On the design of optimal and sub-optimal control systems†

Though dynamic programming is an excellent and versatile algorithm, there are many cases in which we cannot apply this algorithm favourably to control system design because of the limited capacity of a digital computer and other difficulties.

Considering these facts, the successively optimizing method and the combined algorithm of dynamic programming and successively optimizing method are presented in this paper.

One of the objects of this paper is to present a practical method for obtaining the optimal solution of control systems with an integral form performance index by the use of dynamic programming.

The other is to present sub-optiinal solutions by the successively optimizing method and the combined algorithm.

These methods are very easy when compared with dynamic programming. Furthermore, these sub-optimal solutions can be made to converge to the optimum by using the recurrent equations repeatedly.     

On “Superstable” discrete systems

For the class of nonstationary discrete linear systems whose parameters have only multiple estimates, the sufficient condition for their robust stability was obtained on the basis of a discrete analog of the Lyapunov method. If satisfied at each time instant, this condition can be interpreted as the sufficient stability condition for systems with “frozen” coefficients. Robust stability of the stationary systems of the specified class follows from it as a special case.     

On the conjunction practical stability and controllability of large-scale impulsive control systems

1Introduction Asystem may be defined as a series of programs whichare designed to run sequentially.With the development ofscience,the scale of the systems with complicated structureis increasing.This causes the generationand development ofthe theory oflarge_scale system.Now,thelarge_scale systemhas become one of the important theoretic foundations ofthe systems engineering.There has been a significantdevelopment in the theory of impulsive differentialequations inthe past10years[1~3].If we wan…     

On the stability of interconnected systems†

This paper investigates the stability of systems which can be regarded as composed of interconnected subsystems. A sufficient condition for inputs-output L ^p stability, in terms of the L ^p gains of the subsystems and their interconnections is derived. For the case of L ² stability, it is compared with other criteria for asymptotic stability, obtained by Lyapunov techniques, and shown to give better results for a certain class of systems.     

On delay-dependent robust stability of neutral systems

The delay-dependent robust stability of uncertain linear neutral systems with delays is investigated. Both discrete-delay-dependent/neutral-delay-independent and neutral-/discrete- delay-dependent stability criteria will be developed. The proposed stability criteria are formulated in the form of linear matrix inequalities and it is easy to check the robust stability of the considered systems. By introducing certain Lyapunov-Krasovskii functional the mathematical development of our result avoids model transformation and bounding for cross terms, which lead to conservatism. Finally, numerical example is given to indicate the improvement over some existing results.     

On Hamiltonian realization of time-varying nonlinear systems

This paper investigates Hamiltonian realization of time-varying nonlinear (TVN) systems, and proposes a number of new methods for the problem. It is shown that every smooth TVN system can be expressed as a generalized Hamiltonian system if the origin is the equilibrium of the system. If the Jacopian matrix of a TVN system is nonsingu-lar, the system has a generalized Hamiltonian realization whose structural matrix and Hamiltonian function are given explicitly. For the case that the Jacobian matrix is singular, this paper provides a constructive decomposition method, and then proves that a TVN system has a generalized Hamiltonian realization if its Jacobian matrix has a non-singular main diagonal block. Furthermore, some sufficient (necessary and sufficient) conditions for dissipative Hamiltonian realization of TVN systems are also presented in this paper.     

On ℓ∞ and L∞ gains for positive systems with bounded time-varying delays

This paper is devoted to the analysis of the ?_∞ and L_∞ gains for positive linear systems with interval time-varying delays. Through exploiting the monotonicity of the state trajectory, we first prove that for positive systems with constant delays, the ?_∞ and L_∞ gains are fully governed by the system matrices but independent of the delay size. Moreover, for positive systems with bounded time-varying delays, by comparing with the nominal systems with constant delays, it turns out that the ?_∞ and L_∞ gains are exactly the same as that of the constant delay systems. The results in this paper reveal that the ?_∞ and L_∞ gains of positive linear systems are not sensitive to the magnitude of time delays and hence the computation of ?_∞ and L_∞ gains of positive systems with bounded time-varying delays can be reduced to computing the ?_∞ and L_∞ gains of the corresponding delay-free systems. Both continuous-time and discrete-time cases are considered in this paper.     

