Integral Quadratic Constraints Research Articles

Output-feedback controllers with guaranteed ℒ2-gain for continuous-time Lur'e systems using noncausal Zames–Falb multipliers

This paper focuses on the problem of dynamic output feedback control for continuous-time Lur'e systems with slope-bounded nonlinearities, particularly addressing the closed-loop L 2 gain. The synthesis framework is derived using integral quadratic constraints. The following contributions are highlighted: (i) assessment of stability and L 2 gain through causal, anticausal, and noncausal Zames–Falb multipliers; (ii) technical enhancements to improve efficiency and to reduce the conservativeness of two different classes of multipliers regarding the L 1 norm evaluation; (iii) the synthesis conditions are presented in terms of an iterative procedure based on linear matrix inequalities, allowing arbitrary order for both the controller and the multipliers. The proposed achievements are validated through numerical examples, demonstrating the efficacy and flexibility of the approach compared to existing methods in the literature.

Exponential input-to-state stability for Lur’e systems via Integral Quadratic Constraints and Zames–Falb multipliers

Abstract Absolute stability criteria that are sufficient for global exponential stability are shown, under a Lipschitz assumption, to be sufficient for the a priori stronger exponential input-to-state stability property. Important corollaries of this result are as follows: (i) absolute stability results obtained using Zames–Falb multipliers for systems containing slope-restricted nonlinearities provide exponential input-to-state-stability under a mild detectability assumption; and (ii) more generally, many absolute stability results obtained via Integral Quadratic Constraint methods provide, with the additional Lipschitz assumption, this stronger property.

Analysis of Gradient Descent with Varying Step Sizes using Integral Quadratic Constraints

Analysis of Gradient Descent with Varying Step Sizes using Integral Quadratic Constraints

Connections between integral quadratic constraints and dissipativity

Connections between integral quadratic constraints and dissipativity

Event-Triggered Control Based on Integral Quadratic Constraints

Event-Triggered Control Based on Integral Quadratic Constraints

Stability Analysis of Feedback Systems with ReLU Nonlinearities via Semialgebraic Set Representation

This paper is concerned with the stability/instability analysis problem of feedback systems with rectified linear unit (ReLU) nonlinearities. Such feedback systems arise when we model dynamical (recurrent) neural networks (NNs) and NN-driven control systems where all the activation functions of NNs are ReLUs. In this study, we focus on the semialgebraic set representation characterizing the input-output properties of ReLUs. This allows us to employ a novel copositive multiplier in the framework of the integral quadratic constraint and, thus, to derive a new linear matrix inequality (LMI) condition for the stability analysis of the feedback systems. However, the infeasibility of this LMI does not allow us to obtain any conclusion on the system's stability due to its conservativeness. This motivates us to consider its dual LMI. By investigating the structure of the dual solution, we derive a rank condition on the dual variable certificating that the system at hand is unstable. In addition, we construct a hierarchy of dual LMIs allowing for improved instability detection. We illustrate the effectiveness of the proposed approach by several numerical examples.

Hydrazine Concentration Estimation in the Secondary Circuit of a Nuclear Power Plant: a IQC LPV approach

This paper addresses the design of a new linear parameter varying estimator, to estimate the hydrazine molar concentration in the secondary circuit of a nuclear power plant. The method uses dynamic shaping filters of arbitrary order, to formulate the estimation performance objectives. The theory is based on L2-gain synthesis, based on uncertainty characterization via static integral quadratic constraint multipliers. It is shown that the problem can be formulated as a semi definite programming problem, that can be solved efficiently using numerical solvers. The proposed approach is applied to real data, coming from a nuclear power plant located in France. The obtained results demonstrates the efficiency of the proposed approach.

CONVEXITY OF REACHABLE SETS OF QUASILINEAR SYSTEMS

This paper investigates convexity of reachable sets for quasilinear systems under integral quadratic constraints. Drawing inspiration from B.T. Polyak's work on small Hilbert ball image under nonlinear mappings, the study extends the analysis to scenarios where a small nonlinearity exists on the system's right-hand side. At zero value of a small parameter, the quasilinear system turns into a linear system and its reachable set is convex. The investigation reveals that to maintain convexity of reachable sets of these systems, the nonlinear mapping's derivative must be Lipschitz continuous. The proof methodology follows a Polyak's scheme. The paper's structure encompasses problem formulation, exploration of parameter linear mapping and image transformation, application to quasilinear control systems, and concludes with illustrative examples.

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.

Finite Horizon Worst Case Analysis of Linear Time-Varying Systems Applied to Launch Vehicle

This article presents an efficient approach to compute the worst case gain of the interconnection of a finite time horizon linear time-varying system and a perturbation. The input/output behavior of the uncertainty is described by integral quadratic constraints (IQCs). A condition for the worst case gain of such an interconnection can be formulated using dissipation theory as a parameterized Riccati differential equation, which depends on the chosen IQC multiplier. A nonlinear optimization problem is formulated to minimize the upper bound of the worst case gains over a set of admissible IQC multipliers. This problem can be efficiently solved with a custom-tailored logarithmically scaled, adaptive differential evolution algorithm. It provides a fast alternative to similar approaches based on solving semidefinite programs. The algorithm is applied to the worst case aerodynamic load analysis of an expendable launch vehicle (ELV). The worst case load of the uncertain ELV is calculated under wind turbulence during the atmospheric ascend and compared to results from nonlinear simulation.

