- Email: [email protected]
Substructural control of fuzzy nonlinear flexible structures
ARTICLE IN PRESS
Journal of the Franklin Institute 344 (2007) 646–657 www.elsevier.com/locate/jfranklin
Substructural control of fuzzy nonlinear flexible structures M. Sunara,, O. Tokerb a
Mechanical Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia b Systems Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia Received 7 February 2006; accepted 7 February 2006
Abstract It is advantageous to use the substructural and/or decentralized techniques in structural control to save on computations and time. In this paper, a generalized substructural approach is presented in the control of fuzzy nonlinear flexible structures with discrete sensors/actuators. The substructural control scheme is developed using the static condensation technique together with the LQG control method. The subcontrollers and subobservers designed at substructure levels are used to assemble the global controller and observer for the whole structure. Nonlinear effects are included in the structural formulations and a fuzzy methodology is adopted for handling the imprecision present in the structure modeling. The nonlinear and fuzzy schemes are applied to one structural control problem to illustrate the accuracy and capability of the substructural control technique. r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Substructural control; Nonlinear; Flexible structure; Fuzzy; LQG
1. Modelling and Control Numerical techniques, such as the finite element method (FEM), are frequently used in the modeling and control of flexible structures. However, for large flexible structures, the FEM results in a large number of ordinary differential equations often not practical to manipulate. Several substructural and decentralized control techniques have been Corresponding author. Tel.: +966 3 8604976; fax: +966 3 8602949.
E-mail address: [email protected] (M. Sunar). 0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2006.02.010
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
647
developed in the past to deal with the problem of efficiently designing controllers for large flexible structures at substructure levels [1,2]. In a substructural control method [3], the linear quadratic Gaussian (LQG) control approach was applied to flexible structures for the computation of feedback and observer matrices from substructure levels. Nonlinear effects may be present in flexible structures due to geometry, inertia, damping and material. The unmodeled nonlinear behavior can cause the flexible structures to fail without warning [4] and hence care should be taken to account for such a behavior. A modal superposition technique was used for analyzing the dynamic response of nonlinear structures [5]. A nonlinear approach was used to control the speed of a vehicle [6]. The nonlinear effects were included in the control of large flexible structures through the LQG control scheme [7]. The parameter variations introduced by the analysis model, the fuzzy and imprecise material and model properties, and the uncertain exogenous undesired environmental inputs may adversely influence the stability and performance characteristics of a control system. The concept of fuzzy set theory was presented for solving a general structural optimization problem involving multiple objectives [8]. Control of nonlinear structures using fuzzy control approach was considered in the active control of hysteretic structures subjected to environmental loads [9]. In this work, the concept of designing feedback controllers and observers for linear/ nonlinear flexible structures from substructure levels using the LQG control method [3,7] is extended to include the fuzziness in structural modeling. The proposed fuzzy substructural control scheme is tested using a structural control example. 2. Introduction Suppose that the equation of motion for a flexible structure is given as M x€ þ C d x_ þ Kx ¼ Du u þ Dw w,
(1)
where M, Cd and K are the mass, damping and stiffness matrices; Du and Dw are the input matrices due to controller and disturbance; x, u and w are the displacement, controller input and disturbance vectors, respectively. The flexible structure model in state-space form is expressed as [10] z_ ¼ Az þ Bu þ Gw, y ¼ Cz þ v,
(2)
where A, B, C and G are the state, controller input, output and disturbance matrices; z, y, and v are the state, output and sensor noise vectors, respectively. Various entries in the above equation are given as " # " # " # 0 I 0 x ; A¼ ; B¼ , z¼ M 1 K M 1 C d M 1 Du x_ " # 0 G¼ , ð3Þ M 1 Dw where I is the identity matrix. For the LQG control approach, it is well-known that the solution to the optimal u is obtained with a controller (feedback) gain matrix found
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
648
as [10] u ¼ K c z ¼ R1 BT Pc z,
(4)
where Pc satisfies the following Riccati equation: AT Pc þ Pc A Pc BR1 BT Pc þ Q ¼ 0.
