Research Papers

Using Constrained Bilinear Quadratic Regulator for the Optimal Semi-Active Control Problem

[+] Author and Article Information
I. Halperin

Department of Civil Engineering,
Faculty of Engineering,
Ariel University,
Ariel 40700, Israel
e-mail: idoh@ariel.ac.il

G. Agranovich

Department of Electrical
and Electronics Engineering,
Faculty of Engineering,
Ariel University,
Ariel 40700, Israel
e-mail: agr@ariel.ac.il

Y. Ribakov

Department of Civil Engineering,
Faculty of Engineering,
Ariel University,
Ariel 40700, Israel
e-mail: ribakov@ariel.ac.il

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received August 31, 2016; final manuscript received June 21, 2017; published online August 9, 2017. Assoc. Editor: Douglas Bristow.

J. Dyn. Sys., Meas., Control 139(11), 111011 (Aug 09, 2017) (8 pages) Paper No: DS-16-1424; doi: 10.1115/1.4037168 History: Received August 31, 2016; Revised June 21, 2017

Semi-active systems provide an attractive solution for the structural vibration problem. A useful approach, aimed to simplify the control design, is to divide the control system into two parts: an actuator and a controller. The actuator generates a force that tracks a command which is generated by the controller. Such approach reduces the complexity of the control law design as it allows for complex properties of the actuator to be considered separately. In this study, the semi-active control design problem is treated in the framework of optimal control theory by using bilinear representation, a quadratic performance index, and a constraint on the sign of the control signal. The optimal control signal is derived in a feedback form by using Krotov's method. To this end, a novel sequence of Krotov functions which suits the multi-input constrained bilinear-quadratic regulator problem is formulated by means of quadratic form and differential Lyapunov equations. An algorithm is proposed for the optimal control computation. A proof outline for the algorithm convergence is provided. The effectiveness of the suggested method is demonstrated by numerical example. The proposed method is recommended for optimal semi-active feedback design of vibrating plants with multiple semi-active actuators.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Morales-Beltran, M. , and Paul, J. , 2015, “ Technical Note: Active and Semi-Active Strategies to Control Building Structures Under Large Earthquake Motion,” J. Earthquake Eng., 19(7), pp. 1086–1111. [CrossRef]
Scruggs, J. T. , 2004, “ Structural Control Using Regenerative Force Actuation Networks,” Ph.D. thesis, California Institute of Technology, Pasadena, CA. http://thesis.library.caltech.edu/2347/
Karnopp, D. , 1990, “ Design Principles for Vibration Control Systems Using Semi-Active Dampers,” ASME J. Dyn. Syst. Meas. Control, 112(3), pp. 448–455. [CrossRef]
Patten, W. N. , Kuo, C. C. , He, Q. , Liu, L. , and Sack, R. L. , 1994, “ Seismic Structural Control Via Hydraulic Semi-Active Vibration Dampers (SAVD),” First World Conference on Structural Control, Los Angeles, CA, Aug. 3–5, Vol. FA2, pp. 83–89.
Sadek, F. , and Mohraz, B. , 1998, “ Semiactive Control Algorithms for Structures With Variable Dampers,” J. Eng. Mech., 124(9), pp. 981–990. [CrossRef]
Yuen, K. V. , Shi, Y. , Beck, J. L. , and Lam, H. F. , 2007, “ Structural Protection Using MR Dampers With Clipped Robust Reliability-Based Control,” Struct. Multidiscip. Optim., 34(5), pp. 431–443. [CrossRef]
Robinson, W. D. , 2012, “ A Pneumatic Semi-Active Control Methodology for Vibration Control of Air Spring Based Suspension Systems,” Ph.D. thesis, Iowa State University, Ames, IA. https://pdfs.semanticscholar.org/87a8/670b0a7308fdeb56d9283c710f20eece7cb3.pdf
