Frequency domain conditions for strictly positive real functions

Frequency domain conditions for strictly positive real (SPR) functions which appear in literature are often only necessary or only sufficient. This point is raised in [1], [2], where necessary and sufficient conditions in thes-domain are given for a transfer function to be SPR. In this note, the points raised in [1], I2] are clarified further by giving necessary and sufficient conditions in the frequency domain for transfer functions to be SPR. These frequency-domain conditions are easier to test than those given in thes-domain or time domain [1], [2].     

Simplified adaptive control via improved robust positive real conditions

A new method is derived for embedding plants in a robust state-feedback scheme, to achieve strictly positive realness of the resulting augmented plants. A state-feedback gain is derived that guarantees the strictly positive realness of the closed-loop in presence of polytopic type, possibly time-varying, parameter uncertainties in the model that describes the plant. This is achieved by assigning different Lyapunov functions to each of the vertices of the uncertainty polytope. The obtained feedback gain is used to apply existing methods for robust simplified adaptive control on systems with possibly time-varying polytopic uncertainties.     

A behavioral characterization of the positive real and bounded real characteristic values in balancing

In passivity preserving and bounded realness preserving model reduction by balanced truncation, an important role is played by the so-called positive real (PR) and bounded real (BR) characteristic values. Both for the positive real as well as the bounded real case, these values are defined in terms of the extremal solutions of the algebraic Riccati associated with the system, more precisely as the square roots of the eigenvalues of the product matrix obtained by multiplying the smallest solution with the inverse of the largest solution of the Riccati equation. In this paper we will establish a representation free characterization of these values in terms of the behavior of the system. We will consider positive realness and bounded realness as special cases of half line dissipativity of the behavior. We will then show that both for the PR and the BR case, the characteristic values coincide with the singular values of the linear operator that assigns to each past trajectory in the input-output behavior its unique maximal supply extracting future continuation. We will explain that the term ‘singular values’ should be interpreted here in a generalized sense, since in our setup the future behavior is only an indefinite inner product space.     

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     

Strictly positive real transfer functions revisited

The two most commonly used definitions of strictly positive real (SPR) transfer functions are reviewed. Contrary to what has been suggested in the literature, it is proven that the least restrictive (weak) definition of the two is clearly related to the Yacubovich-Kalman lemma. The relationship between time- and frequency-domain conditions pertaining to the weak definition of SPR systems is established     

Equivalence between positive real and norm-bounded uncertainty

This paper focuses on a linear time-invariant system with positive real uncertainty and shows an equivalence between positive real and norm-bounded uncertainty resulting in transformation of a proper transfer function with positive real uncertainty directly to a strictly proper transfer function with norm-bounded uncertainty which is different from the result obtained by using bilinear transform. This paper also shows a secondary result, relations between the quadratic and robust stability of a system with positive real uncertainty     

Strictly positive real matrices and the Lefschetz-Kalman-Yakubovichlemma

The authors give necessary and sufficient conditions in the frequency domain for rational matrices to be strictly positive real. Based on this result, the matrix form of the Lefschetz-Kalman-Yakubovich lemma is proved, which gives necessary and sufficient conditions for strictly positive real transfer matrices in the state-space realization form     

The positive real lemma and construction of all realizations of generalized positive rational functions

We here extend the well known positive real lemma (also known as the Kalman-Yakubovich-Popov lemma) to a complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. All state space realizations are partitioned into subsets, each is identified with a set of matrices satisfying the same Lyapunov inclusion. Thus, each subset forms a convex invertible cone, and is in fact is replica of all realizations of positive functions of the same dimensions. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, this approach enables us to characterize systems which can be brought, through a static output feedback, to be generalized positive.     

Time-optimal control of certain plants with positive real eigenvalues

This note considers the time-optimal control problem for plants with positive real eigenvalues and a single saturable control input. It is demonstrated that a previous analysis (Fuller 1973, 1974 a) of plants with negative real eigenvalues in simple ratio is also applicable, with certain restrictions, to plants with positive real eigenvalues in the same simple ratio. For a second-order plant a region of controllability in the state plane is algebraically determined. For third and higher-order plants, a region of controllability is partially defined.     

