Uncertainty Polytope Research Articles

Robust Static Output-Feedback Control of Stochastic Continuous Time-Delayed Systems

In this article, a descriptor method is introduced for designing static output-feedback controllers for linear, continuous-time, retarded, stochastic systems that achieves a prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty } {\kern2.84526pt}$</tex-math></inline-formula> performance. A design solution is obtained for the uncertain case, where the parameters of the system matrices reside in a given polytope. The latter solution enables the derivation of the required constant output gain by solving a set of linear matrix inequalities that correspond to the vertices of the uncertainty polytope. The theory developed is also extended to the gain-scheduling case, and it is demonstrated by two examples. The first example compares various solution methods and the second is one of robust pitch control of an aircraft.

Optimizing subscriber migrations for a telecommunication operator in uncertain context

We consider a telecommunications company expanding its network capacity to face an increasing demand. The company can also invest in marketing to incentivize clients to shift to more recent technologies, hopefully leading to cheaper overall costs. To model the effect of marketing campaigns, previous works have relied on the Bass model. Since that model only provides a rough approximation of the actual shifting mechanism, the purpose of this work is to consider uncertainty in the shifting mechanism through the lens of robust optimization. We thus assume that the (discrete) shifting function can take any value in a given polytope and wish to optimize against the worst-case realization. The resulting robust optimization problem possesses integer recourse variables and non-linear dependencies on the uncertain parameters. We address these difficulties as follows. First, the integer recourse is tackled heuristically through a piece-wise constant policy dictated by a prior partition of the uncertainty polytope. Second, the non-linearities are handled by a careful analysis of the dominating scenarios. The scalability and economical relevance of our models are assessed through numerical experiments performed on realistic instances. In particular, we choose one of these instances to perform a case study with simulations illustrating the possible benefit of using robust optimization.

Retarded state-multiplicative stochastic systems - Robust H ∞ and H 2 vertex-dependent filtering

We consider linear continuous-time retarded systems with multiplicative noise and polytopic type parameter uncertainty and we address the problems of and general filtering of these systems. These problems are solved by applying a modified version of the descriptor method that leads to a considerable reduction in the over-design associated with the ”quadratic” solution that assigns a single Lyapunov function to all the points in the uncertainty polytope. The theory is extended to the robust gain-scheduling case where on-line parameter measurement is exercised in the estimation process. Two examples are given that demonstrate the superiority of our solution method. The first example relates to a simple second order system whereas the second example is one of altitude estimation taken from the field of aerospace engineering.

Modelling and Compensation of Uncertain Time-delays in Networked Control Systems with Plant Uncertainty Using an Improved Robust Model Predictive Control Method

Control systems with digital communication between sensors, controllers and actuators are called Networked Control Systems (NCSs). In general, NCSs encounter with some problems such as packet dropouts and network induced delays. When plant uncertainty is added to the aforementioned problems, the design of the robust controller that is able to guarantee the stability becomes more complex. In this paper, a method based on Robust Model Predictive Control (RMPC) is proposed to overcome model uncertainty together with unknown delay caused in NCSs. The previous RMPC methods, called Normal RMPC, was proposed to compensate delay or model uncertainty, individually. Hereby, we propose a method, named Improved RMPC, to compensate the effects of delay and model uncertainty on NCSs, simultaneously. The proposed method is based on uncertainty polytope and LMIs (Linear Matrix Inequlities). Also, an experimental evaluation of time-delays in MODBUS networked control systems is proposed. The simulation results show the ability of the proposed method. For comparison, the Normal RMPC is applied as well. The results show that the Normal RMPC has an acceptable performance for time delay compensation, but its performance gets destroyed or even get unstable when the model uncertainty is also considered.

A computational study of exact approaches for the adjustable robust resource-constrained project scheduling problem

We study the robust resource-constrained project scheduling problem under budgeted uncertainty polytope. The problem can be seen as a very challenging variant of the resource-constrained project scheduling problem, where the objective function minimises the worst-case makespan, assuming that activity durations are subject to interval uncertainty. The model allows to control the level of robustness by means of a protection factor related to the risk aversion of the decision maker. The paper introduces two exact decomposition approaches to tackle the solution of this difficult problem. An extensive computational experimentation, on standard benchmark instances from the literature, is carried out to assess and compare the performance of the proposed methods, also with respect to the state-of-the-art exact solution approach.

