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The eigenvalue assignment problem of second-order singular system is investigated by using acceleration–velocity–displacement feedback. The conditions are established to ensure the solvability of partial eigenvalue assignment problem of second-order singular system. The derived results are extended to complete eigenvalue assignment problem of second-order singular system. The presented solvability conditions are easily tested. Then, the methods are given to solve the eigenvalue assignment problem of second-order singular systems. Finally, several examples are given to validate our results and algorithms. [ABSTRACT FROM AUTHOR]

The eigenvalues assignment problems for descriptor linear systems with state and its derivative feedbacks are considered herein. Necessary and sufficient conditions for the existence of solutions to the problems are established. The Euler and Tustin approximations of the continuous-time systems are analyzed. Procedures for computation of the feedbacks are given and illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Fractional time-invariant compartmental linear systems are introduced. Controllability and observability of these systems are analyzed. The eigenvalue assignment problem of compartmental linear systems is considered and illustrated with a numerical example.

The eigenvalues assignment problems for descriptor linear systems with state and its derivative feedbacks are considered herein. Necessary and sufficient conditions for the existence of solutions to the problems are established. The Euler and Tustin approximations of the continuous-time systems are analyzed. Procedures for computation of the feedbacks are given and illustrated by numerical examples.

Further elaborations on the modified Ackermann’s method (MAM) for eigenvalue assignment are considered in this paper. Additional results concerning the incomplete eigenvalue assignment (IEA) are stated, verified, and commented. The advantages of IEA are pursued even further in this study. The study proposes two newly appended approaches based on MAM; named spectral and truncated methods. They are grounded on IEA, which fundamentally exemplify a hybridized approach to eigenvalue assignment. Necessary and sufficient conditions for stability of the truncated hybridized method are established, proved, and validated by examples. All results obtained apply equally-well to identical eigenvalue assignment, complex eigenvalue assignment, as well as to uncontrollable systems. Besides, they lead to simplified state feedback matrix determination. Three numerical examples are fully worked out to substantiate the nature of the IEA and the two hybridized methods. Simulation and visualization using MATLAB demonstrate the flexibility of the proposed methods.

The celebrated method of Ackermann for eigenvalue assignment of single-input controllable systems is revisited in this paper, contributing an elegant proof. The new proof facilitates a compact formula which consequently permits an extension of the method to what we call incomplete assignment of eigenvalues. The inability of Ackermann’s formula to deal with uncontrollable systems is considered a weakness inherent in the method. The notion of incomplete assignment leads to a straightforward generalization of our method to eigenvalue assignment of uncontrollable systems, thus mitigating such a drawback of a popular method. Further results concerning the incomplete assignment are stated, verified, and commented. Such results reveal the trace of the state matrix $A$ a worthy feature pertinent to an open loop system. Finally, four numerical examples are worked out to demonstrate cases of incomplete, and uncontrollable eigenvalues assignment. The examples consider a case where the structure of the feedback matrix can be easily simplified. The paper ends with a commentary brief concerning some commonly used MATLAB commands for eigenvalue assignment.

Analysis of structural vibrations presents challenges due to the coupling between different degrees of freedom, often preventing the derivation of analytical expressions for quantities such as poles, vibration amplitude, etc. While numerical methods are commonly employed to tackle such challenges, their purely numerical nature prevents achieving a comprehensive understanding of the system behaviour. This paper builds on a novel recursive transfer function formulation recently proposed by the authors to analytically characterise the dynamics of generic undamped mass-chain systems. This approach provides fundamental properties and analytical insights that cannot be obtained through current numerical approaches, representing a significant breakthrough in the understanding of vibrations and offering valuable insights for analysing complex vibrating systems. These results presented in this paper allow for a rapid assessment of the minimum phase nature of the vibration system through straightforward stability assessment, and present analytical expressions for sums and multiplications of natural frequencies squared by induction and Vieta's formulas. Furthermore, it derives conditions for existence of repeated roots using direct analytical derivation of the characteristics equations, including purely imaginary roots and repeated roots at the origin, showing that repeated roots can only occur when some forms of spatial symmetries exist, and confirms bounds for the arithmetic mean of natural frequencies by using inequalities between geometric, arithmetic and quadratic means. Additionally, analytical conditions are established for systems with different degrees of freedom to share the same roots, providing a theoretical foundation for eigenvalue assignment in the time domain or pole placement in the frequency domain. This foundation enables the determination of desired natural frequencies from scratch using transparent analytical conditions, without relying on complex algorithms typically required by current inverse methodologies involving passive structural modifications or active controllers. Moreover, this paper also explores the relationship between independent and coupled vibrations, providing insights for comparing natural frequencies. Overall, this paper contributes to a deeper understanding of structural vibrations through analytical exploration and offers practical guidance for eigenvalue assignment and natural frequency analysis. [ABSTRACT FROM AUTHOR]

