Liang-An Zheng

Robust-optimal active vibration controllers design of flexible mechanical systems via orthogonal function approach and genetic algorithm

By integrating the orthogonal-functions approach (OFA), the hybrid Taguchi-genetic algorithm (HTGA) and a robust stabilizability condition, an integrative method is presented in this paper to design the robust-optimal active vibration controller such that (i) the flexible mechanical system with elemental parametric uncertainties can be robustly stabilized, and (ii) a quadratic finite-horizon integral performance index for the nominal flexible mechanical system can be minimized. The robust stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). Based on the OFA, an algebraic algorithm only involving the algebraic computation is derived for solving the nominal flexible mechanical feedback dynamic equations. By using the OFA and the LMI-based robust stabilizability condition, the robust-finite-horizon-optimal active vibration control problem for the uncertain flexible mechanical dynamic systems is transformed into a static constrained-optimization problem represented by the algebraic equations with constraint of LMI-based robust stabilizability condition; thus greatly simplifying the robustoptimal active vibration control design problem. Then, for the static constrained-optimization problem, the HTGA is employed to find the robust-optimal active vibration controllers of the uncertain flexible mechanical systems. Two design examples are given to demonstrate the applicability of the proposed integrative approach.

A Mixed Robust/Optimal Active Vibration Control for Uncertain Flexible Structural Systems with Nonlinear Actuators Using Genetic Algorithm

In this article, a mixed robust/optimal control approach is proposed to treat the active vibration control (or active vibration suppression) problems of flexible structural systems under the effects of mode truncation, linear time-varying parameter perturbations and nonlinear actuators. A new robust stability condition is derived for the flexible structural system which is controlled by an observer-based controller and is subject to mode truncation, nonlinear actuators and linear structured time-varying parameter perturbations simultaneously. Based on the robust stability constraint and the minimization of a defined H 2 performance, a hybrid Taguchi-genetic algorithm (HTGA) is employed to find the optimal state feedback gain matrix and observer gain matrix for uncertain flexible structural systems. A design example of the optimal observer-based controller for a simply supported beam is given to demonstrate the combined application of the presented sufficient condition and the HTGA.

LMI condition for robust stability of linear systems with both time‐varying elemental and norm‐bounded uncertainties

Abstract In this paper, the stability robustness of linear systems with both time‐varying elemental (structured) and norm‐bounded (unstructured) uncertainties is investigated. A sufficient condition is proposed in terms of linear matrix inequalities (LMIs) for ensuring that the linear systems with both time‐varying elemental and norm‐bounded uncertainties are asymptotically stable. Two examples are given to show that the proposed sufficient condition is less conservative than the existing one reported recently.

Stability robustness of linear output feedback systems with both time-varying structured and unstructured parameter uncertainties as well as delayed perturbations

This paper investigates the stability robustness of linear output feedback systems with both time-varying structured (elemental) and unstructured (norm-bounded) parameter uncertainties as well as delayed perturbations by directly considering the mixed quadratically coupled uncertainties in the problem formulation. Based on the Lyapunov approach and some essential properties of matrix measures, two new sufficient conditions are proposed for ensuring that the linear output feedback systems with delayed perturbations as well as both time-varying structured and unstructured parameter uncertainties are asymptotically stable. The corresponding stable region, that is obtained by using the proposed sufficient conditions, in the parameter space is not necessarily symmetric with respect to the origin of the parameter space. Two numerical examples are given to illustrate the application of the presented sufficient conditions, and for the case of only considering both the delayed perturbations and time-varying structured parameter uncertainties, it can be shown that the results proposed in this paper are better than the existing one reported in the literature.

Robust Kalman-Filter-Based Frequency-Shaping Optimal Active Vibration Control of Uncertain Flexible Mechanical Systems

This paper presents a time-domain control methodology, which is named as the robust Kalman-filter-based frequency-shaping optimal feedback (KFBFSOF) control method, to treat the active vibration control (or active vibration suppression) problem of flexible mechanical systems under simultaneously high frequencies unmodelled dynamics, residual modes, linear time-varying parameter perturbations in both the controlled and residual parts, noises (input noise and measurement noise),and noise uncertainties. Two robust stability conditions are proposed for the flexible mechanical system, which is controlled by a KFBFSOF controller and subject to mode truncation, noise uncertainties, and linear structured time-varying parameter perturbations simultaneously. The advantage of the presented KFBFSOF control methodology is that it can make the controlled closed-loop system to obtain both good robustness at high frequencies and good performance at low frequencies. Besides, the proposed robust stability criteria guarantee that the designed KFBFSOF controller can make the controlled flexible mechanical system to avoid the possibilities of both spillover-induced instability and time-varying-parameter-perturbation-induced instability. Two examples are given to illustrate the application of the presented control methodology to the active vibration control problems of a simply supported flexible beam and of a flexible rotor system.

