Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 2, pp. 405-423, Warsaw 2007 MODEL BASED PREDICTIVE CONTROL OF GUYED MAST VIBRATION Bartłomiej Błachowski Institute of Fundamental of Technological Research PAS, Warsaw e-mail: bblach@ippt.gov.pl Thepurpose of thiswork is to present analgorithm for optimal vibration control of guyedmasts and an example of its application to a numerical simulation. The objective of the proposed control system is tominimize amplitudes of transverse vibrations of the top of a mast induced by wind pressure acting on the structure. Control forces are assumed to be physically realized through changes of tension in guy cables, supporting themast.The only requiredmeasurements are velocities of guy cables at the anchor-points.On the basis of those, a complete state of deformation of the structure is obtained by using the Kalman filter. The Davenport spectral density function is adopted as a model of the stochastic action of the wind. Key words: structural vibrations, vibration control, guyed masts, wind fluctuations 1. Introduction Guyedmasts applied in radio, television and cellular phone industry, belong to the class of vibration-prone structures. Both their height (which can be even a few hundredmeters) and their location in open spaces make them exposed to actions of strong windblasts of different velocities. In winter, masts are often covered with icing. It causes an increase, not only of the structure weight, but also of the surfaces of elements exposed to wind pressure. Ice, together with wind pressure, are the most common reasons of mast failures. Dynamic analysis of a guyed mast is nonlinear, because of nonlinear be- haviour of guy cables (McCaffrey and Hartmann, 1972). During recent years, many control methods have been developed, but their practical application to mast-like structures still remains limited. One of the recent approaches was presented by Preumont (2002). Replacing cables with massless strings, he proposed an Integral Force Feedback controller to reduce vibration of a 406 B. Błachowski truss structure. Another control technique of cable structures, called Active Stiffness Control, was proposed by Fujino et al. (1993) and was applied to cable-stayed bridges. Recently, the problem of damping cable vibrations by semi-active control was investigated by Spencer (2002). The objective of this paper is the analysis and simulation of the Model Based Predictive Control (Goodwin et al., 2001) of guyedmast vibrations. 2. The model of a mast In deriving equations of motion, the following assumptions on the dynamics of a mast are made: • for requirement of proper functioning of the equipment attached to the top of the mast, only vibrations of small amplitudes are taken into ac- count • guy cables are tightly stretched and do not carry bending loads • deformations of structural members are linear elastic. Fig. 1. Guyedmast and its feedback control system The column of themast is represented by a prismatic truss of a triangular cross section. One cable end is attached to themast and the second one to an anchored mechanism, allowing for control of cable tension. Model based predictive control of guyed mast vibration 407 For the purpose of simulation, the mast is discretized according to the Finite Element Method (FEM). Every guy cable is represented by a chain of rods. It is assumed that each chain node has three degrees of freedom. The whole structure, i.e. the mast and its supporting guy cables, is subjected to the action of stochastic wind gusts. Thedynamicanalysis of themast isprocededby its static analysis under its dead load andprestressed forces in the cables.Thisway, the initial deformation of the structure is obtained, locations of nodes are determined and the global stiffness matrix is updated. The equations of motion of the N-degree of freedom (DOF) structure can be written as follows Mq̈(t)+Dq̇(t)+Kq(t)=Bff(t)+Buu(t) (2.1) where M – N×N mass matrix, D – N×N dampingmatrix, K – N×N stiffness matrix, q – N-vector representing displacements of the structure, f – P-vector of wind velocity fluctuations, u – R-vector of control forces, Bu – control inputmatrix of proper dimension, Bf – wind inputmatrix of proper dimension Bu =    0 0 0 0 0 0 ... ... ... ... ... ... 