The protection of civil structures, including. Structural control technology FLEXIBLE STRUCTURES

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Edmund Bridges
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1 Structural control technology System identification and control of flexible structures by A. Forrai, S. Hashimoto, H. Funato and K. Kamiyama The main trend in structural technology is to change from conventional earthquake resistant structures, which are designed not to collapse even under the strongest earthquake to structural controlled buildings, which are designed to suppress the vibrations itself. Indications are that control methods will be able to make a genuine contribution to this problem area, which is of great economic and social importance. Current researches in the field of active structural control focus on: system identification, model order reduction, limited control authority, available measurements etc. This article reviews some of the recent developments in the area of controlled civil structures and highlights a robust control perspective in active vibration suppression control of flexible structures, considering a grey box approach in the system identification process. The protection of civil structures, including their human occupants and their material contents, is without doubt a worldwide priority. Such protection may range from reliable operation and comfort, on the one hand, to survivability on the other. Examples of such structures include buildings, offshore rigs, towers, roads, bridges and pipelines. Events that cause the need for such protective measures include: earthquakes, winds, waves, traffic, lightning, and today, regrettably deliberate acts. Minimising the damage to high-rise buildings from large earthquakes is very important for maintaining urban functions, especially if buildings contain critical facilities. Although passive isolation systems have been used in civil structures, if an earthquake of great magnitude occurs passive systems are ineffective, therefore active vibration isolation may be useful. Significant progress has been made over the last decades toward making active structural control a viable technology for enhancing structural functionality and safety against natural hazards, such as earthquakes and strong winds. 1 The first full-scale application of active control to a building was accomplished by the Kajima Corporation, Japan, in 1989 (the Kyobashi Seiwa building in Tokyo 11 storeys, 33 1m). The role of the active system is to reduce building vibration under strong winds and moderate earthquake excitations and consequently to increase comfort of the occupants. Since construction of Kyobashi Seiwa building, a number of serious challenges remain to be resolved before feedback control technology can gain general acceptance by the engineering and construction profession at large. These challenges include: 5 reducing capital cost and maintenance 5 eliminating the reliance on external power 5 increasing system reliability and robustness 5 gaining acceptance of non-traditional technology. Hybrid and semi-active control strategies are particularly promising in addressing a number of challenges to this technology. This article discusses some of the hybrid and semi-active control technologies with the main focus on COMPUTING & CONTROL ENGINEERING JOURNAL DECEMBER

Table 1 Structural control technologies technology control device advantages disadvantages hybrid mass tuned mass damper (TMD) compact, efficient, external power, damper + easy to implement stability

2 Table 1 Structural control technologies technology control device advantages disadvantages hybrid mass tuned mass damper (TMD) compact, efficient, external power, damper + easy to implement stability is essential (HMD) active mass damper (AMD) hybrid base passive base isolation simple, reliable, external power, isolation + cost-effective non-linearity of active base isolation base isolation active vibration suppression systems. Structural control technologies Hybrid isolation technologies A hybrid control system is typically defined as one that employs a combination of passive and active devices. Research in the area of hybrid control systems has focused primarily on two classifications of systems: (i) hybrid mass damper systems and (ii) hybrid base isolation. (i) Hybrid mass damper technology: Hybrid mass damper (HMD) is the most common control device employed in full-scale civil engineering applications. The HMD is a combination of a tuned mass damper (TMD) and an active mass damper (AMD). Many researchers have made significant contributions toward development of hybrid mass damper technology that are compact, efficient and easy to implement in practice. One example is the Yokohama Land Mark Tower, completed in Yokohama, Kanagawa, Japan, in 1993 (see Fig. 1). For this 296m tall, 70 storey building two hybrid mass dampers (each having a mass of 170 tonnes) have been installed and actuated by servo motors, in order to suppress the vibrations caused by strong winds and medium-sized earthquakes. The process of constructing the Land Mark Tower provides another interesting and attractive application of active control, which is associated with the way in which construction cranes were used during its erection. Active position control of the cranes has been carried out by two fans. These fans prevented excessive displacement and rotation of the building panels while hoisting and installing them, even under strong winds. Moreover, the overall efficiency of the crane work was significantly improved, and resulted in reduced construction time for the Yokohama Land Mark Tower. Researchers have investigated various control methods for hybrid mass dampers, for example optimal control methods, gain scheduling techniques (in which the Fig. 1 Yokohama Land Mark Tower control gains vary with the excitation level to account for stroke and control force limitations), sliding mode control etc. (ii) Hybrid base isolation technology: Another class of hybrid control system that has been investigated is the hybrid base isolation system, consisting of a passive base isolation system combined with an active control system to improve the effects of the base isolation system. Base isolation systems have been implemented on civil engineering structures worldwide for a number of years because of their simplicity, reliability and effectiveness. However, base isolation systems are passive systems and are limited in their ability to adapt to changing demands for structural response reduction. With the addition of an active control device a higher level of performance can be achieved without substantial increase in the cost, which is very appealing from a practical viewpoint. Since the base isolation by itself can reduce the inter-storey drift and absolute base displacement, the combination with active control is able to achieve both low inter-storey drift and, at the same time, limit the maximum base displacement with a single set of control forces. Because base isolation systems exhibit nonlinear behaviour, researchers have developed various nonlinear control strategies, including robust nonlinear control. semi-active orifice fluid dampers, cannot destabilise, cannot achieve the isolation variable stiffness devices, can operate performance of fully controllable friction devices on battery power active systems Semi-active isolation technology Control strategies based on semi-active devices appear to combine the best features of both passive and active control systems and to offer the greatest likelihood for near-term acceptance of control technology. The attention received in recent years can be attributed to the fact that semi-active devices offer the adaptability of active control devices without requiring the associated large power source. In fact many can operate on battery power, which is critical during seismic events when the main power source to the structure may fail. According to the presently accepted definitions, a semi-active control device is one that cannot inject mechanical energy into the controlled structural system, but has properties that can be controlled to optimally reduce the response of the system. Therefore, in contrast to active control devices, semiactive control devices do not have the potential to destabilise (in the bounded-input/bounded-output sense) 258 COMPUTING & CONTROL ENGINEERING JOURNAL DECEMBER 2001

