Strictly Positive Real Research Articles

On the Equivalent Relationship Between Generalized Performance, Robust Stability, and Quadratic Stability

This technical note addresses the equivalent relationship between notions of generalized performance, robust stability, and quadratic stability for the feedback connection , where is the transfer matrix of a nominal system and describes the set of uncertainty. By defining the three notions in a more general setting, the conventional equivalent relationship between robust stability and quadratic stability with respect to the norm-bound uncertainty (respectively, the positive real uncertainty) and the corresponding performance (respectively, the extended strict positive realness) has been proved only special case of the results derived in the technical note. A version of the Kalman-Yakubovich-Popov lemma, which plays a crucial role in establishing the equivalence between the generalized performance and the quadratic stability, is also presented.

Decentralized adaptive output feedback fuzzy controller for a class of large-scale nonlinear systems

In the previous work of Huang et al., a coordinated decentralized hybrid adaptive output feedback fuzzy control scheme of large-scale nonlinear systems is obtained predicated upon this prerequisite assumption that the local controllers can share the a priori information about their individual reference models. In this note, we concentrate in the absence of the coordination assumption on developing a classical decentralized combined indirect and direct adaptive fuzzy controller for a class of uncertain large-scale nonlinear systems. The output feedback and adaptation mechanisms proposed for each subsystem hinges just upon its individual output, regardless of any other output reference. Neither the famous strictly positive real (SPR) condition nor a high-gain observer (HGO) is required to realize the overall output feedback algorithm. The tracking errors of the closed-loop large-scale system are shown to converge to tunable neighborhoods of the origin. Simulation results on correlated inverted pendulums verify the validity of the decentralized controller modification.

Strict positive realness of descriptor systems in state space

In this article, we give necessary and sufficient spectral conditions for various notions of strict positive realness for single-input single-output descriptor systems. These conditions only require calculation of eigenvalues of a single matrix. A characterisation of a Kalman–Yacubovich–Popov-like lemma for descriptor systems is also derived, and its implications for the stability of a class of switched descriptor systems are briefly discussed.

On the Characterization of Strict Positive Realness for General Matrix Transfer Functions

We present conditions which are necessary and sufficient for a transfer function (or transfer function matrix) to be strictly positive real. A counter-example is given to illustrate that the conditions presented here differ from those previously presented in the literature. The proof of our results differs from previous related proofs in that it only uses properties of analytic functions and matrices and does not require state-space realizations. Also, the results are not restricted to rational transfer functions with real coefficients.

Síntese de Sistemas Estritamente Reais Positivos através do Critério de Routh-Hurwitz

Dada uma planta Gol(s) linear e invariante no tempo com uma entrada e q saídas, sendo q > 1, é proposto um método, baseado no Critério de Estabilidade de Routh-Hurwitz, para a obtenção de uma matriz constante F ? Rq em série com a saída, de modo que o sistema FGol(s) seja de fase mínima. A partir desta solução, o sistema FGol(s) é representado em espaço de estados por {A,B,FC} e é obtida uma matriz constante de realimentação da saída Ko ? R¹, de modo que o sistema realimentado {A-BKoC,B,FC} seja Estritamente Real Positivo (ERP). O procedimento proposto fornece condições necessárias e suficientes para os dois problemas. Inicialmente, é analisado o caso geral, com q genérico. Em seguida, são estudados os casos particulares q = 2 e q = 3.

Design of Gain-Scheduled Strictly Positive Real Controllers Using Numerical Optimization for Flexible Robotic Systems

The design of gain-scheduled strictly positive real (SPR) controllers using numerical optimization is considered. Our motivation is robust, yet accurate motion control of flexible robotic systems via the passivity theorem. It is proven that a family of very strictly passive compensators scheduled via time- or state-dependent scheduling signals is also very strictly passive. Two optimization problems are posed; we first present a simple method to optimize the linear SPR controllers, which compose the gain-scheduled controller. Second, we formulate the optimization problem associated with the gain-scheduled controller itself. Restricting our investigation to time-dependent scheduling signals, the signals are parameterized, and the optimization objective function seeks to find the form of the scheduling signals, which minimizes a combination of the manipulator tip tracking error and the control effort. A numerical example employing a two-link flexible manipulator is used to demonstrate the effectiveness of the optimal gain-scheduling algorithm. The closed-loop system performance is improved, and it is shown that the optimal scheduling signals are not necessarily linear.

