Finite-dimensional Linear Systems Research Articles

Robust Stabilization by Linear Output Delay Feedback

The main result of this paper establishes that if a controller $C$ (comprising of a linear feedback of the output and its derivatives) globally stabilizes a (nonlinear) plant $P$, then global stabilization of $P$ can also be achieved by an output feedback controller $C[h]$, where the output derivatives in $C$ are replaced by an Euler approximation with sufficiently small delay $h>0$. This is proved within the conceptual framework of the nonlinear gap metric approach to robust stability. The main result is then applied to finite dimensional linear minimum phase systems with unknown coefficients but known relative degree and known sign of the high frequency gain. Results are also given for systems with nonzero initial conditions.

Carleman Linearization for Control and for State and Disturbance Estimation of Nonlinear Dynamical Processes

In this paper, techniques based on Carleman linearization are presented which are applicable to the design of controllers and state estimators for nonlinear dynamical systems. Originally, Carleman linearization has been developed to transform sets of polynomial ordinary differential equations into infinite dimensional linear system representations. After choosing suitable finite orders for truncation of Carleman linearization, finite dimensional linear and bilinear system representations are obtained for which the design of controllers and state estimators can be performed in a straightforward way. To visualize the practical applicability of the presented procedures, state as well as disturbance estimation for a linear axis driven by pneumatic muscle actuators is studied.

Consistent estimation of real NMP zeros in stable LTI systems of arbitrary complexity

In this contribution we show that under certain conditions it is possible to estimate a non-minimum phase zero consistently using a very simple 2 parameter finite impulse response model, for arbitrarily complex finite dimensional stable linear time invariant systems.

Stochastic Passivity and its Application in Adaptive Control

A positive-real like lemma for finite dimensional linear time invariant uncertain systems with state multiplicative noise is derived. The system uncertainties are assumed to be of either polytopic or Markov jump type. Passivity conditions for both cases are derived in terms of linear matrix inequalities. The results are used to the design a direct adaptive controller for a tracking system.

Well-Posedness of Systems of Linear Elasticity with Dirichlet Boundary Control and Observation

An open-loop system of linear elasticity with Dirichlet boundary control and collocated observation is considered. The main result we obtained states that this system is well-posed in the sense of Salamon. The result deduces the exponential stability of the closed-loop system under proportional output feedback. Hence it answers positively an open question proposed by Liu and Krstić [IMA J. Appl. Math., 65 (2000), pp. 109–121]. Moreover, the well-posedness, together with the regularity of the open-loop system in the sense of Weiss that was obtained in our separate paper [S. G. Chai and B. Z. Guo, Feedthrough Operator for Linear Elasticity System with Boundary Control and Observation, Preprint, School of Computational and Applied Mathematics, University of the Witwatersrand, South Africa, 2008], makes this infinite-dimensional system parallel in many ways to a linear finite-dimensional system in the framework of well-posed and regular linear infinite-dimensional systems.

Weakly coprime factorization and state-feedback stabilization of discrete-time systems

The LQ-optimal state feedback of a finite-dimensional linear time-invariant system determines a coprime factorization NM −1 of the transfer function. We show that the same is true also for infinite-dimensional systems over arbitrary Hilbert spaces, in the sense that the factorization is weakly coprime, i.e., Nf, $${Mf \in \mathcal {H}^2 \Longrightarrow f \in \mathcal {H}^2}$$ for every function f. The factorization need not be Bezout coprime. We prove that every proper quotient of two bounded holomorphic operator-valued functions can be presented as the quotient of two bounded holomorphic weakly coprime functions. This result was already known for matrix-valued functions with the classical definition gcd(N, M) = I, which we prove equivalent to our definition. We give necessary and sufficient conditions and further results for weak coprimeness and for Bezout coprimeness. We then establish a variant of the inner–outer factorization with the inner factor being “weakly left-invertible”. Most of our results hold also for continuous-time systems and many are new also in the scalar-valued case.

Rational streams coalgebraically

We study rational streams (over a field) from a coalgebraic perspective. Exploiting the finality of the set of streams, we present an elementary and uniform proof of the equivalence of four notions of representability of rational streams: by finite dimensional linear systems; by finite stream circuits; by finite weighted stream automata; and by finite dimensional subsystems of the set of streams.

Unique continuation property near a corner and its fluid-structure controllability consequences

We study a non standard unique continuation property for the biharmonic spectral prob- lem Δ 2 w = −λΔw in a 2D corner with homogeneous Dirichlet boundary conditions and a supplemen- tary third order boundary condition on one side of the corner. We prove that if the corner has an angle 0 <θ 0 < 2π, θ0 � π and θ0 � π/2, a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing a Stokes fluid. The proof of the main result is based in a power series expansion of the eigenfunctions near the corner, the resolution of a coupled infinite set of finite dimensional linear systems, and a result of Kozlov, Kondratiev and Mazya, concerning the absence of strong zeros for the biharmonic operator (Math. USSR Izvestiya 34 (1990) 337-353). We also show how the same methodology used here can be adapted to exclude domains with corners to have a local version of the Schiffer property for the Laplace operator. Mathematics Subject Classification. 35B60, 35B37.

