A genetic approach to the identification of linear dynamical systems with static nonlinearities

This paper investigates the use of genetic algorithms in the identification of linear systems with static nonlinearitites. Linear systems with static nonlinearities at the input known as the Hammerstein model, and linear systems with static nonlinearities at the output known as the Wiener model are considered in this paper. The parameters of the Hammerstein and the Wiener models are estimated using genetic algorithms from the input-output data by minimizing the error between the true model output and the identified model output. Using genetic algorithms, the Hammerstein and the Wiener models with known nonlinearity structure and unknown parameters can be identified. Moreover, systems with non-minimum phase characteristics can be identified. Extensive simulations have been used to study the convergence properties of the proposed scheme. Simulation examples are included to demonstrate the effectiveness and robustness of the proposed identification scheme.     

A class of linear interval programming problems and its application to portfolio selection

This paper discusses a class of linear programming problems with interval coefficients in both the objective functions and constraints. The noninferior solutions to such problems are defined based on two order relations between intervals, and can be found by solving a parametric linear programming problem. Considering the uncertain returns of assets in capital markets as intervals, we propose a model for portfolio selection based on the semiabsolute deviation measure of risk, which can be transformed to a linear interval programming model studied in the paper. The method is illustrated by solving a simplified portfolio selection problem.     

Qualitative, semiqualitative and interval algebras, and their application to engineering problems

This paper proposes the use of semiqualitative modelling for reasoning about the behaviour of complex physical systems. Semiqualitative modelling is a generalization of qualitative modelling which refines the set of intervals that values may be expressed in. Semiqualitative algebras are introduced, their most important features discussed, and related to qualitative algebras. The advantages that semiqualitative modelling offers over qualitative modelling are demonstrated by the solution of an example from the field of biotechnology. Finally, interval algebras are introduced as a generalization of semiqualitative algebras, and it is proved that it is possible to switch between different interval algebras in the course of computation in order to preserve the greatest possible degree of precision.     

On the application of a hybrid ellipsoidal-rectangular interval arithmetic algorithm to interval Kalman filtering for state estimation of uncertain systems

Modelling uncertainty is a key limitation to the applicability of the classical Kalman filter for state estimation of dynamic systems. For such systems with bounded modelling uncertainty, the interval Kalman filter (IKF) is a direct extension of the former to interval systems. However, its usage is not yet widespread owing to the over-conservatism of interval arithmetic bounds. In this paper, the IKF equations are adapted to use an ellipsoidal arithmetic that, in some cases, provides tighter bounds than direct, rectangular interval arithmetic. In order for the IKF to be useful, it must be able to provide reasonable enclosures under all circumstances. To this end, a hybrid ellipsoidal-rectangular enclosure algorithm is proposed, and its robustness is evidenced by its application to two characteristically different systems for which it provides stable estimate bounds, whereas the rectangular and ellipsoidal approaches fail to accomplish this in either one or the other case.     

An approach to interval programming problems with left-hand-side stochastic coefficients: An application to environmental decisions analysis

An interval programming with stochastic coefficients (IPSC) model is developed for planning of regional air quality management. The IPSC model incorporates stochastic coefficients with multivariate normal distributions within an interval parameter linear programming (ILP) framework. In IPSC, system uncertainties expressed as stochastic coefficients and intervals are addressed. Since stochastic coefficients are the left-hand-side (LHS) parameters of the constraints in IPSC, a left-hand-side chance-constrained programming (LCCP) method is developed to solve the problem. The developed IPSC model is applied to a regional air quality management system. Uncertainties in both abatement efficiencies expressed as stochastic coefficients and environmental standards expressed as intervals are reflected. Interval solutions associated with different violation probability levels and/or different environmental standards have been obtained. Air quality managers can thus analyze the solutions with appropriate combinations of the uncertainties and gain insight into the tradeoffs between the abatement costs and the risks of violating different environmental standards.     

