Proceedings of the Institution of Civil Engineers Structures & Buildings 152 May 2002 Issue 2 Pages 175^191 Paper 12191 Received 02/11/1999 Accepted 09/01/2001 Keywords: design methods & aids/materials technology
Hamid Reza Mirdamadi Assistant Professor,Civil Engineering Department, Isfahan University of Technology, Isfahan, Iran
Ali Akbar Golafshani Assistant Professor,Civil Engineering Department, Sharif University of Technology,Tehran, Iran
Adaptive control of structures by LMS algorithm: a comparative study A. A. Golafshani and H. R. Mirdamadi By using the normalised least mean squared (NLMS) algorithm, a semi-active multi-variable adaptive controller is designed for a seismically excited structure. There is no need for a large power supply. A number of valves and battery size low-power supplies will suffice. The valves control the amount of flow of a fluid through bypass on-off orifice channels in installed energy dissipating mechanisms. Each mechanism is composed of a piston attached to a ¸-shaped chevron wind-bracing on each floor and to a cylinder attached to the upper floor. Adaptive controller parameters are estimated by the LMS optimiser, in order to search for optimal nonclassical damping coefficients of the dissipating systems. They are equivalent to changing the orifice size of the bypass channels. Two versions of the LMS algorithm are used for comparative purposes: the filtered-x LMS and the classical (non-filtered) LMS. For the filtered-x version; two cases are distinguished for a more refined comparison: first, required structural transfer functions for estimating filtered-x signals are assumed to be exact and, second, these transfer functions are assumed not to be exact. Simulation results show that structural responses obtained by filtered-x LMS are more optimised than that of classical LMS, even in the case when the estimation of filtered-x signals is assumed to be poor. 1. INTRODUCTION Semi-active control schemes have been considered as preferred 1 alternatives for the passive and active control of structures. Semi-active control offers advantages that may be observed by comparison. In active control, the system under control is both fully intelligent and energetic. The controller receives the required information about both the behaviour of the structure under control and environmental excitations by sensors. It is able to use the external energy provided by a conceptually 2 unlimited power supply for regulating the responses as well. In a semi-active scheme, the controller is fully informative but not fully energetic. It receives the information it requires but 3 there would be a limited source of external energy. In this strategy, only limited energy is required, i.e. several orders of magnitude less than that required for active control. To change the internal dynamic characteristics of a structure by on-off switching or by adjusting the diameter of an orifice in a Structures & Buildings 152 Issue 2
variable-orifice damper, little energy will be needed. In passive control, the controller is neither informative nor energetic. It 3 neither knows nor changes its conditions. Control strategies based on semi-active devices are able to combine the best features of both passive and active controls. They offer the greatest likelihood for the acceptance of control technology as a means of protecting structures against an earthquake.
2. VARIABLE-ORIFICE ENERGY DISSIPATING MECHANISM One means of achieving a variable damping device for dissipating seismic energy is to use a number of controllable, variable-orifice valves for altering resistance to the flow of a 4 hydraulic fluid. The construction of such a system is highly expensive and requires a high level of technology. Another method is by converting a passive-type damper 5 device, similar to a typical Taylor device, to a hybrid damper, and this is the method proposed by the authors. This is accomplished by adding a number of bypass oil intakes that can be switched on and off through the use of control signals. In the proposed mechanism, there are some closed containers into which the oil can flow. There are a number of orifices in the head of the pistons that make the direct passive-type channels for the oil flow, similar to its passive-type counterpart. So, there is always some passive energy dissipation by oil, which is flowing due to the movement of pistons relative to the containers when some drift occurs between adjacent floors. However, in addition to this passive energy dissipation mechanism, oil discharge is regulated by some controlled valves through the bypass oil intake semiactive mechanism. These valves are powered by battery-size power supplies and are controlled by signals from adaptive filters. Mathematically, a person is changing the damping coefficients of the above defined energy dissipating system automatically. So, that person is designing an optimal controller to do the regulation adaptively. A schematic of the 5 Taylor device, first designed and tested by Constantinou et al. 6 and afterwards by Symans et al., is shown in Fig. 1. The hybrid version of this device proposed by the authors is shown in Fig. 2. The function of this hybrid on-off orifice damper is interpreted in a similar way to a variable-orifice
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distributed and nonviscous damping property of the material. (b) The non-classical damping part represents the physical local viscous property of the oil flowing through valves. A structure including non-classical damping is formulated more conveniently in a state– 7 space time-domain.
Fig. 1. Passive viscous fluid damper of Taylor device
8,9
Fig. 2. Semi-active on-off orifice viscous fluid damper (proposed by the authors)
A multi-variable, threeinput, three-output, adaptive controller based on both filtered-x and classical (nonfiltered) versions of the normalised least mean square 10 (LMS) optimiser is designed. By using a heuristic control law, the control forces obtained from this adaptive strategy are converted into some change in the characteristics of the non-classical damping property through control of the orifice diameters. In the case of the filtered-x version of the LMS algorithm, it is also assumed that:
device, by the following reasoning. By installing three on-off orifice dampers in a three-floor framed structure, and by devising six semi-active oil intakes for each device, it is possible to have 26 ¼ 64 distinct values for the amount of nonclassical damping coefficient of each damper at each computational time-step. By increasing the number of semiactive oil intakes, the number of accessible damping values increases very rapidly. Therefore, it may be assumed that step variation of the device damping coefficient is continuous and this on-off orifice device is acting in an approximately similar way to a variable-orifice system. 3. THE STRUCTURAL MODELLING A three-storey framed-structure that has rigid floors and ¸shaped chevron wind-bracings is considered for this case study. The vertices of the bracing systems are attached rigidly to pistons that are free to move horizontally through some cylinders. The cylinders are fixed rigidly to the upper floors, as shown in Fig. 3. A rough finite element representation for modelling the structure is used. The finite element model consists of structural stiffness, consistent mass and damping matrices. The damping matrix is composed of two distinct parts. Fig. 3. Physical structure (a) The proportional Rayleigh damping part represents 176
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(a) for some simulations the required structural transfer functions are assumed to be known exactly (b) for others, the estimations are assumed to be not exact. It is worth noting that owing to the adaptive nature of LMS 11, 12 algorithms, satisfactory results are attained in both cases. 4. ANALYTICAL FORMULATION The horizontal vibrations of a three-storey structure are modelled by the first six modes of the natural system vibration. There are three principal degrees of freedom (DOF) for the floors and another three for the vertices of the ¸-shaped chevron wind-bracing, as is shown in Figs 3 and 4. The second-order matrix equation of the reduced model may 13 be written as follows 1
M€x þ Cp x_ þ Cb (t) x_ þ Kx ¼ Mrg €xg (t)
where M, Cp , Cb (t) and K are the mass, proportional Rayleigh damping, non-classical damping and stiffness matrices, respectively. The x and its derivatives are relative displacement, velocity and acceleration vectors and xg is the ground motion displacement applied at foundation relative to a reference inertial frame. The rg is a location vector to show the extent and distribution of excitation on each DOF.