The pole-placement issue for linear multi-input multi-output (MIMO) dynamic systems with uncertain parameters has been addressed in this article. A static feedback matrix has been designed for minimizing variances of closed-loop poles (CLPs) and for assigning poles to the nominal system at the desired places. It is assumed that the joint probability density function (PDF) of uncertain parameters is known and the system has more than one input. A new unknown vector is used like an eigenvector for a stochastic closed-loop system matrix to state the problem. The variances of poles are considered as cost functions, and the means of poles are termed constraints. This form of the problem statement has helped us to simply find a solution. In the first step, the optimization problem with constraints was handled by solving the equality constraint, and then, the problem was converted to a classic extended eigenvalue optimization problem. Later, the eigenvalue optimization problem was solved by the Rayleigh quotient and the feedback matrix was accomplished. Finally, this approach was simulated and validated using the MATLAB simulations, and the results were compared with a robust pole-placement method, which MATLAB control toolbox uses. The Monte Carlo simulations showed lower covariance for CLPs around the mean poles as compared to the robust pole-placement method. [ABSTRACT FROM AUTHOR]

The problem of pole placement/eigenvalue assignment for full-state feedback is investigated. A new parametrisation for the whole solution space is proposed and proved, in which the dimension of the search space is minimised to ensure improved optimisation performance. The advantages of the proposed approach is demonstrated in numerical experiments involving 11 benchmark examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents an adaptive filtering output feedback control architecture for multivariable nonlinear systems with mismatched uncertainties enabled by an eigenvalue assignment method. A piecewise constant adaptive law updates the adaptive parameters which represent the uncertainty estimates by solving the error dynamics between the output predictor and the real system with the neglection of unknowns. By employing a computationally efficient eigenvalue assignment method, the multivariable nonlinear system is transformed into Frobenius canonical form. A novel filtering control law which allows the desired system to be nonminimum-phase and does not require dynamic inversion of the desired system is designed to compensate the nonlinear uncertainties and track a given trajectory, following a performance determined by the eigenvalues assigned to the controller. The uniform performance bounds are derived for the system state and control input as compared to the corresponding signals of a bounded virtual reference system, which defines the best performance that can be achieved by the closed-loop system. Numerical examples are provided to illustrate the effectiveness of the eigenvalue assignment enabled control law, comparisons between the proposed controller and funnel controller are carried out. [ABSTRACT FROM AUTHOR]

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In this paper, we propose a method for eigenvalue assignment using linear control systems containing uncertain elements. Uncertain systems are systems described by state equations which depend on uncertain parameters. In this paper, uncertainty is modeled with interval numbers. The proposed method assigns prescribed eigenvalues to a state feedback control system. Also, we introduce two interval operations to be used in our method use them. Some numerical experiments are presented to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents an optimization-based perspective for incorporating disturbance decoupling constraints into controller synthesis, which paves the way for utilizing numerical optimization tools. We consider the constraints arising from the following sets of static state feedback: (i) The set of all disturbance decoupling controllers; (ii) The set of all disturbance decoupling and stabilizing controllers. To inner approximate these sets by means of matrix equations or inequalities, we provide a unifying review of the relevant results of the geometric control theory. The approximations build on the characterization of controlled invariant subspaces in terms of the solvability of a linear matrix equation (LME) involving the state feedback. The set (i) is inner approximated through the LME associated with any element of an upper semilattice generated by controlled invariant subspaces. The set (ii) is inner approximated through a bilinear matrix inequality (BMI) and the LME associated with any element of a different upper semilattice generated by internally stabilizable controlled invariant subspaces. However, the resulting inner approximations depend on the subspaces chosen from the semilattices. It is shown that a specific (internally stabilizable) self-bounded controlled invariant subspace, which is the best choice regarding eigenvalue assignment, yields the largest inner approximation for both of the sets among (internally stabilizable) self-bounded controlled invariant subspaces. The inner approximations exactly characterize the controller sets under particular structural conditions. We have been driven by two primary motivations in investigating inner approximations for the sets above: (i) Enable the formulation of a variety of equality (and inequality) constrained optimization problems, where cost functions, such as a norm of the state feedback, can be minimized over a large subset of the set of all disturbance decoupling (and stabilizing) controllers; (ii) Introduce the disturbance decoupling constraints to members of the control systems community who might not be quite familiar with the elegant geometric state-space theory, similar to the authors themselves. This can add another dimension to research endeavors in resilient control of networked multi-agent systems. [ABSTRACT FROM AUTHOR]

The problem of eigenvalue assignment in fractional descriptor discrete-time linear systems is considered. Necessary and sufficient conditions for the existence of a solution to the problem are established. A procedure for computation of the gain matrices is given and illustrated by a numerical example.

