Performance improvement of PI controller with nonlinear error shaping function: IDA-PBC approach

The general dilemma faced in a conventional linear proportional-integral (PI) controller is to achieve the best transient performance (i.e. fast rise time and low overshoot level) at the same time. However, fast response is usually accompanied by high overshoot level. On the other hand, very stable control without overshoot is usually achieved at the expense of a more sluggish response to set point changes and load disturbances. Therefore, compromise between fast response and low overshoot level should be made. In this paper, to overcome these contradictions and limitations, nonlinear error shaping function (ESF) is introduced to amplify gain at low error level but reduce gain at high error level. Firstly, interconnection and damping structure for the closed-loop system composed of PI controller and first-order plant is revealed based on the port-controlled hamiltonian with dissipation (PCHD) formation. Secondly, passivity analysis is performed by the interconnection and damping assignment (IDA) passivity-based control (PBC) algorithm. In simulation studies, several nonlinear error shaping functions are examined and compared to verify performance improvements.     

Modelling and Control of a Hydraulic Actuated Large Scale Manipulator

This contribution deals with the modelling and control of an elastic manipulator. The arm is actuated by a hydraulic ram due to the significant weight. The overall goal is to achieve good tracking of the tip, as well as to reject disturbances, which act on the flexible arm. The controller design is based on two approaches. A flatness‐based feedforward control takes care of the tracking behaviour, and a passivity‐based feedback law stabilizes the trajectories and suppresses the elastic vibrations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust state estimation for Markov jump genetic regulatory networks based on passivity theory

In this article, the robust state estimation problem for Markov jump genetic regulatory networks (GRNs) based on passivity theory is investigated. Moreover, the effect of time‐varying delays is taken into account. The focus is on designing a linear state estimator to estimate the concentrations of the mRNAs and the proteins of the GRNs, such that the dynamics of the state estimation error can be stochastically stable while achieving the prescribed passivity performance. By applying the Lyapunov–Krasovskii functional method, delay‐dependent criteria are established to ensure the existence of the mode‐dependent estimator in the form of linear matrix inequalities. Based on the obtained results, the parameters of the desired estimator gains can be further calculated. Finally, a numerical example is given to illustrate the effectiveness of our proposed methods. © 2015 Wiley Periodicals, Inc. Complexity 21: 214–223, 2016     

Dynamical Modeling and Flatness Based Control of a Belt Drive System

Belt driven systems are part of many industrial applications, like computerized numerical control (CNC) machines in particular cutting machines and 3D-printers. In this paper the dynamical modeling and a flatness based controller design for belt driven systems are proposed. Due to the special kinematics, the stiffness of the belt is nonlinear, leading to nonlinear equations of motion. By neglecting some minor dynamical effects, the resulting system simplifies to a differentially flat one. This allows to calculate nominal feed-forward control torques by using the flat output of the system. To stabilize the error dynamics, an additional PD control law is introduced. The proposed method is compared with a controller, where elastic deflections for the feed forward part are neglected and elastic deformations are compensated by modifying the desired trajectories in a model-based manner. The tracking performance of both methods is evaluated in certain simulations and experiments. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Various Feedforward Schemes and a Passivity Based Backstepping Feedback Control of an Elastic Robot

Lightweight constructions in industry plants lead to elastic deflections causing vibrations and a loss in tracking precision. In order to keep the tracking error for these elastic multibody systems low, the proposed control strategies are combinations of feedforward and feedback schemes. In this work various implemented strategies for computing a feedforward control are proposed and compared, which can be calculated with some simplifications from the mathematical model of the elastic multibody system. Some of these are considering the elastic deflections. The stabilization of the error dynamics is achieved by a simple PD-joint control or passivity based backstepping. In this algorithm the system is split into subsystems and for these subsystems simple control concepts can be applied. The feedback control law of the total system is obtained by means of backstepping theory, considering the internal energy flows in the system. Experimental results are presented to verify the control strategies. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Passivity preserving parametric model-order reduction for non-affine parameters

Parametric model-order reduction (pMOR) has become a well-established technology for analysing large-scale systems with multiple parameters. However, the treatment of non-affine parameters is still posing significant challenges, because projection-based order-reduction methods cannot be applied directly. A common remedy is to establish affine parameter-dependencies approximately, but present extraction methods do not take important system properties, such as passivity, into account. This article proposes a new order-reduction approach that preserves passivity, reciprocity and causality and applies to a wide class of linear time-invariant (LTI) systems. We present the theory of the suggested method and demonstrate its practical usefulness by numerical examples taken from computational electromagnetics.     

