μSynthesis Control of a Seismic Excited Building
Abstract
In structural earthquake engineering, the earthquake protection of structures has considerably increased these last three decades since structural control appears as an efficient way for the energy dissipation due to dynamic loads. A number of papers investigated this field proving the potential of the active control to realize a significant reduction of the structures’ response that conduct to an important rising of the human safety and seismic structures protection. However, the several control approaches applied did not automatically consider the uncertainties that exist in the realistic case. That means that there are some variations in the coefficients of the structure model as mass, stiffness or damping and even in the structure dynamics which should be tacked into account since it could affect the controller design. The motivation of this study is to perform the µsynthesis, a robust control successfully applied in many other fields, for its ability to explicitly include all these aspects in the same time in the control design procedure. The quantification of the uncertainties as well as the interpretation of the performance requirements in appropriate weighting functions are one of the great advantages of this robust control technique. Hence, this paper presents through simulations the response of a three floors building structure subjected to a seismic excitation, modeled by Kanai Tajimi Filter, where the active component consists in an active bracing system (ABS) attached to the second floor. A robust controller is designed with considering parametric and dynamic uncertainties (structured uncertainties). The designed µcontroller demonstrates its efficiency to considerably attenuate the building response by providing a significant disturbance reduction over low control effort energy when considering the parametric and dynamic (structured) uncertainties acting simultaneously in the control system. The robustness of the controller is evaluated by varying the considered uncertainties to their worst case variation.
1 Introduction
The structural active control represents a field in permanent evolution such it knows these last few years the investigation of several papers using a new generation of controllers based on robust control methods as LQG, H _{ ∞, } adaptive control,…etc. [1, 2, 3, 4, 5]. Indeed, the earthquake protection of structures still constitutes a serious problem that looks for a way more and more efficient to dissipate the energy produced by a seismic load and raise the human safety and the structures protection even in presence of eventual uncertainties. This conducted the researchers to design controllers able to maintain the performance in the real conditions of implementation which implied to include in the control design procedure the maximum of practical considerations and model variations that often occur simultaneously. This means to take into account various aspects as the effect of sensors noise, time delay (induced by a mechanical force), physical actuator limits, and also errors in the parameters and dynamics of the structure and actuator models. Hence, the motivation of this work is to use the µsynthesis, a robust control strategy, to explicitly incorporate the described aspects as robustness in stability and performance requirements. Based on the structured singular value ssv technique and the H _{ ∞ } bound computations, this control allows accounting explicitly for robustness to dynamic uncertainties, in the present case modeling errors in the actuator dynamic and to parametric uncertainties as, in the considered case, variations in mass stiffness and damping coefficients of the second floor of the structure model. In addition to robustness considerations, the µsynthesis problem formulation poses the performance objectives as minimizing norm of some weight transfer functions [4, 5]. However, the quantification of the uncertainties as well as the interpretation of the performance requirements in appropriate weighting functions represent the critical step in the problem formulation of this multivariable control technique [6]. To evaluate the efficiency of the designed µcontroller, simulations are conducted on a building structure of three floors where the active component consists in an active bracing system (ABS) attached to the second floor. The seismic excitation is modelled by Kanai Tajimi filter attacked by a white noise [7, 8]. The resulting controller achieves closely similar performance in nominal and worst case of uncertainties variation and presents a great benefit of costing low control energy. However, the μsynthesis technique presents the convenient of generating too high order controllers. By using a model reduction method as the balanced realization algorithm, a lower order controller is provided to achieve the same level of robust performances.
2 µSynthesis Control Theory
The μsynthesis is a robust control technique since the formulation of its law explicitly takes into account the uncertainties in the system. It is based on the combination of the application of the μanalysis which is in fact a stability robustness criterion, in one hand and H _{ ∞ } bound computations of some weight functions representing the performance specifications and limitations in another hand. The μsynthesis is derived so that the algorithm called “DK iterations” constitutes the effective mean to the computation of the controller. To truly model uncertainties, one should distinguish them from their origins. This leads to classify them into: parametric (real) uncertainty and dynamic (frequency dependent) uncertainty [4, 5]. The first class encompasses the uncertainties in system physical parameters such as: time constant, natural frequency…etc., in the structure case it is question of the mass, stiffness and damper. While the second one refers to those met by simplifying a complex model or in system neglected high frequency dynamics. There is an interesting formulation of uncertainties called structured uncertainties and formed by the combination of a block of real uncertainties and another of dynamic uncertainties. This formulation is particularly adopted in μsynthesis theory because it is possible to make a link between the uncertainties and the physical system by means of the computation of a very useful tool called structured singular value μ which can be used in both stability analysis and synthesis of the control law.
\( \Delta \left( {\text{s}} \right) \) represents the structured uncertainties block; \( \bar{\sigma } \) is the maximum singular value; M(s) is the transfer matrix of the feedback structure system and s(= jω) is the Laplace variable in the frequency domain.
The cost function Eq. (5) is no convex in regard to D and K, that’s why the µsynthesis is executed by means of an algorithm called DK iteration to find a local minimum. Then, the µanalysis procedure can be derived once the controller K is fixed, and constitutes a convex problem. However, if D is fixed the problem becomes convex and we are in face of an H _{ ∞ } optimal controller design. If the minimized μ value is less than 1, the obtained K is a robust stabilizing controller. To facilitate computation of μ controllers, an efficient procedure called DK iteration, resumed in the following algorithm, is executed [10].
3 Formulation of the Building Design
4 Formulation of the Synthesis Design Problem
The example consists on a three floors building simulated by a three massspringdamper system and considers uncertainties in the structure parameter coefficients of the second floor since each variation on these parameters affect directly the two other floors. The variations is about 30% in mass, damper and stiffness of the second floor where the active bracing system (ABS) is placed to produce the force controlled by the feedback. As the displacement measurements are not so expensive to achieve than in the last years, they are used in this example for the control feedback.

