A Method for Determining the Optimal Location of a Distributed Sensor/Actuator

Byeongsik Ko Benson H. Tongue Andrew Packard Active Noise Control Laboratory Department of Mechanical Engineering University of California at Berkeley Berkeley, CA 94720 A Method for Determining the Optimal Location of a Distributed Sensor/Actuator The optimal location problem of distributed sensor/actuator for observation and control of a flexible structure is investigated. Using a property of controllability and observability grammian matrices, this approach employs a nonlinear optimization technique to determine the optimal placement of a distributed sensor/actuator. The effect of unimportant modes that do not strongly affect the structural behavior of a system is minimized and the effect of important modes is maximized. The final objective function is expressed as the combinational form of two different objective functions. This technique is applied to several types of beam support conditions and the corresponding optimal locations are determined. © 1994 John Wiley & Sons, Inc. INTRODUCTION A major avenue for sound generation is through the action of vibrating structural surfaces. One example is the low-frequency interior noise (called boom) that occurs in automobile passenger compartments. This is a particularly complicated problem because the sound comes from the road, engine, and power train excitations transmitted through the structure and into the cabin. The sound is transmitted due to the structural vibration of the surrounding surfaces of the enclosure, and is especiaBy strong when the structural vibration modes and acoustic modes of the enclosure are coupled. This phenomenon occurs between 20 and 60 Hz in most passenger cars. Boom noise has become a critical issue as the automobile body has become lighter in order to save weight, thus leading to increased noise levels. The coupling effect is a critical problem in automobile design because it so strongly affects booming noise, due to the structural-acoustic coupling of an automotive cabin. There currently exist two methods by which one can control acoustics in an active manner. The first is to inject "antisound" by means of speakers, that is, sound that destructively interferes with the offending noise. The second approach is to use a distributed sensor/actuator and implement a control strategy to control the structural vibrations that affect acoustic modes. If the structural vibration modes can be controlled, the acoustic response will be suppressed. The control of flexible structure has been studied for several years now. Piezoelectric material was first used to measure the vibration of a mechanical system by Lee and Moon (1990), who along with other researchers (Chiang, and O'Sullivan, 1991) tailored a piezoelectric sensor to measure a single vibration mode. The piezoelectric sensor was designed to be very sensitive to a desired mode while remaining insensitive to the remaining modes. In this way, the sensor filtered out the contribution of the undesired modes. Because each sensor is tailored to detect a single mode, if mUltiple modes are to be measured, a single sensor is needed for each mode. To mea- Received September 12, 1993; Revised October 14, 1993 Shock and Vibration, Vol. 1, No.4, pp. 357-374 (1994) © 1994 John Wiley & Sons, Inc. CCC 1070-9622/94/040357-18 357 358 Ko, Tongue, and Packard sure a single mode while rejecting the remaining modes, the width of the sensor must be proportional to the second derivative of the desired mode shape. Because it is very difficult to shape the piezoelectric sensor in accordance with this second derivative, an alternative approach is needed. To date, little work has been done to develop a systematic approach for finding the optimallocation of sensor/actuators, but much effort has been put into creating a definition of controllability and observability (Arbel, 1981; Hamden and Nayfeh, 1989). Lim (1992) proposed a method for finding optimal sensor and actuator locations. This approach is based on projecting eigenvectors into the intersection subspace of the controllability and observability subspaces of each collocated sensor-actuator pair. Maghami and Joshi (1993) suggested the optimization of a function of the singular values of the Hankel matrix as a criterion for sensor/actuator placement. Each of these methods applies to the case of a discrete actuator and sensor pair. In addition to the problem of optimal sensor location, it is also necessary to find the optimal location of a distributed actuator. The distributed actuator is well-suited for the control of vibration modes because it is space efficient, light-weight, and requires no external supports. Bailey and Hubbard (1987) first applied a distributed actuator to control the first vibration mode of a clamped-free beam. They demonstrated a drastic change in the decay rate of the impulse response when a distributed actuator is used as an active damper. Crawley and de Luis (1986) experimentally evaluated the effectiveness of using a piezoelectric actuator as a control source. In this article we consider the design problem of locating a distributed sensor/actuator to maximize performance indices with regard to measurement and control of structural vibrations in a low frequency range. One objective of this work is to find the optimal location of a uniform piezoelectric sensor to sense effectively the specific vibration modes of a flexible structure. The method used is based on the properties of the observability grammian matrix because this matrix represents the degree of observability of a system's modes. A second objective of this work is to find the optimal location of a piezoceramic actuator to control effectively the specific vibration modes of a flexible structure. The method used is based on the properties of the controllability grammian matrix, which represents the degree of controllability of the modes. In this work, the model is assumed to be a linear, second-order dynamic system. Another assumption is that we shall predetermine which are the important vibration modes that must be controlled via optimal feedback control. CONSTITUTIVE EQUATIONS OF PI EZOElECTRIC MATERIAL In this section, the governing equations for a pi- ezoelectric material are derived. The constitutive equations for piezoelectric materials can be expressed in terms of the piezoelectric constants that couple the mechanical strain, stress, electric field, and electric displacement based on the IEEE standard on piezoelectricity (1987): (Iij = CVklekl Di = eiklekl ekijEk + E.'fkEk (1) (2) where (I ij is the stress tensor, ekl is the strain tensor, Ek is the electric field, Di is the electric displacement (or electric flux density, the charge distributed uniformly over a surface), and Cijkf, ekij, and Eik indicate the stiffness matrix, the piezoelectric stress/charge constant matrix, and the piezoelectric permittivity constant of a piezoelectric material. Because the stress and strain matrices (I ij and ekl are symmetr (...truncated)