Closed-loop Eigenvectors Research Articles

The Adjugate Method: Reassignment with System Classification

Further elaborations on the adjugate method for eigenvalue–eigenvector assignment are discussed in this paper. Additional results concerning the admissible pair introduced recently are stated, verified, and commented on. Properties of the adjugate method concerning eigenvalue reassignment of controllable and uncontrollable systems are revealed and pursued further in this study A reformulation of how calculations should be carried out is investigated, pointing out merits, bounds, and limitations. It has been found that reassignment can involve a case where the closed-loop eigenvector companion zi is zi=0 and the associated eligible closed-loop eigenvectors wi are open-loop ones. This can be considered an advantage in the sense of knowing zi and wi beforehand. This also applies to the case of uncontrollable eigenvalue reassignment, where enlarged closed-loop eigenvector subspaces are uncovered, enabling more flexible designs. Assessments of the traditional method compared to the adjugate method are commented on whenever appropriate. Numerical issues are pointed out where Leverrier’s algorithm is adapted for efficient matrix null space and matrix adjugate determination. A feature of the adjugate method is a result where wU=0, indicating that the eigenvalue assigned is uncontrollable. Such a distinctive feature enables a classification procedure for systems regarding controllability and observability. The various concepts pointed out have been demonstrated and authenticated through carefully selected examples.

Fully distributed strain-based output feedback for enhanced sensitivity in damage diagnosis of structures

The closed-loop damage diagnosis method has shown significant improvement in recent years, while closed-loop eigenvectors are seldomly used in previous damage diagnosis research due to the lack of adequate measurement points, by which a better damage diagnosis effect can be achieved. With the maturity of fully distributed optical fiber sensing technology, strain data acquisition with many (high-density) measurement points can be realized; in this manner, fully distributed strain-based output feedback for sensitivity enhancement is first investigated in this study, while the main contribution is to first develop a calculation algorithm of closed-loop strain modal shape sensitivity with high-density strain measurements. First, the desired eigenstructure assignment scheme is constructed, and a reasonable range of closed-loop pole selection is determined. On this basis, a sensitivity-based damage diagnosis method using closed-loop strain modal shapes is proposed, with sensitivity and robustness selected as the optimization goals. The optimal gain is obtained, which can benefit the closed-loop damage identification accuracy and stability. The effectiveness of the proposed method is validated and compared with the closed-loop eigenvalue-based method by numerical simulation, and the results indicate that the proposed method can effectively enhance the performance of damage diagnosis and improve system robustness.

Eigenstructure through Matrix Adjugates and Admissible Pairs

The traditional eigenstructure assignment method is reconsidered in this paper, shedding light on its nature and some of its properties. The approach is through matrix adjugates which enable a more eloquent decomposition compared to the lumped solution usually obtained through matrix null spaces. Based on the new approach, the admissible pair (w,z) is formalized. The new approach enables an alternative methodology to the determination of the permissible closed loop eigenvector subspaces and the might be termed the companion input-subspace. Such approach renders the method a formulae-based method as w and z are now obtained formula-wise as opposed to being extracted out of a lumped solution. Compared to the traditional method, the study reveals information such as w and z are independently and explicitly determined. Moreover, newly assigned eigenvalues always result in z≠0, the input matrix B does not influence z and a fundamental feature of the open loop characteristic polynomial is exposed. Furthermore, closed loop eigenvectors associated with repeated eigenvalues can be explicitly computed by means of differentiation as opposed to null space determination. A special form of system representation is highlighted, which considerably eases the calculations. The myriad concepts pointed out have been demonstrated and authenticated through carefully selected examples involving real, complex, repeated eigenvalues, and two practical systems of an electrical nature.

