Robust LQG Controller Design

Robust LQG Controller Design - KSME International Journal,...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: KSME International Journal, Vol. 12, No. 2, pp. 199~205, 1998 199 ’ Robust LQG Controller Design for Lightly Damped Uncertain Linear Systems Kyung—Soo Kim* and Youngiin Park” (Received December 21, 1996) In this note, we consider lightly damped uncertain linear systems with natural frequency variations. This type of uncertainty has multiple uncertain parameters with multiple rank structure. It is well known that the conventional LQG or LQG/LTR methods can not be applied to such systems because of the instability and the design conservatism. To overcome such shortcomings, we propose a systematic method to design robust LQG controllers. The proposed method requires only the LQG tuning parameters and the structure information of uncertainty. It will be shown that our approach can be effectively applied to flexible structure control design problems. Key Words: Lightly Damped Uncertain Systems, Robust LQG Controller, Real Parameter Uncertainty, Quadratic Stability, General I/O Decomposition 1. Introduction In the active vibration control, one of the most important issues has been the robustness problem to the natural frequency variation, specially in the lightly damped systems. For example, the conven— tional, LQG control method cannot be applied to the flexible structures such as flexible beam or plate systems because of the poor robustness to the system parameter variations. LQG/LTR method can be an alternative for the robust design method. However, it is well known that LQG/ LTR generates conservative.controllers to obtain the desired robustness property. Recently, Ha, control theory is frequently applied to the robust compensator design (Doyle et al, 1989; Zhou et al., 1996). However, the robust performance in the time domain can not be considered by the * Graduate research assistant, Center for Noise and Vibration Control (NoViC) , Dept. of Mech. Eng, Korea Advanced Institute of Science and Tech- nology (KAIST) “Associate professor, Center for Noise and Vibra- tion Control(NoViC), Dept. of Mech. Eng, Korea Advanced Institute of Science and Tech- nology (KAIST) theory (Zhou et aL, I994). A notable method to deal with the robust performance in the time domain is the quadratic stabilization technique in view of the robust Hz control theory (e. g. see Bernstein and Haddad, 1989; Petersen, 1995). In the references, the authors have defined an aux- iliary quadratic cost index, which is the upper bound of the quadratic cost index of interest, and have minimized the auxiliary performance. Such frameworks are known to be systematic in for- mulating the robust performance in the time domain. However, previous results are limited to the structured one—block uncertainty. Therefore, the previous works can not be employed for treating the real parameter uncertainty which has multiple uncertain parameters with multiple rank structure. In this note, we investigate a systematic design method to deal with the real parameter uncertainty for the control of lightly damped flexible structures. By considering the general input—Output (I/O) decomposition of real param- eter uncertainty(Kim, 1995; Kim and Park, 1995), we extend the Bernstein and Haddads Riccati approach. A design example will be given to show the effectiveness of the proposed approach. 