Stability Certificates Research Articles

Nonlinear Robust Approaches to Study Stability and Postcritical Behavior of an Aeroelastic Plant

Two approaches to tackle the nonlinear robust stability problem of an aerospace system are compared. The first employs a combination of the describing function method and $\mu $ analysis, while the second makes use of integral quadratic constraints (IQCs). The model analyzed consists of an open-loop wing’s airfoil subject to freeplay and linear time-invariant parametric uncertainties. The key steps entailed by the application of the two methodologies and their main features are critically discussed. Emphasis is put on the available insight on the nonlinear postcritical behavior known as limit cycle oscillation. It is proposed a strategy to apply IQCs, typically used to find absolute stability certificates, in this scenario, based on a restricted sector bound condition for the nonlinearity. Another contribution of this paper is to understand how the conservatism usually associated with the IQCs multipliers selection can be overcome by using information coming from the first approach. Nonlinear time-domain simulations showcase the prowess of these approaches in estimating qualitative trends and quantitative response’s features.

Bounding Errors Due to Switching Delays in Incrementally Stable Switched Systems

Time delays pose an important challenge in networked control systems, which are now ubiquitous. Focusing on switched systems, we introduce a framework that provides an upper bound for errors caused by switching delays. Our framework is based on approximate bisimulation, a notion that has been previously utilized mainly for symbolic (discrete) abstraction of state spaces. Notable in our framework is that, in deriving an approximate bisimulation and thus an error bound, we use a simple incremental stability assumption (namely δ-GUAS) that does not itself refer to time delays. That this is the same assumption used for state-space discretization enables a two-step workflow for control synthesis for switched systems, in which a single Lyapunov-type stability certificate serves for two different purposes of state discretization and coping with time delays. We demonstrate the proposed framework with a boost DC-DC converter, a common example of switched systems.

Sum of Squares Certificates for Stability of Planar, Homogeneous, and Switched Systems

We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of squares (sos) and that the negative of its derivative is also a sum of squares. This result is extended to show that such sos-based certificates of stability are guaranteed to exist for all stable switched linear systems. For this class of systems, we further show that if the derivative inequality of the Lyapunov function has an sos certificate, then the Lyapunov function itself is automatically a sum of squares. These converse results establish cases where semidefinite programming is guaranteed to succeed in finding proofs of Lyapunov inequalities. Finally, we demonstrate some merits of replacing the sos requirement on a polynomial Lyapunov function with an sos requirement on its top homogeneous component. In particular, we show that this is a weaker algebraic requirement in addition to being cheaper to impose computationally.

On Existence of Separable Contraction Metrics for Monotone Nonlinear Systems

Finding separable certificates of stability is important for tractability of analysis methods for large-scale networked systems. In this paper we consider the question of when a nonlinear system which is contracting, i.e. all solutions are exponentially stable, can have that property verified by a separable metric. Making use of recent results in the theory of positive linear systems and separable Lyapunov functions, we prove several new results showing when this is possible, and discuss the application of to nonlinear distributed control design via convex optimization.

Ultimate boundedness and regions of attraction of frequency droop controlled microgrids with secondary control loops

In this paper we study theoretical properties of the frequency control problem in inverter-based microgrids with primary and decentralised/distributed secondary control loops. Stability of these microgrids has been the subject of a number of recent studies. Conventional approaches based on standard hierarchical control rely on time-scale separation between primary and secondary control loops to show local stability of equilibria. In this paper we show that (i) frequency regulation can be ensured without assuming time-scale separation and, (ii) ultimate boundedness of the trajectories starting inside a region of attraction is guaranteed under a condition on the power mismatch between demand and generation at each inverter bus. An estimate of the region of attraction is obtained from which an ultimate bound set for the state trajectories can be determined by recursive iterations of a nonlinear mapping. The derived results provide a certificate of the overall stability and performance of the controlled microgrid.

Robustness of distributed averaging control in power systems: Time delays & dynamic communication topology

Distributed averaging-based integral (DAI) controllers are becoming increasingly popular in power system applications. The literature has thus far primarily focused on disturbance rejection, steady-state optimality and adaption to complex physical system models without considering uncertainties on the cyber and communication layer nor their effect on robustness and performance. In this paper, we derive sufficient delay-dependent conditions for robust stability of a secondary-frequency-DAI-controlled power system with respect to heterogeneous communication delays, link failures and packet losses. Our analysis takes into account both constant as well as fast-varying delays, and it is based on a common strictly decreasing Lyapunov–Krasovskii functional. The conditions illustrate an inherent trade-off between robustness and performance of DAI controllers. The effectiveness and tightness of our stability certificates are illustrated via a numerical example based on Kundur’s four-machine-two-area test system.