(5)
The state estimation to Eq. (2) is accomplished by the optimal state observer, Kalman filter, through an observer gain matrix Kf given by K f ¼ Pf C T V 1 ,
(6)
where Pf satisfies a dual Riccati equation of Pf AT þ APf Pf C T V 1 CPf þ GW GT ¼ 0.
(7)
The eigenvalues of the matrices [A–BKc] and [A–KfC] are the eigenvalues for the combined controller/observer scheme. 3. Substructural control technique The LQG-based substructural control technique used in this study is very powerful for large flexible structures containing many members. The mathematical details of the technique can be found in Ref. [3]. The iterative procedure for the controller and observer design of a flexible structure using the substructural control technique is briefly outlined below. First, the structure is decomposed into substructures, which are numbered in an arbitrary manner. In the first iteration, the subcontroller (Kc) and subobserver (Kf) are designed or computed for the first substructure assuming no boundary forces due to subcontrollers of surrounding substructures. The boundary forces due the subcontroller of the first substructure are condensed to its boundary degrees of freedom. Subcontrollers and subobservers are designed for the remaining substructures with the inclusion of subcontroller forces condensed to their boundary degrees of freedom from previous substructures. The subcontrollers and subobservers are then assembled to obtain the global feedback and observer matrices Kc and Kf for the whole structure. The computation of eigenvalues for [A–BKc] and [A–KfC] completes the first iteration. The subsequent iterations are terminated when these eigenvalues cease changing appreciably. 4. Nonlinear effects In general, for a nonlinear flexible structure, the mass, damping and stiffness matrices given in Eq. (1) are not linear. The proposed substructural control technique assumes a linear structure and hence these matrices must be linearized in some fashion. The schematic of the substructural control technique for the controller and observer designs of the nonlinear flexible structure is shown in Fig. 1 [7]. The basic steps are explained below. 1. As mentioned before, the flexible structure is decomposed into r substructures and the substructures are numbered in some suitable manner. 2. The nonlinear M, Cd and K matrices are linearized to obtain the linear matrices M ‘ , C d‘ and K ‘ at the current state of z. This would be the initial state of z(0) at the beginning of iterations.
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
649
Decompose the structure into r substructures z = z(0) i=0 i = i+1 Yes
i
Set all subcontroller forces to zero k=0 k = k+1 Design subcontroller and subobserver for the kth substructure by considering subcontroller forces at its boundary Find subcontroller forces fB k and fB k+1 No
k=r?
Yes Assemble global controller and observer Compute closed-loop eigenvalues
E.values converged?
No
Yes z=z+Δz
z converged?
No
Yes Stop Fig. 1. Substructural control technique for a nonlinear flexible structure.
3. The global controller and observer are designed for the nonlinear structure using these linearized matrices in the substructural control technique whose steps are briefed in the previous section. 4. The state vector is incremented by z ¼ z þ Dz and the nonlinear iterative scheme continues with step 2 until the convergence on z is achieved. The number n in Fig. 1
ARTICLE IN PRESS 650
M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
indicates that the gain matrices for the controller (feedback) and observer are not calculated at every iteration of the nonlinear scheme, rather they are computed at every n iterations. Hence, an approach similar to the gain scheduling has been utilized.
5. Inclusion of fuzziness The fuzziness in the structure can be due to modeling errors, uncertainties in material properties and damping, and imprecisely known effects of reduced order models. However, some upper and lower limits for the fuzziness can be assumed. The effect of fuzziness on nonlinear structure analysis can then be found by structural response and the LQG controller can be designed using these upper and lower limits. This is the approach taken in this study. The procedure on the substructural LQG controller and observer designs for a nonlinear fuzzy structure is implemented as follows. The upper and lower limits of fuzziness are noted for the subsequent nonlinear analysis. The nonlinear structural responses are found at these limits. The LQG control law for the nonlinear structure is then carried out using the nonlinear structural responses at these limits. The nonlinear controller (feedback) and observer designs are carried out as described in the previous section. This solution procedure is clearly implemented in the next section on a numerical example. 6. Numerical example The two-bay truss shown in Fig. 2 is taken as the numerical example. The truss is made up of aluminum with a modulus of elasticity of 69 GPa and mass density of 2770 kg/m3. The cross-sectional areas of all the members of the structure are taken as 6:45 104 m2 . The actuators and sensors are collocated at nodes 1 and 3. Actuators can generate forces and sensors can measure displacements in x and y directions at these nodes. When the substructural control technique is used, the structure is decomposed into two substructures as shown in Fig. 3. The nonlinearity of the structure is assumed to be due to the nonlinear
1.27m
1.27m
5
3 1
3
5
7
10 2
1
1.27m
8
9
y 6
4 6
2 4
Fig. 2. Two-bay truss.