200 2. Preliminaries We use the following definition of the qua- dratic stability for considering the robust stabil- ity. Definition _2.1 The system, i=f (i, x),__ is _ quadratically stable if there exists a quadratic Lyapunov function such that V=xTPx for a positive definite matrix P. I It is noted that the quadratic stability, which implies the asymptotic stability of the system, plays an important role to describe the robust stability of uncertain systems. Recently, the rela- tion between the small gain theorem and the quadratic stability is known as given in the fol- lowing lemma. Lemma 2.1 (Khargonekar et al., 1990) Con- sider the uncertain system givenby i=[A0+AA(t)]x, AA(t)=DF(t)E (1) where D and F are the known constant matrices, and F( a ) is a time—varying uncertain parameter matrix satisfying F(t) ’F(t)_<_1 for all tefRfi Then, The system (1) is quadratically stable for all F(t) IF(l) TF(t)SIVt if and only if the following conditions hold: (i) A, is asymptotically stable. (ii) ll E0201 —Ao)“D'IIa< 1. l .3. Robust LQG Controller Consider the uncertain system described by x=(A+AA(t))x+Bu+Fw (2a) y=Cx+v (2b) where xeéfi", uEER’", yEER’, E[w(t)uI(T)T]= W8(t-r), E[v(t)v(r)’]= V8(t*2'), W20, V>0 for uncorrelated noises. The uncertainty is the real parameter uncertainty which may be time ~varying as follows: a. ? AA(t)=§r18i(t)Ei. teams (3) where E,- is the known constant matrix with rank (Ei)=q,- and 81% ' ) is a Lebesgue—measurable K yang—Soc Kim and Youngiin Park scalar function. Note that the uncertainty in the natural frequencies of the lightly damped flexible structure can be effectively described by the real parameter uncertainty defined in (3). An illustra- tive example is given in the following. Example Consider the stiffness matrix such as K=‘[k'1:lk2 [81ng In this case, the uncer- tainty can be expressed as AK: (Ski-[i +8k2-[2 in the form of (3). It is known that the uncertainty in (3) has the infinite number of [/0 decomposition form as follows: Definition 3.1(Kim and Park,‘ 1995) Let us define a set F i "'J Fr], detiFi) *0. Regiq’xq‘} Then, given the uncertainty, it is always pos- sible to construct the 1/0 decomposition form as follows: (48) AA”) = (MF)A(t) (IHN) for any ['E 35‘ (4b) where ME[M1, m, Mr], NTEUVF, Ni], 1‘“qu A(t) —=— ' EER’W' 8r I‘erIIr for the minimal rank decomposition of E, such that E,=M,~N,~, rank (Mi) =rank (Ni) =qi. In this case, (4) is said to be the general I/O decom- position of (3). The purpose of Definition 3.1 is to express all the possible I /O decomposition of the uncertainty in (3) by a design variable F. Because of introducing a scaling matrix variable, nonunique- ness of M, and N,- in the minimal rank decompo- sition is not a concern, any more. Without loss of generality, we follow all the basic assumptions on the nominal system as in the standard LQG theory. By using the observer Robust LQG Controller Design for Lightly Damped Uncertain Linear Systems ' 201 -based controller 5:249? +Bu+Kf(y—-Cf), u=—ch (5) where KCEERmx”, Kfemn“, the closed loop can be written as follows: 2:: e: (Ae‘i'AAe) Xe+ FeWe where I l x l [w] i Xe: A ng= 7 x—x v A___[A—BKC BKC ] e 0 A—Kfc ’ _ AA 0] “[F 0 ] AAe[AAO,Fe F‘Kf, E[we(t) weT(2')]= We6(t~r) for We: [I3] It is noted that the augmented un- certainty has the general l/O decomposition as follows: AAe=(MeF)A(t) (F‘lNe) for any [7635(7) Where Me =[MT, MT], Ne=[N, Owen]. From (6) and (7), we can derive the following lemma. Lemma 3.1 The system in (6) is quadratically stable if the following conditions hold: (i) A2 is asymptotically stable. (ii) II F‘1Ne(ij—Ae)"MeF I|w<1 for some F685. (Proof) It is evident from Lemma 2.1. The necessity cannot be proven because of the diago— nal structure of uncertainty A(-). (Q. E. D) Note that Lemma 3.1 is a block—diagonally “scaled small gain condition. If we do not adopt the general I /O decomposition, the quadratic