Semi-decentralized Strategies in Structural Vibration Control

Transcription

1 Semi-decentralized Strategies in Structural Vibration Control By: Josep M. Rossell Francisco Palacios Control, Dynamics and Applications (CoDAlab) Department of Applied Mathematics III Universitat Politècnica de Catalunya (UPC), Barcelona, Spain The Engineering Conference Southern Norway University of Grimstad, Norway, May 2, 20

2 Where is our University placed?

3 Some indicators of the UPC The Universitat Politècnica de Catalunya (UPC) is composed by: More than students 3000 teaching and research staff campuses and regional centers located in 8 cities 42 departments 83 research groups

4 CoDAlab research group Our research group is Control, Dynamics and Applications (CoDAlab), which belongs to the Department of Applied Mathematics III. We wish to emphasize that: It is active in an interdisciplinary intersection of applied mathematics, systems and control theory and engineering. The group is composed by people with backgrounds in mathematics, physics, control engineering and civil engineering. It is composed by 25 researches, at the present time.

5 EPSEM-Manresa, Barcelona, Spain

6 Outline Introduction Motivation Control methods and devices Objectives Control strategies The Inclusion Principle Overlapping controllers Multi-overlapping controllers A five-story building model Numerical simulations Future works

7 Introduction: Motivation Everyday, taller and taller buildings are designed and built

8 Introduction: Motivation A large variety of medium-size and small-size strategic buildings must also be protected, such as: Communication and command centers Emergency service facilities Hospitals Emergency power plants Nuclear plants Etc...

9 Introduction: Motivation Structures such as buildings, bridges and towers can be affected by external vibrations which may compromise its stability and the safety of its occupants. The implementation of vibrational control systems can reduce the negative effects of such external stimuli.

10 Introduction: Motivation More and more destruction caused by earthquakes...

11 Control methods and devices: base isolation

12 Control methods and devices: semi-active hydraulic dampers

13 Main objective OBJECTIVE To keep the response of the structure within the limits defined by safety and human comfort. To design simple but efficient controllers in order to satisfy the first objective.

14 Control strategy The control strategy will be based on: The Inclusion Principle (overlapping / multi-overlapping) A five-story building as model An state-feedback LQR control Three different kinds of controllers: Two types of actuators: centralized control overlapping control multi-overlapping control direct actuation inter-story actuation

15 k 5 c 5 k 4 c 4 k 3 c 3 k 2 c 2 Five-story building model m 5 m 4 m 3 m 2 m w k c Five-story building model

16 Two different actuation schemes (a) Direct actuation u 5 (b) Inter-story actuation -u 5 u 4 u 5 -u 4 u 3 u 4 -u 3 u 2 u 3 -u 2 u u 2 -u Actuation schemes for a five-story building

17 Information structure constraints Wireless transmission range required by the multi-overlapping controllers

18 Overlapping controllers via the Inclusion Principle A large variety of systems are compose by subsystems sharing common parts. For these kinds of systems may be convenient to design semi-decentralized controllers by using overlapping information sets. S 3 S (2) S 2 S S ()

19 What does the Inclusion Principle say? The Inclusion Principle (Ikeda and Siljak 980 s) establishes a mathematical framework for two dynamic systems with different dimensions, in which solutions of system S (virtual) include solutions of S (real). S Expansion S S S S 2 S 2 Contraction

20 Expansion-contraction process The expansion-contraction process, by using the Inclusion Principle, is based on the following process: S K S Expansion S K D Contraction K D S 2 K 2

21 Structure of the system S Assume that the system S has the following structure: A= A A 2 0 A 2 A 22 A 23 0 A 32 A 33, B = B B 2 0 B 2 B 22 B 23 0 B 32 B 33,

22 Expansion-contraction transformations In order to obtain disjoint expanded subsystems we can use the linear expansion transformation V and R given by: I n I V = n2 0 0 I n2 0, R = 0 0 I n3 I m I m2 0 0 I m I m3 Their corresponding pseudoinverse matrices U= ( V T V ) V T and Q= ( R T R ) R T have the form U = [ In In 2 2 In I n3 ], Q = [ Im Im 2 2 Im I m3 ].

23 Expanded system S The expanded matrices Ā and B have the following structure: [ Ā Ā Ā = V AU = 2 Ā 2 Ā 22 ]= B = BV Q =[ B B2 B 2 B22 ]= A 2 A 2 A 2 2 A 22 A 2 2 A A 32 B 2 B 2 B 2 2 B 22 B 2 2 B B 32 2 A A 22 A 23 2 A 22 A 23 2 A 32 A 33 2 B B 22 B 23 2 B 22 B 23 2 B 32 B 33,.

24 Complementary matrices In the context of the Inclusion Principle, the role of the complementary matrices is crucial. In our case, we consider the following matrices: M = 0 2 A 2 2 A A 22 2 A A 22 2 A A 32 2 A 32 0, N = 0 2 B 2 2 B B 22 2 B B 22 2 B B 32 2 B 32 0.

