Min-max Model Research Articles

A Min–Max Model Predictive Control Approach to Robust Power Management in Ambulatory Wireless Sensor Networks

This paper addresses the problem of transmission power control within a network of resource-constrained wireless sensors that operate within a particular ambient healthcare environment. Sensor data transmitted to a remote base station within the network arrive subject to node location, orientation, and movement. Power is optimally allocated to all channels using a novel resource efficient algorithm. The proposed algorithm is based on a computationally efficient min-max model predictive controller that uses an uncertain linear state-space model of the tracking error that is estimated via local received signal strength feedback. An explicit solution for the power controller is computed offline using a multiparametric quadratic solver. It is shown that the proposed design leads to a robust control law that can be implemented quite readily on a commercial sensor node platform where computational and memory resources are extremely limited. The design is validated using a fully IEEE 802.15.4 compliant testbed using Tmote Sky sensor nodes mounted on fully autonomous MIABOT Pro miniature mobile robots. A repeatable representative selection of scaled ambulatory scenarios is presented that is quite typical of the data that will be generated in this space. The experimental results illustrate that the algorithm performs optimal power assignments, thereby ensuring a balance between energy consumption and a particular outage-based quality of service requirement while robustly compensating for disturbance uncertainties such as channel fading, interference, quantization error, noise, and nonlinear effects.

Viscoelastoplastic Model of FCC Monocrystals Deformation: Identification of Parameters

This article considers the description of elastoviscoplastic deformation for FCC crystallite. The deformation model for crystallite is necessary for building the model of a polycrystal. The elastoviscoplastic model is used to consider the crystalline structure using crystalline directions and also using the continuum mechanics approach – averaging by the representative macrovolume. FCC crystalline plastic deformations are considered as a slip of dislocations on slipping systems. А pure fcc cuprum is used аs a material. The algorithm for description of the proposed measures of the stress-strain state is considered, and hardening laws are analyzed. The problem of stochastic optimization of inelastic crystalline deformation is solved. Hardening modules and initial critical stresses are identified and accepted stochastically. The squared difference between the computational and experimental stress intensity for several points have been used as the target function. To solve the optimizing problem, the authors have used the min-max model considering the solution with the least favorable conditions for controlling vector. Nelder-Mead method (numerical method for direct search) is applied to solve the optimization problem. We have provided a description and analysis of the numerical results, recommendations for choosing the values of the identified parameters.

Off-Line Robust MPC Algorithm for VAV Air-Conditioning Systems

The efficient temperature control of variable air volume (VAV) air-conditioning system can greatly reduce the energy consumption. In order to due with system constraints and uncertainties, and guarantee the closed-loop stability, an off-line min-max robust model predictive control algorithm is presented for the temperature control of a VAV system. As the first-order-plus-time-delay model of system is described by a polyhedral described uncertain model, the off-line ellipsoidal invariant set based robust MPC algorithm is employed for the controller design. Simulation results show efficiency of the proposed control algorithm implemented in the temperature control of a VAV system, which can enhance robustness, and satisfy the constraints.

Cost Efficient Deployment and Reliable Routing Modeling Based Multi-Objective Optimization for Dynamic Wireless Body Sensor Networks Topology

Wireless Body Sensor Networks (WBSNs), like any other sensor networks, suffer from limited energy and are highly distributed network, where its nodes organize by themselves and each of them has the flexibility of collecting and transmitting patient biomedical information to a sink. When a knowledge was sent to a sink from a path that doesn't have a definite basis, the routing is a crucial challenge in Wireless Body Area Sensor Networks. Furthermore, reliability and routing delay are the considerable factors in these types of networks. More attention should be given to the energy routing issue and frequent topology's change in WBSNs. That increases the dynamics of network topology, and complicates the relay selection process in cooperative communications. Unreliable communication over the wireless channel complicates communication protocols and results in low data yield (Stathopoulos 2005). The deployment sensors step is a crucial and complex task due to several independent objectives and constraints. This paper presents a Min-Max multi-commodity flow model for WBSNs which allows preventing sensor node saturation and taking best action against reliability and the path loss, by imposing an equilibrium use of sensors during the routing process. This model is based on the authors' optimal sensors deployment method for WBSNs. Simulations results show that the algorithm balances the energy consumption of nodes effectively and maximize the network lifetime. It will meet the enhanced WBSNs requirements, including better delivery ratio, less reliable routing overhead.

