Data-driven Estimation for a Region of Attraction for Transient Stability Using the Koopman Operator

This paper presents a novel Koopman Operator based framework to estimate the region of attraction for power system transient stability analysis.The Koopman eigenfunctions are used to numerically construct a Lyapunov function.Then the level set of the function is utilized to estimate the boundary of the region of attraction.The method provides a systematic method to construct the Lyapunov function with data sampled from the state space,which suits any power system models and is easy to use compared to traditional Lyapunov direct methods.In addition,the constructed Lyapunov function can capture the geometric properties of the region of attraction,thus providing useful information about the instability modes.The method has been verified by a simple illustrative example and three power system models,including a voltage source converter interfaced system to analyze the large signal synchronizing instability induced by the phase lock loop dynamics.The proposed method provides an alternative approach to understanding the geometric properties and estimating the boundary of the region of attraction of power systems in a data driven manner.Index Terms-Koopman operator,lyapunov function,power system transient stability,region of attraction.

An on-line constraint softening strategy to guarantee the feasibility of dynamic controller in double-layered MPC

A double-layered model predictive control(MPC), which is composed of a steady-state target calculation(SSTC)layer and a dynamic control layer, is a prevailing hierarchical structure in industrial process control. Based on the reason analysis of the dynamic controller infeasibility, an on-line constraints softening strategy is given. At first, a series of regions of attraction(ROA) of the dynamic controller is calculated according to the softened constraints;then a minimal ROA containing the current state is chosen and the corresponding softened constraint is adopted by the dynamic controller. Note that, the above measures are performed on-line because the centers of the above ROA are the steady-state targets calculated at each instant. The effectiveness of the presented strategy is illustrated through two examples.

Anti-windup-based Dynamic Controller Synthesis for Lipschitz Systems under Actuator Saturation

This paper presents a new method for simultaneous synthesis of dynamic controller and static anti-windup compensator for saturated Lipschitz systems. Thanks to the reformulated Lipschitz property, the Lipschitz systems can be transformed into LPV(linear parameter-varying) systems whose system matrices are affine in a parameter matrix. Based on the modified sector condition dealing with saturation nonlinearity, the design of a nonlinear anti-windup-based controller leads to the solvability of a set of bilinear matrix inequalities(BMI) on the vertices of a bounded convex set which can be solved by the so-called iterative linear matrix inequality(ILMI) algorithm. A numerical example is presented to illustrate the effectiveness of the proposed method.

Global Stability in Dynamical Systems with Multiple Feedback Mechanisms

A class of n-dimensional ODEs with up to n feedbacks from the n’th variable is analysed. The feedbacks are represented by non-specific, bounded, non-negative C1 functions. The main result is the formulation and proof of an easily applicable criterion for existence of a globally stable fixed point of the system. The proof relies on the contraction mapping theorem. Applications of this type of systems are numerous in biology, e.g., models of the hypothalamic-pituitary-adrenal axis and testosterone secretion. Some results important for modelling are: 1) Existence of an attractive trapping region. This is a bounded set with non-negative elements where solutions cannot escape. All solutions are shown to converge to a “minimal” trapping region. 2) At least one fixed point exists. 3) Sufficient criteria for a unique fixed point are formulated. One case where this is fulfilled is when the feedbacks are negative.

Flexible Robotic Grasping Strategy with Constrained Region in Environment

Grasping is a significant yet challenging task for the robots. In this paper, the grasping problem for a class of dexterous robotic hands is investigated based on the novel concept of constrained region in environment, which is inspired by the grasping operations of the human beings. More precisely, constrained region in environment is formed by the environment, which integrates a bio-inspired co-sensing framework. By utilizing the concept of constrained region in environment, the grasping by robots can be effectively accomplished with relatively low-precision sensors. For the grasping of dexterous robotic hands, the attractive region in environment is first established by model primitives in the configuration space to generate offline grasping planning. Then, online dynamic adjustment is implemented by integrating the visual sensory and force sensory information, such that the uncertainty can be further eliminated and certain compliance can be obtained. In the end, an experimental example of BarrettHand is provided to show the effectiveness of our proposed grasping strategy based on constrained region in environment.

Brockett's Second Example:A FAS Approach Treatment

In this paper,several equivalent forms of the well-known Brockett's second example system are firstly presented.The stabilization of the system is then treated in the fully actuated system approach.A simple continuous time-invariant sub-stabilizing controller is designed,and the corresponding region of attraction is characterized.As a result,all trajectories of the system starting from the characterized region of attraction are driven exponentially to the origin.Since the region of attraction is very large,the designed sub-stabilizing controller can be directly useful in many practical situations.In cases where the initial values are indeed needed to be chosen out of the region of attraction,extremely simple pre-controllers can be designed,which drive the system trajectories into the designed region of attraction.A simulation of the designed control system is carried out to show the effect of the proposed approach.

