147 results found
Nie Y, Kerrigan EC, 2020, External constraint handling for solving optimal control problems with simultaneous approaches and interior point methods, IEEE Control Systems Letters, Vol: 4, Pages: 7-12, ISSN: 2475-1456
Inactive constraints do not contribute to the solution but increase theproblem size and burden the numerical computations. We present a novel strategyfor handling inactive constraints efficiently by systematically removing theinactive constraints and redundant constraint sets under a mesh refinementframework. The method is tailored for interior point-based solvers, which areknown to be very sensitive to the choice of initial points in terms offeasibility. In the example problem shown, the proposed scheme achieves morethan 40% reduction in computation time.
Nie Y, Kerrigan E, 2020, Efficient and more accurate representation of solution trajectories in numerical optimal control, IEEE Control Systems Letters, Vol: 4, Pages: 61-66, ISSN: 2475-1456
We show via examples that, when solving optimal control problems, representing the optimal state and input trajectory directly using interpolation schemes may not be the best choice. Due to the lack of considerations for solution trajectories in-between collocation points, large errors may occur, posing risks if this solution is to be applied. A novel solution representation method is proposed, capable of yielding a solution of much higher accuracy for the same discretization mesh. This is achieved by minimizing the integral of the residual error for the overall trajectory, instead of forcing the errors to be zero only at collocation points. In this way, the requirement for mesh resolution can be significantly reduced, leaving the problem dimensions relatively small. This particular formulation also avoids some of the drawbacks found in the earlier work of integrated residual minimization, leading to more efficient computations.
McInerney I, Kerrigan E, Constantinides G, Modeling round-off error in the fast gradient method for predictive control, 2019 IEEE 58th Conference on Decision and Control (CDC), Publisher: IEEE
We present a method for determining the smallestprecision required to have algorithmic stability of an imple-mentation of the Fast Gradient Method (FGM) when solvinga linear Model Predictive Control (MPC) problem in fixed-point arithmetic. We derive two models for the round-off errorpresent in fixed-point arithmetic. The first is a generic modelwith no assumptions on the predicted system or weight matrices.The second is a parametric model that exploits the Toeplitzstructure of the MPC problem for a Schur-stable system. Wealso propose a metric for measuring the amount of round-offerror the FGM iteration can tolerate before becoming unstable.This metric is combined with the round-off error models tocompute the minimum number of fractional bits needed forthe fixed-point data type. Using these models, we show thatexploiting the MPC problem structure nearly halves the numberof fractional bits needed to implement an example problem. Weshow that this results in significant decreases in resource usage,computational energy and execution time for an implementationon a Field Programmable Gate Array.
Brown J, Su D, Kong H, et al., Improved noise covariance estimation in visual servoing using an autocovariance least-squares approach, Joint 12th IFAC Conference on Control Applications in Marine Systems, Robotics, and Vehicles 1st IFAC Workshop on Robot Control
For pose estimation in visual servoing, by assuming the relative motion over one sample period to be constant, many existing works adopt a linear time invariant (LTI) dynamic model. Since the standard feature point transformation is nonlinear, extended Kalman filtering (EKF) has become popular due to its simplicity. Thus, the problem at hand becomes filtering of an LTI system with a time-varying output matrix. To obtain satisfactory performance, accurate knowledge of the noise covariances is essential. Various methods have been proposed on how to adaptively update their values to improve performance. However, these techniques cannot guarantee the positive semidefiniteness (PSD) of the covariance estimates. In this paper, we propose to apply the autocovariance least-squares (ALS) approach to covariance identification in pose estimation. The ALS approach can provide reliable estimates of the covariance matrices while maintaining their PSD and imposing desired structural constraints. Our tests show that using the covariance estimates from the ALS method in EKF can reduce the average pose estimation error by more than 30% in simulation, and the average position estimation error by about 30% using experimental data, respectively, compared to a hand-tuned EKF.
Zhang L, Kerrigan E, Pal B, 2019, Optimal communication scheduling in the smart grid, IEEE Transactions on Industrial Informatics, ISSN: 1551-3203
This paper focuses on obtaining the optimal communication topology in the smart grid architecture, i.e. what is the optimal communication setup of smart meters in a smart building. The fact that smart meters also consume energy, more often than not, gets ignored by researchers and engineers. In this paper, we will show that smart meter networks can consume significantly less energy with optimal scheduling. Numerical results show that the overall energy consumption can be reduced by implementing the optimal communication architecture and transmission rate setup, rather than implementing a straightforward communication architecture with uniform channel bandwidth.
