Joaquín Carrasco

Reset Control for Passive Bilateral Teleoperation

Communication delays are problematic for teleoperated systems. They give rise to a tradeoff between speed and robustness, which cannot be overcome by means of linear controllers. In order to solve this problem, in this paper, we present a novel approach that combines passivity-based techniques and reset-control principles. In this way, it is possible to obtain simultaneously the robust stability properties of passive control and the performance improvement enabled by reset strategies. Experimental and simulation results are presented, which confirm the good behavior achieved with this method.

A passivity-based approach to reset control systems stability

The stability of reset control systems has been mainly studied for the feedback interconnection of reset compensators with linear time-invariant systems. This work gives a stability analysis of reset compensators in feedback interconnection with passive nonlinear systems. The results are based on the passivity approach to L2-stability for feedback systems with exogenous inputs, and the fact that a reset compensator will be passive if its base compensator is passive. Several examples of full and partial reset compensations are analyzed, and a detailed case study of an in-line pH control system is given.

Reset Times-Dependent Stability of Reset Control Systems

Reset control systems are a special type of hybrid systems in which the time evolution depends both on continuous dynamics between resets and the discrete dynamics corresponding to the reset instants. In this work, stability of reset control systems is approached by using an equivalent (time-varying) discrete time system, introducing necessary and sufficient stability conditions that explicitly depends on the reset times. These conditions have been simplified for the case in which the linear base control system is stable, resulting in a sufficient condition that only depends on a lower bound of the reset intervals.

Second-order counterexamples to the discrete-time Kalman conjecture

The Kalman conjecture is known to be true for third-order continuous-time systems. We show that it is false in general for second-order discrete-time systems by construction of counterexamples with stable periodic solutions. We discuss a class of second-order discrete-time systems for which it is true provided the nonlinearity is odd, but false in general. This has strong implications for the analysis of saturated systems.

Comments on “On the Existence of Stable, Causal Multipliers for Systems With Slope-Restricted Nonlinearities”

The above technical note presents a novel convex search within the Zames-Falb multiplier class. The aim of this correspondence is to correct the misuse of a relaxation on one condition in the search. This relaxation leads to nontrivial numerical errors in all six examples discussed in the technical note. Correct application of the conditions still gives an improvement over absolute stability criteria in the literature for at least one example, but some of the claims for lack of conservativeness in the above technical note should be moderated.

Zames–Falb multipliers for absolute stability: From O׳Shea׳s contribution to convex searches

Abstract Absolute stability attracted much attention in the 1960s. Several stability conditions for loops with slope-restricted nonlinearities were developed. Results such as the Circle Criterion and the Popov Criterion form part of the core curriculum for students of control. Moreover, the equivalence of results obtained by different techniques, specifically Lyapunov and Popov׳s stability theories, led to one of the most important results in control engineering: the KYP Lemma. For Lurye 1 systems this work culminated in the class of multipliers proposed by O׳Shea in 1966 and formalized by Zames and Falb in 1968. The superiority of this class was quickly and widely accepted. Nevertheless the result was ahead of its time as graphical techniques were preferred in the absence of readily available computer optimization. Its first systematic use as a stability criterion came 20 years after the initial proposal of the class. A further 20 years have been required to develop a proper understanding of the different techniques that can be used. In this long gestation some significant knowledge has been overlooked or forgotten. Most significantly, O׳Shea׳s contribution and insight is no longer acknowledged; his papers are barely cited despite his original parameterization of the class. This tutorial paper aims to provide a clear and comprehensive introduction to the topic from a user׳s viewpoint. We review the main results: the stability theory, the properties of the multipliers (including their phase properties, phase-equivalence results and the issues associated with causality), and convex searches. For clarity of exposition we restrict our attention to continuous time multipliers for single-input single-output results. Nevertheless we include several recent significant developments by the authors and others. We illustrate all these topics using an example proposed by O׳Shea himself.

LMI searches for anticausal and noncausal rational Zames?Falb multipliers

Abstract Given a linear time-invariant plant, the search for a suitable multiplier over the class of Zames–Falb multipliers is a challenging problem which has been studied for several decades. Recently, a new linear matrix inequality search has been proposed over rational and causal Zames–Falb multipliers. This letter analyzes the conservatism of the restriction to causality on the multipliers and presents a complementary search for rational and anticausal multipliers. The addition of a Popov multiplier to the anticausal Zames–Falb multiplier is implemented by analogy with the causal search. As a result, a search over a noncausal subset of Zames–Falb multipliers is obtained. A comparison between all the search methods proposed in the literature is given.

Reset Control of an Industrial In-Line pH Process

This work presents a reset/hybrid control application of an in-line pH process. The nonlinear process dynamic is linearized around different operating points, and as a result a second-order plus dead time plant with uncertain gain and delay is obtained for control purposes. A standard PI compensator and reset compensator are designed and tuned. The main aim of this work was to compare the performance of both compensators by practical experimentation. The results show the reset compensator is able to overcome fundamental limitations of linear and time invariant control with a faster tracking response and overall improvement in disturbance rejection.

Reset times-dependent stability of reset control with unstable base systems

Reset control systems are a special type of hybrid systems in which the time evolution depends both on continuous dynamics between resets and the discrete dynamics corresponding to the reset instants. In this work, stability of reset control systems is approached by using an equivalent (time-varying) discrete time system, and necessary and sufficient stability conditions that explicitly depends on the reset times are used. These conditions have been applied for the case in which the linear base control system is unstable.

A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers

We propose a convex search for a subclass of discrete-time Zames-Falb multipliers. Specifically we search for noncausal multipliers with FIR (finite impulse response) structure of arbitrary order. The subclass is shown to be phase-equivalent to the class of discrete-time rational noncausal Zames-Falb multipliers. The search can be expressed as an LMI (linear matrix inequality) whose number of parameters increases quadratically with model order and whose number of linear constraints increases linearly with model order. The search may be used both for the case where the nonlinearity is slope-restricted and for the case where the nonlinearity is odd and slope-restricted. We report favourable results with respect to those in the literature.