Theory Of Integral Quadratic Constraints Research Articles

Robustness analysis of uncertain time‐varying systems with unknown initial conditions

SummaryThis paper focuses on the robustness analysis of discrete‐time, linear time‐varying (LTV) systems subject to various uncertainties, such as static and dynamic, time‐invariant and time‐varying, linear perturbations, and unknown initial conditions. The proposed approach is based on integral quadratic constraint theory and allows for a potentially more accurate characterization of the set in which the initial state resides by imposing separate constraints on the initial values of the state variables as opposed to simply requiring the initial state to lie in some ellipsoid. The adopted problem formulation facilitates the analysis of uncertain LTV systems subject to disturbance inputs that are bounded pointwise in time, and the developed results enable determining useful pointwise bounds on the performance outputs given such inputs. The main analysis result is given for eventually periodic nominal systems, which include linear time‐invariant, finite horizon, and periodic systems as special cases. The analysis conditions are expressed as linear matrix inequalities. Two additional results stemming from the main analysis theorem are provided that can be used to determine overapproximated ellipsoidal reachable sets. Finally, the utility of the proposed approach is demonstrated in an illustrative example.

Transient Growth of Accelerated Optimization Algorithms

Optimization algorithms are increasingly being used in applications with limited time budgets. In many real-time and embedded scenarios, only a few iterations can be performed and traditional convergence metrics cannot be used to evaluate performance in these nonasymptotic regimes. In this article, we examine the transient behavior of accelerated first-order optimization algorithms. For convex quadratic problems, we employ tools from linear systems theory to show that transient growth arises from the presence of nonnormal dynamics. We identify the existence of modes that yield an algebraic growth in early iterations and quantify the transient excursion from the optimal solution caused by these modes. For strongly convex smooth optimization problems, we utilize the theory of integral quadratic constraints to establish an upper bound on the magnitude of the transient response of Nesterov’s accelerated algorithm. We show that both the Euclidean distance between the optimization variable and the global minimizer and the rise time to the transient peak are proportional to the square root of the condition number of the problem. Finally, for problems with large condition numbers, we demonstrate tightness of the bounds that we derive up to constant factors.

On Integral Quadratic Constraints

This article is concerned with robust stability analysis of feedback interconnections of nonlinear systems using integral quadratic constraints (IQCs). Its main purpose involves reconciling hard (a.k.a. unconditional) IQC and soft (a.k.a. conditional) IQC-based analysis. In particular, it is shown that the hard IQC stability result can be recovered from the soft IQC theory. The hard IQC approach is closely related to the dissipativity theory, whereas the soft IQC approach makes use of homotopies that are continuous in the gap distance measure.

Proximal gradient flow and Douglas–Rachford splitting dynamics: Global exponential stability via integral quadratic constraints

Many large-scale and distributed optimization problems can be brought into a composite form in which the objective function is given by the sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal gradient method and its variants, thereby generalizing gradient descent to a nonsmooth setup. In this paper, we view proximal algorithms as dynamical systems and leverage techniques from control theory to study their global properties. In particular, for problems with strongly convex objective functions, we utilize the theory of integral quadratic constraints to prove the global exponential stability of the equilibrium points of the differential equations that govern the evolution of proximal gradient and Douglas–Rachford splitting flows. In our analysis, we use the fact that these algorithms can be interpreted as variable-metric gradient methods on the suitable envelopes and exploit structural properties of the nonlinear terms that arise from the gradient of the smooth part of the objective function and the proximal operator associated with the nonsmooth regularizer. We also demonstrate that these envelopes can be obtained from the augmented Lagrangian associated with the original nonsmooth problem and establish conditions for global exponential convergence even in the absence of strong convexity.

