Integral Quadratic Constraint Framework Research Articles

Stabilizing model predictive control synthesis using integral quadratic constraints and full‐block multipliers

SummaryIn this paper, we discuss how to synthesize stabilizing model predictive control (MPC) algorithms based on convexly parameterized integral quadratic constraints (IQCs), with the aid of general multipliers. Specifically, we consider Lur'e systems subject to sector‐bounded and slope‐restricted nonlinearities. As the main novelty, we introduce point‐wise IQCs with storage in order to accordingly generate the MPC terminal ingredients, thus enabling closed‐loop stability, strict dissipativity with regard to the nonlinear feedback, and recursive feasibility of the optimization. Specifically, we consider formulations involving both static and dynamic multipliers, and provide corresponding algorithms for the synthesis procedures. The major benefit of the proposed approach resides in the flexibility of the IQC framework, which is capable to deal with many classes of uncertainties and nonlinearities. Moreover, for the considered class of nonlinearities, our method yields larger regions of attraction of the synthesized predictive controllers (with reduced conservatism) if compared to the standard approach to deal with sector constraints from the literature.

IQC based analysis and estimator design for discrete-time systems affected by impulsive uncertainties

We propose novel quadratic performance tests for linear discrete-time impulsive systems based on viewing these systems as feedback interconnections of some non-impulsive linear system with an impulsive operator. In order to systematically analyze such interconnections, we employ the framework of integral quadratic constraints and propose novel constraints of this kind for capturing the behavior of the involved impulsive operator. As a major benefit, the modularity of this framework permits seamless extensions to interconnections affected by heterogeneous uncertainties in a straightforward fashion. This contrasts with alternative approaches which are based on capturing the system’s impulsive behavior by means of a clock. Building upon the developed analysis criteria, we characterize the existence of non-impulsive estimators for such impulsive interconnections in a lossless fashion and in terms of linear matrix inequalities. Finally, our approach is illustrated by means of several numerical examples.

A fixed step distributed proximal gradient push‐pull algorithm based on integral quadratic constraint

AbstractIn order to solve the distributed optimization problem with smooth + nonsmooth structure of the objective function on unbalanced directed networks, this article uses the proximal operator to deal with the nonsmooth part of the objective function, and designs and analyzes the fixed step proximal gradient Push‐Pull (PG‐Push‐Pull) algorithm. Firstly, the Integral Quadratic Constraint (IQC) suitable for proximal gradient Push‐Pull algorithm is given. When the smooth part of the objective function is strongly convex and the gradient satisfies the Lipchitz condition, the convergence of the algorithm is proved, and the convergence analysis is transformed into solving a linear matrix inequality by using this IQC framework. Its feasibility can ensure that the proposed algorithm has linear convergence rate, which is the same as that of Push‐Pull gradient algorithm. Then, the upper bound of convergence rate can be found by solving a Non‐Linear Programming problem. Finally, an example is given to analyze the upper bound of the convergence rate and verify the effectiveness of the proposed algorithm.

Robustness Analysis of Nonlinear Systems Along Uncertain Trajectories

The paper presents a novel approach for robustness analysis of nonlinear dynamic systems in the vicinity of a reference trajectory. The approach linearizes the system with respect to a nominal trajectory and calculates a guaranteed upper bound on the worst-case gain. In contrast to existing methods rooted in linear time-varying systems analysis, the approach accurately includes perturbations that drive the system away from the reference trajectory. The approach further includes a bound for the error associated with the time-varying linearization. Hence, the results obtained in the linear framework provide a valid upper bound for the worst-case performance of the nonlinear system. The calculation of the upper bound relies on the dissipation inequalities formulated in the framework of integral quadratic constraints. It is therefore computationally much cheaper than sample-based methods such as Monte Carlo simulation. The feasibility of the approach is demonstrated on a numerical example.

Structure Preserving Uncertainty Modelling and Robustness Analysis for Spatially Distributed Dissipative Dynamical Systems

The paper deals with uncertainty modelling, robust stability and performance analysis of multi-input multi-output (MIMO) reduced order spatially distributed dissipative dynamical systems. While researching the topic of modern robust control of such systems, two key findings were discovered: (i) systematic modelling of the uncertainty and model order reduction (MOR) at the level of a subsystem gives both modelling freedom and the ability for obtaining less conservative uncertainties on the level of a subsystem; (ii) for a special class of interconnected dissipative dynamical systems, uncertainty conservatism at the subsystem level can be reduced—a novel, structure preserving algorithm employing subsystem partitioning and subsystem MOR by means of balanced truncation method (BTM) is used to obtain low-order robustly stable interconnected systems. Such systems are suitable for practical decentralized and distributed robust controller synthesis. Built upon a powerful framework of integral quadratic constraints (IQCs), this approach gives uncertainty modelling flexibility to perform robustness analysis of real world interconnected systems that are usually affected by multiple types of uncertainties at once. The proposed uncertainty modelling procedure and its practical application are presented on the numerical example. A spatially discretized vibration dynamical system comprised of a series of simply supported Euler beams mutually interconnected by springs and dampers is examined. Spatial discretization of the mathematical model is carried out using the finite element method (FEM).

