Optimal LQR Control of Structures using Linear Moal Moel I. Halperin,2, G. Agranovich an Y. Ribakov 2 Department of Electrical an Electronics Engineering 2 Department of Civil Engineering Faculty of Engineering, Ariel University Center of Samaria, Ariel, 47, Israel Abstract Accoring to the seismic moal approach, optimal moal control esign shoul erive a control signal affecting only the ominant moal coorinates. Aitionally, story rifts are the main issue for many seismic esign problems where the esign goal is to reuce rifts in excite structures. This stuy formulates optimal moal feeback esign base on the infinite horizon LQR metho. New state-space similarity transformations are introuce. Efficiency of the metho is emonstrate in a numerical example, presenting a moal LQR esign of a 2 story builing constructe in a seismic area. keywors: moal control, LQR, optimal structural control. Introuction In the last ecaes, there is an increasing interest in the fiel of structural control. It suggests the use of approaches an tools of control theory for analysis an manipulation of structures ynamic behavior. A ynamic response of a structure can be escribe by the isplacements in a selecte set of egrees of freeom, also known as noal coorinates []. These isplacements are calculate by solving the corresponing equations of motion. In civil engineering, the ifference between the isplacements of two ajacent floors is usually enote as story rift [2, 3]. Story rifts are the main issue for many seismic esign problems, where the esign goal is to reuce rifts in the excite structure. In most cases, the control forces entering the structure are inter-story forces. Therefore, from engineering viewpoint, a rift moel is more attractive than a noal one as it highlights the esign scope an allows a more intuitive choice of quaratic optimization inex coefficients, use for optimal seismic esigns base on LQR criteria. The use of story rifts instea of noal isplacements was suggeste in the esign of supplemental ampers for structural control [4]. The equations of motion in the noal coorinates an the rifts coorinates are internally couple through the structure inertia, amping an stiffness properties. Moreover, structural moels are quite large, compare to other controlle systems. It creates computational ifficulties in fining an efficient control esign for large structures. A common solution to this problem is known as moal ecomposition [5]. It leas to a
simpler an convenient esign process [] an it is also physically interprete an aequate from structural engineering viewpoint. The contribution of each moal coorinate to the overall ynamic response is ifferent, therefore, coorinates with small contribution may be neglecte [5, 6, 7, 8]. For structures, subjecte to earthquake excitation, it is known that a seismic esign, taking in account merely the ominant low frequency moal coorinates set is mostly aequate. An optimal control esign that relies on the structure moal properties is enote as optimal moal control. Accoring to the seismic moal approach, optimal moal control esign shoul erive a control signal that affects only the ominant moal coorinates [9]. The problem is that in general case, when a control esign is one, each single control signal generates multiple moal control signals an therefore couples the structural moal coorinates []. In other wors, the control signals affect multiple moes simultaneously. This coupling also happens for feeback control signals. It is unwante since the moal control esign tries to increase the control efficiency by manipulating only the ominant moes. In inepenent moal space control (IMSC) strategy a feeback control signal is esigne in such a way that only the esire moes are affecte. Accoring to this metho, moal control signals are esigne in the moal space for each moal coorinate separately. Next, the implemente control is erive from the moal control signals []. Though in the general case the feeback couples the moes of the system, in the case of inepenent moal space control, where a moal control esign eals merely the esigne moal coorinates, the close loop remains ecouple [2]. A problem that can be troublesome is that a feeback, esigne by the IMSC approach might require estimation of the moal coorinates. Another problem is that the use of IMSC might affect the uncontrolle moes such that their contribution to the overall response might increase [8]. An approach, requiring no moal state estimation is irect feeback control, whereby the sensors are collocate with the actuators an a given actuator force is a linear function of the sensor output at the same point [3]. A Moifie IMSC was recently propose [8]. However, the metho is questionable. It requires pseuo-inversion of matrices which provie the esigner with an approximate solution, because the pseuo-inverse is a solution to the least squares problem. Aitionally, for the case of MIMSC esign which controls less than half of the moes, the propose pseuo-inverse is singular. Design of ampers, base on moal LQR was also stuie [6]. The ominance of the first set of moes from energy viewpoint was emphasize an control esign approach of the ominant moes by LQR metho was suggeste. 4-22