Stepsize selection for tolerance proportionality in explicit Runge–Kutta codes

The potential for adaptive explicit Runge–Kutta (ERK) codes to produce global errors that decrease linearly as a function of the error tolerance is studied. It is shown that this desirable property may not hold, in general, if the leading term of the locally computed error estimate passes through zero. However, it is also shown that certain methods are insensitive to a vanishing leading term. Moreover, a new stepchanging policy is introduced that, at negligible extra cost, ensures a robust global error behaviour. The results are supported by theoretical and numerical analysis on widely used formulas and test problems. Overall, the modified stepchanging strategy allows a strong guarantee to be attached to the complete numerical process. This revised version was published online in June 2006 with corrections to the Cover Date.     

Dissipative sampled‐data control of uncertain nonlinear systems with time‐varying delays

This article investigates the delay‐dependent robust dissipative sampled‐data control problem for a class of uncertain nonlinear systems with both differentiable and non‐differentiable time‐varying delays. The main purpose of this article is to design a retarded robust control law such that the resulting closed‐loop system is strictly (Q, S, R)‐dissipative. By introducing a suitable Lyapunov–Krasovskii functional and using free weighting matrix approach, some sufficient conditions for the solvability of the addressed problem are derived in terms of linear matrix inequalities. From the obtained dissipative result, we deduce four cases namely, H_∞ performance, passivity performance, mixed H_∞, and passivity performance and sector bounded performance of the considered system. From the obtained result, it is concluded that based on the passivity performance it is possible to obtain the controller with less control effort, and also the minimum H_∞ performance and the maximum allowable delay for achieving stabilization conditions can be obtained via the mixed H_∞ and passivity control law. Finally, simulation studies based on aircraft control system are performed to verify the effectiveness of the proposed strategy. © 2015 Wiley Periodicals, Inc. Complexity 21: 142–154, 2016     

New delay‐dependent global robust passivity analysis for stochastic neural networks with Markovian jumping parameters and interval time‐varying delays

The problem of passivity analysis for stochastic neural networks with Markovian jumping parameters and interval time‐varying delays is investigated in this article. By constructing a novel Lyapunov–Krasovskii functional based on the complete delay‐decomposing idea and using improved free‐weighting matrix method, some improved delay‐dependent passivity criteria are established in terms of linear matrix inequalities. Numerical examples are also given to show the effectiveness of the proposed methods. © 2015 Wiley Periodicals, Inc. Complexity 21: 167–179, 2016     

Adaptive-order rational Arnoldi-type methods in computational electromagnetism

In this paper, we will consider moment-matching methods for the application of model order reduction to Maxwell’s equations. Since the main difficulty of moment-matching methods results from the adequate determination of a set of expansion points, we will introduce an adaptive expansion point selection on the basis of a suitable approximation of an upper bound of the output moment-matching error. Furthermore, we give some remarks about structure- and passivity preserving adaptive-order rational Arnoldi methods for Maxwell’s equations. Numerical examples indicate the reliability of the proposed algorithm.     

Approximately linear tracking control of nonlinear systems

An elegant approach to control nonlinear state-space systems is the exact input-to-state linearization, where a nonlinear change of coordinates combined with a nonlinear feedback law yields a linear controllable system. In this contribution, we treat the single-input case, where input-to-state linearizability is equivalent to flatness. Sufficient and necessary existence conditions are well-known, but quite restrictive. We propose the design of a tracking controller for flat single-input systems, where the explicit knowledge of the flat output is not required. Our approach is based on a series expansion of the tracking error along a given reference trajectory. The controller gain can even be computed for non-flat systems with regular controllability matrix. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical and numerical solutions of mathematical biology models: The Newell-Whitehead-Segel and Allen-Cahn equations

In this paper, we combine the unified and the explicit exponential finite difference methods to obtain both analytical and numerical solutions for the Newell-Whitehead-Segel–type equations which are very important in mathematical biology. The unified method is utilized to obtain various solitary wave solutions for these equations. Numerical solutions of the specific case studies are investigated by using the explicit exponential finite difference method ensures the accuracy and reliability of the proposed scheme. After obtaining the approximate solutions, convergence analysis and error estimation (the error norms and absolute errors) are presented by comparing these results with the analytical obtained solutions and other methods in the literature through tables and graphs. The obtained analytical and numerical results are in good agreement.     