x
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
substructure 1 5
substructure 2 3 3
5
651
1
7
10 2
1
9
8
6 6
4
2 4
Fig. 3. Substructures for two-bay truss.
spring behavior of the 5th member of the truss whose linearized stiffness is given as k5 ¼
aE 5 A5 x¯ 5 , L5
(8)
where E5, A5 and L5 are the modulus of elasticity, cross-sectional area and length, x¯ 5 is the current horizontal displacement for the member at node 3, and a is added as a parameter for tuning the amount of nonlinearity. The structural response is found at the two limits of fuzziness on the modulus of elasticity and the average (steady-state) values of responses are noted. The LQG controller and observer are then designed for the nonlinear structure using these average displacement values on the nonlinear member 5. The displacements for the 5th member in the horizontal direction at these two limits are calculated as 1:016 104 m (d1) and 1:524 104 m (d2). The fuzzy control logic is implemented using two approaches. In one approach, the feedback controller and observer matrices are found via the substructural control technique using the above displacement values for the nonlinear (but linearized) member 5. Two controllers and observers are found, one corresponding to the displacement d1 (LQG 1) and the other to d2 (LQG 2). The following linear membership functions (membership function 1) are assumed: 8 0 if dpd 2 ; > > < dd 2 if d 2 odod 1; (9) mLQG1 ¼ d1 d2 > > : 1 if dXd 1 ;
mLQG2
8 0 > > < d d 1 ¼ > d1 d2 > : 1
if dXd 1 ; if d 2 odod 1 ; if dpd 2 ;
where d denotes the horizontal displacement of member 5 at node 3.
(10)
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
652
In the other fuzzy approach, three variations of modulus of elasticity are assumed and the corresponding steady-state displacements are found from the structural response. The displacements are noted as d1 and d2 like before, and a third displacement of 1:27 104 m (d3), all of which correspond to a controller and observer design called LQG 3. The following linear membership functions (membership function 2) are taken: 8 0 if dpd 3 þ e; > > < d d e 3 if d 1 eodod 3 þ e; (11) mLQG1 ¼ d 1 d 3 2e > > : 1 if dXd 1 e;
mLQG2 ¼
mLQG3
8 0 > > > > d1 d e > > > < d d 2e 1
if dXd 1 e or dpd 2 þ e; if d 3 þ eodod 1 e;
3
d d2 e > > > > > d 3 d 2 2e > > :1
8 0 > > < d d e 3 ¼ d d 2 2e > 3 > : 1
(12) if d 2 þ eodod 3 e; if d 3 epdpd 3 þ e; if dXd 3 e; if d 2 þ eodod 3 e;
(13)
if dpd 2 þ e;
where e represents a value taken to represent a small neighborhood. For the LQG control law, the Q, R, W and V are taken as diagonal matrices with appropriate dimensions whose diagonal terms are assumed to be 106, 1, 10 and 1,
1
x 10 -4 complete substructural
Displacement (m)
0.5
0
-0.5
-1
-1.5
-2
0
2
4
6
8
10
12
14
16
18
Time (s) Fig. 4. Closed-loop response at node 1 in x direction (1).
20
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x 10-4
1.5
complete substructural
1
Displacement (m)
653
0.5
0
-0.5
-1
-1.5
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 5. Closed-loop response at node 1 in x direction (2).
4
x 10-4 complete substructural
3.5
Displacement (m)
3 2.5 2 1.5 1 0.5 0
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 6. Closed-loop response at node 1 in y direction (1).
respectively. The time step for iterations is chosen as 0.01 s, but the feedback and observer gain matrices are calculated at the initial step and at every 0.25 s afterwards to speed up the iterative process. Hence the number n in Fig. 1 is 25.