2 Aims an Scope This stuy aims to fin a verifie optimal moal control base on the well known LQR control esign metho for Rayleigh ampe moels. Aitionally it is expecte that the moal approach will provie a more efficient an intuitive optimal control esign of structures. 2 Equations of Motion in Noal Moel Form 2. Secon Orer ODE The equation of motion for a controlle structure, subjecte to seismic excitation is: M z (+ Cz (+ Kz( -M γ z (+ Wu( () = g where z( is a vector function of the noal isplacements; M, C, K are the structural mass, amping an stiffness matrices, respectively; t represents the earthquake groun acceleration signal; u( is a vector of the forces prouce by the actuators; W is the ampers istribution matrix that relates the input forces in the ampers to the forces in the noal coorinates; γ is the groun acceleration istribution vector. It shoul be note that in this stuy, the amping of the structure is moele as Rayleigh amping [5]. z g 2.2 State-Space Form an LQR Control In orer to use the infinite horizon LQR metho, a state-space formulation for Eq. () is use. As LQR esign oes not take in account isturbance signals, the groun acceleration is omitte. In the state space Eq. () takes the following form []: z( x ( Ax( Bu( ; x( (2) z ( where x( is the states vector of the moel an A, B are the state matrices: M K I M C B M W A ; An infinite horizon LQ cost function [2], for the moel escribe by Eq. (2), is: T T J( x, u) x Qx u Ru t (3) 2 The time epenency notation in this equation was remove for simplicity. Accoring the infinite horizon LQR esign metho, the optimal close loop control signals vector is [4]: where G is erive by u( Gx( G R B T P
an P is the solution of the algebraic matrix Riccati equation [4]: T PA A P Q P( BR 2.3 Moal Form of the Unampe Moel B ) P O T It is known that a real an nonsingular matrix Φ that iagonalizes M an K, such that Φ T MΦ an Φ T KΦ are iagonal matrices, can be foun by solving the unampe structure eigenproblem [5]: The matrix Φ is efine by ( M Φ 2.4 State Space Moal Decomposition K I φ. i ) [ φ φ2 l The noal state space moel can be transforme from Eq. (2) into a moal state space moel, compose from l internally inepenent moal subsystems, by using a state space similarity transformation. Consier the block iagonal matrix i φ ] Φ iag( Φ, ) (4) Φ Denote e k as the k th n n column in the ientity matrix I R. A permutation matrix T is: The similarity transformation matrix is T T e e2 en e2 e4 e n T Φ T The transformation into the moal moel is given by x( Tq( ; A T A AT A 2 ; A l B T B B B B where l is the number of moal coorinates. The block iagonal form of A implies that it is compose of l internally inepenent subsystems whose ynamics are goverene by the blocks A i. 2 l 2.4 State Space Drift Transformation Structural control esign, base on story rifts instea of groun relate isplacements, was suggeste in various stuies. The transformation of the system from noal to rift space can be one in the state space
by using a non-singular similarity transformation matrix where T elements are efine by T T, ij T T i j otherwise The similarity transformation from the noal to the rift space is: x( t ) Tx ( t ); A T AT ; The rift moel can be ecompose into corresponing moal moel by using Φ T Φ instea of Φ in Eq. (4). A close loop system, esigne in the rift space, has the form B T x ( ( A B G ) x ( ; G GT B 3 Reuce Moal LQR Design Since seismic control esign can be carrie out only for the ominant moes, it can be applie to reuce moel by taking in account only the ominant moes. Consier A A A 2 ; k l A k which is the leaing block from A. It escribes the ynamics of the moes to be controlle. Here k is the number of moes consiere by the control esign. Denote k n B R 2 u as the corresponing block from B. It escribes how the control inputs affect the selecte moes. Therefore, the algebraic Ricatti's equation for the reuce moel is P A ( A ) T P Q P ( B R B T ) P O (5) k k Here R 2 Q 2 is the moes weighting an R is the same as in Eq. (3). By solving Eq. (5), the full moel gain matrix can be erive An the control signals are G T R B P OT z( u( Gx( 2 2 z ( G G G z( G z ( ) t