Networked min–max model predictive control of constrained nonlinear systems with delays and packet dropouts

This article investigates a class of constrained nonlinear networked control systems (NCSs) subject to external disturbances, input and state constraints and network-induced constraints. From a practical perspective, the network-induced constraints considered include the time delays and packet dropouts on both the sensor-to-controller (S-C) channel and the controller-to-actuator (C-A) channel simultaneously. The min–max model predictive control method is proposed to design the control packets by incorporating the external disturbances into the optimisation problem. Moreover, the input-to-state practical stability of the resulting nonlinear NCS is established by constructing a novel Lyapunov function. Finally, the simulation results and the comparison studies are presented to demonstrate the effectiveness and improvement of the proposed method.

A note: Minmax due-date assignment problem with lead-time cost

All three major classes of due-date assignment models (CON, SLK and DIF) have been solved in the literature for a minsum setting, and only two of them (CON and SLK) have been solved for a minmax setting. In this note we introduce a solution for the missing minmax model of DIF. Specifically, we study a single-machine scheduling and due-date assignment problem, in which job-dependent lead-times are considered. Three cost components for each job are assumed: earliness cost, tardiness cost, and the cost for delaying the due-date (beyond its lead-time). The goal is to schedule the jobs and to assign due-dates, such that the maximum cost among all the jobs is minimized. We introduce an O(nlog2n) solution algorithm (where n is the number of jobs).

Computationally efficient nonlinear Min–Max Model Predictive Control based on Volterra series models—Application to a pilot plant

The mathematical model used in Min–Max MPC (MMMPC) to predict the future trajectory of the system explicitly considers disturbances and uncertainties. Based on the future trajectory, the control sequence is computed minimizing the worst case cost with respect to all possible trajectories of the disturbances and uncertainties. This approach leads to a more robust control performance but also complicates the practical implementation of MMMPC due to the high computational burden required to solve the optimization problem. This computational burden is even worse if a nonlinear prediction model is used. In fact, to the best of the authors’ knowledge, there have not yet been reported any applications of nonlinear MMMPC to real processes. In this paper a nonlinear MMMPC strategy based on a second order Volterra series model is presented. The particular structure of the used prediction model allows to obtain an explicit formulation of the worst case cost and its computation in polynomial time. Real time applications with typical prediction and control horizons are possible because of the reduced complexity of the proposed control strategy. Furthermore, input-to-state practical stability for the proposed control strategy is guaranteed under certain conditions. The MMMPC strategy is implemented and validated in experiments with a continuous stirred tank reactor whose temperature dynamics are approximated by a second order Volterra series model. The control performance of the proposed MMMPC strategy is illustrated by the obtained experimental results.

Liner ship route capacity utilization estimation with a bounded polyhedral container shipment demand pattern

This paper aims to estimate capacity utilization of a liner ship route with a bounded polyhedral container shipment demand pattern, arising in the liner container shipping industry. The proposed maximum and minimum liner ship route capacity utilization problems are formulated as a linear programming model and a min–max model, respectively. We examine two fundamental properties of the min–max model. These two nice properties enable us to develop two ε-optimal global optimization algorithms for solving the min–max model, which find a globally ε-optimal solution by iteratively cutting off the bounded polyhedral container shipment demand set with a cut. The latter algorithm overcomes non-convexity of the remaining feasible demand set generated by the former algorithm via a novel hyperplane cut. Each hyperplane cut can assure that the current vertex of the polyhedral demand set is cut off, whereas solutions that may improve the current one by more than a factor of ε are retained. Extensive numerical experiments for problems larger than those encountered in real applications demonstrate the computational efficacy of the latter algorithm.

The Explicit Constrained Min-Max Model Predictive Control of a Discrete-Time Linear System With Uncertain Disturbances

In this technical brief, we develop an algorithm to determine the explicit solution of the constrained min-max model predictive control problem. For a discrete-time linear system with bounded additive uncertain disturbance, the control law is determined to be piecewise affine from a quadratic cost function and the state space is partitioned into corresponding polyhedral cones. By moving the on-line implementation to an off-line explicit evaluation, the computational burden is decreased and the applicability of min-max optimization is broadened. The results of this approach are shown via computer simulations.