Rc f c (t) ¼ Cb (t) x_
3
where the f c (t) is the assumed virtual control force vector and the Rc is a location matrix. One may imagine that, instead of having internal forces in the structure, produced by an energy dissipating mechanism, there are some internal virtual actuators that are able to apply some external forces to the structure equivalent to those internal forces. Therefore, this semi-active control problem is converted conceptually to a conventional active control problem by using, for example, an adaptive control law. An adaptive forward feed control strategy is selected due to the non-stationary nature of the disturbance process and the quick response of forward feed 14 compensators. It is noted that, mathematically, an adaptive forward feed controller is equivalent to a non-adaptive 14, 15 feedback controller. The continuous time-state variable vector of the structure is defined by z ¼ bx T
4
The first-order equations are written in this form " z_ ¼
Rearranging the equation by transferring internal non-classical damping forces to the rhs 2
5
M€x þ Cp x_ þ Kx ¼ Mrg €xg (t) Cb (t) x_
The last term of the above equation may be interpreted as a virtual control input vector
x_ T cT
0 n3n
I n3 n
#
"
0 n31
#
€xg (t) zþ M 1 K M 1 Cp rg " # " # 0 n33 0 n33 þ f c (t) þ ø(t) M 1 Rc Rn
Or more compactly as 6
z_ ¼ Az þ bg €xg (t) þ Bc f c (t) þ Eø(t)
This equation defines a dynamic system under a random seismic excitation and three control actions. A white noise excitation ø(t) is added, in order to incorporate local structural non-linearities, actuator dynamics and other unmodelled 16 dynamics in the system, indirectly. As shown in Fig. 5, three structural sensors for the three floors and an array of ground sensors near foundation are installed. For each storey, one adaptive filter, corresponding to one ¸shaped chevron wind-bracing is implemented. The output measurement equation for the sensors is
7
Fig. 4. Finite-element model
Structures & Buildings 152 Issue 2
Y(t) ¼ Cz(t) þ d€xg (t) þ V(t)
where Y(t) is a vector of structural responses that should be controlled, d is an influence coefficient of an earthquake—it is a zero vector for this study but is not zero where one deals with soil-structure interaction—and V(t) is the measurement noise vector that also can incorporate echo effects from 17, 18 structural response to ground motion. Adaptive control of structures by LMS algorithm
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Fig. 5. Complete dynamic system
As an example, if the designer’s objective is controlling base shear, C is defined by 8
C ¼ bK
Cp þ Cb (t)c
The derivation of a three-input, three-output version of the filtered-x LMS algorithm is presented in Appendix 1. A block diagram of the LMS algorithm, for the second floor, is shown in Fig. 6. 5. SEMI-ACTIVE CONTROL LAW Up to this point, the assumptions have been compatible with the conventional active controllers that consume a large
amount of external energy and need a large power supply. However, it should be remembered from equation (3) that, by designing f c (t), one may actually find the optimal values for Cb (t), the non-classical damping matrix of energy dissipating systems. The equations (5) and (6) are expanded in order to see the facts more clearly
9
8 > <
10
> :
z_ ¼ Az þ bg €xg (t) þ
3 X
b cm f cm (t) þ Eø(t)
m ¼ 1, 2, 3
m¼1
y m ¼ C m z þ d m €xg (t) þ V m (t) Y ¼ [ y1 , y2 , y3 ] T
From equations (3), (5), (6) and (9) r cm f cm (t) ¼ C bm (t) x_ m 11
m ¼ 1, 2, 3
where the matrices Rc and Cb (t) have been partitioned to the corresponding vectors rcm and the corresponding matrices Cbm (t). Consequently, the following scalar equation is obtained
f cm (t) ¼ C bm (t) 12
3 [ x_ m x_ mþ n ] m ¼ 1, 2, 3
Fig. 6. A block diagram for the second floor
178
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Adaptive control of structures by LMS algorithm
The index n is the total number of dissipating systems, and the C bm (t) is a measure of the damping coefficient of that system. Golafshani • Mirdamadi
From reasons of stability, they are not allowed to become negative
13
f cm (t) x_ m (t) x_ mþ n (t)
m ¼ 1, 2, 3, Normalised
C bm (t) ¼
7·8159
x_ m x_ mþ n 6¼ 0
Half power bandwidth line (HP BW) 3·908
Owing to the practical constraints in construction, the C bm variations are limited between two upper and lower practical limits 0
C lower < C bm (t) < C upper 8t
14
Actually, a parametrically excited non-linear vibration problem is being run, in which the sources of parametric excitations are the internal change of dynamic characteristics of the system.
2
3
4 5 6 Frequency: Hz
7
8
28
32
9
10
31·5704
Percent of g
Elcentro earthquake, USA, 1940 Whittier earthquake, USA, 1987 Manjil earthquake, Iran, 1990 Landers earthquake USA, 1992.
