Erik Verriest

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$Research paper thumbnail of Stability and Realization of Difference Equations Over $$\mathbb {Z}$$ and $$\mathbb {R}$$$

Stability and Realization of Difference Equations Over $$\mathbb {Z}$$ and $$\mathbb {R}$$

Springer proceedings in mathematics & statistics, 2024

Robust stability of systems with delayed feedback

Circuits Systems and Signal Processing, Jun 1, 1994

Some issues in the stability of differential delay systems in the linear and the nonlinear case a... more Some issues in the stability of differential delay systems in the linear and the nonlinear case are investigated. In particular, sufficient robustness conditions are derived under which a system remains stable, independent of the length of the delay(s). Applications in the design of delayed feedback systems are given. Two approaches are presented, one based on Lyapunov theory, the other on a transformation to Jordan form. In the former, sufficient conditions are obtained in the form of certain Riccati-type equations.

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Transitions for the rimless wheel on flat terrain

2021 60th IEEE Conference on Decision and Control (CDC), Dec 14, 2021

This paper investigates transitions between periodic behaviors for the rimless wheel on flat terr... more This paper investigates transitions between periodic behaviors for the rimless wheel on flat terrain. Control is achieved by an impulsive kick by the spoke touching the ground. We describe several periodic orbits, describable by either rolling or involving a flight phase where the wheel performs a jump. We consider a simple method to transfer between orbit types.

Linear systems over the perspective field as a class of nonlinear systems for which a “Laplace” transform can be defined

A class of fields, the perspective fields, of which the field of the reals is a special case, is ... more A class of fields, the perspective fields, of which the field of the reals is a special case, is introduced. The linear algebra over this field and the theory of linear systems over the field has some remarkable properties: when coordinized by the real field, a class of nonlinear systems appears. The underlying linear structure leads to simple characterizations of stability, reachability, observability and realization. The feasibility of defining a useful extension of the Laplace transform is explored. Finally we show that many more nonlinear systems can possess an inherent linear structure based on formal group laws.

Optimal obstacle avoidance trajectory generation using the root locus principle

This paper details a framework to create a set of trajectories, all guaranteed to avoid known obs... more This paper details a framework to create a set of trajectories, all guaranteed to avoid known obstacles, using the generalized root locus (GRL). The creation of harmonic potential fields followed by generalized root locus analysis leads to a set of parameterized curves, whose gradient dynamics uniquely generate the curves. Next, the curve generation is addressed by considering three reparameterizations of the GRL which propagate at different speeds. After defining the set of admissible paths via the GRL, the paper covers a simpler method to solve the minimum time problem in the new framework with and without the state dependent speed constraints along the path. An example of the minimum time problem with two different constraints is included at the end.

Linear systsems with rational distributed delay: Reduction and stability

A reduction method for the class of distributed delay systems with rational kernel is studied. It... more A reduction method for the class of distributed delay systems with rational kernel is studied. It transforms the delay system to one with fixed crisp delays. Applications in stability analysis and realization with integrators and (fixed) delay lines is shown. The second part illustrates the use of algebraic methods to obtain necessary and sufficient stability conditions for a subclass of systems. These conditions are based on the known NASC for the scalar delay system and involve the location of a finite number of generalized poles in C.

Vibrational control of nonlinear time lag systems with bounded delay: averaging theory, stabilizability, and transient behavior

IEEE Transactions on Automatic Control, May 1, 1994

This paper develops the theory of vibrational control Of "lhJear time Systems with arbitrarily la... more This paper develops the theory of vibrational control Of "lhJear time Systems with arbitrarily large but bounded delay. theory for fast Oscillating, differential delay equations is presented and then applied to vibrational control. conditions B1p given which ensure the e*ne ot parametric vibrations that stabilize nonlinear time lag systems. Transient behavior is also discussed. Illustrative examples are given which Show 1) the f-bfiv Ofthe theory to hP0-t aPPfiatiOm and 2, the differences in the presented and the eKisw known theory for vibrational control of ordinary differential equations. the vibrations depended only on time (and not on the value of the state), there no longer was a need to take measurements of concentration, thus reducing the cost of the reaction even more. For similar reasons, the vibrational control described by [61 has many benefits. A number of practical, important systems, however, are best described by including time delays in their states. In particular, if the exothermic reaction vibrationally controlled in [5] includes a recycle stream, as in [7], the model must

format_quoteExperimental verification shows vibrating flow rates in CSTR improves conversion rates, enhancing reaction efficiency significantly.format_quote