Dissipativity and Integral Quadratic Constraints: Tailored Computational Robustness Tests for Complex Interconnections

A central notion in systems theory is dissipativity, which was introduced by Jan Willems with the explicit goal of arriving at a fundamental understanding of the stability properties of feedback interconnections. In robust control, the framework of integral quadratic constraints (IQCs) builds on the seminal contributions of Yakubovich and Zames in the 1960s. It provides a technique for analyzing the stability of an interconnection of some linear system in feedback with a whole class of systems, also referred to as uncertainty. This article surveys the key ideas of exploiting dissipativity and IQCs to systematically construct computational tests for robust stability and performance of uncertain interconnections in terms of linear matrix inequalities. The article focuses on the recently introduced notion of finite-horizon IQCs with a terminal cost, which is shown to provide a seamless link between dissipativity theory and absolute stability theory based on dynamic IQCs.

Robustness analysis of uncertain time‐varying systems using integral quadratic constraints with time‐varying multipliers

SummaryThis article presents a dissipativity approach for robustness analysis using the framework of integral quadratic constraints (IQCs). The derived results apply for linear time‐varying nominal systems with uncertain initial conditions. IQC multipliers are used to describe the sets of allowable uncertainty operators, and signal IQC multipliers are used to describe the sets of allowable disturbance signals. The novel concepts of dichotomic nodes and their corresponding factorizations are introduced, which allow for the aforementioned multipliers to be general time‐varying operators. The results are illustrated via the robustness analysis of a flight controller for an unmanned aircraft system tasked to perform a Split‐S maneuver.

Stability analysis and output-feedback synthesis of hybrid systems affected by piecewise constant parameters via dynamic resetting scalings

We develop novel criteria for robust stability of a class of hybrid systems that are affected by piecewise constant uncertain parameters with dwell-time constraints. These criteria result from clock-dependent Lyapunov arguments that are based on several new robust stability conditions for continuous-time linear time-invariant systems which are affected by parametric passive uncertainties and do not admit any discrete dynamics. In contrast to other approaches, our stability analysis results rely on separation techniques from robust control involving dynamic resetting scalings. Moreover, these techniques are expected to pave the way for covering much more general uncertainties within the framework of integral quadratic constraints. Building upon the new insights on robust stability, we are able to propose an output-feedback gain-scheduling design methodology similarly to our recently obtained synthesis result involving dynamic resetting D-scalings. Most of the obtained conditions are expressed as infinite-dimensional (differential) linear matrix inequalities which can be numerically solved by using relaxation methods based on, e.g., linear splines or matrix sum-of-squares. We apply our results to obtain novel stability criteria and to design distributed controllers for networked systems over undirected switching topologies. Finally, the approach is illustrated with several numerical examples.

Invariance With Dynamic Multipliers

This paper focusses on the robust or absolute stability analysis of feedback interconnections consisting of a linear time-invariant system and an uncertain linear or nonlinear component. Building on the framework of integral quadratic constraints, we establish a novel local stability result using general dynamic multipliers that enables the verification of hard time-domain state and output constraints. This is achieved by merging frequency-domain descriptions of uncertainties with time-domain dissipation arguments as they typically arise in Lyapunov theory. The general applicability of the results provided in this paper is illustrated by means of several examples that demonstrate the effectiveness of the presented approach, even if compared to techniques that are tailored to particular uncertainty classes.

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.

A Swapping Lemma for Switched Systems

Abstract Robust stability of a class of linear time-invariant systems affected by piecewise constant parameters with dwell-time constraints is considered. In contrast to other approaches the proposed result relies on separation techniques within the framework of integral quadratic constraints and is based on a novel version of the swapping lemma. In particular, our result allows to take additional knowledge on the variation of the parameter into account. The obtained conditions are expressed as infinite-dimensional linear matrix inequalities which can be solved, e.g., by using sum-of-squares relaxation methods. We illustrate the proposed approach with a numerical example.

An IQC theorem for relations: Towards stability analysis of data-integrated systems

Abstract We consider feedback interconnections composed of a linear system whose input-output behavior is constrained by a relation. It is shown how to analyze stability of such interconnections by suitably refining the general framework of integral quadratic constraints. For the special case of static multivalued monotone nonlinearities, we reveal that one can employ Zames-Falb multipliers in order to substantially reduce conservatism if compared with passivity-based approaches and as illustrated with a numerical example. Our results are expected to open up an avenue towards a systematic stability analysis of data-integrated systems.

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.

A General Integral Quadratic Constraints Theorem with Applications to a Class of Sampled-Data Systems

In the framework of integral quadratic constraints a more general stability theorem is derived by relaxing the assumptions on the uncertainty. In contrast to existing results, input-output relations are only required to hold on a subspace of the full signal space. This result is employed to incorporate uncertainties exhibiting sampling behavior into the framework of integral quadratic constraints. The arising infinite dimensional stability test is exactly reduced to a finite dimensional linear matrix inequality. Advantages of this approach are illustrated for the specific case of an interconnection including a pulse-width modulator.