3 the structural system. Preliminary studies indicate that appropriately implemented semi-active systems perform significantly better than passive devices and have the potential to achieve the majority of the performance of fully active systems. Examples of such devices are: orifice fluid dampers, variable stiffness devices, controllable friction devices, controllable tuned liquid dampers and controllable impact dampers. Some of the advantages and disadvantages of the presented vibration suppression technologies are summarised in Table 1. q 1 q 3 q 2 c 2 c 1 f c m 3 c 3 m 2 m 1 k 1 m a k 2 k 3 accel. sensor q a accel. sensor host computer PC-AT DSP TMS320C31 D/A motor driver A/D amp. Active vibration suppression control Mathematical modelling In high-performance active control it is a requirement to establish an accurate model of the structure to avoid instability of the control system. First, the equations of motion for an N-storey idealised flexible structure are presented. In this idealisation the beams and floor systems are rigid in flexure, and several factors are neglected: axial deformation of the beams and columns, and the effect of axial force on the stiffness of the columns. The mass is distributed throughout the building, but is idealised as being concentrated at the floor levels. The assumption is generally appropriate for multi-storey buildings because most of the building mass is indeed at the floor levels. For simplicity each storey of the structure has one degree of freedom in the horizontal direction. 2 The linearised mathematical model of an N-storey flexible structure can be written as: Mq + Cq + Kq = p (1) where q =[q 1,,qN]T is the relative displacement vector to the base, M = diag[m 1,,m N] is the mass matrix, C is the damping matrix, K is the stiffness matrix and p is the applied force vector p =[p 1,,pN] T. During the identification the external force to the flexible structure will be applied at the Nth storey by the active mass driver. In this case the applied force vector becomes p = [0,, m aq a] T, where m a is the active mass, and q a is the active mass acceleration. In order to solve eqn. 1, for identification purposes, when the external force is applied at the Nth storey, the Laplace operator is applied. Let us note: H(s)=Ms 2 + Cs + K (2) The transfer functions related to the flexible structure, Fig. 2 Investigated flexible structure q g when the input force is applied at the Nth storey by the active mass driver, and the output signal is the relative displacement of ith story, can be written: G Ni(s)= δni(s) (3) deth(s) It is easy to demonstrate that the δni(s) have the following forms: δni(s)=α N+i 2s N+i 2 + +α 1s + α 0 (4) that is a N + i-2 order polynomial in s. It can be also demonstrated that deth(s) has the form: deth(s)=β2ns 2N + +β1s + β0 (5) where the coefficients α and β can be computed when the M, C and K matrices are known. Furthermore, it can be demonstrated that if the M, C and K matrices are positive semi-definite (which is true in case of flexible structures), then the poles of the characteristic equation are complex conjugate pole pairs with negative real part (the poles lie in the left hand plane). These preliminary results are very useful, during the grey box identification process. System identification grey box approach In this section we will particularise the previous general equations for the investigated experimental setup, a three-storey flexible structure, shown in Fig. 2. In the framework of system identification the basic issue for input signal design can be formulated as: to achieve the desired input spectrum for a signal with as COMPUTING & CONTROL ENGINEERING JOURNAL DECEMBER

gain, db 0 10 20 30 40 50 60 70 80 90 10 0 30th order model: black box identification 6th order model: grey box identification Fig.

