An algorithm for inverting rational matrices

We propose an algorithm for computing the inverses of rational matrices and in particular the inverses of polynomial matrices. The algorithm is based on minimal state space realizations of proper rational matrices and the matrix inverse lemma and is implemented as a MATLAB¹ function. Experiments show that the algorithm gives accurate results for typical rational matrices that arise in analysis and design of linear multivariable control systems. Illustrative examples are given.     

降阶函数观测器的参数化

给出了线性系统函数观测器的一个新的参数化表示,并分析了其物理意义及状态空 间解释.本文得到的函数观测器的阶数比已往的结果降低了系统的输出数     

Design of minimal order stable observers for linear functions of the state via realization theory

Abstract--The design of a minimal order stable observer and a minimal order observer with arbitrary poles that estimates a vector linear function of the state of a multivariable system is discussed. It is shown that both problems can be solved in a straightforward manner using partial realization theory, and several new results are given. These include a strong bound for the dimension of the minimal order stable observer and a simple necessary condition to design the minimal order observer with arbitrary poles that estimates a vector linear function of the state of a multiple-output system. Necessary and sufficient conditions are given for designing a minimal order observer with arbitrary poles for the case of estimating a vector linear function of the state of a single-output system and the case of estimating a scalar linear function of the state of a multiple-output system. A procedure to carry out the design in each of these cases is described. No restrictions whatsoever (except stability) are placed on the possible values of the observer poles. A significant observation of this paper is that the dynamics of the observer are constrained (in all cases) only by the gain matrix in the feedback law to be estimated and the output structure of the given system.     

On the realization of periodic functions

In this article, canonical and spectrally minimal infinite-dimensional state space realizations for periodic functions are considered. It is shown that the periodic functions having ℓ₁ Fourier coefficients are precisely those realizable by a Riesz spectral system (RSS) in which the system operator generates a periodic strongly continuous semigroup and the observation operator is bounded. This realization can easily be converted to a canonical and spectrally minimal form. It is shown how the use of RSS and Cesáro sums of Fourier series allows the construction of a state space realization for a given periodic function merely integrable over its period. Simple finite-dimensional approximations with error bounds are derived for the RSS realization. Regular well-posed linear systems (WPLS) are used to construct a Fuhrmann-type realization for a given periodic function integrable over its period. It is shown that the RSS realizations and WPLS realizations are precisely equally good at coping with the possible ill behavior of a given bounded periodic function integrable over its period, but the WPLS realization is not always spectrally minimal or canonical.     

Generalized linear systems with lumped or distributed parameters and differential vector spaces

By employing differential vector spaces, we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices. Questions such as the characterization of input and output, input-output equivalence, invertibility, feedback decoupling and realization are investigated.     

Minimal realization of transfer function matrices A comparative study of different methods

A comparative study is made of three basically different methods for obtaining minimal-order realizations of linear multivariable systems in the form of state equations for specified rational transfer function matrices. Computational efficiency and suitability for practical implementation are the main criteria used for the comparison. The possibilities of direct realization in canonical forms suitable for special applications are also examined.     

Existence and uniqueness of minimal realizations of nonlinear systems

In the theory of finite dimensional linear systems, it is well known that every input-output map that can be realized by one such system can also be realized by a system which is minimal, i.e. both controllable and observable. Moreover, the minimal realization of a given map is unique up to isomorphism. It is shown here that similar results hold for the class of all systems whose state space is a real analytic manifold, whose dynamics is given by a family of complete real analytic vector fields, and whose output is an arbitrary real analytic function on the state space.Research partially supported by NSF Grant GP-37488.     

非线性系统的实现与最小阶问题

在系统的微分域及相应的微分向量空间上定义了一个非交换的多项式环(算子环), 并利用这个环定义非线性系统的传递函数.用微分向量空间为工具,讨论单输入/单输出非线 性系统的实现问题.主要结果回答了:1)在什么条件下,不同的输入/输出微分方程有相同(等 价)的实现;2)在未知实现的条件下,如何确定输入/输出微分方程最小实现的阶数.覆盖了线 性系统理论的相关结果.     

A global parametrization of asymptotically stable linear systems

In this paper we construct a homeomorphism from the set of p × m transfer functions of McMillan degree n onto the open subspace of asymptotically stable linear systems. This homeomorphism yields a one-to-one correspondence between

1. (i) canonical forms for state space equivalence of minimal systems and

2. (ii) canonical forms for state space equivalence of asymptotically stable minimal systems.

Implications for the topology of various sets of asymptotically stable systems are given.     

含有分数阶有色关联噪声的分数阶系统的卡尔曼滤波器设计

提出基于Tustin生成函数的分数阶卡尔曼滤波器设计方法,以解决含有相互关联的分数阶有色过程噪声和分数阶有色测量噪声的连续时间线性分数阶系统的状态估计问题.通过Tustin生成函数方法,对连续时间线性分数阶系统进行离散化,将分数阶系统的微分方程转化为差分方程.利用增广向量法,将分数阶状态方程和分数阶有色噪声作为新的增广状态向量,从而将分数阶有色噪声转化为高斯白噪声.然后,提出一种基于Tustin生成函数的分数阶卡尔曼滤波算法,有效地实现对含有相互关联的分数阶有色过程噪声和分数阶有色测量噪声的连续时间线性分数阶系统的状态估计.与基于Grddotunwald-Letnikov差分的离散化方法相比,所提出的基于Tustin生成函数的卡尔曼滤波算法得到的状态估计精度更高,状态估计效果更好.最后,通过仿真结果验证所提出算法的有效性.     

