Nonsingular Transformation Research Articles

Aizerman's problem in absolute stability theory for regulated systems

A new method for investigating the absolute stability of regulated systems with limited resources is proposed. It is based on estimating improper integrals along the solution of the system. A nonsingular transformation is obtained which allows information about the nonlinearity properties to be taken into account. A class of regulated systems is distinguished for which Aizerman's problem is solvable. For this class a necessary and a sufficient condition for absolute stability are found. The proposed method for investigating absolute stability differs from the other available methods by the fact that conditions for absolute stability are derived without using the Lyapunov function and the frequency theorem. For systems with limited resources the phase variables are bounded, uniformly continuous functions. These properties were used in deriving a condition for stability and in estimating improper integrals. The estimate obtained allows the domain of absolute stability in the space of constructive parameters of the system to be greatly extended by comparison with the earlier known results, and in a number of cases a necessary and a sufficient condition for absolute stability can be obtained. Bibliography: 15 titles.

Static Output Feedback Stabilization of Positive Polynomial Fuzzy Systems

This paper deals with the static output feedback stabilization of positive polynomial fuzzy-model-based control systems. The positive polynomial fuzzy model does not need to share the same premise membership functions with the static output feedback polynomial fuzzy controller. Unlike the state feedback control case, the static output feedback control usually leads to nonconvex stability conditions. To circumvent the problem, an approach is employed to transform the nonconvex stability conditions into convex ones by introducing a nonsingular transformation matrix. Initially, the conditions guaranteeing the resultant closed-loop systems to be positive and asymptotically stable are obtained. Moreover, the divisional approximated membership functions that carry the local information of the membership functions are employed to facilitate the stability analysis and controller synthesis. The relaxed stability conditions in terms of sum of squares are obtained based on Lyapunov stability theory. Finally, a simulation example is given to testify the validity of the analysis result.

Estimator design of switched linear systems under unknown persistent excitation

In this study, the robust estimation problem is investigated for discrete-time switched linear systems with unknown persistent excitation. The robust current estimator is proposed to estimate the real state of the switched systems without the stability restriction. The estimator input is constructed based on the current and previous information of a switching signal and output measurement. Owing to the reference system and its estimator, the exponential stability of the estimation error systems is presented under the average dwell time switching and arbitrary switching. The upper bound or the property of attenuation is unnecessary for the unknown input considered here. In addition, neither the conventional decoupling principle nor non-singular transformation is employed. At last, the effectiveness of established results is illustrated by a numerical example and a DC–DC boost converter.

Guaranteed‐cost consensus for multiagent networks with Lipschitz nonlinear dynamics and switching topologies

SummaryGuaranteed‐cost consensus for high‐order nonlinear multiagent networks with switching topologies is investigated. By constructing a time‐varying nonsingular matrix with a specific structure, the whole dynamics of multiagent networks is decomposed into the consensus and disagreement parts with nonlinear terms, which is the key challenge to be dealt with. An explicit expression of the consensus dynamics, which contains the nonlinear term, is given and its initial state is determined. Furthermore, by the structure property of the time‐varying nonsingular transformation matrix and the Lipschitz condition, the impacts of the nonlinear term on the disagreement dynamics are linearized, and the gain matrix of the consensus protocol is determined on the basis of the Riccati equation. Moreover, an approach to minimize the guaranteed cost is given in terms of linear matrix inequalities. Finally, the numerical simulation is shown to demonstrate the effectiveness of theoretical results.

A Generalized Matrix Inverse That Is Consistent with Respect to Diagonal Transformations

A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general solution to a longstanding open problem relevant to a wide variety of applications in robotics, tracking, and control systems. The new inverse complements the Drazin inverse (which is consistent with respect to similarity transformations) and the Moore--Penrose inverse (which is consistent with respect to unitary/orthogonal transformations) to complete a trilogy of generalized matrix inverses that exhausts the standard family of analytically important linear system transformations. Results are generalized to obtain unit-consistent and unit-invariant matrix decompositions, and examples of their use are described.

