Positive Real Lemma Research Articles

A Faster Passivity Enforcement Method via Chordal Sparsity

Passivity enforcement methods entail the solution of optimization problems characterized by significant computational complexity. Optimal solutions require the use of positive-real lemma constraints which do not rely on linearization but scale rapidly in size owing to auxiliary slack variables. This paper presents how a chordal sparsity pattern can be used to solve this passivity enforcement problem more efficiently. The proposed algorithm uses graph theory and convex optimization to achieve a suboptimal solution that reduces the computation time required for the solution of the positive-real lemma passivity constraints by constraining the auxiliary Lyapunov variable to have a banded pattern. We provide numerical results using actual power system equipment to show that the proposed method gives faster suboptimal solutions.

Model reduction techniques for the computation of extended Markov parameterizations for generalized Langevin equations

The generalized Langevin equation is a model for the motion of coarse-grained particles where dissipative forces are represented by a memory term. The numerical realization of such a model requires the implementation of a stochastic delay-differential equation and the estimation of a corresponding memory kernel. Here we develop a new approach for computing a data-driven Markov model for the motion of the particles, given equidistant samples of their velocity autocorrelation function. Our method bypasses the determination of the underlying memory kernel by representing it via up to about twenty auxiliary variables. The algorithm is based on a sophisticated variant of the Prony method for exponential interpolation and employs the positive real lemma from model reduction theory to extract the associated Markov model. We demonstrate the potential of this approach for the test case of anomalous diffusion, where data are given analytically, and then apply our method to velocity autocorrelation data of molecular dynamics simulations of a colloid in a Lennard-Jones fluid. In both cases, the velocity autocorrelation function and the memory kernel can be reproduced very accurately. Moreover, we show that the algorithm can also handle input data with large statistical noise. We anticipate that it will be a very useful tool in future studies that involve dynamic coarse-graining of complex soft matter systems.

On a Fejér–Riesz factorization of generalized trigonometric polynomials

Function theory on the unit disc proved key to a range of problems in statistics, probability theory, signal processing literature, and applications, and in this, a special place is occupied by trigonometric functions and the Fejer-Riesz theorem that non-negative trigonometric polynomials can be expressed as the modulus of a polynomial of the same degree evaluated on the unit circle. In the present note we consider a natural generalization of non-negative trigonometric polynomials that are matrix-valued with specified non-trivial poles (i.e., other than at the origin or at infinity). We are interested in the corresponding spectral factors and, specifically, we show that the factorization of trigonometric polynomials can be carried out in complete analogy with the Fejer-Riesz theorem. The affinity of the factorization with the Fejer-Riesz theorem and the contrast to classical spectral factorization lies in the fact that the spectral factors have degree smaller than what standard construction in factorization theory would suggest. We provide two juxtaposed proofs of this fundamental theorem, albeit for the case of strict positivity, one that relies on analytic interpolation theory and another that utilizes classical factorization theory based on the Yacubovich-Popov-Kalman (YPK) positive-real lemma.

Positive real lemmas for singular fractional‐order systems: the case

This study investigates the positive real lemmas for singular fractional-order linear time-invariant systems with the fractional order α ∈ ( 0 , 1 ) . Firstly, a novel condition for the stability and extended strictly positive realness of fractional-order systems is derived in terms of linear matrix inequalities. Secondly, a novel condition for the admissibility and extended strictly positive realness of singular fractional-order systems is obtained with two complex variables. Then, another novel condition for the admissibility and extended strictly positive realness of singular fractional-order systems is established with only one complex variable. Thirdly, the positive real controller synthesis problems are analysed and the corresponding positive real controllers are designed based on the positive real lemmas. Finally, four numerical examples are provided to show the effectiveness of the proposed results in this study.

Positive Realness of Second-order and High-order Descriptor Systems

This paper is concerned with the positive realness problem for second-order and high-order descriptor systems. First, without any linearization, necessary and sufficient conditions are established under which the second-order descriptor systems are strictly positive real and extended strictly positive real, respectively. Applying the relations between the positive realness and the optimal control theory, the solutions of the proposed positive real lemma equations can be represented by the symmetric positive semi-definite solutions for the second-order generalized Riccati equations. Then, employing polynomial matrix decomposition techniques, the extended strictly positive real lemma of high-order descriptor systems is also presented based on the original coefficient matrices of the system. Furthermore, linear matrix inequality conditions are given that can effectively test the positive realness and the extended strictly positive realness of the system. Finally, three numerical examples are provided to verify the effectiveness of the developed theoretical results.

