Two parametric approaches for eigenstructure assignment in second-order linear systems

This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closed-loop eigenvalues to be set undetermined and sought via certain optimization procedures. An example shows the effectiveness of the proposed approaches. Keywords Second-order linear systems - Eigenstructure assignment - Proportional plus derivative feedback - Parametric solution - Singular value decompoition - Right factorization This work was supported in part by the Chinese Outstanding Youth Foundation (No.69504002).

Design of Luenberger function observer with disturbance decoupling for matrix second-order linear systems-a parametric approach

A simple method for disturbance decoupling for matrix second-order linear systems is proposed directly in matrix second-order framework via Luenberger function observers based on complete parametric eigenstructure assignment. By introducing the H2 norm of the transfer function from disturbance to estimation error, sufficient and necessary conditions for disturbance decoupling in matrix second-order linear systems are established and are arranged into constraints on the design parameters via Luenberger function observers in terms of the closed-loop eigenvalues and the group of design parameters provided by the eigenstructure assignment approach. Therefore, the disturbance decoupling problem is converted into an eigenstructure assignment problem with extra parameter constraints. A simple example is investigated to show the effect and simplicity of the approach.

Unified parametric approaches for high-order integral observer design for matrix second-order linear systems

A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parametric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the fight factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.

Observer design for matrix second-order linear systems: a parametric approach

The issue of designing a type of generalized Luenberger observers for matrix second-order linear (MSOL) systems was addressed in the matrix second-order framework. By introducing the concept of stable matrix pair for MSOL systems, sufficient and necessary conditions for the design of the type of generalized Luenberger observers were given under the assumption of controllability and observability of the MSOL system. Based on the proposed conditions and the right coprime factorization of the system, a parametric approach to the design of such type of observers was presented. The proposed approach provides all the degrees of design freedom, which can be further utilized to achieve additional system specifications. A spring-mass system was utilized to show the effect of the proposed method.

Parametric Approach for the Normal Luenberger Function Observer Design in Second-order Descriptor Linear Systems

In this paper, the normal Luenberger function observer design for second-order descriptor linear systems is considered. It is shown that the main procedure of the design is to solve a so-called second-order generalized Sylvester-observer matrix equation. Based on an explicit parametric solution to this equation, a parametric solution to the normal Luenberger function observer design problem is given. The design degrees of freedom presented by explicit parameters can be further utilized to achieve some additional design requirements.

Eigenstructure Assignment in a Class of Second-order Descriptor Linear Systems: A Complete Parametric Approach

In this paper eigenstructure assignment via proportional-plus-derivative feedback is investigated for a class of second-order descriptor linear systems. Under certain conditions, simple, general and complete parametric solutions of both finite closed-loop eigenvector matrices and feedback gain matrices are derived. The parametric approach utilizes directly original system data, involves manipulations only on n-dimensional matrices, and reveals all the design degrees of freedom which can be further utilized to achieve certain additional system specifications. A numerical example shows the effect of the proposed approach.

Distributed Containment Control of Networked Nonlinear Second-order Systems With Unknown Parameters

In this paper, we study the containment control problem for nonlinear second-order systems with unknown parameters and multiple stationary/dynamic leaders. The topologies that characterize the interaction among the leaders and the followers are directed graphs. Necessary and sufficient criteria which guarantee the control objectives are established for both stationary leaders(regulation case) and dynamic leaders(dynamic tracking case) based protocols. The final states of all the followers are exclusively determined by the initial values of the leaders and the topology structures. In the regulation case, all the followers converge into the convex hull spanned by the leaders,while in the dynamic tracking case, not only the positions of the followers converge into the convex hull but also the velocities of the followers converge into the velocity convex hull of the leaders.Finally, all the theoretical results are illustrated by numerical simulations.

STABILITY AND FEEDBACK CONTROL OF A CLASS OF SECOND-ORDER LINEAR SYSTEMS IN BANACH SPACE

In this paper. we are concerned with the stability and control problem for aclass of second-order linear systems in Banach space. First. a criterion for the exponentialstability of a first-order linear system is presented. Then. the exponential stability as wellas some properties of a class of second-order linear systems is proved. At last. the feedbackcontrol of this class of systems is investigated.

Hilbert-Schmidt-Hankel norm model reduction for matrix second-order linear systems

This paper considers the optimal model reduction problem of matrix second-order linear systems in the sense of Hilbert-Schmidt-Hankel norm, with the reduced order systems preserving the structure of the original systems. The expressions of the error function and its gradient are derived. Two numerical examples are given to illustrate the presented model reduction technique.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

Optimal Control of Nonlinear Systems Using Experience Inference Human-Behavior Learning

Safety critical control is often trained in a simulated environment to mitigate risk.Subsequent migration of the biased controller requires further adjustments.In this paper,an experience inference human-behavior learning is proposed to solve the migration problem of optimal controllers applied to real-world nonlinear systems.The approach is inspired in the complementary properties that exhibits the hippocampus,the neocortex,and the striatum learning systems located in the brain.The hippocampus defines a physics informed reference model of the realworld nonlinear system for experience inference and the neocortex is the adaptive dynamic programming(ADP)or reinforcement learning(RL)algorithm that ensures optimal performance of the reference model.This optimal performance is inferred to the real-world nonlinear system by means of an adaptive neocortex/striatum control policy that forces the nonlinear system to behave as the reference model.Stability and convergence of the proposed approach is analyzed using Lyapunov stability theory.Simulation studies are carried out to verify the approach.

