Strictly Positive Real Research Articles

SOME RESULTS ON CONTROL SYSTEMS WITH MIXED PERTURBATIONS

The paper considers control systems with parametric as well as unstructured uncertainty. Parametric uncertainty is modelled by a transfer function whose numerator and denominator polynomials are independent uncertain polynomials of the form of P(s, q) = l0(q) + l1(q)s + ………. + ln(q)sn where the coefficients depend linearly on q = [q1, q2, …, qq]T and the uncertainty box is Q={q:qi∈[qi¯,qi¯],i=1,2,…,q}. The unstructured uncertainty is modelled as H∞ norm bounded perturbations and perturbations consisting of a family of nonlinear sector bounded feedback gains. Using the geometric structure of the value set of P(s, q), some results are presented for determination of the robust small gain theorem, robust performance, strict positive realness and absolute stability problem of control systems with parametric as well as unstructured uncertainty.

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.

ADAPTIVE SPR SPEED/POSITION CONTROL OF INDUCTION MOTOR

This paper proposes an adaptive speed/position tracking control of an induction motor subject to unknown load torque via strictly positive real (SPR) analysis. The controller is developed under a special nonlinear coordinate transform such that either speed or position control objective can be fulfilled. The underlying design concepts are to endow the close-loop system while under lack of knowledge of some key system parameters, such as the rotor resistance, motor inertia and motor damping coefficient. The proposed control scheme comes along with a thorough proof derived based on Lyapunov stability theory. The experimental results are also given to validate the effectiveness of the presented control scheme.

EXTENDED CIRCLE CRITERION: STABLE FUZZY CONTROL OF A MILLING PROCESS.

In this work, the Extended Circle Criterion and fuzzy approaches are combined to develop a practical technique for checking stability of Fuzzy Control Systems (FCS). The controller is designed on the basis of skilled operator criteria and engineering knowledge about the process. Afterwards, stability is verified by means of the so-called Extended Circle Criterion, under the assumption that the static characteristic of the controller is sector-bounded (nonlinear portion) and the process (linear portion) can be rendered Strictly Positive Real (SPR). The method is tested in a FCS applied to the milling process. Theoretical and experimental results show that the closed loop system is stable.

ROBUST STRICTLY POSITIVE REAL SYNTHESIS BASED ON GENETIC ALGORITHM

In this paper, a new numerical method based on Genetic Algorithm (GA) for robust Strictly Positive Real (SPR) synthesis is presented. The algorithm works well in coefficient space of continuous-time systems and is computationally efficient for some types of polynomial families, such as polynomial segments, interval polynomials and polytopic polynomials et al. The method can be easily extended to the discrete-time systems. Illustrative examples are provided showing that the method is rather effective for arbitrary given high-order systems.

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.

Stability of feedback error learning scheme

This paper aims to establish control theoretical validity of the feedback error learning scheme proposed as an architecture of brain motor control with deep physiological root in computational neuroscience. The feedback error learning method is formulated as a two-degree-of-freedom adaptive control. The stability of the adaptive control law is proved based on the strict positive realness, under the assumption that the plant is stable and stably invertible. Extension to non-invertible cases is also discussed. Some simulation results are given to illustrate the effectiveness of the method.

Fault Diagnosis for a Class of Nonlinear Systems with Unknown Parameters

In this paper, the fault diagnosis problem for a class of nonlinear systems with unknown parameters which depend on both states and inputs is discussed. Under some geometric conditions, the system is transformed into two different subsystems. One is not affected by actuator faults, so the nonlinear adaptive observer and the parameter adaptive law can be designed based on the assumption of the strictly positive realness (SPR). The other whose states can be measured is affected by the faults. Actuator fault identification is based on estimations of both the state and the unknown parameters with good accuracy. Discussions on release of SPR requirement and extension to sensor fault case are also made. Finally, a series DC motor example is included to illustrate the applicability of the proposed methods.

Algebraic solution to the robust SPR problem for two polynomials

In this paper we provide an algebraic method for the design of an appropriate transfer function making simultaneously two polynomials strictly positive real (SPR), in the so-called robust SPR problem for two polynomials. This problem arises in the design of recursive estimation algorithms used for identification purposes and in active noise control (ANC) applications. In the former a given transfer function involving the poles of the unknown transfer function must be made SPR; in the latter, the phase of a transfer function must be matched within a range of (− π /2, π /2) . Robust SPR design methods try to fulfill the phase requirements for all possible transfer functions belonging to a given uncertainty description. Here we focus on uncertainties described by two polynomials.

Anderson's claim on fourth-order SPR synthesis is true

In this paper, we give a constructive proof of Anderson's claim: for the fourth-order stable interval polynomials, there always exists a fixed fourth-order polynomial such that their ratio is strict positive realness (SPR) invariant.

Synthesis of robust strictly positive real systems with l/sub 2/ parametric uncertainty

The problem of designing filters ensuring strict positive realness of a family of uncertain polynomials over an assigned region of the complex plane is frequently investigated issue in the analysis of absolute stability of nonlinear Lur'e systems and the design of adaptive schemes. This paper addresses the problem of designing a continuous-time rational filter when the uncertain polynomial family is assumed to be an ellipsoid in coefficient space. It is shown that the stability of all the polynomials of such a family is a necessary and sufficient condition for the existence of the filter. More importantly, contrary to the results available for the case of a polyhedral uncertainty set in coefficient space, it turns out that the filter is a proper rational function with degree smaller than twice the degree of the uncertain polynomials. Furthermore, a closed form solution to the filter synthesis problem based on polynomial factorization is derived.

