Partial Eigenstructure Assignment Research Articles

Adaptive Optimal Terminal Sliding Mode Control for T-S Fuzzy-Based Nonlinear Systems

This study utilizes the Takagi–Sugeno fuzzy model to represent a subset of nonlinear systems and presents an innovative adaptive approach for optimal dynamic terminal sliding mode control (TSMC). The systems under consideration encompass bounded uncertainties in parameters and actuators, as well as susceptibility to external disturbances. Performance evaluation entails the design of an adaptive terminal sliding surface through a two-step process. Initially, a state feedback gain and controller are developed using Linear Matrix Inequality (LMI) techniques, grounded on H2-performance and partial eigenstructure assignment. Dynamic sliding gain is subsequently attained via convex optimization, leveraging the derived state feedback gain and the designed terminal sliding mode (TSM) controller. This approach diverges from conventional methods by incorporating control effort and estimating actuator uncertainty bounds, while also addressing sliding surface and TSM controller design intricacies. The TSM controller is redefined into a strict feedback form, rendering it suitable for addressing output-tracking challenges in nonlinear systems. Comparative simulations validate the effectiveness of the proposed TSM controller, emphasizing its practical applicability.

Minimum norm partial eigenstructure assignment approach for vibration system with time delay

AbstractIn this paper, the minimum norm partial eigenstructure assignment problems (PESAPs) in vibration system with time delay is studied via acceleration‐velocity‐displacement active control. A new explicit solution is given for solving PESAPs using orthogonality relations of symmetric vibration system. A new constructive technique is presented to solve the PESAPs in vibration system with time delay by singular value decomposition method. An optimization algorithm is proposed to solve the minimum norm PESAPs by a quasi‐Newton optimization method. The proposed algorithm works without using the knowledge of receptance matrices and without solving the Sylvester equation. The effectiveness of the proposed algorithm is verified by the results of two numerical examples.

Output Tracking Control of a Nonlinear System Based on Takagi-Sugeno Fuzzy Model: Generalized Partial Eigenstructure Assignment Approach

Output Tracking Control of a Nonlinear System Based on Takagi-Sugeno Fuzzy Model: Generalized Partial Eigenstructure Assignment Approach

Partial Eigenstructure Assignment for Linear Time-Invariant Systems via Dynamic Compensator

This article studies the partial eigenstructure assignment (PEA) problem for a type of linear time-invariant (LTI) system. By introducing a dynamic output feedback controller, the closed-loop system is similar to a given arbitrary constant matrix, so the desired closed-loop eigenstructure can be obtained. Different from the normal eigenstructure assignment, only a part of the left and right generalized eigenvectors is assigned to the closed-loop system to remove complicated constraints, which reflects the partial eigenstructure assignment. Meanwhile, based on the solutions to the generalized Sylvester equations (GSEs), two arbitrary parameter matrices representing the degrees of freedom are presented to obtain the parametric form of the coefficient matrices of the dynamic compensator and the partial eigenvector matrices. Finally, an illustrative example and the simulation results prove the excellent effectiveness and feasibility of parametric method we proposed.

Parametric design method for partial eigenstructure assignment of second-order linear systems via observer-based state feedback

In this article, a parametric method of observer-based partial eigenstrunt in second-order linear time-invariant (LTI) systems is explored. First, an observer is constructed to approximate the full sate vector when the entire state vector is not available for measurement. Meanwhile, the separation principle of the first-order linear systems is extended to second-order linear systems. Second, by partitioning the system matrix of the open-loop system into two parts, the satisfactory and unsatisfactory eigenstructures are selected. On the premise of retaining the satisfactory part in the open-loop system, the unsatisfactory eigenstructure assignment problem can be converted into the solutions of a type of generalized Sylvester equations (GSE). Third, by introducing a group of arbitrary parameters, complete parametric expressions of the proportional plus derivative (PD) state feedback controller can be obtained. The main advantage of the proposed method is that it provides all the design degrees of freedom and reduces the design complexity of the controller. Finally, the correctness of the proposed method is verified by two examples and simulation results. Using the degrees of freedom of the parameters, the gain matrices are optimized through the optimization index provided by the optimization function, and better performance is obtained.

Partial eigenstructure assignment for descriptor high-order linear systems via proportional plus derivative state feedback: A parametric approach

In this paper, we present a parametric method to design proportional plus derivative (PD) state feedback for the problem of partial eigenstructure assignment (PESA) in a type of descriptor high-order linear time-invariant (LTI) systems. By dividing the original eigenstructure into replaced and remaining parts and using the solutions to the high-order generalized Sylvester equations (HGSEs), complete parameterized expressions of PD feedback controller and the replaced part of the right eigenvector matrix are established. With the proposed approach, the unsatisfactory eigenstructure in the open-loop system is changed into the expected eigenstructure in the closed-loop system, while the satisfactory part in the open-loop system is retained. Meanwhile, the regularity of the closed-loop system can also be well-guaranteed by the selection of arbitrary parameters. Finally, a numerical example and a flexible-joint robot system are presented to verify the feasibility and effectiveness of the proposed method, respectively.