1
Fig. 8. The periodogram (Elcentro)
6. NUMERICAL RESULTS AND DISCUSSION In the numerical simulations, four earthquake signals with different frequency contents are applied to a three-storey flexible structure for the first 20 s part of the strong motion. The earthquakes are (a) (b) (c) (d)
0
0
19
34·8318
Percent of g
–22·2571
4
8
12
16 20 24 Time: s
36
40
3·082
HP BW
1·541
0
0
0
Fig. 9. Whittier earthquake, 1987
Normalised
The time histories and the periodograms of these earthquake signals are shown in Figs 7–Fig 14. The ground accelerations have been scaled relative to the gravitational field near the earth (9:81 m=s2 ). These values have been re-scaled for this study. A periodogram is a multiplication of the FFT (Fast Fourier Transform) of the seismic signal by itself, to account for an experimental auto PSD (auto power spectral density 23 function) of the random signal. Auto PSD is an index of the relative energy contained in each frequency content of that 19 signal. These values have been normalised by the window 23 length used (in this study, the Hanning window).
0
1
2
3
4 5 6 Frequency: Hz
7
8
9
10
Fig. 10. The periodogram (Whittier) –26·8196
0
5
10
15
20
25 30 Time: s
Fig. 7. Elcentro earthquake, 1940
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35
40
45
50
In addition, each seismic disturbance is applied to this structure, equipped with three different versions of the LMS optimiser. Two of them are the filtered-x and classical (unfiltered) forms of the normalised LMS (NLMS). In contrast to
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14·7567
Normalised
Percent of g
56·1162
0
–45·9939
0
5
10
15
20
25 30 Time: s
35
40
45
0
50
Fig. 11. Manjil earthquake, 1990
Normalised
0
1
2
3
4 5 6 Frequency: Hz
7
8
9
10
Fig. 14. The periodogram (Landers)
the stiffness matrix used for formulating system TFs is estimated to be twice as large as the true values in the actual simulated structure. In addition, no damping matrix has been estimated or included in the system TFs for the third algorithm. For a simple reference, these algorithms are abbreviated as follows
7·7388
HP BW
3·8694
(a) filtered-x normalised LMS algorithm (FLMS) (b) inexact structure-transfer function estimate version of the filtered-x normalised LMS algorithm (ITFE) (c) classical (unfiltered) normalised LMS algorithm (CLMS). 0
0
1
2
3
4 5 6 Frequency: Hz
7
8
9
10
Fig. 12. The periodogram (Manjil)
50·2661
Percent of g
HP BW
7·3783
It should be stressed that the variations examined in this study are related to controller software not to the physical structure itself. Actually, there are two different finite-element models of a structure in the memory of the computer laboratory in the civil engineering department of Sharif University of Technology. It is assumed one of these models is an exact mathematical model of a hypothetical physical structure while the other model is a rough estimation of that hypothetical structure. In the MATLABTM graphs that follow (Figs 15– 20–25 37), only the simulation results of the semi-active scheme for the filtered-x NLMS version of the control algorithm and
0
14·883
–60·2885
Cm
4·5719
0
10
20
30
40 Time: s
50
60
70
0
80 –4·8158
Fig. 13. Landers earthquake, 1992 –15·8566
the filtered-x version, there is no need for estimating structural system transfer functions (TFs) in the classical form of the NLMS. The third algorithm is a filtered-x version in which the structural system TF estimates are not exact. It is assumed that 180
Structures & Buildings 152 Issue 2
0
2
4
6
8
10 12 Time: s
14
16
18
20
Fig. 15. Third floor relative displacement
Adaptive control of structures by LMS algorithm
Golafshani • Mirdamadi
43·6248
3rd floor max displacement: %
3rd floor max displacement: %
61·0767
19·389 0
–26·3642
–67·2527
0
2
4
6
8
10 12 Time: s
14
16
18
0
2
4
6
8
10 12 Time: s
14
16
18
20
29·9844
18·4104 12·6947 Percent of g
3rd floor max displacement: %
–8·9126
Fig. 19. Drift bandwidth, third and second floors
37·9591
0
0
–18·2071 –16·6428
–40·6935
0
2
4
6
8
10 12 Time: s
14
16
18
–28·1225
20
Fig. 17. First floor relative displacement
0
2
4
6
8
10 12 Time: s
14
16
18
20
16
18
20
Fig. 20. First floor absolute acceleration
37·9446
29·0214
19·9313
10·3706
Percent of g
3rd floor max displacement: %
0
–44·9209
20
Fig. 16. Second floor relative displacement
11·6566
0 –12·7165
0
–14·4846 –36·528
0
2
4
6
8
10 12 Time: s
14
16
18
20
Fig. 18. Drift bandwidth, second and first floors
the passive scheme are shown. The numerical results for all three versions of semi-active control algorithms and the passive system are then tabulated. These values are given for each of the seismic signal inputs listed above. Structures & Buildings 152 Issue 2
–22·3592
0
2
4
6
8
10 12 Time: s
14
Fig. 21. Second floor absolute acceleration
The first six natural frequencies and proportional modal damping properties of the system are given in Table 1. It must be emphasised that there are two simulations
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23·7056
15·6792
kJ
21·9207
Percent of g
54·3974
0
–23·435
–52·9059
0 0
2
4
6
8
10 12 Time: s
14
16
18
20
Fig. 22. Third floor absolute acceleration
0
4
6
8
10 12 Time: s
14
16
18
20
14
16
18
20
14
16
18
20
Fig. 25. Earthquake input energy
8·3818
15·8674
0 kJ
Building weight: %
2
5·4481
–11·1537 –18·2071
0
0
12·6947
0
2
4
6
8
3rd floor max displacement: %
Fig. 23. Base shear displacement 1 (controlled)
Fig. 26. Potential and kinetic energy
