Linear Differential Inclusions Research Articles

Adaptive optimal controller design for a class of LDI-based neural network systems with input time-delays

In this paper, a new online adaptive optimal controller design scheme is studied for a class of nonlinear systems with input time-delays. First, we linearize the original nonlinear systems by means of linear differential inclusion technique. Then the adaptive optimal controller of the linearized systems with input time-delays is obtained by online policy iteration algorithm. It also proves the convergence of the designed adaptive optimal control algorithm. Finally, the effectiveness of the proposed method is verified by two simulation examples.

Intelligent fuzzy algorithm for nonlinear discrete-time systems

This paper is concerned with the stability analysis and the synthesis of model-based fuzzy controllers for a nonlinear large-scale system. In evolved fuzzy NN (neural network) modeling, the NN model and LDI (linear differential inclusion) representation are established for the arbitrary nonlinear dynamics. The evolved bat algorithm (EBA) is first incorporated in the controlled algorithm of stability conditions, which could rapidly find the optimal solution and raise the control performance. This representation is constructed by taking advantage of sector nonlinearity that converts the nonlinear model to a multiple rule base linear model. A new sufficient condition guaranteeing asymptotic stability is implemented via the Lyapunov function in terms of linear matrix inequalities. Subsequently, based on this criterion and the decentralized control scheme, an evolved model-based fuzzy H infinity set is synthesized to stabilize the nonlinear large-scale system. Finally, a numerical example and simulation is given to illustrate the results.

Gain-Scheduled Control of Linear Differential Inclusions Subject to Actuator Saturation

This paper focuses on the problem of robust stabilization of linear differential inclusions subject to actuator saturation. A continuous set of nonlinear controllers is constructed by a parameter-dependent convex hull Lyapunov function to avoid actuator saturation for known worst-case disturbances. The controller, which can achieve the best closed-loop performance while complies the saturation bound, is selected at each time, based on the closed-loop states. Thanks to the application of the continuous dynamic gain-scheduled control law, the internal stability and guaranteed disturbance attenuation can be obtained simultaneously. A quarter-car active suspension system is studied to demonstrate the benefit of the proposed method.

Online policy iterative-based H∞ optimization algorithm for a class of nonlinear systems

A novel policy iterative scheme for the design of online H∞ optimal laws for a class of nonlinear systems is presented. First, neural network-based linear differential inclusion techniques with two multi-layered perceptions are applied to linearize the nonlinear terms. Then, an online partially model-free policy iterative scheme is applied to the linearized system to obtain the design the H∞ optimal control law. The iterative scheme for the linear H∞ control problem consists of policy evaluation and policy improvement by means of algebraic Riccati equations. We establish the convergence of the novel policy iterative scheme to the optimal control law. Numerical simulations demonstrating the feasibility and applicability of our design algorithm are provided.

Adaptive Optimal Control for a Class of Nonlinear Systems: The Online Policy Iteration Approach.

This paper studies the online adaptive optimal controller design for a class of nonlinear systems through a novel policy iteration (PI) algorithm. By using the technique of neural network linear differential inclusion (LDI) to linearize the nonlinear terms in each iteration, the optimal law for controller design can be solved through the relevant algebraic Riccati equation (ARE) without using the system internal parameters. Based on PI approach, the adaptive optimal control algorithm is developed with the online linearization and the two-step iteration, i.e., policy evaluation and policy improvement. The convergence of the proposed PI algorithm is also proved. Finally, two numerical examples are given to illustrate the effectiveness and applicability of the proposed method.

Evolved Fuzzy NN Control for Discrete-Time Nonlinear Systems

To guarantee the asymptotic stability of discrete-time nonlinear systems, this paper proposes an Evolved Bat Algorithm (EBA) fuzzy neural network (NN) controller. In the evolved fuzzy NN modeling, an NN model and linear differential inclusion (LDI) representation are established for arbitrary nonlinear dynamics. This representation is constructed by the use of sector nonlinearity to convert a nonlinear model to the multiple rule base of the linear model, and a new sufficiency condition to guarantee asymptotic stability in the Lyapunov function is implemented in terms of linear matrix inequalities. The proposed method is an enhancement of existing methods which produces good results.