An Industrial-Based Framework for Distributed Control of Heterogeneous Network Systems

In this paper, a novel control strategy for synchronization of heterogeneous network systems in industrial applications is proposed. Nonidentical nodes are adopted to describe the different industrial processes. The target trajectory is the output of an autonomous linear time-invariant system. The designed controller for each nonidentical node is distributed and calculated on the local information. Each distributed controller composes by the reference generator (RG) and the regulator (RE), where RG can copy the dynamics of the target trajectory and RE can ensure the synchronization. Limited to communication constraints, the time-varying sampled-data control strategy is utilized to reduce the updated frequency of the controller and communication burden of the network. The upper bound of the sampling instants is calculated by the theory of small gain theorem and the theory of integral quadratic constraints. Finally, a practical numerical example is presented to illustrate the effectiveness of the distributed controller design strategy.

Robust and structure exploiting optimisation algorithms: an integral quadratic constraint approach

We consider the problem of analysing and designing gradient-based discrete-time optimisation algorithms for a class of unconstrained optimisation problems having strongly convex objective functions with Lipschitz continuous gradient. By formulating the problem as a robustness analysis problem and making use of a suitable adaptation of the theory of integral quadratic constraints, we establish a framework that allows to analyse convergence rates and robustness properties of existing algorithms and enables the design of novel robust optimisation algorithms with prespecified guarantees capable of exploiting additional structure in the objective function.

Global exponential stability of the Douglas-Rachford splitting dynamics

Many modern optimization problems admit a composite form in which the objective function is given by the sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via proximal methods and their variants, including the Douglas-Rachford (DR) splitting algorithm. In this paper, we view the DR splitting flow as a dynamical system and leverage techniques from control theory to study its global stability properties. In particular, for problems with strongly convex objective functions, we utilize the theory of integral quadratic constraints to prove global exponential stability of the ordinary differential equation that governs the evolution of the DR splitting flow. In our analysis, we use the fact that this algorithm can be interpreted as a variable-metric gradient method on the DR envelope and exploit structural properties of nonlinear terms that arise from composition of reflected proximal operators.

An oracle for the discrete-time integral quadratic constraint problem

In the theory of integral quadratic constraints (IQCs), the condition for robust performance is given in terms of a frequency-domain inequality. This inequality can be transformed into an equivalent finite-dimensional linear matrix inequality by the application of the Kalman–Yakubovich–Popov (KYP) lemma and the introduction of an auxiliary matrix variable. Thus, the IQC problem can be expressed as a semi-definite program (SDP) and solved using convex optimization tools such as general-purpose SDP solvers based on interior point methods. However, by introducing the auxiliary variable, the size of the IQC problem significantly increases, and the aforementioned solvers may fail due to the high computational cost of the SDP. To address this issue, specialized SDP solvers have been developed to efficiently solve the SDP by exploiting its structure. An alternative approach consists of using cutting plane methods to directly solve the frequency-domain inequality without resorting to the KYP lemma. Cutting plane methods require an oracle that confirms if a given candidate solution satisfies the frequency-domain inequality. If the candidate solution is not feasible, the oracle generates a hyperplane that separates the candidate solution from the set of feasible solutions. In this paper, the analytic center cutting plane method is applied to the discrete-time IQC problem. The oracle for the discrete-time IQC problem is based on the spectrum of an extended symplectic matrix pencil and does not require an invertibility assumption that holds in continuous-time but not necessarily in discrete-time. An illustrative example is provided that shows the efficacy of the proposed method and points out directions for future research.

Networked Delay Control for 5G Wireless Machine-Type Communications Using Multiconnectivity

Automatic control using ultrareliable and low latency communication is one of the potential applications of the new fifth-generation wireless systems. A remaining challenge is then to guarantee a low end-to-end delay with low jitter over combined internet and wireless interfaces that are packet switched and capacity optimized. The main novelty of this paper is to introduce stringent delay control to meet this challenge, over simultaneous multiple data paths. The proposed multiple-input-multiple-output (MIMO) cascade control system is nonlinear since the dwell times of the transmission node queues used as actuators cannot be negative. Stability analysis based on integral quadratic constraint theory is, therefore, applied to characterize the global stability of the controller. The practical performance is evaluated with experiments using product like test bed C++ code. It is stressed that the proposed controller does not require internode time synchronization.