Seismic excitation filter, so called Kanai Tajimi whose output has the frequency peak corresponding to the maximum energy in a set of near fault earthquakes, and input is white noise excitation. Hence f _{ 2 } is a disturbance modelled as a normalized signal d and shaped by a weighting transfer function W _{ dist } [10].

The controller measures the displacements z _{ 1 }, z _{ 2 }, z _{ 3 } of the mass m _{ 1 }, m _{ 2 } and m _{ 3 } and applies the control force f _{ 1 } via the actuator of G _{act} transfer function.

The seismic disturbance is modelled by Kanai Tajimi second order filter attacked by a white noise dist and shaped by W _{ dist } [7, 8]. The normalised disturbance signal \( f_{2} \) is applied at the system input.

The performance objective is to attenuate the disturbances by a factor of 80 below the frequency 8.5 rd/s.
5 Results of Numerical Simulations
Using the Matlab software computing [10], we developed a controller on the base of μ synthesis theory with displacement measurements feedback to actively control the described structure. This technique is able to treat the real case of a structure model affected by parametric and dynamic uncertainties commonly named structured uncertainties. The parametric uncertainty considered in this paper concerns stiffness, damper and mass of the second floor where the actuator is placed.
Hence, we treat particularly the parametric uncertainties of this floor by considering 30% in k _{ 2 }, c _{ 2 } and m _{ 2 } respectively. The second uncertainty is inherent to the nature of the actuator command. The ABS actuator produces a mechanical control that introduces a non negligible dynamic, which constitutes an important aspect to include in our design to better perform the control of the structure and produce an inaccurate control force. Thus, the possible variations in the actuator model represent a dynamic uncertainty treated as input multiplicative error and modelled by W _{ unc }.
The controller achieves closely similar performance in nominal and worst case which traduces a maintaining of performances even in presence of severe degradation in the structure model.
For more detailed evaluation, the μ value corresponding to how large the gain from disturbancetooutput norm get for the specified system uncertainties is computed. As μ is equal to 1.4, it is steel not far from 1.2 that confirms the robustness of the designed controller.
6 Conclusions

This control is particularly adapted to consider in the control design, weighting functions able to closely match the desired performance and compromise.

With the mean of structured uncertainties, the μcontroller takes naturally into account the parametric and dynamic uncertainties affecting the structure, and provides similar vibration attenuation even in the worst case of uncertainties variation.

This approach shows a great advantage of producing low control energy comparing to several control approaches in literature already performed while providing a significant reduction of structure response, average 70%–78%.
Finally, experimental tests shall be carried out in a future work to certify the actual simulations results and verify the usefulness of the designed µcontroller, which since a long time represents the principal criterion for evaluating the efficiency of an approach.
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