Brockett’s First Example: An FAS Approach Treatment

In this note,the well-known Brockett’s first example system is treated with the fully actuated system(FAS)approach.Firstly,it is shown that the system can be exponentially substabilized by a smooth controller in the sense that,except those starting from initial values on the z0-axis of the initial value space,all trajectories of the designed system as well as the control signals decay to zero exponentially.Secondly,global stabilization is realized through a way of enabling the trajectories starting from initial values on the z0-axis also to go to the origin.The idea is to firstly move an initial point on the z0-axis away from the axis using a pre-controller,and then to take over by the designed exponentially sub-stabilizing controller.

Stabilization via Fully Actuated System Approach:A Case Study

In this note,a benchmark example system which is not stabilizable by a smooth state feedback controller is considered with the fully actuated system(FAS)approach.It is shown that a smooth controller exists which drives the trajectories starting from a large domain in the initial value space to the origin exponentially.Such a result brings about a generalization of Lyapunov asymptotical stability,which is termed as global exponential sub-stability.The region of attraction is allowed to be an unbounded open set of the initial values with closure containing the origin.This sub-stability result may be viewed to be superior to some local stability results in the Lyapunov sense because the region of attraction is much larger than any finite ball containing the origin and meanwhile the feasible trajectories are always driven to the origin exponentially.Based on this sub-stabilization result,globally asymptotically stabilizing controllers for the system can be provided in two general ways,one is through combination with existing globally stabilizing controllers,and the other is by using a pre-controller to first move an initial point which is not within the region of attraction into the region of attraction.

Analysis of Fractional-order Linear Systems with Saturation Using Lyapunov's Second Method and Convex Optimization

In this paper,local stability and performance analysis of fractional-order linear systems with saturating elements are shown,which lead to less conservative information and data on the region of stability and the disturbance rejection.Then,a standard performance analysis and global stability by using Lyapunov’s second method are addressed,and the introduction of Lyapunov’s function candidate whose sub-level set provide stability region and performance with a restricted state space origin is also addressed.The results include both single and multiple saturation elements and can be extended to fractional-order linear systems with any nonlinear elements and nonlinear noise that satisfy Lipschitz condition.A noticeable application of these techniques is analysis of control fractional-order linear systems with saturation control inputs.

Exponential Nonlinear Observer Based on the Differential State-dependent Riccati Equation

This paper presents a novel nonlinear continuous-time observer based on the differential state-dependent Riccati equation (SDRE) filter with guaranteed exponential stability. Although impressive results have rapidly emerged from the use of SDRE designs for observers and filters, the underlying theory is yet scant and there remain many unanswered questions such as stability and convergence. In this paper, Lyapunov stability analysis is utilized in order to obtain the required conditions for exponential stability of the estimation error dynamics. We prove that under specific conditions, the proposed observer is at least locally exponentially stable. Moreover, a new definition of a detectable state-dependent factorization is introduced, and a close relation between the uniform detectability of the nonlinear system and the boundedness property of the state-dependent differential Riccati equation is established. Furthermore, through a simulation study of a second order nonlinear model, which satisfies the stability conditions, the promising performance of the proposed observer is demonstrated. Finally, in order to examine the effectiveness of the proposed method, it is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines. The simulation results verify how effectively this modification can increase the region of attraction and the observer error decay rate.

Envelope protection for aircraft encountering upset condition based on dynamic envelope enlargement

Upset condition encountered by an aircraft in flight could pose great threat to flight safety, which is of chief importance in civil aviation. The causal factors have the nonlinear and multiple characteristics, and as a result the conventional envelope protection system cannot successfully do with the condition. So dynamic envelope based on differential manifold theory, which can take more coupling factors into account, is proposed as a basis to design a novel envelope protection system. Then the relationship between the dynamic envelope and the control coefficient or pilot command is obtained, and the result shows that the dynamic envelope can be enlarged with the change of control coefficient. Furthermore, quantification of flight security is realized via defining relative distance between flight state and dynamic envelope, which can detect whether there is a risk for an aircraft in flight. Finally, an envelope protection system based on dynamic envelope enlargement is proposed on the basis. NASA＇s Generic Transport Model encountering hazard gust wind in climbing phase is taken as an example to verify the system＇s feasibility. The result shows that the system can give a better operation encountering upset condition and to a certain extent reduce the number of accidents or incidents.