Observable State Space Research Articles

Finite-time converging jump observer for switched linear systems with unknown inputs

This work addresses the state observation problem for a class of switched linear systems with unknown inputs. The proposed high-order sliding-mode observer provides a finite-time converging estimate of the continuous system’s state vector in spite of the presence of unknown inputs. The design procedure, which assumes knowledge of the discrete state of the switched system, is based on the principles of disturbance decoupling and hierarchical observer output injection. In order to cope with the switching nature of the plant under observation, jumps in the observer state space are enforced intentionally. The implementation of an additional observer allows for the reconstruction of the unknown inputs, which may be important in the framework of fault detection. Numerical examples illustrate the effectiveness of the suggested technique.

Finite-Time Converging Jump Observer for Linear Switched Systems with Unknown Inputs

Abstract The observation problem for linear switched systems with unknown inputs is addressed and solved via sliding mode technique. The proposed observer provides a finite time converging estimate of the state vector despite the presence of unknown inputs. The design procedure, which requires the knowledge of the discrete state of the switched system, is based on the principles of disturbance decoupling and hierarchical observer output injection. In order to cope with the switching nature of the plant under observation, state jumps in the observer state space are enforced intentionally. The implementation of an additional observer allows the reconstruction of the unknown inputs, which may be important in the framework of fault detection. A numerical example illustrates the effectiveness of the suggested technique.

The Identifiability of Covarion Models in Phylogenetics

Covarion models of character evolution describe inhomogeneities in substitution processes through time. In phylogenetics, such models are used to describe changing functional constraints or selection regimes during the evolution of biological sequences. In this work the identifiability of such models for generic parameters on a known phylogenetic tree is established, provided the number of covarion classes does not exceed the size of the observable state space. ;Generic parameters' as used here means all parameters except possibly those in a set of measure zero within the parameter space. Combined with earlier results, this implies both the tree and generic numerical parameters are identifiable if the number of classes is strictly smaller than the number of observable states.

How Equilibrium Prices Reveal Information in Time Series Models with Disparately Informed, Competitive Traders

Accommodating asymmetric information in a dynamic asset pricing model is technically challenging due to the problems associated with higher-order expectations. That is, rational investors are forced into a situation where they must forecast the forecasts of other agents (i.e., form higher-order expectations). In a dynamic setting, this problem telescopes into the infinite future and the dimension of the relevant state space approaches infinity. By employing the frequency domain approach of Whiteman (1983) and Kasa (2000), this paper demonstrates how information structures previously believed to lead to disparate expectations in equilibrium (e.g., Singleton (1987)) converge to a symmetric equilibrium. The revealing aspect of the price process lies in the invertibility of the observed state space, which makes it possible for agents to inferthe economically fundamental shocks, thus eliminating the need to forecast the forecasts of others.

How equilibrium prices reveal information in a time series model with disparately informed, competitive traders

Accommodating asymmetric information in a dynamic asset pricing model is technically challenging due to the problems associated with higher-order expectations. That is, rational investors are forced into a situation where they must forecast the forecasts of other agents. In a dynamic setting, this problem telescopes into the infinite future and the dimension of the relevant state space approaches infinity. By using the frequency domain approach of Whiteman [Linear Rational Expectations Models: A User's Guide, University of Minnesota Press, Minneapolis, 1983] and Kasa [Forecasting the forecasts of others in the frequency domain, Rev. Econ. Dynam. 3 (2000) 726–756], this paper demonstrates how information structures previously believed to preserve asymmetric information in equilibrium, converge to a symmetric information, rational expectations equilibrium. The revealing aspect of the price process lies in the invertibility of the observed state space, which makes it possible for agents to infer the economically fundamental shocks and thus eliminating the need to forecast the forecasts of others.

Controllable, observable and stable state space representations of a generalized order-up-to policy

We outline a framework for developing state space representations of production and inventory control policies from their transfer functions. The particular replenishment rules that we consider are members of the inventory and order based production control system family that have previously been shown to be a generalization of the periodic review Order-Up-To policy. Traditionally, this model has been studied with transfer functions in the frequency domain. In this paper, we focus on the discrete time case and derive state space representations that are both controllable and observable. The state space approach is then used to determine the stability boundary of the production ordering system based on the eigenvalues of the state matrix.

