Strictly Positive Real Condition Research Articles

Establishment of Strictly Positive Real Condition for Tuning MIMO Feedforward Control

The two-degree-of-freedom structure enables us to achieve stabilization and tracking independently using feedback (FB) and feedforward (FF) controls, respectively. The latter usually undergoes a model uncertainty, which is resolved by online tuning. A tuning law guarantees zero convergence of the output error for any target signals, under a certain strictly positive real (SPR) condition; however, it is noticeably restrictive. This study proposes a method for guaranteeing the SPR condition by designing an FF filter and modifying the error signal for tuning in the multi-input multi-output case. The proposed method is illustrated through a numerical example.

Design and experimental implementation of observer-based adaptive neural network steering control for automated vehicles

In this paper, an observer-based adaptive neural network controller is developed for the Steer-by-Wire (SbW) system of automated vehicles with uncertain nonlinearity and unmeasured state. An observer is introduced to estimate the angular velocity of the front wheels, so the hardware cost and the complexity of mechanical structure and electronic circuits are reduced. Then, an observer-based adaptive neural network controller is proposed for the SbW system to achieve excellent steering precision. A radial basis function (RBF) neural network is used to model the uncertain nonlinearity, which mainly includes self-aligning torque and unknown friction torque with strong nonlinearity. The adaptive law of the RBF neural network designed by fuzzy basis functions rather than by its filtering is derived by Lyapunov stability theory and the Strictly Positive Real (SPR) condition. The tracking error and the observation error can be guaranteed to converge asymptotically to zero. Simulation and experimental results for two paths highlight the effectiveness of the proposed control algorithm.

Observer-based adaptive neural iterative learning control for a class of time-varying nonlinear systems

In this paper an adaptive iterative learning control scheme is presented for the output tracking of a class of nonlinear systems. An observer is designed to estimate the tracking errors. A mixed time domain and s-domain representation is constructed to derive an error model with relative degree one for our purpose. And time-varying radial basis function neural network is employed to deal with system uncertainty. A new signal is constructed by using a first-order filter, which removes the requirement of strict positive real (SPR) condition and identical initial condition of iterative learning control. Based on property of hyperbolic tangent function, the system tracing error is proved to converge to the origin as the iteration tends to infinity by constructing Lyapunov-like composite energy function, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

A robust optimal design for strictly positive realness in recursive parameter adaptation

SummaryThis paper provides an optimization‐based approach to assure the strict positive real (SPR) condition in a set of recursive parameter adaptation algorithms (PAA). The developed algorithms and tools enable a multiobjective formulation of the SPR problem, creating new controls of the stability and parameter convergence in PAAs. In addition to assuring the SPR condition for global stability in PAAs, we provide an algorithmic solution for uniform convergence under performance constraints in PAAs. Several new aspects of parameter convergence were observed with the adoption of the algorithm in a narrow‐band identification problem. The proposed algorithm is verified in simulation and experiments on a precision motion control platform in advanced manufacturing.

Observer-based robust composite adaptive fuzzy control by uncertainty estimation for a class of nonlinear systems

This paper presents an Observer–based Robust Composite Adaptive Fuzzy controller (ORCAFc) for a class of nonlinear systems with uncertainties and external disturbance. Based on the adaptive fuzzy method and modeling error between the system states observer and serial–parallel estimator, an observer-based Composite Adaptive Fuzzy (CAF) scheme is proposed that uses feedback error function as input to approximate and adaptively compensate the unknown uncertainties and external disturbance of the system. Also, an H∞ controller term is employed to compensate the both fuzzy approximation errors and observer errors. The proposed method guarantees the stability of the closed–loop system based on the Strictly Positive Real (SPR) condition and Lyapunov theory.

Output Feedback Adaptive Neural Control Without Seeking SPR Condition

AbstractFor output‐feedback adaptive control of affine nonlinear systems based on feedback linearization and function approximation, the observation error dynamics usually should be augmented by a low‐pass filter to satisfy a strictly positive real (SPR) condition so that output feedback can be realized. Yet, this manipulation results in filtering basis functions of approximators, which makes the order of the controller dynamics very large. This paper presents a novel output‐feedback adaptive neural control (ANC) scheme to avoid seeking the SPR condition. A saturated output‐feedback control law is introduced based on a state‐feedback indirect ANC structure. An adaptive neural network (NN) observer is applied to estimate immeasurable system state variables. The output estimation error rather than the basis functions is filtered and the filter output is employed to update NNs. Under given initial conditions and sufficient control parameter constraints, it is proved that the closed‐loop system is uniformly ultimately bounded stable in the sense that both the state estimation errors and the tracking errors converge to small neighborhoods of zero. An illustrative example is provided to demonstrate the effectiveness of this approach.