19·3463
8·9394
0
kJ
Building weight: %
10 12 Time: s
1·9231
–20·8615 –40·6935
0
37·9591
0 0
2
4
6
3rd floor max displacement: %
Fig. 24. Base shear displacement 1 (no control)
performed with each seismic signal and each controller algorithm. The first group of simulations is termed uncontrolled or passive, because during those simulations, damper coefficients of dissipation systems are kept fixed as 182
Structures & Buildings 152 Issue 2
8
10 12 Time: s
Fig. 27. Potential energy
in a passive control scheme. This case is equivalent to a passive design. The second group of simulations is termed controlled, because the damper coefficients are adjusted continuously online, between two practical limits, based on
Adaptive control of structures by LMS algorithm
Golafshani • Mirdamadi
13·9693
kJ
cb2/kb2
0·20
5·0645 0·05
0
0 0
2
4
6
8
10 12 Time: s
14
16
18
20
Fig. 28. Kinetic energy
2
4
6
8
10 12 Time: s
14
16
18
20
Fig. 31. Second floor damper coefficient
19·3463
1·7663
8·3818 0 1st floor
Building weight: %
0
0
–11·1537
–20·8615
0
2
4
6
8
10 12 Time: s
14
16
18
–3·2605
20
Fig. 29. Total base shear force
2
4
6
8
10 12 Time: s
14
16
18
20
Fig. 32. Adaptive controller weights (first floor)
35·1616
2nd floor
17·7078
Building weight: %
0
0
0
–14·6102
0
2
4
6
8
10 12 Time: s
14
16
18
20
Fig. 30. First floor control force
0
2
4
6
8
10 12 Time: s
14
16
18
20
Fig. 33. Adaptive controller weights (second floor)
one of the three semi-active controller softwares explained previously. The simulation results are obtained by MATLABTM and Structures & Buildings 152 Issue 2
–8·7327
20–25
SIMULINK TM and their toolboxes. The relative displacements and drifts between floors are normalised relative to the peak absolute value of the third-floor displacement of the uncontrolled (passive) structure, the floor absolute
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183
0
1·7355
3rd floor
3rd floor adaptive controller
–20 –40 –60 –80 –100 –120 –140 –167·9808
0
2
4
6
8
10 12 Time: s
14
16
18
20
Fig. 34. Adaptive controller weights (third floor)
0
1st floor adaptive controller
8 10 12 14 16 18 Time index (sample No n)
20 22 24
Natural frequencies: Hz
Damping ratio: %
0·6903 1·5716 2·2028 3·5588 4·6115 6·3837
1·0 2·0 3·0 4·0 5·0 6·0
–1
–2
Table 1. Dynamic properties of a simulated structure
0
0
2
4
6
8 10 12 14 16 18 Time index (sample No n)
20 22 24
Fig. 35. Discrete pulse response at 20 s
lower limits of each graph are reserved for maximum and minimum values of uncontrolled responses. For a clear comparison, a numerical factor called the RRF (response reduction factor) is used. The RRF is a percentage relative to the reduction of responses between the uncontrolled and controlled systems. A brief comment and discussion now follows on the figures and tables. The only points shown in the figures are the responses obtained by considering the application of Manjil, the earthquake of Iran in 1990, to the filtered-x version of the NLMS algorithm, whose transfer functions are to be assumed exact. However, in Tables 2–13, the major responses for all of the algorithms and earthquakes mentioned previously are listed and compared by RRFs.
30 25 2nd floor adaptive controller
6
1 2 3 4 5 6
–3
20 15 10 5 0 –5 0
2
4
6
8 10 12 14 16 18 Time index (sample No n)
20 22 24
Fig. 36. Discrete pulse response at 20 s
accelerations relative to the gravity and the force responses relative to the building weight. The numerical values of maximum and minimum controlled responses are marked by dotted horizontal straight lines in graphs, while the upper and 184
4
Fig. 37. Discrete pulse response at 20 s
Mode no. 1
2
Structures & Buildings 152 Issue 2
Figure 15 is third-floor displacement relative to the foundation. From Table 8, it is observed that the maximum absolute value of the third-floor displacement for FLMS is larger than those of ITFE and CLMS. Although, its difference with ITFE is minor. In fact, this is almost the case for the other excitations. It may be deduced that an ITFE has no considerable effect on the behaviour of FLMS. Nevertheless, FLMS is superior to CLMS for the reduction of maximum displacements. RRF for this response and for earthquake excitation, together with FLMS, is 69·63%. Figs 16 and 17 show the same quantity for the second and first floors. Figures 18 and 19 show drifts between second and first, and third and second floors, respectively. FLMS has reduced the
Adaptive control of structures by LMS algorithm
Golafshani • Mirdamadi
Fl.
Filtered-x NLMS algorithm
3 2 1 Average
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
10·97 38·39 19·48
50·87 41·98 50·84 47·90
11·33 38·38 19·53
49·22 41·99 50·70 47·30
11·21 38·59 19·65
49·79 41·68 50·42 47·30
No control
22·32 66·17 39·62
Table 2. Maximum relative displacements (Elcentro earthquake, 1940)
Fl.
3 2 1 Average
Filtered-x NLMS algorithm
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
37·62 20·09 21·48
41·01 38·94 30·17 36·71
31·44 19·46 21·24
50·71 40·88 30·97 40·85
31·34 19·32 21·21
50·86 41·29 31·05 41·07
No control
63·77 32·91 30·77
Table 3. Maximum absolute accelerations (Elcentro earthquake, 1940)
Fl.
3 2 1 Average
Filtered-x NLMS algorithm
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
6·612 11·90 13·45
48·66 35·28 48·38 44·11
6·627 11·65 13·69
48·55 36·62 47·46 44·21
6·605 11·72 13·76
48·72 36·25 47·18 44·05
No control
12·88 18·38 26·05
Table 4. Maximum cumulative storey shears (Elcentro earthquake, 1940)
Fl.