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x 10-4
4
complete substructural
3.5
Displacement (m)
3 2.5 2 1.5 1 0.5 0
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 7. Closed-loop response at node 1 in y direction (2).
1.5
x 10-4 complete substructural
Displacement (m)
1
0.5
0
-0.5
-1
-1.5
0
2
4
6
8
10
12
14
16
18
20
Time(s) Fig. 8. Closed-loop response at node 3 in x direction (1).
The results are shown in Figs. 4–11. The results of the first fuzzy approach are shown as (1) and those of the second approach are referred as (2) in these figures. Reference or input forces are assumed to be step forces acting at nodes 1 and 3 in positive horizontal and vertical directions (+x and y) with 100 N magnitudes. It is clear from the figures that the
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
6
655
x 10-5 complete substructural
4
Displacement (m)
2 0 -2 -4 -6 -8 -10 -12 -14
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 9. Closed-loop response at node 3 in x direction (2).
1.6
x 10-4
1.4 complete substructural
Displacement (m)
1.2 1 0.8 0.6 0.4 0.2 0
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 10. Closed-loop response at node 3 in y direction (1).
results of the complete and substructural control techniques are close to each other. Both techniques converge to the steady-states in a similar manner. However, some small oscillations around the steady-state are evident for the substructural control technique
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1.5
x 10-4
Displacement (m)
complete substructural
1
0.5
0
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 11. Closed-loop response at node 3 in y direction (2).
especially in y direction at node 3 due to one of its controller eigenvalues being slightly close to the imaginary axis compared with the complete structural technique. These oscillations are less in the case of the second fuzzy approach (2) compared with the first one (1). 7. Conclusions A substructural control technique is presented for the optimal feedback controller and observer designs of fuzzy nonlinear flexible structures using the LQG control method. The substructural control technique developed in this work can be used for a wide variety of structures having nonlinear effects and fuzziness in parameters. The accuracy of the substructural control technique as compared to the complete structural control technique is shown thorough a numerical example. It is true that with fast developments in computational hardware and software many control problems can now be run with no difficulty. But, it is also true that the advancements in technology bring systems which are much more complicated and complex than before. The control applications to these systems keep demanding more powerful computers. Hence it is advantageous to use the substructural control techniques to save on computations and time. It is hoped that the substructural control technique developed in this work will do just that. Acknowledgment The authors acknowledge the support provided by King Fahd University of Petroleum & Minerals through the university project ME/FUZZY/197.
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References [1] T.-J. Su, V. Babuska, R.R. Craig Jr., Substructure-based controller design method for flexible structures, J. Guidance Control Dyn. 18 (1995) 1053–1061. [2] G.C. Archer, Technique for the reduction of dynamic degrees of freedom, Earthquake Eng. Struct. Dyn. 30 (2001) 127–145. [3] M. Sunar, S.S. Rao, Optimal structural control by substructure synthesis, AIAA J 38 (2000) 2191–2194. [4] A.S. Day, Justification of the design of non-linear structures, Struct. Eng. Rev. 6 (1994) 237–243. [5] B. Kuran, H.N. Ozguven, Modal superposition method for non-linear structures, J. Sound Vib. 189 (1996) 315–339. [6] R. Outbib, A. Rachid, Control of vehicle speed: a nonlinear approach, Proc. IEEE Conf. Decision Control 1 (2000) 462–463. [7] M. Sunar, S.A. Al-Kaabi, O. Toker, Substructural control of nonlinear structures using LQG control, Proceedings of ESDA 2002: ASME Sixth Biennial Conference on Engineering Systems Design and Analysis, Istanbul, Turkey, July 8–11, 2002. [8] S.S. Rao, Multi-objective optimization of fuzzy structural systems, Int. J. Numer. Methods Eng. 24 (1987) 1157–1171. [9] F. Casciati, L. Faravelli, T. Yao, Control of nonlinear structures using the fuzzy control approach, Nonlinear Dyn. 11 (1996) 171–187. [10] J.M. Maciejowski, Multivariable Feedback Design, Addison-Wesley, Reading, MA, 1989 (Chapter 5).