On a method of control synthesis for a class of discrete systems with constraints

Finite-dimensional linear stationary systems with discrete time and bounded control are considered. The class of 0-controllable systems, for which thei dimensions of the state vector and control vector coincide. For a given family of systems, an approach is proposed, allowing us to form a bounded control as a function of the current state in the problem of achieving the origin for a finite time.

Generalization of all stabilizing compensators for finite-dimensional linear systems

For finite-dimensional linear systems, the Youla–Kucera parameterization (YKP) with a Q parameter over RH ∞ is assumed to satisfy the Diophantine identity. However, the stability is guaranteed if the Diophantine equation is the “ U( RH ∞)“ equality, but not if it is the “identity” equality. However, Vidyasagar's structure with an H parameter over U( RH ∞) is an observer–controller configuration that satisfies the Diophantine equation. This study discusses the deficiency of the Diophantine identity; expands the YKP using an H parameter over U( RH ∞), and expands the Vidyasagar's structure using a Q v parameter over RH ∞ so that both of the expanded parameterizations satisfy the Diophantine equation and are equivalent for all stabilizing compensators. Moreover, an equation that relates to Q, Q v , and H will be introduced to establish relationships among the YKP, Vidyasagar's structure and both expanded parameterizations.

Analysis of the energetic radiation intensity of a linear antenna array radiating nonsinusoidal signals

We present the energetic radiation intensity (ERI) as the quadratic form of the family of integral operators on a finite interval. The kernel of each operator is the autocorrelation function of the signal, which is radiated in the given direction. Spectral representation of the operators gives a fast-converging series representation of the ERI. For the signals, whose Fourier transforms are rational functions of the frequency, spectral analysis of the operators is reduced to finite-dimensional linear systems. Moreover, for such signals we express the ERI as the linear combination of the monochromatic directivity diagrams, evaluated in the complex poles of the signal’s Fourier transform. For the isotropic array elements and the most important amplitude distributions the ERI is obtained explicitly. We consider in detail a signal given by a truncated decaying exponent. Bibliography: 32 titles.

The Carathéodory–Fejér–Pisarenko Decomposition and Its Multivariable Counterpart

When a covariance matrix with a Toeplitz structure is written as the sum of a singular one and a positive scalar multiple of the identity, the singular summand corresponds to the covariance of a purely deterministic component of a time-series whereas the identity corresponds to white noise-this is the Caratheacuteodory-Fejeacuter-Pisarenko (CFP) decomposition. In the present paper we study multivariable analogs for block-Toeplitz matrices as well as for matrices with the structure of state-covariances of finite-dimensional linear systems (which include block-Toeplitz ones). To this end, we develop theory which addresses questions of existence, uniqueness and realization of multivariable power spectra, possibly having deterministic components. We characterize state-covariances which admit only a deterministic input power spectrum, and we explain how to realize multivariable power spectra which are consistent with singular state covariances via decomposing the contribution of the singular part. We then show that multivariable decomposition of a state-covariance in accordance with a "deterministic component + white noise" hypothesis for the input does not exist in general. We finally reinterpret the CFP-dictum and consider replacing the "scalar multiple of the identity" by a covariance of maximal trace which is admissible as a summand. The summand can be either (block-)diagonal corresponding to white noise or have a "short-range correlation structure" corresponding to a moving average component. The trace represents the maximal variance/energy that can be accounted for by a process at the input (e.g., noise) with the aforementioned structure, and this maximal solution can be computed via convex optimization. The decomposition of covariances and spectra according to the range of their time-domain correlations is an alternative to the CFP-dictum with potentially great practical significance

Zero dynamics modeling and boundary feedback design for parabolic systems

In this work the authors introduce a notion of zero dynamics for distributed parameter systems governed by linear parabolic equations on bounded domains with controls implemented through first order linear boundary conditions. The idea of zero dynamics presented here is motivated by classical root-locus constructs from finite dimensional linear systems theory. In particular for a scalar proportional output feedback trajectories of the closed loop system converge to those of the zero dynamics system as the gain tends to infinity. The main contribution of this work is a simple modeling and design mechanism for solving certain tracking and disturbance rejection problems. This mechanism provides an alternative to the geometric approach involving regulator equations. Our approach is based on the fact that for certain boundary control systems it is very easy to model the system’s zero dynamics, which, in turn, provides a simple systematic methodology for solving certain problems of output regulation. As special cases we describe dynamic and static controllers from the associated zero dynamics system for set-point and harmonic tracking. In order to provide a simple roadmap to using this methodology we present numerical examples for a one dimensional heat equation.