4 gain, db th order model: black box identification 6th order model: grey box identification Fig. 3 System identification-bode plot frequency, rad/s small a crest factor as possible. 3 A most common approach is to generate a pseudorandom binary signal (PRBS), which is a deterministic signal with white-noise like properties. The PRBS is generated by a linear feedback shift register, which consists of n registers. It can be proved that the maximum period length of the output signal is M =2 n -1. In accord with the previous notations the frequency range of the PRBS is: 1 2π ω 2π (6) MT s 2T s where T s is the sampling time related to the PRBS. Like white random binary noise, PRBS has an optimal crest factor. The considered flexible structure is identified based on measured input/output data, using the well known 2 Auto-Regressive exogenous (ARX) model. 3 The input signal is generated by the active mass driver and the output is the relative displacement between the base and the 3rd storey. The identified mathematical model is a 30th-order model; therefore model order reduction 10 3 is performed based on balanced truncation technique and physical model. Checking the pole-zero map as well as the root-locus related to the reduced order model, we will find that the identified plant is a non-minimum phase plant (plant with right half-plane zeros). There are two common physical situations that manifest themselves as right half-plane zeros in a system model. Right half-plane zeros arise due to time delays in a system and they arise when the measurement of important variables cannot or does not occur. For active vibration suppression of flexible structures the right half-plane zeros arise from non-collocation of actuators and sensors, but in the considered case the sensors and the actuators are collocated. Owing to its non-minimal phase realisation, grey box identification of the flexible structure is performed, in order to find out the parameters of the physical model. 4 The mathematical model presented previously (based on physical equations) will represent the starting point during the grey box identification. Estimation method of mass damping and stiffness matrices are presented. According to eqn. 5 for a three-storey flexible structure the characteristic equation is written as: deth(s)=β 6s 6 + +β 1s + β 0 (7) Let us note the characteristic equation of the identified flexible structure, after model order reduction with: deth^(s)=β^6s 6 + +β^1s + β^0 (8) Fig. 4 Experimental set-up It is noted that the parameters β 6,, β0 in the equation are represented symbolically; on the other hand β^6,, β^0 are represented numerically in terms of mass, damping and stiffness constants. From eqns. 7 and 8 the following cost function is defined: 260 COMPUTING & CONTROL ENGINEERING JOURNAL DECEMBER 2001

5 J = Σi=1 6 (βi β^i) 2 (9) By minimising the cost function J via an optimisation method, mass, damping w and stiffness coefficients are estimated. The restrictions imposed during the r minimisation are that: c 1 = c 2 = c 3 = c as u well as k 1 = k 2 = k 3 = k, which is true in the case of the investigated structure. Furthermore, the optimisation is performed in the frequency range defined by the PRBS. Comparing the models identified by grey box and black box identification methods (Fig. 3, Bode plot), we find that they are in good agreement, especially at the resonant frequencies. Line 2 corresponds to the high-order model (30th-order model), based on black box identification, and the line 1 corresponds to the model identified by grey box identification (6th-order model). Robust vibration suppression control This section focuses on robust vibration suppression control of the investigated flexible structure; the experimental set-up s photography is presented in Fig. 4. The controller design employs the well-known H synthesis approach considered to be one of the most robust control methods. Based on the identified black box and grey box model it is easy to derive the model structured (parametric) and unstructured (unmodelled dynamics) uncertainties. The augmented plant is presented in Fig. 5, where G p(s) is the plant and G c(s) is the controller transfer function, with the following notations: r is the reference input; u x is the control input; y is the controlled output; e is the control 4 error; and w is the external disturbance. Furthermore, W S(s) 3 and W T(s) are the well-known 2 weighting functions related to the sensitivity S(s) and complementary 1 sensitivity functions T(s). 0 The main objective of the disturbance rejection problem is to -1 minimise the M(s) transfer function -2 between the w(s) and y(s) over a wide frequency range, which can be -3 expressed as: -4 M(s) < γ (10) displacement, m + e Fig. 5 Augmented plant + + augmented plant G p (s) G c (s) y W S (s) W T (s) The robust performance problem is to design a proper controller G c(s) so that the feedback system for the nominal plant is internally stable and the inequality holds: W S(s)S(s) + W T(s)T(s) < 1 (11) This inequality is called the mixed-sensitivity cost function, because it penalises both the sensitivity S(s) and complementary sensitivity T(s) functions. 5 Another important issue is related to the control constraint imposed on the control signal. Controller design for systems with hard constraints is a quite vivid area of research, due to the fact that most practical control problems are dominated by constraints on the control signal. 6 natural damping active isolation y 1a y 1b y 2 where γ is a constant, or in a more general case a frequency dependent function. time, s Fig. 6 Time response for the El Centro disturbance COMPUTING & CONTROL ENGINEERING JOURNAL DECEMBER

