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The presence of nonsmooth nonlinear characteristics such as dead-zone, backlash, hysteresis and piecewise-linearity is common in actuators and sensors, such as mechanical connections, hydraulic actuators and electric servomotors [47], [51], [91]. In most cases, they are caused by imperfections of system component characteristics. A dead-zone is a static “memoryless” nonlinearity which describes the component’s insensitivity to small signals. Backlash and hysteresis include delays in addition to insensitivity and are dynamic in nature. The presence of these nonlinearities severely limits system performance, giving rise to undesirable inaccuracy or oscillations or even leading to instability. For example, backlash prevents accurate positioning and may lead to chattering and limit cycles. The resultant wear and tear of gears increases backlash. These nonlinearities are usually poorly known and may vary with time. Also, in mass production, they vary from component to component. There are ways to overcome some of these problems such as the use of anti-backlash gears, however, they are quite costly. The same applies to materials with low hysteresis being costly. A desirable feedback control system should be able to compensate for such uncertainties.

Consider the three-block sandwich system with a nonlinear block (·) between two dynamic blocks () and (): where is the differentiation operator: []() = (), or the Laplace transform operator as the case may be, () is the accessible control input, () is the measured output, () and () are both rational transfer functions with their denominator degrees being and respectively, and (·) is the nonsmooth nonlinearity such as a dead-zone, backlash or hysteresis characteristic. Note that (·) includes both static memoryless nonsmooth nonlinearities such as a dead-zone as well ones with memory such as hysteresis or backlash that are dynamic in nature. As a comparison, we note that the cases when () = 1 were considered in [56], [55], [70], {cx90|[90]}, [29], [32], [41], while the case when () = 1 was considered in [70]. The blocks sandwiching the nonlinearity may be nonlinear as well. For example, () or () or both may be nonlinear. In that case, the problem will be essentially more challenging. The control schemes will be resultantly different than if both the blocks are linear. In this book, primarily, control of sandwich systems where () and () both are linear time-invariant dynamic subsystems, is considered, while nonlinear dynamics are also addressed (see Chapter 8).

An ideal control design should be such that the system preferably overcomes the sandwiched nonlinearity as quickly as possible, or the nonlinearity effect is canceled effectively in a short time by adaptive schemes. It is proposed to use the inverse models of the sandwiched nonlinearities. The compensator for the sandwiched nonlinearity, () = (()), is () = (()), where (·) is the desired inverse of (·) and () is the control input which would achieve the control objective in absence of the sandwiched (·). When the inverse of the sandwiched nonlinearity is exact, then () = ((())) = () achieves the control objective as if (·) were absent. This scheme works for systems which have nonsmooth nonlinearities at the input (() = 1) or at the output (() = 1). But if it is sandwiched between () and (), this may not work. In this case, if the () subsystem can be approximately canceled by state feedback or output feedback around it and if () is preceded by the inverse of the sandwiched nonlinearity, then it may approximately cancel (·) following the () block [83], [82].

In continuous-time designs [83] described in Chapter 3, a dead-zone inverse was used in three different control schemes for a sandwich nonlinear system with a sandwiched dead-zone. It was seen that the dynamics of () limit the effectiveness of a dead-zone inverse on the sandwiched dead-zone. In those schemes, partial state feedback, dynamic output feedback and a full state feedback with a dynamic phase difference compensator acting on the reference input, were used respectively in three different schemes to nullify this effect so that the sandwiched dead-zone can be effectively canceled by a fixed dead-zone inverse. For a general case (2.1), if we can make the () system (controlled by a feedback loop) fast or cancel the dynamics of (), the dead-zone inverse can directly cancel the sandwiched dead-zone. However, such fastness or cancellation is difficult to characterize in continuous time and the desired inversion is difficult to achieve. This is the main motivation for the hybrid control approach that is developed in this chapter.

In Chapter 4, hybrid control schemes were developed for a general sandwich nonlinear system with nonsmooth nonlinearity between two linear dynamics, using an inner-loop discrete-time controller and an outer-loop continuous-time model reference controller along with an exact nonlinearity inverse. The use of an exact nonlinearity inverse was possible as the sandwiched nonlinearity was known. In case of unknown sandwiched nonlinearity, the inverse has to be adaptively estimated. Depending upon the availability of the measurement of the input to the known nonlinearity (·), (), or the output of (·), (), two hybrid control schemes have been developed in Chapter 4.

