Closed-loop Eigenstructure Research Articles

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.

Synthesis and Piloted Evaluation of Advanced Rotorcraft Response Types Using Robust Sliding Mode Control

Sliding mode control (SMC) is a promising technique for robust control synthesis with desirable properties. This paper describes the synthesis and piloted evaluation of advanced helicopter response types using the SMC technique. The required closed-loop response characteristics are specified as ideal, lower order, axial transfer functions that conform to predicted level 1 handling qualities. Two-loop, full-authority, output-tracking SMC laws are then synthesized to enforce the closed-loop performance and accurately track pilot commands. Analytical proofs for SMC gain tuning are given for the closed-loop performance to remain robust to unknown but bounded uncertainties in the input channels and the effects of rotor modes on closed-loop stability. The closed-loop eigenstructure is nearly identical to the specified closed-loop performance and has good modal decoupling. Furthermore, a frequency domain analysis with a nonlinear helicopter model shows good stability margins and disturbance rejection characteristics. Finally, the paper reports on simulation testing conducted with four experimental test pilots in a rotorcraft simulation environment. The simulation results indicate improved mission task performance and handling qualities ratings and a substantial reduction in pilot workload for the SMC-based advanced response types compared to the bare-airframe responses.

Rapid Nonovershooting Control for Simultaneous Infusion of Anesthetics and Analgesics

We propose a rapid nonovershooting tracking controller for the continuous infusion of anesthetics and analgesics to prevent overdosing and other harmful side effects on patients. The controller utilizes a state feedback control design methodology for multi-input multi-output systems to achieve a closed-loop eigenstructure that yields a nonovershooting transient response. The method is combined with a global optimization method to achieve a rapid nonovershooting response. The controller uses an extended Kalman filter to estimate system states from measurable outputs, and integral control is added to achieve robust tracking. The performance of the method is simulated on 20 patient models in two groups, and the results are compared against another recent study from the biomedical control literature.

Eigenvalue sensitivity of sampled time systems operating in closed loop

Abstract The use of feedback to create closed-loop eigenstructures with high sensitivity has received some attention in the Structural Health Monitoring field. Although practical implementation is necessarily digital, and thus in sampled time, work thus far has center on the continuous time framework, both in design and in checking performance. It is shown in this paper that the performance in discrete time, at typical sampling rates, can differ notably from that anticipated in the continuous time formulation and that discrepancies can be particularly large on the real part of the eigenvalue sensitivities; a consequence being important error on the (linear estimate) of the level of damage at which closed-loop stability is lost. As one anticipates, explicit consideration of the sampling rate poses no special difficulties in the closed-loop eigenstructure design and the relevant expressions are developed in the paper, including a formula for the efficient evaluation of the derivative of the matrix exponential based on the theory of complex perturbations. The paper presents an easily reproduced numerical example showing the level of error that can result when the discrete time implementation of the controller is not considered.

Eigenstructure Assignment for Componentwise Ultimate Bound Minimization in Discrete-Time Linear Systems

This technical note presents a feedback design approach to minimise state ultimate-bounds (UBs) in linear time-invariant discrete-time systems subject to non-vanishing bounded disturbances. A target closed-loop eigenstructure is first constructed to achieve the lowest possible UBs for one or more state components of the system. Structural conditions on the system matrices are then derived to guarantee that the target eigenstructure can be assigned by state feedback. When the required structural conditions are satisfied, an eigenvector assignment procedure can be applied to simultaneously minimise multiple UBs in systems with multiple inputs. The results are illustrated by a numerical example drawn from a distributed voltage control problem in inverter-based microgrids.

Robust microscale grasping through a multimodel design: synthesis and real time implementation

This paper deals with robust force control at the microscale for safe manipulation of deformable soft materials. Since mechanical properties of micrometer sized objects are highly uncertain, instability often occurs during a gripping task. This leads to object damage or destruction due to excessive gripping force. In this paper we propose the design of a robust dynamic output feedback controller that is able to insure desired performances for a set of 65 soft and resilient microspheres whose diameter ranges from 40μm to 80μm and stiffness varies from 2.8N/m to 15.7N/m. The degrees of freedom of the controller are managed by the use of a set of elementary observers. Robustness with respect to parametric uncertainties is satisfied thanks to an iterative procedure alternating between multimodel closed loop eigenstructure assignment and worst case analysis. The developed controller is of low order and can be implemented in real time. Robust gripping force control is for the first time demonstrated experimentally when dealing with the manipulation of a large number of variable deformable soft materials at the microscale. Both simulations and experimental results validate the interest of such control design approach.

