Exponential Stability of Nonlinear Time-Varying Differential Equations and Partial Averaging

In this paper we formulate, within the Liapunov framework, a sufficient condition for exponential stability of a differential equation. This condition gives rise to a new averaging result referred to as “partial averaging”: exponential stability of a system , with α sufficiently large, is implied by exponential stability of a time-varying system . Date received: March 28, 2000. Date revised: March 7, 2001.     

On Assigning the Derivative of a Disturbance Attenuation Control Lyapunov Function

We consider feedback design for nonlinear, multi-input affine control systems with disturbances and present results on assigning, by choice of feedback, a desirable upper bound to a given control Lyapunov function (clf) candidate's derivative along closed-loop trajectories. Specific choices for the upper bound are motivated by ℒ₂ and ℒ_∞ disturbance attenuation problems. The main result leads to corollaries on “backstepping” locally Lipschitz disturbance attenuation control laws that are perhaps implicitly defined through a locally Lipschitz equation. The results emphasize that only rough information about the clf is needed to synthesize a suitable controller. A dynamic control strategy for linear systems with bounded controls is discussed in detail. Date received: March 27, 1998. Date revised: March 11, 1999.     

Decay Rates for a Beam with Pointwise Force and Moment Feedback

We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0,π) with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if ξ/π is a rational number with coprime factorization ξ/π=p/q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler–Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator. Date received: October 30, 2000. Date revised: December 20, 2001.     

Input-to-State Stability for Nonlinear Time-Varying Systems via Averaging

 We introduce two definitions of an averaged system for a time-varying ordinary differential equation with exogenous disturbances (“strong average” and “weak average”). The class of systems for which the strong average exists is shown to be strictly smaller than the class of systems for which the weak average exists. It is shown that input-to-state stability (ISS) of the strong average of a system implies uniform semi-global practical ISS of the actual system. This result generalizes the result of [TPA] which states that global asymptotic stability of the averaged system implies uniform semi-global practical stability of the actual system. On the other hand, we illustrate by an example that ISS of the weak average of a system does not necessarily imply uniform semi-global practical ISS of the actual system. However, ISS of the weak average of a system does imply a weaker semi-global practical “ISS-like” property for the actual system when the disturbances w are absolutely continuous and . ISS of the weak average of a system is shown to be useful in a stability analysis of time-varying cascaded systems. Date received: April 6, 1999. Date revised: April 11, 2000.     

System Equivalence for AR-Systems over Rings—with an Application to Delay-Differential Systems

In this paper the notion of autoregressive systems over an integral domain ? is introduced, as a generalization of AR-systems over the rings ℝ[s] and ℝ[s,s ⁻¹]. The interpretation of the dynamics represented by a matrix over ? is fixed by the choice of a module ℳ over ?, consisting of all time-trajectories under consideration. In this setup the problem of system equivalence is studied: when do two different AR-representations characterize the same behavior? This problem is solved using a ring extension of ?, that explicitly depends on the choice of the module ℳ of all time-trajectories. In this way the usual divisibility conditions on the system defining matrices can be recovered. The results apply to the class of delay-differential systems with (in)commensurable delays. In this particular application, the ring extension of ? is characterized explicitly. Date received: May 13, 1998. Date revised: December 22, 1998.     

An Asymptotic Scaling Analysis of LQ Performance for an Approximate Adaptive Control Design

We consider the adaptive tracking problem for a chain of integrators, where the uncertainty is static and functional. The uncertainty is specified by L ²/L ^∞ or weighted L ²/L ^∞ norm bounds. We analyse a standard Lyapunov-based adaptive design which utilises a function approximator to induce a parametric uncertainty, on which the adaptive design is completed. Performance is measured by a modified LQ cost functional, penalising both the tracking error transient and the control effort. With such a cost functional, it is shown that a standard control design has divergent performance when the resolution of a “mono-resolution” approximator is increased. The class of “mono-resolution” approximators includes models popular in applications. A general construction of a class of approximators and their associated controllers which have a uniformly bounded performance independent of the resolution of the approximator is given. Date received: 20 April 1999. Date revised: 29 October 2001.     

Observability and Forward–Backward Observability of Discrete-Time Nonlinear Systems

 In this paper we study the observability properties of nonlinear discrete-time systems. Two types of contributions are given. First we present observability criteria in terms of appropriate codistributions. For particular, but significant, classes of systems we provide criteria that require only a finite number of computations. Then we consider invertible systems (which includes discrete-time models obtained by sampling continuous-time systems) and prove that the weaker notion of forward–backward observability is equivalent to the stronger notion of (forward) observability. Date received: January 19, 2001. Date revised: May 14, 2002.     

