Additional Equality Constraints Research Articles

A Moving Horizon Estimation for a Class of Soft Continuum Manipulators

Abstract This work proposes a formulation for the moving horizon estimation technique considering both states and parameters obtained simultaneously with reduced estimation issues. The parameters can be either constant or time-varying, while states are related to complex output trajectories related to underactuated systems. Particularly, the proposed formulation considers additional equality constraints as a counterpart of the dynamics tube-model predictive control. Thus, it becomes less dependent on probabilistic information, such as probability density function and covariance of the process noise. In addition, the calibration of the method parameters has less sensitivity and driven by the tube constraints. The proposed approach can be applied in different systems; however, here it is detailed for a class of soft continuum manipulators with fluidic actuation through a variable flowrate and demonstrated with numerical simulations in planar motion. Results indicate robustness of the algorithm estimation in a challenging scenario arising from underactuation as well as in the presence of uncertainty and external disturbance, while simultaneously states and a vector of structural parameters are coherently estimated.

Supervised Feature Selection via Collaborative Neurodynamic Optimization.

As a crucial part of machine learning and pattern recognition, feature selection aims at selecting a subset of the most informative features from the set of all available features. In this article, supervised feature selection is at first formulated as a mixed-integer optimization problem with an objective function of weighted feature redundancy and relevancy subject to a cardinality constraint on the number of selected features. It is equivalently reformulated as a bound-constrained mixed-integer optimization problem by augmenting the objective function with a penalty function for realizing the cardinality constraint. With additional bilinear and linear equality constraints for realizing the integrality constraints, it is further reformulated as a bound-constrained biconvex optimization problem with two more penalty terms. Two collaborative neurodynamic optimization (CNO) approaches are proposed for solving the formulated and reformulated feature selection problems. One of the proposed CNO approaches uses a population of discrete-time recurrent neural networks (RNNs), and the other use a pair of continuous-time projection networks operating concurrently on two timescales. Experimental results on 13 benchmark datasets are elaborated to substantiate the superiority of the CNO approaches to several mainstream methods in terms of average classification accuracy with three commonly used classifiers.

Optimal control using flux potentials: A way to construct bound-preserving finite element schemes for conservation laws

To ensure preservation of local or global bounds for numerical solutions of conservation laws, we constrain a baseline finite element discretization using optimization-based (OB) flux correction. The main novelty of the proposed methodology lies in the use of flux potentials as control variables and targets of inequality-constrained optimization problems for numerical fluxes. In contrast to optimal control via general source terms, the discrete conservation property of flux-corrected finite element approximations is guaranteed without the need to impose additional equality constraints. Since the number of flux potentials is less than the number of fluxes in the multidimensional case, the potential-based version of optimal flux control involves fewer unknowns than direct calculation of optimal fluxes. We show that the feasible set of a potential-state potential-target (PP) optimization problem is nonempty and choose a primal–dual Newton method for calculating the optimal flux potentials. The results of numerical studies for linear and nonlinear conservation laws in 2D demonstrate the superiority of the new OB-PP algorithms to closed-form flux limiting under worst-case assumptions.

A Single-Loop Framework for the Reliability-Based Control Co-Design Problem in the Dynamic System

When solving the control co-design (CCD) problem using the simultaneous strategy in a deterministic manner, the uncertainty stemming from the stochastic design variables is ignored, and might have a negative influence on the performance of the dynamic system. In attempting to overcome the undesirable effect of the uncertainty, this research investigates the reliability-based control co-design (RB-CCD) problem and presents a single-loop framework for RB-CCD based on the modified RB-CCD model and single-loop approach (SLA). Specifically, the modified model is deduced by introducing additional design variables and equality constraints (state equations and algebraic equality constraints) so as to transform the probabilistic constraints into inequality constraints. Meanwhile, to enhance the solution efficiency, SLA transforms the modified RB-CCD model into an equivalent single-loop deterministic CCD model by incorporating the approximate reliability information of the stochastic design variables into the deterministic optimization. Finally, a numerical example and an engineering example are implemented to verify the feasibility and effectiveness of the single-loop RB-CCD optimization framework. The results demonstrate that the suggested single-loop framework dramatically improves the reliability of the dynamic system, and significantly increases the solving efficiency without compromising accuracy.