Properties of strictly positive real functions: products and compositions

In this work necessary conditions on one side are sufficient conditions on another side given such that the product of strictly positive real (SPR) functions to be an SPR0 function. Also, sufficient conditions are given for the composition of SPR functions to be an SPR function. As a consequence, a result on robust SPR transfer functions, under non-linearly correlated perturbations in the coefficients, induced by the composition and product with SPR functions is presented. Based on these facts, we give an application on preservation of robust and simultaneous stabilisation of single-input single-output (SISO), linear time-invariant (LTI) systems. The applicability of the results is shown through two examples.

Fault Diagnosis Observer-based Output Feedback Fault Tolerant Control Design

For adaptive fault diagnosis observer design,the error dynamics is needed to satisfy the strictly positive real (SPR) condition,and uncertain systems containing output disturbances cannot been dealt with well.So a novel augmented fault diagnosis observer design is proposed not only to broaden application scopes of adaptive fault diagnosis observer evidently,but also to cope with system disturbances well.Then,based on the obtained fault information,a dynamical output feedback fault tolerant control design is proposed to avoid design difficulties caused by observer-based state feedback fault tolerant control.Meanwhile,the fault diagnosis observer and output feedback fault tolerant controller are designed separately and their performances are considered simultaneously to simplify design procedures.Finally,simulation results are presented to show the effectiveness of the proposed method.

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.

Design of discrete time adaptive PID control systems with parallel feedforward compensator

This paper deals with the design of an adaptive PID control system with a parallel feedforward compensator (PFC) for discrete-time SISO systems and its application to water level control of a 3-tank system. The proposed method utilizes the characteristics of almost strict positive realness (ASPR) of the controlled plant. A conventional design scheme of a PFC which realizes an ASPR augmented controlled plant is also proposed. Further it is shown that the introduction of an internal model improves the control performance of the control system with the PFC. The effectiveness of the proposed method is confirmed through water level control experiments on a three-tank SISO system.

Adaptive fuzzy output feedback control for a class of uncertain nonlinear systems with unknown backlash-like hysteresis

An output feedback controller is proposed for a class of uncertain nonlinear systems preceded by unknown backlash-like hysteresis, where the hysteresis is modeled by a differential equation. The unknown nonlinear functions are approximated by fuzzy systems based on universal approximation theorem, where both the premise and the consequent parts of the fuzzy rules are tuned with adaptive schemes. The proposed approach does not need the availability of the states, which is essential in practice, and uses an observer to estimate the states. An adaptive robust structure is used to cope with lumped uncertainties generated by state estimation error, approximation error of fuzzy systems and external disturbances. Due to its adaptive structure the bound of lumped uncertainties does not need to be known and at the same time the chattering is attenuated effectively. The strictly positive real (SPR) Lyapunov synthesis approach is used to guarantee asymptotic stability of the closed-loop system. In order to show the effectiveness of the proposed method simulation results are illustrated.

Variable-Structure Control Design of Switched Systems With an Application to a DC–DC Power Converter

This paper is concerned with the design of state-feedback variable-structure controllers (VSCs) for a class of continuous-time switched plants. In particular, a new VSC design method based on Lyapunov-Metzler (LM) inequalities and the properties of strictly positive real (SPR) systems is suggested. The method is applied to the control of a DC-DC power converter. The performance of the resulting control system is superior to that afforded by a recently proposed alternative sliding-mode control technique. The definition of LM-SPR systems and its direct application to establish a procedure for designing the switching rule and VSC input are presented.

Preservation of properties in discrete‐time systems under substitutions

AbstractFor discrete‐time systems, using substitutions of the variable z by 𝕄1 functions, we give results on preservation of stability, stabilization, positive real (PR), strictly positive real (SPR), bounded real (BR), and strictly bounded real (SBR) functions, along with the ℋ︁∞‐norm. These results can be interpreted in the sense of robust control or uniform systems. Based on these properties, we present results about preservation of the SPR Lemma and absolute stability for discrete‐time descriptor systems using linear matrix inequalities (LMIs) and substitutions. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

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.

Blind equalization of single-input single-output fir channels for chaotic communication systems

Recently we have developed a simplified recursive adaptive blind channel equalization method for Single-Input Single-Output (SISO) chaotic communication systems. Even though the simplified recursive algorithm gives superior results compared to the state of the art chaotic blind channel equalization algorithms, it has a very important limitation: convergence of the adaptive algorithm is ensured for only Strictly Positive Real (SPR) channels. In this study, we propose a non-recursive chaotic blind channel equalization algorithm that works regardless of whether the channel is SPR or not. First, a statistically optimum fixed filter is designed assuming that the channel is known. Then, it is shown via computer simulations that its performance is very close to that of the statistically optimum fixed filter. Furthermore, it gives better results especially for non-SPR channels compared to the well-known minimum nonlinear prediction error method and the simplified recursive algorithm developed in our previous work. The method is computationally simple and does not impose any restrictions on the channel other than being a finite impulse response filter. Since the instantaneous gradient is used to derive the adaptive algorithm, the proposed method works for slowly and smoothly varying linear channels as well.

