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Journal of Vibration and Control
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DOI: 10.1177/1077546310395968
2012 18: 924 originally published online 12 September 2011
Journal of Vibration and Control
Shinn-Horng Chen, Wen-Hsien Ho, Jyh-Horng Chou and Fang Lu
the hybrid Taguchi-genetic algorithm
trajectory sensitivity for the uncertain flexible rotor systems via the orthogonal-functions approach and
Design of robust-stable and quadratic finite-horizon optimal active vibration controllers with low
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Design of robust-stable and quadratic
finite-horizon optimal active vibration
controllers with low trajectory sensitivity
for the uncertain flexible rotor systems via
the orthogonal-functions approach and
the hybrid Taguchi-genetic algorithm
Shinn-Horng Chen
1, Wen-Hsien Ho
2, Jyh-Horng Chou
3,4and
Fang Lu
1Abstract
By studying 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 rotor 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 rotor 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 in this paper for solving the nominal flexible rotor system 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 rotor system is transformed into a static constrained-optimization problem represented by the algebraic equations with constraint of LMI-based robust stabilizability condition. Then, for the static constrained-optimization problem, the HTGA is employed to find the robust-stable and quadratic finite-horizon optimal active vibration controllers of the uncertain flexible rotor system. An example is given to demonstrate the applicability of the proposed integrative approach.
Keywords
Flexible rotor system, hybrid Taguchi-genetic algorithm (HTGA), linear matrix inequality (LMI), orthogonal-functions approach (OFA), parameter uncertainties, stability robustness, trajectory sensitivity
Received: 14 April 2010; accepted: 22 July 2010
1. Introduction
A flexible rotor is a high-order system composed of many vibration modes. The control system of a flexible rotor is generally designed for a reduced-order model which is restricted to a few critical modes. They are often described by the distributed-parameter models and so are essentially infinite dimensional. Thus, it is impractical or impossible to implement the infinite-dimensional feedback controllers based on complete models of the flexible rotor system. Hence, instead of
1
Institute of Mechanical and Precision Engineering, National Kaohsiung University of Applied Sciences, Taiwan, Republic of China
2
Department of Medical Information Management, Kaohsiung Medical University, Taiwan, Republic of China
3
Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, Taiwan, Republic of China
4
Department of Electric Engineering, National Kaohsiung University of Applied Sciences, Taiwan, Republic of China
Corresponding author:
Jyh-Horng Chou, Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, Taiwan, Republic of China
Email: [email protected]
Journal of Vibration and Control 18(7) 924–940
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using the original infinite-dimensional distributed parameter model, many researchers used the finite-dimensional model to approximate the original infi-nite-dimensional model for designing the vibration con-trollers (see, for example, Balas, 1978a,b, 1982; Balas et al., 1988; Chen, 2003; Chen et al., 2000; Chou et al., 1998; Khot and Heise, 1994; Lin et al., 1990; Nonami et al., 1992; Zheng, 2004). Those researchers divided the finite-dimensional model into two parts: the controlled part and the residual part. The controlled part, which is used for designing the vibration controllers, is com-posed of those critical modes that have large contribu-tion to the elastodynamic response, and the residual part is composed of the remainder modes of the finite-dimensional model. The residual part may lead to control and observation spillover that can desta-bilize one or more of the poorly damped modes. Consequently, those researchers proposed some meth-ods in the above-mentioned literature to investigate the problem of spillover suppression to avoid instability.
