MUNAWWAROH, DITA ANIES

Sutrisno, Sutrisno

English

Diponegoro University Institutional Repository

APPLICATION OF MODEL PREDICTIVE CONTROL TO REDUCETHE VIBRATION OF TWO CONNECTED BUILDINGSSutrisno1 and Dita Anies M.2Department of Mathematics, Diponegoro University,1tresno.math@undip.ac.id, tresnowijoyo@gmail.com, 2dita.anies.m@gmail.comAbstract. In this paper, model predictive control (MPC) will be applied to reducethe vibration of two connected buildings (TCB) that can be caused by, for example,seismic activities. Mathematical model of TCB can be formulated as a linear timeinvariant state space equation. The state of this plant is the displacement of eachbuilding measured from the origin. We will regulate or bring the state to the originas fast as possible with optimal control input. This optimal control input will bedetermined using MPC. We give a numerical simulation to observe how this controlmethod is working. From the simulation result, MPC reduces the vibration of TCBand bring the displacement of each building to the origin faster than uncontrolledsystem.Key words and Phrases: model predictive control, vibration control, two connectedbuildings.1. INTRODUCTIONBuildings are susceptible to be dam-aged if some seismic activities attack them.To prevent it, we can apply some controllerto reduce the vibration of the buildings byregulating or bringing the displacement ofeach building to the origin. Some buildingstructures have two buildings connectedby a bridge, said two connected build-ings (TCB). The mathematical model rep-resenting the vibration (displacement) ofTCB can be written as a linear time invari-ant state space equation appeared in [1].Several control methods have been de-veloped such as linear quadratic regulator,H2 and H∞ based on linear matrix inequal-ity (LMI) that were applied to design an ac-tive vibration controller that were given by[1, 2, 3, 4, 5, 6]. One of several optimal con-trol method alternatives is model predictivecontrol (MPC) which is famous for prac-titioners or researchers. Model predictivecontrol (MPC) is an optimization basedcontrol method and it is applicable for regu-lating and tracking problems as mentionedby [7]. MPC predicts the state and inputvariables along horizon prediction, substi-tuting these predictions into some objectivefunction and solving it using optimizationmethod to determine the optimal controlaction. MPC control method can be usedto control a discrete state space system withcontrollability and observability character-istic. The studies about this state spacewere appeared in [8] including it’s control-lability and observability characteristic. Tosimulate the system’s response, we can usedMPC toolbox for MATLAB that was ap-peared in [9].In this paper, we apply MPC to reducethe vibration of two connected buildingsthat can be caused by some seismic activi-ties. We discretise the continuous model ofthis plant, design the controller using MPCand simulate it using MATLAB.2. RESULT2.1 Mathematical Model of Two ConnectedBuildingsDynamical model of two connectedbuildings will be formulated using basicdynamical model of mass-spring-damper.Consider a mass-spring-damper systemgiven by Figure (1) which has one degree offreedom (d.o.f). Let m be the mass of theplant (kg), k be the spring constant (N/m)and c be the damper coefficient (Ns/m).Let x denote the displacement of the massmeasured from the origin and F be the ex-ternal force. By using Newton’s laws, thedifferential equation of this system is fol-lows which is linear differential equationwith two order that was appeared in [1],md2x(t)dt2+ cdx(t)dt+ kx(t) = F. (1)Furthermore, consider two buildings con-nected by a bridge given by Figure (2).This system contains two buildings saidbuilding-A and building-B which were builtusing base isolation at the bottom as themass-spring-damper