On conservative sequences and their application to ergodic multiplier problems

The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n \in \mathbb{Z}$ such that $T^n A \cap A \not = \varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no finite invariant measure $\{T_i\}$, there exists a rank-one transformation $S$ such that $T_i \times S$ is not ergodic for all $i$. Moreover, $S$ can be chosen to be rigid or have infinite ergodic index. We establish similar results for $\mathbb{Z}^d$ actions and flows. Then, we find sufficient conditions on rank-one transformations $T$ that guarantee the existence of a rank-one transformation $S$ such that $T \times S$ is ergodic, or, alternatively, conditions that guarantee that $T \times S$ is conservative but not ergodic. In particular, the infinite Chacon transformation satisfies both conditions. Finally, for a given ergodic transformation $T$, we study the Baire categories of the sets $E(T)$, $\bar{E}C(T)$ and $\bar{C}(T)$ of transformations $S$ such that $T \times S$ is ergodic, ergodic but not conservative, and conservative, respectively.

On some random densities for random maps

Let be a probability vector, be a sequence of nonsingular transformations defined on a probability space and let be a sequence of densities in . We prove that the random density is invariant for the random map if and only if all functions are identical and equal to some invariant density for the Frobenius–Perron operator defined by T.

Distributed Global Output-Feedback Control for a Class of Euler–Lagrange Systems

This published paper investigates the distributed tracking control problem for a class of Euler-Lagrange multi-agent systems when the agents can only measure the positions. In this case, the lack of the separation principle and the strong nonlinearity in unmeasurable states pose severe technical challenges to global output-feedback control design. To overcome these difficulties, a global nonsingular coordinate transformation matrix in the upper triangular form is firstly proposed such that the nonlinear dynamic model can be partially linearized with respect to the unmeasurable states. And, a new type of velocity observers is designed to estimate the unmeasurable velocities for each system. Then, based on the outputs of the velocity observers, we propose distributed control laws that enable the coordinated tracking control system to achieve uniform global exponential stability (UGES). Both theoretical analysis and numerical simulations are presented to validate the effectiveness of the proposed control scheme. Followed by the original paper, a typo and a mistake is corrected.

Effects of Randomization on Asymptotic Periodicity for Nonsingular Transformations

It is known that the Perron–Frobenius operators of piecewise expanding C2 transformations possess an asymptotic periodicity of densities. On the other hand, external noise or measurement errors are unavoidable in practical systems; therefore, all realistic mathematical models should be regarded as random iterations of transformations. The aim of this paper is to discuss the effects of randomization on the asymptotic periodicity of densities.

Feedback Linearization of Continuous and Discrete Multidimensional Systems

The problem of feedback linearization (FL) of continuous and discrete nonlinear MIMO systems is considered. The idea of FL method consists in converting the original nonlinear system into a linear one by means of feedback. Then, the methods of control theory for linear systems are used for system design. A widespread approach to FL design is based on the method of normal form, that uses a nonsingular transformation of system state variables z = T(x). In order to obtain a normal form of the nonlinear system in the neighbor of a some point, it is necessary to determine a special function - the system virtual output, for which a relative degree (in the case of single input) or a vector relative degree (in the case of multiple input) is determined. Applicability of the normal form method for FL is provided by the conditions of controllability and involutivity for the considered nonlinear system, which are not always true. Moreover, when developing a linearizing control law, the main difficulty lies in the transition from transformed variables z to state variables x of the original system. In this paper, we propose another approach, based on representing the original nonlinear system into a state-dependent coefficient form and applying the canonical similarity transformation z = T(x)x, that allow getting the system to canonical form, that considerably simplifies the FL problem. Such similarity transformation allow accomplishing linearization of system without determining of the virtual system output. Another advantage of the proposed method is that the technique of the transition from the transformed variables z to the state variables x of the original system is simpler. The results are illustrated by examples for continuous and discrete nonlinear systems.