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ... ... ... ... ... ... 0 0 0 0 0 0       Allocation of the actuators Nonzero elements in the control input matrix represents the allocation of the actuators. The formof thewind inputmatrixwill be explained in the next section. Additionally, measured and performance outputs are introduced in the following form y(t)=Cqq(t)+Cvq̇(t)+v(t) z(t)=Cpq(t) (2.2) where 408 B. Błachowski y(t) – S-vector of measured node displacements and velocities, Cq,Cv – the allocation of the displacement and velocity sensors, Cp – degrees of freedom to be controlled Cq =0 Cv =    0 · · · 1 0 0 0 0 0 · · · 0 0 · · · 0 1 0 0 0 0 · · · 0 0 · · · 0 0 1 0 0 0 · · · 0 0 · · · 0 0 0 1 0 0 · · · 0 0 · · · 0 0 0 0 1 0 · · · 0 0 · · · 0 0 0 0 0 1 · · · 0    Cp =    0 · · · 1 0 0 0 0 · · · 0 0 · · · 0 1 0 0 0 · · · 0 0 · · · 0 0 1 0 0 · · · 0 0 · · · 0 0 0 1 0 · · · 0 0 · · · 0 0 0 0 1 · · · 0    v(t) is an S-vector characterising themeasurement noise which is assumed to be completely random. Moreover, it is assumed that measurement errors do not depend on wind disturbances E(v(t))= 0 ∀t E(v(t)v(τ)⊤)=Rδ(t− τ) z(t) is a T-vector of the performance output and Cp is a Boolean matrix of proper dimensions selecting the degrees of freedom significant for the perfor- mance output. Themodel of the mast, in the configuration space, entails long simulation times of the mast dynamics due the large number of degrees of freedom. To avoid this difficulty, a modal transformation is performed, and the number of dynamic degrees of freedom is reduced to the first Nc mode shapes q(t)=Θη(t)=Θcηc(t)+Θrηr(t)≈Θcηc(t) (2.3) where Θc – N×Nc matrix of the first Nc mode shapes, η – N-vector of modal amplitudes, ηc,ηr – vectors of controlled and residual modal amplitudes of di- mensions Nc and Nr, respectively. It is assumed that Nc ≪N. Nc dependson the requiredaccuracyof the control. Themode shapes are normalised with respect to the mass matrix M, that is Θ ⊤ c MΘc = I Model based predictive control of guyed mast vibration 409 which yields Θ⊤c KΘc =Ω 2 c where Ωc is an Nc×Nc matrix with the diagonal containing successive eigen- frequencies of the structure. Next, adopting the model of viscous damping, one obtains Θ ⊤ c DΘc =2ΞcΩc where Ξc is an Nc×Nc modal dampingmatrix. Performing modal reduction and normalising the mass matrix (Thomson, 1981), the equations ofmotion,measuredoutput y(t) andperformanceoutput z(t) take the following forms η̈c(t)+2ΞcΩcη̇c(t)+Ω 2 cηc(t)=Θ ⊤ c Bff(t)+Θ ⊤ c Buu(t) y(t)=CqΘcηc(t)+CvΘcη̇c(t)+v(t) (2.4) z(t)=CpΘcηc(t) Now, the state vector of the structure is introduced in the form x⊤m(t)= [η ⊤ c (t), η̇ ⊤ c (t)] which, after substitution into (2.4), yields ẋm(t)=Amxm(t)+Bm1f(t)+Bm2u(t) z(t)=Cm1xm(t) (2.5) y(t)=Cm2xm(t)+v(t) where Am = [ 0 I −Ω2c −2ΞcΩc ] Bm1 = [ 0 Θ⊤c Bf ] Bm2 = [ 0 Θ⊤c Bu ] Cm1 = [ CpΘc 0 ] Cm2 = [ CqΘc CvΘc ] Finally, it can be observed that the modal truncation does not affect the dimension of the measured output y(t). 3. Wind modelling Themast is exposed to action of wind, and therefore experiences time-varying loads. Initially, wind pressure was only modelled as a static loading. Later, 410 B. Błachowski (Iannuzzi and Spinelli, 1987) more realistic wind models were introduced in the formof trigonometric series. Currently, in themodelling of stochastic wind pressure, linear filters are frequently used (Gawronski et al., 1994). The overall wind velocity can be decomposed into its average value f and velocity fluctuation f(t) F(t)= f+f(t) (3.1) It is assumed that the average value f exerts a static load. The fluctuation component f(t) is a random with the zero mean, which in the frequency domain is characterized by the spectral density function called theDavenport spectrum