6 Fig. 7 Frequency response gain, db without damping natural damping passive isolation active isolation frequency, rad/s Controller designs that take into account the saturation effect a priori are usually separated into two categories: 5 designs that prevent saturation of the control signal and therefore enjoy a linear framework (as long as plant and controller are linear) 5 methods that allow saturation and therefore facing a nonlinear set-up. The considered approach focuses on the first category (saturation avoiding) philosophy. The H controller design is approached from linear matrix inequality (LMI) methods, which are particularly well suited for solution with modern computational tools, such as the LMI Control Toolbox in Matlab or other software packages. The controller s continuous transfer function is discretised with the sampling time T s = 0 001s and is implemented on the TMS 320C31 DSP board. The most common criteria on which the controllers are evaluated can be summarised as: 5 ability to minimise the root-mean-square displacement and acceleration of the ith storey 5 ability to minimise the non-dimensional peak displacement and acceleration 5 ability to minimise the inter-storey drift. The effectiveness of the controller taking into account the constraint on the control signal is evaluated through experiments comparing the open- and closed-loop time histories of the 3rd storey, when the structure is excited with a scaled historical (El Centro) earthquake record, shown in Fig. 6. Fig. 7 shows the frequency response of the investigated flexible structure for undamped, naturally damped as well as passively and actively isolated structures. Concluding remarks Protecting civil structures from natural and other types of unwanted dynamic influences is continuing to move steadily up the list of high-priority needs of the world community. The structures alone represent a huge investment of resources; moreover there are platforms that carry within them very expensive equipment, irreplaceable records and priceless human life. Full-scale buildings are being controlled successfully and attention is turning toward the features of a whole new family of actuators, especially those of semi-active type. In summary, the modern thrust toward control of civil structures is providing a new opportunity for control engineers to make their work more understandable to the public, while at the same time making significant technical, economic and social contributions. References 1 SPENCER, B. F., JUN., and SAIN, M. K.: Controlling buildings: a new frontier in feedback, IEEE Control Systems Magazine on Emerging Technology, 1997, 17, pp CHOPRA, A. K.: Dynamics of structures theory and applications of earthquake engineering (Prentice-Hall, New York, 1995) 3 LJUNG, L.: System identification theory for the users (Prentice- Hall, New York, 1987) 4 FORRAI, A., HASHIMOTO, S., ISOJIMA, A., FUNATO, H., and KAMIYAMA, K.: Grey box identification of flexible structures: application to robust active vibration suppression control, Journal of Earthquake Engineering and Structural Dynamics, 2001, 30, pp DOYLE, J. C., FRANCIS, B. A., and TANNENBAUM, A. R.: Feedback control theory (Macmillan, New York, 1992) 6 REINELT, W.: H loop shaping for systems with hard bounds, Proc. on Quantitative Feedback Theory and Robust Frequency Domain Methods, Durban, South Africa, 1999, pp IEE: 2001 The authors are with Satellite Venture Business Laboratory, Department of Electrical & Electronic Engineering, Utsunomiya University, Yoto, Utsunomiya-shi, Tochigiken, , Japan. Tel: ; Fax ; Sandor.Forrai@altavista.net 262 COMPUTING & CONTROL ENGINEERING JOURNAL DECEMBER 2001

Kiran K. Shetty & Krishnamoorthy Department of Civil Engineering, Manipal Institute of Technology, Manipal , Karnataka, India

Electronic Journal of Structural Engineering 12(1) 212 Response of space frame structure resting on non-linear rubber base isolation system Kiran K. Shetty & Krishnamoorthy Department of Civil Engineering,