Christophe Tricaud

An approximate method for numerically solving fractional order optimal control problems of general form

In this article, we discuss fractional order optimal control problems (FOCPs) and their solutions by means of rational approximation. The methodology developed here allows us to solve a very large class of FOCPs (linear/nonlinear, time-invariant/time-variant, SISO/MIMO, state/input constrained, free terminal conditions etc.) by converting them into a general, rational form of optimal control problem (OCP). The fractional differentiation operator used in the FOCP is approximated using Oustaloups approximation into a state-space realization form. The original problem is then reformulated to fit the definition used in general-purpose optimal control problem (OCP) solvers such as RIOTS_95, a solver created as a Matlab toolbox. Illustrative examples from the literature are reproduced to demonstrate the effectiveness of the proposed methodology and a free final time OCP is also solved.

Time-Optimal Control of Systems with Fractional Dynamics

We introduce a formulation for the time-optimal control problems of systems displaying fractional dynamics in the sense of the Riemann-Liouville fractional derivatives operator. To propose a solution to the general time-optimal problem, a rational approximation based on the Hankel data matrix of the impulse response is considered to emulate the behavior of the fractional differentiation operator. The original problem is then reformulated according to the new model which can be solved by traditional optimal control problem solvers. The time-optimal problem is extensively investigated for a double fractional integrator and its solution is obtained using either numerical optimization time-domain analysis.

D-optimal trajectory design of heterogeneous mobile sensors for parameter estimation of distributed systems

An approach is proposed to joint optimization of trajectories and measurement accuracies of mobile nodes in a sensor network collecting measurements for parameter estimation of a distributed parameter system. The problem is cast as maximization of the log-determinant of the information matrix associated with the estimated parameters over the set of all feasible information matrices, which yields a formulation in terms of convex optimization. This makes it possible to employ powerful tools of optimum experimental design theory to characterize the optimal solution and adapt the Wynn-Fedorov algorithm to construct its numerical approximation. As a crucial subtask in each iteration, a nontrivial optimal control problem must be solved, which is accomplished using the MATLAB PDE toolbox and the RIOTS_95 optimal control toolbox which handles various constraints imposed on the sensor motions. The effectiveness of the method is illustrated with a numerical example regarding a two-dimensional diffusion equation.

Optimal mobile sensing and actuation policies in cyber-physical systems

Introduction.- Distributed Parameter Systems: Controllability, Observability and Identification.- Optimal Heterogeneous Mobile Sensing for Parameter Estimation of Distributed Parameter Systems.- Optimal Mobile Remote Sensing Policies.- On-line Optimal Mobile Sensing Policies: Finite-horizon Control Framework.- Optimal Mobile Actuation/Sensing Policies for Parameter Estimation off Distributed Parameter Systems.- Optimal Mobile Sensing with Fractional Sensor Dynamics.- Optimal Mobile Remote Sensing Policy for Downscaling and Assimiliation Problems.- Conclusions and Future Work.- Appendices: Notation RIOTS Tutorial Implentations.

Resource-Constrained Sensor Routing for Parameter Estimation of Distributed Systems

Abstract We consider a setting where mobile nodes with sensing capacity form a network whose mission is to collect measurements for parameter estimation of a distributed parameter system (DPS). Two techniques to optimize node motions are presented which constitute a trade-off between the achievable accuracy of parameter estimates and limited motion resources of sensor network nodes. The framework is based on the use of the D-optimality criterion defined on the Fisher information matrix associated with the estimated parameters as a measure of the information content in the measurements. Restrictions on maximal distances traveled by sensor nodes are imposed so as to guarantee realizable solutions. The approach is to convert the problem to a canonical optimal control one in Mayer form, in which both the control forces of the sensors and the initial sensor positions are optimized. Numerical solutions are then obtained using the M atlab PDE toolbox and the RIOTS_95 optimal control toolbox which handles various constraints imposed on the node motions. Illustrative numerical experiments with the proposed techniques are presented.

Optimal mobile actuator/sensor network motion strategy for parameter estimation in a class of cyber physical systems

This paper introduces a framework to solve the problem of determining optimal sensors and actuators trajectories so as to estimate a set of unknown parameters in what constitutes a cyber-physical system (CPS). Given a distributed systems set of partial differential equation describing its dynamic behavior, the optimal steering of a team of sensors and actuators is obtained by minimizing the D-optimality criteria associated with the expected accuracy of the obtained parameter values. The problem is then reformulated into an optimal control one, whose solution can be computed by readily available commercial softwares. A numerical example is used to demonstrate the feasibility of the proposed method.

Solution of fractional order optimal control problems using SVD-based rational approximations

This paper introduces a new direction to approximately solving fractional order optimal control problems (FOCPs). A general methodology is described that can potentially solve any type of FOCPs (linear/nonlinear, time-invariant/time-variant, SISO/MIMO, state/input constrained, free terminal conditions etc.). The method uses a rational approximation of the fractional derivative operator obtained from the singular value decomposition of the Hankel data matrix of the impulse response. The FOCP is then reformulated to be solved by RIOTS_95, a general-purpose optimal control problem (OCP) solver in the form of a MATLAB toolbox. Illustrative examples from the literature are reproduced to demonstrate the effectiveness of the propose methodology and a free final time OCP is also demonstrated.

Great Salt Lake Surface Level Forecasting Using FIGARCH Model

In this paper, we have examined 4 models for Great Salt Lake level forecasting: ARMA (Auto-Regression and Moving Average), ARFIMA (Auto-Regressive Fractional Integral and Moving Average), GARCH (Generalized Auto-Regressive Conditional Heteroskedasticity) and FIGARCH (Fractional Integral Generalized Auto-Regressive Conditional Heteroskedasticity). Through our empirical data analysis where we divide the time series in two parts (first 2000 measurement points in Part-1 and the rest is Part-2), we found that for Part-2 data, FIGARCH offers best performance indicating that conditional heteroscedasticity should be included in time series with high volatility.Copyright

Cooperative Control of Water Volumes of Parallel Ponds Attached to An Open Channel Based on Information Consensus with Minimum Diversion Water Loss

Decision making through information sharing for a canal irrigation system is discussed in this paper. A consensus-based decision algorithm is used to manage water distribution into a parallel ponds network and achieve good quality of service and minimum water-loss. The algorithm is tested in a simulation software reproducing the major dynamics of the system for different scenarios. While not taking into account some of the non-linearities and focusing on feasibility, the algorithm shows interesting results under ideal flow control conditions especially regarding convergence. Robustness is estimated by Monte Carlo evaluation of the effects of delay uncertainty, and the strategy maintains convergence.

Optimal trajectories of mobile remote sensors for parameter estimation in distributed Cyber-Physical Systems

In this paper, we present a method to obtain the optimal trajectories of a team of robots monitoring a distributed parameter system located in a different domain. The mobile robots are equipped with remote sensors capable of measuring the considered systems state from a separate space. The purpose of trajectory planning of the team of robots is to obtain measurements so as to estimate the parameters of the considered system. From a given set of partial differential equation the dynamics of the distributed systems behavior, the optimal path and steering of the team of mobile nodes is obtained by minimizing the D-optimality criteria associated with the expected accuracy of the obtained parameter values. From this original problem, an optimal control problem is derived with the advantage of being solvable by readily available commercial softwares. The method is illustrated on a diffusive distributed syste.