Filtered-x NLMS algorithm Control
RRF
Inexact TF estimate version Control
RRF
Classical NLMS algorithm Control
No control
the position of the valves a fewer number of times than FLMS during the occurrence of a seismic event. In fact, CLMS has performed suboptimally. This is the deficiency of the algorithm that the sudden change of valve positions, commanded by the control signal, will develop larger absolute accelerations. Figures 23 and 24 show force–displacement curves. Force is base shear and displacement is drift between the first floor and foundation. Hysteresis loops for a controlled structure are much wider than those are for an uncontrolled structure. It is observed how large a damping force would be developed in a controlled structure, as compared to an uncontrolled one. Figure 25 is the amount of external energy injected from the seismic signal to the structures. Seismic input energy is more for an uncontrolled structure. In addition, it is shown in Fig. 26 that absorbed dynamic energy is more for an uncontrolled structure. The reason is that much more energy is damped out in the semi-active dampers of a controlled structure. Figs 27 and 28 show potential and kinetic energy time-histories of the structural systems.
RRF
Figure 29 shows base-shear time-history. As mentioned 3 4·42 55·23 4·406 55·37 4·387 55·57 9·874 before, the cost function of 2 32·61 57·13 33·17 56·39 33·29 56·22 76·06 1 19·58 53·07 18·57 55·48 18·85 54·82 41·71 an adaptive optimiser has Average 55·14 55·75 55·54 been selected based on the error signal obtained from the base-shear response. Table 5. Maximum relative displacements (Whittier earthquake, 1987) From Table 10 (Manjil earthquake) it can be seen that the RRF is largest (46·53%) for FLMS. Again, this is almost the case for the other kinematic quantities (i.e. the drifts) related to the potential earthquakes. It may be possible to deduce a specific conclusion energy, more than the other two algorithms. by comparing the values obtained for FLMS and CLMS, and by considering all four excitations. FLMS optimises cost function Figures 20–Fig 22 are floor absolute accelerations. On average, better than CLMS. In addition, it is clear that an inexact TF CLMS has reduced third floor absolute accelerations more than estimate is not a major problem for an algorithm, the reason FLMS. This may be related to the fact that CLMS has changed Structures & Buildings 152 Issue 2
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185
Fl.
3 2 1 Average
Filtered-x NLMS algorithm
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
15·82 10·26 11·90
49·46 48·19 40·93 46·19
15·21 9·553 12·21
51·41 51·74 39·38 47·51
13·88 10·29 13·46
55·67 48·04 33·19 45·63
No control
31·31 19·80 20·14
Table 6. Maximum absolute accelerations (Whittier earthquake, 1987)
Fl.
3 2 1 Average
Filtered-x NLMS algorithm
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
3·274 5·517 6·669
46·85 34·15 45·89 42·30
2·786 5·305 6·931
54·77 36·68 43·77 45·07
2·91 5·347 7·201
52·76 36·18 41·57 43·50
No control
6·16 8·378 12·33
Table 7. Maximum cumulative storey shears (Whittier earthquake, 1987)
Fl.
3 2 1 Average
Filtered-x NLMS algorithm
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
4·816 26·36 18·21
69·63 60·80 55·26 61·90
4·844 27·61 19·03
69·45 58·95 53·24 60·55
5·986 32·09 19·29
62·25 52·28 52·61 55·71
No control
15·86 67·25 40·69
Table 8. Maximum relative displacements (Manjil earthquake, 1990)
Fl.
3 2 1 Average
Filtered-x NLMS algorithm
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
23·71 19·93 18·41
56·42 31·32 38·60 42·11
22·09 18·05 18·65
59·40 37·81 37·80 45·00
19·37 14·72 16·64
64·39 49·29 44·50 52·73
No control
54·40 29·02 29·98
a first-floor virtual actuator. If the system was an active controller, this force should have been applied to the system using transduction devices and so consuming lots of external energy. However, for this semi-active controller, these virtual control forces are divided by relative velocities to obtain measures for the semi-active damper parameters. Variation of the resulting non-classical damping coefficient for the variable-orifice damper, installed on the second floor, is shown in Fig. 31, between two practical limits. Figures 32–34 show the time evolution of adaptive filter weights of each floor. Convergence for the controller installed at the third floor is rapid (not shown). However, for first floor, the rate of convergence is low. Graphs showing time evolution of adaptive controller weights are a direct measure of the performance of an active controller. The time evolution of nonclassical damping coefficients of semi-active dampers are obtained by dividing virtual active control forces taken from LMS by relative velocities between each floor and lower bracing. Therefore, convergence (or divergence) of weights may be an indirect measure of stability, robustness and performance of an algorithm for a semiactive damper.
Figures 35–37 show FIR parameters of adaptive filters of all the floors for 20 s after Table 9. Maximum absolute accelerations (Manjil earthquake, 1990) the simulation is initialised. These parameters are the same as the discrete pulse response of an adjustable filter at that time. The number of parameters for all filters is being that an algorithm is adaptive. This is an inherent selected to truncate after 25 samples. Although the pulse performance of all adaptive systems—they are robust against response of a first-floor filter has not gone to zero for the unmodelled dynamics or even incorrect modelling. twenty-fifth parameter, the performance of the filter and the LMS are satisfactory. In addition, from Fig. 36, it may be Figure 30 shows the virtual control force obtained by LMS for 186
Structures & Buildings 152 Issue 2
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Golafshani • Mirdamadi
Fl.
Filtered-x NLMS algorithm
3 2 1 Average
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
4·576 9·065 11·15
57·56 29·71 46·53 44·60
4·284 7·903 11·71
60·26 38·72 43·85 47·61
4·07 7·235 11·65
62·25 43·90 44·16 50·10
No control
10·78 12·90 20·86
Table 10. Maximum cumulative storey shears (Manjil earthquake, 1990)
Fl.
Filtered-x NLMS algorithm
3 2 1 Average
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
7·071 26·46 15·34
64·98 64·87 61·22 63·69
7·085 27·11 15·55
64·91 64·01 60·69 63·20
9·10 35·73 19·83
54·93 52·58 49·88 52·46
No control
20·19 75·34 39·57
Table 11. Maximum relative displacements (Landers earthquake, 1992)
Fl.
