Online Determination of LQR Controller Weight Matrix Using Fuzzy Supervisor under Artificial Earthquake Records

Document Type : Research Article

Authors

1 Ph.D. Candidate, Department of Civil Engineering, Shahrood University of Technology, Shahrood, Iran

2 M.Sc. Graduate, Department of Civil Engineering, University of Maragheh, Maragheh, Iran

3 Associate Professor, Department of Civil Engineering, Shahrood University of Technology, Shahrood, Iran

Abstract

The linear quadratic regulator algorithm (LQR) is one of the common algorithms in control engineering that is used in many studies due to its simplicity. This controller is used to achieve optimal system performance by minimizing the cost function related to the state vector and the control input vector. In LQR control, the control gain is determined based on the weight matrices Q and R. One of the main issues in using this algorithm is adjusting the values of its weight matrices, which are determined based on trial and error or the use of meta-heuristic optimization algorithms. The improper determination of the weight matrices can reduce the performance of the controller. For example, increasing Q or decreasing R leads to a larger control gain matrix, which ultimately increases the control force and decreases system states (or responses). Therefore, if the control gain is designed using a very large Q matrix and its value remain constant during the simulation process, the possibility of producing a control force with the maximum capacity is very high. If this happens, the force generated by the actuator during the simulation process may be more than the required force, and the differences between the designed control force and the applied control force can in some cases reduce the control performance significantly or consume more energy than required. On the other hand, it is appropriate to assume the values of these two matrices to be constant in the case of sensors without noise and structure without uncertainty, and LQR control has the ability to reduce the response of the structure under various external vibrations. However, when the information coming from the sensors include noises and the system has uncertainty, the LQR control with constant gain has little ability to reduce the system response properly. In order to solve this problem in this study, a hybrid Fuzzy-LQR controller is proposed in which a fuzzy supervisor is used to change the LQR gain matrix online. In general, fuzzy logic has the ability to deal with noise and system uncertainty due to the lack of need for an accurate mathematical model of the system. Therefore, by combining LQR control and fuzzy control, the advantages of both controllers can been used. This fuzzy supervisor adjusts the pre-designed initial gain according to the created conditions by changing R weight matrix, so that the control force applied to the structure is proportional to the required control force and does not exceed the allowable actuator capacity. This change is implemented also with the aim of minimizing the amount of energy consumed by the actuator. To evaluate the performance of the proposed controller, three-story and eight-story buildings are used, all stories of which are equipped with the active tendon actuator. This structure has been subjected to the vibrations of two artificial earthquakes with a risk level of 10% and 2% in 50 years and different responses of the structure including the maximum value of responses and root of their mean squares have been determined. Then, the high capability of the Fuzzy-LQR controller, which even in the presence of parametric uncertainty and noise in the sensors can reduce the response by up to 90%, can be proved by comparing the results of the proposed controller and LQR controller based on different meta-heuristic algorithms. For example, the responses of displacement, velocity and acceleration of stories are reduced by an average of 81%, 83% and 71%, respectively using the proposed controller and under earthquake with a risk level of 10%. While using only the LQR controller, these values are 71%, 72% and 64%, respectively. The following results show that the root mean square of the responses has decreased more than their maximum value, which indicates the proper performance of the proposed algorithm during the excitation. 
Finally, it can be concluded that the proposed controller has a robust and stable behavior against various vibration and system uncertainties.

Keywords

Main Subjects

References

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History

  • Receive Date: 28 February 2021
  • Revise Date: 30 August 2021
  • Accept Date: 26 September 2021

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