Stochastic problems in <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> and <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

We consider stochastic control systems subjected simultaneously to stochastic and determinate perturbations. Stochastic perturbations are assumed to be state-multiplicative stochastic processes, while determinate perturbations can be any processes with finite energy on infinite time interval. The results of the determinate H _∞-theory are compared to their stochastic analogs. The determinate and stochastic theories are linked together by the lemma that establishes the equivalence between the stability and boundness of the ‖L‖_∞ < γ norm of the perturbation operator L, from one side, and the solvability of certain linear matrix inequalities (LMIs), from the other side. As soon as the stochastic version of the lemma is proven, the γ-controller analysis and design problems are solved, in general, identically in the frame of the united LMI methodology.     

Approximate Pattern Matching with the <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> Metrics

Given an alphabet Σ={1,2,…,|Σ|} text string T∈Σ ⁿ and a pattern string P∈Σ ^m , for each i=1,2,…,n−m+1 define L _p (i) as the p-norm distance when the pattern is aligned below the text and starts at position i of the text. The problem of pattern matching with L _p distance is to compute L _p (i) for every i=1,2,…,n−m+1. We discuss the problem for d=1,2,∞. First, in the case of L ₁ matching (pattern matching with an L ₁ distance) we show a reduction of the string matching with mismatches problem to the L ₁ matching problem and we present an algorithm that approximates the L ₁ matching up to a factor of 1+ε, which has an O(\frac1e²nlogmlog|S|)O(\frac{1}{\varepsilon^{2}}n\log m\log|\Sigma|) run time. Then, the L ₂ matching problem (pattern matching with an L ₂ distance) is solved with a simple O(nlog m) time algorithm. Finally, we provide an algorithm that approximates the L _∞ matching up to a factor of 1+ε with a run time of O(\frac1enlogmlog|S|)O(\frac{1}{\varepsilon}n\log m\log|\Sigma|) . We also generalize the problem of String Matching with mismatches to have weighted mismatches and present an O(nlog ⁴ m) algorithm that approximates the results of this problem up to a factor of O(log m) in the case that the weight function is a metric.     

Design of optimal and robust control with <Emphasis Type="Italic">H</Emphasis> <Subscript>∞</Subscript>/<Emphasis Type="Italic">γ</Emphasis> <Subscript>0</Subscript> performance criterion

For a double-input single-output system, this paper defines a disturbance attenuation level (called H_∞/γ₀ norm) as the maximum-value L₂ norm of the output under an unknown disturbance with a bounded L₂ norm supplied to the first input and an impulsive disturbance in the form of the product of an unknown vector and the delta function supplied the second input, where the squared L₂ norm of the former disturbance plus the quadratic form of the impulsive disturbance vector does not exceed 1. Weight matrix choice in the H_∞/γ₀ norm yields a trade-off between the attenuation level of the L₂ disturbance and the attenuation level of the impulsive disturbance in corresponding channels. For the uncertain systems with dynamic or parametric uncertainty in the feedback loop, a robust H_∞/γ₀ norm is introduced that includes the robust H_∞ and γ₀ norms as special cases. All these characteristics or their upper bounds in the uncertain system are expressed via solutions of linear matrix inequalities. This gives a uniform approach for designing optimal and robust control laws with the H_∞/γ₀, H_∞ and γ₀ performance criteria.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for linear infinite-dimensional systems

In this paper, we consider mixed H ₂/H _∞ control problems for linear infinite-dimensional systems. The first part considers the state feedback control for the H ₂/H _∞ control problems of linear infinite-dimensional systems. The cost horizon can be infinite or finite time. The solutions of the H ₂/H _∞ control problem for linear infinitedimensional systems are presented in terms of the solutions of the coupled operator Riccati equations and coupled differential operator Riccati equations. The second part addresses the observer-based H ₂/H _∞ control of linear infinite-dimensional systems with infinite horizon and finite horizon costs. The solutions for the observer-based H ₂/H _∞ control problem of linear infinite-dimensional systems are represented in terms of the solutions of coupled operator Riccati equations. The first-order partial differential system examples are presented for illustration. In particular, for these examples, the Riccati equations are represented in terms of the coefficients of first-order partial differential systems.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> DOF control of stochastic systems with time-varying delay and <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> disturbance

