现代控制理论的自适应控制理论.pdf

第第 4 章章 Adaptive Fuzzy Control 教学内容 教学内容 本章将现代控制理论的自适应控制理论应用于模糊控制技术中，以期提高模糊控制系统的动态、静态 性能，增强模糊控制系统对于模型的参数、结构变化的鲁棒性以及控制系统不同相应阶段对控制其性能的 不同要求，提高控制系统的综合品质。主要内容 1. 直接自适应控制和间接自适应控制的基本概念； 2. 模糊模型参考学习控制（FMRLC）的原理、结构； 3. FMRLC 的设计和实现。 教学重点 教学重点 模糊模型参考学习控制（FMRLC）的原理和结构 。 教学难点 教学难点 对 FMRLC 中学习机制的准确把握和理解,关键是模糊逆模型的设计。 教学要求 教学要求 本章的学习需要预先掌握一定的自适应控制、自校正调节器的基础知识、概念。要求掌握模糊模型参 考学习控制（FMRLC）的原理和结构。 4.1 Overview The design process for fuzzy controllers that is based on the use of heuristic ination from human experts has found success in many industrial applications. Moreover, the approach to constructing fuzzy controllers via numerical -output data is increasingly finding use. Regardless of which approach is used, however, there are certain problems that are encountered for practical control problems, including the following 1 The design of fuzzy controllers is pered in an ad hoc manner so it is often difficult to choose at least some of the controller parameters. For example, it is sometimes difficult to know how to pick the membership functions and rule-base to meet a specific desired level of perance. 2 The fuzzy controller constructed for the nominal plant may later per inadequately if significant and unpredictable plant parameter variations occur, or if there is noise or some type of disturbance or some other environmental effect. Hence, it may be difficult to per the initial synthesis of the fuzzy controller, and if the plant changes while the closed-loop system is operating we may not be able to maintain adequate perance levels. As an example, we showed how our heuristic knowledge can be used to design a fuzzy controller for the rotational inverted pendulum. However, we also showed that if a bottle half-filled with water is attached to the endpoint, the perance of the fuzzy controller degraded. While we certainly could have tuned the controller for this new situation, it would not then per as well without a bottle of liquid at the endpoint. It is for this reason that we need a way to automatically tune the fuzzy controller so that it can adapt to different plant conditions. Indeed, it would be nice if we had a that could automatically per the whole design task for us initially so that it would also synthesize the fuzzy controller for the nominal condition. In this chapter we study systems that can automatically synthesize and tune direct fuzzy controllers. There are two general approaches to adaptive control, the first of which is depicted in Figure 4.1. In this approach the “adaptation mechanism“ observes the signals from the control system and adapts the parameters of the controller to maintain perance even if there are changes in the plant. Sometimes, the desired perance is characterized with a “reference model,“ and the controller then seeks to make the closed-loop system behave as the reference model would even if the plant changes. This is called “model reference adaptive control“ MRAC. In Section 4.2 we use a simple example to introduce a for direct model reference adaptive fuzzy control where the controller that is tuned is a fuzzy controller. Next, we provide several design and implementation case studies to show how it compares to conventional adaptive control for a ship steering application, how to make it work for a multi- multi-output MIMO fault-tolerant aircraft control problem. Following this, in Section 4.4 we show several ways to “dynamically focus“ the learning activities of an adaptive fuzzy controller. A simple magnetic levitation control problem is used to introduce the s, and we compare the perance of the s to a conventional adaptive control technique. Design and implementation case studies are provided for the rotational inverted pendulum with a sloshing liquid in a bottle at the endpoint. Figure 4.1 direct adaptive controls. In the second general approach to adaptive control, which is shown in Figure 4.2, we use an on-line system identification to estimate the parameters of the plant and a “controller designer“ module to subsequently specify the parameters of the controller. Figure 4.2 indirect adaptive controls. If the plant parameters change, the identifier will provide estimates of these and the controller designer will subsequently tune the controller. It is inherently assumed that we are certain that the estimated plant parameters are equivalent to the actual ones at all times this is called the “certainty equivalence principle“. Then if the controller designer can specify a controller for each set of plant parameter estimates, it will succeed in controlling the plant. The overall approach is called “indirect adaptive control“ since we tune the controller indirectly by first estimating the plant parameters as opposed to direct adaptive control, where the controller parameters are estimated directly without first identifying the plant parameters. In Section 4.6 we explain how to use the on-line estimation techniques, coupled with a controller designer, to achieve indirect adaptive fuzzy control for nonlinear systems. We discuss two approaches, one based on feedback linearization and the other we name “adaptive parallel distributed compensation“ since it builds on the parallel distributed compensator. 