Iftikhar S, Faqir O, Kerrigan E, Nonlinear model predictive control of an overhead laboratory-scale gantry crane with obstacle avoidance, 2019 IEEE Conference on Control Technology and Applications (CCTA), Publisher: IEEE
Gantry cranes are complex nonlinear electrome- chanical systems representing a challenging control problem. We propose an optimization-based controller for guiding the crane through arbitrary obstacles. Solving path planning problems with obstacles typically requires a two-stage approach. First, a path is found that is feasible w.r.t. system dynamics and obstacles. The path is then interpreted as a series of set points by a lower-level controller that guides the system. We instead generate a path, and the associated control input to move along that path, from a single optimization problem using a nonlinear model predictive control framework. In doing so, we generate a trajectory that is locally optimal and feasible w.r.t. system dynamics and obstacles. Multiple obstacle avoidance constraint formulations are proposed as smooth, differentiable functions. Objects are approximated either as the union of a set of smooth shapes or as smooth indicator functions. The formulations presented in this work are applicable to (non-)convex problems in 2-D or 3-D spaces. Numerical methods are used to solve the proposed problems for both 2-D (fixed string length) and 3-D (varying string length) models of the gantry crane, resulting in consistently lower costs than nodal or sampling based algorithms.
Zhang L, Kerrigan E, Pal B, Optimal Communication Scheduling in the Smart Grid, IEEE Transactions on Industrial Informatics, ISSN: 1551-3203
McInerney I, Kerrigan EC, Constantinides GA, 2019, Bounding computational complexity under cost function scaling in predictive control, Publisher: arXiv
We present a framework for upper bounding the number of iterations requiredby first-order optimization algorithms implementing constrained LQRcontrollers. We derive new bounds for the condition number and extremaleigenvalues of the primal and dual Hessian matrices when the cost function isscaled. These bounds are horizon-independent, allowing for their use withreceding, variable and decreasing horizon controllers. We considerably relaxprior assumptions on the structure of the weight matrices and assume only thatthe system is Schur-stable and the primal Hessian of the quadratic program (QP)is positive-definite. Our analysis uses the Toeplitz structure of the QPmatrices to relate their spectrum to the transfer function of the system,allowing for the use of system-theoretic techniques to compute the bounds.Using these bounds, we can compute the effect on the computational complexityof trading off the input energy used against the state deviation. An examplesystem shows a three-times increase in algorithm iterations between the twoextremes, with the state 2-norm decreased by only 5% despite a greatlyincreased state deviation penalty.
Faqir OJ, Kerrigan EC, Gunduz D, 2018, Information transmission bounds in mobile communication networks, UKACC 12th International Conference on Control (CONTROL), Publisher: IEEE, Pages: 99-99
Faqir OJ, Kerrigan EC, Gunduz D, 2018, Energy-optimal control in mobile aerial relay-assisted networks, UKACC 12th International Conference on Control (CONTROL), Publisher: IEEE, Pages: 100-100
Nie Y, Kerrigan EC, 2018, Efficient Implementation of Rate Constraints for Nonlinear Optimal Control, 2018 UKACC 12th International Conference on Control (CONTROL), Publisher: IEEE
Khusainov B, Kerrigan EC, Suardi A, et al., 2018, Nonlinear predictive control on a heterogeneous computing platform, Control Engineering Practice, Vol: 78, Pages: 105-115, ISSN: 0967-0661
We propose an implementation of an interior-point-based nonlinear predictive controller on a heterogeneous processor. The workload can be split between a general-purpose CPU and a field-programmable gate array to trade off the contradicting design objectives of control performance and computational resource usage. A new way of exploiting the structure of the KKT matrix yields significant memory savings. We report an 18x memory saving, compared to existing approaches, and a 6x speedup over a software implementation with an ARM Cortex-A9 processor. We also introduce a new release of Protoip, which abstracts low-level details of heterogeneous programming and allows processor-in-the-loop verification.