In fact, in many cases it is very difficult, if not impos-sible, to obtain the accurate values of some system parameters. This is due to the inaccurate measurement, unaccessibility to the system parameters, or variation of the parameters. The parametric uncertainties can be viewed to take different forms like elemental and norm-bounded. Elemental parametric uncertainties are those for which the elemental information of the uncertain matrix is utilized and bounds on the individ-ual elements of the uncertain matrix are considered, whereas norm-bounded parametric uncertainties are those for which only a norm bound on the uncertain matrix is considered (Chou, 1995). If the elemental information of the parametric uncertain matrices is considered, the results will be less conservative than those results that do not utilize the elemental informa-tion of the parametric uncertain matrices (Chou, 1995; Ho et al, 2007). In a flexible rotor system, the system parameters are often subject to parameter perturba-tions due to inaccuracies in the calculaperturba-tions of the fre-quencies and damping due to approximations in the structural model, material properties, mass, damping, and so forth. These parameter perturbations can degrade the system performance, and it is sometimes possible to destabilize the system (Lin et al., 1990; Khot and Heise, 1994). However, the problem of parameter perturbations in the active control design of a flexible rotor system has not received as much attention as the spillover problem. Thus, recently, some articles (Chen, 2003; Chou et al., 1998; Khot and Heise, 1994; Lin et al., 1990; Zheng, 2004) have discussed the robustness analysis of active vibration control of flexible rotor system with both the residual modes and the linear elemental parameter perturba-tions. On the other hand, the plant parameters may
deviate from their nominal values and the system designed to be optimal for one set of parameter values no longer remains optimal for a different set of values. Hence, the trajectories may deviate considerably from the trajectories corresponding to the optimal con-trol due to changes in parameter values, unless care is taken to design the control taking into account consid-erations of sensitivity (Eslami, 1994; Gopal, 1987).
Trajectory (state vector) sensitivity is a very impor-tant measure of sensitivity which we will also consider in this paper. Besides, only robust stability is often not enough in control design. In control system design, it is often of interest to synthesize an optimal controller such that the control objective of minimizing a qua-dratic integral performance criterion is achieved (Goodwin et al., 2001).
Besides, for some practical problems, we need to deal with the finite-horizon (i.e., finite-time) optimal control problems (Friedland, 1986). Most of the LQ-based and related optimal control methods of active vibration control (see, for example, Balas, 1978a,b, 1982; Balas et al., 1988; Chen, 2003; Chen et al., 2000; Chou et al., 1998; Khot and Heise, 1994; Lin et al., 1990; Nonami et al., 1992; Zheng, 2004) belong to quadratic-infinite-horizon-optimal control approaches, and thus can only guarantee the steady-state performance. But the quadratic-finite-horizon-optimal control approaches can ensure both transient and steady-state performances. However, to the authors’ best knowledge, there is no literature to study the issue of designing the robust-stable and quadratic finite-horizon optimal active vibration con-troller with low trajectory sensitivity such that (i) the flexible rotor system with elemental parametric uncer-tainties can be robustly stabilized, and (ii) a quadratic finite-horizon integral performance index including a quadratic trajectory sensitivity term for the nominal flexible rotor system can be minimized. Very recently, Ho and Chou (2007) have proposed a computational optimization method, which integrates the orthogonal-functions approach (OFA) and the genetic algorithm, to design the optimal fuzzy controllers. Since the method proposed by Ho and Chou (2007) only involves the algebraic computation and is straightforward and well-adapted to computer implementation, the design procedures of the optimal fuzzy controllers may be either greatly reduced or much simplified accordingly. Ho and Chou (2007) have also shown that the computational optimization method integrat-ing the OFA and the genetic algorithm may obtain better results than those existing approaches reported in the literature.
Summing up the above statements and reasons, the purpose of this paper is to propose an integrative opti-mization method to design the robust-stable and
quadratic finite-horizon optimal active vibration regu-lator and observer gains for the uncertain flexible rotor system. The proposed integrative method complemen-tarily fuses the OFA, the hybrid Taguchi-genetic algo-rithm (HTGA) and the robust stabilizability condition, where the robust stabilizability condition is derived in this paper for ensuring that the closed-loop uncertain flexible rotor system can be stabilized. In this paper, by using the OFA and the robust stabilizability condition, the robust-stable and quadratic finite-horizon optimal active vibration control problem for the uncertain flexible rotor system is transformed into a static param-eter constrained-optimization problem represented by the algebraic equations with constraint of a robust sta-bilizability condition; thus greatly simplifying the robust-stable and quadratic finite-horizon optimal active vibration control with low trajectory sensitivity problem of the uncertain flexible rotor system. Then, for the static constrained-optimization problem, the HTGA is employed to find the robust-stable and qua-dratic finite-horizon optimal active vibration regulator and observer gains of the uncertain flexible rotor system. The reason why the HTGA is applied in this paper is that Chou and his associates have shown that the HTGA may obtain both better and more robust results than those existing improved genetic algorithms reported in the literature (Tsai et al., 2004). An illustrative example is also given to demon-strate the applicability of the proposed integrative opti-mization method.
