Modeling the system of suboptimal robust tracking under unknown upper bounds on the uncertainties and external disturbances

This paper addresses a problem of suboptimal robust tracking for a discrete-time plant under unknown upper bounds on external disturbances. The transfer function of the nominal plant model is assumed to be known, whereas the upper bounds on the external disturbance, measurement noise, and coprime factor perturbations are assumed to be unknown. The results of numerical modeling of the algorithm of suboptimal robust tracking based on relaxed verification of the estimates of upper bound in the closed loop and the use of control criterion associated with such verification as the identification criterion are presented and analyzed. The results of numerical modeling illustrate efficiency of the proposed method of control design.     

Lower and upper bounds for a capacitated plant location problem with multicommodity flow

In this paper, we introduce a capacitated plant location problem with multicommodity flow. Given a set of potential plant sites and a set of capacitated arcs linking plants, transshipment points and customers, the aim is to determine where to locate plants and how to move flows from open plants to customers through a set of transshipment points. This model extends the classical capacitated plant location problem by introducing a multicommodity flow problem in the distribution issue. The combination of the location problem and the flow distribution problem is reasonable and realistic since both of them belong to strategic planning horizons. We propose a Lagrangean-based method, including a Lagrangean relaxation, a Lagrangean heuristic and a subgradient optimization, to provide lower and upper bounds of the model. Then, we employ a Tabu search to further improve upper bounds provided by the Lagrangean procedure. The computational results demonstrate that our solution method is effective since gaps between the upper and lower bound are on average around 2%.     

Robust performance optimization based on stochastic model errors: the stable case

Robust control aims to account for model uncertainty in design. Traditional methods for robust control typically assume knowledge of hard bounds on the system frequency response. However, this does not match well with system identification procedures which typically yield statistical confidence bounds on the estimated model. This paper explores a new procedure for obtaining a better match between robust control and system identification by using stochastic confidence bounds for robust control design. Given a nominal design, we set up an optimization problem which is aimed at reducing the statistical variability, measured in a mean square sense, from the nominal sensitivity. The proposed procedure is straightforward and leads to an easily computable solution for the final robust controller in the case of a stable plant and modest plant uncertainty. An illustrative example is provided which shows the advantages of the method. Copyright © 2002 John Wiley & Sons, Ltd.     

基于多胞体双重滤波的系统状态估计方法

针对未知但有界噪声离散时间状态空间系统,提出一种基于多胞体双重滤波的系统状态估计方法.首先,采用有界误差方法对测量噪声和状态预测过程进行分析,利用正多胞体预测状态集包裹后离散成初始约束条件;然后,根据更新最小边,全对称多胞体经过正多胞体紧致包裹后离散成约束条件,与测量方程约束条件组成3重约束;最后,通过求解线性规划问题得到全部状态的上下界,并获得包裹状态可行集的最紧致正多胞体.仿真示例验证了该方法估计离散状态空间系统状态的有效性和准确性.     

A two-valued state feedback control for a second order plant with low switching rate

In this work a two-valued state feedback control is developed for a plant of second order with known constant coefficients and no disturbance, and for a plant with unknown varying coefficients and additive disturbance with known upper and lower bounds. A reference model is used in order to avoid large commutation rate of the control input. In the first part of the present work, the controller is developed by means of the Lyapunov-like function method achieving the convergence of the tracking error to a residual set whose size is user-defined. In the second part, the Lyapunov function at zero time is inside a predefined set and the developed controller forces the Lyapunov function to dwell inside such set. The size of this set is chosen such that the tracking error asymptotically converges to a residual set of user-defined size.     