Controllable and Uncontrollable Poles and Zeros of nD Systems

 We use the behavioural approach to define and characterize controllable and uncontrollable poles and zeros of multidimensional (nD) linear systems. We show a strong relationship between controllable poles and zeros and properties of the transfer function matrix, and we give characterizations of uncontrollable poles and zeros, in particular demonstrating that these have an input decoupling property. Date received: December 13, 1999. Date revised: October 9, 2000.     

Orthogonal Embeddings of Linear Time-Varying Systems

 Given a contractive linear system T which is represented as a causal linear contractive operator on a Hilbert space H, we study the question: When does there exist an isometric or unitary causal extension of T to H⊕H? Here causality is formulated as a lower triangularity condition and we assume the same causality structure for the extension. This question is often referred to as the generalized Darlington synthesis problem for linear time-varying systems. Date received: November 26, 2001. Date revised: June 19, 2002.     

Exponential Stability of Slowly Time-Varying Nonlinear Systems

Let be a time-varying vector field depending on t containing a regular and a slow time scale (α large). Assume there exist a k (τ)≥1 and a γ(τ) such that ∥x _τ(t, t ₀, x ₀)∥≤k(τ) e ^{−γ(τ)(t−t0)}∥x ₀∥, with x _τ(t, t ₀, x ₀) the solution of the parametrized system with initial state x ₀ at t ₀. We show that for α sufficiently large is exponentially stable when “on average”γ(τ) is positive. The use of this result is illustrated by means of two examples. First, we extend the circle criterion. Second, exponential stability for a pendulum with a nonlinear slowly time-varying friction attaining positive and negative values is discussed. Date received: January 22, 2000. Date revised: April 14, 2001.     

Formal Elimination for Multidimensional Systems and Applications to Control Theory

Following Douglas's ideas on the inverse problem of the calculus of variations, the purpose of this article is to show that one can use formal integrability theory to develop a theory of elimination for systems of partial differential equations and apply it to control theory. In particular, we consider linear systems of partial differential equations with variable coefficients and we show that we can organize the integrability conditions on the coefficients to build an “intrinsic tree”. Trees of integrability conditions naturally appear when we test the structural properties of linear multidimensional control systems with some variable or unknown coefficients (controllability, observability, invertibility, …) or for generic linearization of nonlinear systems. Date received: June 29, 1998. Date revised: January 29, 2000.     

Nonsmooth Robust Stabilization of a Class of Two-Dimensional Nonlinear Systems

In this paper we consider a class of parameterized families of nonlinear systems which cannot be robustly asymptotically stabilized by means of C ¹ feedback. We construct C ⁰ state feedback laws which are smooth away from the origin and which robustly asymptotically stabilize these families of systems. We then show that, in some cases, the regularity of the obtained robust asymptotic stabilizers is “maximum” in the sense that the considered families of systems do not admit any Lipschitz continuous robust asymptotic stabilizer. Date received: June 27, 1997. Date revised: July 28, 1998.     

Orthonormal Basis Functions for Continuous-Time Systems and Lp Convergence

In this paper, model sets for linear-time-invariant continuous-time systems that are spanned by fixed pole orthonormal bases are investigated. These bases generalize the well-known Laguerre and two-parameter Kautz cases. It is shown that the obtained model sets are everywhere dense in the Hardy space H ₁(Π) under the same condition as previously derived by the authors for the denseness in the (Π is the open right half plane) Hardy spaces H _p(Π), 1<p<∞. As a further extension, the paper shows how orthonormal model sets, that are everywhere dense in H _p(Π), 1≤p<∞, and which have a prescribed asymptotic order, may be constructed. Finally, it is established that the Fourier series formed by orthonormal basis functions converge in all spaces H _p(Π) and (D is the open unit disk) H _p(D), 1<p<∞. The results in this paper have application in system identification, model reduction, and control system synthesis. Date received: June 16, 1998. Date revised February 4, 1999.     