Design of Non-Singular Two-Dimensional Layered Cloaks Mapped from Small Areas

A singular coordinate transformation leads to extreme materials in a transformation-based cloak. If a non-singular transformation is used, the perfect invisibility is lost. The invisibility is also influenced by the homogenization process. In this paper, we investigate these two influences in 2-D circular cylindrical cloaks, obtained from different non-singular mappings. Considering a further deterministic optimization, we also conclude that it is better to use a non-singular cloak based on a mapping scale up to 20%, since it has lower scattering and simpler materials when compared with cloaks based on other mappings.

Dynamic Learning from Adaptive Neural Control of Uncertain Robots with Guaranteed Full-State Tracking Precision

A dynamic learning method is developed for an uncertain n-link robot with unknown system dynamics, achieving predefined performance attributes on the link angular position and velocity tracking errors. For a known nonsingular initial robotic condition, performance functions and unconstrained transformation errors are employed to prevent the violation of the full-state tracking error constraints. By combining two independent Lyapunov functions and radial basis function (RBF) neural network (NN) approximator, a novel and simple adaptive neural control scheme is proposed for the dynamics of the unconstrained transformation errors, which guarantees uniformly ultimate boundedness of all the signals in the closed-loop system. In the steady-state control process, RBF NNs are verified to satisfy the partial persistent excitation (PE) condition. Subsequently, an appropriate state transformation is adopted to achieve the accurate convergence of neural weight estimates. The corresponding experienced knowledge on unknown robotic dynamics is stored in NNs with constant neural weight values. Using the stored knowledge, a static neural learning controller is developed to improve the full-state tracking performance. A comparative simulation study on a 2-link robot illustrates the effectiveness of the proposed scheme.

Nonlinear Reduced-Order Observer-Based Predictive Control for Diving of an Autonomous Underwater Vehicle

The attitude control and depth tracking issue of autonomous underwater vehicle (AUV) are addressed in this paper. By introducing a nonsingular coordinate transformation, a novel nonlinear reduced-order observer (NROO) is presented to achieve an accurate estimation of AUV’s state variables. A discrete-time model predictive control with nonlinear model online linearization (MPC-NMOL) is applied to enhance the attitude control and depth tracking performance of AUV considering the wave disturbance near surface. In AUV longitudinal control simulation, the comparisons have been presented between NROO and full-order observer (FOO) and also between MPC-NMOL and traditional NMPC. Simulation results show the effectiveness of the proposed method.

Fault tolerant control for a class of nonlinear non-Gaussian singular stochastic distribution systems

New fault diagnosis and active fault tolerant control (FTC) algorithms are proposed for a class of nonlinear singular stochastic distribution control (SDC) systems in this paper. Different from general SDC systems, in singular SDC systems, the relationship between the weights and the control input is expressed by a singular state space model, which increases the difficulty in design of fault diagnosis and fault tolerant control. A non-singular state transformation is made to transform the singular dynamic system into a differential-algebraic system. An adaptive nonlinear observer is designed to estimate the size of the fault occurring in the system. Furthermore, the linear matrix inequality (LMI) approach is applied to establish sufficient conditions for the existence of the observer. Based on the estimated fault information, the active fault tolerant controller is designed to make the post-fault probability density function (PDF) still track the given distribution. At last, an illustrated example is given to demonstrate the effectiveness of the proposed algorithm, and satisfactory results have been obtained.

Fault tolerant control for a class of nonlinear non-Gaussian singular stochastic distribution systems

New fault diagnosis and active fault tolerant control (FTC) algorithms are proposed for a class of nonlinear singular stochastic distribution control (SDC) systems in this paper. Different from general SDC systems, in singular SDC systems, the relationship between the weights and the control input is expressed by a singular state space model, which increases the difficulty in design of fault diagnosis and fault tolerant control. A non-singular state transformation is made to transform the singular dynamic system into a differential-algebraic system. An adaptive nonlinear observer is designed to estimate the size of the fault occurring in the system. Furthermore, the linear matrix inequality (LMI) approach is applied to establish sufficient conditions for the existence of the observer. Based on the estimated fault information, the active fault tolerant controller is designed to make the post-fault probability density function (PDF) still track the given distribution. At last, an illustrated example is given to demonstrate the effectiveness of the proposed algorithm, and satisfactory results have been obtained.