Journal of the Franklin Institute 344 (2007) 646–657 www.elsevier.com/locate/jfranklin
Substructural control of fuzzy nonlinear flexible structures M. Sunara,, O. Tokerb a
Mechanical Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia b Systems Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia Received 7 February 2006; accepted 7 February 2006
Abstract It is advantageous to use the substructural and/or decentralized techniques in structural control to save on computations and time. In this paper, a generalized substructural approach is presented in the control of fuzzy nonlinear flexible structures with discrete sensors/actuators. The substructural control scheme is developed using the static condensation technique together with the LQG control method. The subcontrollers and subobservers designed at substructure levels are used to assemble the global controller and observer for the whole structure. Nonlinear effects are included in the structural formulations and a fuzzy methodology is adopted for handling the imprecision present in the structure modeling. The nonlinear and fuzzy schemes are applied to one structural control problem to illustrate the accuracy and capability of the substructural control technique. r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Substructural control; Nonlinear; Flexible structure; Fuzzy; LQG
1. Modelling and Control Numerical techniques, such as the finite element method (FEM), are frequently used in the modeling and control of flexible structures. However, for large flexible structures, the FEM results in a large number of ordinary differential equations often not practical to manipulate. Several substructural and decentralized control techniques have been Corresponding author. Tel.: +966 3 8604976; fax: +966 3 8602949.
E-mail address: [email protected] (M. Sunar). 0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2006.02.010
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
647
developed in the past to deal with the problem of efficiently designing controllers for large flexible structures at substructure levels [1,2]. In a substructural control method [3], the linear quadratic Gaussian (LQG) control approach was applied to flexible structures for the computation of feedback and observer matrices from substructure levels. Nonlinear effects may be present in flexible structures due to geometry, inertia, damping and material. The unmodeled nonlinear behavior can cause the flexible structures to fail without warning [4] and hence care should be taken to account for such a behavior. A modal superposition technique was used for analyzing the dynamic response of nonlinear structures [5]. A nonlinear approach was used to control the speed of a vehicle [6]. The nonlinear effects were included in the control of large flexible structures through the LQG control scheme [7]. The parameter variations introduced by the analysis model, the fuzzy and imprecise material and model properties, and the uncertain exogenous undesired environmental inputs may adversely influence the stability and performance characteristics of a control system. The concept of fuzzy set theory was presented for solving a general structural optimization problem involving multiple objectives [8]. Control of nonlinear structures using fuzzy control approach was considered in the active control of hysteretic structures subjected to environmental loads [9]. In this work, the concept of designing feedback controllers and observers for linear/ nonlinear flexible structures from substructure levels using the LQG control method [3,7] is extended to include the fuzziness in structural modeling. The proposed fuzzy substructural control scheme is tested using a structural control example. 2. Introduction Suppose that the equation of motion for a flexible structure is given as M x€ þ C d x_ þ Kx ¼ Du u þ Dw w,
(1)
where M, Cd and K are the mass, damping and stiffness matrices; Du and Dw are the input matrices due to controller and disturbance; x, u and w are the displacement, controller input and disturbance vectors, respectively. The flexible structure model in state-space form is expressed as [10] z_ ¼ Az þ Bu þ Gw, y ¼ Cz þ v,
(2)
where A, B, C and G are the state, controller input, output and disturbance matrices; z, y, and v are the state, output and sensor noise vectors, respectively. Various entries in the above equation are given as " # " # " # 0 I 0 x ; A¼ ; B¼ , z¼ M 1 K M 1 C d M 1 Du x_ " # 0 G¼ , ð3Þ M 1 Dw where I is the identity matrix. For the LQG control approach, it is well-known that the solution to the optimal u is obtained with a controller (feedback) gain matrix found
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
648
as [10] u ¼ K c z ¼ R1 BT Pc z,
(4)
where Pc satisfies the following Riccati equation: AT Pc þ Pc A Pc BR1 BT Pc þ Q ¼ 0.
(5)
The state estimation to Eq. (2) is accomplished by the optimal state observer, Kalman filter, through an observer gain matrix Kf given by K f ¼ Pf C T V 1 ,
(6)
where Pf satisfies a dual Riccati equation of Pf AT þ APf Pf C T V 1 CPf þ GW GT ¼ 0.
