arXiv subjects

Sei Zhen Khong

Publications and source records attributed to Sei Zhen Khong.

At least 37 records · Page 2Linked to original sources

arXiv2022

A Phase Theory of MIMO LTI Systems

In this paper, we define the phase response for a class of multi-input multi-output (MIMO) linear time-invariant (LTI) systems whose frequency responses are (semi-)sectorial at all frequencies. The newly defined phase concept subsumes the well-known notions of positive real systems and negative imaginary systems. We formulate a small phase theorem for feedback stability, which complements the celebrated small gain theorem. The small phase theorem lays the foundation of a phase theory of MIMO systems. We also discuss time-domain interpretations of phase-bounded systems via both energy signal analysis and power signal analysis. In addition, a sectored real lemma is derived for the computation of MIMO phases, which serves as a natural counterpart of the bounded real lemma.

eess.SY↗

arXiv2021

On the Necessity and Sufficiency of Discrete-Time O'Shea-Zames-Falb Multipliers

This paper considers the robust stability of a discrete-time Lurye system consisting of the feedback interconnection between a linear system and a bounded and monotone nonlinearity. It has been conjectured that the existence of a suitable linear time-invariant (LTI) O'Shea-Zames-Falb multiplier is not only sufficient but also necessary. Roughly speaking, a successful proof of the conjecture would require: (a) a conic parameterization of a set of multipliers that describes exactly the set of nonlinearities, (b) a lossless S-procedure to show that the non-existence of a multiplier implies that the Lurye system is not uniformly robustly stable over the set of nonlinearities, and (c) the existence of a multiplier in the set of multipliers used in (a) implies the existence of an LTI multiplier. We investigate these three steps, showing the current bottlenecks for proving this conjecture. In addition, we provide an extension of the class of multipliers which may be used to disprove the conjecture.

eess.SY↗

arXiv2021

A Frequency-Domain Approach to Nonlinear Negative Imaginary Systems Analysis

In this study, we extend the theory of negative imaginary (NI) systems to a nonlinear framework using a frequency-domain approach. The extended notion is completely characterized via a finite-frequency integration over a "kernel function" on energy-bounded input and output signal pairs. The notion is closely related to and carefully contrasted with the well-studied extension of negative imaginariness -- the theory of counterclockwise dynamics. A condition for feedback stability of the proposed nonlinear NI systems is then developed based on the technique of integral quadratic constraints. Examples and simulations on feedback interconnections of typical nonlinear systems are provided to demonstrate the effectiveness.

eess.SY↗

arXiv2021

On the Necessity and Sufficiency of the Zames-Falb Multipliers for Bounded Operators

This paper analyzes the robust feedback stability of a single-input-single-output stable linear time-invariant (LTI) system against four different classes of nonlinear systems using the Zames-Falb multipliers. The contribution is fourfold. Firstly, we present a generalised S-procedure lossless theorem that involves a countably infinite number of quadratic forms. Secondly, we identify a class of uncertain systems over which the robust feedback stability implies the existence of an appropriate Zames-Falb multiplier based on the generalised S-procedure lossless theorem. Meanwhile, we show that the existence of such a Zames-Falb multiplier is sufficient for the robust feedback stability over a smaller class of uncertain systems. Thirdly, when restricted to be static (a.k.a. memoryless), the second class of systems coincides with the class of sloped-restricted monotone nonlinearities, and the classical result of using the Zames-Falb multipliers to ensure feedback stability is recovered. Lastly, when restricted to be LTI, the second class is demonstrated to be a subset of the third, and the existence of a Zames-Falb multiplier is shown to be sufficient but not necessary for the robust feedback stability.