McInerney I, Kerrigan EC, 2018, Automated project-based assessment in a predictive control course, UKACC 12th International Conference on Control (CONTROL), Publisher: IEEE, Pages: 443-443
Written assessments, such as book problems and exams, have customarily been used in control courses to measure student progress, but usually only gauge their knowledge of the theoretical concepts. More complicated control methods, such as predictive control, benefit from gauging student progress through implementation projects. We present a set of automatically marked project-based assessments that test student knowledge on concepts ranging from the derivation of physics models to the creation of a closed-loop predictive controller. We present a simulation framework that allows for the students to utilize any predictive control concepts that they decide to use in their implementation. The framework then automatically tests the student solutions against multiple constraint sets and conditions to provide quantitative data for marking the assessment.
McInerney I, Constantinides G, Kerrigan EC, 2018, A Survey of the implementation of linear model predictive control on FPGAs, 6th IFAC Conference on Nonlinear Model Predictive Control, Publisher: IFAC Secretariat, Pages: 381-387, ISSN: 2405-8963
Over the past 20 years, great strides have been made in the real-time implementationof linear MPC on FPGA devices. Starting from initial work, which demonstrated the benefits ofembedding linear MPC onto FPGAs, recent work has shown sampling rates of more than 1 MHzare possible with FPGA-based implementations. This work surveys FPGA implementationsof linear MPC, with a focus on the computational architecture. This includes the choice ofnumber representation, the parallelizations exploited and the memory architecture. We discussthe transferability of those design choices to the FPGA implementation of nonlinear MPC, andprovide some future research directions related to the implementation of MPC on FPGAs.
Khusainov B, Kerrigan EC, Constantinides G, 2018, Automatic software and computing hardware co-design for predictive control, IEEE Transactions on Control Systems Technology, Vol: 27, Pages: 2295-2304, ISSN: 1063-6536
Model predictive control (MPC) is a computationally demanding control technique that allows dealing with multiple-input and multiple-output systems while handling constraints in a systematic way. The necessity of solving an optimization problem at every sampling instant often 1) limits the application scope to slow dynamical systems and/or 2) results in expensive computational hardware implementations. Traditional MPC design is based on the manual tuning of software and computational hardware design parameters, which leads to suboptimal implementations. This brief proposes a framework for automating the MPC software and computational hardware codesign while achieving an optimal tradeoff between computational resource usage and controller performance. The proposed approach is based on using a biobjective optimization algorithm, namely BiMADS. Two test studies are considered: a central processing unit and field-programmable gate array implementations of fast gradient-based MPC. Numerical experiments show that the optimization-based design outperforms Latin hypercube sampling, a statistical sampling-based design exploration technique.
Nie Y, Kerrigan EC, 2018, How should rate constraints be implemented in nonlinear optimal control solvers?, 6th IFAC Conference on Nonlinear Model Predictive Control, Publisher: IFAC Secretariat, ISSN: 2405-8963
This paper investigates the problem of implementing rate constraints when solvingnonlinear optimal control problems with direct transcription methods. We generalize theapproach of directly implementing rate constraints on the discretization mesh to all typesof collocation methods (h,pandhp), for both state and input variables. This “on mesh”implementation replaces the additional dynamic equations and nonlinear path constraints inclassical implementations with linear equations. Thus, there is no contribution to the Hessianand the contribution to the Jacobian can be precomputed, enabling faster iterations. Throughan example, the benefits of the proposed approach are demonstrated, both in terms of obtainingsingular arc-free solutions, as well as reductions in computation time of more than 20%.
Faqir O, Nie Y, Kerrigan E, et al., 2018, Energy-efficient communication in mobile aerial relay-assisted networks using predictive control, 6th IFAC Conference on Nonlinear Model Predictive Control, Publisher: IFAC Secretariat, ISSN: 2405-8963
Energy-efficient communication in wireless networks of mobile autonomous agentsmandates joint optimization of both transmission and propulsion energy. In Faqir et al. (2017) wedeveloped communication-theoretic data transmission and Newtonian flight mechanics models toformulate a nonlinear optimal control problem. Here we extend the previous work by generalizingthe communication model to include UAV-appropriate slow fading channels and specificallyinvestigate the potential from joint optimization of mobility and communication over a multipleaccess channel. Numerical results exemplify the potential energy savings available to all nodesthrough this joint optimization. Finally, using the slow fading channel problem formulation,we generate a chance-constrained nonlinear model predictive control scheme for control of aterrestrial network served by a single UAV relay. Closed-loop simulations are performed subjectto uncertainties in both transmission and mobility models.
