Characterization and Attenuation of Sandwiched Deadband Problem Using Describing Function Analysis and Its Application to Electro-hydraulic Systems Controlled by Closed-center Valves

Sandwiched deadbands can be seen in a wide variety of systems, such as electro-hydraulic systems controlled by closedcenter valves. In such a system, the deadband is between the plant and actuator dynamics and therefore can not be compensated directly like an input deadband. Though this sandwiched deadband problem may be attenuated to certain degree through sophisticated advanced control techniques, the increased cost and the necessity of actuator state feedback prohibit their widespread application in the industry. An economical and popular method is to add an inverse deadband function in the controller to cancel or compensate the highly nonlinear behavior of the deadband. However, such a solution requires that the dynamics before the deadband (eg. the valve dynamics) is fast enough to be neglected a requirement that can not be met in reality unless the closed loop bandwidth of the overall system is limited very low. To raise the achievable closed loop bandwidth for a much improved control performance, it is essential to be able to precisely characterize the effect of this sandwiched deadband on the stability and performance of the overall closed-loop system, which is the main focus of the paper. Specifically, a describing function based nonlinear analysis will be conducted to predict when the instability will occur and how the resulting limit cycle depends on the actuator dynamics and the targeted closed-loop bandwidth. Based on the analysis, the optimal closed-loop bandwidth can be determined to maximize the achievable overall system perfor∗Address all correspondence to this author. 1 DB u Sandwiched Deadband System C(s) A(s) P(s) Controller Actuator Dynamics Plant Dynamics DBI _ r y u p v e u c Figure 1. Sandwiched Deadband System with Direct Compensation. mance. The technique is applied to an electro-hydraulic system controlled by closed-center valves to optimize the controller design. INTRODUCTION Sandwiched deadbands, as shown in Fig. 1, are common nuisances that exist in many systems, such as electro-hydraulic systems controlled by closed center valves, where the deadband of the valve is sandwiched by the valve and plant dynamics. Compared with the input deadband, which can be cancelled or compensated with an inverse deadband function, the sandwiched deadband is more difficult to deal with. The actuator dynamics in a sandwiched deadband system is usually not high enough to be neglected, otherwise, the sandwiched deadband problem can be simplified into the input deadband problem. There are several solutions to this problem. One is to use a feed-forward controller to boost the actuator dynamics response, so that it would be sufficiently fast to be neglected, like that Bu Copyright c © 2004 by ASME and Yao have done in [1]. Once the actuator dynamics was neglected, the deadband became the input deadband, which could be compensated by an inverse deadband function. However, the success of such a strategy depends on the accuracy of the actuator dynamics. It can only achieve some limited improvements in practice due to the unavoidable uncertainties in any physical systems. Another way to solve this problem is to use a local high gain feedback controller at the actuator level to attenuate the sandwiched deadband [1–4], like what Tao, etc. and Bu and Yao have done. However, to apply this technique, the feedback of the actuator states or output is required, which may significantly increase the system cost. For example, to apply the feedback compensation to electro-hydraulic systems controlled by closed-center valves, the feedback of the valve-spool position is required. Although the spool position feedback is available in some valves, it is not a general valve configuration and would definitely increase the system cost. In addition, the spool position measurement is normally too noisy to help increasing the actuator bandwidth significantly [1, 5]. A practical and industry-favorite solution is to simply neglect the actuator dynamics and use an inverse deadband function to compensate the deadband, shown in Fig. 1. This solution is simple, easy to implement, and is used in a lot of applications like [6]. Such a deadband compensation usually results in a very conservative controller design because the closed loop bandwidth has to be tuned very low to guarantee that the actuator dynamics can be safely neglected. When the closed loop bandwidth is increased to certain level, limit cycle is expected because the actuator dynamics makes the deadband compensation not perfect. Limit cycle is potentially very dangerous to systems with neglected vibration modes, because it may excite the vibration modes or even destabilize the system. This paper focuses on the practical solution of the sandwiched deadband problem and uses describing function to systematically analyze and characterize the closed loop system. The analysis would give us a rough idea about when limit cycle would occur so that we can maximize the closed loop bandwidth without exciting limit cycle. Although this technique would not increase the theoretically achievable performance, it is helpful to make the system less conservative and to actually achieve the achievable performance in implementation. CHARACTERIZATION OF SANDWICHED DEADBAND SYSTEM USING DESCRIBING FUNCTION ANALYSIS A closed loop system with sandwiched deadband DB(·) between the linear actuator dynamics A(s) and the plant dynamics P(s), and a linear feedback controller C(s) along with a direct deadband compensation DBI(·), shown in Fig. 1, is analyzed in this section. The analysis is not limited to linear systems and linear controllers as explained as follows. The limit cycle happens, 2 K(a) f(y) e