Robust Synchronization in Multi-Agent Networks With Unstable Dynamics

The problem of synchronization in heterogenous linear time-invariant networks is investigated. An approach blending the theory of integral quadratic constraints (IQCs) with the gap metric is proposed to study the problem within a unifying framework, where both the agents and communication channels can be dynamic, infinite-dimensional, unstable, and uncertain. Structural properties of the uncertainty are described by IQCs and exploited in the analysis to reduce conservatism. The homotopy employed in IQC analysis is defined with respect to the graph topology induced by the gap metric, whereby open-loop unstable dynamics are accommodated. Sufficient conditions for synchronism are provided, extending recent developments which have been shown to unify several existing synchronization analysis results in the literature.

Zames–Falb Multipliers for Invariance

This letter provides a comprehensive framework for local stability analysis of uncertain feedback interconnections within the integral quadratic constraints theory using general dynamic multipliers. It is shown how so-called hard and soft constraints can be effectively combined in order to (locally) capture the action of the uncertainty. This is illustrated for Zames–Falb multipliers, where it is proven that the subclasses of causal and anticausal multipliers can easily be factorized into hard constraints and thus individually be incorporated into the framework without introducing conservatism.

Full‐block multipliers for repeated, slope‐restricted scalar nonlinearities

SummaryThis paper provides a comprehensive treatment of full‐block multipliers within the integral quadratic constraints framework for stability analysis of feedback systems containing repeated, slope‐restricted scalar nonlinearities. We develop a novel stability result that offers more flexibility in its application because it allows for the inclusion of general Popov and Yakubovich criteria in combination with the well‐established Circle and Zames‐Falb stability tests within integral quadratic constraint theory. A particular focus lies on the formulation of stability criteria in terms of full‐block multipliers, some of which are new, and thus typically involve less conservatism than current methods. Furthermore, a new asymptotically exact parametrization of full‐block Zames‐Falb multipliers is given that allows to exploit the complete potential of this stability test. Copyright © 2017 John Wiley & Sons, Ltd.

An IQC Approach to Robust Stability of Aperiodic Sampled-Data Systems

Conditions for robust stability of sampled-data systems with non-uniform sampling patterns and structural uncertainties are derived. The problem is tackled under the integral quadratic constraint (IQC) framework, where the “aperiodic sampling operation” is modelled by an “delay-integration” operator. Characterization based on integral quadratic constrains (IQC) is identified for this operator and the IQC theory is applied to derive convex stability criteria. Compared to the dominating Lyapunov approach where the candidate Lyapunov-Krasovskii functionals or looped functionals need to be tailored for the systems under consideration and therefore the stability conditions need to be re-derived whenever additional uncertainties are considered, the proposed approach has the advantage of avoiding such endeavor. Numerical examples are given to illustrate this main point and effectiveness of the proposed approach.

A Unifying Framework for Robust Synchronization of Heterogeneous Networks via Integral Quadratic Constraints

A general framework for analysing robust synchronization in large-scale heterogenous networks is proposed based on the theory of integral quadratic constraints (IQCs). Dynamic agents are represented as linear time-invariant single-input-single-output systems. The agents exchange information according to a sparse dynamical interconnection operator in order to achieve synchronization, where their outputs are steered to the same, possibly time-varying, signal. The main technical hindrance to applying IQCs in this context lies with the presence of the marginally stable dynamics which define the trajectory to which the agents' outputs synchronize. It is shown that by working with conditions defined on modified signal spaces of interest and exploiting the graph structure underlying the connections between the dynamic systems, IQC methods can be applied directly to synchronization analysis without recourse to loop transformations, which may obscure the inherent structural properties of the multi-agent networked systems. Decentralized and scalable conditions for synchronization are proposed within this setting. The IQC framework is demonstrated to unify and generalize some of the existing results in the literature, including certain Nyquist-type consensus certificates for time-delay systems. Moreover, it allows the role of feedback in robustness against uncertainty to be better manifested within the context of synchronization.

Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints

This manuscript develops a new framework to analyze and design iterative optimization algorithms built on the notion of Integral Quadratic Constraints (IQC) from robust control theory. IQCs provide sufficient conditions for the stability of complicated interconnected systems, and these conditions can be checked by semidefinite programming. We discuss how to adapt IQC theory to study optimization algorithms, proving new inequalities about convex functions and providing a version of IQC theory adapted for use by optimization researchers. Using these inequalities, we derive numerical upper bounds on convergence rates for the gradient method, the heavy-ball method, Nesterov's accelerated method, and related variants by solving small, simple semidefinite programming problems. We also briefly show how these techniques can be used to search for optimization algorithms with desired performance characteristics, establishing a new methodology for algorithm design.

A Less Conservative LMI Condition for Stability of Discrete-Time Systems With Slope-Restricted Nonlinearities

Many conditions have been found for the absolute stability of discrete-time Lur'e systems in the literature. It is advantageous to find convex searches via LMIs where possible. In this technical note, we construct two less conservative LMI conditions for discrete-time systems with slope-restricted nonlinearities. The first condition is derived via Lyapunov theory while the second is derived via the theory of integral quadratic constraints (IQCs) and noncausal Zames-Falb multipliers. Both conditions are related to the Jury-Lee criterion most appropriate for systems with such nonlinearities, and the second generalizes it. Numerical examples demonstrate a significant reduction in conservatism over competing approaches.

On Stability of Systems With Aperiodic Sampling Devices

This technical note is concerned with stability analysis of aperiodic sampled-data systems. The stability problem is tackled from a pure discrete-time point of view, where the at-sampling behavior of the system is modelled as the response of a nominal discrete-time LTI system in feedback interconnection with a structured uncertainty. Conditions under which the uncertainty is positive real (PR) are identified. Based on the PR property, a number of integral quadratic constraints (IQC) are derived and the IQC theory is applied to derive stability conditions. Numerical examples are given to illustrate the effectiveness of the proposed approach.

Factorization of multipliers in passivity and IQC analysis

Multipliers are often used to find conditions for the absolute stability of Lur’e systems. They can be used either in conjunction with passivity theory or within the more recent framework of integral quadratic constraints (IQCs). We compare the use of multipliers in both approaches. Passivity theory requires that the multipliers have a canonical factorization and it has been suggested in the literature that this represents an advantage of the IQC theory. We consider sufficient conditions on the nonlinearity class for the associated multipliers to have a canonical factorization.

A parameter set division and switching gain-scheduling controllers design method for time-varying plants

This paper presents a new technique to design switching gain-scheduling controllers for plants with measurable time-varying parameters. By dividing the parameter set into a sufficient number of subsets, and by designing a robust controller to each subset, the designed switching gain-scheduling controllers achieve a desired L 2 -gain performance for each subset, while ensuring stability whenever a controller switching occurs due to the crossing of the time-varying parameters between any two adjacent subsets. Based on integral quadratic constraints theory and Lyapunov stability theory, a switching gain-scheduling controllers design problem amounts to solving optimization problems. Each optimization problem is to be solved by a combination of the bisection search and the numerical nonsmooth optimization method. The main advantage of the proposed technique is that the division of the parameter region is determined automatically, without any prespecified parameter set division which is required in most of previously developed switching gain-scheduling controllers design methods. A numerical example illustrates the validity of the proposed technique.

Multipliers for model predictive control with structured input constraints

The stability and robustness of input-constrained model predictive control can be analyzed using the theory of integral quadratic constraints. We demonstrate the existence of improved multipliers when there are only staged input or box input constraints. This can significantly reduce the conservatism of any stability analysis, and we illustrate the improved performance with a simple numerical example.