There have been many applications of neural networks (NN) in controls. Algorithms that effectively tune the weights on-line have been developed. NN applications in control can be broadly classified into two sorts: identification and control. In this chapter, it is intended to use NN for closed-loop control of a system with a dead-zone sandwiched in between two dynamic blocks. An adaptive version of the hybrid control scheme presented in Chapter 4, will be developed using a NN to compensate the unknown sandwiched dead-zone. As the controller structure is hybrid and has neural networks, we term it as a neural-hybrid controller. The proposed neural-hybrid controller consists of an inner loop discrete-time feedback structure incorporated with an adaptive inverse using NN for the unknown nonlinearity (·), in the present case, a dead-zone, and an outer-loop continuous-time feedback control law for achieving desired output tracking [86]. The dead-zone compensator consists of two NN’s, one used as an estimator of the sandwiched dead-zone function and the other for the compensation itself. To approximate jump functions such as a dead-zone inverse, it is found that the NN that uses smooth activation functions should be augmented with extra nodes containing a jump function approximation basis set of discontinuous activation functions. This is necessary as otherwise to approximate such jump functions using smooth activation functions, many NN nodes and many training iterations are required [40], [60].

There is an increased growth in mechanical systems that must be capable of precisely controlled maneuvering, to which friction poses a challenge. Friction is a natural occurrence that affects all objects in motion. It is present in servo-mechanisms, hydraulic systems, pneumatic systems and most other mechanical systems. It results from a complex microscopic phenomena dependent on surface material, characteristics of lubrication between the surfaces, forces normal to the direction of motion and the dynamics of the rubbing or rolling motion. In the control of a mechanical system, failing to compensate for friction can lead to tracking errors when velocity reversals are demanded and oscillations when very small motions are required. Thus, it is important to design control strategies that can alleviate the performance deterioration due to friction. In addition to the disruptive nature of friction and the lack of an universal model of friction, friction compensation is further complicated by the fact that friction parameters vary with temperature and age.

In this chapter, a more challenging problem than the one solved in Chapter 7 is considered: friction is sandwiched between linear and nonlinear blocks instead of two linear dynamic blocks. One such system is a single-link two-body system such as a manipulator arm with friction affecting the motion of the load having nonlinear dynamics driven by a motor, where the connection between the two bodies is through a flexible rod. Friction is acting on the load side and the control input has to pass through the joint flexibility and damping block before it can affect the load. This system can be interpreted as a linear block followed by joint flexibility and damping, and then friction acting on a nonlinear block representing the load. Thus, the friction is sandwiched in between linear and nonlinear dynamics. The joint flexibility and damping also contribute to what we can call as sandwiched dynamics. For such a nonlinear system that has sandwiched friction and dynamics, an adaptive feedback friction compensating controller is developed with the requirement that the system be feedback linearizable and minimum phase whenever load velocity is not zero. Whenever load velocity is zero, with the friction being discontinuous, the system is no longer feedback linearizable and a different strategy of maximum-magnitude control is applied [81].

Numerous practical systems refered in this book, such as aircraft and spacecraft flight control systems, process control systems, and power systems that have actuator nonlinearities present at the input or nonlinearities sandwiched in between two dynamic blocks, are also prone to actuator failures. Adaptive control of such systems in presence of nonlinearities as well as failures, presents a new challenging problem. This research topic is practically important as many of these systems are critical and actuator failures and non-linearities if not compensated, may lead to disasters, for instance, aircraft accidents have been caused by actuator failures. The compensation of actuator nonlinearities is equally important as they limit both static and dynamic performance of control systems. These nonlinearities are typically nonsmooth nonlinearities such as dead-zone, hysteresis and backlash.

In this chapter, an optimal and nonlinear control solution is proposed for control of multi-body, multi-input and multi-output nonlinear systems with joint backlash, flexibility and damping, represented by a gun turret-barrel model which consists of two subsystems: two motors driving two loads (turret and barrel) coupled by nonlinear dynamics. The key feature of such systems is that the backlash is between two dynamic blocks. Optimal control schemes are employed for backlash compensation and nonlinear feedback control laws are used for control of nonlinear dynamics. When one load is in contact phase and the other load is in backlash phase, a feedback linearization design decouples the multivariable nonlinear dynamics so that backlash compensation and tracking control can be both achieved. Nonlinear zero dynamics systems caused by joint damping are bounded-input, bounded state stable so that feedback linearization control designs ensure that all closed-loop signals are bounded and asymptotic tracking is achievable.