Dyke, S. J. , Spencer, B. F., Jr. , Sain, M. K. , and Carlson, J. D. , 1996, “ Modeling and Control of Magnetorheological Dampers for Seismic Response Reduction,” Smart Mater. Struct., 5(5), pp. 565–575. [CrossRef]
Aguirre, N. , Ikhouane, F. , and Rodellar, J. , 2011, “ Proportional-Plus-Integral Semiactive Control Using Magnetorheological Dampers,” J. Sound Vib., 330(10), pp. 2185–2200. [CrossRef]
Krotov, V. F. , Bulatov, A. V. , and Baturina, O. V. , 2011, “ Optimization of Linear Systems With Controllable Coefficients,” Autom. Remote Control, 72(6), pp. 1199–1212. [CrossRef]
Bruni, C. , DiPillo, G. , and Koch, G. , 1974, “ Bilinear Systems: An Appealing Class of ‘Nearly Linear’ Systems in Theory and Applications,” IEEE Trans. Autom. Control, 19(4), pp. 334–348. [CrossRef]
Aganovic, Z. , and Gajic, Z. , 1994, “ The Successive Approximation Procedure for Finite-Time Optimal Control of Bilinear Systems,” IEEE Trans. Autom. Control, 39(9), pp. 1932–1935. [CrossRef]
Lee, S. H. , and Lee, K. , 2005, “ Bilinear Systems Controller Design With Approximation Techniques,” J. Chungcheong Math. Soc., 18(1), pp. 101–116. http://www.mathnet.or.kr/mathnet/kms_tex/982630.pdf
Halperin, I. , Agranovich, G. , and Ribakov, Y. , 2016, “ A Method for Computation of Realizable Optimal Feedback for Semi-Active Controlled Structures,” Sixth European Conference on Structural Control (EACS), Sheffield, England, July 11–13, pp. 1–11. https://figshare.com/articles/EACS_2016_paper_-_A_METHOD_FOR_COMPUTATION_OF_REALIZABLE_OPTIMAL_FEEDBACK_FOR_SEMI-ACTIVE_CONTROLLED_STRUCTURES/4206111/1
Ribakov, Y. , Gluck, J. , and Reinhorn, A. M. , 2001, “ Active Viscous Damping System for Control of MDOF Structures,” Earthquake Eng. Struct. Dyn., 30(2), pp. 195–212. [CrossRef]
Ribakov, Y. , 2009, “ Semi-Active Pneumatic Devices for Control of MDOF Structures,” Open Constr. Build. Technol. J., 3(1), pp. 141–145. [CrossRef]
Agrawal, A. , and Yang, J. , 2000, “ A Semi-Active Electromagnetic Friction Damper for Response Control of Structures,” Structures Congress, Philadelphia, PA, May 8–10, pp. 1–8.
Halperin, I. , and Agranovich, G. , 2014, “ Optimal Control With Bilinear Inequality Constraints,” Funct. Differ. Equations, 21(3–4), pp. 119–136. http://functionaldifferentialequations.com/index.php/fde/article/view/570
Krotov, V. F. , 1988, “ A Technique of Global Bounds in Optimal Control Theory,” Control Cybern., 17(2–3), pp. 115–144.
Khurshudyan, A. Z. , 2015, “ The Bubnov–Galerkin Method in Control Problems for Bilinear Systems,” Autom. Remote Control, 76(8), pp. 1361–1368. [CrossRef]
Krotov, V. F. , 1995, Global Methods in Optimal Control Theory (Chapman & Hall/CRC Pure and Applied Mathematics), Marcel Dekker, New York.
Spencer, B., Jr. , Christenson, R. , and Dyke, S. , 1998, “ Next Generation Benchmark Control Problems for Seismically Excited Buildings,” Second World Conference on Structural Control, Kyoto, Japan, Vol. 2, pp. 1335–1360.
Craig, R. , 1981, Structural Dynamics: An Introduction to Computer Methods, Wiley, New York.
Leavitt, J. , Jabbari, F. , and Bobrow, J. E. , 2007, “ Optimal Performance of Variable Stiffness Devices for Structural Control,” ASME J. Dyn. Syst. Meas. Control, 129(2), pp. 171–177. [CrossRef]


Grahic Jump Location
Fig. 1

CBQR—Successive improvement of control process

Grahic Jump Location
Fig. 2

A scheme of the analyzed building and the actuators location

Grahic Jump Location
Fig. 3

Performance index values for each iteration

Grahic Jump Location
Fig. 4

Control signals for cases C and D

Grahic Jump Location
Fig. 5

Case B—control force versus actuator velocity

Grahic Jump Location
Fig. 6

Case D—control force versus actuator velocity

Grahic Jump Location
Fig. 7

Roof displacements

Grahic Jump Location
Fig. 8

Peak DOFs accelerations

Grahic Jump Location
Fig. 9

Nonlinear frequency response function



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In