Slack Variables Research Articles

An LMI-Based Algorithm to Compute Robust Stabilizing Feedback Gains Directly as Optimization Variables

This article addresses the problem of robust stabilization of uncertain linear systems by static state- and output-feedback control laws. The uncertainty is supposed to belong to a polytope and both continuous- and discrete-time systems are investigated. Contrarily to the main stream of the linear matrix inequality (LMI)-based robust stabilization methods available in the literature, where the product between the Lyapunov (or slack variable) and control gain matrices is transformed in a new variable, this article proposes a change of paradigm, avoiding the standard change of variables and providing synthesis conditions that deal directly with the control gain as an optimization variable. The design procedure is formulated in terms of a locally convergent iterative algorithm based on LMIs, having as main novelties the following points: Both the Lyapunov and the closed-loop dynamic matrices appear affinely in the conditions; the iterative scheme involves the slack variables only, avoiding the classic alternation between the Lyapunov matrix and the control gain; an extra degree of freedom is created in terms of an additional scalar variable, which also appears affinely in the conditions and represents a scaling on the closed-loop dynamic matrix; a smart termination for the algorithm through a stability analysis condition. Numerical experiments based on exhaustive simulations show that all these features combined provide a robust stabilization technique capable to outperform the best available methods in the literature in terms of effectiveness, being specially suitable to address static output-feedback and decentralized control problems.

Improved Model Predictive-Based Underwater Trajectory Tracking Control for the Biomimetic Spherical Robot under Constraints

To improve the autonomy of the biomimetic sphere robot (BSR), an underwater trajectory tracking problem was studied. Considering the thrusters saturation of the BSR, an improved model predictive control (MPC) algorithm that features processing multiple constraints was designed. With the proposed algorithm, the kinematic and dynamic models of the BSR are combined in order to establish the predictive model, and a new state-space model is designed that is based on an increment of the control input. Furthermore, to avoid the infeasibility of the cost function in the MPC controller design, a new term with a slack variable is added to the objective function, which enables the constraints to be imposed as soft constraints. The simulation results illustrate that the BSR was able to track the desired trajectory accurately and stably while using the improved MPC algorithm. Furthermore, a comparison with the traditional MPC shows that the designed MPC-based increment of the control input is small. In addition, a comparative simulation using the backstepping method verifies the effectiveness of the proposed method. Unlike previous studies that only focused on the simulation validations, in this study a series of experiments were carried out that further demonstrate the effectiveness of the improved MPC for underwater trajectory tracking of the BSR. The experimental results illustrate that the improved MPC is able to drive the BSR to quickly track the reference trajectory. When compared with a traditional MPC and the backstepping method used in the experiment, the proposed MPC-based trajectory is closer to the reference trajectory.

An analytic solution for real-time bus holding subject to vehicle capacity limits

This study focuses on single variable optimization approaches which determine the holding time of a vehicle when it is ready to depart from a bus stop. Up to now, single variable optimization methods resort to rule-based control logics to equalize the inter-departure headways or adhere to the target headway values. One of them is the two-headway-based control logic which determines the holding time of a bus based on its headway with its preceding and following bus without addressing other implications, such as overcrowding. To rectify this, we introduce a new model for the single variable bus holding problem that considers the passenger demand and vehicle capacity limits. Then, we reformulate this problem to an easier-to-solve program with the use of slack variables and introduce an analytic solution that can determine the holding time of a vehicle at the respective bus stop. Our analytic solution does not add a computational burden to the two-headway-based control logic and can be applied in real time. The operational benefit of our bus holding approach compared to other analytic solutions that do not consider the vehicle capacity is investigated using actual data from bus line 302 in Singapore.

Slack-variable models versus component-proportion models for mixture experiments: Literature review, evaluations, and recommendations

A mixture experiment (ME) combines components in various proportions and observes the values of one or more responses for each mixture. The proportions of the q components in each mixture must sum to 1.0. Two statistical approaches for modeling MEs have been discussed in the literature. The slack-variable (SV) approach uses proportions of all but one (q − 1) of the components varied in a mixture (the SV). The component-proportion (CP) approach uses the relative proportions of all q components varied in a mixture. Several articles have claimed advantages and recommended the SV modeling approach over the CP modeling approach. In contrast, a 2009 article recommended the CP modeling approach for several reasons. Our article reviews the literature, evaluates the literature justifications for using the SV modeling approach, and uses literature examples to compare the CP and SV modeling approaches. Recommendations regarding when to use the CP and SV modeling approaches are provided.

