we envision large-scale semidefinite feasibility optimisation programs including sum-of-squares SOS programs. We present numerical examples from SOS optimisation showcasing the properties of this decomposition. KTH Royal Institute of Technology Keywords: Network analysis and control , Stochastic systems , Applications in neuroscience Abstract: This paper presents the notion of stochastic phase-cohesiveness based on the concept of recurrent Markov chains and studies the conditions under which a discrete-time stochastic Kuramoto model is phase-cohesive.

It is assumed that the exogenous frequencies of the oscillators are combined with random variables representing uncertainties. A bidirectional tree network is considered such that each oscillator is coupled to its neighbors with a coupling law which depends on its own noisy exogenous frequency. In addition, an undirected tree network is studied. For both cases, a sufficient condition for the common coupling strength and a necessary condition for the sampling-period are derived such that the stochastic phase-cohesiveness is achieved.

The analysis is performed within the stochastic systems framework and validated by means of numerical simulations. Necessary and sufficient conditions for the task accomplishment are investigated and a design procedure is provided. Special attention is also devoted to limit the in-degree and out-degree of the controller by possibly perturbing the target equilibrium. An application to a multispecies biological system, with logistic growth and mutual interactions of the species, is worked out in order to show both the relevance of the considered problem and effectiveness of the proposed solution.

A key issue is to control the phage's replication process: indeed, phage may switch either towards lytic state or lysogeny state during its reproduction due to some conditions. However, only lytic state is relevant for bacteria elimination. An algebraic state-space representation method is adopted to determine all possible families of reachable sets.

Necessary and sufficient conditions for the stabilization of SBCN to a given equilibrium point under arbitrary switching signals are presented, where the control input is switching signal-dependent. Moreover, all possible switching signal-dependent state feedback controllers are obtained based on all complete families of reachable sets. Results are relevant for both the development of SBCN theory and for practical applications. For a given graph clustering of an original complex network, we construct a simplified network consisting of fewer nodes, where the edge weights are to be determined.

An optimal weight assignment procedure is proposed to select suitable edge weights of the reduced network, aiming for the minimum H2 approximation error between the original network and the reduced-order network model. The effectiveness of the proposed method is illustrated by means of an example. This paper proposes a broad generalisation of the DeGroot--Friedkin model by allowing each individual's self-appraisal process to be distorted by behavioural characteristics such as humility.

We establish the generalised dynamical model for the evolution of individuals' social power a measure of an individual's contribution to each topic discussion. For some types of individuals, whom we term "humble" and "unreactive", results are provided on the existence of equilibria, whether such equilibria are unique, and convergence to said equilibria. Simulations are used to illustrate that networks of "emotional" individuals, who at times act like humble individuals and at other times like arrogant individuals, can have at least two attractive equilibria. The main motivating applications are modeling infectious waterborne diseases such as cholera, SARS, and amoebiasis hand-to-mouth.

We present the model and its derivation, explore the equilibria of the model, and analyze the healthy equilibria. We illustrate the behavior of the model via simulation, and demonstrate how the proposed model captures the behavior of Dr. John Snow's seminal cholera dataset. Our system uses as sensors a single mono-camera to recognize lane markings on the road ahead and traditional Electronic Stability Control ESC sensors e. The optimal lateral control algorithm calculates on vehicle states estimation base a steering angle command that is tracked by a low level steering wheel position control in a nested control loop framework.

The results of developed solution on hereafter experimental lane change are satisfying and lead to a maximum lateral error of less than 20 cm. The proposed motion planning works directly on a depth map which represents partial surrounding environment due to limited field of view FOV of the stereo camera.

The depth map updates at each sampling time. An optimal local target is selected from a set of local targets to which line-of-sight paths are collision-free. Then an optimal collision-free trajectory satisfying kinematic constraints is generated. The simulation results demonstrate the effectiveness of the proposed motion planning approach. Air Force Research Lab Keywords: Autonomous systems , Cooperative control , Aerospace Abstract: A pursuit-evasion differential game with simple motion where an additional player, a Defender, is tasked with intercepting the pursuer before the latter can capture the evader is addressed.