Solving continuous min max problem for single period portfolio selection with discrete constraints by DCA

In this article, we consider the application of a continuous min–max model with cardinality constraints to worst-case portfolio selection with multiple scenarios of risk, where the return forecast of each asset belongs to an interval. The problem can be formulated as minimizing a convex function under mixed integer variables with additional complementarity constraints. We first prove that the complementarity constraints can be eliminated and then use Difference of Convex functions (DC) programming and DC Algorithm (DCA), an innovative approach in non-convex programming frameworks, to solve the resulting problem. We reformulate it as a DC program and then show how to apply DCA to solve it. Numerical experiments on several test problems are reported that demonstrate the accuracy of the proposed algorithm.

Improved load balancing and resource utilization for the Skill Vehicle Routing Problem

The Skill Vehicle Routing Problem (Skill VRP) considers vehicle routing under the assumption of skill requirements given on demand nodes. These requirements have to be met by the serving vehicles. No further constraints, like capacity or cost restrictions, are imposed. Skill VRP solutions may show a tendency to have a bad load balancing and resource utilization. In a majority of solutions only a subset of vehicles is active. Moreover, a considerable share of demand nodes is served by vehicles that have a skill higher than necessary. A reason for that solution behavior lies in the model itself. As no resource restrictions are imposed, the Skill VRP tends to produce TSP-like solutions. To obtain better balanced solutions, we introduce two new approaches. First we propose a minmax model that aims at minimizing the maximal vehicle tour length. Second we suggest a two-step method combining the minmax approach with a distance constrained model. Our experiments illustrate that these approaches lead to improvements in load balancing and resource utilization, but, with different impact on routing costs.

Robust optimisation approach to the design of service networks for reverse logistics

In this paper we discuss a typical service network of a company providing repair and refurbishment services for its products. To optimise the network we propose a mixed-integer linear programming (MILP) model that minimises the total cost subject to flow balance and logical constraints. The solution of the model determines which facilities are to be used in the network and the associated flows. The model is validated against data obtained from a computer manufacturer and real-life sources. The optimal network configuration depends on several factors such as the number of products returned, percent of faulty products and warranty fraction of modules, which are uncertain in nature. We also address the issue of identifying these uncertain parameters, and their levels to be considered. A min-max robust optimisation model is then developed and solved to address these uncertainties.

A weighted min–max model for balanced freight train routing problem with fuzzy information

A multi-objective freight train routing problem with fuzzy information is investigated in this article. To handle the fuzziness in the railway transportation system, the measure ℳλ (i.e. the convex combination of a possibility measure and a necessity measure) is first introduced. Then, a min–max chance-constrained programming model is constructed to obtain optimal train routing plans. In order to solve the model, a potential route algorithm, fuzzy simulation and tabu search algorithm are integrated as a hybrid algorithm. Finally, some numerical experiments are performed to show the applications of the model and the algorithm.

Computational burden reduction in min–max MPC

Min–max model predictive control (MMMPC) is one of the strategies used to control plants subject to bounded uncertainties. The implementation of MMMPC suffers a large computational burden due to the complex numerical optimization problem that has to be solved at every sampling time. This paper shows how to overcome this by transforming the original problem into a reduced min–max problem whose solution is much simpler. In this way, the range of processes to which MMMPC can be applied is considerably broadened. Proofs based on the properties of the cost function and simulation examples are given in the paper.

The Robust Model of Continuous Transportation Network Design Problem with Demand Uncertainty

In the urban traffic network, the trip generations of every origin and trip attractions of every destination are closely related to the economic level of the traffic zone, land use, population, and the other factors, generally accurately forecast, although the demands between the origin and the destination are difficult to accurately predict. In this paper, we assume that the trip generations of every origin and trip attractions of every destination are deterministic, but the demands between every origin and destination are uncertain and belong to a bounded interval. We propose a min-max model of continuous network design with demand uncertainty under user equilibrium with robust optimizations. For our proposed model, we apply sensitivity analysis combined with the methods of sequence average algorithm to solve the robust model. Numerical examples demonstrate that the robust solutions of the continuous network design are more reliable than the deterministic solutions.