stability becomes very conservative for the multi- ple uncertain parameters with multiple rank struc- ture. The importance of the general I/O decompo» sition in the robust full state feedback problem has been pointed out in some materials(Kim, 1995; Savkin and Petersen, 1995). Now, we consider the stationary LQG perfor- mance index as follows: Jazmin lim otl(xTQx+uTRu)dt] for KcyK! firm a given AA (8) where Q20, R >0. For the structured one~block uncertainty, Bernstein and Haddad(l988) has shown that f“ is bounded as follows: 1AA tr[FeWeFeT P] for any admissible AA (9) Cu I where OSPEWMZ" and P satisfies A3P+PAe+Qe+ 11T(P)=0 (10) ‘ .. ' ~ T __ T v for Qe=[Q_+I§}?RI§C KI? R156] and any con- stant function y(-) such that AAEP-l-PAAeS W (P). We propose the following theorem to extend the Bernstein and Haddads Riccati approach (1988, 1989) for treating the real parameter uncer- tainty. Theorem 3.1 The given system (6) is qua- dratically stable if there exist a symmetric matrix PEWW“ and an Xe{ 1‘1"] F635}, for a Kc EER’M" and a Kfeflim“, satisfying A§P+PA9+ Qe+NJXrlNe +PMeXMZP=0 (11) such that Ae+ MeXMeTP is asymptotically stable. Further more, for all the admissible uncertainty, [arsz min tr[FeWeFZP] (12) X .Kch/ (Proof) Since Ae+ MaX M; P is asymptotically stable, so is Ag. By the bounded real lemma (Green and Limebeer, 1995), (l 1) can be equivalently -1/2 X <ij eAe) e. It implies that n X_1/2Ne(ij—Ae)—1MeX1/2iiw< 1. Since X “2 is a real positive definite matrix, there always exists a PE SE and a real orthogonai matrix U such that X 1’2: PU, UU T: I . There- fore, by the invariance of Hw~norm for the unitar- y transformation, the inequality is identical to the condition (ii) in Lemma 3.1. Also, the upper bound of cost index can be obtained by applying Bernstein and Haddads results described in (9) and (10) after showing AA§P+PAAe= (NJF“T)A(FTM§P) + (PMZF)A(F"N2) SNZX“1Ne+PMeXMeTP = ZNP) (13) (Q. E. D) written as“ [ <1- 202 Kyung—Sao Kim and Youngjin Park Based on Theorem 3.1, we define an optimiza- tion problem for the robust LQG controller as follows: Definition 3.2 Consider the pair of gains, (Kém’c, K/RLQG)=arg min tr[FeWeFeTP}, x.Kc,K, where arg( .) denotes the optimal values of variables. Then; the observer based—controller in’ (5) with the gains (KORLQG, Kfmoc) is called as the robust LQG controller. I In the case of no uncertainty, that is, M = N = ’0, the robust LQG controller becomes the nominal LQG controller. The robust LQG con- troller guarantees the robust stability for all the allowable uncertainty and confines the LQG cost index, j“, which depends on the uncertainty, to a certain bound. Such a concept for designing the robust controllers is referred to as the guaranteed cost controllers or the robust H2 controllers (Bernstein and Haddad, 1989; Petersen, 1995). It is noted that the Riccati solution matrix P in (11) is an implicit function of the scaling matrix as well as the controller gains. Therefore, the inclusion of scalings enlarges the feasible control- ler set for the robust LQG controller so that the upper bound of the cost index in (12) can be made to be tight. In the most of literatures which deal with real parameter uncertainty, the scaling matrix is not included or used in the limited fashion. Moreover, scalings is not treated as a design variable but selected by the trial and error method in the step of determining the I /O decom- position (e. g. Bernstein and Haddad, 1988, 1989; Jabbari and Schmitendorf, 1993). In the practical point of view, the robust LQG controller suffers from the numerical difficulty because the constraint (11) is nonlinear with respect to the variables. Even the feasibility can not be globally checked with the current numeri~ cal algorithms such as LMI method. Such a numerical difficulty is well known in the area of the robust H2 control theory(Khou et al., 1994) and developing the solving methods remains an open problem. In this note, the gradient search with Lagrange multiplier(Bazaraa et al., 1993), which is one of nonlinear programming tech— niques and commonly used in many nonlinear optimization problems, is applied for solving the robust LQG controller. 