Vibration control for structural damage mitigation

Substantial developments have taken place in the areas of structural health monitoring, wherein, the objective is not only to detect damage but also to determine its size and location across the structure. Often, the damaged structures do not become obsolete unless the damage is severe. Thus, it is important to mitigate the growth of damage and prevent the possibilities of a structural failure. Vibration is an important source of damage growth. Therefore, it is important to control vibrations such that the damage growth is mitigated. Supposing an effect of damage in the structure is said to be mitigated when the vibration response of the damaged structure with control is the same as the vibration response of the undamaged structure, then one of the requirements is that the closed loop eigenstructure of the damaged structure needs to match with the open loop eigenstructure of the undamaged structure. In this framework, this paper uses a linear algebraic technique to assign the undamaged eigenvalues and eigenvectors for the damaged structure with a stiffness loss using a state feedback controller. Through this technique, it is shown that the vibration control using an actuator load can help to mitigate damage growth by reducing the vibration response magnitudes at the damaged structural degrees of freedom, while these magnitudes at other nodes are allowed to increase due to the actuator loads. Vibration control in this sense for structural damage mitigation is illustrated using discrete and continuous structures.

Robust Eigenstructure Assignment in Geometric Control Theory

In this paper we employ the Rosenbrock system matrix pencil for the computation of output-nulling subspaces of linear time-invariant systems which appear in the solution of a large number of control and estimation problems. We also consider the problem of finding friends of these output-nulling subspaces, i.e., the feedback matrices that render such subspaces invariant with respect to the closed-loop map and output-nulling with respect to the output map, and which at the same time deliver a robust closed-loop eigenstructure. We show that the methods presented in this paper offer considerably more robust eigenstructure assignment than the other commonly used methods and algorithms.

Optimal Feedback Control of Power Systems Using Eigenstructure Assignment and Particle Swarm Optimization

AbstractThe US Navy has a continuing interest and investment in basic and applied research in the area of automation and control. The potential naval applications for this research are numerous and wide ranging. The need for advances in control and automation systems exists from missile defense, to shipboard auxiliary systems, to naval aircraft, and virtually everywhere in between. This research is performed in industry, academia, and in naval laboratories across the nation. This paper will detail particular research in control theory being performed in the area of automation and controls in the naval laboratories. A particle swarm optimization algorithm is used to manipulate the state and control weighting matrices of a linear quadratic regulator to achieve an optimal control for a desired eigenstructure. The algorithm is demonstrated on a nonlinear power system model, and is found to be highly effective in the stabilization of the system output performance, showing both rapid convergence and a closed loop eigenstructure very close to the specified eigenstructure.

An Optimal Sensitivity-Enhancing Feedback Control Approach via Eigenstructure Assignment for Structural Damage Identification

The concept of using sensitivity-enhancing feedback control to improve the performance of frequency-shift-based structural damage identification has been recently explored. In previous studies, however, the feedback controller is designed to alter only the closed-loop eigenvalues, and the effect of closed-loop eigenvectors on the sensitivity enhancement performance has not been considered. In this research, it is shown that the sensitivity of the natural frequency shift to the damage in a multi-degree-of-freedom structure can be significantly influenced by the placement of both the eigenvalues and the eigenvectors. A constrained optimization problem is formulated to find the optimal assignment of both the closed-loop eigenvalues and eigenvectors, and then an optimal sensitivity-enhancing control is designed to achieve the desired closed-loop eigenstructure. Another advantage of this scheme is that the dataset of frequency measurement for damage identification can be enlarged by utilizing a series of closed-loop controls, which can be realized by activating different combinations of actuators in the system. Therefore, by using this proposed idea of multiple sensitivity-enhancing feedback controls, we can simultaneously address the two major limitations of frequency-shift-based damage identification: the low sensitivity of frequency shift to damage effects and the deficiency of frequency measurement data. A series of case studies are performed. It is demonstrated that the sensitivity of natural frequency shift to stiffness reduction can be significantly enhanced by using the designed sensitivity-enhancing feedback control, where the optimal placement of closed-loop eigenvectors plays a very important role. It is further verified that such sensitivity enhancement can directly benefit the damage identification accuracy and robustness.

Robust eigenstructure assignment with minimum subspace separation

AbstractA computation method for delivering a state feedback controller that assigns closed‐loop eigenstructure with a minimum subspace separation and is robust towards unstructured perturbations is presented. The subspace separation is measured via the difference between two subspace projections, while the eigenstructure robustness is achieved by maximizing an upper bound on the allowable perturbation size such that the closed‐loop system remains stable. By exploiting the Sylvester equation parametrization, the constraint of pole assignment can be satisfied intrinsically and the robust eigenstructure assignment design essentially becomes an unconstrained optimization task. Analytical gradient formulas of the objective functions to be handled are developed. The design approach is demonstrated through numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.

Robust magnetic bearing control via eigenstructure assignment dynamical compensation

This paper deals with the robust control of a basic current-controlled magnetic bearing by an output dynamical compensator. Based on a parametric approach for eigenstructure assignment and a linearized model of the magnetic bearing system, a general explicit parametric expression for all the first-order dynamical compensators assigning desired nondefective closed-loop eigenstructure is obtained. This parametric expression is expressed in terms of the closed-loop eigenvalues and a free parameter. Through optimizing this free parameter and the closed-loop eigenvalues, a dynamical compensator is obtained which gives insensitive closed-loop eigenvalues, effectively attenuates the effect of the disturbance, and uses small control effort. To ensure desired closed-loop dynamical performance, the closed-loop eigenvalues are optimized within some desired regions on the left-half complex plane. Measures are also taken to ensure the robust stability of the closed-loop system in the case of large system parameter perturbations. The proposed approach is applied to a flywheel which is supported by four current-controlled active magnetic bearings. Nonlinear simulation and experimental results show the effect of the proposed approach.