This paper considers the problem of H _∞ dynamic output feedback (DOF) control for a class of stochastic systems with time-varying delay and L _∞ disturbance. A new delay-dependent sufficient condition for the existence of the DOF controller is derived and the controller design method is given in the form of bilinear matrix inequalities (BMIs). Moreover, a variable step size path-following algorithm is proposed to solve the BMI problem. Numerical examples are given to illustrate the effectiveness of the proposed methods.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> filtering for a class of high-speed sampling uncertain systems

The problem of mixed H2/H∞ filtering for polytopic Delta operator systems is investigated. The aim is to design a linear asymptotically stable filter which guarantees that the filtering error system has different performances in different filtering channels. Based on a parameter-dependent Lyapunov function, a new mixed H2/H∞ performance criterion is presented. Upon this performance criterion, a sufficient condition for the full-order mixed H2/H∞ filter is derived in terms of linear matrix inequalities. The filter can be obtained from the solution of a convex optimization problem. The proposed filter design procedure is less conservative than the strategy based on the quadratic stability notion. A numerical example is given to illustrate the feasibility of the proposed approach.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for a class of nonlinear networked control systems

In this paper, the mixed H ₂/H _∞ control problem is investigated for a class of nonlinear discrete-time networked control systems with random network-induced delays, stochastic packet dropouts and probabilistic sensor faults. The packet dropouts process is modeled as a homogeneous Markov chains taking values in a finite state space. Network-induced delays occur in a random way with known upper bound. A set of stochastic variables are exploited to describe sensor faults with different probabilistic density functions. By using a delay-dependent Lyapunov functional, a mode-dependent mixed H ₂/H _∞ controller is designed to guarantee both stochastic stability of the closed-loop system and the prescribed H2, H¥ control performances. Sufficient conditions for the existence of the mixed H ₂/H _∞ controller are presented in terms of a series of LMIs. If these LMIs are feasible, then the modedependent mixed H ₂/H _∞ controller can be obtained. A numerical example is given to demonstrate the effectiveness of the developed method.     

Simultaneous <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> stabilization for chemical reaction systems based on orthogonal complement space

This paper presents a simultaneous H₂/H_∞ stabilization problem for the chemical reaction systems which can be modeled as a finite collection of subsystems. A single dynamic output feedback controller which simultaneously stabilizes the multiple subsystems and captures the mixed H₂/H_∞ control performance is designed. To ensure that the stability condition, the H₂ characterization and the H_∞ characterization can be enforced within a unified matrix inequality framework, a novel technique based on orthogonal complement space is developed. Within such a framework, the controller gain is parameterized by the introduction of a common free positive definite matrix, which is independent of the multiple Lyapunov matrices. An iterative linear matrix inequality (ILMI) algorithm using Matlab Yalmip toolbox is established to deal with the proposed framework. Simulation results of a typical chemical reaction system are exploited to show the validity of the proposed methodology.     

FIR-type robust <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of discrete linear time-invariant polytopic systems via memory state-feedback control laws

In this paper, robust H ₂ and H _∞ control problems for discrete linear time-invariant (LTI) systems with polytopic uncertainties are addressed. The so-called finite impulse response (FIR) controller incorporating the states over several samples from the past to the present is adopted to design robust control laws with improved performances. For the closed-loop stability, parameter-dependent quadratic Lyapunov functions (PD-QLFs) are employed. Sufficient controller synthesis conditions are derived in the form of linear matrix inequalities (LMIs). Finally, examples are given to demonstrate the usefulness of the proposed methods.     