eess.SY↗

arXiv2021

Converse Theorems for Integral Quadratic Constraints

A collection of converse theorems for integral quadratic constraints (IQCs) is established for linear time-invariant systems. It is demonstrated that when a system interconnected in feedback with an arbitrary system satisfying an IQC is stable, then the given system must necessarily satisfy the complementary IQC. These theorems are specialized to derive multiple versions of converse passivity results. They cover standard notions of strict passivity as well as passivity indices that characterize the trade-offs between passivity surplus and deficit. Converse frequency-weighted small-gain and passivity theorems are also established.

math.OC↗

arXiv2021

Phase of Nonlinear Systems

In this paper, we propose a definition of phase for a class of stable nonlinear systems called semi-sectorial systems, from an input-output perspective. The definition involves the Hilbert transform as a critical instrument to complexify real-valued signals since the notion of phase arises most naturally in the complex domain. The proposed nonlinear system phase, serving as a counterpart of $\mathcal{L}_2$-gain, quantifies the passivity and is highly related to the dissipativity. It also possesses a nice physical interpretation which quantifies the tradeoff between the real energy and reactive energy. A nonlinear small phase theorem is then established for feedback stability analysis of semi-sectorial systems. Additionally, its generalized version is proposed via the use of multipliers. These nonlinear small phase theorems generalize a version of the classical passivity theorem and a recently appeared linear time-invariant small phase theorem.

eess.SY↗

arXiv2020

Low Phase-Rank Approximation

In this paper, we propose and solve a low phase-rank approximation problem, which serves as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. More specifically, a nonzero complex number can be specified by its gain and phase, and while it is generally accepted that the gains of a matrix may be defined by its singular values, there is no widely accepted definition for its phases. In this work, we consider sectorial matrices, whose numerical ranges do not contain the origin, and adopt the canonical angles of such matrices as their phases. Similarly to the rank of a matrix defined to be the number of its nonzero singular values, we define the phase-rank of a sectorial matrix as the number of its nonzero phases. While a low-rank approximation problem is associated with matrix arithmetic means, as a natural parallel we formulate a low phase-rank approximation problem using matrix geometric means to measure the approximation error. A characterization of the solutions to the proposed problem is then obtained, when both the objective matrix and the approximant are restricted to be positive-imaginary. Moreover, the obtained solution has the same flavor as the Schmidt-Mirsky theorem on low-rank approximation problems. In addition, we provide an alternative formulation of the low phase-rank approximation problem using geodesic distances between sectorial matrices. The two formulations give rise to the exact same set of solutions when the involved matrices are additionally assumed to be unitary.

math.RA↗

arXiv2020

On Spectral Properties of Signed Laplacians with Connections to Eventual Positivity

Signed graphs have appeared in a broad variety of applications, ranging from social networks to biological networks, from distributed control and computation to power systems. In this paper, we investigate spectral properties of signed Laplacians for undirected signed graphs. We find conditions on the negative weights under which a signed Laplacian is positive semidefinite via the Kron reduction and multiport network theory. For signed Laplacians that are indefinite, we characterize their inertias with the same framework. Furthermore, we build connections between signed Laplacians, generalized M-matrices, and eventually exponentially positive matrices.

eess.SY↗

arXiv2020

A second order primal-dual method for nonsmooth convex composite optimization

We develop a second order primal-dual method for optimization problems in which the objective function is given by the sum of a strongly convex twice differentiable term and a possibly nondifferentiable convex regularizer. After introducing an auxiliary variable, we utilize the proximal operator of the nonsmooth regularizer to transform the associated augmented Lagrangian into a function that is once, but not twice, continuously differentiable. The saddle point of this function corresponds to the solution of the original optimization problem. We employ a generalization of the Hessian to define second order updates on this function and prove global exponential stability of the corresponding differential inclusion. Furthermore, we develop a globally convergent customized algorithm that utilizes the primal-dual augmented Lagrangian as a merit function. We show that the search direction can be computed efficiently and prove quadratic/superlinear asymptotic convergence. We use the $\ell_1$-regularized model predictive control problem and the problem of designing a distributed controller for a spatially-invariant system to demonstrate the merits and the effectiveness of our method.