Asymptotic stability and stabilization of discrete‐time switched systems with time‐varying delay

SummaryThis article deals with the problems of stability and stabilization for a class of discrete‐time switched systems with time‐varying delay under arbitrary switching signal. First, sufficient conditions guaranteeing the asymptotic stability of the unforced system are developed, by using the switched Lyapunov‐Krasovskii functional method. Then, based on the obtained results, a state feedback controller is designed in order to ensure the asymptotic stability of the closed‐loop system. Furthermore, slack variables are introduced for more relaxation. The proposed approach is proved to have some less conservative results than some existing works. Finally, numerical examples are provided to illustrate the effectiveness and the merit of the proposed method and to compare the obtained results with some previous works in the literature.

Robust Control of Synchronous Reluctance Motors by Means of Linear Matrix Inequalities

This article proposes an alternative for robust control of synchronous reluctance motors using linear matrix inequalities to obtain fixed gains that ensure current and speed regulation, under parameter uncertainties and variations. The proposed design procedure is based on a polytopic model, which takes into account: i) uncertainties and variations of the mechanical and electrical parameters of the motor, ii) one step delay from digital implementation of the control law, and iii) internal model based controllers to guarantee reference tracking. Less conservative linear matrix inequalities, relying on slack variables, ensure robust pole location and certify the closed-loop stability for the entire domain of uncertain and time-varying motor parameters, based on parameter-dependent Lyapunov functions. The linear matrix inequalities ensure a fast and systematic computation of the control gains, that are also easy to be implemented, not adding computational complexity when compared, for instance, to usual strategies based on PI controllers. Experimental results for a 2.2 kW commercial motor are provided, validating the proposed procedure, and illustrating good tracking of currents and speed references, with suitable dynamic responses and reduced cross-coupling effects, when compared to sliding mode and PI controllers.

Urban sustainability assessment: The evaluation of coordinated relationship between BRTS and land use in transit-oriented development mode using DEA model

The present study represents the novel methodology for quantitative evaluation of the coordinated relationships (CR) between BRTS and land use under transit-oriented development (TOD) mode by using the data envelopment analysis (DEA) model. By using the most robust indicators of the BRTS and land use as input-output. Ranking assigned to the TOD area according to their coordinated development score. The superefficiency DEA method is used as a ranking methodology to rank the extremely efficient TOD area. With the help of DEA projection theory by utilizing the slack variables, the target value is assigned to inefficient TOD area to make it efficient. This methodology applies to 16 TOD area in Bhopal city, India. It is an adaptive method to find out the CR between BRTS and land use. Measuring CR is useful for efficient and sustainable urban planning. This study will provide sustainable station level planning for urban planners and experts.

Relaxed State and Fault Estimation for Vehicle Lateral Dynamics Represented by T–S Fuzzy Systems

This paper deals with the problem of observer design of the vehicle model which is represented by Takagi–Sugeno (T–S) fuzzy systems with the presence of uncertainties. A relaxed observer design is presented to estimate the unmeasurable states and the faults of the vehicle lateral dynamics model, simultaneously. The vehicle model is transformed into a system with unknown inputs. Then, it is rebuilt by adding the default to the system state equation. Based on the Lyapunov function approach and the introduction of some slack variables, sufficient conditions of unknown input observer design are formulated as Linear Matrix Inequalities (LMIs). Finally, the simulation section clearly shows the importance and effectiveness of the proposed strategy.

Improved multiobjective switching gain‐scheduled controller synthesis exploiting inexact scheduling parameters

SummaryIn this article, the design problem of switching gain‐scheduled output‐feedback (SGSOF) controller with mixed performance exploiting inexact scheduling parameters for continuous‐time linear parameter‐varying (LPV) systems is tackled. The absolute and proportional uncertainties on the scheduling parameters are considered. All the system matrices of the LPV plant are supposed to have polynomial dependency of arbitrary degree on the scheduling parameters. By exploiting hysteresis switching law and utilizing two independent families of parameter‐dependent Lyapunov matrices and a family of slack variables, introducing additional degrees of freedom, performance improvement is achieved. A design approach, involving a two‐step algorithm, is proposed in terms of solutions to a set of parameter‐dependent linear matrix inequalities (LMIs) including parameter searches for two scalar values. The merit of the proposed method lies in its less conservativeness in comparison with the available approaches in the literature. Numerical studies are carried out to demonstrate the effectiveness of the presented method.