(7)
The eigenvalues of the matrices [A–BKc] and [A–KfC] are the eigenvalues for the combined controller/observer scheme. 3. Substructural control technique The LQG-based substructural control technique used in this study is very powerful for large flexible structures containing many members. The mathematical details of the technique can be found in Ref. [3]. The iterative procedure for the controller and observer design of a flexible structure using the substructural control technique is briefly outlined below. First, the structure is decomposed into substructures, which are numbered in an arbitrary manner. In the first iteration, the subcontroller (Kc) and subobserver (Kf) are designed or computed for the first substructure assuming no boundary forces due to subcontrollers of surrounding substructures. The boundary forces due the subcontroller of the first substructure are condensed to its boundary degrees of freedom. Subcontrollers and subobservers are designed for the remaining substructures with the inclusion of subcontroller forces condensed to their boundary degrees of freedom from previous substructures. The subcontrollers and subobservers are then assembled to obtain the global feedback and observer matrices Kc and Kf for the whole structure. The computation of eigenvalues for [A–BKc] and [A–KfC] completes the first iteration. The subsequent iterations are terminated when these eigenvalues cease changing appreciably. 4. Nonlinear effects In general, for a nonlinear flexible structure, the mass, damping and stiffness matrices given in Eq. (1) are not linear. The proposed substructural control technique assumes a linear structure and hence these matrices must be linearized in some fashion. The schematic of the substructural control technique for the controller and observer designs of the nonlinear flexible structure is shown in Fig. 1 [7]. The basic steps are explained below. 1. As mentioned before, the flexible structure is decomposed into r substructures and the substructures are numbered in some suitable manner. 2. The nonlinear M, Cd and K matrices are linearized to obtain the linear matrices M ‘ , C d‘ and K ‘ at the current state of z. This would be the initial state of z(0) at the beginning of iterations.
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
649
Decompose the structure into r substructures z = z(0) i=0 i = i+1 Yes
i
Set all subcontroller forces to zero k=0 k = k+1 Design subcontroller and subobserver for the kth substructure by considering subcontroller forces at its boundary Find subcontroller forces fB k and fB k+1 No
k=r?
Yes Assemble global controller and observer Compute closed-loop eigenvalues
E.values converged?
No
Yes z=z+Δz
z converged?
No
Yes Stop Fig. 1. Substructural control technique for a nonlinear flexible structure.
3. The global controller and observer are designed for the nonlinear structure using these linearized matrices in the substructural control technique whose steps are briefed in the previous section. 4. The state vector is incremented by z ¼ z þ Dz and the nonlinear iterative scheme continues with step 2 until the convergence on z is achieved. The number n in Fig. 1
ARTICLE IN PRESS 650
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indicates that the gain matrices for the controller (feedback) and observer are not calculated at every iteration of the nonlinear scheme, rather they are computed at every n iterations. Hence, an approach similar to the gain scheduling has been utilized.
5. Inclusion of fuzziness The fuzziness in the structure can be due to modeling errors, uncertainties in material properties and damping, and imprecisely known effects of reduced order models. However, some upper and lower limits for the fuzziness can be assumed. The effect of fuzziness on nonlinear structure analysis can then be found by structural response and the LQG controller can be designed using these upper and lower limits. This is the approach taken in this study. The procedure on the substructural LQG controller and observer designs for a nonlinear fuzzy structure is implemented as follows. The upper and lower limits of fuzziness are noted for the subsequent nonlinear analysis. The nonlinear structural responses are found at these limits. The LQG control law for the nonlinear structure is then carried out using the nonlinear structural responses at these limits. The nonlinear controller (feedback) and observer designs are carried out as described in the previous section. This solution procedure is clearly implemented in the next section on a numerical example. 6. Numerical example The two-bay truss shown in Fig. 2 is taken as the numerical example. The truss is made up of aluminum with a modulus of elasticity of 69 GPa and mass density of 2770 kg/m3. The cross-sectional areas of all the members of the structure are taken as 6:45 104 m2 . The actuators and sensors are collocated at nodes 1 and 3. Actuators can generate forces and sensors can measure displacements in x and y directions at these nodes. When the substructural control technique is used, the structure is decomposed into two substructures as shown in Fig. 3. The nonlinearity of the structure is assumed to be due to the nonlinear
1.27m
1.27m
5
3 1
3
5
7
10 2
1
1.27m
8
9
y 6
4 6
2 4
Fig. 2. Two-bay truss.
