Use of Pseudo-Observability Indices In Identification of Continuous-Time Multivariable Models

This chapter studies null controllability of continuous-time linear systems with bounded controls. Null controllability of a system refers to the possibility of steering its state to the origin in a finite time by an appropriate choice of the admissible control input. If the system is linear and is controllable, then any state can be steered to any other location, for example, the origin, in the state space in a finite time by a control input. This implies that any controllable linear system is null controllable from any point in the state space. The control inputs that are used to drive certain states to the origin, however, might have to be large in magnitude. As a result, when the control input is limited by the actuator saturation, a linear controllable system might not be globally null controllable. In this situation, it is important to identify the set of all the states that can be steered to the origin with the bounded controls delivered by the actuators. This set is referred to as the null controllable region and is denoted as C

In this paper a basis is developed for the systematic design of linear multivariable continuous-time tracking systems for plants with inaccessible states which thus extends the results for continuous-time plants with accessible states previously obtained by Porter and Bradshaw (1974). The design method is illustrated by the presentation of the results of computer simulation studies.

This paper is concerned with the property of asymptotic hyperstability of a continuous-time linear system under a class of continuous-time nonlinear and perhaps time-varying feedback controllers belonging to a certain class with two main characteristics; namely, (a) it satisfies discrete-type Popov’s inequality at sampling instants and (b) the control law within the intersample period is generated based on its value at sampling instants being modulated by two design weighting auxiliary functions. The closed-loop continuous-time system is proved to be asymptotically hyperstable, under some explicit conditions on such weighting functions, provided that the discrete feed-forward transfer function is strictly positive real.

This technical note has been motivated by the need to assess the preservation of polyhedral Lyapunov functions for stable continuous-time linear systems under numerical discretization of the transition matrix. This problem arises when discretizing linear systems in such a manner as to preserve a certain type of stability of the discrete time approximation. Our main contribution is to show that a continuous-time system and its Padé discretization (of any order and sampling) always share at least one common piecewise linear (polyhedral) Lyapunov function.

A methodology for the synthesis of time-varying static output-feedback gains, capable of stabilising continuous-time linear time-varying systems, is proposed in this paper. The method is based on the computation of output-feedback gains that stabilise a family of periodic discrete-time subsystems, related to the original model, followed by a transformation that yields the desired stabilising gains for the continuous-time system. Based on such strategy, three different approaches to compute stabilising periodic discrete-time gains, as well as the necessary transformations to generate the continuous-time output-feedback gains, are proposed. Numerical examples are presented in order to compare and analyze the efficiency of the proposed approaches.

The shuffle algorithm is applied to analysis of the fractional descriptor Roesser type continuous-time linear systems. Using the shuffle algorithm the fractional descriptor linear system is reduced to the equivalent standard system and the system is decomposed into dynamic and static parts. Procedure for computation of the matrices of equivalent standard system and of the dynamical and static parts of the system is proposed.

An alternative method was proposed for identification of linear continuous time system with noisy output. Using Gaussian modulating filters, by numerical integration, a discrete equivalent identification model that was parametrized directly in continuous time model parameters was obtained, and the LS estimation algorithm was developed. The LS estimator is biased even with white measurement output noise, and the bias is determined by variance of output noise. On the basis of obtained output measurement noise variance, the bias was compensated and the consistent estimates of system parameters were given. Simulation which compares the standard LS with BCLS algorithm indicates that the proposed algorithm is an efficient method for noisy output continuous time system identification.

This work presents a two-dimensional (2-D) system theory based iterative learning control (ILC) method for linear continuous-time multivariable systems. We demonstrate that a 2-D continuous-discrete model can be successfully applied to describe both the dynamics of the control system and the behavior of the learning process. We successfully exploited the 2-D continuous-discrete Roesser's linear model by extending the ILC technique from discrete control systems to continuous control systems. Three learning rules for ILC are derived. Necessary and sufficient conditions are given for convergence of the proposed learning rules. Compared to the learning rule suggested by Arimoto et al. (1984), our developed learning rules are less restrictive and have wider applications. The third learning rule proposed ensures the reference output trajectory can be accurately tracked after only one learning trial. Three numerical examples are used to illustrate the proposed control procedures.

In this note, the novel representation is proposed for a linear periodic continuous-time system with T-periodic real-valued coefficients. We prove that a T-periodic real-valued factor and two real-valued matrix exponential functions can be extracted from a state transition matrix, while, in the well-known Floquet representation theorem, a 2T-periodic real-valued factor and a real-valued matrix exponential function are extracted from the state transition matrix. Then we also proved that any T-periodic system can be transformed to a system with T-periodic real-valued trigonometric coefficients using a T-periodic real-valued coordinate transformation, while, in the well-known Lyapunov reducibility theorem, a 2T-periodic real-valued coordinate transformation is utilized to transform the given periodic system into a time-invariant system with real coefficients. This new information can be useful for designing a T-periodic control law.

In this brief note, a probability theory approach is proposed to investigate the stability analysis issue for a class of networked sampled-data systems with packet dropouts. The continuous-time linear system under consideration is first transformed into a discrete-time system, where the system matrix of the discrete-time system is characterized by stochastic characteristic. Then, by the matrix exponential computation, an equivalent discrete-time stochastic system is established. Based on this, a controller is designed by the law of total expectation, which guarantees the stochastic stability of the closed-loop stochastic system. Finally, a numerical example and an example using power grid with IEEE 4-bus line are used to verify the effectiveness and applicability of the proposed design algorithm.

In present work we consider a multidimensional linear system with additive inputs (controls) and noise governed by a Brownian motion. We allow parameters of system to change in time. In addition we assume that exact form of coefficients is not known, rather intervals within which these coefficients fluctuate are given. In addition, we do not know exact state of system, rather we observe output correlated with this state. The set of allowable controls are feedback controls based on observation of output and an ideal model dynamics. With each control we associate a performance functional characterizing the quality of tracking process and objective is to construct a robust feedback control which minimizes maximal performance functional within class of systems with nonstochastic coefficients fluctuating in same intervals as those of our problem.

We study continuous-time switched linear systems with unobserved and exogenous mode signals. We analyze the observability of the initial state and initial mode under arbitrary switching, and characterize both properties in both the autonomous and non-autonomous cases.

SUMMARY The LQ+ problem, i.e. the finite-horizon linear quadratic optimal control problem with nonnegative state constraints, is studied for positive linear systems in continuous time and in discrete time. Necessary and sufficient optimality conditions are obtained by using the maximum principle. These conditions lead to a computational method for the solution of the LQ+ problem by means of a corresponding Hamiltonian system. In addition, the necessary and sufficient conditions are proved for the LQ+-optimal control to be given by the standard LQ-optimal state feedback law. Then sufficient conditions are established for the positivity of the LQ-optimal closed-loop system. In particular, such conditions are obtained for the problem of minimal energy control with penalization of the final state. Moreover, a positivity criterion for the LQ-optimal closed-loop system is derived for positive discrete-time systems with a positively invertible (dynamics) generator. The main results are illustrated by numerical examples. Copyright q 2010 John Wiley & Sons, Ltd.

In this chapter, the sliding mode control (SMC) design for a class of continuous-time linear uncertain systems is considered based on the parametric approaches utilizing the Lyapunov (or Riccati-like) inequalities. SMC for the continuous-time linear systems is one of the well-known issues in control theory. However, this chapter addresses a new aspect of SMC for linear uncertain systems and establishes a systematic procedure to design a sliding hyperplane having multiple design objectives.

OBSERVABILITY OF DISCRETE TIME LINEAR SWITCHING SYSTEMS1