A scalable and generic approach to range joins

Analytical database systems provide great insights into large datasets and are an excellent tool for data exploration and analysis. A central pillar of query processing is the efficient evaluation of equi-joins, typically with linear-time algorithms (e.g. hash joins). However, for many use-cases with location and temporal data, non-equi joins, like range joins, occur in queries. Without optimizations, this typically results in nested loop evaluation with quadratic complexity. This leads to unacceptable query execution times. Different mitigations have been proposed in the past, like partitioning or sorting the data. While these allow for handling certain classes of queries, they tend to be restricted in the kind of queries they can support. And, perhaps even more importantly, they do not play nice with additional equality predicates that typically occur within a query and that have to be considered, too. In this work, we present a kd-tree-based, multi-dimension range join that supports a very wide range of queries, and that can exploit additional equality constraints. This approach allows us to handle large classes of queries very efficiently, with negligible memory overhead, and it is suitable as a general-purpose solution for range queries in database systems. The join algorithm is fully parallel, both during the build and the probe phase, and scales to large problem instances and high core counts. We demonstrate the feasibility of this approach by integrating it into our database system Umbra and performing extensive experiments with both large real world data sets and with synthetic benchmarks used for sensitivity analysis. In our experiments, it outperforms hand-tuned Spark code and all other database systems that we have tested.

Minimum variance constrained estimator

This paper is concerned with the problem of state estimation for discrete-time linear systems in the presence of additional (equality or inequality) constraints on the state (or estimate). By use of the minimum variance duality, the estimation problem is converted into an optimal control problem. Two algorithmic solutions are described: the full information estimator (FIE) and the moving horizon estimator (MHE). The main result is to show that the proposed estimator is stable in the sense of an observer. The proposed algorithm is distinct from the standard algorithm for constrained state estimation based upon the use of the minimum energy duality. The two are compared numerically on the benchmark batch reactor process model.

A Binary Integer Programming Method for Optimal Wind Turbines Allocation

The present study introduces a Binary Integer Programming (BIP) method to minimize the number of wind turbines needed to be installed in a wind farm. The locations of wind turbines are selected in a virtual grid which is constructed considering a minimum distance between the wind turbines to avoid the wake effect. Additional equality constraints are also included to the proposed formulation to prohibit or enforce the installation of wind turbines placement at specific locations of the wind farmland. Moreover, a microscopic wind turbine placement considering the local air density is studied. To verify the efficiency of this proposal, a square site was subdivided into 25 square cells providing a virtual grid with 36 candidate placement locations. Moreover, a virtual grid with 121 vertices related with a Greek island is also tested. All simulations conducted considering the area of geographical territory, the length of wind turbine blades, as well as the capacity of each turbine.

An efficient algorithm for large-scale extended trust-region subproblems with non-intersecting linear constraints

In this paper, we study the extended trust-region subproblem in which the trust-region intersects the ball with m linear inequality constraints (m-eTRS). We assume that the linear constraints do not intersect inside the ball. We show that the optimal solution of m-eTRS can be found by solving one TRS, computing the local non-global minimizer of TRS if it exists and solving at most two TRSs with an additional linear equality constraint (1-eqTRS). Both TRS and (1-eqTRS) are polynomially and efficiently solvable, thus the new algorithm significantly improves over the SOCP/SDP relaxation of Burer and Yang [Math Program 149(1-2):253–264, 2015]. on two classes of test problems, the efficiency of the proposed approach is compared with the SOCP/SDP relaxation and branch and bound algorithm of Beck and Pan [J Global Optim 69(2):309–342, 2017].

A modified optimization method for robust partial quadratic eigenvalue assignment using receptances and system matrices

In this paper, we focus on the robust partial quadratic eigenvalue assignment problem for vibrating structures by active feedback control. This problem is reformulated as an minimization problem, where the cost function can measure both the sensitivity of the closed-loop eigenvalues and the feedback norms. This can be seen as a modified version of the minimization problem proposed by Bai et al. (2016) [5], where there exist additional equality constraints involving determinants of linear functions of the parameter matrix. By using the receptance measurements, the system matrices and a few undesired open-loop eigenvalues and associated eigenvectors, we propose a modified gradient-based optimization method for solving the minimization problem, where the explicit gradient formula of the cost function is derived. To implement our method in real operation, the real form of our method is also presented. Finally, some numerical examples are given to illustrate the validity of the proposed method.

A Two-Timescale Duplex Neurodynamic Approach to Mixed-Integer Optimization.