Algebraic solution to the robust SPR problem for two polynomials

In this paper we provide an algebraic method for the design of an appropriate transfer function making simultaneously two polynomials strictly positive real (SPR), in the so-called robust SPR problem for two polynomials. This problem arises in the design of recursive estimation algorithms used for identification purposes and in active noise control (ANC) applications. In the former a given transfer function involving the poles of the unknown transfer function must be made SPR; in the latter, the phase of a transfer function must be matched within a range of (− π/2, π/2) . Robust SPR design methods try to fulfill the phase requirements for all possible transfer functions belonging to a given uncertainty description. Here we focus on uncertainties described by two polynomials.

On the strengthened robust SPR problem for discrete-time systems

The well-known necessary and sufficient condition to find a solution to the robust strictly positive real (SPR) problem, namely, that the phase difference of the polynomials involved must be less than π, was found to be also necessary and sufficient in the strengthened robust SPR problem by Anderson and Landau in a recent work, only for the continuous-time case. In this brief communication, we derive a necessary condition in the discrete-time case, which cannot be formulated in phase terms, and which is more stringent than the condition of the robust SPR problem, thereby establishing a fundamental difference between both continuous-time and discrete-time domains.

Corrections and further comments to 'Equivalence relations between learnability, output-dissipativity and strict positive realness'

This correspondence presents corrections and further comments to our previous paper titled 'Equivalence relations between learnability, output-dissipativity and strict positive realness' (Arimoto and Naniwa 2000). It corrects (1) the previous definition of input?output 'invertibility' by assuming an additional continuity property of the system inverse operator, redefines exactly (2) the 'output-dissipativity' by adding a condition of invertibility, and presents (3) a further comment on the proof of Lemma 1. A diagram is given in order to clearly show interrelationships between these concepts.

SPR criteria for uncertain rational matrices via polynomial positivity and Bernstein's expansions

The main purpose of this brief is to convert the strict positive real (SPR) conditions for rational matrices to conditions involving only positivity of polynomials. The polynomial formulation provides efficient SPR criteria for matrices with uncertain interval parameters. To establish the robust SPR property, it is sufficient to test positivity of only three uncertain polynomials regardless of the order of the matrix. The most interesting feature of the proposed polynomial formulation is that the coefficients of uncertain matrices are allowed to have polynomic uncertainty structure. This generality is easily handled by using the Bernstein expansion algorithm. The efficiency of the proposed polynomial approach is illustrated by testing absolute stability of a MIMO Lur'e-Postnikov system having interval parameters.

Adaptive observers for unknown general nonlinear systems

Several neural network (NN) models have been applied successfully for modeling complex nonlinear dynamical systems. However, the stable adaptive state estimation of an unknown general nonlinear system from its input and output measurements is an unresolved problem. This paper addresses the nonlinear adaptive observer design for unknown general nonlinear systems. Only mild assumptions on the system are imposed: output equation is at least C(1) and existence and uniqueness of solution for the state equation. The proposed observer uses linearly parameterized neural networks (LPNNs) whose weights are adaptively adjusted, and Lyapunov theory is used in order to guarantee stability for state estimation and NN weight errors. No strictly positive real (SPR) assumption on the output error equation is required for the construction of the proposed observer.

New results for strict positive realness and feedback stability

Proposes an algorithm which enables the authors to characterize positive realness and three different classes of strict positive realness in terms of Riccati equations and inherent integrations. It has been an open question whether or not the feedback system consisting of a weakly strict positive realness and a nonlinear passivity is asymptotically stable. The paper also shows that the statement is indeed the case.

Positive real analysis and synthesis of uncertain discrete time systems

The positive real analysis and synthesis problems for a class of uncertain discrete-time systems are considered. The uncertainties are expressed in a linear fractional form. The analysis problem is to characterize conditions for a which the uncertain discrete-time system is positive real for all admissible uncertainties. A linear matrix inequality (LMI)-based analysis result is derived. The synthesis problem is to design linear state-feedback and dynamic output-feedback controllers that robustly stabilize the uncertain discrete-time system and achieve the strict positive realness for the closed-loop transfer functions. It has been established that the problems above can be cast into a class of parameterized strict positive real control problems without uncertainties.

Equivalence relations between learnability, output-dissipativity and strict positive realness

This paper was originally motivated by aiming at explicating why a simple iterative learning control scheme for complicated robot dynamics with strong non-linearities works well in acquiring any given desired motion over a finite or infinite time duration or any periodic motion. To gain a physical insight into the problem, a class of linear dynamical systems with specified input and output of the same dimension is treated by defining two properties: output-dissipativity and learnability. It is then shown that the former implies the latter and furthermore, for a class of linear systems with single input and single output, they are equivalent to each other and each of them is also equivalent to strict positive realness of input-output transfer function. For a class of MIMO (multiple inputs and multiple outputs) systems, it is possible to prove that each of these properties is equivalent to strict positive realness of the input-output transfer function matrix if it is strictly proper or otherwise its direct term from input to output satisfies an extra condtion.

Composite Interval Control Systems: Some Strong Kharitonov-Like Properties

The interval model plays an important role in parametric robust control. This paper deals with the strict positive realness of a class of composite interval control systems, and establishes a strong Kharitonov-like extreme point criterion. We also discuss the robust strict positive realness of the shifted interval plant-controller family, and improve some previous results in the literature. Furthermore, for the collection of Popov plots of an interval transfer function family, we prove that a large portion of its outer boundary comes from the sixteen Kharitonov transfer functions. Finally, we study the beta boundedness of interval systems. A strong Kharitonov-like vertex result is established, namely, that all members in the interval system family are beta bounded if and only if the sixteen Kharitonov critical vertices are beta bounded.