Nie Y, Faqir O, Kerrigan EC, ICLOCS2: Solve your optimal control problems with less pain, 6th IFAC Conference on Nonlinear Model Predictive Control, Publisher: IFAC Secretariat, ISSN: 2405-8963
ICLOCS2 is the new version of ICLOCS (pronounced `eye-clocks') and is a comprehensive software suite for solving nonlinear optimal control problems (OCPs) in Matlab and Simulink. The toolbox builds on a wide selection of numerical methods and automated tools to assist the design and implementation of OCPs. The aim is to reduce the requirements on the experience of the user, by providing a first port of call to solve a variety of OCPs. ICLOCS2 may not be the fastest solver for some problems, but it might work where other solvers fail.
Vemuri H, Bosworth R, Morrison JF, et al., 2018, Real-time feedback control of 3D Tollmien-Schlichting waves using a dual-slot actuator geometry, Physical Review Fluids, Vol: 3, ISSN: 2469-990X
The growth of Tollmien-Schlichting (TS) waves is experimentally attenuated using a single-inputand single-output (SISO) feedback system, where the TS wave packet is generated by a surfacepoint source in a flat-plate boundary layer. The SISO system consists of a single wall-mountedhot wire as the sensor and a miniature speaker as the actuator. The actuation is achieved througha dual-slot geometry to minimise the cavity near-field effects on the sensor. The experimentalset-up to generate TS waves or wave packets is very similar to that used by Li and Gaster . Theaim is to investigate the performance of the SISO control system in attenuating single-frequency,two-dimensional disturbances generated by these configurations. The necessary plant models areobtained using system identification, the controllers are then designed based on the models andimplemented in real-time to test their performance. Cancellation of the rms streamwise velocityfluctuation of TS waves is evident over a significant domain.
Thammawichai M, Baliyarasimhuni SP, Kerrigan EC, et al., 2018, Optimizing communication and computation for multi-UAV information gathering applications, IEEE Transactions on Aerospace and Electronic Systems, Vol: 54, Pages: 601-615, ISSN: 0018-9251
Typical mobile agent networks, such as multi-unmanned aerial vehicle (UAV) systems, are constrained by limited resources: energy, computing power, memory and communication bandwidth. In particular, limited energy affects system performance directly, such as system lifetime. Moreover, it has been demonstrated experimentally in the wireless sensor network literature that the total energy consumption is often dominated by the communication cost, i.e., the computational and the sensing energy are small compared to the communication energy consumption. For this reason, the lifetime of the network can be extended significantly by minimizing the communication distance as well as the amount of communication data, at the expense of increasing computational cost. In this paper, we aim at attaining an optimal tradeoff between the communication and the computational energy. Specifically, we propose a mixed-integer optimization formulation for a multihop hierarchical clustering-based self-organizing UAV network incorporating data aggregation, to obtain an energy-efficient information routing scheme. The proposed framework is tested on two applications, namely target tracking and area mapping. Based on simulation results, our method can significantly save energy compared to a baseline strategy, where there is no data aggregation and clustering scheme.
Faqir O, Kerrigan EC, Gunduz D, 2018, Joint optimization of transmission and propulsion in aerial communication networks, IEEE Conference on Decision and Control, Publisher: IEEE
Communication energy in a wireless network of mobile autonomous agents should be considered as the sum of transmission energy and propulsion energy used to facilitate the transfer of information. Accordingly, communication-theoretic and Newtonian dynamic models are developed to model the communication and locomotion expenditures of each node. These are subsequently used to formulate a novel nonlinear optimal control problem (OCP) over a network of autonomous nodes. It is then shown that, under certain conditions, the OCP can be transformed into an equivalent convex form. Numerical results for a single link between a node and access point allow for comparison with known solutions before the framework is applied to a multiple-node UAV network, for which previous results are not readily extended. Simulations show that transmission energy can be of the same order of magnitude as propulsion energy allowing for possible savings, whilst also exemplifying how speed adaptations together with power control may increase the network throughput.