Small‐signal stability analysis and control of stochastic time‐variant power system through differential inclusion theory

Here, small-signal stability issue associated with multi-scale uncertainty excitation is investigated by using differential inclusion theory. Specifically, a polytopic linear differential inclusion (PLDI) model for large-scale uncertainty system is developed, in which a modified non-sequence Monte Carlo method is introduced to identify a series of time-variant operation states. Additionally, a simplified small-signal model of renewable energy resources (RES) is proposed, the outputs of RES are modelled as time-varying elements in PLDI model to reflect the uncertainty in the linearised matrix. The stability criterion for the stochastic time-varying system is mathematically deduced based on convex hull Lyapunov function (CHLF). The criterion is then utilised to design robust stabilisers for stochastic system. Simulation, utilising two-area four-machine system and New England 39-bus test system, demonstrates the benefits of the proposed model in describing system stochastic characteristics, designing additional controllers and reducing computational burden.

Small‐signal stability analysis of Energy Internet through differential inclusion theory

In this study, small-signal stability issue associated with uncertainty excitation is investigated by using differential inclusion theory. Specifically, a polytopic linear differential inclusion (PLDI) model for Energy Internet is developed, in which a modified non-sequence Monte Carlo method is introduced to identify a series of time-variant operation states. Additionally, a simplified small-signal model of renewable energy sources (RES) with virtual synchronous generator control is proposed, and the outputs of RES are modelled as time-varying elements in PLDI model to reflect the inner stochastic excitation in the linearised matrix. The stability criterion for the stochastic time-varying system is mathematically deduced based on convex hull Lyapunov function. Simulation demonstrates the benefits of the proposed model in describing system stochastic characteristics and reducing computational burden.

Coordinated Control of Multiple Virtual Synchronous Generators in Mitigating Power Oscillation

Virtual synchronous generators (VSGs) present attractive technical advantages and contribute to enhanced system operation and reduced oscillation damping in dynamic systems. Traditional VSGs often lack an interworking during power oscillation. In this paper, a coordinated control strategy for multiple VSGs is proposed for mitigating power oscillation. Based on a theoretical analysis of the parameter impact of VSGs, a coordinated approach considering uncertainty is presented by utilizing polytopic linear differential inclusion (PLDI) and a D-stable model to enhance the small-signal stability of system. Subsequently, the inertia and damping of multiple VSGs are jointly exploited to reduce oscillation periods and overshoots during transient response. Simulation, utilizing a two-area four-machine system and a typical microgrid test system, demonstrates the benefits of the proposed strategy in enhancing operation stability and the anti-disturbing ability of multiple VSGs. The results conclusively confirm the validity and applicability of the method.

H∞ output tracking control of Electric-motor-driven aerodynamic Load Simulator with external active motion disturbance and nonlinearity

The work presented in this paper seeks to address the tracking problem for an Electric-motor-driven aerodynamic Load Simulator (ELS). The tracking performance of ELS is mainly affected by the actuator'sactive motion disturbance, friction nonlinear, and parametric uncertainties. Most of the previous studies on ELS focus on the actuator's active motion disturbance, while deemphasizing the other two factors. This paper concerns ELS as a motion loading system. A new high performancecontrol scheme is developed to deal with the three factors based on the established nonlinear mode of ELS. The control scheme is based on neural networks and the Linear Difference Inclusions (LDI) model. Aparallel distributed compensation (PDC) structure and H∞ performance criterion are used to attenuate external disturbances. The stability of the whole closed-loop model is investigated using the well-known quadratic Lyapunovfunction. Both asimulation and an experiment are performed to validate the effectiveness of the developed algorithm.

Step-By-Step Averaging of Linear Differential Inclusions of Variable Dimension on a Finite Interval

We consider linear differential inclusions of variable dimension and substantiate the possibility of their step-by-step averaging on a finite interval.

Optimization of Lagrange problem with higher order differential inclusions and endpoint constraints

In the paper minimization of a Lagrange type cost functional over the feasible set of solutions of higher order differential inclusions with endpoint constraints is studied. Our aim is to derive sufficient conditions of optimality for m th-order convex and non-convex differential inclusions. The sufficient conditions of optimality containing the Euler-Lagrange and Hamiltonian type inclusions as a result of endpoint constraints are accompanied by so-called ?endpoint? conditions. Here the basic apparatus of locally adjoint mappings is suggested. An application from the calculus of variations is presented and the corresponding Euler-Poisson equation is derived. Moreover, some higher order linear optimal control problems with quadratic cost functional are considered and the corresponding Weierstrass-Pontryagin maximum principle is constructed. Also at the end of the paper some characteristic features of the obtained result are illustrated by example with second order linear differential inclusions.