Improving adaptive feedforward vibration compensation by using “ Integral+Proportional ” adaptation

Analysis of various adaptive feedforward vibration compensation schemes has shown that a strictly positive real (SPR) condition has to be satisfied in order to guarantee the stability of the whole system (Jacobson, Johnson, Cormick, & Sethares, 2001; Landau, Airimiţoaie, & Alma, 2011a; Landau, Alma, & Airimiţoaie, 2011b; Wang & Ren, 1999). Filters have to be implemented in order to satisfy this condition. The problem becomes even more crucial in the presence of the internal mechanical coupling between the compensator system and the reference source (a correlated measurement with the disturbance) since some information is not available when adaptation starts (see Landau et al., 2011b). It is therefore very important to relax the SPR condition at least in the initial phase, and in the same time to improve the adaptation transients. It is shown, in this paper, that adding a proportional adaptation to the standard integral type parametric adaptation, the SPR condition can be relaxed and the adaptation transients are improved. Theoretical developments are enhanced by real time experimental results obtained on an active vibration control (AVC) system.

A practical projective synchronization approach for uncertain chaotic systems with dead-zone input

In this paper, a practical projective synchronization problem of master–slave chaotic systems is investigated. More specifically, a fuzzy adaptive slave chaotic system subject to dead-zone nonlinearity in the input channel is proposed using only the measurable output of the master system thanks to a suitable observer. A practical projective synchronization between the master and slave systems is achieved by an adequate fuzzy adaptive control system. The underlying parameter adaptation design as well as stability analysis are carried out using a Lyapunov based approach. Unlike the previous works, in the design of the proposed synchronization scheme, we do not require to know the uncertainties function and that the dynamics of the original synchronization error are strictly positive real (SPR). In fact, herein, the uncertainties function is estimated by a fuzzy adaptive system and the dynamics of the original synchronization error are augmented by a low pass filter designed to satisfy the SPR condition. Simulation results are given to show the effectiveness of the proposed practical projective synchronization scheme.

Necessary and sufficient condition for generalized passivity, passification and application to multivariable adaptive systems

AbstractRecently, a generalized passivity concept for linear multivariable systems was obtained which allows circumventing the restrictiveness of the usual passivity concept. The latter is associated with the classical SPR (Strictly Positive Real) condition whereas the new concept of passivity is associated with the so called WSPR condition and its advantage in multivariable systems is that it does not require a restrictive symmetry condition of SPR systems. As a result, it allows the design of multivariable adaptive control that, unlike some existing factorization approaches, does not imply in additional overparameterization of the adaptive controller. In this paper, we complete a previously established WSPR sufficient condition and prove that it is also necessary. We also propose some methods of passification by either premultiplying the system output tracking error vector or the system input vector by an adequate passifying matrix multiplier, so that the resulting input/output transfer function becomes WSPR. The efficiency of our proposals are illustrated by simulation utilizing a well known robotics adaptive visual servoing problem.

Decentralised static output feedback stabilisation and synchronisation of networks

In this paper global stabilisation of a complex network is attained by applying local decentralised output feedback control to a minimum number of nodes of the network. The stabilisation of the network is treated as a rank constrained problem. Strict positive realness conditions on the node level dynamics allow nonlinearities/uncertainties which satisfy the sector conditions to be considered. A network of Chua oscillators with 75 nodes is considered to demonstrate the efficacy of the approach.

Static output feedback stabilisation and synchronisation of complex networks with ℋ︁2 performance

In this paper global stabilisation of a complex network is obtained by applying local decentralised static output feedback control, ensuring guaranteed bounds on quadratic performance. Necessary conditions for stabilisation of a complex network with quadratic performance are derived as a convex LMI representation. In addition, a centralised static output feedback controller is designed and its quadratic performance is compared with that of the decentralised one. Strict positive realness conditions on the node level dynamics allow nonlinearities/uncertainties, which satisfy sector conditions, to be considered. Two examples are used to demonstrate the efficacy of the approach. The first one is a randomly generated academic example with 10 nodes. The second example is based on a network formed from the interconnection of 20 Chua circuits. In both cases controllers are developed with ℋ︁2 bounded performance using only measured output information. Copyright © 2009 John Wiley & Sons, Ltd.

Further result on a dynamic recurrent neural-network-based adaptive observer for a class of nonlinear systems

In Kim et al. [(1997) A dynamic recurrent neural-network-based adaptive observer for a class of nonlinear systems. Automatica 33(8), 1539–1543], authors present an excellent neural network (NN) observer for a class of nonlinear systems. However, the output error equation in their paper is strictly positive real (SPR) which is restrictive assumption for nonlinear systems. In this note, by introducing a vector b 0 and Lyapunov equation, the observer design is obtained without requiring the SPR condition. Thus, our observer can be applied to a wider class of systems.

Multi-periodic repetitive control system: a Lyapunov stability analysis for MIMO systems

A multi-input–multi-output (MIMO) repetitive control problem of tracking and disturbance rejection is considered when both reference and disturbance signals are finite linear combinations of periodic but not necessarily sinusoidal signals. Lyapunov stability analyses under a positive real condition (and a natural relaxation) and exponential stability under a strict positive real condition are provided together with bounds on the induced L 2 and RMS gains of the closed loop system. It is shown that similar Lyapunov stability results apply when the plant is a positive real state-delay system. Extension of the analyses to a class of non-linear systems is discussed and indicates a good degree of robustness in the design.