Partial Eigenstructure Assignment Using Incomplete Measured Modal Data

Partial Eigenstructure Assignment Using Incomplete Measured Modal Data

Numerical approach for partial eigenstructure assignment problems in singular vibrating structure using active control

In the paper, the partial eigenstructure assignment problems are investigated using acceleration–velocity–displacement active control in a singular vibrating structure. The problems are transformed into solving matrix equations using the receptance matrix method. Iterative sequences are constructed, and the iterative feasibility is presented for solving the matrix equations. The partial eigenvectors of the closed-loop system are reassigned by imposing modal constraints. An algorithm is proposed to get numerical solutions of the derived matrix equations. The initial value condition is discussed to obtain the minimum norm solution of the partial eigenstructure assignment problems. The designed acceleration–velocity–displacement active control can solve the partial eigenstructure assignment problems depending only on original vibrating structure information. The proposed numerical algorithm can obtain the minimum norms of controller gain, which implies minimum energy consumption. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Receptance-based partial eigenstructure assignment by state feedback control

The partial eigenvalue assignment problem has drawn much attention for decades because of its usefulness and fascinating appeal in vibration control. It is also a mathematically challenging problem, particularly when the mathematical description is cast in the second-order formulation framework. Previous methods require either the system matrices, such as mass, damping or stiffness matrix, or the eigenvectors of the original system. Those data, especially the system matrices which are usually obtained from a finite element model, are not easy to obtain or very accurate. To overcome this drawback, a new partial eigenvalue assignment method by multi-input active control is proposed in this paper. Only part of the receptance matrix of the original system is required in this method. In addition, this method can simultaneously assign eigenvalues and the associated eigenvectors, including assigning nodes at desired locations. Three numerical examples are used to validate the proposed method and demonstrate the role of the control efforts. The robustness of the proposed method is analysed through a Monte-Carlo simulation. Numerical results show that this method is robust for the four-DoF damped system where there are 5% variations of the needed receptance matrix elements. This paper reports the first attempt to make partial eigenstructure assignment in the second-order eigenvalue framework using only the receptances.

Robust minimum norm partial eigenstructure assignment approach in singular vibrating structure via active control

Robust minimum norm partial eigenstructure assignment approach in singular vibrating structure via active control

Minimum norm partial eigenstructure assignment problems in high‐order system via feedback control

AbstractIn this article, the minimum norm partial eigenstructure assignment problem (MNPESAP) is considered in a symmetric high‐order system using feedback control law with the highest‐order derivative term. Two new orthogonality relations are constructed using the system matrices and undetermined feedback gains. The parametric expressions of the feedback controller and the necessary and sufficient condition for solving the partial eigenstructure assignment problem are derived. The Sylvester matrix equations and gradient formula are presented for solving MNPESAP. An algorithm is proposed to obtain the minimum norm solutions of feedback gains using gradient‐based optimization method. Numerical examples are provided to verify the effectiveness of given results.

A homotopy transformation method for interval-based model updating of uncertain vibrating systems

This paper proposes a novel indirect, two-stage approach for model updating in linear vibrating systems, exploiting measured natural frequencies, antiresonances and uncertain or incomplete data of the mode shapes. In the first stage, the technique relies on the partial eigenstructure assignment paradigm and recasts model updating into a non-linear, non-convex minimization that simultaneously updates the mass and stiffness matrices. Uncertainty on mode shapes is formulated through interval (bounds) where they should belong. The inverse eigenvalue problem is solved through homotopy optimization, improved through variables lifting and McCormick's constraints. The unknown parameters are normalized introducing Jacobian matrices, computed through the complex step derivatives, to improve the numerical conditioning of the problem and speed up the computation. In the second stage, the damping matrix is identified through the generalized formulation of proportional damping provided by the Caughey's series.Two challenging experimental test-cases are solved and prove the method effectiveness: the linearized multibody model of a flexible manipulator and a structure made by a cantilever beam plus a lumped spring-mass system.

A parametric approach of partial eigenstructure assignment for high-order linear systems via proportional plus derivative state feedback

<abstract><p>In this paper, a partial eigenstructure assignment problem for the high-order linear time-invariant (LTI) systems via proportional plus derivative (PD) state feedback is considered. By partitioning the open-loop system into two parts (the altered part and the unchanged part) and utilizing the solutions to the high-order generalized Sylvester equation (HGSE), complete parametric expressions of the feedback gain matrices of the closed-loop system are established. Meanwhile, a group of arbitrary parameters representing the degrees of freedom of the proposed method is provided and optimized to satisfy the stability of the system and robustness criteria. Finally, a numerical example and a three-axis dynamic flight motion simulator system example with the simulation results are offered to illustrate the effectiveness and superiority of the proposed method.</p></abstract>