Parametric control to quasi-linear systems based on dynamic compensator and multi-objective optimization

This paper considers a parametric approach for quasi-linear systems by using dynamic compensator and multi-objective optimization. Based on the solutions of generalized Sylvester equations, we establish the more general parametric forms of dynamic compensator and the left and right closed-loop eigenvector matrices, and give two groups of arbitrary parameters. By using the parametric approach, the closed-loop system is converted into a linear constant one with a desired eigenstructure. Meanwhile, it also proposes a novel method to realize multi-objective design and optimization. Multiple performance objectives, containing overall eigenvalue sensitivity, $H_2$ norm, $H_\infty$ norm and low compensation gain, are formulated by arbitrary parameters, then robustness and low compensation gain criteria are expressed by a comprehensive objective function which contains each performance index weighted. By utilizing degrees of freedom (DOFs) in arbitrary parameters, we can optimize the comprehensive objective function such that an optimized dynamic compensator is found to satisfy the robustness and low compensation gain criteria. Finally, an example of attitude control of combined spacecrafts is presented which proves the effectiveness and feasibility of the parametric approach.

Receptance-based robust eigenstructure assignment

The robustness of receptance-based active control techniques to uncertain parameters in fitted transfer function matrices is considered. Variability in assigned closed-loop poles, which arises from uncertain open-loop poles, zeros, and scaling parameters, is quantified by means of analytical sensitivity formulae, which are derived in this research. The sensitivity formulae are shown to be computationally efficient, even for a large number of random parameters, and require only measured receptances, thereby preserving the model-free superiority of receptance-based techniques. An evolution-based global optimisation procedure is used to perform eigenstructure assignment so that the robustness, as defined by a metric, is maximised. The robustness metric is designed to scale the relative importance of each closed-loop pole and their respective real and imaginary parts. The proposed technique is tested numerically on a multi-degree-of-freedom system. It is shown that, in both single- and multiple-input systems, it is possible to increase the robustness by optimally selecting a set of closed-loop poles. However, it is determined that the closed-loop eigenvectors of the system play a significant role in the propagation of uncertainty and hence, since multiple-input systems may independently assign both closed-loop poles and eigenvectors, multiple-input systems are able to reduce the uncertainty propagation to a greater extent.

The minimum norm multi-input multi-output receptance method for partial pole placement

Abstract A closed-form analytical solution is developed for the first time that fully addresses the problem of choosing feedback gains that minimize the control effort required for partial pole placement in multi-input, multi-output systems. The norm of the feedback gain matrix is shown to take the form of an inverse Rayleigh quotient, such that the optimal closed-loop system eigenvectors are given as a function of the dominant (highest) eigenvectors of the matrix in the quotient. The feedback gains that deliver the required pole placement with minimum effort may then be determined using standard procedures. The original formulation by the receptance method proposed an arbitrary choice of the closed loop eigenvectors that assigned the poles exactly but was generally wasteful of control effort that might otherwise be conserved or put to good use in satisfying additional control objectives. The analytical solution is validated against a set of numerical examples.

Parametric control to a type of quasi-linear high-order systems via output feedback

This paper investigates the design problem of output feedback control to a type of quasi-linear high-order systems containing time-variant coefficient matrices which include time-variant parameters and system variables. Based on the solution to a class of high-order generalized Sylvester matrix equation, we can design the left and right closed-loop eigenvectors and obtain two groups of arbitrary parameters, further establish a general complete parametrized expression for a quasi-linear high-order output feedback controller. By the presented parametric approach, the closed-loop system can be transformed into a linear time-invariant one with the desired eigenstructure. Finally, a nonlinear feedback synchronization problem of Genesio-Tesi and Coullet chaotic systems is presented to prove the effectiveness of the proposed approach.