2. Problem statement
The flexible rotor model shown in Figure 1 is a four-mass approximate model (Nonami et al., 1992).
The masses m1, m2, m3and m4 are concentrated masses among which m1and m4are connected to springs k1and
k2, respectively. Assume that the control inputs f1and f2 are applied to m1 and m4, x1ð Þ, xt 2ð Þ, xt 3ð Þt and x4ð Þt are the general coordinates.
Using the influence coefficient method and modal analysis, we can obtain the state space model for this flexible rotor without gyroscopic effects as following (Nonami et al., 1992): _ xcð Þ ¼t Acxcð Þ þt Bcf tð Þ, ð1Þ _ xrð Þ ¼t Arxrð Þ þt Brf tð Þ, ð2Þ y tð Þ ¼Ccxcð Þ þt Crxrð Þ,t ð3Þ where xcð Þ ¼t ½z1ð Þt z2ð Þt _z1ð Þt _z2ð ÞtTdenotes the con-trolled mode state vector, xrð Þ ¼t ½z3ð Þt z4ð Þt _z3ð Þt _z4ð Þt T denotes the residual mode state vector, f tð Þ ¼ f1ð Þt f2ð Þt
T
denotes the control input vector, y tð Þ ¼½x1ð Þt x4ð ÞtT denotes the observation vector, Note that z tð Þ ¼½z1ð Þt z2ð Þt z3ð Þt z4ð ÞtT¼1
x1ð Þt x2ð Þt x3ð Þt x4ð Þt
T
in which is a normalized modal transformation matrix. The system matrices Ac, Bc, Cc, Ar, Br, Cr are given as following (Nonami et al., 1992): Ac¼ 0 0 1 0 0 0 0 1 !21 0 21!1 0 0 !2 2 0 22!2 2 6 6 6 4 3 7 7 7 5, Ar¼ 0 0 1 0 0 0 0 1 !2 3 0 23!3 0 0 !24 0 24!4 2 6 6 6 4 3 7 7 7 5, Bc¼ 0 0 0 0 11 41 12 42 2 6 6 6 4 3 7 7 7 5, Br¼ 0 0 0 0 13 43 14 44 2 6 6 6 4 3 7 7 7 5, Cc ¼ 11 12 0 0 41 42 0 0 , and Cr¼ 13 14 0 0 43 44 0 0
in which iis the i-th damping ratio, !i is the i-th nat-ural frequency, and ijis the ij-th element of the modal matrix .
Due to the limitation on the computation speed of the computer for realizing the control algorithm, we use the reduced-order model (1), which is obtained by neglecting the residual model (2), for the design of the control system. In equation (1), reduced-order model including only the first and second modes, neglect the third and fourth modes. Note that, in this paper, the
Figure 1. The system configuration of the flexible rotor system.
residual mode matrix Ar is assumed to be a stable matrix. It is noted that if there are finite unstable eigen-values in the uncontrolled flexible rotor system, we can arrange Ac so that it contains those nominal unstable eigenvalues and the critical modes to be controlled, with Ar containing the remaining modes.