A new approach to obtain algebraic solution of interval linear systems

In this paper, an algebraic solution of interval linear system involving a real square matrix and an interval right-hand side vector is obtained. A new approach to solve such systems based on the new concept “inclusion linear system” is proposed. Moreover, new necessary and sufficient conditions are derived for obtaining the unique algebraic solution. Furthermore, based on our method, an algorithm is proposed and numerically demonstrated. Finally, we compare the result obtained by our method with that obtained by interval Gauss elimination procedure.     

Legendre series approach to identification and analysis of linear systems

In this paper the Legendre operational matrix of integration is introduced, and it is subsequently used for the identification and analysis of linear systems. The algorithms proposed are easily implementable in a digital computer, and they are analogous to those already derived for other types of orthogonal functions.     

Simultaneous confidence bands for all contrasts of three or more simple linear regression models over an interval

Simultaneous confidence intervals are used in Scheffé (1953) to assess any contrasts of several normal means. In this paper, the problem of assessing any contrasts of several simple linear regression models by using simultaneous confidence bands is considered. Using numerical integration, Spurrier (1999) constructed exact simultaneous confidence bands for all the contrasts of several regression lines over the whole range (−∞,∞) of the explanatory variable when the design matrices of the regression lines are all equal. In this paper, a simulation-based method is proposed to construct simultaneous confidence bands for all the contrasts of the regression lines when the explanatory variable is restricted to an interval and the design matrices of the regression lines may be different. The critical value calculated by this method can be as close to the exact critical value as required if the number of replications in the simulation is chosen sufficiently large. The methodology is illustrated with a real problem in which sizes of the left atrium of infants in three diagnostic groups (severely impaired, mildly impaired and normal) are compared.     

Optimal interval enclosures for fractionally-linear functions,and their application to intelligent control

One of the main problems of interval computations is, given a functionf(x ₁, ...,x _n ) andn intervals x₁, ..., x _n , to compute the range y=f(x₁, ..., x _n ). This problem is feasible for linear functionsf, but for generic polynomials, it is known to be computationally intractable. Because of that, traditional interval techniques usually compute theenclosure of y, i.e., an interval that contains y. The closer this enclosure to y, the better. It is desirable to describe cases in which we can compute theoptimal enclosure, i.e., the range itself. In this paper, we describe a feasible algorithm for computing the optimal enclosure forfractionally linear functionsf. Applications of this result tointelligent control are described.     

Two-level stochastic control for the linear quadratic problem related to a static system

Two-level, hierarchical, partially decentralized control is considered for a large-scale system composed of M linear static subsystems with an interaction and a quadratic performance index. The coordinator and the local controllers, using their different information, realize the control resulting from the global and local optimization, respectively. It is seen that the optimal control laws are linear functions of some estimates and are determined by the formulated two-level certainty equivalence principle. A discussion on the usefulness of the application of particular local controllers is described.     

On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach

This paper deals with the computation of control invariant sets for constrained nonlinear systems. The proposed approach is based on the computation of an inner approximation of the one step set, that is, the set of states that can be steered to a given target set by an admissible control action. Based on this procedure, control invariant sets can be computed by recursion.We present a method for the computation of the one-step set using interval arithmetic. The proposed specialized branch and bound algorithm provides an inner approximation with a given bound of the error; this makes it possible to achieve a trade off between accuracy of the computed set and computational burden. Furthermore an algorithm to approximate the one step set by an inner bounded polyhedron is also presented; this allows us to relax the complexity of the obtained set, and to make easier the recursion and storage of the sets.     

解系统控制问题的递推方法

本文对系统控制及其他领域中的一大类问题,介绍一种递推解法,使当数据增加时,递推解以概率1收敛到真解.这方法主要以3个步骤来实现:首先,把欲解的问题参数化;然后,选择适用的递推参数估计算法;最后,证明递推估计收敛到笔者想要的值.文中列举了具体的系统控制问题,展示如何把它们参数化.作为递推估计算法,文中介绍了扩展截尾的随机逼近算法(SAAWET),给出了它的一般收敛定理.随后,以两个系统控制问题,展示该方法的具体实现,所附模拟计算实例证实了该方法.实际上,这种方法已成功地解决了系统控制及其他相关领域中的一系列问题.     