Design of strictly positive real systems using constant outputfeedback

The authors present a linear matrix inequality (LMI) approach to the strictly positive real (SPR) synthesis problem: find an output feedback K such that the closed loop system T(s) is SPR. The authors establish that if no such constant output feedback K exists, then no dynamic output feedback with a proper transfer matrix exists to make the closed-loop system SPR. The existence of K to guarantee the SPR property of the closed-loop system is used to develop an adaptive control scheme that can stabilize any system of arbitrary unknown order and unknown parameters     

On the solvability of the positive real lemma equations

In this paper, we consider the classical equations of the positive real lemma under the sole assumption that the state matrix A has unmixed spectrum: σ(A)∩σ(−A)=. Without any other system-theoretic assumption (observability, reachability, stability, etc.), we derive a necessary and sufficient condition for the solvability of the positive real lemma equations.     

On necessary conditions for real robust Schur-stability

A coefficient diameter of real Schur-stable interval polynomials is bounded using meromorphic functions. The resulting bound is the best possible. A known bound for the ratio of the two leading coefficients is improved for large diameters.     

Low frequency positive real control for delta operator systems

A strictly positive real control problem for delta operator systems in a low frequency range is presented by using the generalized Kalman–Yakubovic?–Popov lemma. The objective of the strictly positive real control problem is to design a controller such that the transfer function is strictly positive real and the resulting closed-loop system is stable. Sufficient conditions for the low frequency strictly positive real controller of the closed-loop delta operator systems are presented in terms of solutions to a set of linear matrix inequalities. A numerical example is given to illustrate the effectiveness and potential for the developed techniques.     

Finite frequency positive real control for singularly perturbed systems

In this paper, strictly positive real control for singularly perturbed systems in (semi)finite frequency ranges is studied. For the general linear systems, necessary and sufficient conditions for the existence of a stabilizing state feedback controller are given based on the generalized KYP lemma, and use the results to study singularly perturbed systems, a composite state feedback controller is constructed, which preserves the stability and positive real property.     

Consensus of positive real systems cascaded with a single integrator

We study output consensus in a network of interconnected non‐identical positive real systems cascaded with a single integrator. Assuming undirected, diffusive interconnections, sufficient conditions are provided to ensure output consensus for two cases: (i) the individual systems are weakly strictly positive real systems cascaded with a single integrator, and (ii) the individual systems are positive real systems cascaded with a single integrator. We illustrate our results with the example of a load frequency control network of synchronous generators. Copyright © 2013 John Wiley & Sons, Ltd.     

On satisfying strict positive real condition for convergence by overparameterization

Strict positive real (SPR) conditions for global convergence of stochastic adaptive control and recursive identification schemes have appeared in a number of analyses using different approaches. It is shown that this strict positive real condition on the unknown operator associated with the plant can be removed in every case by a sufficient overparameterization of the parameter vector.     

Comments on “strictly positive real transfer functionsrevisited”

In the original paper, by R. Lozano-Leal and S. Joshi (lEEE Trans. Automat. Contr., vol. 35, no. 11, p. 1243-5, 1990), the distinction between weak and strong strictly positive real (SPR) functions was addressed, and the feedback interconnection of a weak SPR system and a passive one was shown to be stable. The purpose of this note is to show that the proof of this lemma is actually incorrect     

Solution to the positive real control problem for lineartime-invariant systems

In this paper we study the problem of synthesizing an internally stabilizing linear time-invariant controller for a linear time-invariant plant such that a given closed-loop transfer function is extended strictly positive real. Necessary and sufficient conditions for the existence of a controller are obtained. State-space formulas for the controller design are given in terms of solutions to algebraic Riccati equations (or inequalities). The order of the constructed controller does not exceed that of the plant