In this paper, the linear time-varying elemental parameter perturbations should be also considered in the problem of actively controlling the vibration of a flexible rotor system, where the parameter perturbations exist in both the controlled and residual parts. Let the matrices AcðtÞ, BcðtÞ, CcðtÞ, ArðtÞ, BrðtÞ and CrðtÞ denote, respectively, the linear time-varying ele-mental parameter perturbations of the nominal matrices Ac, Bc, Cc, Ar, Br and Cr in the controlled and residual parts due to the variations of the natural frequencies and damping ratios due to data errors, changes in operating points, changes in environmental conditions, aging, and/ or other modeling inaccuracies, and so forth. Since it is not practical to control all vibrational modes of the flex-ible rotor system (1) with elemental parametric uncer-tainties, we assume that only some critical modes are considered to be controlled, and thus the above distrib-uted parameter system (1) with elemental parametric uncertainties can be expressed in the following two par-titioned finite-dimensional state-space forms:
(i) controlled part:
_
xcðtÞ ¼ Að cþAcðtÞÞxcðtÞ þ Bð cþBcðtÞÞf ðtÞ, ð4aÞ ycðtÞ ¼ Cð cþCcðtÞÞxcðtÞ, ð4bÞ (ii) residual part:
_
xrðtÞ ¼ Að rþArðtÞÞxrðtÞ þ Bð rþBrðtÞÞf ðtÞ, ð4cÞ yrðtÞ ¼ Cð rþCrðtÞÞxrðtÞ: ð4dÞ and
yðtÞ ¼ ycðtÞ þ yrðtÞ, ð5Þ We assume that the parameter uncertain matrices AcðtÞ, BcðtÞ, CcðtÞ, ArðtÞ, BrðtÞ and CrðtÞ may be of the form
AcðtÞ ¼ Xm i¼1 "iðtÞAci, BcðtÞ ¼ Xm i¼1 "iðtÞBci;CcðtÞ ¼X m i¼1 "iðtÞCci, ð6aÞ and ArðtÞ ¼ Xm i¼1 "iðtÞAri, BrðtÞ ¼ Xm i¼1 "iðtÞBri;CrðtÞ ¼X m i¼1 "iðtÞCri, ð6bÞ in which Aci, Bci, Cci, Ari, Bri and Cri (i ¼ 1, 2, . . . , m) are given constant matrices, and "iðtÞ("i"iðtÞ "iand i ¼1, 2, . . . , m) are uncertain parameters.
The modes of the controlled and residual parts can be separated from each other as required by standard mode-reduction techniques (Balas, 1982; Lin et al., 1990), which allow for an additional freedom in the choice of the two parts. Without loss of generality, an assumption is first made with respect to the controlled dynamics: the matrix pair ðAc, CcÞis completely observ-able and the pair ðAc, BcÞis completely controllable. It is noted that if there are finite unstable poles in the uncontrolled flexible rotor systems, we can arrange Ac so that it contains those nominal unstable poles and the critical modes to be controlled, with Ar containing the remaining modes.
Consider the controlled part of the flexible rotor system in equations (4a) and (4b) with the initial state vector xcð0Þ, and the observer-based controller has the form
_^xcðtÞ ¼ Acx^cðtÞ þ Bcf ðtÞ þ Kc½yðtÞ Ccx^cðtÞ, ^xcð0Þ ¼ 0, ð7Þ
and
f ðtÞ ¼ Fcx^cðtÞ, ð8Þ in which ^xcðtÞis the 2N 1 estimator state vector, and Kc is the 2N P estimator gain matrix and Fc is the M 2N feedback gain matrix.
Combining equations (4), (5), (7) and (8) the actual closed-loop uncertain flexible rotor dynamic system is given by the following composite matrix form:
_xðtÞ ¼ A þ AðtÞxðtÞ, ð9Þ where xðtÞ ¼ x cTðtÞ, x^TcðtÞ, xTrðtÞT, A ¼ Ac BcFc 0 KcCc AcBcFcKcCc KcCr 0 BrFc Ar 2 4 3 5, Chen et al. 927
and
In the nominal closed-loop system, the terms BrFc and KcCr are the control and observation spillover, respectively, which may cause a destabilizing effect such that the nominal eigenvalues of the system will shift away from those of AcBcFc, AcKcCc and Ar (Balas, 1978a,b). From equation (9), we can see that the destabilizing effect may be raised from the spill-over terms and/or the linear time-varying elemental parameter perturbation matrices AcðtÞ, BcðtÞ, CcðtÞ, ArðtÞ, BrðtÞand CrðtÞ.
For the closed-loop uncertain flexible rotor dynamic system in equation (9), the problem of robust stabiliz-ability analysis is, under the condition that the feedback gain matrix Fc and the regulator gain matrix Kc in equations (7) and (8), respectively, have been specified in advance, to derive a robust stabilizability criterion for checking whether the closed-loop uncertain flexi-ble rotor dynamic system in equation (9) can be stabi-lized by the specified gain matrices Fc and Kc or not. Hence, in what follows, we present a linear-matrix-inequality-based (LMI-based) robust stabilizability criterion to analyze whether the closed-loop uncertain flexible rotor dynamic system in equation (9) can be stabilized by the specified gain matrices Fc and Kc or not, where the gain matrices Fc and Kc have been s-pecified in advance.