Filtered-x NLMS algorithm
3 2 1 Average
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
29·45 16·65 14·12
36·62 48·27 28·95 37·95
29·10 15·75 14·36
37·36 51·06 27·75 38·72
24·58 18·33 14·38
47·10 43·03 27·63 39·25
No control
46·46 32·18 19·87
Table 12. Maximum absolute accelerations (Landers earthquake, 1992)
Fl.
Filtered-x NLMS algorithm
3 2 1 Average
Inexact TF estimate version
Classical NLMS algorithm
Control
RRF
Control
RRF
Control
RRF
4·717 9·239 12·81
50·52 46·44 47·73 48·23
4·508 9·215 12·77
52·71 46·58 47·88 49·06
5·269 10·82 14·62
44·72 37·26 40·34 40·77
No control
9·532 17·25 24·51
the hypothetical cylindrical semi-active damper with internal radius R, whose cross-sectional design is shown in Fig. 38. The piston head is characterised by its radius Rp and axial length Lp . The chambers are filled with an incompressible viscous fluid obeying Newton’s Viscosity Law and having dynamic viscosity and density r. Meanwhile, the piston rod is assumed to move in the axial direction with velocity V, forcing the fluid through the annular passage of width h ¼ R Rp and, thus, producing a pressure differential across the head. Assume that an accumulator is present to compensate for the volumetric changes associated with the piston rod. There are two groups of oil intakes. The first group is designed as the annular passage around the main piston head within the cylinder and for a minimum passive action to be available all the time. The second group is provided through bypass passages outside the main cylinder for semi-active action, possibly by the same mechanism of annular passages around some of the piston heads. The time deviations of non-classical damping coefficients are developed in these bypass intakes. If it is assumed that there are six bypass intakes, all having ¼ 0:26 kg=(m=s), Lps ¼ 0:5 m and Rps ¼ 0:1 m, the time evolution of the annular passage width is as shown in Fig. 38.
For this calculation, it has been assumed that h R, and a high viscosity fluid, small gaps, long flow passages and the Navier–Stokes equations for a planar uniaxial flow are all applicable. The 5 simplified formula is
Table 13. Maximum cumulative storey shears (Landers earthquake, 1992)
possible to identify dynamic characteristics of an adaptive filter installed in the second floor by fewer parameters, (for example, by three parameters). It is possible to show how a semi-active damping coefficient time history may be interpreted as time evolution of an orifice size in the mechanism. For the purpose of illustration, consider Structures & Buildings 152 Issue 2
Adaptive control of structures by LMS algorithm
C b ¼ 3Lp
Rp h
3
Golafshani • Mirdamadi
187
APPENDIX 1. DERIVATION OF THREE-CHANNEL FILTERED-X LMS For the following, the state-space, z domain and backwardshift operator calculus, q 1 , or delay operator, in discrete time26–28 domain are used as descriptions of dynamic systems. If f (k) is a digital sequence, in which t k ¼ kTs , and Ts is the sampling period for converting a continuous-time signal f (t k ) 28 to f (k), then one-sided z-transformation of f (k) is defined 27 by Fig. 38. Time evolution of the annular passage width F(z)
15
1 X
f (k)z k
k¼0
In Tables 2–13 more responses are shown and which have been compared. It can be observed that, on average, the FLMS algorithm with exact TF estimates is the most effective algorithm for reduction of major responses of structure.
7. CONCLUSIONS A semi-active multi-variable adaptive forward feed controller was designed for a seismically excited structure, based on filtered-x and classical (unfiltered) NLMS algorithms. This filtering action represents a major difference to the two versions of the algorithm. In addition, the filtered-x version was implemented by an inexact transfer function for filtering seismic input. The hardware of the controller comprised a variable-orifice, energy dissipation mechanism, powered by battery-size power supplies. For each floor, one of these mechanisms was installed between that floor and the bracing underneath. The software of the controller comprised FIR adaptive filters whose parameters were identified by filtered-x and classical NLMS optimisers. The advantage of using this technology was that the need for large power supplies had been omitted. However, numerical results showed a good performance of the semiactive dynamic compensator in the reduction of desired structural responses. Major results of three different optimisation algorithms were summarised. These algorithms were compared for the responses of uncontrolled passive and controlled semi-active controllers. Simulations were performed and compared for four earthquakes with different frequency contents and they were Elcentro (1940), Whittier (1987), Manjil (1990) and Landers (1992) earthquakes. In conclusion, in this case study it was demonstrated that, on average, the filtered-x NLMS algorithm performed better than the classical NLMS (no filtering action) algorithm. In addition, the filtered-x NLMS optimiser was not very sensitive to system transfer function estimates, which had been used in the estimation and computation of filtered seismic input. As a whole, the results are encouraging and bring hope of a future application of this technology for special structures, such as nuclear power plants. 188
Structures & Buildings 152 Issue 2
The z and q variables are not exactly the same. In many system theory textbooks, z is used for the shift operator (i.e. instead of q) as well as for the complex variable in z transformation. However, it is convenient to have different notations of the two notions. This is the same separation that is normally used between the complex variable s in Laplace transformation and 28, 29 the differential operator p ¼ d=dt. The shift operator q in discrete time is a counterpart of the differential operator p in continuous time. The backward-shift operator has a time28 shifting property. q 1 f (k) ¼ f (k 1)
16
The backward-shift operator causes one sampling time delay in sequence. The goal of the backward-shift operator and the algebraic system theory is to convert manipulations of 28 differential equations to purely algebraic problems. This is similar to the objective of the Laplace transformation operator, s, in (continuous time) differential equations. Owing to linearity assumptions for structure dynamics, the error signal at discrete time k, measured at ith floor ei (k), is defined as a superposition of structural floor responses to 9, 14, 15 earthquake and to three control inputs
17
e i (k) ¼ d i (k) þ
3 X
yim (k)
i ¼ 1, 2, 3
m¼1
where sequence, di (k) is the desired response for sampling time t k , di (k) is the structural response due to the earthquake and yim is ith floor response to mth floor control action, as shown in Fig. 6. The ‘desired response’ or ‘training signal’ is a signal that should be followed by the output of adaptive filters. Parameters of adaptive filters are adjusted to cause their output to agree as closely as possible with the desired response signal. This is accomplished by comparing this output with the desired response in order to obtain an ‘error signal’ and then by 18 adjusting or optimising parameters to minimise this signal. The desired response is simply some objective output that the adaptive filter must be designed to replicate in an opposite 15 phase.