Robust Stability Analysis of Grid-Connected Converters Based on Parameter-Dependent Lyapunov Functions

This paper deals with the problem of robust stability analysis of grid-connected converters with LCL filters controlled through a digital signal processor and subject to uncertain grid inductance. To model the uncertain continuous-time plant and the digital control gain, a discretization procedure, described in terms of a Taylor series expansion, is employed to determine an accurate discrete-time model. Then, a linear matrix inequality-based condition is proposed to assess the robust stability of the polynomial discrete-time augmented system that includes the filter state variables, the states of resonant controllers and the delay from the digital control implementation. By means of a parameter-dependent Lyapunov function, the proposed strategy has as main advantage to provide theoretical certification of stability of the uncertain continuous-time closed-loop system, circumventing the main disadvantages of previous approaches that employ approximate discretized models, neglecting the errors. Numerical simulations illustrate the benefits of the discretization technique and experimental results validate the proposed approach.

PMU-Based Model-Free Approach for Real-Time Rotor Angle Monitoring

In this letter, we present an algorithm for rotor angle stability monitoring of power system in real-time. The proposed algorithm is model-free and can make use of high resolution phasor measurement units (PMUs) data to provide reliable, timely information about the system's stability. The theoretical basis behind the proposed algorithm is adopted from dynamical systems theory. In particular, the algorithm approximately computes the system's Lyapunov exponent (LE), thereby measuring the exponential convergence or divergence rate of the rotor angle trajectories. The LE serves as a certificate of stability where the positive (negative) value of the LE implies exponential divergence (convergence) of nearby system trajectories, hence, unstable (stable) rotor angle dynamics. We also show the proposed model-free algorithm can be used for the identification of the coherent sets of generators. The simulation results are presented to verify the developed results in the paper on the modified IEEE 162-bus system.

Stability assessment in ligand-binding assays: a critical parameter for data integrity.

Stability assessment in ligand-binding assays: a critical parameter for data integrity.

Lyapunov Functions Family Approach to Transient Stability Assessment

Analysis of transient stability of strongly nonlinear post-fault dynamics is one of the most computationally challenging parts of Dynamic Security Assessment. This paper proposes a novel approach for assessment of transient stability of the system. The approach generalizes the idea of energy methods, and extends the concept of energy function to a more general Lyapunov Functions Family (LFF) constructed via Semi-Definite-Programming techniques. Unlike the traditional energy function and its variations, the constructed Lyapunov functions are proven to be decreasing only in a finite neighborhood of the equilibrium point. However, we show that they can still certify stability of a broader set of initial conditions in comparison to the traditional energy function in the closest-UEP method. Moreover, the certificates of stability can be constructed via a sequence of convex optimization problems that are tractable even for large scale systems. We also propose specific algorithms for adaptation of the Lyapunov functions to specific initial conditions and demonstrate the effectiveness of the approach on a number of IEEE test cases.

ЭКСПРЕСС-МЕТОД АТТЕСТАЦИИ ВРЕМЕННО́Й СТАБИЛЬНОСТИ МАГНИТОВ (NdDySm)(FeCo)B

ЭКСПРЕСС-МЕТОД АТТЕСТАЦИИ ВРЕМЕННО́Й СТАБИЛЬНОСТИ МАГНИТОВ (NdDySm)(FeCo)B

Linear matrix inequalities for analysis and control of linear vector second‐order systems

SummaryMany dynamical systems are modeled as vector second‐order differential equations. This paper presents analysis and synthesis conditions in terms of LMI with explicit dependence in the coefficient matrices of vector second‐order systems. These conditions benefit from the separation between the Lyapunov matrix and the system matrices by introducing matrix multipliers, which potentially reduce conservativeness in hard control problems. Multipliers facilitate the usage of parameter‐dependent Lyapunov functions as certificates of stability of uncertain and time‐varying vector second‐order systems. The conditions introduced in this work have the potential to increase the practice of analyzing and controlling systems directly in vector second‐order form. Copyright © 2014 John Wiley & Sons, Ltd.

On Complexity of Lyapunov Functions for Switched Linear Systems

We show that for any positive integer d, there are families of switched linear systems—in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤ d, or (ii) a polytopic Lyapunov function with ≤ d facets, or (iii) a piecewise quadratic Lyapunov function with ≤ d pieces. This implies that there cannot be an upper bound on the size of the linear and semidefinite programs that search for such stability certificates. Several constructive and non-constructive arguments are presented which connect our problem to known (and rather classical) results in the literature regarding the finiteness conjecture, undecidability, and non-algebraicity of the joint spectral radius. In particular, we show that existence of a sum of squares Lyapunov function implies the finiteness property of the optimal product.