Theorem: The system in equation (9) will be stable, if the matrix A is an asymptotically stable matrix and if there exists a symmetric positive definite matrix Psuch that the following LMIs are satisfied
UTjP þ PUj5 0, ð10Þ where j ¼ 1, 2, . . . , 2m, and Uj¼ Xm i¼0 "iðtÞEi "iðtÞ¼"ior "i ,
in which E0 ¼ A, and "0ðtÞ ¼ "0¼ "0¼1, and
Ei¼ Aci BciFc 0 KcCci AciBciFcKcCci KcCri 0 BriFc Ari 2 4 3 5: Proof:
Using equation (6), we can rewrite equation (9) as _xðtÞ ¼ A þ AðtÞxðtÞ ¼ A þ X m i¼1 "iðtÞEi ! xðtÞ ¼X m i¼0 "iðtÞEixðtÞ, ð11Þ
where E0¼ A, and "0ðtÞ ¼ "0¼ "0¼1, and Ei¼ Aci BciFc 0 KcCci AciBciFcKcCci KcCri 0 BriFc Ari 2 4 3 5:
From the book of Roberts and Varberg (1973), we can see thatP
m
i¼0
"iðtÞEiis a polytope and is the convex hull of its extreme points Uj (j ¼ 1, 2, . . . , 2m), where Uj (j ¼ 1, 2, . . . , 2m) are constant matrices defined as
Uj¼ Xm i¼0 "iðtÞEi "iðtÞ¼"ior "i , for j ¼ 1, 2, . . . , 2m:
That is to say that A þP m i¼1 "iðtÞEi¼P m i¼0 "iðtÞEican be represented in a convex combination form (Chou et al., 2003) A þX m i¼1 "iðtÞEi¼ Xm i¼0 "iðtÞEi¼ Xq j¼1 jðtÞUj, where jðtÞ 0, Pq j¼1 jðtÞ ¼1 and q ¼2m. Then equation (11) becomes
_xðtÞ ¼ X q j¼1 jðtÞUj ! xðtÞ: ð12Þ
Let VðtÞ ¼ xTðtÞ P xðtÞ be the Lyapunov function of the system in equation (12) (i.e., in equation (9)). Then, differentiating the Lyapunov function, we obtain AðtÞ ¼ AcðtÞ BcðtÞFc 0 KcCcðtÞ AcðtÞ BcðtÞFcKcCcðtÞ KcCrðtÞ 0 BrðtÞFc ArðtÞ 2 4 3 5:
_ VðtÞ ¼X q j¼1 jðtÞ xTðtÞ UTjP þ PUj xðtÞ : ð13Þ
It is obvious that _VðtÞ5 0, if there exists a symmetric positive definite matrix Psuch that
UTjP þ PUj5 0, for j ¼ 1, 2, . . . , q, ð14Þ So, from the results mentioned above, we can obtain that the linear uncertain system in equation (10) is asymptotically stable if there exists a symmetric positive definite matrix Psuch that the LMIs in equation (10) is satisfied. Thus the proof is completed. Q.E.D. Remark 1: From above Theorem, the problem of deter-mining the robust stability of the flexible rotor system having time-varying parameter uncertainties in equa-tion (12) can be considered as LMI feasibility problems which are convex and can be effectively solved by cor-responding LMI software (e.g., Gahinet et al., 1995).
From equation (14), it can be seen that the proposed robust stability criterion becomes finding a common P such that the LMIs in equation (10) hold, which is sim-ilar to the problem of finding a common Pfor a set of Lyapunov inequalities in fuzzy control system design (Tanaka and Wang, 2001).
In addition, the trajectory sensitivity analysis problem is an important engineering problem which we shall study in this section. Let the parameters of a system be repre-sented by a vector ¼ ½1, 2, . . . , rT where r denote the number of parameters. Consider the controlled part of the flexible rotor system as the following form:
_
xcðtÞ ¼ AðÞxcðtÞ þ BðÞ f ðtÞ, ð15aÞ
and
yðtÞ ¼ CcxcðtÞ ð15bÞ with the initial state vector xcð0Þ. The nominal param-eter vector of the model (16a) will be denoted by 0 in the sequel, and the nominal values of the system matrix AðÞ and input matrix BðÞ are, respectively, Ac¼Að0Þand Bc¼Bð0Þ. The problem of trajectory sensitivity analysis consists in determining the deviation in xcðtÞ about its nominal value xc0ðt, 0Þ due to the variations in about its nominal value 0.