An introduction to the shift realization for finite dimensional continuous time systems

A new approach is presented for the realization of continuous-time finite dimensional linear systems. Using standard results on Laplace transforms our results are also used to present a new derivation of Fuhrmann's shift realization for rational matrix functions.     

Canonical forms for the identification of multivariable linear systems

The advantage of using a unique parameterization in a numerical procedure for the identification of a system from operating records has been well established. In this paper several sets of canonical forms are described for state space models of deterministic multivariable linear systems; the members of these sets having therefore the required uniqueness property within the equivalence classes of minimal realizations of the system. In the identification of a stochastic system, it is shown how the problem depends also upon determining a unique factorization of the spectral density matrix of the system, and the sets of canonical forms obtained for the deterministic system are extended to this case.     

Balanced realizations of regime-switching linear systems

In this work, we establish a framework for balanced realization of linear systems subject to regime switching modulated by a continuous-time Markov chain with a finite state space. First, a definition of balanced realization is given. Then a ρ-balanced realization is developed to approximate the system of balancing equations, which is a system of time-varying algebraic equations. When the state space of the Markov chain is large, the computational effort becomes a real concern. To resolve this problem, we introduce a two-time-scale formulation and use decomposition/aggregation and averaging techniques to reduce the computational complexity. Based on the two-time-scale formulation, further approximation procedures are developed. Numerical examples are also presented for demonstration.     

A nonlinear time-delay realization for gastroparesis in patients with diabetes

The realization problem is fundamental in control systems theory and is well understood for linear systems. The situation is quite different for nonlinear systems with or without time delays. New results are given herein either as an abstract characterization of the class of delay–free nonlinear systems which are known to admit a realization, or by characterizing the existence of a realization for a wider class of retarded type or neutral type time delay nonlinear systems. The problem is motivated by the very practical modelling and realization problem of diabetes with or without gastroparesis. Up to 4% to 12% patients with diabetes are affected by the so-called gastroparesis which delays the digestion process. Several concurrent state space mathematical models do exist for diabetes and require a re-evaluation based on physiologic facts. These are reviewed herein to derive a (minimal) realization.     

Structural transformations of probabilistic finite state machines

Probabilistic finite state machines have recently emerged as a viable tool for modelling and analysis of complex non-linear dynamical systems. This paper rigorously establishes such models as finite encodings of probability measure spaces defined over symbol strings. The well known Nerode equivalence relation is generalized in the probabilistic setting and pertinent results on existence and uniqueness of minimal representations of probabilistic finite state machines are presented. The binary operations of probabilistic synchronous composition and projective composition, which have applications in symbolic model-based supervisory control and in symbolic pattern recognition problems, are introduced. The results are elucidated with numerical examples and are validated on experimental data for statistical pattern classification in a laboratory environment.     

Approximated modeling and minimal realization of transfer function matrices with multiple time delays

The modeling and minimal realization techniques for a specific multiple time-delay continuous-time transfer function matrix with a delay-free denominator and a multiple (integer/fractional) time-delay numerator matrix have been developed in the literature. However, this is not the case for a general multiple time-delay continuous-time transfer function matrix with multiple (integer/fractional) time delays in both the denominator and the numerator matrix. This paper presents a new approximated modeling and minimal realization technique for the general multiple time-delay transfer function matrices. According to the proposed technique, an approximated discrete-time state-space model and its corresponding discrete-time transfer function matrix are first determined, by utilizing the balanced realization and model reduction methods with the sampled unit-step response data of the afore-mentioned multiple time-delay (known/unknown) continuous-time systems. Then, the modified Z-transform method is applied to the obtained discrete-time transfer function matrix to find an equivalent specific multiple time-delay continuous-time transfer function matrix with multiple time delays in only the inputs and outputs, for which the existing control and design methodologies and minimal realization techniques can be effectively applied. Illustrative examples are given to demonstrate the effectiveness of the proposed method.     

Cascade decomposition and realization of multivariable systems via block-pole and block-zero placement

A technique is presented for state-feedback and state-feedforward block decomposition of a class of multivariable systems into a block-cascaded form structure having the assigned block poles and block zeros. A state-space technique is also presented for minimal realization of a multivariable compensator, represented by a voltage transfer function matrix, using the decoupled RC cascaded networks and summers.     

On the problems of constructing minimal realizations for two-dimensional filters

The input-output behavior of a two-dimensional linear filter is defined by a formal power series in two variables. If the power series is rational the dynamics of the filter is described by updating equations on finite dimensional local state spaces. The class of realizations considered in this paper is constituted by doubly indexed dynamical systems of reduced structure. The construction of the class of minimal realizations is based on matrix representation techniques of noncommutative power series.     

Realizations of Nonlinear Control Systems on Time Scales

Nonlinear partially defined systems on an arbitrary unbounded time scale are studied. They include continuous-time and discrete-time systems. The main problem is to find necessary and sufficient conditions for an abstract input/output map to have a realization as a nonlinear system of a specific class on the time scale. The obtained results extend criteria of realizability of continuous-time polynomial systems. A simple construction of a realization is provided.