x
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
substructure 1 5
substructure 2 3 3
5
651
1
7
10 2
1
9
8
6 6
4
2 4
Fig. 3. Substructures for two-bay truss.
spring behavior of the 5th member of the truss whose linearized stiffness is given as k5 ¼
aE 5 A5 x¯ 5 , L5
(8)
where E5, A5 and L5 are the modulus of elasticity, cross-sectional area and length, x¯ 5 is the current horizontal displacement for the member at node 3, and a is added as a parameter for tuning the amount of nonlinearity. The structural response is found at the two limits of fuzziness on the modulus of elasticity and the average (steady-state) values of responses are noted. The LQG controller and observer are then designed for the nonlinear structure using these average displacement values on the nonlinear member 5. The displacements for the 5th member in the horizontal direction at these two limits are calculated as 1:016 104 m (d1) and 1:524 104 m (d2). The fuzzy control logic is implemented using two approaches. In one approach, the feedback controller and observer matrices are found via the substructural control technique using the above displacement values for the nonlinear (but linearized) member 5. Two controllers and observers are found, one corresponding to the displacement d1 (LQG 1) and the other to d2 (LQG 2). The following linear membership functions (membership function 1) are assumed: 8 0 if dpd 2 ; > > < dd 2 if d 2 odod 1; (9) mLQG1 ¼ d1 d2 > > : 1 if dXd 1 ;
mLQG2
8 0 > > < d d 1 ¼ > d1 d2 > : 1
if dXd 1 ; if d 2 odod 1 ; if dpd 2 ;
where d denotes the horizontal displacement of member 5 at node 3.
(10)
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In the other fuzzy approach, three variations of modulus of elasticity are assumed and the corresponding steady-state displacements are found from the structural response. The displacements are noted as d1 and d2 like before, and a third displacement of 1:27 104 m (d3), all of which correspond to a controller and observer design called LQG 3. The following linear membership functions (membership function 2) are taken: 8 0 if dpd 3 þ e; > > < d d e 3 if d 1 eodod 3 þ e; (11) mLQG1 ¼ d 1 d 3 2e > > : 1 if dXd 1 e;
mLQG2 ¼
mLQG3
8 0 > > > > d1 d e > > > < d d 2e 1
if dXd 1 e or dpd 2 þ e; if d 3 þ eodod 1 e;
3
d d2 e > > > > > d 3 d 2 2e > > :1
8 0 > > < d d e 3 ¼ d d 2 2e > 3 > : 1
(12) if d 2 þ eodod 3 e; if d 3 epdpd 3 þ e; if dXd 3 e; if d 2 þ eodod 3 e;
(13)
if dpd 2 þ e;
where e represents a value taken to represent a small neighborhood. For the LQG control law, the Q, R, W and V are taken as diagonal matrices with appropriate dimensions whose diagonal terms are assumed to be 106, 1, 10 and 1,
1
x 10 -4 complete substructural
Displacement (m)
0.5
0
-0.5
-1
-1.5
-2
0
2
4
6
8
10
12
14
16
18
Time (s) Fig. 4. Closed-loop response at node 1 in x direction (1).
20
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
x 10-4
1.5
complete substructural
1
Displacement (m)
653
0.5
0
-0.5
-1
-1.5
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 5. Closed-loop response at node 1 in x direction (2).
4
x 10-4 complete substructural
3.5
Displacement (m)
3 2.5 2 1.5 1 0.5 0
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 6. Closed-loop response at node 1 in y direction (1).
respectively. The time step for iterations is chosen as 0.01 s, but the feedback and observer gain matrices are calculated at the initial step and at every 0.25 s afterwards to speed up the iterative process. Hence the number n in Fig. 1 is 25.
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M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
x 10-4
4
complete substructural
3.5
Displacement (m)
3 2.5 2 1.5 1 0.5 0
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 7. Closed-loop response at node 1 in y direction (2).
