Dongkun Han

Robust Consensus for a Class of Uncertain Multi-Agent Dynamical Systems

This paper investigates robust consensus for a class of uncertain multi-agent dynamical systems. Specifically, it is supposed that the system is described by a weighted adjacency matrix whose entries are polynomial functions of an uncertain vector constrained in a semi-algebraic set. For this uncertain topology, we provide necessary and sufficient conditions for ensuring robust first-order consensus and robust second-order consensus, in both cases of positive and non-positive weighted adjacency matrices. Moreover, we show how these conditions can be investigated through convex programming by using standard software. Some numerical examples illustrate the proposed results.

Control synthesis for non-polynomial systems: A domain of attraction perspective

This paper studies a control synthesis problem to enlarge the domain of attraction (DA) for non-polynomial systems by using polynomial Lyapunov functions. The basic idea is to formulate an uncertain polynomial system with parameter ranges obtained form the truncated Taylor expansion and the parameterizable remainder of the non-polynomial system. A strategy for searching a polynomial output feedback controller and estimating the lower bound of the largest DA is proposed via an optimization of linear matrix inequalities (LMIs). Furthermore, in order to check the tightness of the lower bound of the largest estimated DA, a necessary and sufficient condition is given for the proposed controller. Lastly, several methods are provided to show how the proposed strategy can be extended to the case of variable Lyapunov functions. The effectiveness of this approach is demonstrated by numerical examples.

Formal Analysis of Drum-Boiler Units to Maximize the Load-Following Capabilities of Power Plants

The load-following capabilities of power plants became increasingly important in recent years as a means of ensuring a reliable operation of future power systems. In this work, we propose a generic approach, based on reachability analysis, to rigorously verify the safety of critical components that often pose limitations on the flexibility of conventional power plants to perform fast load changes. The proposed reachability algorithm makes it possible to compute the bounds of all possible trajectories for a range of operating conditions while simultaneously meeting the practical requirements of a real power plant. As an example, we consider the verification of the water level inside a drum unit. In contrast to previous work, our results are based on measurement data of a realistic configuration of a boiler system located within a 450 MW combined cycle plant in Germany. We use an abstract model which considers the modelling errors to ensure that all dynamic behaviors of the process are replicated by the abstraction. Through the implementation of our abstract model, we formally guarantee that the water level inside the drum always remains within safe limits for load changes equivalent to 40 MW which, as a result, exploits the power plants adaptability and load-following capabilities.

Robust Synchronization via Homogeneous Parameter-Dependent Polynomial Contraction Matrix

Robust synchronization problem is a key issue in chaotic circuits and nonlinear systems. This paper is concerned with robust synchronization problem of polynomial nonlinear system affected by time-varying uncertainties on topology, i.e., structured uncertain parameters constrained in a bounded-rate polytope. Via partial contraction analysis, novel conditions, both for robust exponential synchronization and for robust asymptotical synchronization, are proposed by using parameter-dependent contraction matrices. In addition, for polynomial nonlinear system, this paper introduces a new class of contraction matrix, i.e., homogeneous parameter-dependent polynomial contraction matrix (HPD-PCM), by which tractable conditions of linear matrix inequalities (LMIs) are provided via affine space parametrizations. Furthermore, the variant rate margin for robust asymptotical synchronization is, for the first time, proposed and investigated via handling generalized eigenvalue problems (GEVPs). A set of representative examples demonstrate the effectiveness of proposed method.

Synchronization Conditions for Multiagent Systems With Intrinsic Nonlinear Dynamics

This brief studies local and global synchronization in multiagent systems with nonlinear dynamics with respect to equilibrium points and periodic orbits. For local synchronization, a method is proposed based on the transformation of the original system into an uncertain polytopic system and on the use of homogeneous polynomial Lyapunov functions. For global synchronization, another method is proposed based on the search for a suitable polynomial Lyapunov function. The proposed methods exploit linear matrix inequalities and have several advantages. In particular, the proposed methods require the solution of convex optimization problems. Also, the proposed methods exploit more complex Lyapunov functions than the quadratic Lyapunov functions typically considered in the literature and included in this brief as a special case.