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Maximum Persistence of Periodic Behavior leads to Functional Equations

IFAC-PapersOnLine, 2021

Minimum Time Path with Smooth and Non-smooth Slope-Dependent Speed

2019 American Control Conference (ACC), 2019

We solve a problem in slope-dependent path planning, as an illustration for a more general proble... more We solve a problem in slope-dependent path planning, as an illustration for a more general problem in optimal control for a class of hybrid systems, where the modes are determined by the control and states. It is shown that optimal solutions (namely trajectories that follow the boundaries) may be lost if one naively solves the optimal control problem. The correct procedure involves the simultaneous satisfaction of the necessary conditions for different modes, having complementary constraints. We identify optimal path segments in various cases for smooth and non-smooth speed-velocity characteristic. The hybrid character stems from this non-smoothness, which in this paper is lack of differentiability.

ON A HYPERBOLIC PDE DESCRIBING THE FORWAII U EVOLrl'rI0,V OF A CLASS OF RANDOMLY ALTERNATING SYSTEMS

For a class of randomly switched linear systems a transition functional is introduced. It is show... more For a class of randomly switched linear systems a transition functional is introduced. It is shown that the expectation of this function at time t, satisfies a HYPERBOLIC partial differential equation, which plays a similar role as the backwards Kolmogorov equation for diffusions. Its formal adjoint leads to the forward equation, and their complexity is determined by the Lie algebra associated with the set of values assumable by the dynamic matrix A(t,w). The usual PARABOLIC Kolmogorov equation is derived from this as a limiting case. The result leads to Monte-Carlo simulation methods for solving hyperbolic PDE's.

A Unified Approach to State Space Realization*

2020 American Control Conference (ACC), 2020

A new spin is given on the old problem of realizing a system, abstractly defined by its verificat... more A new spin is given on the old problem of realizing a system, abstractly defined by its verification operation a(ℒ)y = b(ℒ)u, where ℒ is an arbitrary surjective operator from a infinite dimensional vector space $\left( {V,{\mathbb{F}}} \right)$ of functions to itself. In contrast to LTI-system theory governed by differential equations (where ℒ is the differential operator), this basic operator ℒ may itself describe a system and hence possess a nullspace of higher dimension. The formulation leads to new insights and generalizations: e.g., the realization of a network of (identical) systems. Likewise, other operators, such as shifts (for discrete time problems), and/or their frequency representation may be formulated in the same abstract framework. We start with a definition of a process and a functional state, which lead to the usual notion of a state (in a finite dimensional vector space) as a sufficient statistic replacing missing functional information. In addition, the notion of ...

Robust stability and stabilization of deterministic and stochastic time-delay systems

Recently obtained sufficient conditions for stability of deterministic and stochastic delay syste... more Recently obtained sufficient conditions for stability of deterministic and stochastic delay systems based on the Lyapunov-Krasovskii theory are exploited to derive stabilizability conditions under state and under output feedback. The existence proofs are constructive, and hence lead to the prescription of a whole class of stabilizing gains. Sufficient conditions for robust stability of deterministic delay systems of neutral type are derived.

Simultaneous input-state estimation with direct feedthrough based on a unifying MMSE framework with experimental validation

Mechanical Systems and Signal Processing, 2021

This paper addresses the simultaneous input-state estimation problem for discrete-time linear sto... more

format_quoteFIC estimator outperformed AKF and WLS in simulations, effectively eliminating low-frequency drift error with only acceleration data.format_quote

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Range-only Position Estimation with Delay Correction via Lagrange-Bürman Inversion. * *E.I. Verriest gratefully acknowledges the support by NSF CPS-1544857