Load Balancing in Low-Voltage Distribution Network via Phase Reconfiguration: An Efficient Sensitivity-Based Approach

Operational performance in the low-voltage distribution network (LVDN) can be undermined by its inherent unbalances, which may become worse as the penetration of rooftop solar continuously increases. To address this issue, load balancing via phase-reconfiguration devices (PRDs), which can change phase positions of residential customers as required, provides a cost-efficient option. However, most reported approaches to control PRDs require that demands of all residential customers are available, which are not viable for many LVDNs without smart meters or advanced metering infrastructure (AMI) installed. To bridging the gap in this field, this paper proposes a novel method to control PRDs purely based on measurable data from PRDs, and its controller. Based on limited information, sensitivity analysis in the network with PRDs is studied, followed by the optimization model that comprehensively considers operational requirements in the network. Moreover, slack variables are introduced to the model, and penalized in the objective function to assure either a strategy that is secure or with minimized violations can always be provided. The model is a challenging mixed-integer non-convex programming (MINCP) problem, which is reformulated as an efficient solvable mixed-integer second-order cone programming (MISOCP) based on exact reformulations or accurate linear approximations. Simulations based on two modified IEEE systems, and a real system in Australia demonstrate that an efficient strategy can be provided to mitigate unbalances in the network.

A soft-margin convex polyhedron classifier for nonlinear task with noise tolerance

As a special form of piecewise linear classifier, the convex polyhedron classifier is simple to implement and achieves rapid response in real-time classification. However, it usually performs badly in the case of high noise where severe boundary intrusion exists. Inspired by the scheme of soft margin in support vector machine, in this paper we propose a soft-margin convex polyhedron classifier for nonlinear classification task. The base (linear) classifier is first generalized to its soft-margin version through kernelization process and slack variables. In each local region, the soft-margin base classifier learns a decision hyperplane with noise tolerance. Then, a series of learned hyperplanes are structurally integrated into a convex polyhedron classifier, which is essentially a convex polyhedron that encloses one class and excludes the other class outside. Experimental results on fifteen benchmark datasets show the proposed soft-margin convex polyhedron classifier is comparable to linear support vector machine and four piecewise linear classifiers, but does not perform as well as the support vector machine with radial basis function kernel in general. When random noises are added to datasets, the soft-margin convex polyhedron classifier achieves similar or better accuracies with the well-known classifiers used for comparison, implying its promising ability of noise tolerance.

Density-weighted support vector machines for binary class imbalance learning

In real-world binary classification problems, the entirety of samples belonging to each class varies. These types of problems where the majority class is notably bigger than the minority class can be called as class imbalance learning (CIL) problem. Due to the CIL problem, model performance may degrade. This paper presents a new support vector machine (SVM) model based on density weight for binary CIL (DSVM-CIL) problem. Additionally, an improved 2-norm-based density-weighted least squares SVM for binary CIL (IDLSSVM-CIL) is also proposed to increase the training speed of DSVM-CIL. In IDLSSVM-CIL, the least squares solution is obtained by considering 2-norm of slack variables and solving the primal problem of DSVM-CIL with equality constraints instead of inequality constraints. The basic ideas behind the algorithms are that the training datapoints are given weights during the training phase based on their class distributions. The weights are generated by using a density-weighted technique (Cha et al. in Expert Syst Appl 41(7):3343–3350, 2014) to reduce the effects of CIL. Experimental analyses are performed on some interesting imbalanced artificial and real-world datasets, and their performances are measured using the area under the curve and geometric mean (G-mean). The results are compared with SVM, least squares SVM, fuzzy SVM, improved fuzzy least squares SVM, affinity and class probability-based fuzzy SVM and entropy-based fuzzy least squares SVM. Similar or better generalization results indicate the efficacy and applicability of the proposed algorithms.