This paper presents an approach for eigenvalue assignment in second-order linear systems with no spillover property. Second-order differential equations arise from dynamical modeling of vibrating structures by finite element or lumped parameter first principles approach in several practical problems. Certain structures can face practical issues when subjected to external perturbation forces, as resonance or flutter type vibrations. The control of excessive vibrations can be attempted by techniques of active vibration control using linear feedback. To change only a few eigenvalues and eigenvectors that cause excessive vibrations, the requirement of no spillover property is a somewhat attractive issue. Furthermore, only the part of the eigenstructure whose eigenvalues must be reassigned is necessary to be known for an efficient parametrization of the feedback matrices. Brauer's theorem, a milestone result of linear algebra, as well as some recent related results, is applied here to achieve partial eigenvalue assignment using dense force actuator (influence) matrices. The proposal can be applied to general systems with no restriction on the mass, damping, and stiffness with symmetry or definiteness. The procedures to implement the proposal are synthesized in a step-by-step form, and some numerical examples are given to illustrate its application. [ABSTRACT FROM AUTHOR]

This study develops an approach of controller design, on the basis of Lambert W function structure for Internet-based bilateral teleoperation systems. Actually, time-delay terms in bilateral teleoperation systems lead to an infinite number of characteristic equation roots making difficulty in analysis of systems by classical methods. As delay differential equations (DDEs) have infinite eigenspectrums, all closed-loop eigenvalues are not feasible to locate in desired positions by using classical control methods. Therefore, this study suggests a new feedback controller for assignment of eigenvalues, in compliance with Lambert W function. In this regard, an adaptive controller is accurately employed in order to provide the controller with updated predicted time-delay and robust the system against the time-delay. This novel control approach causes the rightmost eigenvalues to locate exactly in desired positions in the stable left hand of the imaginary axis. The simulation results show strong and robust closed-loop performance and better tracking in constant and time-varying delay. [ABSTRACT FROM AUTHOR]

Stability and transparency play key roles in a bilateral teleoperation system with communication latency. This study developed a new method of controller design, based on the Lambert W function for the bilateral teleoperation through the Internet. In spite of the time-delay in the communication channel, system disturbance, and modeling errors, this approach causes the slave manipulator tracks the master appropriately. Time-delay terms in the bilateral teleoperation systems result in an infinite number of characteristic equation roots making difficulty in the analysis of systems by traditional strategies. As delay differential equations have infinite eigenspectrums, it is not possible to locate all closed-loop eigenvalue in desired positions by using classical control methods. Therefore, this study suggested a new feedback controller for assignment of eigenvalues, in compliance with Lambert W function. Lambert W function causes the rightmost eigenvalues to locate exactly in desired possible positions in the stable left hand of the imaginary axis. This control method led to a reduction in the undesirable effect of time-delay on the communication channel. The simulation results showed great closed-loop performance and better tracking in case of different time-delay types. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a new problem of establishing feasibility conditions for regional eigenvalue assignment of positive observers. Previous results on regional eigenvalue assignment for observers of linear time-invariant positive systems are also improved. Unlike observable dynamical systems whose closed-loop eigenvalues can be assigned arbitrarily, eigenvalues of positive observers are indicated unable to be assigned into any arbitrary region. We derive feasibility conditions in the form of constrained convex programming under which the regional eigenvalue assignment is possible. Moreover, we propose a new method for solving the regional eigenvalue problem of positive observers once the feasibility conditions are satisfied. Numerical examples are given to show the efficacy of the proposed method. [ABSTRACT FROM AUTHOR]