4. Numerical Example Consider a flexible structure modeled by a five —mass_ system connected by four uncertain springs as shown in Fig. l. The variations of spring constants imply the perturbation of the natural frequencies. We assumed the spring constants can vary within :596 from the nominal values. The force actuator is located on the first mass and only the position of the fifth mass is measured. State vector is defined as x’=[x1,*xz, x3, x4, x5, gel, 352, x3, :54, 355]. The used LQG parameters are given as Q=diag[0, 0, 0, 0, 1, 0, 0, 0, 0, 0], R: 0.01, F=B, W21 and V: 10. To construct an I/O decomposition in (4b), we used the singular value decomposition method such that Eiz U.a.-V.T=(U.-JB?) - wait/2’) =M.N. by a MAT- LAB function svd. Since ran/A E,)=1 for i=1, m, 4, the uncertainty and the scaling matrix have the following structure 629 839 64], deidgifi, 72, 7’3, 7’4]: (7i*0) By using a gradient search with Lagrange multiplier, we solved the nonlinear optimization problem in Definition 3.1 and obtained the cost bound such that f M $138.6. The obtained robust LQG controller and the nominal LQG controller are as follows: ~~ 4.8093 —0.0040 45.3590 1.4594 1.3041 T 0.0585 —0.8688 0.1594 0.9946 1.5671 RLQG: moo: KC 2.5265 ’ K’ 0.0873 2.2706 —1.3098 —0.3433 2.4092 1.0144 —2.3647 0.4977 1.4715 Robust LQG Controller Design for Lightly Damped Uncertain Linear Systems 203 7.3759 0.3490 3.0323 0.3578 —0.5755 7 0.3334 0.1494 0.4390 —0.0327 0.0572 LQG: LOG: K” 3.8408 ’ K’ 0.0113 9.8055 0.0144,. g, 10.1419 0.0256 9.9402 0.0497 10.0102 0.2159 In fact, we could not find any feasible set of gains by conventional approaches (Bernstein and Haddad, 1989; Petersen, 1995) because we should repeat the laborious modification of the 1/0 decomposition by the trial and error method. Figure 2 shows the quadratic performance 182(Kc, Kf)EE{gE[71;—’/O‘U(XTQX + uTRu) dt] (14) obtained by the given controller (Kc, K,), with respect to the variation of 62 in the case of 81:83 264:0. The nominal LQG controller cannot guarantee the robust stability for the uncertainty k,. = 1+0.056i(t), |6,.(2)| s 1 Fig. 1 A five—mass system connected by four uncertain springs. Cos! Fig. 2 Quadratic performance index 132(KC, K,) in the case of 81:63:84=0- while the quadratic performance of the robust LQG controller is insensitive to the parameter variation. Figure 3 shows the control perfor- mances of two controllers in the presence of parameter uncertainties. Consequently, we can observe the robustness of our approach to param- , eter uncertainties. Figure 4 shows the sensitivity and complementary sensitivity functions defined as S(s)={I+P(s)K(s) }“ and T<s> ={I+P(s)K(s)}"P(s)K(s) (15) where P( o ) and K ( o ) denote the nominal plant and the controller, respectively. As can be seen in Fig. 4(b), the robust LQG controller makes the notches, which correspond to the natural fre. quencies, deeper than that of LQG controller. w (5., 5:. 5:- QM °~ 0. 0. 0) ...... .. (5“ 5r 5',5‘)=( 0. 1. 1. 0 _.___. (5" 5133,5941! 1, -1, 1) ........ .V(5"51,5v5‘)=(1,1, 1.1) Y“. 0 10 20 30 40 50 60 (a) Performance of LQG controller __ (5" 52' 5r 5):( 0, 0. 0. 