Robust discrete-time consensus of multi-agent systems with uncertain interaction

This paper addresses robust discrete-time consensus problem of multiple agents with uncertain structure, where the network coupling weights are supposed polynomial functions of an uncertain vector constrained in a semialgebraic set. Based on the Lyapunov stability theory, a necessary and sufficient condition for robust discrete-time consensus is proposed. Then, we investigate the robust discrete-time consensus with positive weighted network, and a necessary and sufficient condition is also provided based on the property of an uncertain matrix. Corresponding sufficient conditions for robust discrete-time consensus are derived by solving a linear matrix inequality (LMI) problem built by exploiting sum-of-squares (SOS) polynomials. Some examples illustrate the proposed results.

Estimating the domain of attraction based on the invariance principle

Estimating the domain of attraction (DA) of an equilibrium point is a long-standing yet still challenging issue in nonlinear system analysis. The method using the sublevel set of Lyapunov functions is proven to be efficient, but sometimes conservative compared to the estimate via invariant sets. This paper studies the estimation problem of the DA for autonomous polynomial system by using the invariance principle. The main idea is to estimate the DA via sublevel sets of a positive polynomial, which characterizes the boundary of invariant sets. This new type of invariant sets admits the condition that the derivative of Lyapunov functions is non-positive, which generalizes the sublevel set method via Lyapunov functions. An iterative algorithm is then proposed for enlarging the estimate of the DA. Finally, the effectiveness of the proposed method is illustrated by numerical examples.

Estimating the region of attraction via forward reachable sets

We propose and implement an algorithm based on reachability analysis to estimate the region of attraction (ROA) of an equilibrium point for nonlinear systems. The stability region is obtained via the computation of forward reachable sets. We compare our results with well-established techniques in this area. In particular, we consider the optimization of the Lyapunov function (LF) sub-level set using sum-of-squares (SOS) decomposition, and the computation of backward reachable sets of a target set using the viscosity solution of a time-dependant Hamilton-Jacobi-Isaacs (HJI) formulation. Our method can overcome many limitations imposed on the applicability of Lyapunov-based approaches, such as conservatism in estimating the stability region, and difficulties associated with choosing a suitable LF. This is due to the fact that our reachability algorithm does not require a LF in order to provide an estimate of the ROA. Various numerical examples show that our proposed approach can estimate the exact ROA quite accurately, and more importantly, scales moderately with the system dimension compared to alternative techniques.

Power systems transient stability analysis via optimal rational Lyapunov functions

Transient stability analysis is a traditional yet significant topic in power systems. In order to obtain the stability domain of the post-fault equilibrium point, the Lyapunov method is proven to be effective and efficient once a Lyapunov function has been found. The main innovation of this paper consists in the use of rational Lyapunov functions to compute the largest estimate of the Region of Attraction (ROA) of an equilibrium point for power systems. Firstly, the non-polynomial power systems are reconstructed to uncertain differential algebraic systems via the multi-variate truncated Taylor expansion. An iteration procedure is proposed to compute the largest estimate of the ROA by exploiting the Sum of Squares (SOS) technique and the Squared Matrix Representation (SMR). A classical power system with transfer conductances is studied to demonstrate the effectiveness of the proposed approach.

Synchronization seeking in multi-agent dynamic systems with communication uncertainties

This paper addresses robust consensus problems among multiple agents with uncertain parameters constrained in a given set. Specifically, the network coefficients are supposed polynomial functions of an uncertain vector constrained in a set described by polynomial inequalities. First, the paper provides a necessary and sufficient condition for robust first-order consensus based on the eigenvalues of the uncertain Laplacian matrix. Based on this condition, a sufficient condition for robust first-order consensus is derived by solving a linear matrix inequality (LMI) problem built by exploiting sum-of-squares (SOS) polynomials. Then, the paper provides a necessary and sufficient condition for robust second-order consensus through the uncertain expanded Laplacian matrix and Lyapunov stability theory. Based on this condition, a sufficient condition for robust second-order consensus is derived by solving an LMI problem built by exploiting SOS matrix polynomials. Some numerical examples illustrate the proposed results.