Minimax Control Research Articles

Minimax Multivariate Control Chart Using a Polynomial Function

Minimax control chart uses the joint probability distribution of the maximum and minimum standardized sample means to obtain the control limits for monitoring purpose. However, the derivation of the joint probability distribution needed to obtain the minimax control limits is complex. In this paper the multivariate normal distribution is integrated numerically using Simpson’s one third rule to obtain a non-linear polynomial (NLP) function. This NLP function is then substituted and solved numerically using Newton Raphson method to obtain the control limits for the minimax control chart. The approach helps to overcome the problem of obtaining the joint probability distribution needed for estimating the control limits of both the maximum and the minimum statistic for monitoring multivariate process.

Quasi-Model-Based Control of Type 1 Diabetes Mellitus

Glucose-insulin models appeared in the literature are varying in complexity. Hence, their use in control theory is not trivial. The paper presents an optimal controller design framework to investigate the type 1 diabetes from control theory point of view. Starting from a recently published glucose-insulin model a Quasi Model with favorable control properties is developed minimizing the physiological states to be taken into account. The purpose of the Quasi Model is not to model the glucose-glucagon-insulin interaction precisely, but only to grasp the characteristic behavior such that the designed controller can successfully regulate the unbalanced system. Different optimal control strategies (pole-placement, LQ, Minimax control) are designed on the Quasi Model, and the obtained controllers' applicability is investigated on two more sophisticated type 1 diabetic models using two absorption scenarios. The developed framework could help researchers engaging the control problem of diabetes.

Quasi Model Based Optimal Control of Type 1 Diabetes Mellitus

AbstractThe paper presents an optimal controller design framework to investigate the type 1 diabetes from control theory point of view. The starting point of the problem is based on a recently published glucose-insulin model (Liu and Tang [2008]). Based on this a Quasi Model (a type 1 diabetes linear model) with favorable control properties is developed minimizing the physiological states to be taken into account. Consequently, different optimal control strategies (LQ, minimax control) can be designed on the Quasi Model, and the obtained controller applicability is investigated on more sophisticated type 1 diabetic models. It has to be remarked that it is not the purpose of the Quasi Model to model precisely the glucose-glucagon-insulin interaction, only to grasp the characteristic behavior such that the designed controller can successfully regulate the unbalanced system. In order to validate the applicability of this method, absorption scenarios taken from the model of Dalla Man et al. [2007] are generated and the controller is tested on the modified Liu-Tang model (Liu and Tang [2008]) as well as on the modified Sorensen model (Parker et al. [2000]).

Seminar “Modern navigation and motion control methods”

On April 1, 2010, the Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences hosted the seminar “Modern Navigation and Motion Control Methods.” It was the fifth such seminar in ICS RAS. A history and a topical survey of these seminars are given in [1]. In 2010, the seminar had a thematic tagline “Information Processing Models and Methods in Motion Control Problems” and was devoted to the 80th birthday of Rudolf Emil Kalman and the 50th anniversary of his influential paper [2] that determined the development of filtering theory in automated control problems for a long time and began a huge series of works devoted to solving the linear–quadratic problem in presence of random white noise. The seminar traditionally consisted of two sessions, morning and evening. There were 11 talks presented at the seminar, with more than 200 participants. The morning session was opened by the head of the Section of mechanical engineering and control processes of the OEMMPU RAS, academician E.A. Fedosov, who emphasized the importance of this topic and wished the participants a successful seminar. Then, director of the Trapeznikov Institute of Control Sciences of the RAS, academician S.N. Vasilyev took the floor. In his address, Vasilyev presented the most important dates of R.E. Kalman’s life and work, listed his major achievements and recognitions. In particular, it was noted that Kalman is an honorary member of the Russian academy of sciences. The vice director of the Trapeznikov Institute of Control Sciences of the RAS E.Ya. Rubinovich delivered a talk entitled “50 years of the Kalman filter.” The talk contained a historical survey of Kalman linear filtering and the theory of solving the linear–quadratic Gaussian (LQG) control problem. Results of A.N. Kolmogorov and N. Wiener that foreshadowed Kalman theory were discussed. The talk also touched upon the geometry of linear estimation, recursive estimation, the Kalman–Bucy filter for both discrete and continuous time. The talk also considered generalizations of the Kalman–Bucy filter and especially emphasized the generalizations made over the last 40 years in the ICS RAS. All generalizations considered in the talk dealt with the equations that describe the control process and did not concern the changes in the quality criterion. The latter remained an integral of some quadratic form throughout the entire process. The report of A.P. Kurdyukov (ICS RAS) “From LQG theory to minimax filtering and control,” on the other hand, was devoted to constructing a control under changing quality criteria in the problem of minimizing the influence of external disturbances. This report outlined quality criteria suggested for the problem of minimizing the influence of external disturbances which have gathered popularity in the 50 years that have passed since the solution of the LQG problem was published. For instance, the report included solutions to the LEQG (linear–exponential–quadratic Gaussian) problem, risk sensitivity problem, and H∞ optimization problem. The talk also touched upon advantages and disadvantages of each of these problems in case the assumptions concerning the