Least squares KNN-based weighted multiclass twin SVM

K-nearest neighbor (KNN) based weighted multi-class twin support vector machines (KWMTSVM) is a novel multi-class classification method. In this paper, we propose a novel least squares version of KWMTSVM called LS-KWMTSVM by replacing the inequality constraints with equality constraints and minimized the slack variables using squares of 2-norm instead of conventional 1-norm. This simple modification leads to a very fast algorithm with much better results. The modified primal problems in the proposed LS-KWMTSVM solves only two systems of linear equations whereas two quadratic programming problems (QPPs) need to solve in KWMTSVM. The proposed LS-KWMTSVM, same as KWMTSVM, employed the weight matrix in the objective function to exploit the local information of the training samples. To exploit the inter class information, we use weight vectors in the constraints of the proposed LS-KWMTSVM. If any component of vectors is zero then the corresponding constraint is redundant and thus we can avoid it. Elimination of redundant constraints and solving a system of linear equations instead of QPPs makes the proposed LS-KWMTSVM more robust and faster than KWMTSVM. The proposed LS-KWMTSVM, commensurate as the KWMTSVM, all the training data points into a “1-versus-1-versus-rest” structure, and thus our LS-KWMTSVM generate ternary output {-1,0,+1} which helps to deal with imbalance datasets. Numerical experiments on several UCI and KEEL imbalance datasets(with high imbalance ratio) clearly indicate that the proposed LS-KWMTSVM has better classification accuracy compared with other baseline methods but with remarkably less computational time.

Bidirectional smart charging of electric vehicles considering user preferences, peer to peer energy trade, and provision of grid ancillary services

Electric vehicles (EVs) are witnessing increased utilization throughout the world as an alternative to fossil–fueled vehicles. The extensive deployment of EVs can bring challenges to the grid if not properly integrated. Such challenges, however, could be exploited as opportunities if the huge unused capacity of the battery storage in millions of EVs are utilized for ancillary services to the grid and peer–to–peer (PtP) energy trade. Given that there is at least one human user per vehicle, human input must be considered to improve the scheduling process. To that end, this paper presents a new algorithm for bidirectional smart charging of EVs considering user preferences, PtP energy trade, and provision of ancillary services to the grid. The preferences of an EV user as input to the model are embedded into the scheduling process enabling the model to be adaptive to various conditions. Optimization slack variables are utilized for optimal management of EV battery SOC and energy allocation for multiple services. New indices are developed and introduced for quantification of the EV participation in ancillary services and PtP transactions. Real–world data has been collected and utilized for model specification and simulation to make the assumptions more realistic. The efficacy and feasibility of the proposed model are validated using numerical studies. The results indicate that incorporating users’ preferences into the scheduling process would improve the aggregated revenue generated by the EV scheduling model which in turn could offset the charging costs by up to 100%. Further, an increase of about 90% in peer-to-peer energy transactions among EVs and 11% in ancillary services provision to the grid are achieved through the developed user-centric smart charging model.

Adaptive and efficient high-order rating distance optimization model with slack variable

The classical high-order rating distance model which aims to minimize not only (i) the difference between the estimated and real ratings of the same user–item pair (i.e., the first-order rating distance), but also (ii) the difference between the estimated and real rating difference of the same user across different items (i.e., the second-order rating distance), and use stochastic gradient descent to solve this convex optimization problem in recommender systems, has good performance in prediction accuracy. However, when the manually set parameter for the second-order rating difference is fixed, this model will not converge as the size of dataset increasing, and its performance on efficiency is slow compared with the matrix factorization method. Aiming at improving such model’s adaptability and efficiency, we propose an improved high-order rating distance model with omitting rules based on slack variable, in which the static parameter used to balance the first-order rating distance and the second-order rating distance is replaced by a data-scale sensitive function. We choose Newton method to solve the convex recommendation optimization problem defined in this paper instead of stochastic gradient descent. Our model not only achieves the adaptability by eliminating several static parameters for module balancing, reduces the computation complexity, but also accelerates the optimization function convergence speed. We provide solid theoretical support and conduct comprehensive experiments on four real-world datasets. Experimental results show the proposed model has good performance in terms of prediction accuracy and efficiency.

Model predictive control‐based optimal voltage regulation of active distribution networks with OLTC and reactive power capability of PV inverters

This study presents a model predictive control (MPC)-based centralised control approach for maintaining the voltages of the nodes within permissible limits in the presence of high photovoltaic (PV) penetration. The proposed control scheme coordinates the actions of on-load tap changer (OLTC) and PV inverters optimally to fulfil the desired objectives. The objectives are minimisation of change in control variables, slack variables, energy loss, and voltage error. These objectives are weighted to form the overall objective function. Three rules are formulated based on the severity of voltage magnitudes. The weights of the objectives are adjusted according to these pre-defined rules. Simulations are performed in an active distribution network (ADN) integrated with and without microgrids, where both demands and generations vary hourly over the day. As power is injected by the microgrids during most of the time of the day, the excursions of node voltages are slightly higher in the microgrids integrated ADN. Moreover, the incorporation of the proposed rule-based MPC drastically reduces the energy loss due to active power loss in distribution networks, as is evident from the simulation results obtained by comparing the proposed approach with an existing MPC-based approach.