Thammawichai M, Kerrigan EC, 2017, Energy-efficient real-time scheduling for two-type heterogeneous multiprocessors, Real-Time Systems, Vol: 54, Pages: 132-165, ISSN: 0922-6443
We propose three novel mathematical optimization formulations that solve the same two-type heterogeneous multiprocessor scheduling problem for a real-time taskset with hard constraints. Our formulations are based on a global scheduling scheme and a fluid model. The first formulation is a mixed-integer nonlinear program, since the scheduling problem is intuitively considered as an assignment problem. However, by changing the scheduling problem to first determine a task workload partition and then to find the execution order of all tasks, the computation time can be significantly reduced. Specifically, the workload partitioning problem can be formulated as a continuous nonlinear program for a system with continuous operating frequency, and as a continuous linear program for a practical system with a discrete speed level set. The latter problem can therefore be solved by an interior point method to any accuracy in polynomial time. The task ordering problem can be solved by an algorithm with a complexity that is linear in the total number of tasks. The work is evaluated against existing global energy/feasibility optimal workload allocation formulations. The results illustrate that our algorithms are both feasibility optimal and energy optimal for both implicit and constrained deadline tasksets. Specifically, our algorithm can achieve up to 40% energy saving for some simulated tasksets with constrained deadlines. The benefit of our formulation compared with existing work is that our algorithms can solve a more general class of scheduling problems due to incorporating a scheduling dynamic model in the formulations and allowing for a time-varying speed profile.
Cantoni M, Farokhi F, Kerrigan EC, et al., 2017, Structured computation of optimal controls for constrained cascade systems, International Journal of Control, ISSN: 0020-7179
Constrained finite-horizon linear-quadratic optimal control problems are studied within the context of discrete-time dynamics that arise from the series interconnec- tion of subsystems. A structured algorithm is devised for computing the Newton-like steps of primal-dual interior-point methods for solving a particular re-formulation of the problem as a quadratic program. This algorithm has the following properties: (i) the computation cost scales linearly in the number of subsystems along the cascade; and (ii) the computations can be distributed across a linear proces- sor network, with localized problem data dependencies between the processor nodes and low communication overhead. The computation cost of the approach, which is based on a fixed permutation of the primal and dual variables, scales cubically in the time horizon of the original optimal control problem. Limitations in these terms are explored as part of a numerical example. This example involves application of the main results to model data for the cascade dynamics of an automated irrigation channel in particular.
Bachtiar V, Manzie C, Kerrigan EC, 2017, Nonlinear model-predictive integrated missile control and Its multiobjective Tuning, Journal of Guidance Control and Dynamics, Vol: 40, Pages: 2961-2970, ISSN: 1533-3884
Shukla H, Khusainov B, Kerrigan EC, et al., Software and hardware code generation for predictive control using splitting methods, IFAC World Congress 2017, Publisher: IFAC
This paper presents SPLIT, a C code generation tool for Model Predictive Control (MPC)based on operator splitting methods. In contrast to existing code generation packages, SPLIT iscapable of generating both software and hardware-oriented C code to allow quick prototyping ofoptimization algorithms on conventional CPUs and field-programmable gate arrays (FPGAs). AMatlab interface is provided for compatibility with existing commercial and open-source softwarepackages. A numerical study compares software, hardware and heterogeneous implementationsof splitting methods and investigates MPC design trade-offs. For the considered testcases thereported speedup of hardware implementations over software realizations is 3x to 11x.
Khusainov B, Kerrigan EC, Suardi A, et al., Nonlinear predictive control on a heterogeneous computing platform, IFAC World Congress 2017, Publisher: IFAC
Nonlinear Model Predictive Control (NMPC) is an advanced control technique that often relieson computationally demanding optimization and integration algorithms. This paper proposesand investigates a heterogeneous hardware implementation of an NMPC controller based onan interior point algorithm. The proposed implementation provides flexibility of splitting theworkload between a general-purpose CPU with a fixed architecture and a field-programmablegate array (FPGA) to trade off contradicting design objectives, namely performance andcomputational resource usage. A new way of exploiting the structure of the Karush-Kuhn-Tucker(KKT) matrix yields significant memory savings, which is crucial for reconfigurable hardware.For the considered case study, a 10x memory savings compared to existing approaches anda 10x speedup over a software implementation are reported. The proposed implementationcan be tested from Matlab using a new release of the Protoip software tool, which isanother contribution of the paper. Protoip abstracts many low-level details of heterogeneoushardware programming and allows quick prototyping and processor-in-the-loop verification ofheterogeneous hardware implementations.