Continuous-Time Blind Channel Deconvolution Using Laguerre Shifts

The objective of this paper is to study the problem of continuous-time blind deconvolution of a pulse amplitude modulated signal propagated over an unknown channel and perturbed by additive noise. The main idea is to use so-called Laguerre filters to estimate a continuous-time model of the channel. Laguerre-filter-based models can be viewed as an extension of finite-impulse-response (FIR) models to the continuous-time case, and lead to compact and parsimonious linear-in-the-parameters models.? Given an estimate of the channel, different symbol estimation techniques are possible. Here, the shift property of Laguerre filters is used to derive a minimum mean square error estimator to recover the transmitted symbols. This is done in a way that closely resembles recent FIR-based schemes for the corresponding discrete-time case.? The advantage of this concept is that physical a priori information can be incorporated in the model structure, like the transmitter pulse shape. Date received: July 23, 1998. Date revised: August 26, 1999.     

Algebraic Construction of Normalized Coprime Factors for Delay Systems

We introduce an algebraic approach to the problem of constructing normalized coprime factorizations for retarded delay systems. A parametrization is given of all the possible factorizations that can be obtained by solving algebraic equations over the field generated by s and e ^−s. This enables us to provide a means of determining when such factors can be calculated by solving such algebraic equations, and to show that in general they cannot. Some illustrative examples are given. Date received: August 12, 1999. Date revised: October 12, 2001.     

Hankel Singular Values and Vectors of a Class of Infinite-Dimensional Systems: Exact Hamiltonian Formulas for Control and Approximation Problems

This paper derives exact formulas for singular values and vectors of Hankel operators whose symbol is a product of a single-input single-output inner function and a multi-input multi-output rational function. This class of Hankel operators arises from the sensitivity minimization H ^∞ control problem with a rational weight function and the approximation problem of transfer functions having rational outer parts. It is shown that there is a Hamiltonian transcendental equation characterizing singular values which leads to a matrix function formula for singular vectors. Date received: May 12, 1998. Date revised: May 14, 1999.     

A Trajectory-Based Approach for the Stability Robustness of Nonlinear Systems with Inputs

 We show, for two different definitions of semiglobal practical external stability, that the stability property holds on semi-infinite time intervals if and only if it holds on arbitrarily long but finite time intervals. These results have immediate applications in the analysis of the stability properties of highly oscillatory systems with inputs using averaging or for systems with inputs that are slowly varying. Results are stated for general flows and the stability is given with respect to arbitrary (not necessarily compact) sets. Date received: May 11, 2001. Date revised: March 14, 2002.     

Optimal Average Case Estimation in Hilbert Norms

In contrast to the worst case approach, the average case setting provides less conservative insight into the quality of estimation algorithms. In this paper we consider two local average case error measures of algorithms based on noisy information, in Hilbert norms in the problem element and information spaces. We define the optimal algorithm and provide formulas for its two local errors, which explicitly exhibit the influence of factors such as information, information (measurement) errors, norms in the considered spaces, a subset where approximations are allowed, and “unmodeled dynamics.” Based on the error expression, we formulate in algebraic language the problem of selecting the optimal approximating subspace. The solution is given along with the specific formula for the error, which depends on the eigenvalues of a certain matrix defined by information and norms under consideration. Date received: November 25, 1999. Date revised: May 30, 2000.     

On Local Structural Stability of Differential 1-Forms and Nonlinear Hypersurface Systems on a Manifold with Boundary

In this paper we consider smooth differential 1-forms and smooth nonlinear control-affine systems with (n−1)-inputs evolving on an n-dimensional manifold with boundary. These systems are called hypersurface systems under the additional assumption that the drift vector field and control vector fields span the tangent space to the manifold. We locally classify all structurally stable differential 1-forms on a manifold with boundary. We give complete local classification of structurally stable hypersurface systems on a manifold with boundary under static state feedback defined by diffeomorphisms, which preserve the manifold together with its boundary. Date received: March 30, 2000. Date revised: October 30, 2000.     

Singular Linear Behaviors and Their AR-Representations

A linear behavior is defined essentially as a family of finite but sufficiently long trajectories that is invariant with respect to restriction operators. This may be singular in the sense that it may contain trajectories that cannot be continued to the right. An AR-model is defined essentially as a pair (D,R), where D is a nonsingular rational matrix and R is a full row rank polynomial matrix such that D ⁻¹ R is proper. A canonical one-to-one correspondence is established between linear behaviors and equivalence classes of AR-models. The result is a natural generalization of a well-known result due to Willems. Date received: April 19, 1999. Date revised: April 18, 2000.