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.

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.

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).

An explicit solution to right factorization with applicationin eigenstructure assignment

Based on the well-known Leverrier algorithm, a simple explicit solution to right factorization of a linear system is established. This solution is expressed by the controllability matrix of the given system and a symmetric operator matrix. Applications of this solution to a type of generalized Sylvester matrix equatiorls and the problem of parametric eigenstructure assignment by state feedback are investigated,and general complete parametric solutions to these two problems are deduced. These new solutions are simple, and possess desirable structural properties which render the solutions readily implementable. An example demonstrates the effect of the proposed results.

Eigenstructure assignment of lower-order dynamical compensatorsfor linear systems with unknown inputs

Presents a systematic design method of reduced order dynamical compensator via the parametric representations of eigenstructure assignment for linear system, which provides maximum degree of freedom, and can be easily used for the design of a linear system with unknown inputs under some conditions. Even when these conditions are not satisfied, the lower order dynamical compensator can also be designed under some relaxed conditions. Some examples illustrate that the method is neat, simple and effective.

RESOLUTION PERFORMANCE OF THE EIGENSTRUCTURE APPROACH

The resolution performance for a variety of eigenstructure approaches is analyzedand compared.The expressions of SNR thresholds to resolve two uncorrelated equal powersignals for several typical eigenstructure approaches are given.The simulation results agree withthe theoretical analysis.

A NEW METHOD FOR EIGENSTRUCTURES EXTRACTION AND ITS NEURAL NETWORKS IMPLEMENTATION

The cost function for eigenstructures extraction is discussed in detail in this paper, one can obtain the largest eigenvector by minimizing the cost function. In order to obtain other eigenvectors, a covariance matrix series is constructed. If one compares the cost function with the energy function of a neural networks, the neural networks can be easily introduced to extract the eigenvectors. Theoretical analysis and computer simulations show that the proposed method is reasonable and feasible.

Array Shape Estimation Using Partitioned Eigenstructure Method with Sources in Unknown Localizations

Advanced array processing approaches require accurate knowledge of the location of individual element in a sensor array.Most array shape estimation methods require the directions of sources.In this paper,an array shape estimation method based on eigen-decomposition is presented.The directions of sources do not need to be considered in advance and optimal array shape is generated through a series of iterations.To further improve the accuracy of this algorithm,a partitioned eigenstructure method is introduced.Numerical simulations using non-partitioned and partitioned method are conducted to verify the performance of the proposed technique.

Updating Mass and Stiffness Matrices Using Eigenstructure Assignment Methods

A novel numerical method is presented to update mass and stiffness matrices simultaneously with measured vibration data by means of the combined acceleration and displacement output feedback.By the method,the required displacement and acceleration output feedback gain matrices are determined,and thus the optimal approximation mass matrix and stiffness matrix which satisfy the required orthogonality relation and eigenvalue equation are found.The proposed method is computationally efficient and the updated mass and stiffness matrices are also symmetric and have the compact expressions.The numerical example shows that the proposed method is reliable and attractive.

EIGENSTRUCTURE ASSIGNMENT VIA DYNAMICOUTPUT FEEDBACK

In this papers the problem of eigenstructure assignment via dynamic output feedback is studied. First, the necessary and sufficient assignability conditions via dynamic output feedback are given. Then, an approach to determine the structure and the minimal order of the dynamic output feedback based on the assigned eigenstructure is proposed.Finally, the controllability and the observability of the compensated system are analyzed.

Three-axis coupled flight control law design forflying wing aircraft using eigenstructure assignment method

Due to elimination of horizontal and vertical tails,flying wing aircraft has poor longitudinal and directional dynamic characteristics.In addition,flying wing aircraft uses drag rudders for yaw control,which tends to generate strong three-axis control coupling.To overcome these problems,a flight control law design method that couples the longitudinal axis with the lateraldirectional axes is proposed.First,the three-axis coupled control augmentation structure is specified.In the structure,a‘‘soft/hard"cross-connection method is developed for three-axis dynamic decoupling and longitudinal control response decoupling from the drag rudders;maneuvering turn angular rate estimation and subtraction are used in the yaw axis to improve the directional damping.Besides,feedforward control is adopted to improve the maneuverability and control decoupling performance.Then,detailed design methods for feedback and feedforward control parameters are established using eigenstructure assignment and model following technique.Finally,the proposed design method is evaluated and compared with conventional method by numeric simulations.The influences of control derivatives variation of drag rudders on the method are also analyzed.It is demonstrated that the method can effectively improve the dynamic characteristics of flying wing aircraft,especially the directional damping characteristics,and decouple the longitudinal responses from the drag rudders.