Let sjð Þt (j ¼ 1, 2, . . . , r) denote the trajectory sensi-tivity vector with respect to the parameter j, i.e.,
sjð Þ ¼t @xcð Þt
@j
: ð16Þ
Then the corresponding trajectory sensitivity equa-tion is given by _sjðtÞ ¼ AcsjðtÞ þ AjxcðtÞ þ Bjf ðtÞ, sjð0Þ ¼ 0; for j ¼ 1, 2, . . . , r, ð17Þ where Aj¼ @AðÞ @j 0 and Bj¼ @BðÞ @j 0 :
However, only robust stabilizability is often not enough in control design. The control objective with low trajectory sensitivity of minimizing a quadratic finite-horizon integral performance criterion including a quadratic trajectory sensitivity term for the nominal dynamic systems is also considered in many practical control engineering applications (Friedland, 1986; Goodwin et al., 2001). On the other hand, before we are able to synthesize a controller with low trajectory sensitivity such that the good control performance including a quadratic trajectory sensitivity term for a given dynamic system can be efficiently achieved, it is necessary that the given dynamic system can be stabi-lized by the controller (Nise, 2000; Yu, 2002). Therefore, the problem considered in this paper is how to specify the estimator gain matrix Kc and the feedback gain matrix Fc in equations (8) and (9), respectively, such that the constraint of LMI-based robust stabilizability condition in equation (10) for the closed-loop uncertain flexible rotor dynamic system in equation (9) can be satisfied, and such that the optimal control performance including a quadratic trajectory sensitivity term for the nominal closed-loop flexible rotor dynamic system
_ xðtÞ ¼ ~AxðtÞ, ð18aÞ and xð0Þ ¼ xT cð0Þ, x^ T cð0Þ, s T 1ð0Þ, s T 2ð0Þ, . . . , s T rð0Þ T , ð18bÞ where xðtÞ ¼ xT cðtÞ, x^TcðtÞ, s1TðtÞ, sT2ðtÞ, . . . , sTrðtÞ T , and ~ A ¼ Ac BcFc 0 0 KcCc AcBcFcKcCc 0 0 Ac1 Bc1Fc Ac 0 .. . .. . .. . . . . .. . Ac r Bc rFc 0 Ac 2 6 6 6 6 6 4 3 7 7 7 7 7 5 , Chen et al. 929
and quadratic finite-horizon optimal active vibration controller with low trajectory sensitivity of the uncer-tain flexible rotor system.
5. Conclusions
In this paper, the active robust-stable and quadratic finite-horizon optimal vibration controllers with low trajectory sensitivity design of the uncertain flexible rotor system is investigated. A new sufficient condition is derived in terms of LMIs for ensuring that the flexible rotor system under mode truncation and linear time-varying elemental parameter uncertainties is asymptot-ically stable. Based on the OFA, an algorithm for the optimal control design has been presented in this paper. The integration of the presented algorithm and the HTGA is used to design the robust-stable and qua-dratic finite-horizon optimal active vibration control-lers with low sensitivity for uncertain flexible rotor system under robust stability constraint and the mini-mization of a quadratic performance index included a quadratic trajectory sensitivity term. Since the pro-posed algorithm only involves algebraic computation, the design procedure of the robust-stable and quadratic finite-horizon optimal controller with low sensitivity for the flexible rotor system with linear time-varying ele-mental parameter perturbations can be either greatly reduced or much simplified accordingly. A design example is given to demonstrate the applicability of the proposed approach, which integrates the LMI tech-nique, OFA and the HTGA. The computational exper-iment shows that the proposed approach can obtain satisfactory results. The designed robust-stable and quadratic finite-horizon optimal active vibration con-trollers with low trajectory sensitivity by using the pro-posed integrative approach not only can actively significantly suppress the vibration of the flexible rotor system, but also can avoid the possibilities of both spillover-induced instability and time-varying-parameter-perturbation-induced instability.
Funding
This work is supported by the National Science Council, Taiwan, Republic of China, under Grants NSC99-2221-E151-009 and NSC99-2221-E327-043-MY3.
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