Fault‐tolerant mixed /passive synchronization for delayed chaotic neural networks with sampled‐data control

This article is concerned with the fault‐tolerant mixed /passive synchronization problem for chaotic neural networks by sampled‐data control scheme. The objective is focused on the design of a reliable controller such that the mixed /passivity performance level of the resulting synchronization error system is ensured in the presence of actuator failures. A time‐dependent Lyapunov functional and an improved reciprocally convex approach combined with a novel integral inequality are applied to optimize the availability of the information on the actual sampling pattern. Then, some sufficient conditions of mixed /passivity performance analysis for the synchronization error systems are derived. A desired reliable sampled‐data controller is designed by solving the optimization problems. Finally, to demonstrate the effectiveness of the proposed method, a practical chaotic neural networks is provided. © 2015 Wiley Periodicals, Inc. Complexity 21: 246–259, 2016     

Combining passivity and classical frequency-domain methods: An insight into decentralised control

A novel approach to tackle passivity-related issues in the frequency domain for linear multiple-input multiple-output (MIMO) cross-coupled systems is given. The aim is to design passivity-based stabilising diagonal controllers within the framework of Individual Channel Analysis and Design (ICAD). Two main results are presented. First, the ICAD is reinterpreted in terms of the passivity-related properties of either the channels or the closed-loop system. The notion of practical passivity is introduced. Second, for linear MIMO systems, a novel frequency-domain passification procedure is proposed. This procedure is used in the design process of the diagonal controllers. Furthermore, an indicator of how far the system is from being passive is defined. This indicator is stated in terms of gain and phase margins, with the consequent statement of robustness. Such a passivity indicator has not been established so far, and for practical applications can be more useful than setting the passivity of the system. Classical frequency-domain control techniques based on Bode and Nyquist plots are used. The results are applied to a 2-input-2-output system modelling an induction motor.     

Constrained trace-optimization of polynomials in freely noncommuting variables

The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry. In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation scheme is governed by flatness, i.e., a rank-preserving property for associated dual SDPs. If flatness is observed, then optimizers can be extracted using the Gelfand–Naimark–Segal construction and the Artin–Wedderburn theory verifying exactness of the relaxation. To enforce flatness we employ a noncommutative version of the randomization technique championed by Nie. The implementation of these procedures in our computer algebra system NCSOStoolsis presented and several examples are given to illustrate our results.     

On monoids over which all generators satisfy a flatness property

This paper addresses conditions under which all generators in the category of right S-acts (where S is a monoid) satisfy a flatness property. There are characterizations for monoids over which all generators satisfy a flatness property α where α can stand for freeness, projectivity, strong flatness, Condition (P), principal weak flatness and torsion freeness. To our knowledge, the problem has not been studied for other flatness properties such as weak flatness, Condition (E) and regularity. The present paper addresses this gap.     

Passivity-preserving model reduction of differential-algebraic equations in circuit simulation

We present an extension of the positive real balanced truncation model reduction method for differential-algebraic equations that arise in circuit simulation. This method is based on balancing the solutions of the projected generalized algebraic Riccati equations. Important properties of this method are that passivity is preserved in the reduced-order model and that there exists an approximation error bound. Numerical solution of the projected Riccati equations using the special structure of circuit equations is also discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation and implementation of transformation based feedback laws for distributed parameter systems

A novel approach to the efficient numerical approximation and implementation of transformation based state feedback originating from backstepping or flatness based approaches is applied to a flatness based feedback law for the so called heavy rope system. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Properties of the Nonlinear Ship Equations of Motion

Nonlinear ship control systems can be designed by exploiting system properties like passivity and dissipativeness in nonlinear system. The nonlinear ship model is written in a vectorial setting with emphasis placed on matrix properties like positiveness, symmetry and skew-symmetry. As a result of energy conservation the ship dynamics can be considered as two interconnected systems. The first system describes the dissipative motion of the rigid-body (ship) while the second system represents the forces due to potential theory generated by the ambient water particles. It is shown that for a stable ship, both subsystems are passive as well as the interconnected system. For an unstable ship, the ambient water system is input feedforward passive with shortage of passivity and therefore the ship must be stabilized by positive feedback. The structural properties of the nonlinear equations of motion are exploited in the Lyapunov analysis when designing ship control systems.