Partial eigenstructure assignment problem for vibration system via feedback control

AbstractThe partial eigenstructure assignment problem is to replace a small number of eigenvalues and eigenvectors of system while the remaining eigenvalues and corresponding eigenvectors are kept unchanged. The paper studies partial eigenstructure assignment problem for vibration system via acceleration–velocity feedback control. Two new orthogonality relations are established to construct a feedback control law which can solve the proposed partial eigenstructure assignment problem. The solvability condition and parametric expressions of feedback gains are derived for solving this problem. The minimum norm partial eigenstructure assignment problem is further considered by using gradient‐based optimization method. An optimization algorithm is proposed for solving the minimum norm partial eigenstructure assignment problem. Numerical examples are provided to show the effectiveness of the algorithm.

Application of partial eigenvalue assignment techniques to dampen electromechanical oscillations in multi-machine power systems

Abstract In this study, a methodology for partial eigenstructure assignment (PEVA) is applied to dampen electromechanical oscillations in electrical multi-machine power systems. The approach is anchored in allocating a small number of undesirable eigenvalues, for example, which are poorly damped, preserving the other eigenvalues in the system - the so-called no-spillover spectrum. The new position of the selected eigenvalues is carried out based on the partial controllability analysis of the system, in order to minimize the control effort. Simulation examples using a system with 68 buses, 16 generators and five areas showed that the presented methodology is efficient in dampening the local and inter-area oscillation modes when compared to the classic power system stabilizers (PSS). The quality of the solution is illustrated through computer simulations, eigenvalues tables and mode-shapes.

Circulation algorithm for partial eigenstructure assignment via state feedback

This paper treats the partial eigenstructure assignment problem for multi-variable linear systems by state feedback. Both the open-loop and the closed-loop eigenvalues are allowed to have arbitrary geometric and algebraic multiplicities. Based on a previously proposed result on partial eigenstructure assignment, a new complete parametric solution is derived, which gives a complete general parameterization of the feedback gain and also provides the general parametric expression for all the admissible eigenvector matrices. The solution involves, instead of a matrix inverse of full-dimension, the inverse of a matrix with a much smaller dimension. By using this new solution repeatedly, a circulation algorithm is proposed, which is simple and dramatically reduces the computational load in the case of large scale systems. Furthermore, two important special cases are especially examined, and simple and direct solutions are proposed, which no longer needs computation of matrix inverses, and suits best in the proposed circulation algorithm. The approach provides all the design degrees of freedom which can be further utilized to achieve additional closed-loop system performance. An illustrative example is thoroughly examined, which demonstrates the procedure and the advantages of the proposed parametric approach.

An algorithm of partial eigenstructure assignment for high‐order systems

We put forward an algorithm for the issue on the partial eigenstructure assignment in a high‐order system, in which some certain eigenpairs, in a given system, can be assigned without changing others. Then a differential equation could be used to model the given systems, so that a multi‐input state feedback control deal with this assignment. Moreover, the algorithm requests the information of a few of the eigenpairs merely. As well some numerical cases are shown to prove the effect of our algorithm.

Static output feedback for partial eigenstructure assignment of undamped vibration systems

A novel method for partial eigenstructure assignment of undamped vibration systems using acceleration and displacement output feedback is presented in this paper. It is based on modifications of mass and stiffness that preserve partial eigenstructure. A numerical algorithm for determining the required control gain matrices of acceleration and displacement output feedback, which assign the desired eigenstructure, is developed. This algorithm is easy to implement, and works directly on the second-order system model. More importantly, the algorithm allows the output matrix and the input matrix to be specified beforehand and also leads naturally to a small norm solution of the gain matrices. Finally, some numerical results are presented to demonstrate the effectiveness and accuracy of the proposed algorithm.

A New Approach for Symmetry Preserving Partial Eigenstructure Assignment of Undamped Vibrating Systems

A new approach for the partial eigenvalue and eigenstructure assignment of undamped vibrating systems is developed. This approach deals with the constant output feedback control with the collocated actuator and sensor configuration, and the output matrix is also considered as a design parameter. It only needs those few eigenpairs to be assigned as well as mass and stiffness matrices of the open-loop vibration system and is easy to implement. In addition, this approach preserves symmetry of the systems. Numerical example demonstrates the effectiveness and accuracy of the proposed approach.

Eigenstructure Assignment Method and Its Applications to the Constrained Problem

A partial eigenstructure assignment method that keeps the open-loop stable eigenvalues and the corresponding eigenspace unchanged is presented. This method generalizes a large class of systems previous methods and can be applied to solve the constrained control problem for linear invariant continuous-time systems. Besides, it can be also applied to make a total eigenstructure assignment. Indeed, the problem of finding a stabilizing regulator matrix gain taking into account the asymmetrical control constraints is transformed to a Sylvester equation resolution. Examples are given to illustrate the obtained results.