math.OC↗

arXiv2020

Stabilization of Cascaded Two-Port Networked Systems with Simultaneous Nonlinear Uncertainties

We introduce a versatile framework to model and study networked control systems (NCSs). An NCS is described as a feedback interconnection of a plant and a controller communicating through a bidirectional channel modelled by cascaded nonlinear two-port networks. This model is sufficiently rich to capture various properties of a real-world communication channel, such as distortion, interference, and nonlinearity. Uncertainties in the plant, controller and communication channels can be handled simultaneously in the framework. We provide a necessary and sufficient condition for the robust finite-gain stability of an NCS when the model uncertainties in the plant and controller are measured by the gap metric and those in the nonlinear communication channels are measured by operator norms of the uncertain elements. This condition is given by an inequality involving "arcsine" of the uncertainty bounds and is derived from novel geometric insights underlying the robustness of a standard closed-loop system in the presence of conelike nonlinear perturbations on the system graphs.

eess.SY↗

arXiv2019

On the Phases of a Complex Matrix

In this paper, we define the phases of a complex sectorial matrix to be its canonical angles, which are uniquely determined from a sectorial decomposition of the matrix. Various properties of matrix phases are studied, including those of compressions, Schur complements, matrix products, and sums. In particular, by exploiting a notion known as the compound numerical range, we establish a majorization relation between the phases of the eigenvalues of $AB$ and the phases of $A$ and $B$. This is then applied to investigate the rank of $I + AB$ with phase information of $A$ and $B$, which plays a crucial role in feedback stability analysis. A pair of problems: banded sectorial matrix completion and decomposition is studied. The phases of the Kronecker and Hadamard products are also discussed.

math.RA↗

arXiv2019

A Structure-Tensor Approach to Integer Matrix Completion with Applications to Differentiated Energy Services

Efficient resource allocation is one of the main driving forces of human civilizations. Of the many existing approaches to resource allocation, matrix completion is one that is frequently applied. In this paper, we investigate a special type of matrix completion problem concerning the class of $(0,1)$-matrices with given row/column sums and certain zeros prespecified. We provide a necessary and sufficient condition under which such a class is nonempty. The condition is stated in the form of the nonnegativity of a structure tensor constructed from the information regarding the given row/column sums and fixed zeros. Moreover, we show that a more general matrix completion problem can be studied in a similar manner, namely that involving the class of nonnegative integer matrices with prescribed row/column sums, predetermined zeros, and different bounds across the rows. To illustrate the utility of our results, we apply them to demand response applications in smart grids. Specifically, we address two adequacy problems in differentiated energy services, namely, the problems of supply/demand matching and minimum purchase profile.

eess.SY↗

arXiv2019

Phase Analysis of MIMO LTI Systems

In this paper, we introduce a definition of phase response for a class of multi-input multi-output (MIMO) linear time-invariant (LTI) systems, the frequency responses of which are cramped at all frequencies. This phase concept generalizes the notions of positive realness and negative imaginariness. We also define the half-cramped systems and provide a time-domain interpretation. As a starting point in an endeavour to develop a comprehensive phase theory for MIMO systems, we establish a small phase theorem for feedback stability, which complements the well-known small gain theorem. In addition, we derive a sectored real lemma for phase-bounded systems as a natural counterpart of the bounded real lemma.

eess.SY↗

arXiv2018

The converse of the passivity and small-gain theorems for input-output maps

We prove the following converse of the passivity theorem. Consider a causal system given by a sum of a linear time-invariant and a passive linear time-varying input-output map. Then, in order to guarantee stability (in the sense of finite L2-gain) of the feedback interconnection of the system with an arbitrary nonlinear output strictly passive system, the given system must itself be output strictly passive. The proof is based on the S-procedure lossless theorem. We discuss the importance of this result for the control of systems interacting with an output strictly passive, but otherwise completely unknown, environment. Similarly, we prove the necessity of the small-gain condition for closed-loop stability of certain time-varying systems, extending the well-known necessity result in linear robust control.