Adaptive control of structures by LMS algorithm
Golafshani • Mirdamadi
In the adaptive control of structures, the desired response is any response of an uncontrolled structure (to a seismic signal) that the designer wishes to be minimised. The uncontrolled structure means the structure without any augmented semiactive control system. The output of adaptive filters is the response of the structure to semi-active control systems when the response to seismic excitation is off. The error signal is the superposition of these two principal effects on the structure under control. The following pulse-transfer functions in a backward-shift operator relate parts of the error signal to the corresponding inputs. Tdei (q) is pulse-transfer function from a seismic disturbance to one part of the error signal i 18
d i (k) ¼ Tdei (q)€xg (k)
i ¼ 1, 2, 3
It is a rational function whose numerator and denominator are polynomials of different powers of the complex variable q 1 . Tcmei (q) is a pulse-transfer function from the control action applied at mth floor to the other part of error signal i 19
yim (k) ¼ T cmei (q) f cm (k)
i ¼ 1, 2, 3
m ¼ 1, 2, 3
Consider the input–output relations between control forces as outputs of the adaptive filters W m (z, k) and seismic input 20
f cm (k) ¼ W m (q, k)€x g (k)
By substituting equations (20) and (22) into (19)
23
yim (k) ¼ T cmei (q)
Lm X
w mj (k)q j €xg (k)
By substituting equations (18) and (23) into equation (17), the error signal is obtained in terms of transfer functions
e i (k) ¼ T dei (q)€xg (k) þ 24
3 X
25
€ ~x mi (k) ¼ T cmei (q)€xg (k)
By substituting the filtered-x signal in equations (25) into the error signal in equation (24)
w m ( j, k)€xg (k j)
m ¼ 1, 2, 3
In the above equation, wm ( j, k) ¼ wmj (k) is the jth weight of the discrete-pulse response of the mth floor adaptive filter, at sampling period k. The objective in adaptive control is to calculate W m (z, k) weights, i.e. wmj at each time step k. The relation between W m (z, k) and its jth parameter, wm ( j, k), in z domain is written as
22
W m (z, k) ¼
Lm X
w mj (k)z j
m ¼ 1, 2, 3
3 X Lm X
€x mi (k j) w mj (k)~
m¼1 j¼0
i ¼ 1, 2, 3
The above relation has been written by time shifting property €x mi (k j) €x mi (k) ¼ ~ q j ~
27
j¼0
m, i ¼ 1, 2, 3
This filtered signal contains only those poles that cause resonance in a structure. Other less important poles of nonfiltered seismic signal are filtered out. This is the core idea of the filtered version of LMS.
Both q and k are used as arguments of W m (q, k) to emphasise the time-varying behaviour of adaptive algorithms.
Lm X
w mj (k)q j €xg (k)
j¼0
A filtered sequence of acceleration of the seismic signal through the transfer functions is defined by
26
f cm (k) ¼
Lm X
i ¼ 1, 2, 3
e i (k) ¼ Tdei (q)€xg (k) þ
21
T cmei (q)
m¼1
m ¼ 1, 2, 3
For comparison, the above equation in z domain (or q 1 domain) is equivalent to the following convolution sum in the time domain
i, m ¼ 1, 2, 3
j¼0
Mean squares of error signals are chosen as the objective function for determining optimal values of wmj (k) parameters of W m (z, k) at time k
28
C(w mj (k)) ¼
3 X
Efe2i (k)g
j ¼ 0, . . ., L m
m ¼ 1, 2, 3
i¼1
Ef:g is the expectation operator. This function has only one global minimum and the solution is unique. The easiest technique to find that minimum is Steepest Descent and by searching for updated values of filter weights in the negative of a gradient, while searching step size should guarantee the 17, 18 stability of the algorithm.