systems. Each build-ing has two d.o.f in the x and y direction.Let xa and ya denote the displacementsof building-A in the x-direction and y-direction respectively. Similarly, let xb andyb denote the displacements of building-Bin the x-direction and y-direction respec-tively.Figure 1. Mass-spring-damper systemFigure 2. Two connected buildingsThen, the differential equation forthese buildings arem1d2xa(t)dt2+c1dxa(t)dt+k1xa(t) = −F, (2)m2d2xb(t)dt2+ c2dxb(t)dt+ k2xb(t) = F. (3)For simplicity, let F = u is the actuatorforce. Then (2)-(3) can be rewritten as alinear time invariant state space equationas follows[ẋ1ẋ2]=[A1 00 A2] [x1x2]+[B1B2]u, (4)y(k) = Cx(k) (5)wherex = [x1, x2]T ,x1 = [ẋa, ẏa, xa, ya]T ,x2 = [ẋb, ẏb, xb, yb]T ,y = [xa, ya, xb, yb]T ,A1 =− c1m10 − k1m100 − c1m10 − k1m11 0 0 00 1 0 0 ,A2 =− c2m20 − k2m200 − c2m20 − k2m21 0 0 00 1 0 0 ,B1 =−f000 , B2 =f000 ,C =0 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1 .Since we apply discrete time MPC tocontrol this system, we discretise (4)-(5) us-ing MATLAB function c2d in the numericalsimulation using time step 1 second.2.2 Model Predictive Control (MPC)Model predictive control (MPC) is anoptimization based control method that itcan be applied for controlling of linear dis-crete system for regulating and trackingproblems. Reducing the vibration of TCBis a regulating control problem, then MPCregulator is applicable to control this plant.By following Maciejowski [7], let k denotesthe time instant. Consider a discrete lineartime invariant system of the formx(k + 1) = Ax(k) + Bu(k), x(0) = x0,(6)y(k) = Cx(k), (7)where x(k) ∈ Rn is the state vector,u(k) ∈ Rp is the input vector, y(k) ∈ Rq isthe output vector, x(0) is the initial stateand A,B and C are real constant matriceswith dimension n × n, n × p and q × nrespectively. Assume that pair (A,B) iscontrollable.Let Hp be the horizon predictionlength. MPC for regulator can be illus-trated by Figure (3). For regulator case,MPC brings the state to the origin by pre-dicting the state and input vector alonghorizon prediction and minimizing the ob-jective function of the quadratic form of thestate and input vectors. At time instant k,let the cost function of MPC has the formJ(k) =Hp−1∑k=0x(k)TQx(k) + u(k)TRu(k)(8)where Q and R are real symmetric matriceswhich have properties positive semi-definiteand positive definite respectively. Let theconstraint has the form umin ≤ u(k) ≤umax. Let [x̂(k + 1), x̂(k + 2), ..., x̂(k + Hp)]be the state prediction and [∆û(k),∆û(k+1), ...,∆û(k +Hp− 1)] be the input predic-tion where ∆û(k) = û(k) − û(k − 1), bysubstituting them into (8) and using somealgebraic manipulation, (8) can be rewrit-ten as the following constrained quadraticoptimizationmin∆U(k)J(k) = −∆U(k)TG + ∆U(k)TH∆U(k)(9)subject to :GΘ∆U(k) ≤ −G [Ψx(0) + Υu(k − 1)]− gwhere G,Θ,Ψ,Υ, and g are real constantmatrices with appropriate dimensions. Op-timization (9) can be solved using active-setalgorithm that was embedded in MATLABfunction quadprog.Figure 3. MPC principleTable 1. Parameter values for numerical simulationm1[ton] m2[ton] k1[N/m] k2[N/m] c1[Ns/m] c2[Ns/m]38531 24098 7.62× 106 3.32× 106 1.08× 106 4.29× 1062.3 Simulation ResultsWe simulate TCB system illustratedby Figure (2) using MPC controller. Pa-rameter values for this plant is summarizedby Table (1) that was taken from Preumontand Seto [1]. The matrix weightings for ob-jective function (8) are Q = I and R = Iwhere I is identity matrix. Then the matri-ces of (4) are given byA1 =−0.0280 0 −0.1978 00 −0.0280 0 −0.19781 0 0 00 1 0 0 ,A2 =−0.1780 0 −13.7771 00 −0.1780 0 −13.77711 0 1 00 1 0 0 ,B1 =−1000 , B2 =1000 ,and matrix C is unchanged. We discretise(4) using MATLAB function c2d with sam-pling time 1 second then the above matricescan be replaced using the following matri-cesA1 =0.8769 0 −0.1887 00 0.8769 0 −0.18870.9549 0 0.9036 00 