EIGENSTRUCTURE ASSIGNMENT FOR TIME-VARYING LINEAR SYSTEMS

The problem of eigenstructure assignment via output feedback for time-varying linearsystems is treated, and the sufficient and necessary condition and two simple effective algorithmsare presented. Two forms of parametric representations of the set of closed loop eigenvectormatrices and the set of feedback gains with respect to the closed loop eigenvalues and twogroups of parameter vectors are established. Without any constraint to the closed loop poles,the obtained results have generalized many previous results even in the constant case, andhave also overcome their defects.

EIGENSTRUCTURE ASSIGNMENT PROBLEM

There have been many discussions about the eigenstructure assignment problem of linear multivariable time-invariant system in the past years. In [1], we studied the problem by polynomial matrix theory for state feedback case and gave two ways to find the state feedback matrix K. But there was a presumption that the given eigenstructure is assignable.

Global stabilizing controller design for linear time-varying systems and its application on BTT missiles

A parametric method for the gain-scheduled controller design of a linear time-varying system is given. According to the proposed scheduling method, the performance between adjacent characteristic points is preserved by the invariant eigenvalues and the gradually varying eigenvectors. A sufficient stability criterion is given by constructing a series of Lyapunov functions based on the selected discrete characteristic points. An important contribution is that it provides a simple and feasible approach for the design of gain-scheduled controllers for linear time-varying systems, which can guarantee both the global stability and the desired closed-loop performance of the resulted system. The method is applied to the design of a BTT missile autopilot and the simulation results show that the method is superior to the traditional one in sense of either global stability or system performance.

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.

Robust Near-Hovering Flight Controller for Model-Scale Helicopters Via Parametric Approach

This paper aims to provide a parametric design for robust flight controller of the model-scale helicopter. The main contributions lie in two aspects. Firstly,under near-hovering condition,a procedure is presented for simplification of the highly nonlinear and under-actuated model of the model-scale helicopter. This nonlinear system is linearized around the trim values of the chosen flight mode,followed by decomposing this high-order linear model into three lower-order subsystems according to the coupling properties among channels.After decomposition,the three subsystems are obtained which include the coupling subsystem between the roll( pitch) motion and the lateral( longitudinal) motion,the subsystem of the yaw motion and the subsystem of the vertical motion. Secondly,by using eigenstructure assignment,the problem of flight controller design can be converted into solving two optimization problems and the linear robust controllers of these subsystems are designed through solving these optimization problems. Besides, this paper contrasts and analyzed the performances of the LQR controller and the parametric controller. The results demonstrate the effectiveness and the robustness against the parametric perturbations of the parametric controller.

Explicit Determination of State Feedback Matrices

Methods which calculate state feedback matrices explicitly for uncontrollable systems are considered in this paper. They are based on the well-known method of the entire eigenstructure assignment. The use of a particular similarity transformation exposes certain intrinsic properties of the closed loop w-eigenvectors together with their companion z-vectors. The methods are extended further to deal with multi-input control systems. Existence of eigenvectors solution is established. A differentiation property of the z-vectors is proved for the repeated eigenvalues assignment case. Two examples are worked out in detail.

Optimal non-fragile controller realization algorithm and application to control problems

To reduce the fragility encountered in controller implementation, which is a measure of extent to describe small perturbations in controller parameters caused by rounding-off errors or component tolerances, and keep the system stability and performance, approaches of weighted eigenvalue sensitivity and stability radii comparison were used for computation and reduction of controller fragility. An algorithm has been derived for the efficient reduction of controller fragility, which used eigenstructure decomposition to obtain the suboptimal solution. The algorithm was tested for different control problems through reducing their fragility by a large margin. Different canonical forms were analyzed for fragility, including controllable canonical form, observable canonical form, modal canonical form, balanced realization and optimal (non-fragile) form. Different realizations were implemented through C language Matlab EXecutable (CMEX) S-function discrete state space block. Double precision calculations were performed. Open and closed loop controller realizations were compared with simulink state space (optimal) block. Results of comparison indicate that the optimal non-fragile controller realization shows better results both in open loop and closed loop realization.

Robust parametric approach for tracking control of an air-breathing hypersonic cruise vehicle

To realize the stabilization and the tracking of flight control for an air-breathing hypersonic cruise vehicle, the linearization of the longitudinal model under trimmed cruise condition is processed firstly. Furthermore, the flight control problem is formulated as a robust model tracking control problem. And then, based on the robust parametric approach, eigenstructure assignment and reference model tracking theory, a parametric optimization method for robust controller design is presented. The simulation results show the effectiveness of the proposed approach.