Feedback Stability of Generalized Positive Real and Negative Imaginary Systems

This paper derives necessary and sufficient conditions for robust stability of a feedback interconnection of linear time-invariant (LTI) systems that exhibit positive realness on finite nonzero frequencies (excluding pole locations) through a weighting function, i.e. a multiplier. This class of LTI systems importantly includes the well-studied negative imaginary systems without poles at the origin as a subset. Under the assumption that the instantaneous gain of the loop transfer function is zero, it is shown that the aforementioned feedback interconnection is stable if and only if the static (a.k.a. DC) loop gain is less than unity. This condition is identical to the one that guarantees feedback stability of negative imaginary systems, but is applicable to a much wider class of systems beyond negative imaginariness. Our proof is based entirely on the multivariable Nyquist stability criterion — it does not make use of realisations of the transfer functions. Comparisons with integral quadratic constraint based robust stability results are also made.

Trajectory‐based robustness analysis for nonlinear systems

AbstractThis article considers the robustness of an uncertain nonlinear system along a finite‐horizon trajectory. The uncertain system is modeled as a connection of a nonlinear system and a perturbation. The analysis relies on three ingredients. First, the nonlinear system is approximated by a linear time‐varying (LTV) system via linearization along a trajectory. This linearization introduces an additional forcing input due to the nominal trajectory. Second, the input/output behavior of the perturbation is described by time‐domain, integral quadratic constraints (IQCs). Third, a dissipation inequality is formulated to bound the worst‐case deviation of an output signal due to the uncertainty. These steps yield a differential linear matrix inequality (DLMI) condition to bound the worst‐case performance. The robustness condition is then converted to an equivalent condition in terms of a Riccati differential equation. This yields a computational method that avoids heuristics often used to solve DLMIs, for example, time gridding. The approach is demonstrated by a two‐link robotic arm example.

Kernel-Based Models for System Analysis

This paper introduces a computational framework to identify nonlinear input-output operators that fit a set of system trajectories while satisfying incremental integral quadratic constraints. The data fitting algorithm is thus regularized by suitable input-output properties required for system analysis and control design. This biased identification problem is shown to admit the tractable solution of a regularized least squares problem when formulated in a suitable reproducing kernel Hilbert space. The kernel-based framework is a departure from the prevailing state-space framework. It is motivated by fundamental limitations of nonlinear state-space models at combining the fitting requirements of data-based modeling with the input-output requirements of system analysis and physical modeling.

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.

H ∞ negative imaginary static output feedback controller for low frequency networked control systems

In this paper, an H ∞ negative imaginary static output feedback (SOF) controller is designed for a low frequency (LF) networked control system (NCS) with time-delays. A difference operator is introduced to deal with the time-delays in a feedback configuration via integral quadratic constraints (IQCs). Both H ∞ performance and negative imaginariness are analysed via LF IQCs and a generalised Kalman–Yakubovich–Popov (GKYP) lemma. A two-stage algorithm is presented to obtain the H ∞ negative imaginary SOF controller for the LF NCS with time-delays. Simulation results are given to show the effectiveness and superiority of the H ∞ negative imaginary SOF controller.

Robustness analysis of gain‐scheduled Model Predictive Control for Linear Parameter Varying systems: An Integral Quadratic Constraints approach

AbstractIn this paper, we evaluate the robustness qualities of Model Predictive Control (MPC) algorithms applied for Linear Parameter Varying (LPV) systems. Specifically, we analyze gain‐scheduled LPV MPC schemes, that is, those that use model predictions based on the LPV scheduling variables available at each sampling instant. Accordingly, we extend previous results on finite‐horizon robustness analysis of linear time‐variant (LTV) systems, employing Integral Quadratic Constraints (IQCs) to describe the input‐output behavior of prediction uncertainties. We provide two main novelties in our formulation: (i) we propose a parameter‐dependent Karush–Kuhn–Tucker (KKT) inequality to describe the existence and feasibility of the LPV MPC control inputs; and (ii) we model the uncertainties that arise due to the unavailability of the scheduling trajectory along the prediction horizon as a bounded interconnection in the form of a Linear Fractional Transformation (LFT). Accordingly, we use dissipativity arguments (‐hard IQCs) in order to compute robust induced gains of the closed‐loop system (specifically, the and ‐to‐Euclidean metrics), taking into account the MPC prediction uncertainties. We also generate the set of reachable states from a given initial condition. A benchmark example is used to illustrate the proposed analysis procedure.

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.

Computation of invariant sets for discrete‐time uncertain systems

AbstractThis article proposes novel methods for the computation of state and output bounding sets for discrete‐time uncertain systems. The systems under consideration are formed from the interconnection of a nominal linear time‐invariant system and an uncertainty operator that is known to lie within a prespecified set. The set of allowable uncertainties is described by a pointwise integral quadratic constraint (IQC), that is, a quadratic constraint that holds for all time‐steps. Examples of pointwise IQCs characterizing common uncertainty sets are provided. The exogenous input to the system is assumed to lie in a given polytope or ellipsoid for all time‐steps. The proposed methods are illustrated via an example.

Integral Quadratic Constraints with Infinite-Dimensional Channels.

Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such input-output analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinite-dimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinite-dimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity.

Data-Driven Inference on Optimal Input–Output Properties of Polynomial Systems With Focus on Nonlinearity Measures

In the context of dynamical systems, nonlinearity measures quantify the strength of nonlinearity by means of the distance of their input-output behaviour to a set of linear input-output mappings. In this paper, we establish a framework to determine nonlinearity measures and other optimal input-output properties for nonlinear polynomial systems without explicitly identifying a model but from a finite number of input-state measurements which are subject to noise. To this end, we deduce from data for the unidentified ground-truth system three possible set-membership representations, compare their accuracy, and prove that they are asymptotically consistent with respect to the amount of samples. Moreover, we leverage these representations to compute guaranteed upper bounds on nonlinearity measures and the corresponding optimal linear approximation model via semi-definite programming. Furthermore, we extend the established framework to determine optimal input-output properties described by time domain hard integral quadratic constraints.