Realization of linear dynamical systems

Realization theory for both time-invariant and time-variable linear systems is developed and its applicability to linear quadratic control and filtering is discussed. For time-invariant systems a review of canonical structure theory is given and various properties such as minimality and equivalence are characterized in terms of the Hankel matrix. Realization theory for such systems is then developed based on the Hankel matrix and a new computational algorithm is presented. For time-variable systems the emphasis is on obtaining physically meaningful realizations and several procedures which accomplish this are detailed. For "constant rank" systems, a generalization of the Hankel matrix approach is also presented.     

Optimal-observer design for linear dynamical systems with uncertain parameters

This paper presents a design technique for the synthesis of robust observers for linear dynamical systems with uncertain parameters. The perturbations under consideration are modelled as unknown but bounded disturbances and an ellipsoidal set-theoretic approach is used to formulate the optimal-observer design problem. The optimal criterion introduced here is the minimization of the ‘size’ of the bounding ellipsoid of the estimation error. A necessary and sufficient condition for this optimal design problem is presented. The results are stated in terms of a reduced-order observer with constant gain matrix, which is then determined by solving a matrix Riccati-type equation. Furthermore, a gradient-search algorithm is presented to find the optimal solution when the free parameter that enters in the construction of the bounding ellipsoids of the estimation error is considered as a design parameter. The effectiveness of the proposed approach is illustrated through a numerical example.     

Stabilizability of continuous-time linear dynamical systems with retarded control

In this paper, conditions for the stabilizability by linear state feedback of a class of linear dynamical systems with retarded control are established using the method of D-partition. Typical transient response characteristics which illustrate these stabilizability conditions arc presented,     

Kirchhoff interconnectability of linear constant dynamical systems

The paper introduces the concept of Kirchhoff interconnection, proper to multi-variable dynamical systems, and characterized by relations which generalize in a certain sense those relations as set down by Kirchhoff's theorems for an electric network with one independent node and an arbitrary number of independent loopa. It also defines the partially reversing system, the conditions of Kirchhoff interconnectability, and some aspects of the output controllability and input observability of Kirchhoff interconnectable and interconnected systems.     

Identifiability of linear and nonlinear dynamical systems

This short paper considers the identification of dynamical systems from input-output data. The problem of parameter identifiability for such systems is approached by considering whether system outputs obtained with different parameter values can be distinguished one from another. The results are stated formally by defining the notion of "output distinguishability." Parameter identifiability is then defined precisely in terms of output distinguishability. Relationships have been developed with the other definitions such as least square identifiability and identifiability from the transfer function. Several results for linear and nonlinear systems are presented with examples.     

Minimal parametrizations of single-input linear dynamical systems

The parametrization of the set of all controllable single-input time-invariant linear dynamical systems is defined as a map π from a parameter set M_π onto a given set of canonical forms 𝒸_π. A parametrization is called ‘minimal’ if it induces a continuous canonical map and the number of parameters is minimal. Some of its general properties, necessary and sufficient conditions, and several new concrete forms suitable for identification are also given.     

New dynamical optimal learning for linear multilayer FNN

This letter presents a new dynamical optimal learning (DOL) algorithm for three-layer linear neural networks and investigates its generalization ability. The optimal learning rates can be fully determined during the training process. The mean squared error (mse) is guaranteed to be stably decreased and the learning is less sensitive to initial parameter settings. The simulation results illustrate that the proposed DOL algorithm gives better generalization performance and faster convergence as compared to standard error back propagation algorithm.     

New multi-level algorithm for linear dynamical systems

A new approach is developed for the solution of linear interconnected dynamical systems. The algorithm is within the class of non-feasible methods. Its main feature is that it allows the transfer of information between the subsystems with each iteration. The convergence properties of this technique are given in detail. It is shown that the proposed methodology may be faster than the well-known prediction algorithm given in the literature since in each iteration better interconnection information is used within the period of the optimization horizon.     

Robustness analysis for linear dynamical systems with linearlycorrelated parametric uncertainties

The authors propose an approach for robust pole location analysis of linear dynamical systems with parametric uncertainties. Linear control systems with characteristic polynomials whose coefficients are affine in a vector of uncertain physical parameters are considered. A design region in complex plane for system pole placement and a nominal parameter vector generating a characteristic polynomial with roots in that region are given. The proposed method allows the computation of maximal domains bounded by linear inequalities and centered at the nominal point in system parameter space, preserving system poles in the given region. The solution of this problem is shown to also solve the problem of testing robot location of a given polytope of polynomials in parameter space. It is proved that for stability problems for continuous-time systems with independent perturbations on polynomial coefficients, this method generates the four extreme Kharitonov polynomials     

Estimators for linear dynamical SISO systems with disturbed input-output measurements

Several types of instrumental variable estimators are proposed for estimating parameters of linear dynamical single-input single-ouput systems with disturbed input-output measurements.No restrictions on the colouring of the distributions are imposed. The availability of extra (disturbed) measurements of input or output data simpifies the choice of the instrumental variable.     