On a differential interception game

We consider an interception game and discuss, using this particular example, aspects of the formalization of differential games that is being developed in Yekaterinburg. We establish both the functional character of the value of a game as a function of the current history of a process and the positional character of optimal strategies. We develop a unification of the original game, thus relating the descriptive theorem on the value and the saddle point of the game to the formalism of Hamilton-Jacobi equations and further to generalized minimax solutions to these equations. We emphasize the relationship between the unified form of a differential game and L.S. Pontryagin’s concept of differential game. On the basis of an auxiliary boundary value problem for a degenerate parabolic equation, we develop and substantiate minimax and maximin controls in a feedback scheme with a stochastic guide. Finally, we discuss an algorithm for calculating the value and the saddle point of a game.

Min–max control using parametric approximate dynamic programming

This study presents a computationally efficient approximate dynamic programming approach to control uncertain linear systems based on a min–max control formulation. The optimal cost-to-go function, which prescribes an optimal control policy, is estimated using piecewise parametric quadratic approximation. The approach requires simulation or operational data only at the bounds of additive disturbances or polyhedral uncertain parameters. This strategy significantly reduces the computational burden associated with dynamic programming and is not limited to a particular form of performance criterion as in previous approaches.

Revisited LQ output-feedback control: minimax controller for a set of initial states

A minimax linear quadratic (LQ) output-feedback controller is introduced which minimises the maximal value of the performance index over all initial states belonging to some set separated out a priori. If the set is an ellipsoid or a polygon, such controllers are synthesised in terms of linear matrix inequalities (LMIs). In particular case when a size of this set tends to zero tightening to a point, the minimax LQ controller approaches the optimal LQ output-feedback controller for the given initial state, while in another extreme case when this size tends to infinity, we have the worst-case LQ output-feedback controller. Numerical results for an inverted pendulum are presented.

Bayesian Robustness in the Control of Gene Regulatory Networks

The errors originating in the data extraction process, gene selection and network inference prevent the transition probabilities of a gene regulatory network from being accurately estimated. Thus, it is important to study the effect of modeling errors on the final outcome of an intervention strategy and to design robust intervention strategies. Two major approaches applied to the design of robust policies in general are the minimax (worst case) approach and the Bayesian approach. The minimax control approach is typically conservative because it gives too much importance to the scenarios which hardly occur in practice. Consequently, in this paper, we formulate the Bayesian approach for the control of gene regulatory networks. We characterize the errors emanating from the data extraction and inference processes and compare the performance of the minimax and Bayesian designs based on these uncertainties.

Discrete control and algorithms for solving antagonistic dynamic games on networks

The concept of antagonistic games for classical discrete control problems is applied and new classes of zero-sum dynamic games on networks are formulated and studied. Polynomial-time algorithms for solving max–min paths problem on networks are proposed and their applications (which might occur within certain financial applications) for solving max–min control problems and determining optimal strategies in zero-sum cyclic games are described. In addition max–min control problems with infinite time horizons which lead to cyclic games are studied and polynomial-time algorithm for solving zero value cyclic games is proposed.

Comparison between Skyhook and Minimax Control Strategies for Semi-active Suspension System

Comparison between Skyhook and Minimax Control Strategies for Semi-active Suspension System

Evaluation of Disturbance Weighting Parameter of Minimax Attenuation Problems

Abstract: This article presents a simple evaluation of the disturbance weighting parameter of well‐known minimax disturbance attenuation problems derived for earthquake‐excited structures. It is concluded from a careful analysis of the derived closed form solutions that linear quadratic regulator (LQR) performance which corresponds to white noise assumption is the best performance of the sub‐optimal minimax control (OMC) although it is also the worst performance of the OMC and the disturbance weighting parameter is as a measure of the system stability. For the completeness of the article, OMC is also compared to LQR and unimplementable optimal control.

Robust forecasts and run-to-run control for processes with linear drifts

Processes experiencing linear drift over time are usually forecasted using a double exponentially weighted moving average (d-EWMA) filter. d-EWMA has incorrectly been claimed as an optimal filter for the integrated moving average (2,2) (IMA(2,2)) process, which is a stochastic equivalent of a process with linear drifts (ramps). It is shown that the optimal filter for such a process has a different structure but can be put in a similar form with same effective tuning parameters. The problem of batch-to-batch process gain variation (with known bounds) has been addressed by using a robust run-to-run control algorithm. This algorithm solves a minimax problem that determines the next run input adjustment by minimizing the worst-case predicted error. The conditions for equivalence of the minimax controller to a nominal model inverse based controller for a simple SISO system based on the type of model used and the nature of bounds have been investigated. An important implication of the equivalence result for nonlinear systems is pointed out. The proposed robust run-to-run controller formulation is tested on a number of examples including a chemical mechanical polishing (CMP) process model.

