Minimum dimension of a functional observer with specifiable dynamical properties

The article considers the construction of a functional observer with a given rate of convergence for the most general case: a vector state functional of a linear dynamical system with a vector output. An upper bound is derived on the minimum dimension of such an observer, which holds for almost all functionals. An algorithm is proposed for constructing an observer that achieves this bound. The algorithm can be used to construct a functional observer for almost all specified spectra (i.e., with the exception of a set of measure zero). The scalar observer method previously developed by the authors and proposed in the present article is based on canonical form Luenberger observability for systems with vector output.     

Multi-particle dynamical systems and polynomials

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived.     

Output Stabilization for Nonminimum-Phase Systems Using Observer Backstepping

The article considers stabilization of the output programmed trajectory in nonminimum-phase dynamical systems of a special kind with unstable linear zero dynamics. An algorithm is proposed for the construction of a stabilizing control by observer backstepping. The backstepping method is generalized to nonminimum-phase nonlinear dynamical systems.     

Synthesis of linear systems with desired equivalent form

A method for feedback synthesis of linear control systems with desired linearly equivalent form of the closed loop system matrix is proposed. The method is based on the serial canonical form of linear multivariable systems which is an alternative to the Luenberger canonical form. A stable computational algorithm for finding the serial canonical form using orthogonal similarity transformations is described. The algorithm for synthesis involves a simple recurrent procedure and gives the possibility to achieve any attainable equivalent form of the closed loop system. The results obtained are extended to the synthesis of reduced order state observers.     

State feedback for uncertain dynamical systems

We consider a linear dynamical system in which the system and input matrices, as well as the input, are uncertain. We present a control system consisting of a linear control to stabilize the nominal system, a nonlinear control to cope with the uncertainties, and an insensitive observer for the state estimation. Practical stability is guaranteed for uncertainties with known bounds. Furthermore, the control system is designed to achieve insensitivity against parameter variations. The theoretical results are illustrated by an application to the suspension control of a maglev vehicle.     

Superfast Iterative Solvers for Linear Matrix Equations

Superfast algorithms for solving large systems of linear equations are developed on the basis of an original method for multistep decomposition of a linear multidimensional dynamical system. Examples of analytical synthesis of iterative solvers for matrices of the general form and for large numerical systems of linear algebraic equations are given. For the analytical case, it is shown that convergence occurs at the second iteration.     

New synthesis methods for optimal feedback systems

The article examines optimal synthesis methods for time-dependent deterministic systems with feedback (the regulator problem and the output problem). The linear problem is solved using the structure of optimal control. Nonlinear problems are reduced to the linear problem by linear and quadratic approximation. Existence and uniqueness theorems are stated and proved. The notion of generalized solution in dynamical programming is utilized.     

Synthesis of minimum-order robust inverters

In the present paper, we consider the inversion problem for dynamical systems, that is, the problem of reconstruction of the unknown input signal ξ(t) of a given system on the basis of known information (about either the complete phase vector or a measurable output of the system). An auxiliary dynamical system forming the desired estimate of the signal ξ(t) is called an inverter.In earlier papers of the authors, attention was mainly paid to the possibility of inversion of a dynamical system in different cases in principle. In this relation, a model of dynamical systems with some stabilizing control was used as an inverter for the solution of the problem; moreover, this control was often designed with the use of an additional dynamical system, an observer of the phase vector of the original system or the system in deviations. Thus, a dynamical system whose dimension either coincides with the dimension of the original system or exceeds it was considered as an inverter.In the solution of practical problems, it is often required to synthesize inverters of minimal order. (This requirement is related to constraints on the complexity, cost, and operation speed of automated control systems.) In the present paper, we consider the problem on the possible reduction of the order of the inverter in various cases and the problem on the construction of inverters of minimal order.     

Stochastic model order reduction in randomly parametered linear dynamical systems

This study focuses on the development of reduced order models for stochastic analysis of complex large ordered linear dynamical systems with parametric uncertainties, with an aim to reduce the computational costs without compromising on the accuracy of the solution. Here, a twin approach to model order reduction is adopted. A reduction in the state space dimension is first achieved through system equivalent reduction expansion process which involves linear transformations that couple the effects of state space truncation in conjunction with normal mode approximations. These developments are subsequently extended to the stochastic case by projecting the uncertain parameters into the Hilbert subspace and obtaining a solution of the random eigenvalue problem using polynomial chaos expansion. Reduction in the stochastic dimension is achieved by retaining only the dominant stochastic modes in the basis space. The proposed developments enable building surrogate models for complex large ordered stochastically parametered dynamical systems which lead to accurate predictions at significantly reduced computational costs.     

Minimum-order functional observers

The article considers the construction of a minimum-order observer for a linear state functional of a linear dynamical system. For a one-output system we derive a necessary and sufficient condition that allows reconstruction of the functional by a k th order observer. This result is generalized to a system with vector output. An algorithm for the construction of a minimal observer is proposed. The problem is solved by the method of decomposition into scalar observers.     