Dynamic structural modification of vibrating systems oriented to eigenstructure assignment through active control: A concurrent approach

Eigenvalues and eigenvectors play a fundamental role in determining the dynamic behavior of a vibrating system. Thus, an important goal in vibration control is to modify the eigenstructure to match the design specifications. Feedback control is a popular approach to this problem, which is however not always satisfactory. In fact, while system controllability suffices to assure that any requirement on the eigenvalues can be met by the controlled system, desired closed-loop eigenvectors cannot be attained in general, due to inherent limitations of active control.To overcome such a problem, this paper proposes a hybrid approach in which the mechanical system and the controller are concurrently designed to improve the attainability of the desired eigenstructure. Indeed, the suitable modification of the system inertial and elastic parameters modifies the set of eigenvectors that can be achieved through active control.In this work is demonstrated that such an objective can be effectively accomplished by minimizing the rank of a certain matrix which depends on the features of the original system and on the desired eigenpairs. Two algorithms for rank minimization are described and adjusted for the problem under consideration.The method is validated with three examples. In the first one, eigenstructure assignment of a 5 degrees of freedom system that previously appeared in the literature is performed, demonstrating the effectiveness of the proposed approach with respect to the state of the art. In the second example the same 5 degrees of freedom system is considered by also including damping, to evaluate the method capability to deal with damped systems. The third one is a 30 degrees of freedom system that enables the comparison of the two algorithms for different choices of the actuation.

Parametric control to a type of quasi-linear second-order systems via output feedback

ABSTRACTThis paper considers the design of output feedback control for a type of quasi-linear second-order systems with the time-varying coefficient matrices containing the state variables and a time-varying parameter vector. Based on the solution to a type of second-order generalised Sylvester matrix equations, general complete parameterisation of a quasi-linear output feedback controller is established with respect to the state variables, the time-varying parameter vector, the constant closed-loop system and another two groups of arbitrary parameters, and also for the left and right closed-loop eigenvectors matrices. With the proposed parametric output feedback control, the closed-loop system can be transformed into a constant linear system with desired eigenstructure. Finally, simulation results are provided to illustrate the convenience and effectiveness of application in the general spacecraft rendezvous problem.

Parametric stabilization for descriptor linear systems via state-proportional and -derivative feedback

This paper considers eigenstructure assignment problem for descriptor linear systems via state-proportional and -derivative feedback. New complete parametric approaches for the problem are proposed in two cases. General complete parametric expressions in direct closed forms for the closed-loop eigenvectors associated with the finite closed-loop eigenvalues are presented. Based on a recently proposed complete parametric solution to the generalized Sylvester matrix equation AV−EVJ=BWDJ+BWP, very simple complete parametric solutions to the feedback gains are also established. The approach guarantees arbitrary assignment of rank[EB] number finite closed-loop eigenvalues with arbitrary given algebraic and geometric multiplicities. It also guarantees the closed-loop regularity and impulse-freeness when several simple constraints on the design parameters are added. A numerical example shows the effectiveness of the proposed approaches.

Characterization of Controllability Based on Continuity of Closed-Loop Eigenvectors: Application to Controller-Driven Sampling Stabilization

This technical note presents a novel characterization of controllability for linear time-invariant finite-dimensional systems. This characterization relates eigenvalue controllability with the continuity of the map that assigns to each closed-loop eigenvalue the smallest subspace containing the set of corresponding closed-loop eigenvectors. Application of the given characterization is illustrated on a specific case of controller-driven sampling stabilization, where the sampled system is interpreted as a discrete-time switched system and stability under arbitrary switching is ensured via simultaneous triangularization (Lie-algebraic solvability).

Time Response Options for Structural Damage Mitigation Using an Actuator Load

It is well known that time responses in a single input system cannot be altered under a state variable based feedback control system design. The closed loop eigenvector options assignable using the linear state feedback controller is fixed in single input systems. As a result, time responses are also fixed. However, when state derivative feedback is augmented, similar to a multiple input system, several closed loop eigenvector options are possible. Hence, it allows time response shaping available in the single input systems. In this paper, a novel procedure to design simultaneous state and state derivative feedback controller for single input systems is presented. Then to illustrate its utility, structural damage growth mitigation (active structural prognosis) using time response shaping under a structural load is illustrated. It is shown that the life of the damaged structure in a dynamic environment is possible to extend using time response options resulting from a typical state/state derivative feedback controller. A discrete dynamic structure is considered to illustrate the underlying principles of active structural prognosis.