This article presents a two-timescale duplex neurodynamic approach to mixed-integer optimization, based on a biconvex optimization problem reformulation with additional bilinear equality or inequality constraints. The proposed approach employs two recurrent neural networks operating concurrently at two timescales. In addition, particle swarm optimization is used to update the initial neuronal states iteratively to escape from local minima toward better initial states. In spite of its minimal system complexity, the approach is proven to be almost surely convergent to optimal solutions. Its superior performance is substantiated via solving five benchmark problems.

Low-Thrust Trajectory Design Using Finite Fourier Series Approximation of Pseudoequinoctial Elements

A new low-thrust trajectory design method is proposed that is based on the finite Fourier series method with pseudoequinoctial elements rather than the more common cylindrical coordinate components. The bijection relation between the elements and control variables is ensured by introducing an additional equality constraint derived from the angular momentum conservation. The guidance law and on-line control variables are obtained by applying inverse dynamics and the framework of inverse simulation technology, respectively. The pseudoequinoctial finite Fourier series method has the advantages of both the Fourier series and the perturbation analysis methods. For two-body problems, three cases were studied: the Earth to Mars, 1989ML, and Tempel-1 missions. Regarding the design of a rendezvous trajectory with a large inclination angle and a high eccentricity rate, this method yields a broader range of feasible results than the traditional Fourier series method. The circular restricted three-body problem was solved for the first time using the pseudoequinoctial finite Fourier series method combined with the patched conics method. The low-thrust Earth-Moon transfer was analyzed, and the results show that this method improves window analysis efficiency and guarantees precision of the initial geocentric trajectory for the low-thrust transfer.

Minimal Characterization of Shannon-Type Inequalities Under Functional Dependence and Full Conditional Independence Structures

The minimal set of Shannon-type inequalities (or elemental inequalities) plays a central role in efficiently determining whether a given inequality is in fact Shannon-type or not and in computing the linear programming bound for network coding capacity. In many cases, random variables under consideration are subject to additional constraints, such as functional dependence and conditional independence constraints. For example, functional dependence constraints are common in many communication problems due to deterministic encoding and decoding constraints. In other situations, the variables involved may form a Markov chain or in general a Markov random field, leading to conditional independence constraint. Subject to additional constraints, the challenge is how to identify the non-redundant inequalities. While one can always numerically determine the non-redundant inequalities (subject to additional linear equality constraints), it will be instrumental and also important if the non-redundant inequalities can be listed explicitly. In this paper, we show that this is achievable under the functional dependence and full conditional independence constraints.

On an optimal control problem with gradient constraints

ABSTRACTUsually, control functions in control-constrained optimal control are chosen from a Lebesgue space. This choice, however, makes it impossible to postulate additional conditions on the control function's slope which is practically relevant in some situations. In order to overcome this disadvantage, a natural assumption would be to demand at least first-order Sobolev regularity for control functions. The present paper is devoted to the study of an elliptic optimal control problem whose control function is chosen from a Sobolev space and has to satisfy additional equality constraints on its weak gradient. Noting that the associated Karush–Kuhn–Tucker conditions do not provide a necessary optimality condition for the underlying optimal control problem in general, one cannot simply solve the problem of interest by considering the system of first-order optimality conditions. Instead a penalization procedure with strong convergence properties for the computational solution is suggested and its computational implementation is studied in detail. Particularly, some essential difficulties arising from the gradient constraints which do not appear in standard optimal control are discussed.

Tools for handling steady‐state under‐frequency regulation in isolated microgrids

This study presents two approaches to handling under-frequency regulation in isolated microgrids. The first one is a governor power flow (PF) model in which the generators steady-state governor equations are modeled into a full Newton PF formulation resulting in an augmented sparse linearised system of equations to be solved at each iteration. As a result, it is possible to evaluate the under frequency problem in the case of microgrid islanding as well as to set the reference bus angle to evaluate the possibility of reconnection to the main grid. The second approach presents an optimal PF (OPF) formulation in which the generators governor equations are considered as additional equality constraints to determine the minimum amount of load shedding to keep the system frequency between specified limits. In both methods, it is also used an equation that provides voltage regulation in a droop-based control. The reactive power generation limits in the OPF approach are bounded by a proposed complementarity constraint-based method adapted for the generators' droop-controlled terminal voltage. The proposed approaches are evaluated and validated through the study of the EHV1 Network, a 61-bus 33/11-kV radial distribution system, and the New England test system.