IQC arguments for analysis of the 3-state Moore-Greitzer compressor system

The Integral Quadratic Constraint (IQC) framework developed by Professor Yakubovich and his co-workers, see Yakubovich et.al. (2004), is one of few available constructive tools for establishing robust stability of nonlinear systems. An explicit format of stability conditions, procedures for computing a Lyapunov function and developed libraries of IQCs for common nonlinearities in dynamics, all together have made the approach unique and at the same time so to say automatic for recovering stability conditions for many applications: in the process of analyzing a dynamical system, an engineer is just required to search for a suficiently rich set of IQCs describing nonlinearities in the dynamics so that such nonlinearities can be substituted in analysis by quadratic constraints, which they satisfy. The power of the methodology becomes also its weak part in an analysis of concrete systems. Searching IQCs is the dificult task in new examples, where a lack of a rich set of IQCs for concrete nonlinearities makes the method inconclusive or too rough to detect (in)-stability. The paper is aimed at a discussion of such an example of a nonlinear dynamical system (the classical 3-state Moore-Greitzer compressor model) augmented with the dynamical feedback controller, whose parameters should be adjusted to meet a stability condition. The closed-loop system has several nonlinearities and searching the corresponding IQCs to meet the stability conditions for this example is rather involved. To overcome the problem, we have previously described by different methods a set of parameters for which any solution of the closed loop system, if bounded, will converge to the origin and that the origin is locally asymptotically stable. However, the proof is incomplete without showing a boundedness of all solutions. To solve the task we have re-used some of the IQC framework ideas, where the method has been utilized and the corresponding IQCs have been found only for unbounded trajectories, if they would be present in closed loop system. The arguments have allowed completing the proof of stability and illustrating deliberate use of the IQC framework aimed at analysis of behavior of specific trajectories.

New results on stability analysis of delayed recurrent neural networks based on the integral quadratic constraints approach

This paper is concerned with the analysis of the stability of delayed recurrent neural networks. In contrast to the widely used Lyapunov–Krasovskii functional approach, a new method is developed within the integral quadratic constraints framework. To achieve this, several lemmas are first given to propose integral quadratic separators to characterize the original delayed neural network. With these, the network is then reformulated as a special form of feedback-interconnected system by choosing proper integral quadratic constraints. Finally, new stability criteria are established based on the proposed approach. Numerical examples are given to illustrate the effectiveness of the new approach.

Design of an electron beam stabilisation controller for a synchrotron

Synchrotrons are used to generate light for academic and industry research by accelerating electrons travelling in a circular path to relativistic speeds. In order to achieve optimum performance, electron beam stability is a crucial parameter for synchrotrons. This paper describes the design of a beam stabilisation controller, using Internal Model Control. Basis functions are used to identify the controllable components of the system and it is demonstrated how by selecting dynamics for each spatial mode, enhanced performance is achieved. The robust stability of the controller in the presence of spatial uncertainties is developed within an Integral Quadratic Constraint framework using two methods of spatial decomposition: Singular Value decomposition and Fourier decomposition. The controller has been implemented at Diamond Light Source, the UK׳s national synchrotron science facility. Results from the controller implementation are presented and it is demonstrated how the controller design and robust stability analysis are used to tradeoff performance and robustness.

IQC-synthesis with general dynamic multipliers

SUMMARY Analogously to the existing μ-synthesis tools, we propose an alternative algorithm for the systematic design of robust controllers based on an iteration of standard nominal controller synthesis and integral quadratic constraint (IQC) analysis with general dynamic multipliers. The suggested algorithm enables us to perform robust controller synthesis for a significantly larger class of uncertainties if compared with the existing methods. Indeed, while the classical approaches are restricted to the use of real/complex time invariant or arbitrarily fast time-varying parametric uncertainties, the IQC framework also offers, for example, the possibility to efficiently handle sector-bounded and slope restricted nonlinearities or time-varying parametric uncertainties and uncertain time-varying time-delays, both with bounds on the rate-of-variation. Secondly, in contrast to the classical approaches, the proposed techniques completely avoid gridding and curve-fitting. We present new insights that allow us to reformulate the robust IQC analysis LMIs into a standard quadratic performance problem. This enables us to generate suitable initial conditions for each subsequent iteration step. Depending on the size of the problem, this can significantly speed up the synthesis process. The results are illustrated by means of two numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.

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.

IQC-Synthesis with General Dynamic Multipliers

AbstractIn this paper we generalize our previous results on the synthesis of robust controllers. A novel controller/scaling algorithm is proposed that allows for the use of arbitrary real-rational Integral Quadratic Constraint (IQC) multipliers with no poles on the extended imaginary axis. In contrast to the classical μ-synthesis approaches, the techniques completely avoid gridding as well as curve-fitting. Moreover, while the classical approaches are restricted to the use of real/complex time-invariant or arbitrarily fast time-varying parametric uncertainties, the IQC framework can be employed for a much larger class of uncertainties involving nonlinearities and bounds on rates of time-varying parametric uncertainties. The results are illustrated through a numerical example.