0.9539 0 0.9036 ,A2 =0.7588 0 1.8303 00 −0.7588 0 1.8303−0.1329 0 −0.7825 00 −0.1329 0 −0.7825 ,B1 =−0.95390−0.48730 , B2 =−0.132900.12940 ,and the pair([A1 00 A2],[B1B2])is con-trollable. The length of the horizon pre-diction is Hp = 10. The simulation time is150s. Finally the control action constraintis −2 ≤ u(k) ≤ 2. The initial state forthis plant is x1(0) = x2(0) = [0, 0, 0, 0]T .The simulation results are given by Figures(4)-(8).Figure 4. Optimal Input Generated By MPCFigure (4) shows the evolution of theoptimal control action generated by MPC.This control action implies the system’s re-sponses given by Figure (5,6) for building-Aand Figure (7,8) for building-B. From Fig-ure (5) and (6), the controller reduces thevibration of the building-A and bring thestate to the origin faster than uncontrolledsystem. From Figure (7) and (8), similarto building-A, the controller reduces thevibration of the building-B and bring thestate to the origin faster than uncontrolledsystem.3. CONCLUDING REMARKS ANDFUTURE WORKSIn this paper, the application of MPCto reduce the vibration of TCB was consid-ered. MPC was applied to determine theoptimal control action of TCB in order toreduce and bring the displacements of TCBto the origin as fast as possible. Numericalsimulation was given to observe how thiscontrol method is working. From the sim-ulation result, the controller was reducedFigure 5. Displacement of building-A in the x-directionFigure 6. Displacement of building-A in the y-directionFigure 7. Displacement of building-B in the x-directionFigure 8. Displacement of building-B in the y-directionthe vibration of the buildings and the dis-placements of the buildings for controlledsystem was reached the origin faster thanuncontrolled system.In the future works, we will compareour results to the existing results to observehow the performance the controller. Andalso, we will apply this control method tocontrol the multiconnected buildings thatcontain three or more connected buildings.REFERENCES[1] Preumont, A., Seto, K., 2008, Active Controlof Structures, John Wiley and Sons, UK.[2] Santos, R. B., Bueno, D. D., Marqui, C. R.,Lopes, V., Active Vibration Control of ATwo-Floors Building Model Based On H2And Hinf , Methodologies Using Linear MatrixInequalities (LMIs), Mechanical EngineeringDepartment, UNESP, Ilha Solteira, Brazil.[3] Koike, Y., Nakagawa, K., Imaseki, M., Mu-rata, T., Shiraki, H., Development of Con-necting Type Actively Controlled VibrationControl Devices and Application to High-rise Triple Buildings, IHI Engineering Review,Vol. 37 No. 1 February 2004.[4] Maebayashi, K., Shiba, K., Mita, A., Inada,Y., Hybrid Mass Damper System for ResponseControl of Building, Earthquake Engineering,Tenth World Conference, 1992, Rotterdam.[5] Obata, T., Hayashikawa, T., Sato, K., Shi-moda, K., Analytical Study of Active Vi-bration Control on Tall Buildings, EleventhWorld Conference on Earthquake Engineer-ing, Paper No. 24, 1996, Elsevier Science.[6] Seto, K., Watanabe, T., Vibration ControlOf Multiple Buildings Connected With Ac-tive Controlled Bridges For Improving Reli-ability Under The Large Seismic Excitation,8th ASCE Specialty Conference on Proba-bilistic Mechanics and Structural Reliability,PMC2000-277.[7] Maciejowski, J.M., 2001, Predictive Controlwith Constraints, Prentice Hall, USA.[8] Elaydi, S., 2005, An Introduction to Differ-ence Equations, 3rd ed., Springer Science andBusiness Media, inc., USA.[9] Wang, L., 2009, Model Predictive Control Sys-tem Design and Implementation Using MAT-LAB, Springer, Australia.

Application of Model Predictive Control to Reduce The Vibrationof Two Connected Buildings

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Application of Model Predictive Control to Reduce The Vibrationof Two Connected Buildings

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