Stabilization of nonholonomic chained systems via nonregular feedback linearization

This paper addresses the problem of feedback stabilization of nonholonomic chained systems within the framework of nonregular feedback linearization. Firstly, the nonsmooth version of nonregular feedback linearization is formulated, and a criterion for nonregular feedback linearization is provided. Then, it is proved that the chained form is linearizable via nonregular feedback control, thus enable us to design feedback control laws using standard techniques for linear systems. The obtained discontinuous control laws guarantee convergence of the closed-loop system with exponential rates. Finally, simulation results are presented to show the effectiveness of the approach.     

On regularizing singular systems by decentralized output feedback

This paper considers linear time-invariant decentralized singular systems which are either nonregular or, if they are regular, they have impulsive modes. It derives algebraic necessary and sufficient conditions for making a singular system both regular and impulse-free by decentralized output feedback control laws and decentralized proportional-plus-derivative output feedback control laws     

Separating the Drift from Affine Control Systems via Feedbacks

This paper investigates the use of state feedbacks to transform an affine control system into a special form with a driftless property. First, we consider the regular dynamic state feedback for an affine control system. Both local and global sufficient and necessary conditions are given on the affine control system such that it admits the driftless property. An algorithm is provided to compute explicitly the desired coordinate transformation. Then, the nonregular cases are studied. Some necessary conditions, as well as sufficient conditions, for nonregular feedback are presented. We illustrate with some examples.     

关于非线性系统非正则线性化的一个充分条件

本文得到关于仿射非线性系统非正则反馈线性问题的一个充分条件，给出并讨论了一个例子。     

Generalizations of the Statement of Compensator Realizability at the System Input

Consideration was given to transformation (realizability) of the equivalent compensator at the input of a linear system into a feedback control law. By using the system embedding technology, unique results were obtained for the matrix quotients which enabled one to formulate and prove new assertions which in part contradict the existing results and, in contrast to them, are valid also for the nonregular (degenerate) compensators, as well as for the plants with redundant (dependent) control inputs. Illustrative examples were given.     

System Embedding. Classes of the Control Laws

Consideration was given to control of the linear systems using the precompensator and controller. By means of the technology of system embedding, the problem was reduced to the linear matrix equations for which the general structures of classes of all were obtained. In doing so, problems using both the regular and nonregular (with nonreversible precompensator) control laws were encompassed. The generally nonformalizable dependence of the structure of controller class on the choice of the precompensator was shown to be the basic distinction of the nonregular laws. The solution was illustrated by an instructional example and an example of control of the aircraft lateral (roll–yaw) motion.     

A necessary and sufficient condition for the existence of finitestate supervisors in discrete-event systems

A supervisor realized by finite automaton is called a finite state supervisor. It is clear that a finite state supervisor exists for every regular controllable language. However, an example of a finite state supervisor for a nonregular one has been reported. A necessary and sufficient condition is proved here, for the existence of a finite state supervisor, and an example of a simple manufacturing system that has a finite state supervisor for a nonregular controllable language is given     

Singularities in complete bipartite graph-type Boltzmann machines and upper bounds of stochastic complexities

It is well known that Boltzmann machines are nonregular statistical models. The set of their parameters for a small size model is an analytic set with singularities in the space of a large size one. The mathematical foundation of their learning is not yet constructed because of these singularities, though they are applied to information engineering. Recently we established a method to calculate the Bayes generalization errors using an algebraic geometric method even if the models are nonregular. This paper clarifies that the upper bounds of generalization errors in Boltzmann machines are smaller than those in regular statistical models.     

Monotonicity properties for the viable control of discrete-time systems

This paper deals with the control of nonlinear systems in the presence of state and control constraints for discrete-time dynamics in finite-dimensional spaces. The viability kernel is known to play a basic role for the analysis of such problems and the design of viable control feedbacks. Unfortunately, this kernel may display very nonregular geometry and its computation is not an easy task in general. In the present paper, we show how monotonicity properties of both dynamics and constraints allow for relevant analytical upper and lower approximations of the viability kernel through weakly and strongly invariant sets. An example on fish harvesting management illustrates some of the assertions.     

Partwise cross-parameterization via nonregular convex hull domains

In this paper, we propose a novel partwise framework for cross-parameterization between 3D mesh models. Unlike most existing methods that use regular parameterization domains, our framework uses nonregular approximation domains to build the cross-parameterization. Once the nonregular approximation domains are constructed for 3D models, different (and complex) input shapes are transformed into similar (and simple) shapes, thus facilitating the cross-parameterization process. Specifically, a novel nonregular domain, the convex hull, is adopted to build shape correspondence. We first construct convex hulls for each part of the segmented model, and then adopt our convex-hull cross-parameterization method to generate compatible meshes. Our method exploits properties of the convex hull, e.g., good approximation ability and linear convex representation for interior vertices. After building an initial cross-parameterization via convex-hull domains, we use compatible remeshing algorithms to achieve an accurate approximation of the target geometry and to ensure a complete surface matching. Experimental results show that the compatible meshes constructed are well suited for shape blending and other geometric applications.     