Real-Time Monitoring of Short-Term Voltage Stability Using PMU Data

We develop a model-free approach for the short-term voltage stability monitoring of a power system. Finite time Lyapunov exponents are used as the certificate of stability. The time-series voltage data from phasor measurement units (PMU) are used to compute the Lyapunov exponent to predict voltage stability in real time. Issues related to practical implementation of the proposed method, such as phasor measurement noise, communication delay, and the finite window size for prediction, are also discussed. Furthermore, the stability certificate in the form of Lyapunov exponents is also used to determine the stability/instability contributions of the individual buses to the overall system stability and for computation of critical clearing time. Simulation results are provided for the IEEE 162-bus system to demonstrate the application of the developed method.

Stability certification of large scale stochastic systems using dissipativity

In this paper, we analyse the stability of large-scale nonlinear stochastic systems, represented as an interconnection of lower-order stochastic subsystems. Stochastic stability in probability and noise-to-state stability are addressed, and sufficient conditions for the latter are provided. The method proposed proves network stability by using appropriate stochastic passivity properties of its subsystems, and the structure of its interactions. Stability properties are established by the diagonal stability of a dissipativity matrix, which incorporates information about the passivity properties of the systems and their interconnection. Next, we derive equilibrium-independent conditions for the verification of the relevant passivity properties of the subsystems. Finally, we illustrate the proposed approach on a class of biological reaction networks.

On wind stability requirements for emergency car warning triangles

This work discusses the wind stability requirements specified by UN Reg. 27 on emergency car warning triangles, which are of mandatory use in many countries. Wind tunnel experiments have been carried out in order to determine aerodynamic coefficients of commercial warning triangles and the friction coefficients between the triangle legs and an asphalt base that fulfils the roughness requirements stated by Reg. 27 for wind stability certification. The wind stability specifications for warning triangles are reviewed, compared with pressure field measurements and discussed. Results of wind tunnel tests and comparison with field measurements reported in the literature show that the requirements could be excessively conservative.

Positive trigonometric polynomials for strong stability of difference equations

We follow a polynomial approach to analyse strong stability of continuous-time linear difference equations with several delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5.

Stability analysis of interconnected Hamiltonian systems under time delays

Sufficient conditions are derived for checking whether interconnected port-Hamiltonian systems are stable in the presence of time delays. It is assumed that the time delay parameters are unknown time-varying functions for which the only available information is about the upper bounds on their magnitude and/or variation. The stability conditions proposed here are established by constructing Lyapunov–Krasovskii stability certificates, based on the Hamiltonians of the individual port-Hamiltonian systems. The forms of the Lyapunov–Krasovskii functionals vary according to the information available on the delay parameter. It is shown how different informations on the delay are utilised to construct stability certificates.

Frozen-time and time-varying robust stability certificates for the coupled pitch/yaw motion of an axial-symmetric launch vehicle

Clearance of Thrust Vector Control laws for launchers demands the capability to provide analytical certificates for the system's robust stability with respect to nonzero roll rates. The current Verification and Validation approach used in industry is limited in as much as it entails evaluation of gain margins for a finite set of conditions and assuming a frozen-time setting. Advanced analysis techniques, such as mu analysis and Integral Quadratic Constraints, have the potential to clear the whole analysis space and take into consideration nonlinear and time-varying aspects. The contribution of this paper stands in exploiting such potential to provide frozen-time and time-varying robust stability certificates for the coupled pitch/yaw attitude motion of an axial-symmetric launch vehicle.

A Robust H2 State Feedback Controller Applied To Boost Converters

This paper proposes the design and the experimental validation of a robust H2 control technique applied to boost converters. Differently from conventional techniques, the proposed controller has a certificate of stability and performance under arbitrary parameter variations. The design procedure adopts a linearized model with a polytopic structure for the converter affected by time-varying parameters. An optimal H2 state feedback controller (under quadratic stability) is synthesized using convex optimization based on linear matrix inequalities. This framework allows the determination of the control gains in a very fast way, using a finite set of inequalities to obtain a controller that ensures robust stability and performance for the entire set of uncertain parameters. An experimental setup of a boost converter is used to validate the controller designed here. The input voltage, the operating point duty cycle and the load are considered as arbitrarily time-varying uncertain parameters. The synthesized control gains produce good simulated and experimental results when compared to those obtained with a classical strategy based on PI controllers, without introducing additional implementation cost. The experimental validation proves the viability of practical application of this important robust control technique for boost converters.