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,     

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.     

A note on the control of uncertain linear dynamical systems with constrained control input

A method is presented to design a stabilizing saturated state feedback controller for linear continuous-time systems with bounded uncertainty acting via the input channel.     

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.     

On control of two-scale stochastic systems with linear dynamics in the fast variables

We consider a problem of optimal control for a stochastic two-scale system that depends on a small positive parameter and where the stochastic differential equations, describing the evolution of the fast variables, degenerate to algebraic equations in the limit when=0. The model is nonlinear, but with the fast components entering linearly. Our main result is to show that, when tends to zero, the optimal value of the cost functional, that also includes the fast variables in the terminal pay-off, converges to the optimal value of a suitable reduced stochastic control problem. As a consequence we also have that a nearly optimal control for the limit problem can be modified to become nearly optimal also for the prelimit problems when. is sufficiently small.The work by Yuri M. Kabanov was performed during a stay in Padova supported by GNAFA/CNR.     

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.     

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.     

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.     

Effect of integral action on the stabilizability of continuous-time linear dynamical systems with retarded control

In this paper, the effect of integral action on the stabilizability of a class of linear dynamical systems with retarded control is investigated using the method of D-partition. This effect is illustrated by the presentation of a number of D-partition diagrams and a selection of transient response characteristics.     

Decentralized control of linear dynamical systems via polynomial matrix methods I: Two interconnected scalar systems

The problem of the decentralized control of a linear interconnected dynamical system is considered. This is done by defining the problems of the decentralized controllability and of the decentralized observability, which are solved for a system consisting of two interconnected scalar systems. The solution is determined using the theory of the linear dynamical systems in the differential operator representation (Wolovich 1974), and provides, besides structural results, a systematic method for the decentralized control of these systems.     

Modeling sensorimotor learning with linear dynamical systems

Recent studies have employed simple linear dynamical systems to model trial-by-trial dynamics in various sensorimotor learning tasks. Here we explore the theoretical and practical considerations that arise when employing the general class of linear dynamical systems (LDS) as a model for sensorimotor learning. In this framework, the state of the system is a set of parameters that define the current sensorimotor transformation-the function that maps sensory inputs to motor outputs. The class of LDS models provides a first-order approximation for any Markovian (state-dependent) learning rule that specifies the changes in the sensorimotor transformation that result from sensory feedback on each movement. We show that modeling the trial-by-trial dynamics of learning provides a substantially enhanced picture of the process of adaptation compared to measurements of the steady state of adaptation derived from more traditional blocked-exposure experiments. Specifically, these models can be used to quantify sensory and performance biases, the extent to which learned changes in the sensorimotor transformation decay over time, and the portion of motor variability due to either learning or performance variability. We show that previous attempts to fit such models with linear regression have not generally yielded consistent parameter estimates. Instead, we present an expectation-maximization algorithm for fitting LDS models to experimental data and describe the difficulties inherent in estimating the parameters associated with feedback-driven learning. Finally, we demonstrate the application of these methods in a simple sensorimotor learning experiment: adaptation to shifted visual feedback during reaching.     

On the control of discrete-event dynamical systems

We study a class of problems related to the supervisory control of a discrete-event system (DES), as formulated by Ramadge and Wonham, and we focus on the computational effort required for their solution. While the problem of supervisory control of a perfectly observed DES may be easily solved by dynamic programming, the problem becomes intractable (in the sense of complexity theory) when imperfectly observed systems are considered. Research supported by the Army Research Office (Grant No. DAAL03-86-K-0171) and by an NSF PYI award, with matching funds from Bellcore Inc.     

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.     

On the properties of the realizations of linear dynamical systems

Summary This paper is concerned with the problem of finding realizations for dynamical systems, with the properties of minimality, stability and boundedness.In particular the paper considers the problems of the existence and uniqueness, within suitable equivalence relations, of these realizations, as well as the problems of the invariance of listed properties.Using a general procedure, the problems of invariance and uniqueness are solved in the case of subsets of minimal and stable (uniformly stable, restrictively stable, asymptotically and exponentially stable) realizations. The same problems are solved in the case of restrictively stable, bounded and minimal realizations.This work was partially supported by C.N.R. and by Fondazione U. Bordoni, Rome, Italy.This paper is a revised version of a contribution bearing the same title which the authors presented at the IFAC Symposium (System Engineering Approach to Computer Control) held at Kyoto in 1970. It contains some results they mentioned in the course of the discussion, as well as others that were subsequently established.     

On absolute stability of Lur''''e control systems with multiple non-linearities using linear matrix inequalities

Necessary and suffcient conditions for the existence of a Lyapunov function in the Lur'e form to guarantee the absolute stability of Lur'e control systems with multiple non-linearities are discussed in this paper. It simplifies the existence problem to one of solving a set of linear matrix inequalities (LMIs), If those LMIs are feasible, free parameters in the Lyapunov function,such as the positive definite matrix and the coefficients of the integral terms, are given by the solution of the LMIs. Otherwise, this Lyapunov function does not exist. Some sufficient conditions are also obtained for the robust absolute stability of uncertain systems.A numerical example is provided to demonstrate the effectiveness of the proposed method.     

On absolute stability of Lur＇e control systems with multiple non-linearities using linear matrix inequalities

Necessary and suffcient conditions for the existence of a Lyapunov function in the Lur‘e form to guarantee the absolute stability of Lur‘e control systems with multiple non-linearities are discussed in this paper. It simplifies the existence problem to one of solving a set of linear matrix inequalities (LMIs). If those LMIs are feasible, free parameters in the Lyapunov function, such as the positive definite matrix and the coefficients of the integral terms, are given by the solution of the LMIs. Otherwise, this Lyaptmov function does not exist. Some sufficient conditions are also obtained for the robust absolute stability of uncertain systems. A numerical example is provided to demonstrate the effectiveness of the proposed method.     

On the selection of variables for qualitative modelling of dynamical systems

Behavioural modelling of physical systems from observations of their input/output behaviour is an important task in engineering. Such models are needed for fault monitoring as well as intelligent control of these systems. The paper addresses one subtask of behavioural modelling, namely the selection of input variables to be used in predicting the behaviour of an output variable. A technique that is well suited for qualitative behavioural modelling and simulation of physical systems is Fuzzy Inductive Reasoning (FIR), a methodology based on General System Theory. Yet, the FIR modelling methodology is of exponential computational complexity, and therefore, it may be useful to consider other approaches as booster techniques for FIR. Different variable selection algorithms: the method of the unreconstructed variance for the best reconstruction, methods based on regression coefficients (OLS, PCR and PLS) and other methods as Multiple Correlation Coefficients (MCC), Principal Components Analysis (PCA) and Cluster analysis are discussed and compared to each other for use in predicting the behaviour of a steam generator. The different variable selection algorithms previously named are then used as booster techniques for FIR. Some of the used linear techniques have been found to be non-effective in the task of selecting variables in order to compute a posterior FIR model. Methods based on clustering seem particularly well suited for pre-selecting subsets of variables to be used in a FIR modelling and simulation effort.     

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.     

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.