1.5
x 10-4 complete substructural
Displacement (m)
1
0.5
0
-0.5
-1
-1.5
0
2
4
6
8
10
12
14
16
18
20
Time(s) Fig. 8. Closed-loop response at node 3 in x direction (1).
The results are shown in Figs. 4–11. The results of the first fuzzy approach are shown as (1) and those of the second approach are referred as (2) in these figures. Reference or input forces are assumed to be step forces acting at nodes 1 and 3 in positive horizontal and vertical directions (+x and y) with 100 N magnitudes. It is clear from the figures that the
ARTICLE IN PRESS M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
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655
x 10-5 complete substructural
4
Displacement (m)
2 0 -2 -4 -6 -8 -10 -12 -14
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 9. Closed-loop response at node 3 in x direction (2).
1.6
x 10-4
1.4 complete substructural
Displacement (m)
1.2 1 0.8 0.6 0.4 0.2 0
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 10. Closed-loop response at node 3 in y direction (1).
results of the complete and substructural control techniques are close to each other. Both techniques converge to the steady-states in a similar manner. However, some small oscillations around the steady-state are evident for the substructural control technique
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M. Sunar, O. Toker / Journal of the Franklin Institute 344 (2007) 646–657
1.5
x 10-4
Displacement (m)
complete substructural
1
0.5
0
0
2
4
6
8
10
12
14
16
18
20
Time (s) Fig. 11. Closed-loop response at node 3 in y direction (2).
especially in y direction at node 3 due to one of its controller eigenvalues being slightly close to the imaginary axis compared with the complete structural technique. These oscillations are less in the case of the second fuzzy approach (2) compared with the first one (1). 7. Conclusions A substructural control technique is presented for the optimal feedback controller and observer designs of fuzzy nonlinear flexible structures using the LQG control method. The substructural control technique developed in this work can be used for a wide variety of structures having nonlinear effects and fuzziness in parameters. The accuracy of the substructural control technique as compared to the complete structural control technique is shown thorough a numerical example. It is true that with fast developments in computational hardware and software many control problems can now be run with no difficulty. But, it is also true that the advancements in technology bring systems which are much more complicated and complex than before. The control applications to these systems keep demanding more powerful computers. Hence it is advantageous to use the substructural control techniques to save on computations and time. It is hoped that the substructural control technique developed in this work will do just that. Acknowledgment The authors acknowledge the support provided by King Fahd University of Petroleum & Minerals through the university project ME/FUZZY/197.
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References [1] T.-J. Su, V. Babuska, R.R. Craig Jr., Substructure-based controller design method for flexible structures, J. Guidance Control Dyn. 18 (1995) 1053–1061. [2] G.C. Archer, Technique for the reduction of dynamic degrees of freedom, Earthquake Eng. Struct. Dyn. 30 (2001) 127–145. [3] M. Sunar, S.S. Rao, Optimal structural control by substructure synthesis, AIAA J 38 (2000) 2191–2194. [4] A.S. Day, Justification of the design of non-linear structures, Struct. Eng. Rev. 6 (1994) 237–243. [5] B. Kuran, H.N. Ozguven, Modal superposition method for non-linear structures, J. Sound Vib. 189 (1996) 315–339. [6] R. Outbib, A. Rachid, Control of vehicle speed: a nonlinear approach, Proc. IEEE Conf. Decision Control 1 (2000) 462–463. [7] M. Sunar, S.A. Al-Kaabi, O. Toker, Substructural control of nonlinear structures using LQG control, Proceedings of ESDA 2002: ASME Sixth Biennial Conference on Engineering Systems Design and Analysis, Istanbul, Turkey, July 8–11, 2002. [8] S.S. Rao, Multi-objective optimization of fuzzy structural systems, Int. J. Numer. Methods Eng. 24 (1987) 1157–1171. [9] F. Casciati, L. Faravelli, T. Yao, Control of nonlinear structures using the fuzzy control approach, Nonlinear Dyn. 11 (1996) 171–187. [10] J.M. Maciejowski, Multivariable Feedback Design, Addison-Wesley, Reading, MA, 1989 (Chapter 5).