4.2 Fuzzy Model Reference Learning Control FMRLC A “learning system“ possesses the capability to improve its perance over time by interacting with its environment. A learning control system is designed so that its “learning controller“ has the ability to improve the perance of the closed-loop system by generating command s to the plant and utilizing feedback ination from the plant. In this section we introduce the “fuzzy model reference learning controller“ FMRLC, which is a direct model reference adaptive controller. The term “learning“ is used as opposed to “adaptive“ to distinguish it from the approach to the conventional model reference adaptive controller for linear systems with unknown plant parameters. In particular, the distinction is drawn since the FMRLC will tune and to some extent remember the values that it had tuned in the past, while the conventional approaches for linear systems simply continue to tune the controller parameters. Hence, for some applications when a properly designed FMRLC returns to a familiar operating condition, it will already know how to control for that condition. Many past conventional adaptive control techniques for linear systems would have to retune each time a new operating condition is encountered. Figure 4.3 Fuzzy model reference learning controller The functional block diagram for the FMRLC is shown in Figure 4.3. It has four main parts the plant, the fuzzy controller to be tuned, the reference model, and the learning mechanism an adaptation mechanism. We use discrete time signals since it is easier to explain the operation of the FMRLC for discrete time systems. The FMRLC uses the learning mechanism to observe numerical data from a fuzzy control system i.e., rkT and ykT where T is the sampling period. Using this numerical data, it characterizes the fuzzy control systems current perance and automatically synthesizes or adjusts the fuzzy controller so that some given perance objectives are met. These perance objectives closed-loop specifications are characterized via the reference model shown in Figure 4.3. In a manner analogous to conventional MRAC where conventional controllers are adjusted, the learning mechanism seeks to adjust the fuzzy controller so that the closed-loop system the map from rkT to ykT acts like the given reference model the map from rkT to ymkT. Basically, the fuzzy control system loop the lower part of Figure 4.3 operates to make ykT track rkT by manipulating ukT, while the upper-level adaptation control loop the upper part of Figure 4.3 seeks to make the output of the plant ykT track the output of the reference model ymkT by manipulating the fuzzy controller parameters. Next, we describe each component of the FMRLC in more detail for the case where there is one and one output from the plant we will use the design and implementation case studies in Section 4.3 to show how to apply the approach to MIMO systems. 4.2.1 The Fuzzy Controller The plant in Figure 4.3 has an ukT and output ykT. Most often the s to the fuzzy controller are generated via some function of the plant output ykT and reference rkT. Figure 4.3 shows a simple example of such a map that has been found to be useful in some applications. For this, the s to the fuzzy controller are the error ekT rkT ykT and change in error e kTe kTT c kT T −− i.e.,a PD fuzzy controller. There are times when it is beneficial to place a smoothing filter between the rkT reference and the summing junction. Such a filter is sometimes needed to make sure that smooth and reasonable requests are made of the fuzzy controller e.g., a square wave for rkT may be unreasonable for some systems that you know cannot respond instantaneously. Sometimes, if you ask for the system to perfectly track an unreasonable reference , the FMRLC will essentially keep adjusting the “gain“ of the fuzzy controller until it becomes too large. Generally, it is important to choose the s to the fuzzy controller, and how you process rkT and ykT, properly; otherwise perance can be adversely affected and it may not be possible to maintain stability. Returning to Figure 4.3, we use scaling gains ge,gc and gu for the error ekT, change in error ckT, and controller output ukT, respectively. A first guess at these gains can be obtained in the following way The gain ge , can be chosen so that the