Robust combinatorial optimization with knapsack uncertainty

We study in this paper min max robust combinatorial optimization problems for an uncertainty polytope that is defined by knapsack constraints, either in the space of the optimization variables or in an extended space. We provide exact and approximation algorithms that extend the iterative algorithms proposed by Bertsimas and Sim (2003). We also study the limitation of the approach and point out NP-hard situations. Then, we approximate axis-parallel ellipsoids with knapsack constraints and provide an approximation scheme for the corresponding robust problem. The approximation scheme is also adapted to handle the intersection of an axis-parallel ellipsoid and a box.

Decomposition for adjustable robust linear optimization subject to uncertainty polytope

We present in this paper a general decomposition framework to solve exactly adjustable robust linear optimization problems subject to poly-tope uncertainty. Our approach is based on replacing the polytope by the set of its extreme points and generating the extreme points on the fly within row generation or column-and-row generation algorithms. The novelty of our approach lies in formulating the separation problem as a feasibility problem instead of a max-min problem as done in recent works. Applying the Farkas lemma, we can reformulate the separation problem as a bilinear program, which is then linearized to obtained a mixed-integer linear programming formulation. We compare the two algorithms on a robust telecommunications network design under demand uncertainty and budgeted uncertainty polytope. Our results show that the relative performance of the algorithms depend on whether the budget is integer or fractional.

Robust estimation of linear switched systems with dwell time

A covariance analysis is introduced for linear switched systems with dwell time constraints. This analysis reduces the conservatism that is encountered in the analysis of switched systems without dwell. An upper-bound on the covariance of the states of the switched system is obtained that depends on the switching signal. The obtained results are then applied to filtering problems in switched systems with driving and measurement with noise signals, where general type filters are considered. These filters assume that the switching signal is not given a priori but it is available in real time. In order to overcome the deficiencies of the standard approaches to robust filtering of systems with large parameter uncertainties, we also present a new method of estimation for non-switched, exponentially stable, uncertain systems. We divide the uncertainty region into several overlapping sub-regions where, for each of these sub-regions, a separate filter should be assigned. The resulting system is then treated as a switched system that switches between the sub-regions. The theory developed is demonstrated by two simple examples. In the first, a filtering solution is obtained for a switched system with error variance that is very close to the best achievable result for a single subsystem. The second example demonstrates the benefit, from the estimation error point of view, of dividing the uncertainty polytope of an uncertain non-switched system into two overlapping subpolytopes.

Polytopic best-mean H ∞ performance analysis

In Boyarski and Shaked [(2005), ‘Robust H ∞ Control Design for Best Mean Performance Over an Uncertain-parameters Box’, Systems and Control Letters, 54, 585–595], a novel best-mean approach to robust analysis and control over uncertain-parameters boxes was presented. This article extends the results of Boyarski and Shaked (2005) to convex uncertainty polytopes of arbitrary shape and arbitrary number of vertices, without an underlying parameters model. The article addresses robust, polytopic, probabilistic H ∞ analysis of linear systems and focuses on the performance distribution over the uncertainty region (rather than just on the performance bound, as is customary in robust control). It is assumed that all system instances over the uncertainty polytope may occur with equal probability; this uniform distribution assumption is common in robust statistical analysis and is known to be conservative. The proposed approach considers different disturbance attenuation levels (DALs) at the vertices of the uncertainty polytope. It is shown that, under the latter assumption, the mean DAL over the polytope is the algebraic average of the DALs at the polytope's vertices. Thus, the mean DAL over the polytope can be optimised by minimising the sum of the DALs at the vertices. The standard deviation of the DAL over the uncertainty polytope is also addressed, and a method to minimise this standard deviation (in order to enforce uniform performance over the polytope) is shown. The example utilises a state-feedback synthesis theorem presented in Boyarski and Shaked (2005). A Monte-Carlo analysis verifies correct statistics of the resulting closed-loop ‘pointwise’ H ∞-norms over the uncertainty region, and highlights the differences from a corresponding bound minimisation: a marginally higher bound, but much better mean and variance.

Robust linear matrix inequality‐based model predictive control with recursive estimation of the uncertainty polytope

The present work is concerned with the recursive estimation of the uncertainty polytope in a robust model predictive control (RMPC) framework. For this purpose, the unknown but bounded error method is employed to update the uncertainty polytope on the basis of sensor measurements at each sampling period. The recursive feasibility and asymptotic stability properties of the proposed approach are demonstrated as an extension of previous results concerning the RMPC formulation. For illustration, a simulated example involving an angular positioning system is presented. The results show that the proposed scheme provides a performance improvement, as indicated by the resulting cost function values.