Note that in the general case, G is a full matrix. It implies that a single control signal epens on both the isplacements an the velocities throughout the moel. In orer to realize such feeback an active control shoul be use. 4 Numerical Example- 2 story moel In orer to verify the suggeste approach, structural seismic response was simulate numerically. An active control was esigne for a ynamic moel accoring to two esign approaches. A rift space moal LQR approach an classical rift space LQR esign. Seismic response to the following four earthquakes was obtaine: El-Centro, Hachinoche, Kobe, an Northrige. The earthquakes acceleration recors were scale to PGA =.3g. Scaling was use in orer to obtain better comparison basis of the ifferent responses. The 2 story moel is base on the evaluation moel escribe by Spencer et. al [5], with slight moifications. The moel originates from a planne north-south moment resisting frame (MRF) of a 2 story, steel builing which is also a typical meium to high-rise builing in the Los Angeles, California region. The moele MRF is 3.48 m in plan an 8.77 m in elevation. It consists of 2 story an 2 basement levels with five bays of 6. m. The floors are composite mae of concrete an steel an the columns are mae of 345 [MPa] steel. Typical story heights are 3.96 m, except for the heights of the two basement levels an the groun floor which are 3.65 m an 5.49 m respectively. The column bases are moele as pinne an secure to the groun. Concrete founation walls an surrouning soil are assume to restrain the structure at the first floor from horizontal isplacement. The floors are composite an Figure. Structural moel consist of 248 [MPa] steel wie-flange beams acting compositely with the floor slab. In accorance with common practice, the floor system, which provies iaphragm action, is assume to be rigi in the horizontal plane. The basement floor beams are simply connecte to the columns. The seismic mass of the structure first level is 5.32 5 kg, for the secon level is 5.65x 5 kg, for the thir level to 2 th level is 5.5 5 kg an for the roof is 5.83 5 kg. The inertial effects of each level are assume to be carrie evenly by the floor iaphragm to each perimeter MRF, hence each frame resists one half of the seismic mass associate with the entire structure. Floor inertial loas are uniformly istribute at the noes of each respective floor assuming a consistent mass formulation. A more etaile
Gain, B escription of the moel can be foun in [5]. Guyan reuction is use to reuce the number of DOF to a manageable size. In each above groun floor one horizontal DOF (at the 6 th column) an two vertical DOF (at the 2 n an 5 th columns), were efine as active DOF. Namely, the teste moel consists of 6 active DOF. The amping matrix correspon Rayleigh amping. The noal moel was transforme into the rift space an then ecompose into corresponing moal moel. The control esign consiere 6 moes out of 6 moes an the moes weights are illustrate in Figure 2. The weights were chosen so that the magnitue of the force in the actuator woul not excee 22 kn for El-Centro response. Figure 2: Moel B, Moal Weighting. The affect of the moal control on the structure frequency response is emonstrate by a Boe iagram in Figure 3. As it follows from the figure, the gains in the resonance frequencies of the selecte moes were reuce. The change in the moel eigenvalues is illustrate in Figure 4. It can be seen that only the controlle eigenvalues move ue to the feeback. ω, Ra/sec Figure 3. Moel B, st story rift earthquake gains for the uncontrolle an the close loop moels.
Im (λ) Re (λ) Figure 4: Moel B, Uncontrolle an moal controlle eigenvalues The peak response an time history analysis of the moel to El-Centro earthquake are presente in Figures 5 an 6. The response to the other earthquakes is similar. Analysis of the control energy shows that the overall energy consume by the actuators is of the same orer for both moal an classical esign approaches. The form of G shows that each control force epens on all the noal velocities. It means that active control approach is require for accurate implementation of such feeback. However, this control esign might provie a platform for approximation of active evices by passive or semi-active control approaches. Comparison of the time require for the esign process in MATLAB shows that the calculation time of the reuce moal esign was 25 times smaller than the calculation time of the classical esign. 5 Conclusions A metho for moal LQR control of lumpe mass structures was formulate an verifie numerically. The metho offers a esign algorithm that reuces the numerical calculation effort require for the esign by controlling only the esire moe shapes. The metho reuces the orer of Riccati s equation that shoul be solve. It was verifie numerically that LQR moal control oes exist an that the close loop eigenvalues iffers from the uncontrolle eigenvalues only for the selecte moe shapes. A flexible 2 story builing was analyze. Response of to 4 scale natural earthquake recors was simulate in orer to verify the propose metho. It was foun that for the 6 DOF moel, the moal esign approach performs 25 times faster than the classical approach while the esign results are similar.
u(, kn Roof accel., g Roof isp., cm (a) (b) (c) Figure 5. Peak response: (a) - Noal Displacements [cm], (b) - Drift Displacements [cm], (c) - Control Forces [ 3 kn]. t, sec Figure 6. Time history 4-34
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