Invariant convex approximations of the minimal robust invariant set for linear difference inclusions

In this article, we propose a new computationally efficient algorithm for computing an outer convex robust positively invariant (RPI) approximation to the minimal robust positively invariant (mRPI) set for polytypic linear difference inclusion (PLDI) systems with additive disturbances. The LDI modelling framework is useful to analyse parametrically uncertain, time-varying linear system or switching linear discrete-time systems. The disturbance which is considered in this paper, is bounded by a polytypic set and acts additively on the state of the system. It is also assumed that the nominal LDI system is absolutely asymptotically stable by a stabilizing linear state feedback. The accuracy of the approximation can be set in advance. The proposed algorithm has far less computational burden in comparison with existing algorithms.

Step average linear differential inclusions of variable dimension

Step average linear differential inclusions of variable dimension

Switching Antiwindup Design on Enlarging the Domain of Attraction for a Supercavitating Vehicle Subject to Actuator Saturation

This paper presents a new switching antiwindup compensation design to maximize the domain of attraction for a supercavitating vehicle subject to actuator saturation. The dive-plane dynamics of the vehicle are considered. By applying the linear differential inclusion expression of saturated feedbacks, conditions under which the compensator locally stabilizes the closed-loop system are then derived. The design of antiwindup gains on maximizing the system's domain of attraction is finally formulated and solved as an iterative optimization problem with linear matrix inequality constraints. Simulations are conducted for systems with magnitude and rate limits to evaluate the effectiveness of the proposed method.

On practical stability of differential inclusions using Lyapunov functions

In this paper we consider the problem of practical stability for differential inclusions. We prove the necessary and sufficient conditions using Lyapunov functions. Then we solve the practical stability problem of linear differential inclusion with ellipsoidal righthand part and ellipsoidal initial data set. In the last section we apply the main result of this paper to the problem of practical stabilization.

Existence of Semi Linear Impulsive Neutral Evolution InclusionsWith Infinite Delay in Frechet Spaces

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct.

Properties of Composite Laplacian Quadratics and Their Applications in Consensus of Linear Differential Inclusions

This technical note introduces a non-quadratic function that is defined in terms of multiple Laplacian quadratics, called the function of composite Laplacian quadratics. The function is mainly used for control synthesis and analysis of multi-agent systems. Some important properties of the function are studied. In particular, the one-level-set dissipation property is established, and the gradient of the function is derived. The function is further applied to solve the consensus problem of linear differential inclusions (LDIs). Finally, a numerical example is given to verify the validity of the derived results.

On the optimal parameter of the composite Laplacian quadratics function

Recently, the composite quadratic Lyapunov function has been extended to study of multi-agent systems, leading to the so-called composite Laplacian quadratics (CLQs) function. Compared with quadratic Lyapunov functions, the CLQs function can yield a larger convergence region and is particularly useful in stabilization of multi-agent systems with complex dynamics, such as differential inclusions. In the definition of the CLQs function, an optimal vector parameter plays a critical role in determining the value of the CLQs function and in constructing stabilization laws derived from the CLQs function. This paper focuses on the properties of the optimal parameter of the CLQs function. The uniqueness of the optimal parameter is established. A distributed computation approach is further proposed, which is useful in computing the optimal parameter. The robustness issue of the optimal parameter is also investigated for a multi-agent system described by linear differential inclusions. Finally, a numerical example is provided to validate the proposed theoretical results.

Reach Control Problem for Linear Differential Inclusion Systems on Simplices

This technical note concerns the reach control problem for the dynamical systems represented by linear differential inclusions. The goal is to derive conditions for the trajectories of a differential inclusion defined on a full-dimensional simplex to reach exit facets in finite time using affine feedback. To achieve this goal, an invariance condition is firstly derived for Lipschitz differential inclusions. As an application of the obtained invariance condition, the reach control problem (RCP) is solved for two classes of differential inclusions: norm-bounded linear differential inclusion and polytypic linear differential inclusion, in the sense of strong and weak reachability, respectively.