Sliding-mode control design for a complex valued sliding manifold

This paper gives precise conditions for the existence of a state feedback sliding-mode controller to achieve a complex, closed-loop eigenstructure. This structure can be advantageous in providing maximum flexibility in specifying closed-loop dynamics. It is found that the controller is inherently dynamic in general, and recovers standard sliding-mode control results when all real elements are chosen for the switching surface matrix. A control design algorithm is presented, and robustness of the resulting closed-loop system is demonstrated. An illustrative example is provided.

Design and Optimisation of Restricted Complexity Controller: A Modal Approach for Reduced-Order Controllers

This article presents a new powerful design technique for reducing an initial high-order controller. For this, the initial closed-loop behavior is studied by modal analysis. Then, this behavior will be reproduced to a desired level of precision by re-designing the controller using the Robust Modal Control. The leading closed-loop eigenstructure of the system is assigned and the dynamic of the reduced-order controller is extracted from the relevant dynamic of the initial controller. This leads to a low-order controller, with a closed-loop behavior similar to the initial one.

Robust Flight Control Design With Eigenstructure Assignment

Although the H∞ control method is a powerful tool to design robust controller, it is difficult to attain the performances in time domain. A new design method of H∞ state feedback controller is proposed which guarantees the robust stability and assigns eigenstructure in the desired regions. Conditions of robust stability and closed loop eigenstructure assignment are solved in LMI form with one nonlinear constraint. This method is validated with numerical example of flight control design.

固有構造の存在範囲を指定したロバスト飛行制御系の設計

Although the H∞ control method is powerful tool to design robust controller, it is difficult to attain the performances in time domain. A new design method of H∞ state feedback controller is proposed which guarantees the robust stability and assigns eigenstructure in the desired regions. The time responses can be shaped with assigning appropriate eigenstructure. Conditions of robust stability and closed-loop eigenstructure assignments are solved in LMI form with one nonlinear constraint. This method is validated with numerical example of flight control design.

Decentralized controllers design for open-channel hydraulic systems via eigenstructure assignment

In this paper, we propose the design of both a proportional (P) and a proportional integral (PI) decentralized controller for open-channel hydraulic systems by assigning the closed-loop eigenstructure. The system dynamic is described by a linear, time-invariant model deduced from the Saint-Venant equations. A constant-volume control law is designed, satisfying the requirement of decentralization, typical of large-scale systems like the hydraulic one herein examined. The synthesis procedure followed in this paper allows us to derive a parametric expression for the set of feedback gains of decentralized controllers which achieve the desired eigenvalue assignment. The free parameters in this parametric expression can be used to assign eigenvectors as close to the desired ones as possible, while achieving the required eigenvalue assignment.

Assignment of reduced sensitivity eigenstructure

The presented technique addresses the problem of maintaining the powerful design simplicity and visibility of eigenstructure assignment, while reducing closed-loop eigenstructure sensitivity to dominant parameter variations. The method results from consideration of the allowable eigenvector subspace as a complete description of the eigenstructure assignment problem. Variations in open-loop system dynamics can thus be visualised as movements of the closed-loop allowable subspace. If a nonlinear model is linearised about two operating points with intersecting subspaces, assignment from the intersection will yield the same closed-loop eigenstructure at both operating points. This hypothesis is proven, and the analysis extended to include the case of non-intersecting subspaces, where it has been shown that reduced sensitivity solutions may also be generated. This reduced subspace design method has been tested on an eighth-order helicopter model that exhibits significant nonlinearity when forward speed is varied. The results demonstrate a considerable reduction in variation of the closed-loop eigenstructure over conventional techniques. The method is found to be a simple variation to the standard projection technique, while offering implicit consideration of dominant parameter variations.

A bilinear formulation for the output feedback problem in linear systems

This note ties together a geometric theory and previous algebraic results for the closed-loop eigenstructure assignment by output feedback in multivariable systems. Necessary and sufficient conditions for the closed-loop stabilization, eigenstructure assignment are presented in terms of a bilinear Sylvester equation, the solutions of which completely characterize the closed-loop eigenstructure problem with output feedback. Some preliminary results for the solution of this bilinear equation are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Genetic design of linear multivariable feedback control systems using eigenstructure assignment

In the design of linear multivariable feedback control systems, performance specifications can frequently be met by assigning appropriate closed-loop eigenstructure. However, although various powerful algorithms for the solution of this problem of eigenstructure assignment have been proposed over the years, no universally applicable and computationally stable methodology has yet emerged. These difficulties motivate the use of genetic algorithms in the solution of the fundamental problem of eigenstructure assignment.