Parametric eigenstructure assignment for descriptor systems via proportional plus derivative state feedback

Eigenstructure assignment for descriptor systems with proportional plus derivative state feedback is studied. Based on a simple complete explicit parametric solution to a group of recursive equations, a parametric approach for eigenstrucure assignment in descriptor systems via proportional plus derivative state feedback is proposed. The proposed approach possesses the following features: 1) it does not impose any condition on the closed-loop eigenvalues, simultaneously assigns arbitrary n finite and infinite eigenvalues to the closed-loop system and guarantees the closed-loop regularity; 2) it is simple and needs less computational work; 3) it gives general complete parametric expressions for the closed-loop eigenvectors, the proportional state feedback gain matrix and the derivative state feedback gain matrix.

MULTI-OBJECTIVE RECONFIGURABLE CONTROLLER DESIGN FOR THREE-TANK SYSTEM USING PARAMETRIC EIGENSTRUCTURE ASSIGNMENT

The reconfigurable controller (RC) design of strongly interacting multi-input multi-output system is a challenging problem for the designer. In this paper, a parametric right eigenstructure assignment based multiobjective dynamic reconfigurable controller through state feedback is proposed against partial actuator failure for a benchmark three-tank system. The practical aspects of controller design require the conflicting objectives of robust stability to parameter perturbation and smaller controller gain to improve the transient response of the system. These design objectives rise to the minimum condition number of closed-loop eigenvector and minimum norm of feedback controller gain of closed-loop system respectively. Fast and elitist multiobjective genetic algorithm known as nondominated sorting genetic algorithm-II is successfully applied to select the pareto-optimal eigenstructure of closed-loop system. Further, RC has been designed to accommodate partial actuator failure which is considered as loss of control effectiveness in actuators. Added value of this work, the controller uses complex-valued chromosomes. The effectiveness of the proposed controller is illustrated for different actuator fault scenarios through simulation and its performance is analysed.

Infinite eigenvalue assignment in descriptor systems via state feedback

The problem of infinite eigenvalue assignment in the descriptor system via state feedback control u = Kx is considered. The problem is related to a group of recursive equations. By proposing a general complete parametric solution to this group of recursive equations, a general complete parametric approach is presented for the proposed infinite eigenvalue assignment problem. General parametric forms of the closed-loop eigenvectors and the feedback gain matrix are given in terms of certain parameter vectors which represent the design degrees of freedom. The approach involves mainly a singular value decomposition of the matrix E and a singular value decomposition of a lower dimension matrix, and thus is very simple and requires less computational work. Moreover, it overcomes the defects of some previous works. An example is given to illustrate the effect of the approach.

Vibration Cancellation in a Plate Using Orthogonal Eigenstructure Control

Orthogonal eigenstructure control is a novel control method that can be used for vibration suppression in flexible structures. The method described in this study does not need defining the desired locations of the closed-loop poles or predetermining the closed-loop eigenvectors. The method, which is applicable to linear multi-input multi-output systems, determines an output feedback control gain matrix such that some of the closed-loop eigenvectors are orthogonal to the open-loop eigenvectors. Using this, the open-loop system’s eigenvectors as well as a group of orthogonal vectors are regenerated based on a matrix that spans the null space of the closed-loop eigenvectors. The gain matrix can be generated automatically; therefore, the method is neither a trial and error process nor an optimization of an index function. A finite element model of a plate is used to study the applicability of the method to systems with relatively large degrees of freedom. The example is also used to discuss the effect of operating eigenvalues on the process of orthogonal eigenstructure control. The importance of the operating eigenvalues and the criteria for selecting them for finding the closed-loop system are also investigated. It is shown that choosing the operating eigenvalues from the open-loop eigenvalues that are farthest from the origin results in convergence of the gain matrix for the admissible closed-loop systems. It is shown that the converged control gain matrix has diagonal elements that are two orders of magnitude larger than the off-diagonal elements, which implies a nearly decoupled control.