25 Structure of the expanded system By adding convenient complementary matrices, the expanded state matrices adopt the form: [ Ã=Ā + M = Ã Ã 2 Ã 2 Ã 22 ]= B = B + N =[ B B2 B 2 B22 ]= A A A 2 A 22 0 A 23 A 2 0 A 22 A A 32 A 33 B B B 2 B 22 0 B 23 B 2 0 B 22 B B 32 B 33,.

26 Decoupled expanded system Now, S can be represented as two interconnected subsystems S : S 2 : x (t) = Ã x (t) + B ũ (t) + Ã2 x 2 (t) + B 2 ũ 2 (t), x 2 (t) = Ã22 x 2 (t) + B 22 ũ 2 (t) + Ã2 x (t) + B 2 ũ (t).

27 Decoupled expanded system Now, S can be represented as two interconnected subsystems S : S 2 : x (t) = Ã x (t) + B ũ (t) + Ã2 x 2 (t) + B 2 ũ 2 (t), x 2 (t) = Ã22 x 2 (t) + B 22 ũ 2 (t) + Ã2 x (t) + B 2 ũ (t). The decoupled subsystems will be S D : S 2 D : x (t) = Ã x (t) + B ũ (t), x 2 (t) = Ã22 x 2 (t) + B 22 ũ 2 (t).

28 Design of local controllers in S D The local control laws will be ũ (t) = K x (t), ũ 2 (t) = K 22 x 2 (t). The gain matrices in the expanded and initial systems have the following form: [ K ] K D = 0 contraction 0 K D =Q K 0 D V =. K 22 0 This kind of controller is called overlapped controller, and the gain matrix K D has a tridiagonal structure.

29 Design of overlapping controllers for a five-story building S (2) S () 5 S (2) D S () D K (2) K () K o Overlapping controller design

30 Overlapping gain matrix By designing an overlapping control (inter-story actuation), the following gain matrix is obtained: [ ] K o (is) = The value of the quadratic index J c corresponding to the overlapping controller is [J c ] (is) K = o

31 Centralized control matrix By designing a centralized controller (inter-story actuation), the following full gain matrix is obtained: K (is) c = 0 8 [ with an optimal cost [ J (is) c ] opt = ],

32 Multi-overlapping decompositions (a) Longitudinal (b) Loop (c) Radial Multi-overlapping decompositions

33 Two-step decomposition for longitudinal multi-overlapping systems S () S (2) Ŝ Ŝ2 Ŝ3 Ŝ4 Ŝ5 S () D Ŝ S (2) D S () S (2) Ŝ2 Ŝ3 Ŝ3 S (2) S (22) Ŝ4 Ŝ5 Ŝ S () D Ŝ2 Ŝ2 S (2) D Ŝ3 Ŝ3 S (2) D Ŝ4 Ŝ4 S (22) D Ŝ5 Decoupling expansion for multi-overlapping decomposition

34 Multi-overlapping controller for a five-story building S (2) S () S (2) D S () D S (22) D S (2) D S (2) D S () D K (22) K (2) K (2) K () K (2) K () Multi-overlapping expansion-contraction process K mo

35 Multi-overlapping control matrix By designing a multi-overlapping control (inter-story actuation), the following decentralized gain matrix is obtained: [ ] K mo (is) = The value of the quadratic index J c corresponding to the multi-overlapping controller is [J c ] (is) K = mo

36 Numerical simulations: El Centro 940 Acceleration (m/s 2 ) Stories El Centro 940 seismic record Time(sec) El Centro 940: Direct Actuation Free response Centralized Overlap. Multi Overlap maximum inter story drifts (cm) Controller Centralized Overlap. Multi Overlap maximum control force (x0 6 N) El Centro 940. Maximum inter-story drifts and control efforts

37 Numerical simulations: Northridge 944 Acceleration (m/s 2 ) Stories Northridge 994 seismic record Time(sec) Northridge 994: Direct Actuation Free response Centralized Overlap. Multi Overlap maximum inter story drifts (cm) Controller Centralized Overlap. Multi Overlap maximum control force (x0 6 N) Northridge 994. Maximum inter-story drifts and control efforts

38 Future works (I) Other complexities can appear in a building model, such as: Uncertainties in the state, inputs, outputs,... Delays in the state, inputs, control,... Non-linearities: saturation in the control, hysteresis, etc. Other kinds of complexities...

39 Future works (II) u 2 5 m 2 5 k 2 5 c 2 5 u 2 4 m 2 4 k 2 4 u 3 u 2 3 ˆk 3 c 2 4 m 3 m 2 3 k 3 ĉ 3 k 2 3 c 3 u 2 ˆk 2 c 2 3 u 2 2 ˆk 2 2 u 3 2 m 2 m 2 2 m 3 2 k 2 ĉ 2 k 2 2 ĉ 2 2 k 3 2 c 2 u ˆk c 2 2 u 2 ˆk 2 c 3 2 u 3 m m 2 m 3 k ĉ k 2 ĉ 2 k 3 ω c ω c 2 ω c 3 Multi-building structures

40 Thanks for your attention!