向量丛动力系统研究註记(Ⅱ)──部份1

过去,向量丛线性动力系统的整体线性性质研究已经显得相当广泛。现在,我们提议研究这种线性系统的扰动性质。我们要考虑的这种扰动系统将不再是线性的,但要研究的性质一般仍是整体性的。再者我们感兴趣的为非一致双曲性。在本文中我们给出了这种扰动的恰当的定义。它虽表现得有几分不太通常,然而它较深地植根于有关微分动力系统理论的典泛方程组中。这里一般的问题是要观察,当扰动发生后,原给系统的何种性质得以保持下来。本文的全部内容是要建立这种类型的一个定理。     

A relaxation method for solving systems with infinitely many linear inequalities

The problem of finding a feasible solution to a linear inequality system arises in numerous contexts. We consider solving linear semi-infinite inequality systems via an extension of the relaxation method for finite linear inequality systems. The difficulties are discussed and a convergence result is derived under fairly general assumptions on a large class of linear semi-infinite inequality systems.     

Design of observers for nonlinear systems with H ∞ performance analysis

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time‐delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H _∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.     

Multicriteria optimization of convex dynamical systems

We suggest an approach to the solution of multicriteria optimization problems for dynamical systems described by differential inclusions. The investigation is restricted to dynamical systems with concave differential inclusion, for which the trajectory tube is convex. Such systems are typical of economic models. We assume that the criteria for the choice of the solution depend on the system state at a given terminal time and are related to it by sufficiently arbitrary functions. The approach is based on the interactive visualization of the Pareto frontier, which is carried out by approximating the reachable set of the dynamical system and the Edgeworth-Pareto set of feasible criteria vectors.     

Variable structure control systems: synthesis of scalar and vector systems by state and output feedback

Modern views are applied to examine synthesis of variable structure systems (VSS) for SISO and MIMO systems of general position in the presence of information on the full phase vector or only on the output vector. In the latter case we consider separately the cases of organizing a sliding mode in plant output coordinates and in arbitrary coordinates. The latter case requires the construction of asymptotic observers for linear systems with uncertainty in the parameters or in inputs, and the corresponding observation theory is sketched. Special attention is given to the case of so-called hyperoutput systems, i.e., multiply connected systems in which the number of outputs exceeds the number of inputs. For hyperoutput systems under fairly general conditions we can arbitrarily choose the observer dynamics, which is important for improving the performance of the control system as a whole. We show that the observer dynamics does not change the VSS stability.     

Invariance and asymptotic stability

The stability of the zero solution of an autonomous non-linear system is considered. The problem of finding the variables in relation to which the solution is asymptotically stable if the Lyapunov function with a sign-definite derivative is known, is formulated and solved. The maximality of the set in relation to which the solution is asymptotically stable is established. The investigation is based on the method of auxiliary functions and clarifies the relation between the properties of invariance and the asymptotic stability of dynamical systems. The constructiveness of the results obtained is demonstrated by an example.     

Linear system identification via an asymptotically stable observer

This paper presents a formulation for identification of linear multivariable systems from single or multiple sets of input-output data. The system input-output relationship is expressed in terms of an observer, which is made asymptotically stable by an embedded eigenvalue assignment procedure. The prescribed eigenvalues for the observer may be real, complex, mixed real and complex, or zero corresponding to a deadbeat observer. In this formulation, the Markov parameters of the observer are first identified from input-output data. The Markov parameters of the actual system are then recovered from those of the observer and used to realize a state space model of the system. The basic mathematical formulation is derived, and numerical examples are presented to illustrate the proposed method.     

Synthesis of controllers for damping oscillations in dynamical systems under uncertainty

We review the modern approaches to the synthesis of robust H _∞ controllers that ensure optimal damping of oscillations in dynamical systems under uncertainty. In the synthesis method based on Riccati equations, these many-parameter equations can be solved only when the parameters are contained in a bounded parallelepiped with given boundaries. The synthesis of a robust H _∞ output control for systems with unknown bounded parameters is reducible to the solution of an optimization problem constrained by a system of linear matrix inequalities. The proposed controller synthesis algorithms are implemented using standard MATLAB procedures. The efficiency of the proposed methods and algorithms is demonstrated in application to optimal damping of oscillations in a parametrically excited pendulum. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 87–104, 2004.     

Synthesis of minimal linear stabilizers

We consider the stabilizer synthesis problem for linear stationary control dynamical systems; attention is mainly focused on the investigation of possibilities to reduce the order of such stabilizers. The problem is considered in two settings. The first setting deals with the construction of stabilizers of given order (in particular, the minimum possible order); no conditions are imposed on the spectrum of the closed system. For scalar (single-input-single-output) control systems, we suggest an approach to the solution of this problem and obtain necessary and sufficient conditions for the existence of a stabilizer of a given order. The second problem is that of synthesizing a stabilizer of minimum order with given dynamic properties (a given spectrum or a given distribution of the spectrum of the closed system, in particular, with a guaranteed convergence rate of the closed system). For this problem, we suggest two approaches that permit one to obtain an upper bound for the dimension of such stabilizers.