Applications of the preview control method to the optimal problem for linear continuous-time stochastic systems with time-delay

This paper discusses the optimal control problem for a class of linear continuous-time stochastic systems with time-delay based on the preview control method. In the design procedure, the time-delay terms are formally eliminated and the augmented error system method is used. The optimal controller of the time-delay stochastic system is obtained based on dynamic programming principle. In the previous approaches, the preview control method is applied to continuous-time deterministic system. Consequently, compared with the existing results, the main contributions of this paper are: (1) the conclusion of the non-singular transformation is improved and (2) the conclusion in deterministic system is extended by using the methods of designing an assistant system and introducing an integrator when building the stochastic augmented error system. The optimal tracking of the stochastic system with time-delay is realized by solving the optimal regulation problem of the stochastic augmented error system. The simulation shows the effectiveness of the controller designed in this paper.

Time‐varying group formation analysis and design for general linear multi‐agent systems with directed topologies

SummaryTime‐varying group formation control problems for general linear multi‐agent systems with directed topologies are studied. Different from the traditional complete formation, where only one formation is realized by the multi‐agent system, in the group formation, there could be multiple time‐varying sub‐formations. Firstly, a time‐varying group formation protocol is constructed by local neighboring relative information. Then nonsingular transformations are applied to the closed‐loop multi‐agent systems. Sufficient conditions for the multi‐agent systems to achieve time‐varying group formation are further presented together with the time‐varying group formation feasibility constraints. Explicit expressions of the subgroup formation reference functions are derived to describe the macroscopic movement of the time‐varying subgroup formations. Moreover, an approach to design the time‐varying group formation protocol is proposed by solving an algebraic Riccati equation. Finally, a numerical example with three subgroups is provided to demonstrate the effectiveness of the obtained theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

Synchronization for a class of coupled linear partial differential systems via boundary control

The synchronization for coupled linear partial differential systems(PDSs) with boundary control is considered. Based on the nonsingular matrix transformation method, we decouple the coupled synchronization error dynamical systems and turn the synchronization control problem into the asymptotical stabilization problem. Then, using the backstepping approach, we present the boundary control strategy to guarantee the synchronization. At last, numerical example is given to illustrate the effectiveness of our result.

Functional observer design using linear matrix inequalities

It is shown that the problem of observer design for estimating a set of linear combinations of state variables of a plant can be formulated in terms of linear matrix inequalities. An algorithm for constructing functional observers is proposed based on a nonsingular transformation of a plant model in the state space by matrix canonization with subsequent solution of the system of linear matrix inequalities.

Control efficacy of complex networks.

Controlling complex networks has become a forefront research area in network science and engineering. Recent efforts have led to theoretical frameworks of controllability to fully control a network through steering a minimum set of driver nodes. However, in realistic situations not every node is accessible or can be externally driven, raising the fundamental issue of control efficacy: if driving signals are applied to an arbitrary subset of nodes, how many other nodes can be controlled? We develop a framework to determine the control efficacy for undirected networks of arbitrary topology. Mathematically, based on non-singular transformation, we prove a theorem to determine rigorously the control efficacy of the network and to identify the nodes that can be controlled for any given driver nodes. Physically, we develop the picture of diffusion that views the control process as a signal diffused from input signals to the set of controllable nodes. The combination of mathematical theory and physical reasoning allows us not only to determine the control efficacy for model complex networks and a large number of empirical networks, but also to uncover phenomena in network control, e.g., hub nodes in general possess lower control centrality than an average node in undirected networks.