Linear differential and difference systems: EGδ- and EGσ- eliminations

Systems of linear ordinary differential and difference equations of the form $A_r (x)\xi ^r y(x) + \ldots + A_1 (x)\xi y(x) + A_0 (x)y(x) = 0,\xi \in \left\{ {\frac{d} {{dx}},E} \right\}$ , where E is the shift operator, Ey(x) = y(x + 1), are considered. The coefficients A _i (x), i = 0, ..., r, are square matrices of order m, and their entries are polynomials in x over a number field K, with A _r (x) and A ₀(x) being nonzero matrices. The equations are assumed to be independent over K[x, ξ]. For any system S of this form, algorithms EG_δ (in the differential case) and EG_σ (in the difference case) construct, in particular, the l-embracing system $\bar S$ of the same form. The determinant of the leading matrix $\bar A_r (x)$ of this system is a nonzero polynomial, and the set of solutions of system $\bar S$ contains all solutions of system S. (Algorithm EG_δ provides also a number of additional possibilities.) Examples of problems that can be solved with the help of EG_δ and EG_σ are given. The package EG implementing the proposed algorithms in Maple is described.     

Order-k α-hulls and α-shapes

We introduce order-k α-hulls and α-shapes – generalizations of α-hulls and α-shapes. Being also a generalization of k-hull (known in statistics as “k-depth contour”), order-k α-hull provides a link between shape reconstruction and statistical depth. As a generalization of α-hull, order-k α-hull gives a robust shape estimation by ignoring locally up to k outliers in a point set. Order-k α-shape produces an “inner” shape of the set, with the amount of “digging” into the points controlled by k. As a generalization of k-hull, order-k α-hull is capable of determining “deep” points amidst samples from a multimodal distribution: it correctly identifies points which lie outside clusters of samples.The order-k α-hulls and α-shapes are related to order-k Voronoi diagrams in the same way in which α-hulls and α-shapes are related to Voronoi diagrams. This implies that order-k α-hull and α-shape can be readily built from order-k Voronoi diagram, and that the number of different order-k α-shapes for all possible values of α is proportional to the complexity of order-k Voronoi diagram.     

On the state estimation for non-linear dynamic systems †

The object of this paper is to present an approximate technique for state estimation of non-linear dynamical systems under noisy observations. The conditional cumulant is introduced, by which the conditional probability density can be characterized. A set of dynamical equations satisfied by conditional cumulants is derived, and an approximate method is proposed for computing the cumulants. The relation of the cumulant method to the stochastic linearization technique is also discussed. Finally the state estimation problem for linear stochaatic system with state-dependent disturbance is solved to illustrate the use of the Gaussian approximation.     

On various types of decoupling for time-varying systems†

This paper deals with the problem of decoupling a class of linear time-varying multi-variable systems, based on the defining property that the impulse response matrix of a decoupled system is diagonal. Depending on the properties of the coefficient matrices of the vector differential equation of the open-loop system, the system may be uniformly or totally decoupled. The necessary and sufficient conditions that permit a system to be uniformly or totally decoupled by state variable feedback are given. The main contribution of this paper is the precise definition of these two classes of decoupling and a rigorous derivation of the necessary and sufficient conditions which show the necessity of requiring that the system be of constant ordered rank with respect to observability. A simple example illustrates the importance of having several definitions of decoupling. Finally, the results are specialized to the case of time invariant systems.     

On the abs algorithms for perturbed linear systems ∗

One of the most important problems in numerical analysis and numerical optimization is to solve a linear system of equations. Sometimes it should be repeated when one of the equations is replaced by a new one. In this paper as a result of theoretical analysis, an algorithm model and a particular algorithm which are based on the ABS class are proposed. After the original linear system has been solved by the ABS class, the algorithms proposed here can efficiently solve the new system which is obtained from the original system by replacing one of its equations by using information obtained in the previous computation. These algorithms can be used continually when some equations of the original system are replaced by new equations successively with less computation effort.     

On improving resource utilization and system throughput of master slave job scheduling in heterogeneous systems

The rapid advances of network technologies shed light on many aspects of the practicability of large scale ubiquitous computing. Grid technology has been recognized as an efficient solution to coordinate large-scale shared resources and execute complex applications in heterogeneous network environments. The problem of resource management and task allocation has always been one of the main challenges. In this paper, we present an efficient task allocation strategy for distributing tasks onto computing nodes in the underlying heterogeneous networks. The contribution of the proposed technique is to minimize average turnaround time by dispatching tasks to processors with smallest communication ratio. System throughput could be also enhanced by dispersing processor idle time. The proposed technique can be applied to heterogeneous cluster systems as well as computational grid environments, in which the communication costs vary in different clusters. Experimental results show that the proposed scheme outperforms other previous algorithms in terms of throughput and turnaround time.