Robust Control of Linear Systems via Switching

The standard approach to the problem of controlling linear systems with large parameter uncertainty is to seek a controller that stabilizes the system and achieves a required performance over the whole polytope of uncertainty. In the case where the latter polytope is large, the design may become very conservative. We present an alternative approach where the uncertainty polytope is divided into overlapping smaller regions and where each of these regions is assigned to a separate subsystem. Assuming that there is online information on which of the regions the parameters of the system move to, a recently developed method for H∞ design of switched system with dwell time is applied. A Lyapunov Function (LF) in a quadratic form, which is non-increasing at the switching instants, is assigned to each subsystem. This function is used to determine the stability and to find a bound on the L2 -gain of the switched system. The obtained results are used to solve the corresponding robust H∞ state-feedback and static output-feedback control problems.

The robust vehicle routing problem with time windows

This paper addresses the robust vehicle routing problem with time windows. We are motivated by a problem that arises in maritime transportation where delays are frequent and should be taken into account. Our model only allows routes that are feasible for all values of the travel times in a predetermined uncertainty polytope, which yields a robust optimization problem. We propose two new formulations for the robust problem, each based on a different robust approach. The first formulation extends the well-known resource inequalities formulation by employing adjustable robust optimization. We propose two techniques, which, using the structure of the problem, allow to reduce significantly the number of extreme points of the uncertainty polytope. The second formulation generalizes a path inequalities formulation to the uncertain context. The uncertainty appears implicitly in this formulation, so that we develop a new cutting plane technique for robust combinatorial optimization problems with complicated constraints. In particular, efficient separation procedures are discussed. We compare the two formulations on a test bed composed of maritime transportation instances. These results show that the solution times are similar for both formulations while being significantly faster than the solutions times of a layered formulation recently proposed for the problem.

Parameter-dependent robust stability of uncertain neural networks with time-varying delay

This paper is concerned with the problem of global robust asymptotic stability for delayed neural networks with polytopic parameter uncertainties and time-varying delay. A delay-dependent and parameter-dependent robust stability criterion for the equilibrium of delayed neural networks in the face of polytopic type uncertainties is presented by using a parameter-dependent Lyapunov functional and taking the relationship between the terms in the Leibniz–Newton formula into account. This criterion, expressed as a set of linear matrix inequalities, requires no matrix variable to be fixed for the entire uncertainty polytope, which produces a less conservative stability result.

Robust MPC for temperature control of air-conditioning systems concerning on constraints and multitype uncertainties

This paper presents a robust model-based predictive control (MPC) strategy for temperature control of an air-conditioning system, which consists of multiple local-loop processes and each process suffers from different dynamics uncertainties or variations. When an appropriate sampling period is chosen to discretise the system for computer control, a state-space discrete model with an uncertainty polytope is developed to describe the mixing uncertainties of these type of systems. The main benefit of the proposed description is that robust model predictive control can be easily used to design a robust controller for such a system while taking account of constraints associated with this system. A linear matrix inequality-based MPC algorithm is employed for control design. Case study was conducted on a dynamic simulation platform of an air-conditioning system, which evaluated the developed strategy in various simulation tests by comparing with the conventional PID control. Results demonstrated that the developed strategy is able to deal with constraints and allows stable and robust control while maintaining acceptable thermal comfort. Although the strategy is illustrated and validated using a constant air volume air-conditioning system, it can be applied to other constraint HVAC processes suffering from similar uncertainties. Practical applications: The main benefit of the developed strategy (including the proposed description and the adopted robust control algorithm) for practical application is that uncertainties and constraints can be dealt with simultaneously in one framework. Constraints in HVAC systems exist due to the application of actuators, for example, the rate limit considered in the paper. Taking account of constraints is useful to prevent unnecessary damages to equipments and maintain the system operating in a safe mode. Most importantly, the control objectives can be achieved in a constraint manner. The consideration of uncertainties, probably due to the changes of operating environment, is useful to release the work of accurate modelling of HVAC processes and online tuning of controllers.