Sf(n)= 4f 2 10κ n X2 (1+X2)4/3 X = 1200n f10 (3.2) where n is the frequency [Hz], f10 is the average velocity at the 10 meter altitude [m/s] and κ – terrain roughness coefficient. The along-wind force acting on both the mast and its guy cables can be decomposed into a static and dynamic part P(t)= 1 2 ρaCdAef 2 +ρaCdAeff(t) (3.3) where ρa is the air density, Cd – drag coefficient, Ae – exposition area, f de- notes the average wind velocity and f is the wind velocity fluctuation. The above relation allows determination of elements of the matrix Bf in (2.4)1 Bf =    ρaCdAe,1f cosα ρaCdAe,1f sinα 0 ρaCdAe,2f cosα ρaCdAe,2f sinα 0 ...    (3.4) where α is the wind direction. The wind load is obtained by applying as an input a purely random pro- cess to a filter that approximates the Davenport spectrum (Davenport, 1961) within a desired bandwidth. TheDavenport filter is presented as a linear dynamical system of the form ẋw(t)=Awxw(t)+Bww(t) (3.5) f(t)=Cwxw(t) Model based predictive control of guyed mast vibration 411 Fig. 2. Davenport filter where w(t) is a purely random sequence of data, E(w(t)) = 0 ∀t (3.6) E(w(t)w(τ)⊤)=Qwδ(t− τ) and the matrices Aw, Bw, Cw are chosen such that the following formula is satisfied Hw(j̟)=Cw(sI−Aw) −1 Bw (3.7) The structure of the Davenport filter ensures that the input w(t) is uncorre- lated with the measurement noise v(t) E ( v(t)w(τ)⊤ ) =0 (3.8) 4. Controllability Before designing a control system, it is important to verify the controllability. The classical criterion for the controllability tells that a dynamical system is controllable if its controllability matrix has the rank 2Nc rank [ Bm2 AmBm2 A 2 mBm2 . . . A 2Nc−1 m Bm2 ] =2Nc (4.1) where Nc is the number of mode shapes. However, in large systems there are numerical difficulties involved in cal- culating this rank. Hence, from the computational point of view, it is much more convenient to use an alternative method. One of them is the concept of modal controllability index. Information on the controllability is here obtained for particular mode shapes, through calculating the lengths of successive rows of the matrix Θ⊤c Bu µj =Θ (j) c ⊤ BuB ⊤ uΘ (j) c j=1,2, . . . ,Nc (4.2) where Θ(j)c is the jth column of the matrix of the mode shapes. 412 B. Błachowski The index equal to zero signifies that the corresponding mode shape is uncontrollable. Additional attention has to be paid to those indices which correspond to multiple eigenfrequencies (Joshi, 1989). For the guyed mast under consideration it was observed that the control forces very weakly affect the symmetrical mode shapes of cables (Fig.4), but they have significant influence on the antisymmetrical ones (Fig.5), i.e. those involving transverse vibrations of the column. It is advantageous because those modes are responsible for transverse vibration of the top of the mast, which is going to be damped. Fig. 3. Dominant mode shapes of column Fig. 4. Dominant axi-symmertical mode shapes of cables Fig. 5. Dominant antisymmetrical mode shapes of cables Model based predictive control of guyed mast vibration 413 5. Control system One of the possible implementations of this idea is a control system which consists of: a velocity sensor, digital controller and hydraulic actuator. The actuator is driven by the controller which gets information from the velocity sensor. As a result, the control forces change the tension in guy cables. Moreover, it is assumed that the control system is collocated (Fig.6), which means that sensor placements coincide with that of actuators. Aditionally, the hydraulic actuator and velocity sensor are supported in such a way that the control forces do not affect measurements. Fig. 6. Control system scheme 6. Control strategy After answering the question of the controllability, one can proceed to choose a suitable control strategy. In this paper, the idea of Model Based Predictive Control (Goodwin et al., 2001) is used. It is an algorithm, based on on-line solving an optimal control problem (Fig.7), which