Absolute stability of nonlinear systems with disc and norm‐bounded perturbations

AbstractAbsolute stability of uncertain nonlinear systems is studied using the celebrated Popov and circle criteria. Uncertainty is assumed to exist in terms of disc and norm‐bounded perturbations in the linear plant. The use of circular arithmetic is proposed as an accurate but computationally more demanding alternative to the already existing approach based on strict positive realness conditions which is easier and faster to implement but gives, in general, conservative results. Numerical examples are given in order to illustrate the salient features of the mathematical developments. Copyright © 2004 John Wiley & Sons, Ltd.

ABSOLUTE STABILITY ANALYSIS IN THE PRESENCE OF DISC UNCERTAINTIES

The famous Popov criterion is used for absolute stability analysis of uncertain nonlinear systems. Uncertainty is assumed to exist in the linear subsystem in terms of coefficient perturbations in complex plane discs. Existing results, which are based on strict positive realness (SPR) conditions, are generalized so as to cover a wider spectrum of systems. All the results are then restated using circular arithmetic which is proposed as an alternative to SPR conditions. The use of circular arithmetic leads to non-conservative results at the expense of more computational effort. This is in contrast with the conservative results yielded by the computationally more efficient approach based on SPR conditions.

Youlaパラメトリゼーションに基づくAnti-Windup制御系の解析と設計

The paper is concerned with the analysis and synthesis of an anti-windup (AW) control system based on the Youla parametrization. The Youla parametrization enables us to derive a general characterization of AW controllers that directly utilizes a coprime factorization of the plant. It is proved by the multi-loop circle criterion that the global L/sub 2/ stability of the closed-loop system is guaranteed if the coprime factor of the plant satisfies a certain strict positive realness condition. On the basis of this stability condition, we develop a method for the AW control synthesis in terms of linear matrix inequalities. We also study the relation between the present AW controller and an IMC-based AW controller. Furthermore, we extend the above results to the AW control for two-degree-of-freedom control systems.

Robust estimation without positive real condition

The strictly positive real (SPR) condition on the noise model is necessary for a discrete-time linear stochastic control system with unmodeled dynamics, even so for a time-invariant ARMAX system, in the past robust analysis of parameter estimation. However, this condition is hardly satisfied for a high-order and/or multidimensional system with correlated noise. The main work in this paper is to show that for robust parameter estimation and adaptive tracking, as well as closed-loop system stabilization, the SPR condition is replaced by a stable matrix polynomial. The main method is to design a "two-step" recursive least squares algorithm with or without a weighted factor and with a fixed lag regressive vector and to define an adaptive control with bounded external excitation and with randomly varying truncation.

双一次変換パラメータをもつ補間コントローラ

A stabilization problem is considered for linear time-invariant systems interpolated with two nominal plants, where the controller is an interpolation of stabilizing controllers for the nominal plants. So far, a linear function is used as the interpolation, and the linearity restricts the class of stabilizable plants. Employing a bilinear function as the interpolation leads to some enlargement of the freedom of the controller design, although the cost of the implementation remains to be almost same as the linear case. Therefore, we consider the stabilization problem, where both the plant and the controller are interpolated bilinearly.In the SISO case, a necessary and sufficient condition for the stabilization of the interpolated plants is derived based on the Nevanlinna-Pick interpolation problem. Furthermore, an LMI description of the condition gives a method to decide a feasible bilinear interpolation. In the MIMO case, a sufficient condition is formed as a strict positive realness condition, and the design method is also proposed with LMI conditions in the state-space form. The effectiveness of the proposed methods is shown with some examples in both cases.

Modified output error identification-elimination of the SPR condition

In this note, we present a modified adaptive output error algorithm for identification. We refer to both the continuous time and the discrete time cases. In the standard algorithm, stability requires that a certain transfer function should be strictly positive real (SPR); however, in the present algorithm, the SPR condition, which is difficult to satisfy, is eliminated. This is achieved by adding a fixed feedback gain. When there exists an a priori partial knowledge about the parameters of the plant, we provide a simple procedure to design that gain. An illustrative example comparing the modified algorithm to the standard one is provided. >

Convergence of the signed output error adaptive identifier

This paper considers an adaptive output error identifier with a signum function in its update kernel. It is shown that the classical strict positive real (SPR) condition required for the convergence of traditional adaptive identifiers does not suffice for the convergence of this signed identifier. Instead, what is needed is a stronger operator condition called the strict dominant passive (SDP) condition. We give an analog of the Kalman-Yakubovic-Popov Lemma for the SDP conditions and, using it, give a convergence proof under the assumptions of persistent excitation and SDP.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>