j¼0
Calculating the gradient of cost function in equation (28) 19
W m (z, k) is an FIR filter that is truncated after Lm time points for an mth floor adaptive controller. The number of parameters of FIR filters implies that the dimension of optimisation space is (L1 þ 1) 3 (L2 þ 1) 3 (L3 þ 1). wmj , at each sampling time k, 30 are design variables. Structures & Buildings 152 Issue 2
29
3 X @C(w mj (k)) @e i (k) ¼ E 2e i (k) @w mj (k) @w mj (k) i¼1
Adaptive control of structures by LMS algorithm
i ¼ 1, 2, 3
m ¼ 1, 2, 3 Golafshani • Mirdamadi
189
9
The approximation of this deterministic gradient, converging theoretically to the Wiener filter solution, by a stochastic gradient, yields the LMS algorithm
30
3 @C(w mj (k)) X @e i (k) ffi 2e i (k) @w @w mj (k) mj (k) i¼1
Using this gradient, the recursive updating formula of Steepest Descent is given by
31
w mj (k þ 1) ¼ w mj (k) ª
@C(w mj (k)) @w mj (k)
This is specialised to the following difference equation for filter weights in LMS
32
w mj (k þ 1) ¼ w mj (k) 2ª
3 X
e i (k)
i¼1
@e i (k) @w mj (k)
The gradient of the error signal relative to filter weights is 29 called the sensitivity derivative. Multiplication shows inherent non-linearity of adaptive controllers. Sensitivity derivatives are calculated by differentiating error signals in equation (26)
33
@e i (k) ¼ ~€x mi (k j) @w mj (k)
m, i ¼ 1, 2, 3
j ¼ 0, K, L m
The result of the filtered-x LMS algorithm is
w mj (k þ 1) ¼ w mj (k) 2ª 34
3 X
e i (k)€~x mi (k j)
i¼1
m, i ¼ 1, 2, 3
j ¼ 0, . . ., L m
It is observed that nine filtered seismic signals and, therefore, nine Tcmei (z) should be estimated for this multi-variable control strategy. It should be noted that the derivation of the classical LMS algorithm is the same, except that the filtered-x seismic signal is replaced by (non-filtered) seismic input. Referring to the block diagram in Fig. 6, it is shown that in the classical LMS algorithm there is no need for filtering seismic disturbance through estimated (overhat) transfer functions. The major difficulty of the filtered-x version of LMS is in estimating transfer functions Tcmei (z) for obtaining filtered seismic signals. This can be accomplished by a rough finite element model of the structure 35
~€x mi (k) ffi T^ cmei (q)€xg (k)
m, i ¼ 1, 2, 3
T^cmei (q) are calculated from finite elements. SISO derivation of LMS using z transformation for active 11 control is due to Burdisso et al. Two-input, two-output derivation using z transformation for active control is due to 190
Structures & Buildings 152 Issue 2
12
Smith et al. and Burdisso. Single channel and multi-channel derivations using convolution and matrix representation for 31 active control were presented by Elliot et al. and Kuo and 32 Morgan. Three-input, three-output derivation using the backward-shift operator (delay operator) for semi-active control is also due to these authors.
REFERENCES 1. Nemir D. C., Lin Y. and Osegueda R. A. Semi-active Motion Control Using Variable Stiffness. Journal of Structural Engineering, ASCE, 1994, 120, No. 4, 1291–1307. 2. Inman D. J. Vibration with Control, Measurement and Stability. Prentice-Hall, New Jersey, 1989. 3. Spencer B. F. and Sain M. K. Controlling Buildings: A New Frontier in Feedback. IEEE Control Systems Magazine, 1997, Dec., 19–35. 4. Housner G. W., Bergman L. A., Caughey T. K., Chassiakos A. G., Claus R. O., Masri S. F., Skelton R. E., Soong T. T., Spencer B. F. and Yao J. T. P. Structural Control: Past, Present, and Future. Journal of Engineering Mechanics, ASCE, Special Issue, 1997, 123, No. 9, 897–971. 5. Soong T. T. and Dargush G. F. Passive Energy Dissipation Systems in Structural Engineering. John Wiley and Sons, Chichester, 1997. 6. Symans M. D. and Constantinou M. C. Seismic Testing of A Building Structure with A Semi-active Fluid Damper Control System. Earthquake Engineering and Structural Dynamics, 1997, 26, No. 7, 759–778. 7. Meirovitch L. Dynamics and Control of Structures. John Wiley and Sons, Chichester, 1990. 8. Skogestad S. and Postlethwaite I. Multi-variable Feedback Control, Analysis and Design. John Wiley and Sons, Chichester, 1996. 9. Smith J. P., Burdisso R. A. and Suarez L. E. An Experimental Investigation of Adaptive Control of Secondary Systems. Proceedings of the First World Conference on Structural Control, Los Angeles, 3–5 Aug. 1994. 10. Vipperman J. S., Burdisso R. A. and Fuller C. R. Active Control of Broadband Structural Vibration Using the LMS Adaptive Algorithm. Journal of Sound and Vibration, 1993, 166, No. (2), 283–299. 11. Burdisso R. A., Suarez L. E. and Fuller C. R. Feasibility Study of Adaptive Control of Structures under Seismic Excitation. Journal of Engineering Mechanics, ASCE, 1994, 120, No. 3, 580–592. 12. Burdisso R. A. Structural Attenuation Due to Seismic Inputs with Active/Adaptive Systems. Proceedings of the First World Conference on Structural Control, Los Angeles, 3–5 Aug. 1994. 13. Soong T. T. Active Structural Control, Theory and Practice. Longman Scientific and Technical, Essex, 1990. 14. Fuller C. R., Elliot S. J. and Nelson P. A. Active Control of Vibration. Academic Press, London, 1996. 15. Clark R. L., Saunders W. R. and Gibbs G. P. Adaptive Structures. John Wiley and Sons, Chichester, 1998. 16. Sastry S. and Bodson M. Adaptive Control, Stability, Convergence and Robustness. Prentice-Hall., New Jersey, 1989.
Adaptive control of structures by LMS algorithm
Golafshani • Mirdamadi
17. Haykin S. Adaptive Filter Theory, 3rd edn. Prentice-Hall, New Jersey, 1996. 18. Widrow B. and Stearns S. D. Adaptive Signal Processing. Prentice-Hall, New Jersey, 1985. 19. Proakis J. G. and Manolakis D. G. Introduction to Digital Signal Processing. Macmillan Publishing. Company, NY, 1989. 20. Matlab. MATLAB Reference Guide: High Performance Numeric Computation and Visualization Software. The MathWorks Inc., MA, 1992. 21. Simulink. SIMULINK User’s Guide: Dynamic System Simulation Software. Version 1.3. The MathWorks Inc., MA, 1994. 22. Ljung L. System Identification TOOLBOX User’s Guide: For Use with MATLAB. The MathWorks Inc., MA, 1993. 23. Matlab. Control System TOOLBOX User’s Guide: For Use with MATLAB. The MathWorks Inc., MA, 1994. 24. Grace A. Optimization TOOLBOX User’s Guide: For Use with MATLAB. The MathWorks Inc., MA, 1994. 25. Kraus T. P., Shure L. and Little J. N. Signal Processing
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Please email, fax or post your discussion contributions to the secretary by 14 November 2002: email:
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Structures & Buildings 152 Issue 2
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191