math.OC↗

arXiv2018

The proximal augmented Lagrangian method for nonsmooth composite optimization

We study a class of optimization problems in which the objective function is given by the sum of a differentiable but possibly nonconvex component and a nondifferentiable convex regularization term. We introduce an auxiliary variable to separate the objective function components and utilize the Moreau envelope of the regularization term to derive the proximal augmented Lagrangian $-$ a continuously differentiable function obtained by constraining the augmented Lagrangian to the manifold that corresponds to the explicit minimization over the variable in the nonsmooth term. The continuous differentiability of this function with respect to both primal and dual variables allows us to leverage the method of multipliers (MM) to compute optimal primal-dual pairs by solving a sequence of differentiable problems. The MM algorithm is applicable to a broader class of problems than proximal gradient methods and it has stronger convergence guarantees and a more refined step-size update rules than the alternating direction method of multipliers. These features make it an attractive option for solving structured optimal control problems. We also develop an algorithm based on the primal-descent dual-ascent gradient method and prove global (exponential) asymptotic stability when the differentiable component of the objective function is (strongly) convex and the regularization term is convex. Finally, we identify classes of problems for which the primal-dual gradient flow dynamics are convenient for distributed implementation and compare/contrast our framework to the existing approaches.

math.OC↗

arXiv2017

Stabilization of Cascaded Two-Port Networked Systems Against Nonlinear Perturbations

A networked control system (NCS) consisting of cascaded two-port communication channels between the plant and controller is modeled and analyzed. Towards this end, the robust stability of a standard closed-loop system in the presence of conelike perturbations on the system graphs is investigated. The underlying geometric insights are then exploited to analyze the two-port NCS. It is shown that the robust stability of the two-port NCS can be guaranteed when the nonlinear uncertainties in the transmission matrices are sufficiently small in norm. The stability condition, given in the form of "arcsin" of the uncertainty bounds, is both necessary and sufficient.

eess.SY↗

arXiv2017

Supervisory observer for parameter and state estimation of nonlinear systems using the DIRECT algorithm

A supervisory observer is a multiple-model architecture, which estimates the parameters and the states of nonlinear systems. It consists of a bank of state observers, where each observer is designed for some nominal parameter values sampled in a known parameter set. A selection criterion is used to select a single observer at each time instant, which provides its state estimate and parameter value. The sampling of the parameter set plays a crucial role in this approach. Existing works require a sufficiently large number of parameter samples, but no explicit lower bound on this number is provided. The aim of this work is to overcome this limitation by sampling the parameter set automatically using an iterative global optimisation method, called DIviding RECTangles (DIRECT). Using this sampling policy, we start with 1 + 2np parameter samples where np is the dimension of the parameter set. Then, the algorithm iteratively adds samples to improve its estimation accuracy. Convergence guarantees are provided under the same assumptions as in previous works, which include a persistency of excitation condition. The efficacy of the supervisory observer with the DIRECT sampling policy is illustrated on a model of neural populations.

math.OC↗

arXiv2017

Robust stability conditions for feedback interconnections of distributed-parameter negative imaginary systems

Sufficient and necessary conditions for the stability of positive feedback interconnections of negative imaginary systems are derived via an integral quadratic constraint (IQC) approach. The IQC framework accommodates distributed-parameter systems with irrational transfer function representations, while generalising existing results in the literature and allowing exploitation of flexibility at zero and infinite frequencies to reduce conservatism in the analysis. The main results manifest the important property that the negative imaginariness of systems gives rise to a certain form of IQCs on positive frequencies that are bounded away from zero and infinity. Two additional sets of IQCs on the DC and instantaneous gains of the systems are shown to be sufficient and necessary for closed-loop stability along a homotopy of systems.

eess.SY↗