Min-Max Model Predictive Controller for Trajectory Tracking of a Wheeled Mobile Robot with Slipping Effects

A min-max model predictive controller is developed in this paper for tracking control of wheeled mobile robots (WMRs) subject to the violation of nonholonomic constraints in an environment without obstacles. The problem is simplified by neglecting the vehicle dynamics and considering only the steering system. The linearized tracking-error kinematic model with the presence of uncertain disturbances is formed in the frame of the robot. And then, the control policy is derived from the worst-case optimization of a quadratic cost function, which penalizes the tracking error and control variables in each sampling time over a finite horizon. As a result, the input sequence must be feasible for all possible disturbance realizations. The performance of the control algorithm is verified via the computer simulations with a predefined trajectory and is compared to a common discrete-time sliding mode control law. The result shows that the proposed method has a better tracking performance and convergence.

A general framework for explaining the results of a multi-attribute preference model

The automatic generation of an explanation of the prescription made by a multi-attribute decision model is crucial in many applications, such as recommender systems. This task is complex since the quantitative models are not designed to be easily explainable. The major limitation of the previous research is that there is no formal justification of the arguments that are selected in the explanation. The goal of this paper is to define a general framework to justify which arguments shall be selected, in the case where the decision model is based on weights assigned to the attributes. Due to the complexity of explaining a preference model based on utility theory, several explanation reasonings are necessary to cover all cases – ranging from situations where the prescription is trivial to situations where the prescription is much more tight. The set of selected arguments is, in this framework, a non-dominated element of a combinatorial structure in the sense of an order relation. Our general approach is instantiated precisely on three models: the probabilistic expected utility model, the qualitative pessimistic minmax model and the concordance rule, which are all constructed from a weight vector.

Input‐to‐state stability of min‐max MPC scheme for nonlinear time‐varying delay systems

AbstractThis paper studies the robustness problem of the min–max model predictive control (MPC) scheme for constrained nonlinear time‐varying delay systems subject to bounded disturbances. The notion of the input‐to‐state stability (ISS) of nonlinear time‐delay systems is introduced. Then by using the Lyapunov–Krasovskii method, a delay‐dependent sufficient condition is derived to guarantee input‐to‐state practical stability (ISpS) of the closed‐loop system by way of nonlinear matrix inequalities (NLMI). In order to lessen the online computational demand, the non‐convex min‐max optimization problem is then converted to a minimization problem with linear matrix inequality (LMI) constraints and a suboptimal MPC algorithm is provided. Finally, an example of a truck‐trailer is used to illustrate the effectiveness of the proposed results.Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Min-Max Fair Power Flow Tracing for Transmission System Usage Cost Allocation: A Large System Perspective

Power flow tracing has been suggested as an approach for evaluating 1) transmission system usage (TSU) cost and 2) loss (MW) cost for generator and load entities in the system. Recently, optimal power flow tracing methods have been proposed to “explicitly” model fairness constraints in the tracing framework. This paper, further, strengthens the tracing-compliant min-max fair cost allocation approach. The min-max model proposed in this paper is robust. It addresses concerns like scalability, numerical stability and termination in a finite number of steps while searching the optimal solution. We also propose a methodology to model DISCOMs and GENCOs as coalition within min-max framework. Case studies on an all India network of 1699 nodes and comparison with average participation and marginal participation methods bring out the better conflict resolution feature of the proposed approach. A method to model HVDC lines within the marginal participation scheme is also proposed. Quantitative and qualitative comparison of various TSU cost allocation methods on such a large system is another noteworthy contribution of the paper.

Feedback min–max model predictive control using robust one-step sets

A solution to the infinite-horizon min–max model predictive control (MPC) problem of constrained polytopic systems has recently been defined in terms of a sequence of free control moves over a fixed horizon and a state feedback law in the terminal region using a time-varying terminal cost. The advantage of this formulation is the enlargement of the admissible set of initial states without sacrificing local optimality, but this comes at the expense of higher computational complexity. This article, by means of a counterexample, shows that the robust feasibility and stability properties of such algorithms are not, in general, guaranteed when more than one control move is adopted. For this reason, this work presents a novel formulation of min–max MPC based on the concept of within-horizon feedback and robust contractive set theory that ensures robust stability for any choice of the control horizon. A parameter-dependent feedback extension is also proposed and analysed. The effectiveness of the algorithms is demonstrated with two numerical examples.