0) ...... t. “"5” 5’, 5):.(0. 1, 1.1) _,_,-.- (5" 51.5.,5‘)=(1, 1, -1. 1) _ ....... «(3" 5!, 8,, 8‘)=(1.1. 1.1) 0 10 20 30 40 50 60 t (b) Performance of robust LQG controller Fig. 3 Regulating performances of the nominal LQG and the robust LQG controllers. 204 K yang-S00 Kim and Youngjin Park 0.01 IE4 lTOw)l lE—G "5.x ........... .. “5—10 0.! I 10 wmdls) (a) Complementary sensitivity function 10 isuwu 0.01 . 0.1 w (rad/a) (b) Sensitivity function Fig. 4 Sensitivity and Complementary sensitiv- ity functions for the nominal system. 5. Concluding Remarks In this note, we have proposed the robust LQG control design method for treating lightly damped systems with multiple natural frequency uncer- tainties. Since such an uncertainty is described by the form of real parameter uncertainty, which contains multiple uncertain parameters with mul- tiple rank structure, the conventional quadratic stabilization technique can not be directly applied. The conventional methods can be im- proved by adopting the general I/O decomposi- tion of the real parameter uncertainty. It is noted that the conventional quadratic stability leads to a block<diagonally scaled small gain condition, which makes it possible to treat the uncertainty of interest, with the help of the general 1/0 decom- position. The effectiveness of our approach was shown by a design example. References Bazaraa, M. S., Sherali, H. D. and Shetty, C. M., 1993, Nonlinear Programming Theory and _ Algorithms, 2'ml Ed., John Wiley & Sons, Inc. Bernstein, D. S. and Haddad, W. M., I988, “The Optimal Projection Equations with Petersen -Hollot Bounds: Robust Stability and Perfor- mance via Fixed—Order Dynamic Compensation for Systems with Structured Real—Valued Parame- ter Uncertainty,” IEEE Trans. on A. C., Vol. 33, No.6, pp. 578~582; 1989, “LQG Control with an H“, performance Bound: A Riccati Equation Approach,” IEEE Trans. on A. C., Vol. 34, No. 3, pp. 293~305. Doyle, J. C., Glover, K., Khargonekar, P. P., and Francis, B. A., 1989, “State—Space Solution to Standard H2 and H“, Control Problems,” IEEE Trans. on A. C., Vol. 34, No.8, pp. 831 ~847. Green, M. and Limebeer, D. J. N., 1995, Linear Robust Control. Prentice—Hall. Khargonekar, P. P., Petersen, I. R. and Zhou, K., 1990, “Robust Stabilization of Uncertain Linear Systems: Quadratic Stabilizability and H °° Control Theory,” IEEE Trans. on A. C., Vol. 35, No.3, pp. 356~361. Jabbari, F. and Schmitendorf, W. E., 1993, “Effects of Using Observers on Stabilization of Uncertain Linear Systems,” IEEE Trans. on A. C., Vol. 38, No. 2, pp. 266~27l. Kim, K.—S., 1995, On the Robust LQR and LQG Control Based On Lyapunov’s Second Method, M. S. Thesis (in English), KAIST, Korea. Kim, K. —S. and Park, Y., 1995, “Robust L2 Optimization For Uncertain Systems,” Proc. of the 10‘” KA CC International Program, pp. 348 ~351. Petersen, I. R., 1995, “Guaranteed cost LQG control of uncertain linear systems,” IEE Proc. ~Control Theory Appl., Vol.142, No.2, pp. 95 ~102. Savkin, A. V. and Petersen, I. R., 1995, “Minimax Optimal Control of Uncertain Systems with Structured Uncertainty,” Int. J. Robust and Robust LQG Controller Design for Lightly Damped Uncertain Linear Systems 205 Nonlinear Contr., Vol. 5, pp. ll9~137. Doyle, J., 1994, “Mixed H2 and H“, Performance Zhou, K, Doyle, J. and Glover, K., 1996, Objective 1: Robust Performance Analysis,” Robust and Optimal Control, Prentice—Hall, Inc. IEEE Trans. on A. C., Vol.39, No.8, pp. 1564 Zhou, K., Glover, K., Bodenheimer, B. and ~1574. ...
View Full Document

Page1 / 7

Robust LQG Controller Design - KSME International Journal,...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online