Probabilistic tubes in linear stochastic model predictive control

This paper considers constrained control of linear systems with additive and multiplicative stochastic uncertainty and linear input/state constraints. Both hard and soft constraints are considered, and bounds are imposed on the probability of soft constraint violation. Assuming the plant parameters to be finitely supported, a method of constraint handling is proposed in which a sequence of tubes, corresponding to a sequence of confidence levels on the predicted future plant state, is constructed online around nominal state trajectories. A set of linear constraints is derived by imposing bounds on the probability of constraint violation at each point on an infinite prediction horizon through constraints on one-step-ahead predictions. A guarantee of the recursive feasibility of the online optimization ensures that the closed loop system trajectories satisfy both the hard and probabilistic soft constraints. The approach is illustrated by a numerical example.     

The stability robustness determination of state space models with real unstructured perturbations

This paper considers the robust stability of a linear time-invariant state space model subject to real parameter perturbations. The problem is to find the distance of a given stable matrix from the set of unstable matrices. A new method, based on the properties of the Kronecker sum and two other composite matrices, is developed to study this problem; this new method makes it possible to distinguish real perturbations from complex ones. Although a procedure to find the exact value of the distance is still not available, some explicit lower bounds on the distance are obtained. The bounds are applicable only for the case of real plant perturbations, and are easy to compute numerically; if the matrix is large in size, an iterative procedure is given to compute the bounds. Various examples including a 46th-order spacecraft system are given to illustrate the results obtained. The examples show that the new bounds obtained can have an arbitrary degree of improvement over previously reported ones. This work has been supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A4396.     

Stable transport of assemblies: pushing stacked parts

This work presents a method to determine stable pushing motions for a planar stack of polygonal parts. The approach consists of solving a series of subproblems where each part in the stack is pushing the parts ahead of it. The solutions to these subproblem an sets of stable motions, and their intersection is the set of stable motions for the entire stack. The motion of multiple parts depends on the exact locations of the centers of mass and the relative masses of the parts. If either or both of these is unknown, it is still possible to calculate a conservative set of motions guaranteed to be stable by using a center of mass uncertainty region. Local-local controllability is also analyzed for single parts and stocks of parts with uncertain centers of mass. Once parts have been brought together in an automated assembly sequence, they typically must be repositioned to complete fastening or welding operations. This can be done with powerful robots capable of grasping and carrying the assembly and may involve a unique fixture to maintain the assembly during transport. A cheaper and more flexible alternative is to use a less powerful robot that can push the assembly along a horizontal surface without the aid of fixtures. This work presents a graphical method that produces conservative bounds on the pushing motions that guarantee the stability of a linear assembly (i.e., a stock of parts) during the push. The main application is in motion planning for assembly sequencing but the results could also he useful, for example, for mobile robots pushing multiple boxes in a warehouse. The method can be made robust to uncertainty in The mass properties of the parts, such as boxes with unknown contents. It is limited to linear stacks of parts where each part pushes no more than one other part In future work, we plan to devise a method to compute stable pushing motions for arbitrary assemblies of parts.     

An approach to stable indirect adaptive control

Stable indirect adaptive control of minimum and nonminimum phase plants is established for cases where a priori bounds on the unknown plant parameters are known and where for each set of parameter values within these bounds the plant has no unstable pole-zero cancellation. By incorporating this partial parameter knowledge in the adaptive law and using a nonminimal representation of the plant, it is shown that the adaptive closed loop control system can be written as an exponentially stable system driven by the identification error. The stability of the adaptive control system is then shown using techniques similar to those known from the model reference adaptive control approach.     

An improved method for bounding stationary measures of finite Markov processes

A new method to compute bounds on stationary results of finite Markov processes in discrete or continuous time is introduced. The method extends previously published approaches using polyhedra of eigenvectors for stochastic matrices with a known lower and upper bound of their elements. Known techniques compute bounds for the elements of the stationary vector with respect to the lower bounds of the matrix elements and another set of bounds with respect to the upper bounds of matrix elements. The resulting bounds are usually not sharp, if lower and upper bounds for the elements are known. The new approach combines lower and upper bounds resulting in sharp bounds which are often much tighter than bounds computed using only one bounding value for the matrix elements.     

