Minimal state-space realization in linear system theory: an overview

In this paper, we show that a minimal state space realization in Jordan canonical form for linear multivariable continuous‐time systems described by rational transfer function matrices could be obtained in a natural and basic way by using the concept of Nerode equivalence. While the minimal state space realization is known, the contribution of this note is to provide an alternative realization procedure which is directly introduced in a simple and self‐contained manner. Both scalar and multivariable cases in the continuous‐time setting are discussed. The presentation also provides a concrete construction of rational vector function with preassigned poles to interpolate any 2‐norm bounded analytical vector function in the open left‐half of the complex plane. The basic idea of Nerode equivalence is that the state can be identified with a corresponding equivalence class of inputs. For a linear finite dimensional time‐invariant continuous‐time system, the zero state is identified with the kernel of certain Hankel operator. This characterization via Nerode equivalence class sheds light on the construction of state for general nonlinear input–output systems.

A moment-based approach to ensemble controllability of linear systems

State space realizations and monomial equivalence for convolutional codes

Identification of structural parameters by using observation data with noise in time domain was studied. Autoregressive-moving-average (ARMA) time-series model of a vibrating structure was established, and the identification problem of structural parameters was transformed into the identification problem of parameters of ARMA mode1. Based on the assumption of the unknown but bounded noise, an interval algorithm for set-membership identification of parameters of linear time-invariant system was used to seek the minimal hyper-rectangle of parameters, which is compatible with the measurements and the bounded noise, so that the structural parameters can be obtained. Then the Taian pumped-storage power station engineering project illustrates its feasibility and effectiveness.

A data-driven fault isolation and estimation approach for unknown linear systems

The standard state space solutions to the /spl Hscr//sub /spl infin// control problem for linear time-invariant systems are generalized to nonlinear time-invariant systems. A class of local nonlinear (output feedback) /spl Hscr//sub /spl infin//-controllers are parameterized as nonlinear fractional transformations on contractive, stable nonlinear parameters. As in the linear case, the /spl Hscr//sub /spl infin// control problem is solved by its reduction to state feedback and output estimation problems, together with a separation argument. Sufficient conditions for /spl Hscr//sub /spl infin//-control problem to be locally solved are also derived with this machinery. >

This paper addresses the dynamic output feedback stabilization problem for linear time-invariant systems where the process disturbance preview is available to the controller via communication networks. A lower bound of data rates of communication channels, above which there exists a feedback control policy to stabilize the unstable plant with unbounded disturbance, is presented. Furthermore, the problem of bandwidth allocation in the communication channel is analyzed based on the system dynamics. Simulation results show the validity of the proposed scheme.

We consider the robust (H???) control problem for linear time-invariant systems over unreliable communication channels that are subject to packet losses from sensors to the controller and from the controller to the actuators. Moreover, the acknowledgments of control packet transmissions are also sent over reverse unreliable communication channels, which is known as the quasi-TCP model. The robust control problem is formulated within a stochastic zero-sum dynamic game framework, from which we obtain an H??? controller that minimizes the worst-case cost function in the presence of an adversary in the dynamical system. We characterize a set of existence conditions of the H??? controller in terms of the disturbance attenuation parameter ?? and packet loss rates. For the least disturbance attenuation scenario (as ?? ??? 8), we show that the H??? control system converges to the corresponding LQG system. We also show that the H??? controller for the TCP- and UDP-cases can be arrived at as appropriate limits of the solution for the quasi-TCP problem. Numerical examples are presented to illustrate the theoretical results.

Exact model matching of linear systems using generalized sampled-data hold functions

This article studies the remote state estimation problem of linear time-invariant systems with stochastic event-triggered sensor schedules in the presence of packet drops between the sensor and the estimator. Due to the existence of packet drops, the Gaussianity at the estimator side no longer holds. It is proved that the system state conditioned on the available information at the estimator side is Gaussian mixture distributed. The minimum-mean-square-error (MMSE) estimator can be obtained from the bank of Kalman filters. Since the optimal estimators require exponentially increasing computation and memory with time, suboptimal estimators to reduce computational complexities by limiting the length and numbers of hypotheses are further provided. In the end, simulations are conducted to illustrate the performance of the optimal and suboptimal estimators.

A matrix Euclidean algorithm induced by state space realization

The problem of minimal state space realization of two-dimensional systems is considered. The approach followed is, initially, to derive a circuit realization of the given transfer function involving a minimum number of delay elements. To facilitate this realization, the transfer function is expanded into a continued fraction. Using this circuit realization, an algorithm is proposed which readily provides the matrices of the state space model. The simplicity in deriving this model is due to a novel state space model introduced, which is of cyclic structure. Several examples are presented to illustrate ihe proposed algorithm.

A framework for analyzing nonlinear eigenproblems and parametrized linear systems

Given a discrete-time linear system C, a shortest basis for C is a set of linearly independent generators for C with the least possible lengths. A basis B is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J, the generators in B whose span is in J is a basis for the subsystem CJ). The dimensions of the minimal state spaces and minimal transition spaces of C are simply the numbers of generators in a shortest basis B that are active at any given state or symbol time, respectively. A minimal linear realization for C in controller canonical form follows directly from a shortest basis for C, and a minimal linear realization for C in observer canonical form follows directly from a shortest basis for the orthogonal system C⊥. This approach seems conceptually simpler than that of classical minimal realization theory.

: This report considers a feedback control law for linear time-varying and time invariant discrete systems based on a receding horizon concept applied to a minimum energy problem with fixed terminal constraints. The control law is shown to be asymptotically stable and to result in a new method for stabilizing linear time-varying systems as well as extending some well known methods for stabilizing time invariant systems. In particular, the stabilizing gains of the feedback control law are obtained from the solution to a discrete Riccati equation over an arbitrary finite time interval, which is relatively easy to compute. The gain matrix reduces to a constant matrix for linear time invariant systems. Some stability results will turn out to be special cases of these results. The results parallel those for linear continuous time systems, although the technical details are tedious and more involved.