can be summarized in the following steps: (i) at each time instant, using past and currentmeasurements, estimate the current state vector 414 B. Błachowski (ii) solve on-line the optimal control problem over some future interval, ta- king into account the current and future constraints (iii) use, as the current control signal, the first step in the computed optimal control sequence. When dealing with an uncertain structure, the control system has to be au- gmented with a model updating procedure. The objective of that procedure is to identify current dynamics of the structure which operates in varying environmental conditions. Fig. 7. Model predictive control of guyedmast 7. Optimal estimator Optimal control of a dynamical system requires knowledge of the state of that system. In practice, individual state variables cannot be determined exactly by direct measurements. Instead, measurements that can be made are func- tions of the state variables, and thesemeasurements contain randomerrors. A Model based predictive control of guyed mast vibration 415 guyedmast itself is also subjected to randomwind disturbances. If the struc- ture is completely observable, and, in collocated systems, this property holds simultaneously with the controllability, a virtual dynamical system called the estimator can be constructed. The goal of the estimator is to assess, on the basis of the measured qu- antities, the current deformation of the structure and the value of the wind force (Fig.8). For that purpose, an augmented dynamical system is introduced which consists of a model of the guyed mast and a model of the wind force represented by the Davenport filter. It takes the form ẋm(t)=Amxm(t)+Bm1Cwxw(t)+Bm2u(t) ẋw(t)=Awxw(t)+Bww(t) z(t)=Cm1xm(t) y(t)=Cm2xm(t)+v(t) (7.1) The above equations can be written uniformly as ẋ(t)=Acx(t)+Bc1w(t)+Bc2u(t) z(t)=Cc1x(t) y(t)=Cc2x(t)+v(t) (7.2) Finally, equations (7.2) are translated into a discrete time state-space formu- lation (Franklin et al., 1990) xk+1 =Axk+B1wk+B2uk zk =C1xk (performance output) yk =C2xk+vk (measured output) (7.3) The design of an optimal estimator depends on probabilistic data, concerning the initial condition of the system, disturbances andmeasurement errors E(x0x ⊤ 0 )=M0 E(wkw ⊤ l )= 1 ∆t Qwδkl =Qδkl (7.4) E(vkv ⊤ l )=Rδkl where ∆t is the sampling time. 416 B. Błachowski Fig. 8. Kalman filter concept Taking into account the above facts, the maximum likelihood estimate of the state xm is given by sequential use of the following procedure (Bryson and Ho, 1975) x̂k+1 =Ax̂k+B2uk+Ke(yk−C2x̂k) x̂0 given Ke =APkC ⊤ 2R −1 k=0,1, . . . ,m Pk =Mk−MkC ⊤ 2 (C2MkC ⊤ 2 +R) −1C2Mk Mk+1 =APkA ⊤+B1QB ⊤ 1 (7.5) To implementMPC, it is required toknownotonly the current, butalso future states of the system. Theory of optimal prediction tells that the best estimate of the future state vector can be obtained by taking the expected value of the forward solution to discrete state equations, calculated on the basis of the current estimate: x̂k =A k x̂0+ k−1∑ m=0 A k−1−m B2um k=1,2, . . . ,n (7.6) Model based predictive control of guyed mast vibration 417 8. Optimal control Knowing the current state estimate, one can propose an optimization proce- dure for the discussed control system. The following cost function is proposed by wishing to drive the current state vector to the smallest possible value over a specified time interval, but without spending toomuch control effort to achieve this goal J = z⊤nΦznzn+ n−1∑ k=0 z ⊤ kzk+u ⊤ kΨuk =x ⊤ nΦnxn+ n−1∑ k=0 x ⊤ kΦxk+u ⊤ kΨuk (8.1) where Φn,Φ and Ψ are weighting matrices of proper dimensions. It is assumed that the control forces uk are several times smaller than the static tensile force in cables. This allows one to use a quadratic cost function as the performance index. Assuming a randomwind fluctuation to be equal to its mean value, which is zero, and using the solution to the discrete state equation, one can arrive at the followingquadraticprogrammingproblem.Additionally, the constraints on the control force amplitude can be expressed in the form of linear inequalities. Finally, one obtains a quadratic programming problemwith the constraints J = J0+U ⊤ ΓU+HU Umin ¬U¬Umax (8.2) where H=2x⊤0Ω ⊤ΦΛ, Γ=Λ ⊤ ΦΛ+Ψ and Λ=    B2 0 0 . . . 