Componentwise asymptotic stability of linear constant dynamical systems

This note deals with a special type of asymptotic stability, namely componentwise asymptotic stability with respect to the vectorgamma(t)(CWASγ) of systemS: dot{x} = Ax + Bu, t geq 0, wheregamma(t) > 0(componentwise inequality) andgamma(t) rightarrow 0ast rightarrow + infty.Sis CWASγ if for eacht_{0} geq 0and for each|x(t_{0})| leq gamma (t_{0}) (|x (t_{0})|with the components|x_{i}(t_{0})|the free response ofSsatisfies|x(t)| leq gamma (t)for eacht geq t_{0}. Forgamma(t){underline { underline delta} } alphae^{-beta t}, t geq 0, withalpha > 0andbeta > 0(scalar), the CWEAS (E= exponential) may be defined.Sis CWAS γ (CWEAS) if and only ifdot{gamma}(t) geq bar{A}gamma(t), t geq 0 (bar{A}alpha < 0); A {underline { underline delta} } (a_{ij})andbar{A}has the elements aijand|a_{ij}|, i neq j. These results may be used in order to evaluate in a more detailed manner the dynamical behavior ofSas well as to stabilizeScomponentwise by a suitable linear state feedback.     

A generalized orthonormal basis for linear dynamical systems

In many areas of signal, system, and control theory, orthogonal functions play an important role in issues of analysis and design. In this paper, it is shown that there exist orthogonal functions that, in a natural way, are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l₂ⁿ . To this end, use is made of balanced realizations of inner transfer functions. The orthogonal functions can be considered as generalizations of, for example, the pulse functions, Laguerre functions, and Kautz functions, and give rise to an alternative series expansion of rational transfer functions. It is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion, i.e., to obtain a good approximation by retaining only a finite number of expansion coefficients. Consequences for identification of expansion coefficients are analyzed, and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients     

On efficient sensor scheduling for linear dynamical systems

Consider a set of sensors estimating the state of a process in which only one of these sensors can operate at each time-step due to constraints on the overall system. The problem addressed here is to choose which sensor should operate at each time-step to minimize a weighted function of the error covariances of the state estimates. This work investigates the development of tractable algorithms to solve for the optimal and suboptimal sensor schedules. A condition on the non-optimality of an initialization of the schedule is developed. Using this condition, both an optimal and a suboptimal algorithm are devised to prune the search tree of all possible sensor schedules. The suboptimal algorithm trades off the quality of the solution and the complexity of the problem through a tuning parameter. The performance of the suboptimal algorithm is also investigated and an analytical error bound is provided. Numerical simulations are conducted to demonstrate the performance of the proposed algorithms, and the application of the algorithms in active robotic mapping is explored.     

Asymptotics of reachable sets of linear dynamical systems with impulsive control

Explicit asymptotic formulas for the reachable sets of linear dynamical systems with constraints on the total impulse of control action under different assumptions concerning the spectrum of the system’s matrix are obtained. It is shown that for large time, the reachable sets can be approximately represented in the form of the product of the scaling matrix and the normalized reachable set, where the matrix is an elementary function of time, and the normalized reachable set depends on time quasi-periodically. Analysis of asymptotic formulas has shown that, generally speaking, there exists a continuum of limit shapes, the aggregate of which produces a multidimensional attractor.     

Invertibility of linear dynamical systems over commutative rings

This correspondence is addressed to the question of invertibility of linear dynamical systems. Specifically, inversion over principal ideal domains is discussed in view of the fact that 2-D systems can be considered as 1-D systems over a principal ideal domain.     

Target path controllability of linear time-varying dynamical systems

The purpose of this paper is to derive necessary and sufficient conditions for target path controllability (TPC) of time-varying dynamical systems, and to relate the results to those of Brockett and Mesarović in 1965. It is shown that conditions for TPC inW^{p}_{1}by means of L^p-controls are easy to verify, but are very strong. TPC implies that the number of target variables is less than or equal to the number of control variables.     

Modeling learning technology systems as business systems

The design of Learning Technology Systems, and the Software Systems that support them, is largely conducted on an intuitive, ad hoc basis, thus resulting in inefficient systems that defectively support the learning process. There is now justifiable, increasing effort in formalizing the engineering of Learning Technology Systems in order to achieve better learning effectiveness as well as development efficiency. This paper presents such an approach for designing Learning Technology Systems and their most popular specialization, the Web-based Learning Systems, by modeling them as business systems, using business-modeling methods. The aim is to provide an in-depth analysis and comprehension of the Learning Technology Systems and Web-based Learning Systems domain, that can be used for improving the systems themselves, as well as for building the supporting software systems. Our work is based upon the Learning Technology Systems Architecture standard of IEEE LTSC, on the empirical results of designing Web-based Learning Systems for university courses and on the practices of the Rational Unified Process and the Unified Modeling Language.     

Learning discrete linear systems with the orthogonal learning algorithm

The orthogonal learning algorithm (OLA), based on the linear least-squares error, provides a useful tool to the problem of generating a good discrete model out of a continuous linear system. It is shown in this paper as the learning capabilities of the OLA can be used to determine such an equivalent model along with the generation of the required data. The procedure has been applied to plants of several types like high-order, integrating and non-minimum phase; all of them including an input delay. Discrete PIDs have also been implemented.