基于通用访问结构的秘密共享的一般性结论

目前对于秘密共享的研究主要集中在具备完善性的访问结构上,且所包含的访问集个数较少;关于份额界的研究主要是以被研究对象服从均匀分布为假设前提,并以份额所需比特位数作为界的度量,从而导致研究成果具有局限性.基于通用访问结构,给出了包含任意多个访问集、适用于完善性与非完善性访问结构的基于信息论的一般性结论,是当前相关研究成果的一般化总结,并可作为更深层次研究的基础和工具.同时,以份额的信息熵作为界的度量,给出了适用于所有份额的通用界和只适用于特定份额的通用界,这些结论同样是对相关研究成果的一般化总结,且均适用于任意概率分布,其中某些界要比许多已知研究结果具有更好的紧致性.     

Control-oriented model validation and errors quantification in the /spl lscr//sub 1/ setup

A priori information required for robust synthesis includes a nominal model and a model of uncertainty. The latter is typically in the form of additive exogenous disturbance and plant perturbations with assumed bounds. If these bounds are unknown or too conservative, they have to be estimated from measurement data. In this paper, the problem of errors quantification is considered in the framework of the /spl lscr//sub 1/ optimal robust control theory associated with the /spl lscr//sub /spl infin// signal space. The optimal errors quantification is to find errors bounds that are not falsified by measurement data and provide the minimum value of a given control criterion. For model with unstructured uncertainty entering the system in a linear fractional manner, the optimal errors quantification is reduced to quadratic fractional programming. For system under coprime factor perturbations, the optimal errors quantification is reduced to linear fractional programming.     

On the use of bifurcation analysis for robust controller tuning for nonlinear systems

This paper presents a new approach to robust controller tuning for nonlinear systems. The main idea is to apply numerical bifurcation analysis to the closed-loop process, using the controller tuning parameters, the set points, and parameters describing model uncertainty (parametric as well as unmodeled dynamics) as bifurcation parameters. By analyzing the Hopf bifurcation and saddle-node bifurcation loci, bounds on the controller parameters are identified. These bounds depend upon the type as well as the degree of mismatch that exists between the plant and the model used for controller design.The method is illustrated by tuning a state feedback linearizing controller for an unstable reactor as well as by comparing a proportional-integral (PI) controller and a state feedback linearizing controller applied to a continuously operated fermenter. The feedback linearizing controller can result in better performance than the PI controller if a very accurate model of the process is known and if the operating conditions vary over a significant range. However, stability properties of systems controlled by feedback linearizing controllers can degrade significantly as the mismatch between the plant and the model increases. This is illustrated in the fermenter example by showing that bounds on the tuning parameter of the feedback linearizing controller are significantly tighter than the ones for the PI controller.     

STATE FEEDBACK CONTROLLER DESIGN OF NETWORKED CONTROL SYSTEMS WITH PARAMETER UNCERTAINTY AND STATE‐DELAY

This paper is concerned with the controller design of networked control systems. The continuous time plant with parameter uncertainty and state delay is studies. A new model of the networked control system is provided under consideration of the nonideal network conditions. In terms of the given model, a controller design method is proposed based on a delay dependent approach. The maximum allowable synthetical bounds related with the discarded data packet and network‐included delay and the feedback gain of a memoryless controller can be derived by solving a set of linear matrix inequalities for the stabilizablity of the networked control system based on Lyapunov functional method. An example is given to show the effectiveness of our method.     