0 AB2 B2 0 . . . 0 A 2 B2 AB2 B2 . . . 0 ... ... ... ... ... A n−1 B2 A n−2 B2 A n−3 B2 . . . B2    Ω=    I A ... A n−1 A n    Ψ=    Ψ 0 0 . . . 0 0 Ψ 0 . . . 0 0 0 Ψ . . . 0 ... ... ... ... ... 0 0 0 . . . Ψ    Φ=    Φ 0 . . . 0 0 0 Φ . . . 0 0 ... ... ... ... ... 0 0 . . . Φ 0 0 0 . . . 0 Φn    U= [ u0 u1 u2 . . . un−1 ]⊤ Constraints can be also imposed on the rate of change of the control forces. This can be particularly important in the case of using hydraulic actuators. 418 B. Błachowski After solution, the first signal in the optimal control sequence is applied, and the whole procedure is repeated at the next time instant U OPT =arg min LU¬b HU+U⊤ΓU (8.3) where L= [ I −I ] b= [ Umax Umin ] I is identity matrix. Closed-loop stability of the system can be verified by Lyapunov theory where the quadratic cost index is chosen as the Lyapunov function. As it was mentioned earlier, the identification of themast parameters, like its mass and stiffness, is performedby an updatingprocedure of themodel. Similarly to the case of the state estimator, the assessment of the mast parameters could be done by aKalman filter whose state is defined by the uncertain parameters of themast.To realize the above idea, it is necessary to assumea certainmeasure for the system of parameter estimation, like covariance of uncertainty of the parameters. 9. Numerical simulation In this section some results of numerical simulations are shown. A 100-meter- high mast is supported by 6 active guy cables. The guy cables are anchored at an angle of 45◦. The column of the mast is a spatial truss structure with a cross-section of shape of an equilateral triangle of 1.5 meter side length. The constitutent elements of the column are circular tubes with following cross-section areas: 0.0074m2 for vertical members, 0.001m2 for horizontal and diagonal members. It is assumed that the structure is made from steel, which has following properties: Young’s modulus E =205GPa, mass density ρ=7500kg/m3. The diameter of guy cables is chosen as d=30mm.The guy cables are prestressed by a force of T =400kN, which correspond to internal stresses of 570MPa. Dynamics of themast ismodelledby32mode shapesof frequencies ranging from 0 to 20rad/s. For the first 10modes, damping equal to 1% of the critical damping is assumed, i.e. D=0.006M+0.0001K In order to verify the controllability of the structure,modal controllability indices are calculated: Model based predictive control of guyed mast vibration 419 Mode no. Frequency Modal controllability ω [rad/s] index µ 1. 0.3438 9.8361E-2 2. 0.3440 9.8499E-2 3. 4.7474 9.0610E-2 4. 4.7492 9.0650E-2 5. 6.6279 7.9389E-9 6. 6.6892 4.2767E-6 7. 6.6892 3.9231E-6 8. 6.6899 1.8798E-7 9. 6.8727 8.8209E-3 10. 6.8747 8.9116E-3 The data characterising wind gusts are following: • air density 1.22kg/m3 • average wind velocity 25m/s • drag coefficient Cd =1 Fig. 9. Scheme of guyedmast and direction of applied wind forces Davenport filter (3.7) has the following transfer function Hw(s)= a0+a1s+a2s 2+a3s 3 b0+ b1s+ b2s2+ b3s3+ b4s4 where the filter parameters are a0 =3.4197 a1 =−686.3151 a2 =230.1426 a3 =3.9021 b0 =0.3538 b1 =22.7788 b2 =224.7118 b3 =38.2997 b4 =0.331 Full spatial correlation of velocity fluctuation is assumed. 