New Results for the Handling of Additional Equality Constraints in One-Shot Optimization

The one-shot approach is often applied for design optimization tasks involving a slowly converging Newton-like solver for the underlying partial differential equations. The state solver is augmented with an adjoint solver to obtain reduced derivatives for an optimization step. The idea of the one-shot approach is to simultaneously pursue state and adjoint feasibility as well as optimality by using a suitable design space preconditioner. In various applications, further equality constraints additional to the partial differential equations describing the state are required. The one-shot approach can be extended to deal with additional equality constraints by introducing an additional update formula for the corresponding constraint multipliers. The choice of a suitable preconditioner for the design update as well as for the constraint multiplier update is important to achieve bounded retardation. We derive conditions for the preconditioners of the extended one-shot approach and propose a suitable preconditioner for the multiplier updates for the additional constraints.

An improved differential evolution algorithm for optimization including linear equality constraints

A differential evolution algorithm (DE) is proposed to exactly satisfy the linear equality constraints present in a continuous optimization problem that may also include additional non-linear equality and inequality constraints. The proposed DE technique, denoted by DELEqC-II, is an extension of a previous method developed by the authors. In contrast to the previous approach, it uses both mutation and crossover strategies that maintain feasibility with respect to the linear equality constraints. Also, a procedure to correct numerical errors detected in the previous approach was incorporated in DELEqC-II. In the numerical experiments, scalable test-problems with linear equality constraints are used to analyze the performance of the new proposal.

A one-shot optimization framework with additional equality constraints applied to multi-objective aerodynamic shape optimization

This paper concerns the implementation and application of the extended one-shot approach including additional equality constraints to achieve a direct transition from simulation to optimization. The approach can be applied for different areas of scientific computing where partial differential equations are treated by using a fixed-point solver. The solver is extended in a semi-automated fashion. In a first step it is augmented with a consistent adjoint solver using algorithmic differentiation. Then the obtained reduced derivative information is directly employed to simultaneously achieve optimality and primal as well as adjoint feasibility. The methodology is implemented in the multi-physics package SU2 and applied for multi-objective aerodynamic shape optimization.

On the Effect of Stabilization Methods for Quaternion Invariants on the Uncertainty in Optimization-based Estimation

Abstract The handling of 3D orientations is a common element in many problems that arise in the estimation and control of dynamic systems. Over-parametrizations such as unit quaternions are commonly used to avoid singularities but come with the property of an invariant which needs to be preserved. By using numerical optimization methods, these invariants are subject to numeric errors and require stabilization. In this work, we adopt methods known from optimal control for the problem of state estimation. We present an optimization-based attitude estimator using the measurements of an inertial measurement unit and evaluate the performance of a first-order stabilization of the invariant by modifying the dynamics. The uncertainties of the estimator are analyzed for different configurations of the proposed stabilization. Finally, we show how the stabilization affects the estimation of parameters and justify the use of an additional equality constraint for the invariant to yield more robust and consistent results.

Integrated online trajectory planning and optimization in distinctive topologies

This paper presents a novel integrated approach for efficient optimization based online trajectory planning of topologically distinctive mobile robot trajectories. Online trajectory optimization deforms an initial coarse path generated by a global planner by minimizing objectives such as path length, transition time or control effort. Kinodynamic motion properties of mobile robots and clearance from obstacles impose additional equality and inequality constraints on the trajectory optimization. Local planners account for efficiency by restricting the search space to locally optimal solutions only. However, the objective function is usually non-convex as the presence of obstacles generates multiple distinctive local optima.The proposed method maintains and simultaneously optimizes a subset of admissible candidate trajectories of distinctive topologies and thus seeking the overall best candidate among the set of alternative local solutions. Time-optimal trajectories for differential-drive and carlike robots are obtained efficiently by adopting the Timed-Elastic-Band approach for the underlying trajectory optimization problem. The investigation of various example scenarios and a comparative analysis with conventional local planners confirm the advantages of integrated exploration, maintenance and optimization of topologically distinctive trajectories.

Augmenting the one-shot framework by additional constraints

The (multistep) one-shot method for design optimization problems has been successfully implemented for various applications. To this end, a slowly convergent primal fixed-point iteration of the state equation is augmented by an adjoint iteration and a corresponding preconditioned design update. In this paper we present a modification of the method that allows for additional equality constraints besides the usual state equation. A retardation analysis and the local convergence of the method in terms of necessary and sufficient conditions are given, which depend on key characteristics of the underlying problem and the quality of the utilized preconditioner.