On positive realness and negative imaginariness of uncertain discrete-time state-space symmetric systems

ABSTRACTThis paper provides an analysis of uncertain discrete-time negative imaginary and positive real systems with state-space symmetry. First, a necessary and sufficient condition is provided for a minimal state-space symmetric system to be discrete-time positive real or discrete-time negative imaginary. Then, based on the discrete-time symmetric negative imaginary and positive real lemmas, bounds on symmetrically structured perturbations that maintain the negative imaginariness and positive realness of a discrete-time state-space symmetric system are derived. Finally, an RL network example is provided to illustrate the main results of the paper.

On Positive Realness for Stochastic Hybrid Singular Systems

This paper deals with the problem of passivity analysis and passivity-based synthesis for continuous-time stochastic hybrid singular systems (SHSSs). First, a set of linear matrix inequalities for the stochastic admissibility and passivity, which is also known as the positive real lemma, is obtained for continuous-time SHSSs. The proposed condition successfully holds a necessary and sufficient condition for the positive realness of SHSSs, whereas the existing papers in the literature have been handled only sufficient conditions. Next, the passivity-based control synthesis problem is also considered based on the new positive real lemma. The passivity-based stabilization criterion for the closed-loop system with its mode-dependent state-feedback control is expressed in terms of matrix inequality. Thus, by introducing an additional slack matrix to the non-convex conditions, the feasible conditions for the control gain are obtained in terms of linear matrix inequalities. Finally, a numerical example is provided to show the effectiveness of the result.

An assumption-free theorem on discrete-time positive real systems

The discrete-time positive real lemma is examined and the assumptions of controllability and observability are removed. An intermediate theorem is proved that removes only the assumption of controllability. This work is intended to be the counterpart—and is inspired by—recent work on the continuous-time positive real lemma, and does not assume asymptotic stability unlike the recent results of Ferrante and Ntogramatzidis (2017).

A positive real lemma for singular hybrid systems

Abstract This paper introduces a positive real lemma for continuous-time singular hybrid systems. The necessary and sufficient condition of stochastic admissibility and strict passivity for the singular hybrid systems is obtained in terms of linear matrix inequalities. The sufficient condition is derived by using mode-dependent Lyapunov function. In this step, two slack variables are inserted to make the proposed condition be necessary and sufficient condition in terms of strict linear matrix inequalities. Then, to prove the necessary condition, the positive real lemma for the hybrid system is proposed. Since the admissible singular system can be reformulated into stable normal system, the positive real lemma for the hybrid system holds. Thus, we give a necessary condition by constructing the solution of the proposed lemma from that of the hybrid systems.

Positive and Generalized Positive Real Lemma for Slice Hyperholomorphic Functions

In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball.

Synchronization in large-scale nonlinear network systems with uncertain links

In this paper, we study the problem of synchronization over a network, with nonlinear components dynamics modeled in Lure form, and linear stochastic interaction among the components. To study this problem we utilize the stochastic version of Positive Real Lemma (PRL), which is used to provide a sufficient condition for synchronization of stochastic network systems. This sufficiency condition is a function of nominal (mean coupling) Laplacian eigenvalues, and the statistics of link uncertainty in the form of coefficient of dispersion (CoD). Robust control-based small-gain interpretation is provided for the derived sufficiency condition which allows us to define the margin of synchronization. The margin of synchronization is used to understand the important tradeoff between the component dynamics, network topology, and uncertainty characteristics for network synchronization. Our results indicate that significant role played by both the largest and the second smallest eigenvalue of the nominal Laplacian in synchronization of stochastic networks. Furthermore, for a special class of network system connected over torus topology we provide an analytical expression for the tradeoff between the number of neighbors and the dimension of the torus. Similarly, by exploiting the identical nature of component dynamics computationally efficient sufficient condition, independent of network size, is provided for general class of network system. Simulation results for network of coupled oscillators with stochastic link uncertainty are presented to verify the developed theoretical framework.

Robust filtering for uncertain two‐dimensional continuous‐discrete state‐delay systems in finite frequency domains

In this work, the problem of robust H ∞ filtering for uncertain two-dimensional (2D) continuous-discrete Roesser systems with state delays in finite frequency ranges is investigated. This study first develops the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality. In particular, the proposed result covers FDIs in finite frequency intervals for 2D continuous/discrete/continuous-discrete settings. Using the result, the existing finite frequency bounded ream lemmas and the finite frequency positive real lemmas have been generalised to uncertain 2D state-delay Roesser systems. Then, a robust finite frequency H ∞ filter design method for uncertain 2D continuous-discrete state delay Roesser systems is given. Finally, examples are provided to clearly demonstrate the effectiveness of the proposed method.