On the design of optimal output feedback control systems†

For a linear control system with quadratic performance index the optimal control takes a feedback form of all state variables. However, if there are some states which are not fed in the control system, it is impossible to obtain the optimal feedback control by using the usual mathematical optimization technique such as dynamic programming or the maximum principle.

This paper presents the optimal control of output feedback systems for a quadratic performance index by using a new parameter optimization technique.

Since the optimal feedback gains depend on the initial states in the output feedback control system, two cases where (1) the initial states are known, and (2) the statistical properties of initial states such as mean and covariance matrices are known, are considered here. Furthermore, the proposed method for optimal output feedback control is also applied to sampled-data systems.     

On stability of switched linear systems with perturbed switching paths

1 Introduction The study of switched and hybrid dynamical systems has attracted much attention since 1990s. A switched linear sys- tem is a hybrid system which consists of several linear time- invariant subsystems and a switching path that orchestrates the switching among them. The importance of the switched linear systems scheme stems from the facts that i) the frame- work represents a wide class of practical systems, ii) the two-level system structure provides an effective multiple- controll…     

On realtime operating systems: How to compare performance ?

To compare realtime operating systems one must often rely on specifications given by the manufacturers. These published data are often measured by so-called benchmarks. But with the help of benchmarks it is not possible to collect sufficient information to determine the system behaviour for worst cases. Therefore it must be clarified which latency times exist and how they arise. Of great significance is the additional latency time of the operating system, which is compulsory for the design of the application. In this paper it is investigated whether and how long a given task, activated by an external interrupt, is delayed until task completion. For that all latency times which can occur are discussed, specifically the worst cases. We are taking a closer look at realtime operating systems which encourage so-called kernel-level threads. The advantage of these kernel-level threads systems in contrast to traditional systems are discussed referring to performance. At the moment we are testing two systems, LynxOS and HP-RT. We want to measure the worst-case latency times and therefore illustrate how this is done.     

On smoothing in linear discrete systems with time delays†

An algorithm is presented for optimal linear fixed-point and fixed-lag smoothing in non-stationary linear discrete systems with multiple time delays. Relations derived previously in the problem of filtering are used directly to obtain the fixed-point and fixed-lag smoothing filters. The smoothing error covariance matrix is obtained via a recursive matrix equation. For the particular case of systems without delay terms, the algorithm defines new smoothing procedures.     

On the controllability of systems with a time-varying delay†

This paper is concerned with the controllability of a system with a time-varying delay. The system to be considered here is described by a linear differential-difference equation of a retarded type, where a time-varying delay is a certain class of a time-varying function. First, the concepts of controllability for linear differential-difference systems with a constant delay, introduced by Weiss (1987), are developed to the system with a time-varying delay. Second, necessary and sufficient conditions for controllability are obtained. Finally, the controllability of a stationary system with a constant delay is treated, including the results of Chyung and Lee (1966).     

The asymptotic analysis of multiple imaginary characteristic roots for LTI delayed systems based on Puiseux–Newton diagram

The paper proposes a novel procedure for the asymptotic expansions of root loci around multiple imaginary roots of an exponential polynomial, which is necessary for the stability analysis of the LTI systems with commensurate delays. With the LTI delay systems given as exponential polynomials (also called quasi-polynomial), we seek to characterise the asymptotic behaviours of the characteristic roots of such systems in an algebraic way and determine whether the imaginary roots cross from one half-plane into another or only touch the imaginary axis. According to the Weierstrass preparation theorem, the quasi-polynomial equation is equivalent to an algebraic equation in the neighbourhood of a singular point. Furthermore, our result gives an explicit expression of the coefficients of the algebraic equation in infinite power series of delay parameter, and the determinations of such power series coefficients refer to the computation of residues of memorphic functions. Subsequently, the classic Puiseux–Newton diagram algorithm can be used to calculate the algebraic expansions of the reduced equation directly. Thus, the asymptotic behaviours of root loci around singular points of the quasi-polynomial equation are obtained. Some illustrative simulations are given to check the validity of the proposed method on asymptotic analysis with a powerful software.