Irreducibility conditions for discrete-time nonlinear multi-input multi-output systems

The purpose of this paper is to present necessary and sufficient condition for irreducibility of discrete-time nonlinear multi-input multi-output system which extends directly the corresponding condition for the linear case. The condition is presented in terms of the greatest common left divisor of two polynomial matrices related to the input-output equations of the system. The basic difference is that unlike the linear case the elements of the polynomial matrices belong to a non-commutative polynomial ring. This condition provides a bases for finding the equivalent minimal irreducible representation of the i/o equations which is a suitable starting point for constructing an observable and accessible state space realization.

State space approach to behavioral systems theory: the Dirac–Bergmann algorithm

This paper develops an approach to behavioral systems theory in which a state space representation of behaviors is utilised. This representation is a first order hybrid representation of behaviors called pencil representation. An algorithm well known after Dirac and Bergmann (DB) is shown to be central in obtaining a constraint free and observable (CFO) state space representation of a behavior. Results and criteria for asymptotic stability, controllability, inclusions and Markovianity of behaviors are derived in terms of the matrices of this representation which involve linear algebraic processes in their computation.

P05 Adequacy of the proportional hazards model using lin, wei and ying plots

In this paper a hybrid state observer with deadbeat-like properties for a class of planar linear and nonlinear continuous time systems is presented. Finite-time convergence This is achieved either by performing a reset exactly when the estimation error has a suitable value, or by properly initializing the observer reset. In the first case the reset time can be determined using a bisection method provided the estimation error has an oscillatory behavior. In the second case the bisection algorithm is applied in the observer state space and does not require oscillatory behavior of the error dynamics. Both approaches can be used to improve the performance of a given local observer. We also discuss a robustified version in the presence of measurement noise.

Observability of discrete event dynamic systems

A finite state automaton is adopted as a model for discrete event dynamic systems (DEDS). Observations are assumed to be a subset of the event alphabet. Observability is defined as having perfect knowledge of the current state at points in time separated by bounded numbers of transitions. A polynomial test for observability is given. It is shown that an observer may be constructed and implemented in polynomial time and space. A bound on the cardinality of the observer state space is also presented. A notion of resiliency is defined for observers, and a test for resilient observability and a procedure for the construction of a resilient observer are presented. >

Persistency of Excitation in Possibly Unstable Continuous Time Systems and Parameter Convergence in Adaptive Identification

We find conditions on the input signal of an output reachable possibly unstable linear system, under which the output is persistently exciting. The conditions are given in both frequency and in time domain versions. Interpreting these results in the context of controllable or observable state space realizations we obtain some interesting facts relating persistency of excitation of the input, state and output signals. To illustrate the importance of our results we propose an adaptive identification scheme with “least squares” update law for multivariable plants with proper transfer function. We conclude that parameter convergence is guaranteed for any stationary piecewise uniformly continuous input with nonzero minimum interdiscontinuity distance and at least 2n + 1 points of strong support of its spectral measure, where n is the McMillan degree of the plant. With covariance resetting the convergence rate is shown to be exponential. Without covariance resetting we prove, that the convergence rate is as 1 / t for sufficiently fast identifiers, and in any case at least as fast as 1 /√t.

Group representations in the Liouville representation and the algebraic approach

Group representations on the Liouville representation spaces are considered. It is shown that the state space ℐ1(H) of trace class operators on Hilbert space H and the observable space ℒ(H) of bounded operators are completely reducible under physically induced representations of compact Hausdorff groups when appropriate topologies are used. For state space ℐ1(H) both the norm topology and the weak topology lead to complete reducibility, while for observable space ℒ(H) the weak-* topology—but not the norm topology—suffices. This leads to conservation laws, selection rules, and Wigner–Eckart theorems for the Liouville representation. It is shown that serious difficulties are encountered when a similar theory is attempted on the observable space and state space used in the algebraic approach.

Invariants and canonical forms for systems structural and parametric identification

The basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled. With reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set. Then the set of polynomial input-output models of the type P( z) y( t)= Q( z) u( t) and the equivalence relation induced by the premultiplication of P and Q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced. Since the two sets of canonical forms share common sets of complete independent invariants, very simple algebraical links between state space and input-output canonical forms can be deduced. The previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input-output sequences generated by a linear, discrete, time-invariant multivariable system. The results obtained in the identification of a real process are then reported.