Picciau A, Inggs G, Wickerson J, et al., 2017, Balancing locality and concurrency: solving sparse triangular systems on GPUs, 23rd IEEE International Conference on High Peformance Computing, Data, and Analytics (HiPC), Publisher: IEEE, Pages: 183-192
Many numerical optimisation problems rely onfast algorithms for solving sparse triangular systems of linearequations (STLs). To accelerate the solution of such equations,two types of approaches have been used: on GPUs, concurrencyhas been prioritised to the disadvantage of data locality, whileon multi-core CPUs, data locality has been prioritised to thedisadvantage of concurrency.In this paper, we discuss the interaction between data localityand concurrency in the solution of STLs on GPUs, and we presenta new algorithm that balances both. We demonstrate empiricallythat, subject to there being enough concurrency available in theinput matrix, our algorithm outperforms Nvidia’s concurrencyprioritisingCUSPARSE algorithm for GPUs. Experimental resultsshow a maximum speedup of 5.8-fold.Our solution algorithm, which we have implemented inOpenCL, requires a pre-processing phase that partitions thegraph associated with the input matrix into sub-graphs, whosedata can be stored in low-latency local memories. This preliminaryanalysis phase is expensive, but because it depends onlyon the input matrix, its cost can be amortised when solving formany different right-hand sides.
Khusainov B, Kerrigan EC, Constantinides GA, 2017, Multi-objective Co-design for Model Predictive Control with an FPGA, European Control Conference 16, Publisher: IEEE, Pages: 110-115
In order to achieve the best possible performanceof a model predictive controller (MPC) for a given set ofresources, the software algorithm and computational platformhave to be designed simultaneously. Moreover, in practicalapplications the controller design problem has a multi-objectivenature: performance is traded off against computational hardwareresource usage, namely time, energy and space. Thispaper proposes formulating an MPC design problem as a multiobjectiveoptimization (MOO) problem in order to explore thedesign trade-offs in a systematic way.Since the design objectives in the resulting MOO problem areexpensive to evaluate, i.e. evaluation requires time consumingsimulations, most of the classical and evolutionary MOOalgorithms cannot be employed for this class of design problems.For this reason a practical MOO algorithm that can deal withexpensive-to-evaluate functions is presented. The algorithm isbased on Kriging and the hypervolume criterion that wasrecently proposed in the expensive optimization literature. Anumerical example for a fast gradient-based controller designshows that the proposed approach can efficiently exploreoptimal performance-resource trade-offs.
Thammawichai M, Kerrigan EC, 2016, Feedback Scheduling for Energy-Efficient Real-Time Homogeneous Multiprocessor Systems, 55th IEEE Conference on Decision and Control, Publisher: IEEE
Real-time scheduling algorithms proposed in theliterature are often based on worst-case estimates of taskparameters and the performance of an open-loop scheme cantherefore be poor. To improve on such a situation, one caninstead apply a closed-loop scheme, where feedback is exploitedto dynamically adjust the system parameters at run-time. Wepropose an optimal control framework that takes advantageof feeding back information of finished tasks to solve a realtimemultiprocessor scheduling problem with uncertainty intask execution times, with the objective of minimizing thetotal energy consumption. Specifically, we propose a linearprogramming-based algorithm to solve a workload partitioningproblem and adopt McNaughton’s wrap around algorithmto find the task execution order. Simulation results for aPowerPC 405LP and an XScale processor illustrate that ourfeedback scheduling algorithm can result in an energy savingof approximately 40% compared to an open-loop method.
Ge M, Kerrigan EC, 2016, Relations between Full Information and Kalman-Based Estimation, 55th IEEE Conference on Decision and Control, Publisher: IEEE
For nonlinear state space systems with additivenoises, sometimes the number of process noise signals couldbe less than the dimension of the state space. In order toimprove the accuracy and stability of nonlinear state estimation,this paper provides for the first time the derivation of thefull information estimator (FIE) for such nonlinear systems.We verify our derivation of the FIE by firstly proving theunbiasedness and minimum-variance of the FIE for linear timevarying (LTV) systems, then showing the equivalence betweenthe Kalman filter/smoother and the FIE for LTV systems.Finally, we prove that the FIE will provide more accurate stateestimates than the extended Kalman filter (EKF) and smoother(EKS) for nonlinear systems.
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