Group-Consensus of Hierarchical Containment Control for Linear Multi-Agent Systems

The existing containment control has been widely developed for several years, but ignores the case for large-scale cooperation. The strong coupling of large-scale networks will increase the costs of system detection and maintenance. Therefore, this paper is concerned with an extensional containment control issue, hierarchical containment control. It aims to enable a multitude of followers achieving a novel cooperation in the convex hull shaped by multiple leaders. Firstly, by constructing the three-layer topology, large-scale networks are decoupled. Then,under the condition of directed spanning group-tree, a class of dynamic hierarchical containment control protocol is designed such that the novel group-consensus behavior in the convex hull can be realized. Moreover, the definitions of coupling strength coefficients and the group-consensus parameter in the proposed dynamic hierarchical control protocol enhance the adjustability of systems. Compared with the existing containment control strategy, the proposed hierarchical containment control strategy improves dynamic control performance. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed hierarchical control protocol.

Two-Parameter Block Triangular Splitting Preconditioner for Block Two-by-Two Linear Systems

This paper proposes a two-parameter block triangular splitting(TPTS)preconditioner for the general block two-by-two linear systems.The eigenvalues of the corresponding preconditioned matrix are proved to cluster around 0 or 1 under mild conditions.The limited numerical results show that the TPTS preconditioner is more efficient than the classic block-diagonal and block-triangular preconditioners when applied to the flexible generalized minimal residual(FGMRES)method.

Event-Triggered Finite-Time H∞ Filtering for Discrete-Time Nonlinear Stochastic Systems

This paper addresses the problem of event-triggered finite-time H∞ filter design for a class of discrete-time nonlinear stochastic systems with exogenous disturbances. The stochastic Lyapunov-Krasoviskii functional method is adopted to design a filter such that the filtering error system is stochastic finite-time stable (SFTS) and preserves a prescribed performance level according to the pre-defined event-triggered criteria. Based on stochastic differential equations theory, some sufficient conditions for the existence of H∞ filter are obtained for the suggested system by employing linear matrix inequality technique. Finally, the desired H∞ filter gain matrices can be expressed in an explicit form.

Cluster consensus of second-order multi-agent systems via pinning control

This paper investigates the cluster consensus problem for second-order multi-agent systems by applying the pinning control method to a small collection of the agents. Consensus is attained independently for different agent clusters according to the community structure generated by the group partition of the underlying graph and sufficient conditions for both cluster and general consensus are obtained by using results from algebraic graph theory and the LaSalle Invariance Principle. Finally, some simple simulations are presented to illustrate the technique.

Eigenstructure assignment in a class of second-order dynamic systems

Eigenstructure assignment using the proportional-plus-derivative feedback controller in a class of secondorder dynamic system is investigated. Simple, general, complete parametric expressions for both the closed-loop eigenvector matrix and the feedback gains are established based on two simple Smith form reductions. The approach utilizes directly the original system data and involves manipulations only on n-dimensional matrices. Furthermore, it reveals all the degrees of freedom which can be further utilized to achieve additional system specifications. An example shows the effect of the proposed approach.

Distributed event-triggered consensus tracking of second-order multi-agent systems with a virtual leader

This paper investigates the consensus tracking problems of second-order multi-agent systems with a virtual leader via event-triggered control. A novel distributed event-triggered transmission scheme is proposed, which is intermittently examined at constant sampling instants. Only partial neighbor information and local measurements are required for event detection. Then the corresponding event-triggered consensus tracking protocol is presented to guarantee second-order multi-agent systems to achieve consensus tracking. Numerical simulations are given to illustrate the effectiveness of the proposed strategy.

Consensus of Second-order Multi-agent Systems with Nonsymmetric Interconnection and Heterogeneous Delays

In this paper, a consensus algorithm of multi-agent second-order dynamical systems with nonsymmetric interconnection and heterogeneous delays is studied. With the hypothesis of directed weighted topology graph with a globally reachable node, decentralized consensus condition is obtained by applying generalized Nyquist criterion. For the systems with both communication and input delays, it is shown that the consensus condition is dependent on input delays but independent of communication delays.

Reference model based consensus control of second-order multi-agent systems

This paper deals with the consensus problem of multi-agent systems with second-order dynamics. The objective is to design algorithms such that all agents will have same positions and velocities. First, a reference model based consensus algorithm is proposed. It is proved that the consensus can be achieved if the communication graph has a spanning tree. Different from most of the consensus algorithms proposed in the literature, the parameters of the control laws are different among agents. Therefore, each agent can design its control law independently. Secondly, it gives a consensus algorithm for the case that the velocities of the agents are not available. Thirdly, the effectiveness of the input delay and the communication delay is considered. It shows that consensus can be achieved if the input delay of every agent is smaller than a bound related to parameters in its control law. Finally, some numerical examples are given to illustrate the proposed results.

Distributed Fault Detection for Consensus in Second-Order Discrete-Time Multiagent Systems with Adversary

This paper is concerned with distributed fault detection of second-order discrete-time multi-agent systems with adversary,where the adversary is regarded as a slowly time-varying signal.Firstly,a novel intrusion detection scheme based on the theory of unknown input observability( UIO) is proposed. By constructing a bank of UIO,the states of the malicious agents can be directly estimated. Secondly,the faulty-node-removal algorithm is provided.Simulations are also provided to demonstrate the effectiveness of the theoretical results.