Robust Model Predictive Control of VAV Air-Handling Units Concerning Uncertainties and Constraints

This paper presents a new strategy for robust temperature control of variable-air-volume (VAV) air-handling units (AHUs) that can deal with dynamic variations and the associated constraints in a straightforward manner. The dynamics of VAV AHUs is described by a first-order-plus-time-delay model, and the dynamic variations of the process gain, time constant, and time delay are described using uncertainty sets. These uncertainty sets are converted into an uncertainty polytope, and then an offline robust model predictive control (MPC) algorithm is employed for robust control design. The design procedure is illustrated step by step. An operating mode identification scheme is also developed based on the operating characteristics of VAV AHUs in order to reduce the size of uncertainty sets and, hence, to improve the control performance. Case studies are performed on a simulated VAV AHU. Results are presented to show that the proposed control strategy is able to enhance robustness without much user intervention, reduce the control activities, and satisfy constraints when implemented in the temperature control of VAV AHUs.

A robust model predictive control strategy for improving the control performance of air-conditioning systems

This paper presents a robust model predictive control strategy for improving the supply air temperature control of air-handling units by dealing with the associated uncertainties and constraints directly. This strategy uses a first-order plus time-delay model with uncertain time-delay and system gain to describe air-conditioning process of an air-handling unit usually operating at various weather conditions. The uncertainties of the time-delay and system gain, which imply the nonlinearities and the variable dynamic characteristics, are formulated using an uncertainty polytope. Based on this uncertainty formulation, an offline LMI-based robust model predictive control algorithm is employed to design a robust controller for air-handling units which can guarantee a good robustness subject to uncertainties and constraints. The proposed robust strategy is evaluated in a dynamic simulation environment of a variable air volume air-conditioning system in various operation conditions by comparing with a conventional PI control strategy. The robustness analysis of both strategies under different weather conditions is also presented.

Use of uncertainty polytope to describe constraint processes with uncertain time-delay for robust model predictive control applications

This paper studies the application of robust model predictive control (MPC) in a constraint process suffering from time-delay uncertainty. The process is described using a transfer function and sampled into a discrete model for computer control design. A polytope is firstly developed to describe the uncertain discrete model due to the process’s time-delay uncertainty. Based on the proposed description, a linear matrix inequality (LMI) based MPC algorithm is employed and modified to design a robust controller for such a constraint process. In case studies, the effect of time-delay uncertainty on the control performance of a standard MPC algorithm is investigated, and the proposed description and the modified control algorithm are validated in the temperature control of a typical air-handling unit.

Robust filter design for piecewise discrete‐time systems with time‐varying delays

AbstractA novel delay‐dependent filtering design approach is developed for a class of linear piecewise discrete‐time systems with convex‐bounded parametric uncertainties and time‐varying delays. The time‐delays appear in the state as well as the output and measurement channels. The filter has a linear full‐order structure and guarantees the desired estimation accuracy over the entire uncertainty polytope. The desired accuracy is assessed in terms of either ℋ︁∞‐performance or ℒ︁2–ℒ︁∞ criteria. A new parametrization procedure based on a combined Finsler's Lemma and piecewise Lyapunov–Krasovskii functional is established to yield sufficient conditions for delay‐dependent filter feasibility. The filter gains are determined by solving a convex optimization problem over linear matrix inequalities. In comparison to the existing design methods, the developed methodology yields the least conservative measures since all previous overdesign limitations are almost eliminated. By means of simulation examples, the advantages of the developed technique are readily demonstrated. Copyright © 2009 John Wiley &amp; Sons, Ltd.

LMI Conditions for Robust Stability of 2D Linear Discrete‐Time Systems

Robust stability conditions are derived for uncertain 2D linear discrete‐time systems, described by Fornasini‐Marchesini second models with polytopic uncertainty. Robust stability is guaranteed by the existence of a parameter‐dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities, formulated at the vertices of the uncertainty polytope. Several examples are presented to illustrate the results.

New Versions of Bounded Real Lemmas for Continuous and Discrete Uncertain Systems

New bounded real lemmas for linear systems with parameter uncertainties belonging to a convex bounded-polyhedral domain are established in this paper. Both continuous- and discrete-time uncertain systems are investigated. The derived conditions are expressed in terms of a set of linear matrix inequalities defined at the vertices of the uncertainty polytope. Numerical comparisons with those appearing recently in the literature show that the proposed conditions provide less conservative results.