Abstract. Universal tracking control is investigated in the context of a class S of M -input,M -output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains -as a prototype subclass -all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary Ê M -valued reference signal r of class W 1,∞ (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that ϕ(t) e(t) < 1 for all t ≥ 0, where ϕ a prescribed real-valued function of class W 1,∞ with the property that ϕ(s) > 0 for all s > 0 and lim infs→∞ ϕ(s) > 0. A simple (neither adaptive nor dynamic) error feedback control of the form u(t) = −α(ϕ(t) e(t) )e(t) is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain α(ϕ(·) e(·) ).Mathematics Subject Classification. 93D15, 93C30, 34K20.
Abstract. An adaptive servomechanism is developed in the context of the problem of approximate or practical tracking (with prescribed asymptotic accuracy), by the system output, of any admissible reference signal (absolutely continuous and bounded with essentially bounded derivative) for every member of a class of controlled dynamical systems modelled by functional differential equations.Key words. adaptive control, nonlinear systems, functional differential equations, practical tracking, universal servomechanism AMS subject classifications. 93C23, 93C10, 93C40, 34K20PII. S03630129003797041. Introduction. A servomechanism problem is addressed in the context of a class of controlled dynamical systems having the interconnected structure shown in the dashed box in Figure 1. In particular, the aim is the development of an adaptive servomechanism which, for every system of the underlying class, ensures practical tracking (in the sense that prespecified asymptotic tracking accuracy, quantified by λ > 0, is assured), by the system output, of an arbitrary reference signal assumed to be locally absolutely continuous and bounded with essentially bounded derivative. (We denote by R the class of such functions and remark that bounded globally Lipschitz functions form an easily recognized subclass.) The system consists of the interconnection of two blocks: The dynamic block Σ 1 , which can be influenced directly by the system input/control u (an R M -valued function), is also driven by the output w from the dynamic block Σ 2 . Viewed abstractly, the block Σ 2 can be considered as a causal operator which maps the system output y (an R M -valued function) to w (an internal quantity, unavailable for feedback purposes).In essence, the underlying system class S consists of infinite-dimensional nonlinear M -input u, M -output y systems (p, f, g, T ), given by a controlled nonlinear functional differential equation of the form (1.1)ẏ (t) = f (p(t), (T y)(t))+g(p(t), (T y)(t), u(t)), y| [−h,0]
In this paper we report the results of an experiment designed to test the hypothesis that when faced with a question involving the inverse direction of a reversible mathematical process, students solve a multiple-choice version by verifying the answers presented to them by the direct method, not by undertaking the actual inverse calculation. Participants responded to an online test containing equivalent multiple-choice and constructed-response items in two reversible algebraic techniques: factor/expand and solve/verify. The findings supported this hypothesis: Overall scores were higher in the multiple-choice condition compared to the constructed-response condition, but this advantage was significantly greater for items concerning the inverse direction of reversible processes compared to those involving direct processes.
Assessment is a key component of all teaching and learning, and for many students is a key driver of their activity. This paper considers automatic computer aided assessment (CAA) of mathematics. With the rise of communications technology this is a rapidly expanding field. Publishers are increasingly providing online support for textbooks with automated versions of exercises linked to the work in the book. There are an expanding range of purely online resources for students to use independently of formal instruction. There are a range of commercial and open source systems with varying levels of mathematical and pedagogic sophistication. History and BackgroundAssessment is a key component of all teaching and learning, and for many students is a key driver of their activity. Computer aided assessment (CAA) has a history going back over half a century, for example (Hollingsworth 1960) reports a "grader" programme which automatically checked some aspects of students' computer programmes. "We are still doing considerable hand filing of punched cards at this stage. This large deck of cards which includes the grader program is then run on the computer. Our largest single run has been 106 student programs covering 9 different exercises." By the mid-1960s computers were being used to teach arithmetic, e.g. (Suppes 1967), and by the 1980s there were a number of separate strands of research in this area including the artificial intelligence (AI) community, e.g. (Sleeman and Brown 1982). The ambitious goals of such artificial intelligence-led systems have only been achieved in confined and specialized subject areas. These difficulties were acknowledged early, for example in their preface to (Sleeman and Brown 1982).
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