A new target escape strategy is investigated. KTH Royal Institute of Technology Tumova, Jana KTH Royal Institute of Technology Keywords: Autonomous systems , Hybrid systems , Autonomous robots Abstract: This paper proposes an approach that combines motion planning and hybrid feedback control design in order to find and follow trajectories fulfilling a given complex mission involving time constraints. The solution builds on three main steps: i using sampling-based motion planning methods and the untimed version of the mission specification in the form of Zone automaton, we find a sequence of waypoints in the workspace; ii based on the clock zones from the satisfying run on the Zone automaton, we compute time-stamps at which these waypoints should be reached; and iii to control the system to connect two waypoints in the desired time, we design a low-level feedback controller leveraging Time-varying Control Barrier Functions.

Illustrative simulation results are included. We propose an intent-aware overtaking controller for the ego car.

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  5. The intention of the lead car is estimated via a combination of active model discrimination and model invalidation algorithms. Then, a safe overtaking controller is designed based on vector fields that take into account the estimated intent, and ensure safety of the overtaking maneuver. Simulation results demonstrate the efficacy of the proposed approach. Worcester Polytechnic Institute Keywords: Autonomous systems , Sensor and mesh networks , Observers for linear systems Abstract: We address planar path-planning for a mobile vehicle to traverse a workspace with minimum exposure to an unknown spatially and temporally varying scalar field, which we call the threat field.

    The threat field is estimated by a finite number of mobile sensors that can take pointwise measurements with finite sampling frequency. We address this problem using an iterative bidirectional interaction between planning and sensor placement phases. Owing to this interaction, sensors are placed at locations where the threat field measurements are of most relevance to the path-planning problem. We provide theoretical conditions under which this iterative sensing-planning algorithm is convergent. This algorithm includes considerations for finite durations of time required for the various computations involved therein.

    Numerical simulation results and prior theoretical results with time-invariant threat fields indicate that not only is the proposed algorithm convergent, but it also produces near optimal paths. Universidad Nacional De Colombia Mojica-Nava, Eduardo Universidad Nacional De Colombia Keywords: Robotics , Agents networks , Autonomous robots Abstract: This paper introduces an ant's behavior based exploration algorithm for multiple agents to explore non-convex environments without requiring connectivity maintenance for each instant of time.

    The employed algorithm emulates the pheromone trails that the real ants employ in order to avoid exploration already done by a group of robots using periodically deployed markers. Also, it includes the possibility that when two or more agents enter within their communication range, they exchange information related to the known map and the markers found.

    Those behaviors are integrated with a frontier-based exploration algorithm, that originally required the connectivity of the network for it to work, removing this requirement and allowing it to work in broader environments. Motivated by the need that robot teams have in many real-world applications of remaining operational for long periods of time, we allow each robot to choose tasks taking into account the energy consumed by executing them, besides the global specifications on the task allocation.

    The tasks are encoded as constraints in an energy minimization problem solved at each point in time by each robot. The prioritization of a task over others—effectively signifying the allocation of the task to that particular robot—occurs via the introduction of slack variables in the task constraints. Moreover, the suitabilities of certain robots towards certain tasks are also taken into account to generate a task allocation algorithm for a team of robots with heterogeneous capabilities. The efficacy of the developed approach is demonstrated both in simulation and on a team of real robots.

    Different from most of the existing related studies on multi-robot formation, which mainly concentrate on motion planning or control algorithm design, this work pay attention to a long-neglected issue that whether the desired formation can be maintained with velocity and angular velocity constraints imposed on individual robots. To solve this problem, a quantitative metric, the feasible set of formation angular velocity, is introduced to characterize the turning ability of a formation.

    Three common formations, namely perfect rigid formation, semi-perfect rigid formation and flexible formation, are discussed in this paper. It has been proven that perfect rigid formation has a rather limited mobility, while flexible formation is more agile than rigid formation in that it can execute more flexible maneuvers than the corresponding rigid formation with the same formation width. Simulations have been performed to illustrate the theoretical analysis.

    During this mission, safety appears to be a key issue, as numerous obstacles are lying in a close neighborhood of the aircraft. They are very different in terms of size, shape and mobility and are often unforeseen. To cope with this highly evolutive environment, it is necessary to design a method sufficiently generic to deal with the obstacles variety and efficient enough to guarantee non collision and avoid classical problems such as local minima, singularities, etc.

    Iterative Learning Control for Deterministic Systems Iterative Learning Control for Deterministic Systems
    Iterative Learning Control for Deterministic Systems Iterative Learning Control for Deterministic Systems
    Iterative Learning Control for Deterministic Systems Iterative Learning Control for Deterministic Systems
    Iterative Learning Control for Deterministic Systems Iterative Learning Control for Deterministic Systems
    Iterative Learning Control for Deterministic Systems Iterative Learning Control for Deterministic Systems

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