Robust stability region of fractional order PIλ controller for fractional order interval plant

In this article, by using the fractional order PI^λ controller, we propose a simple and effective method to compute the robust stability region for the fractional order linear time-invariant plant with interval type uncertainties in both fractional orders and relevant coefficients. The presented method is based on decomposing the fractional order interval plant into several vertex plants using the lower and upper bounds of the fractional orders and relevant coefficients and then constructing the characteristic quasi-polynomial of each vertex plant, in which the value set of vertex characteristic quasi-polynomial in the complex plane is a polygon. The D-decomposition method is used to characterise the stability boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters, which can obtain the stability region by varying λ orders in the range (0,?2). These regions of each vertex plant are computed by using three stability boundaries: real root boundary (RRB), complex root boundary (CRB) and infinite root boundary (IRB). The method gives the explicit formulae corresponding to these boundaries in terms of fractional order PI^λ controller parameters. Thus, the robust stability region for fractional order interval plant can be obtained by intersecting stability region of each vertex plant. The robustness of stability region is tested by the value set approach and zero exclusion principle. Our presented technique does not require sweeping over the parameters and also does not need linear programming to solve a set of inequalities. It also offers several advantages over existing results obtained in this direction. The method in this article is useful for analysing and designing the fractional order PI^λ controller for the fractional order interval plant. An example is given to illustrate this method.     

GENERATION OF EXACT QFT BOUNDS FOR PLANTS WITH AFFINELY DEPENDENT UNCERTAINTIES

This paper presents an efficient method for the generation of exact QFT bounds for robust sensitivity reduction and gain‐phase margin specifications for plants with affinely dependent uncertainties. It is shown that, for a plant with m affinely dependent uncertainties, the exact QFT bounds for robust sensitivity reduction and gain‐phase margin specifications at a given frequency and controller phase can be computed by solving m^2m‐1 bivariate polynomial inequalities corresponding to the edges of the parameter domain box. Moreover, the solution set for each bivariate polynomial inequality can be computed by solving for the real roots of one fourth‐order and six second‐order polynomials. This avoids the unfavorable trade‐off between the computational burden and the accuracy of QFT bounds that has arisen in the application of many existing QFT bound generation algorithms. Numerical examples are given to illustrate the proposed method and its computational superiority.     

An implicit small gain condition and an upper bound for the real structured singular value

In this paper we develop an upper bound for the real structured singular value that has the form of an implicit small gain theorem. The implicit small gain condition involves a shifted plant whose dynamics depend upon the uncertainty set bound and, unlike prior bounds, does not require frequency-dependent scales or multipliers. Numerical results show that the implicit small gain bound compares favorably with real-μ bounds that involve frequency-dependent scales and multipliers.     

On the error behavior in linear minimum variance estimation problems

For linear systems the error covariance matrix for the unbiased, minimum variance estimate of the state does not depend upon any specific realization of the measurement sequence. Thus it can be examined to determine the expected behavior of the error in the estimate before actually using the filter in practice. In this paper, the general linear system that contains both plant and measurement noise is shown to exhibit a decomposition property that permits the derivation of upper and lower bounds upon the error covariance matrix. This decomposition allows systems containing either plant or measurement noise, but not both, to be considered separately. Some general characteristics of these simpler systems are discussed and conditions for the positive definiteness and vanishing of the error covariance matrix are established. It is seen that the presence of plant noise, in general, prevents the error from vanishing. Alternatively, the condition ofq-stage observability is seen to be sufficient to insure that the error covariance matrix asymptotically approaches the zero matrix for systems with noise-free plants. These results are used to establish very specific lower bounds. Through the application of the duality principle, they can be applied directly to the analysis of the linear regulator problem.     

Robustness of supervisors for discrete-event systems

Supervisory control in the context of ω-languages is considered. The nominal supervisor design problem is to find a non-blocking supervisor for a nominal plant such that the closed-loop infinite behavior equals a specified closed-loop behavior. The robustness of solutions to the nominal problem is defined with respect to variations in the plant. It is shown there exists a supervisor solving the nominal problem which maximizes the set of plants for which the closed-loop languages for all other plants in the set satisfy lower and upper bounds in the sense of language containment. Computational issues are discussed and the theoretical results are illustrated with an example