Extreme events in time series aggregation: A case study for optimal residential energy supply systems

To account for volatile renewable energy supply, energy systems optimization problems require high temporal resolution. Many models use time-series clustering to find representative periods to reduce the amount of time-series input data and make the optimization problem computationally tractable. However, clustering methods remove peaks and other extreme events, which are important to achieve robust system designs. This work addresses the challenge of including extreme events. We present a general decision framework to include extreme events in a set of representative periods. We introduce a method to find extreme periods based on the slack variables of the optimization problem itself. Our method is evaluated and benchmarked with other extreme period inclusion methods from the literature for a design and operations optimization problem: a residential energy supply system. Our method ensures feasibility over the full input data of the residential energy supply system although the design optimization is performed on the reduced data set.We show that using extreme periods as part of representative periods improves the accuracy of the optimization results by 3% to more than 75% depending on system constraints compared to results with clustering only, and thus reduces system cost and enhances system reliability.

Dynamic Output Feedback H∞ Control for Linear Parameter-varying Systems with Time-delay

In this paper, we address a synthesis problem of the parameter-dependent output feedback H∞ control for linear parameter-varying systems with time-delay. The scheme adopts the parameter-dependent past state information to construct the dynamic output feedback controller. In this case, on basis of the quadratic Lyapunov functional with parameter-dependence, we analyze the parameter-dependent H∞ stability conditions for the closed-loop time-delayed linear parameter-varying system in terms of linear matrix inequalities. However, this stability condition is of an infinite-dimension. To derive computationally tractable criteria for the dynamic output feedback controller, several slack variables and a convex relaxation technique are employed to have the infinite-dimensional condition of linear matrix inequalities cast into a finite dimensional convex optimization problem. By solving the convex optimization problem, we harvest the dynamic output feedback controller with memory for the time-delayed linear parameter-varying system. Finally, two examples are included to illustrate the effectiveness of the proposed approach.

Weak Disambiguation for Partial Structured Output Learning.

Existing disambiguation strategies for partial structured output learning just cannot generalize well to solve the problem that there are some candidates that can be false positive or similar to the ground-truth label. In this article, we propose a novel weak disambiguation for partial structured output learning (WD-PSL). First, a piecewise large margin formulation is generalized to partial structured output learning, which effectively avoids handling a large number of candidate-structured outputs for complex structures. Second, in the proposed weak disambiguation strategy, each candidate label is assigned with a confidence value indicating how likely it is the true label, which aims to reduce the negative effects of wrong ground-truth label assignment in the learning process. Then, two large margins are formulated to combine two types of constraints which are the disambiguation between candidates and noncandidates, and the weak disambiguation for candidates. In the framework of alternating optimization, a new 2n -slack variables cutting plane algorithm is developed to accelerate each iteration of optimization. The experimental results on several sequence labeling tasks of natural language processing show the effectiveness of the proposed model.

Factorization of Noncommutative Polynomials and Nullstellensätze for the Free Algebra

Abstract This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots ,x_g)$ is $\mathscr{Z}(\,f)=(\mathscr{Z}_n(\,f))_n$, where $\mathscr{Z}_n(\,f)=\{X \in{\operatorname{M}}_{n}({\mathbb{C}})^g \colon \det f(X) = 0\}.$ The 1st main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $\mathscr{Z}_n(\,f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $\mathscr{Z}^{{\operatorname{re}}}(h)=\{X\colon \det h(X,X^*)=0\}$. A polynomial $f$ that depends on $x$ alone (no $x^*$ variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic $f$ but for $h$ dependent on possibly both $x$ and $x^*$, the containment $\mathscr{Z}(\,f) \subseteq \mathscr{Z}^{{\operatorname{re}}} (h)$ is equivalent to each factor of $f$ being “stably associated” to a factor of $h$ or of $h^*$. For perspective, classical Hilbert-type Nullstellensätze typically apply only to analytic polynomials $f,h $, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above “algebraic certificate” does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018): 589–626) obtained such a theorem for special classes of analytic polynomials $f$ and $h$. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form. Finally, the paper gives a Nullstellensatz for zeros ${\mathcal{V}}(\,f)=\{X\colon f(X,X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.