Minimax control for nonstationary linear operator systems

A solution to the minimax linear-quadratic problem of control of an operator system on a semi-infinite time interval is presented. The solution is based on the abstract maximum principle, Willems’ behavioral approach, the direct method of basic operators, and a small gain theorem.

A constructive algorithm for max–min paths problems on energy networks

Max–min paths problems on energy networks are the main center of interest of this article. They typically arose as an auxiliary problem within the study of a special class of discrete min–max control models and within so-called cyclic games. These two classes generalize the well-known combinatorial problem of the shortest and the longest paths in a weighted directed graph. A constructive algorithm for determining the tree of max–min paths in these special networks is proposed. Furthermore, we apply it as a new approach to the solution of special zero value cyclic games. Such a class is not too restrictive. Furthermore we refer to more general models which are very close to real-world examples.

Suboptimal minimax design of constrained parabolic systems with mixed boundary control

The paper concerns minimax control problems for linear multidimensional parabolic systems with distributed uncertain perturbations and control functions acting in mixed (Robin) boundary conditions. The main goal is to design a feedback control regulator that ensures the required state performance and robust stability under any feasible perturbations and minimize an energy-type functional under the worst perturbations from the given area. We design and justify an easily implemented suboptimal structure of the feedback boundary regulator and compute its optimal parameters ensuring the required state performance and robust stability of the nonlinear closed-loop control system on the infinite horizon.

Tolerance Limits Based on the Multivariate MaxMin Chart

The MaxMin EWMA control chart for the univariate case has proven to be an effective control tool for monitoring the process mean and variability (Amin et al., 1999). It has been shown (Amin and Li, 2000) how to utilize the information from an Exponentially Weighted Moving Average (EWMA) control chart, based on the smallest and largest observations within each sample, to obtain smoothed tolerance intervals for the univariate case. This idea here is extended to the multivariate case where p correlated variables are considered, with the goal of establishing multivariate EWMA control charts and of constructing smoothed tolerance intervals on a continual basis. Such multivariate tolerance plots are easy to construct and to use, and they provide an alternative approach for estimating the tolerances for multivariate continuous processes. Some average run length results on the multivariate MaxMin control chart are given.

Robust control of constrained max‐plus‐linear systems

AbstractMax‐plus‐linear (MPL) systems are a class of nonlinear systems that can be described by models that are ‘linear’ in the max‐plus algebra. We provide here solutions to the three types of finite‐horizon min–max control problems for uncertain MPL systems, depending on the nature of the control input over which we optimize: open‐loop input sequences, disturbances feedback policies, and state feedback policies. We assume that the uncertainty lies in a bounded polytope and that the closed‐loop input and state sequence should satisfy a given set of linear inequality constraints for all admissible disturbance realizations. Despite the fact that the controlled system is nonlinear, we provide sufficient conditions that allow one to preserve convexity of the optimal value function and its domain. As a consequence, the min–max control problems can be either recast as a linear program or solved via N parametric linear programs, where N is the prediction horizon. In some particular cases of the uncertainty description (e.g. interval matrices), by employing results from dynamic programming, we show that a min–max control problem can be recast as a deterministic optimal control problem. Copyright © 2008 John Wiley & Sons, Ltd.

Some Results on Guaranteeing Cost Strategies: Full-block Multipliers and Fictitious Games

Guaranteeing cost robust minimax controls are determined for a discrete-time linear system with uncertainty of linear fractional form over infinite horizon by means of the full-block multiplier technique. It is highlighted in what sense this technique can be considered as the extension of other approaches applying a single adjustable parameter in the derivation of the linear matrix inequalities to be solved. Moreover, it is shown how to construct a fictitious game with completely known dynamics and with a cost criterion parametrized by the full-block multipliers in such a way that the minimax strategy and the upper value for it turn out to be the robust minimax strategy and the guaranteed cost for the original uncertain system.

Penalization techniques in L ∞ optimization problems with unbounded horizon

In this work we present a numerical procedure for the ergodic optimal minimax control problem. Restricting the development to the case with relaxed controls and using a perturbation of the instantaneous cost function, we obtain discrete solutions Uek that converge to the optimal relaxed cost U when the relation ship between the parameters of discretization k and penalization e is an appropriate one.

Minimax control of a process in a linear uncertain-stochastic system with incomplete data

Consideration is given to the control problem in a linear stochastic differential system where constant noise intensities in equations of state and observation are prescribed only accurate within the membership of some known sets. For control optimization, an integral root-mean-square performance criterion is used. The problem is solved by the transition to a dual one, which makes it possible to prove the existence of a saddle point of the criterion and obtain an explicit expression for the minimax control operator as functions of the solution to the dual problem. To solve the latter, an iteration algorithm is proposed; the convergence of the algorithm is proved and investigated by a model example.