Automata and processes on multisets of communicating objects

Inspired by P systems initiated by Gheorghe Pãun, we study a computation model over a multiset of communicating objects. The objects in our model are instances of finite automata. They interact with each other by firing external transitions between two objects. Our model, called a service automaton, is intended to specify, at a high level, a service provided on top of network devices abstracted as communicating objects. We formalize the concept of processes, running over a multiset of objects, of a service automaton and study the computing power of both single-process and multiprocess service automata. In particular, in the multiprocess case, regular maximal parallelism is defined for inter-process synchronization. It turns out that single-process service automata are equivalent to vector addition systems and hence can define nonregular processes. Among other results, we also show that Presburger reachability problem for single-process service automata is decidable, while it becomes undecidable in the multiprocess case. Hence, multiprocess service automata can not be effectively simulated by vector addition systems.     

一种地表造型方法

本文提出一种简单而有效的地表造型方法：中点随机扰动插值法，给出了用这个方法在微机上生成的几幅地表图。这个方法也适合其它不规则自然景物的造型。     

More on Nonregular PDL: Finite Models and Fibonacci-like Programs

We continue research on enriching propositional dynamic logic (PDL) with nonregular programs. Previous work indicates that the general problem of characterizing those extensions for which PDL becomes undecidable is probably very hard. After observing that any nonregular extension increases the expressive power of PDL, we concentrate on one-letter extensions. First, we address the issue of finite models: A general condition is formulated, and is proven to be sufficient for a one-letter extension to violate the finite model property. We show the condition to hold in several cases, including all polynomials, sums of primes, and linear recurrences. We then build on a technique of Paterson and Harel, and show that the validity problem for PDL enriched with any Fibonacci-like sequence isΠ¹₁-complete.     

非线性奇异系统非交互控制的反馈实现

研究非线性奇异系统的非交互控制的反馈实现问题．首先给出了非线性奇异系统的向量相对阶与其非交互控制实现的关系；然后对正则非线性奇异系统的反馈控制系统的非交互控制的实现问题，给出了一种可使系统实现非交互控制的反馈律的构造方法。     

Cascaded feedback linearization and its application to stabilization of nonholonomic systems

In this paper, a new cascaded feedback linearization problem is formulated and a set of conditions on the cascaded feedback linearizability are established for a class of two-input affine nonlinear systems. The proposed cascaded feedback linearization method enlarges the classes of nonlinear systems which can be dealt with using the feedback linearization technique. In particular, the proposed design can be applied to address the feedback stabilization problem for a few classes of nonlinear systems which have uncontrollable linearization and do not satisfy the standard feedback linearization conditions. As an illustrative application, the proposed cascade feedback linearization concept is used to solve the feedback stabilization problem of nonholonomic systems within the framework of continuously differentiable state feedback control. Simulation results are provided to illustrate the proposed method.     

Passive stochastic feedback stability-Part II. Applications

In Part I of this paper it was shown that the concept of passivity, used extensively in deterministic feedback stability, could be extended to stochastic feedback systems described through Markov theory to produce stochastic feedback stability theorems. To apply the theory developed in Part I, one is required to do a passivity analysis of the sub-systems of the feedback system. In this paper, six particular types of stochastic systems (sub-systems) are considered and a passivity analysis of each is performed. These systems have the appearance of standardnth-order linear differential systems with a random process entering at some point. The randomness is either white noise, impulsive noise or a Markov jump process. Once the passivity analysis has been made, feedback stability theorems can be derived for many feedback systems containing one or more of these six systems as sub-systems. Two such feedback systems are considered in the paper.     

On asymptotically efficient solutions for a class of supervisorycontrol problems

We consider restricted sequences of forbidden-string and forbidden-state problems and show the limiting problem instance defined by a sequence has a solution if and only if each problem instance in the sequence has a solution. Furthermore, the supervisor that enforces the desired behavior in the limiting problem instance also enforces the desired behavior in each problem instance in the sequence. In essence, the storage requirement for supervisory control of each instance is bounded above by that of the limiting problem instance. In addition, if the limiting problem instance is available, a similar observation can be made regarding the complexity of supervisory control. This observation coupled with recent results on the solvability of the supervisory control problem involving certain classes of nonregular behaviors and finite-representations of infinite-state systems is suggested as an effective procedure to combat the computational and storage issues in supervisory control. We conclude with an incomplete list of future research directions