ESA in high-order descriptor linear systems via output feedback

This paper considers eigenstructure assignment (ESA) in high-order descriptor linear systems via output feedback. Parametric expressions for the left and right closed-loop eigenvectors associated with the finite closed-loop eigenvalues and two simple and complete parametric solutions to the feedback gain matrix are obtained on the basis of the parametric solution to the generalized high-order Sylvester matrix equation. This approach does not impose any restriction on the closed-loop eigenvalues. The numerical example illustrates the effectiveness of the proposed approach.

Orthogonal Eigenstructure Control for Vibration Suppression

Orthogonal eigenstructure control is a novel active control method for vibration suppression in multi-input multi-output linear systems. This method is based on finding an output feedback control gain matrix in such a way that the closed-loop eigenvectors are almost orthogonal to the open-loop ones. Singular value decomposition is used to find the matrix, which spans the null space of the closed-loop eigenvectors. This matrix has a unique property that has been used in this new method. This unique property, which has been proved here, can be used to regenerate the open-loop system by finding a coefficient vector, which leads to a zero gain matrix. Also several vectors, which are orthogonal to the open-loop eigenvectors, can be found simultaneously. The proposed method does not need any trial and error procedure and eliminates not only the need to specify any location or area for the closed-loop eigenvalues but also the requirements of defining the desired eigenvectors. This method determines a set of limited number of closed-loop systems. Also, the elimination of the extra constraints on the locations of the closed-loop poles prevents the excessive force in actuators.

Parametric control systems design with applications in missile control

This paper considers parametric control of high-order descriptor linear systems via proportional plus derivative feedback. By employing general parametric solutions to a type of so-called high-order Sylvester matrix equations, complete parametric control approaches for high-order linear systems are presented. The proposed approaches give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices, and produce all the design degrees of freedom. Furthermore, important special cases are particularly treated. Based on the proposed parametric design approaches, a parametric method for the gain-scheduling controller design of a linear time-varying system is proposed and the design of a BTT missile ai]topilot is carried out. The simulation results show that the method is superior to the traditional one in sense of either global stability or system performance.

Robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems: A new optimization approach

Abstract The partial quadratic eigenvalue assignment problem (PQEVAP) concerns reassigning a few undesired eigenvalues of a quadratic matrix pencil to suitably chosen locations and keeping the other large number of eigenvalues and eigenvectors unchanged (no spill-over). The problem naturally arises in controlling dangerous vibrations in structures by means of active feedback control design. For practical viability, the design must be robust, which requires that the norms of the feedback matrices and the condition number of the closed-loop eigenvectors are as small as possible. The problem of computing feedback matrices that satisfy the above two practical requirements is known as the Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP). In this paper, we formulate the RPQEVAP as an unconstrained minimization problem with the cost function involving the condition number of the closed-loop eigenvector matrix and two feedback norms. Since only a small number of eigenvalues of the open-loop quadratic pencil are computable using the state-of-the-art matrix computational techniques and/or measurable in a vibration laboratory, it is imperative that the problem is solved using these small number of eigenvalues and the corresponding eigenvectors. To this end, a class of the feedback matrices are obtained in parametric form, parameterized by a single parametric matrix, and the cost function and the required gradient formulas for the optimization problem are developed in terms of the small number of eigenvalues that are reassigned and their corresponding eigenvectors. The problem is solved directly in quadratic setting without transforming it to a standard first-order control problem and most importantly, the significant “no spill-over property” of the closed-loop eigenvalues and eigenvectors is established by means of a mathematical result. These features make the proposed method practically applicable even for very large structures. Results on numerical experiments show that the proposed method considerably reduces both feedback norms and the sensitivity of the closed-loop eigenvalues. A study on robustness of the system responses of the method under small perturbations show that the responses of the perturbed closed-loop system are compatible with perturbations.