As friction couples tangential and lateral degrees-of-freedom of a structure at contact interfaces, the resulting asymmetric dynamic system is prone to dynamic instability. Using state-feedback control, such a frictional asymmetric system can be stabilized through assigning the system desirable eigenvalues; but uncertainties in system parameters can cause assigned eigenvalues to deviate from desired locations and thus stability may be lost. This study presents a robust stabilization method that assigns both desirable eigenvalues and their sensitivities and thus render assigned eigenvalues stable and insensitive to perturbations in uncertain contact parameters (the friction coefficient, contact damping, and contact stiffness). This method utilizes receptances of the corresponding symmetric part of the asymmetric system. The optimal control input location is first determined by minimizing the Frobenius norm of the normalized eigen-sensitivity matrix. The normalized eigen-sensitivities indicate that the friction coefficient and contact stiffness intrinsically have similar crucial effects on the stability of the system. To demonstrate the application of the proposed control method, the eigen-sensitivities with respect to only the friction coefficient are assigned. A constrained over-determined least-squares problem is solved to assign both required eigenvalues and eigen-sensitivities. Numerical examples validate the effectiveness of the proposed robust control scheme by Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

This paper considers the eigenvalue distribution of a linear time-invariant (LTI) system with time delays and its application to some controllers design for a delay plant via eigenvalue assignment. First, a new result on the root distribution for a class of quasi-polynomials is developed based on the extension of the Hermite–Biehler theorem. Then, such result is applied to proportional–integral (PI) controller parameter design for a first-order plant with time delay through pole placement. The complete region of PI gains can be obtained so that the rightmost eigenvalues in the infinite eigenspectrum of the closed-loop system with delay plant are assigned to desired positions in the complex plane. Furthermore, on the basis of the previous result, this paper also extended the PI control to the proportional–integral–derivative (PID) control. It is worth pointing out that this work aims to improve the performance of the closed-loop system on the premise of guaranteeing the stability. [ABSTRACT FROM AUTHOR]

A double-step algorithm is presented to solve the pole placement problem for descriptor single-input linear systems using state feedback. The algorithm places two finite poles (eigenvalues) at a time so that complex conjugate pairs of poles are placed using only real arithmetic. Possible infinite poles of the open loop system are replaced by finite poles, therefore making the closed loop system either causal (discrete time) or free of impulsive behavior (continuous time). The numerical stability of the algorithm is proved by doing a backward rounding error analysis. It is worth pointing out that the algorithm is closely related to the double-step QZ algorithm for the generalized eigenproblem. [ABSTRACT FROM AUTHOR]

This paper presents an eigenvalue assignment method for the time-delay systems with feedback controllers. A new form of Runge-Kutta algorithm, generalized from the classical fourth-order Runge-Kutta method, is utilized to stabilize the linear delay differential equation (DDE) with a single delay. Pole placement of the DDEs is achieved by assigning the eigenvalue with maximal modulus of the Floquet transition matrix obtained via the generalized Runge-Kutta method (GRKM). The stabilization of the DDEs with feedback controllers is studied from the viewpoint of optimization, i.e., the DDEs are controlled through optimizing the feedback gain matrices with proper optimization techniques. Several numerical cases are provided to illustrate the feasibility of the proposed method for control of linear time-invariant delayed systems as well as periodic-coefficient ones. The proposed method is verified with high computational accuracy and efficiency through comparing with other methods such as the Lambert W function and the semidiscretization method (SDM). [ABSTRACT FROM AUTHOR]

Proportional-integral-derivative (PID) control has been widely adopted for stable and reliable operation of rigid rotors supported by a pair of active magnetic bearing systems. Conventional centralized PID control methods manipulate the feedback gains in prescribed forms until they achieve the desired control performance. In this study, an eigenvalue assignment for decoupled translational and conical modes is proposed in the complex domain to yield a unique PID controller in a closed form, preserving the isotropic bearing characteristics. The eigenvalue assignment necessitates the constraints required for decoupling of the translational and conical whirl motions from the complex equation of motion written in the center of gravity coordinates of the rigid rotor. The complex equation of motion integrates the rigid rotor and electro-magnetic control force models defined in two different coordinate systems by utilizing complex coordinate transformation relations. A flywheel energy storage system is taken as a simulation example in order to demonstrate the validity and effectiveness of the proposed eigenvalue assignment. The simulation results show that the eigenvalue assignment algorithm is superior to conventional control methods at systematically ensuring sufficient stability margins for the lightly-damped and unstable conical modes. [ABSTRACT FROM AUTHOR]

This paper introduces a new framework for implicit restarting of the Krylov-Schur algorithm. It is shown that restarting with arbitrary polynomial filter is possible by reassigning some of the eigenvalues of the Rayleigh quotient through a rank-one correction, implemented using only the elementary transformations (translation and similarity) of the Krylov decomposition. This framework includes the implicitly restarted Arnoldi (IRA) algorithm and the Krylov-Schur algorithm with implicit harmonic restart as special cases. Further, it reveals that the IRA algorithm can be turned into an eigenvalue assignment method. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