range of values that ekT typically takes on will not make it so that its values will result in saturation of the corresponding outermost membership functions. The gain gc can be determined by experimenting with various s to the fuzzy control system without the adaptation mechanism to determine the normal range of values that ckT will take on. Using this, we choose the gain so that normally encountered values of ckT will not result in saturation of the outermost membership functions. We can choose gu so that the range of outputs that are possible is the maximum one possible yet still so that the to the plant will not saturate for practical problems the s to the plant will always saturate at some value. Clearly, this is a very heuristic choice for the gains and hence may not always work. Sometimes, tuning of these gains will need to be pered when we tune the overall FMRLC. Rule-Base The rule-base for the fuzzy controller has rules of the jlm If e is Eand c is C then u is U where and denote the linguistic variables associated with controller s eeckT and ckT, respectively, denotes the linguistic variable associated with the controller output u, u j E and denote the l C jth lth linguistic value associated with , respectively, denotes the consequent linguistic value associated with . e c m U u Hence, as an example, one fuzzy control rule could be If error is positive-large and change-in-error is negative-small Then plant- is positive-big in this case “error“, e 4 E “positive-large“, etc.. We use a standard choice for all the membership functions on all the universes of discourse, such as the ones shown in Figure 4.4. Hence, we would simply use some membership functions similar to those in Figure 4.4, but with a scaled horizontal axis, for the ckT . e kT Figure 4.4 Membership functions for universe of discourse We will use all possible combinations of rules for the rule-base. For example, we could choose to have 11 membership functions on each of the two universes of discourse, in which case we would have 112 121 rules in the rule-base. At first glance it would appear that the complexity of the controller could make implementation prohibitive for applications where it is necessary to have many s to the fuzzy controller. However, we must remind the reader of the results in Section 2.6 where we explain how implementation tricks can be used to significantly reduce computation time when there are membership functions of the shown in Figure 4.4. Rule-Base Initialization The membership functions are defined to characterize the premises of the rules that define the various situations in which rules should be applied. The membership functions are left constant and are not tuned by the FMRLC. The membership functions on the output universe of discourse are assumed to be unknown. They are what the FMRLC will automatically synthesize or tune. Hence, the FMRLC tries to fill in what actions ought to be taken for the various situations that are characterized by the premises. We must choose initial values for each of the output membership functions. For example, for an output universe of discourse [-1, 1] we could choose triangular-shaped membership functions with base widths of 0.4 and centers at zero. This choice represents that the fuzzy controller initially knows nothing about how to control the plant so it s u 0 to the plant initially well, really it does know something since we specify the remainder of the fuzzy controller a priori. Of course, one can often make a reasonable best guess at how to specify a fuzzy controller that is “more knowledgeable“ than simply placing the output membership function centers at zero. For example, we could pick the initial fuzzy controller to be the best one that we can design for the nominal plant. Notice, however, that this choice is not always the best one. Really, what you often want to choose is the fuzzy controller that is best for the operating condition that the plant will begin in this may not be the nominal condition. Unfortunately, it is not always possible to pick such a controller since you may not be able to measure the operating condition of the plant, so making a best guess or simply placing the membership function centers at zero are common choices. To complete the specification of the fuzzy controller, we use minimum or product to represent the conjunction in the premise and the implication in this book we will use minimum unless otherwise stated and the standard center-of-gravity defuzzification technique. As an alternative, we could use appropriately initialized singleton output membership functions and centeraverage defuzzification. Learning, Memorization, and Controller Choice For some applications you may want to use an integral of the error or other preprocessing