Robust Fractional Adaptive Control Based on the Strictly Positive Realness Condition

Robust Fractional Adaptive Control Based on the Strictly Positive Realness ConditionThis paper presents a new approach to robust adaptive control, using fractional order systems as parallel feedforward in the adaptation loop. The problem is that adaptive control systems may diverge when confronted with finite sensor and actuator dynamics, or with parasitic disturbances. One of the classical robust adaptive control solutions to these problems makes use of parallel feedforward and simplified adaptive controllers based on the concept of positive realness. The proposed control scheme is based on the Almost Strictly Positive Realness (ASPR) property of the plant. We show that this condition implies also robust stability in the case of fractional order controllers. An application to Model Reference Adaptive Control (MRAC) with a fractional order adaptation rule is provided with an implementable algorithm. A simulation example of a SISO robust adaptive control system illustrates the advantages of the proposed method in the presence of disturbances and noise.

An Improved Simple Adaptive Control Applied to Power System Stabilizer

A conventional power system stabilizer (PSS) can improve the steady-state stability margin and increase the system positive damping, but it has some drawbacks, such as the time-consuming tuning and nonoptimal damping in the entire operating process. A proportional--integral (PI) adaptive control law in simple adaptive control (SAC) can overcome these drawbacks. However, it can only be applied to the plants with almost strict positive realness (ASPR), and the computation of adaptive control law is really complex. A new improved SAC (ISAC) based on quadratic performance index was proposed. A new PSS can be designed using the ISAC. The simulation and experimental results show that this stabilizer possesses good dynamic characteristics and regulation precision. They also show that the new PSS can improve the dynamic stability of the power system and suppress the low-frequency oscillation effectively.

NN-based output feedback adaptive variable structure control for a class of non-affine nonlinear systems: A nonseparation principle design

Based on non-separation principle design and the application of Implicit Function Theorem, Mean Value Theorem, neural network parametrization and a simple linear observer, a new adaptive variable structure output feedback control is addressed for a class of non-affine nonlinear systems. In the presented scheme, many restrictive conditions, such as Lipschitz assumption, SPR (strictly positive realness) condition and contracting assumption are not required with the help of neural network parametrization. One of the main purposes of this work is to find out the relationships between the design parameters for observer and controller, and how to choose these design parameters in non-separation principle way. Semi-globally uniformly ultimate boundedness for the steady-state and transient performance is provided, and simulations verify the effectiveness of the proposed scheme.

SPR and ASPR Untangled

The theory, proofs, use and implementation of robust adaptive control algorithms require the understanding of the concepts of Strictly Positive Real (SPR) and Almost Strictly Positive Real (ASPR) plants. Although these concepts are defined in the existing literature, both in time and frequency domains, their grasp is not straight forward for the practicing control engineer that deals with real-world plants. Here, we attempt to present the interpretation and meaning of these concepts in a more intuitive way for the practicing control engineer. That is, we use the Bode, Nyquist, Nichols and Root-Locus domains that may help the control engineer better grasp their implications. The paper is oriented towards the practitioner and therefore any proof that already appears in the literature is only cited. An important result that is formalized and proved is that any stable system can be made ASPR. An example is also given in order to demonstrate these concepts.

Stability of Robotic Obstacle Avoidance and Force Interaction* *The work was partly supported by the Swedish Research Council under the grants: 2005-4182, 2006-5243.

Stability problems associated with robot control with obstacle avoidance are analysed. Obstacle avoidance algorithms based on potential function are revised to accomplish stable re-design. A stability proof using Lyapunov theory and passivity theory is provided for the re-designed obstacle avoidance algorithm. The paper presents a modification of potential functions for obstacle avoidance aiming towards stable obstacle avoidance and force interaction. The input is modified to permit block decomposition into a strict positive real (SPR) block and a passive high-gain block. This modification suggested involves a velocity-dependent potential-like repulsive force action from the obstacle. Stability proofs using Lyapunov theory and passivity theory as well as simulations demonstrate asymptotic stability, eliminating oscillatory or divergent behaviors.