An approach for eigenvalue assignment in strongly controllable and observable linear descriptor systems using dynamic compensators is proposed. Parametric expressions for the controller coefficient matrices are given. The approach assigns the full number of distinct finite closed-loop eigenvalues, guarantees the closed-loop regularity and overcomes the defects of some previous works. In addition, using the proposed eigenvalue assignment approach, a sufficient condition for generic eigenvalue assignability using dynamic compensators is proved. [ABSTRACT FROM AUTHOR]

In this article, the static output feedback problem for linear time-invariant systems is considered. For arbitrary assignability of the roots of the characteristic polynomial by static output feedback, a new necessary and sufficient condition is derived. Although, the proof is based on simple analysis, the known sufficient conditions (derived by techniques of algebraic geometry) are directly covered. Furthermore, an algorithm for the calculation of feedback matrices assigning a desired set of eigenvalues is proposed. This algorithm does not require the desired eigenvalues to be distinct and it explicitly exploits the available degrees of freedom for reducing the feedback gain. The presented approach is illustrated on computational examples. [ABSTRACT FROM PUBLISHER]

An approach for eigenvalue assignment in linear descriptor systems via output feedback is proposed. Sufficient conditions in order that a given set of eigenvalues is assignable are established. Parametric form of the desired output feedback gain matrix is also given. The approach assigns the full number of generalised eigenvalues, guarantees the closed‐loop regularity and overcomes the defects of some previous works. [ABSTRACT FROM AUTHOR]

A new design method is presented for proportional-integral (PI) controllers of first-order plants in the presence of time delays. In general, time delays can limit and degrade the achievable performance of the controlled system, and even induce instability. Thus, the PI gains should be selected carefully considering such effects of time delays. Unlike existing methods, the design method presented in this paper is based on solutions to delay differential equations, which are derived in terms of the Lambert W function. PI controllers for first-order plants with time delays are designed by obtaining the rightmost (i.e., dominant) eigenvalues in the infinite eigenspectrum of time-delay systems and assigning them to desired positions in the complex plane. The process is possible due to a novel property of the Lambert W function. The controllers designed using the presented method can improve the system performance and successfully stabilize an unstable plant. Also, sensitivity analysis of the rightmost eigenvalues is conducted, and the results compare favorably with those of a prediction-based method for eigenvalue assignment. Extension to design of PI-differential controllers is discussed with examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the transfer function of periodic control systems and try to find state feedback matrices such that the minimal and maximal eigenvalues of the leading principal submatrices of the transfer function will be at desired locations. [ABSTRACT FROM AUTHOR]

SUMMARY Substructured and analytical frameworks for the development of numerical-substructure-based and output-based substructuring controllers are proposed. The principle of numerical-substructure-based control systems utilises online parameters and signals related to numerical substructures for the control gain synthesis. Output-based controllers consider only online information related to transfer systems. The resulting linear feedforward and feedback controllers are synthesised using state-space and transfer-function techniques, assuming that the dynamics of tested specimens are to be completely unknown, and thus can cope with nonlinear substructuring tasks. Selection of the controllers for implementation depends on the synchronisation requirement and the dynamic properties of the controllers in real-time conditions. Experimental results of a two-mass-spring system verifying the controller designs are presented, which also show that the addition of feedback controllers can effectively reduce the synchronised errors. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this article, the output regulation problem is solved for singular systems by using dual observer-based compensators. This has the advantage that output regulation can be achieved under weak conditions. Namely, different from previous approaches, an implementable compensator can be directly determined in form of a classical state space model without a transformation into Weierstrass–Kronecker canonical form. Furthermore, the impulse controllability and observability of the singular system is not required and the output to be controlled needs not be measurable. The results of the article are demonstrated by means of a simple mechanical system. [ABSTRACT FROM AUTHOR]

In this article, the finite-dimensional control of distributed-parameter systems with boundary control and unbounded observation is considered. This problem is solved by extending the concept of dual observer-based compensators to infinite dimensions. These compensators have the advantage that, on the one hand, the separation principle can be used to directly determine a finite-dimensional compensator and, on the other hand they can be directly applied to boundary control systems with a time-derivative of the input. In order to achieve a low controller order as well as a desired control performance, a parametric approach is proposed to parameterise the degrees of freedom contained in the controller. Consequently, desired design specifications can be achieved by using a parameter optimisation. The proposed design of finite-dimensional dual observer-based compensators is demonstrated by means of a heat-conducting system with Neumann boundary control and boundary measurements. [ABSTRACT FROM PUBLISHER]