Analyzing linear systems containing strict inequalities via evenly convex hulls

The evenly convex hull of a given set is the intersection of all the open halfspaces which contain such set (hence the convex hull is contained in the evenly convex hull). This paper deals with finite dimensional linear systems containing strict inequalities and (possibly) weak inequalities as well as equalities. The number of inequalities and equalities in these systems is arbitrary (possibly infinite). For such kind of systems a consistency theorem is provided and those strict inequalities (weak inequalities, equalities) which are satisfied for every solution of a given system are characterized. Such results are formulated in terms of the evenly convex hull of certain sets which depend on the coefficients of the system.

State-Feedback Stabilization of Well-Posed Linear Systems

A finite-dimensional linear time-invariant system is output-stabilizable if and only if it satisfies the finite cost condition, i.e., if for each initial state there exists at least one L2 input that produces an L2 output. It is exponentially stabilizable if and only if for each initial state there exists at least one L2 input that produces an L2 state trajectory. We extend these results to well-posed linear systems with infinite-dimensional input, state and output spaces. Our main contribution is the fact that the stabilizing state feedback is well posed, i.e., the map from an exogenous input (or disturbance) to the feedback, state and output signals is continuous in Lloc2 in both open-loop and closed-loop settings. The state feedback can be chosen in such a way that it also stabilizes the I/O map and induces a (quasi) right coprime factorization of the original transfer function. The solution of the LQR problem has these properties.

Memoryless output feedback nullification and canonical forms, for time varying systems

We study the possibility of nullifying time-varying systems with memoryless output feedback. The systems we examine are discrete-time linear single-input single-output finite-dimensional time-varying systems. For generic completely controllable and completely observable discrete-time systems, we show that any state at any time can be steered to the origin within finite time. An algorithm for nullification and an upper bound for nullification time, depending only on the system's dimension, are provided. The algorithm is described using a representation of the system in time-varying controller canonical form. We verify that every completely controllable system has such a representation.

Alternative structures and bi-Hamiltonian systems on a Hilbert space

We discuss transformations generated by dynamical quantum systems which are bi-unitary, i.e. unitary with respect to a pair of Hermitian structures on an infinite-dimensional complex Hilbert space. We introduce the notion of Hermitian structures in generic relative position. We provide few necessary and sufficient conditions for two Hermitian structures to be in generic relative position to better illustrate the relevance of this notion. The group of bi-unitary transformations is considered in both the generic and non-generic case. Finally, we generalize the analysis to real Hilbert spaces and extend to infinite dimensions results already available in the framework of finite-dimensional linear bi-Hamiltonian systems.

H∞ control of parameter-dependent state-delayed systems using polynomial parameter-dependent quadratic functions

This paper presents the issue of robust disturbance attenuation and robust asymptotic stability problem for finite-dimensional linear parameter-dependent state-delayed systems. The use of polynomial parameter-dependent quadratic Lyapunov functions and linear matrix inequalities (LMIs) formulations for robust H ∞ control are considered. It is shown that the state feedback control can be determined to guarantee the stability of the closed-loop system independently of the time-delay. We present an illustrative example to demonstrate the applicability of the proposed design approach.

Using delays and time-varying gains to improve the static output feedback stabilizability of linear systems: a comparison

We address the output feedback stabilization problem of linear finite-dimensional SISO systems. Limitations of static time-invariant output feedback on stabilizability are well known. We investigate and compare the possibilities of two recently proposed simple modifications/generalizations of static time-invariant output feedback to remove such limitations. The first approach consists of introducing a time-delay in the control law, which can be treated as an additional control parameter. The second approach consists of making the gain time-varying. We show that both approaches are complementary. Existing theoretical results are brought together in a unifying framework. Numerical procedures for the construction of the controllers are provided. Robustness w.r.t. both parametric and delay uncertainty are dealt with. As an illustration the stabilizability of all second-order systems is completely determined.

Decentralized Power System Stabilizer Design Using Linear Parameter Varying Approach

In this paper, the power system model is formulated as a finite dimensional linear system whose state-space entries depend continuously on a time varying parameter vector called the scheduling variables. This system is referred to as the linear parameter varying (LPV) system. Although the trajectory of the changing parameters such as load levels and tie line flows is not known in advance, in most situations, they can be measured in real time. The LPV technique is applied to the decentralized design of power system stabilizers (PSS) for large systems. In the approach developed, instead of considering the complete system model with all the interconnections, we develop a decentralized approach where each individual machine is considered separately with arbitrarily changing real and reactive power output in a defined range. These variables are chosen as the scheduling variables. The designed controller automatically adjusts its parameters depending on the scheduling variables to coordinate with change of operating conditions and the dynamics of the rest of the system. The resulting decentralized PSSs give good performance in a large operating range. Design procedures are presented and comparisons are made between the LPV decentralized PSSs and conventionally designed PSSs on the 50-generator IEEE test system.