A behavioral approach to the control of discrete linear repetitive processes

This paper formulates the theory of linear discrete time repetitive processes in the setting of behavioral systems theory. A behavioral, latent variable model for repetitive processes is developed and for the physically defined inputs and outputs as manifest variables, a kernel representation of their behavior is determined. Conditions for external stability and controllability of the behavior are then obtained. A sufficient condition for stabilizability is also developed for the behavior and it is shown under a mild restriction that, whenever the repetitive system is stabilizable, a regular constant output feedback stabilizing controller exists. Next, a notion of eigenvalues is defined for the repetitive process under an action of a closed-loop controller. It is then shown how under controllability of the original repetitive process, an arbitrary assignment of eigenvalues for the closed-loop response can be achieved by a constant gain output feedback controller under the above restriction. These results on the existence of constant gain output feedback controllers are among the most striking properties enjoyed by repetitive systems, discovered in this paper. Results of this paper utilize the behavioral model of the repetitive process which is an analogue of the 1D equivalent model of the dynamics studied in earlier work on these processes.     

An iterative approach to the optimal co-design of linear control systems

This paper investigates the optimal co-design of both physical plants and control policies for a class of continuous-time linear control systems. The optimal co-design of a specific linear control system is commonly formulated as a nonlinear non-convex optimisation problem (NNOP), and solved by using iterative techniques, where the plant parameters and the control policy are updated iteratively and alternately. This paper proposes a novel iterative approach to solve the NNOP, where the plant parameters are updated by solving a standard semi-definite programming problem, with non-convexity no longer involved. The proposed system design is generally less conservative in terms of the system performance compared to the conventional system-equivalence-based design, albeit the range of applicability is slightly reduced. A practical optimisation algorithm is proposed to compute a sub-optimal solution ensuring the system stability, and the convergence of the algorithm is established. The effectiveness of the proposed algorithm is illustrated by its application to the optimal co-design of a physical load positioning system.     

An approach to the construction of approximate solutions of Boolean linear programming problems

Translated from Kibernetika, No. 1, pp. 40–43, January–February, 1988.     

On sampled data approach to parameter identification of continuous linear systems

It is generally preferable to estimate unknown parameters in analog transfer functions by sampled data techniques. This approach calls for the derivation of an appropriate discrete modelD(z)for a given analog systemH(s)with undetermined parameters, and then the fitting of the sampled input-output data byD(z)to estimate these parameters. This correspondence presents a method for obtaining such discrete models. It also provides a more effective procedure for calculating the parameter estimates from the identified discrete transfer functionhat{D}(z).     

A unified approach to the stability of generalized static neural networks with linear fractional uncertainties and delays

In this paper, the robust global asymptotic stability (RGAS) of generalized static neural networks (SNNs) with linear fractional uncertainties and a constant or time-varying delay is concerned within a novel input-output framework. The activation functions in the model are assumed to satisfy a more general condition than the usually used Lipschitz-type ones. First, by four steps of technical transformations, the original generalized SNN model is equivalently converted into the interconnection of two subsystems, where the forward one is a linear time-invariant system with a constant delay while the feedback one bears the norm-bounded property. Then, based on the scaled small gain theorem, delay-dependent sufficient conditions for the RGAS of generalized SNNs are derived via combining a complete Lyapunov functional and the celebrated discretization scheme. All the results are given in terms of linear matrix inequalities so that the RGAS problem of generalized SNNs is projected into the feasibility of convex optimization problems that can be readily solved by effective numerical algorithms. The effectiveness and superiority of our results over the existing ones are demonstrated by two numerical examples.     

Symmetric Laplace transform and its application to parametric identification of linear systems

Properties of bilateral symmetric integral transform, which possesses useful features for identification of linear systems, are investigated. Relationships for covariance functions of stochastic stationary processes of measurement errors and disturbances are determined. The optimal solution is proposed to the identification problem for linear multivariable system with measured state coordinates in the transformation parameter domain, with account for restricted bandwidth of dynamic processes in the system at hand. The problem is solved for unknown initial state of the system and complex measurement errors including correlated noise, constant errors and linear trends.