Design of robust-stable and quadratic finite-horizon optimal controllers with low trajectory sensitivity for uncertain active suspension systems

This paper presents a design method for designing the robust-stable and quadratic-finite-horizon-optimal controllers of uncertain active suspension systems. The method integrates a robust stabilisability condition, the orthogonal functions approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA). Using the integrative computational method, a robust-stable and quadratic-finite-horizon-optimal controller with low-trajectory sensitivity can be obtained such that (i) the active suspension system with elemental parametric uncertainties is stabilised and (ii) a quadratic-finite-horizon-integral performance index including a quadratic trajectory sensitivity term for the nominal active suspension system is minimised. The robust stabilisability condition is proposed in terms of linear matrix inequalities (LMIs). Based on the OFA, an algebraic algorithm only involving the algebraic computation is derived for solving the nominal active suspension feedback dynamic equations. By using the OFA and the LMI-based robust stabilisability condition, the dynamic optimisation problem for the robust-stable and quadratic-finite-horizon-optimal controller design of the linear uncertain active suspension system is transformed into a static-constrained-optimisation problem represented by the algebraic equations with constraint of LMI-based robust stabilisability condition; thus greatly simplifies the design problem. Then, for the static-constrained-optimisation problem, the HTGA is employed to find the robust-stable and quadratic-finite-horizon-optimal controllers of the linear uncertain active suspension systems. A design example is given to demonstrate the applicability of the proposed integrative computational approach.

Stability Robustness of Discrete-Time Singular Systems With Structured Parameter Perturbations

In this paper, the stability robustness of linear discrete-time singular systems with structured parameter perturbations is investigated. A new sufficient criterion is presented for ensuring that the linear discrete-time singular system remains regular, impulse-free, and asymptotically stable under structured parameter perturbations. The presented criterion is proved to be less conservative than the existing one reported recently.

Robust Regional Eigenvalue-Clustering Analysis for Linear Discrete Singular Time-Delay Systems With Structured Parameter Uncertainties

In this paper, the regional eigenvalue-clustering robustness problem of linear discrete singular time-delay systems with structured (elemental) parameter uncertainties is investigated. Under the assumptions that the linear nominal discrete singular time-delay system is regular and causal, and has all its finite eigenvalues lying inside certain specified regions, two new sufficient conditions are proposed to preserve the assumed properties when the structured parameter uncertainties are added into the linear nominal discrete singular time-delay system. When all the finite eigenvalues are just required to locate inside the unit circle, the proposed criteria will become the stability robustness criteria. For the case of eigenvalue clustering in a specified circular region, one proposed sufficient condition is mathematically proved to be less conservative than those reported very recently in the literature. Another new sufficient condition is also proposed for guaranteeing that the linear discrete singular time-delay system with both structured (elemental) and unstructured (norm-bounded) parameter uncertainties holds the properties of regularity, causality, and eigenvalue clustering in a specified region. An example is given to demonstrate the applicability of the proposed sufficient conditions.

Design of robust-stable and quadratic finite-horizon optimal active vibration controllers with low trajectory sensitivity for uncertain flexible mechanical systems using an integrative computational method

By integrating the robust stabilizability condition, the orthogonal-functions approach (OFA), and the hybrid Taguchi-genetic algorithm (HTGA), an integrative method is presented in this paper to design the robust-stable and quadratic finite-horizon optimal active vibration controller with low trajectory sensitivity such that (i) the flexible mechanical system with elemental parametric uncertainties can be robustly stabilized, and (ii) a quadratic finite-horizon integral performance index including a quadratic trajectory sensitivity term for the nominal flexible mechanical system can be minimized. In this paper, the robust stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). Based on the OFA, an algebraic algorithm only involving the algebraic computation is derived for solving the nominal flexible mechanical feedback dynamic equations. By using the OFA and the LMI-based robust stabilizability condition, the robust-stable and quadratic finite-horizon optimal active vibration control problem for the uncertain flexible mechanical dynamic systems is transformed into a static constrained-optimization problem represented by the algebraic equations with constraint of LMI-based robust stabilizability condition; thus greatly simplifying the robust-stable and quadratic finite-horizon optimal active vibration control design problem. Then, for the static constrained-optimization problem, the HTGA is employed to find the robust-optimal active vibration controllers of the uncertain flexible mechanical systems. A design example is given to demonstrate the applicability of the proposed integrative approach.

Robust Frequency-Shaping Optimal Active Vibration Control of Uncertain Flexible Mechanical Systems with Persistent Excitation

Based on the frequency-shaping optimal control method, this paper proposes a time-domain robust disturbance rejection design approach for a class of uncertain flexible mechanical systems subject to persistent disturbances and time-varying parameter perturbations. In the approach, a frequency-shaping output filter is first employed to combine with the mechanical system and a Kalman filter to form an augmented system. Some eigenvalues of the frequency-shaping output filter coincide with the unstable poles of the disturbance dynamics to perform disturbance rejection. Then, for the designed closed-loop system to have asymptotic stability, a new robust stability condition is proposed. It is shown that, using the proposed stability condition, the resulting controller can suppress the persistent disturbance and keep the flexible mechanical system from the possibility of instability caused by spillover and time-varying parameter perturbations. Finally, two examples are given to demonstrate the use of the design approach.