Discrete-time setpoint-triggered reset integrator design with guaranteed performance and stability

According to Bode's gain-phase relationship, in linear time-invariant controllers, introducing an integral action to eliminate the steady-state error has an adverse effect of increased phase delay and overshoot, leading to performance deterioration. Moreover, increasing the bandwidth of the closed-loop system to enhance the low-frequency disturbance rejection invariably amplifies the sensitivity to high-frequency disturbances. Hence, the performance of the linear controllers is always limited due to these fundamental frequency- and time-domain limitations. Motivated by the desire to address the fundamental limitations of linear controllers and improve the time-varying closed-loop performance, we put forward a novel setpoint-triggered reset integrator strategy that varies the integrator cut-off frequency based on the setpoint information. Particularly, to tackle the time-varying disturbances and setpoint profiles, the proposed controller consists of a nominal linear controller and a variable-gain reset integrator. We show the global asymptotic stability of the proposed methodology using positive-real lemma along with the LaSalle's invariance principle and experimentally validate using measured frequency response function. Moreover, the efficacy of the proposed technique compared to that of the linear controller is experimentally demonstrated on a benchmark rotary servo system. Experimental results assessed using the tracking error and cumulative power spectral density substantiate that the proposed control strategy can not only improve the low-frequency disturbance rejection but also augment the high-frequency trajectory tracking performance.

Solvability conditions for the positive real lemma equations in the discrete time

Passivity theory and the positive real lemma have been recognised as two of the cornerstones of modern systems and control theory. As digital control is pervasive in virtually all control applications, developing a general theory on the discrete-time positive real lemma appears to be an important issue. While for minimal realisations the relations between passivity, positive-realness and existence of solutions of the positive real lemma equations is very well understood, it seems fair to say that this is not the case in the discrete-time case, especially when the realisation is non-minimal and no conditions are assumed on left- and/or right-invertibility of the transfer function. The purpose of this study is to present a necessary and sufficient condition for existence of solutions of the positive real equations under the only assumption that the state matrix A is asymptotically stable.

A theory of passive linear systems with no assumptions

We present two linked theorems on passivity: the passive behavior theorem, parts 1 and 2. Part 1 provides necessary and sufficient conditions for a general linear system, described by a set of high order differential equations, to be passive. Part 2 extends the positive-real lemma to include uncontrollable and unobservable state-space systems.

Frequency-selective Vandermonde decomposition of Toeplitz matrices with applications

The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices, which dates back to the early twentieth century, forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric K-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known a priori to lie in certain frequency bands. Numerical examples are also provided.

A Guaranteed and Efficient Method to Enforce Passivity of Frequency-Dependent Network Equivalents

Rational models of frequency-dependent network equivalents (FDNEs) must be passive to ensure numerical stability in time domain simulations. Therefore, passivity enforcement is an essential step in FDNE rational modeling procedures; however, current approaches are either not guaranteed or not efficient. Here, a positive real lemma based semidefinite programming model is first implemented to guarantee the passivity of the result obtained. Second, this paper incorporates an approach to significantly improve efficiency while guaranteeing passivity by exploiting an important feature of rational models: that constituent complex-pole terms corresponding to dominant resonance peaks can be adjusted to be passive through minor changes, and a partition-based scheme is proposed. Numerical investigations demonstrate that the proposed method is efficient while introducing little additional modeling error.

Lyapunov Stability Analysis of Explicit Direct Integration Algorithms Considering Strictly Positive Real Lemma

AbstractIn structural dynamics, direct integration algorithms are commonly used to solve the temporally discretized differential equations of motion. Integration algorithms are categorized into either implicit or explicit. Implicit algorithms require iterations and may face numerical convergence problems when applied to nonlinear structural systems. On the other hand, explicit algorithms do not need iterations by making use of certain approximations, making them appealing for nonlinear dynamic problems. In this paper, a generic explicit integration algorithm is formulated for a nonlinear system governed by a nonlinear function of the restoring force. Based on this formulation, an approach is proposed to investigate the Lyapunov stability of explicit algorithms by means of the strictly positive real lemma. In this approach, the stability analysis is equivalent to investigating the strictly positive realness of the transfer function for the formulated system. Subsequently, a Nyquist plot is used to determin...

Reprint of “The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions”

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.

The ADI method for bounded real and positive real Lur’e equations

We propose an algorithm for the numerical solution of the Lur'e equations in the bounded real and positive real lemma for stable systems. The algorithm provides approximate solutions in low-rank factored form. We prove that the sequence of approximate solutions is monotonically increasing with respect to definiteness. If the shift parameters are chosen appropriately, the sequence is proven to be convergent to the minimal solution of the Lur'e equations. The algorithm is based on the ideas of the recently developed ADI iteration for algebraic Riccati equations (Massoudi et al., SIAM J Matrix Anal Appl, 2016). In particular, the matrices obtained in our iteration express the optimal cost in a certain projected optimal control problem.