420 B. Błachowski Fig. 10. Displacement in x1 direction (dashed line-without control, solid line-with MP control) Fig. 11. Displacement in x2 direciton (dashed line-without control, solid line-with MP control) Fig. 12. Inclination angle in x1x3 plane (dashed line-without control, solid line-with MP control) Model based predictive control of guyed mast vibration 421 InFig.10-Fig.13, dynamicbehaviour of themast is presented.Thedashed line corresponds to the open loop systemwithout control and the solid line to the closed loop system with Model Based Predictive Control. The disturbing forces are applied in the form of random fluctuations with the Davenport spectrum. In the case of the transverse displacement of the top of the mast, a significant reduction of vibration amplitudes can be observed. The control strategy is effective regardless of the wind direction. Fig. 13. Inclination angle in x2x3 plane (dashed line-without control, solid line-with MP control) 10. Conclusions • A 3DFEMmodel of a guyed mast under control forces is proposed. • The external loading is included in the form of a stochastic windmodel with the Davenport spectrum. • Controllability of individual mode shapes of the mast is determined. • Amodel based estimator for the mast is constructed. • It has been demonstrated that a combination of theKalman andDaven- port filters enables prediction of wind forces acting on the guyedmast. • A numerical simulation of the control process is presented, displaying a significant reduction in vibration amplitudes of the top of the mast. 422 B. Błachowski Fig. 14. Control forces in guy cables (Fig.9) 11. Future work The linear dynamicalmodel of a guyedmast the presentwork is based ondoes not reflect such phenomena as e.g. parametric resonance. On the other hand, it was shownbyPreumont (2002) thatActive Damping strategies are effective even in the presence of loads capable of inducing parametric resonance. To verify in this respect the performance of the proposed algorithm based on Model Based Predictive Control, further research in this field is required. Acknowledgements The work described herein has been carried out with the financial support pro- vided by the State Committee for Scientific Research of Poland (KBN) under grant No. 5T07A00123, which is gratefully acknowledged. References 1. Bryson A.E. Jr., Ho Y., 1975,Applied Optimal Control-Optimization, Esti- mation and Control, Hemisphere Publishing Corporation 2. DavenportA.G., 1961,The spectrumof horizontal gustiness near the ground in high winds, J. of the Royal Meterological Society, 87, 194-211 Model based predictive control of guyed mast vibration 423 3. Franklin G.F., Powell G.D., Workman M.L., 1990, Digital Control of Dynamic Systems, Sec. Ed., AddisonWesley 4. Fujino Y., Warnitchai P., Pacheco B.M., 1993, Active stiffness control of cable vibration,ASME Journal of Applied Mechanics, 60 5. GawronskiW., Bienkiewicz B., Hill R.E., 1994,Wind-induced dynamics of a deep space network antenna, J. of Sound and Vibration, 178, 1, 67-77 6. GoodwinG.C.,GraebeS.F., SalgadoM.E., 2001,Control SystemDesign, Prentice Hall 7. Iannuzzi A., Spinelli P., 1987, Artificial wind generation and structural response, J. of Structural Engineering, 113, 12 8. Joshi S.M., 1989,Control of Large Flexible Space Structures, LectureNotes in Control and Information Sciences, vol. 131, Springer, Berlin 9. McCaffreyR.J.,HartmannA.J., 1972,Dynamics of guyed tower,J. Struc- tural Division Proceedings of the ASCE, 98, ST6 10. PreumontA., 2002,VibrationControl of Active Structures –An Introduction, 2nd ed., Kluwer 11. Spencer B.F., 2002, Smart damping technology: Application and opportuni- ties, 34th Solid Mechanics Conference, SolMech 2002, Zakopane 12. Thomson W.T., 1981,Theory of Vibration with Applications, Prentice Hall Predykcyjne sterowanie drganiami masztów z odciągami Streszczenie Celem pracy jest algorytm i jego przykładowe zastosowanie do symulacji nume- rycznej optymalnego sterowania drganiami masztów z odciągami. Zadaniem układu sterowania jest minimalizacja amplitud poprzecznych drgań wierzchołka masztu wy- wołanychoddziaływaniemwiatru.Realizacja sił sterowania odbywa się poprzez zmia- nę naciągu w odciągachmasztu. Estymacja pełnego stanu deformacji konstrukcji na podstawie pomiaru jedynie prędkościw punktach zakotwienia odciągówuzyskana jest poprzez wykorzystanie filtru Kalmana. Do zamodelowania losowego oddziaływania wiatru użyto funkcję gęstości widmowej Davenporta. Manuscript received December 28, 2004; accepted for print October 19, 2006