IFAC-PapersOnLine, 2017

We consider the problem of localization and tracking for mobile units submersed in the ocean. Ass... more We consider the problem of localization and tracking for mobile units submersed in the ocean. Assuming that only range information can be gathered by sonar, we present solutions to the problem taking the sound propagation speed into consideration. Its effect is that range and propagation delay are implicitly related and the design of observers or Kalman filters based on a fixed delay is not correct. Analytic tools, based on the Lagrange-Bürman inversion technique are developed, leading to improved data analysis methods. Direct inversion and an adaptation of the extended Kalman filter are suggested for the one-way and two-way sonar problem.

format_quoteDetails the relationship between signal emission times and delays, using specific mathematical constructs to improve range calculation methodology.format_quote

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The Principle of “Borrowed Feedback” and Application to Control and Observation for Systems with Varying Delay

IFAC-PapersOnLine, 2015

This short paper first explains the general principle of "borrowed feedback" for linear systems w... more This short paper first explains the general principle of "borrowed feedback" for linear systems with constant input delay, and in its dual form of observation with delayed measurements. This is of interest in problems where the external interaction with the system incurs a delay. One application is the control of systems over a network. In this case the delay cannot be considered constant, but depends on a congestion state of the network. It is shown that if the causality constraint, which imposes an upper bound on the rate of change of the delay, does not hold the design method fails to give consistent results.

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Stability of Stochastic Neural Networks with Delays

IFAC Proceedings Volumes, 1998

In this paper the stochastic stability of a network of neurons connected in feedback under the as... more In this paper the stochastic stability of a network of neurons connected in feedback under the asswnption that all neurons exhibit an unknown time delay between stimuli and response is considered. The analysis is based on the Lyapunov•Krasovskii theory for tjme delay systems, and yields sufficient conditions for stochastic stability independent of the delay. These are in the form of the existence of certain positive definite matrices satisfying an algebraic Riccati equation. It is shown that in the presence of random input perturhations an WlStable network may become stable.

format_quoteSufficient conditions for stochastic stability of delay systems are derived using Lyapunov functionals independent of delay time, requiring positive definite matrices fulfilling a Riccati equation.format_quote

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Robust stability and stabilization: From linear to nonlinear

IFAC Proceedings Volumes, 2000

Some ramifications of the Lyapunov-Krasovskii Theory are explored. First, for linear delay system... more Some ramifications of the Lyapunov-Krasovskii Theory are explored. First, for linear delay systems the resulting sufficient conditions for robust asymptotic stability are related to the Schur-Cohn and the Hurwitz stability of the system matrices. Second, ideas towards a constructive parameterization for linear timeinvariant delay systems are presented. Third, new applications for pseudo-linear systems are explored. Finally, some recent advances in the stability and stabilization for nonlinear systems are surveyed, which may be exploited in the arena of delay systems. Copyright©2000 IFAC

format_quoteLyapunov-Krasovskii theory reformulated for linear delay systems, yielding a condition linked to a Riccati equation for robust stability.format_quote

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Linear Systems Over Projective Fields

IFAC Proceedings Volumes, 2001

A class of fields projective fields, of which the field of the reals is a special case, is introd... more A class of fields projective fields, of which the field of the reals is a special case, is introduced. The linear algebra over this field and the theory of linear systems over the field has some remarkable properties: when coordinatized by the real field, a class of nonlinear systems appears. The underlying linear structure leads to simple characterizations of stability, reachability, observability and realization. Both the discrete and the continuous case are analyzed.

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Frequency domain stability criterion for discrete delay systems

IFAC Proceedings Volumes, 2001

This paper extends earlier work on stability of discrete time delay systems, and derives an analo... more

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Delay dependent stability of scalar discrete delay systems

IFAC Proceedings Volumes, 2001

Necessary and sufficient conditions for the delay dependent stability of a linear difference-dela... more Necessary and sufficient conditions for the delay dependent stability of a linear difference-delay equation are derived. In particular, the delay dependent stability of a scalar delay delay system is related to the existence of a positive Markov parameter sequence for an associated fixed second order system.

format_quoteConfirmed that testing the positivity of the response sequence equates to assessing global asymptotic stability for delay systems.format_quote

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