Multicriteria Robust Generalized <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript> and γ<Subscript>0</Subscript> Controllers with Application to Stabilization of a Rotor in Electromagnetic Bearings

For linear plants with unstructured or structured uncertainty of bounded norm, this paper designs Pareto optimal robust controllers in terms of linear matrix inequalities in multicriteria control problems with the generalized H₂ or γ₀ norms. The controller design procedure is based on optimization of a scalar objective function (Germeier convolution) and semi-definite programming. The developed theory is used to design multicriteria robust controllers in the stabilization problem for a rotor in electromagnetic bearings.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> digital redesign for LTI systems

The so-called digital redesign (DR) is a sampled-data (SD) controller design method where an analogue controller is designed firstly, and then transformed to an approximately equivalent digital controller in the sense of state-matching. In this approach, the SD controller is designed by reducing the discrepancy between the discrete-time (DT) counterpart of the closed-loop SD control system and the continuous-time (CT) closed-loop system. In this paper, we develop a DR strategy for CT linear time-invariant systems. More specifically, H _∞ norm of the error dynamic system between the CT and DT plants is minimized for the optimal state-matching performance at every sampling point. The design problem is formulated as linear matrix inequalities which can be efficiently solved by using convex optimization techniques. Finally, an example is given to illustrate the effectiveness of the proposed method.     

Nonnegative matrix factorization by joint locality-constrained and <Emphasis Type="Italic">ℓ</Emphasis><Subscript>2,1</Subscript>-norm regularization

Nonnegative matrix factorization has been widely applied recently. The nonnegativity constraints result in parts-based, sparse representations which can be more robust than global, non-sparse features. However, existing techniques could not accurately dominate the sparseness. To address this issue, we present a unified criterion, called Nonnegative Matrix Factorization by Joint Locality-constrained and ? _2,1-norm Regularization(NMF2L), which is designed to simultaneously perform nonnegative matrix factorization and locality constraint as well as to obtain the row sparsity. We reformulate the nonnegative local coordinate factorization problem and use ? _2,1-norm on the coefficient matrix to obtain row sparsity, which results in selecting relevant features. An efficient updating rule is proposed, and its convergence is theoretically guaranteed. Experiments on benchmark face datasets demonstrate the effectiveness of our presented method in comparison to the state-of-the-art methods.     

A Nash Game Approach to Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Model Predictive Control: Part 3 – Output Feedback Case

In this paper, the state-feedback Nash game based mixed H₂/H_∞ design^{[1, 2]} has been extended for output feedback case. The algorithm is applied to control bioreactor system with a Laguerre-Wavelet Network (LWN)^{[3, 4]} model of the bioreactor. This is achieved by using the LWN model as a deviation model and by successively linearising the deviation model along the state trajectory. For reducing the approximation error and to improve the controller performance, symbolic derivation algorithm, viz., automatic differentiation is employed. A cautionary note is also given on the fragility of the output feedback mixed H₂/H_∞ model predictive controller^{[4, 5]} due to its sensitivity to its own parametric changes.     

Finite-time <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>−<Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> filtering for nonlinear stochastic systems based on a novel stochastic finite-time stability theorem

This paper presents a novel stochastic finite-time stability theorem and gives its application in the finite-time L ₂?L _∞ filter design for nonlinear stochastic systems. Different form the frequently-used stochastic finite-time stability result, the proposed one does not require that all the states have the same fractional order exponent. Based on this result, a sufficient condition is given for nonlinear stochastic systems to possess the finite-time L ₂?L _∞ performance with a prescribed gain. Further, an existence condition of the finite-time L ₂?L _∞ filter with a prescribed disturbance attenuation level is given for nonlinear stochastic systems with external disturbance inputs. The effectiveness of the obtained results is illustrated by an example.     

Fast unsupervised feature selection with anchor graph and <Emphasis Type="Italic">ℓ</Emphasis><Subscript>2,1</Subscript>-norm regularization

Graph-based unsupervised feature selection has been proven to be effective in dealing with unlabeled and high-dimensional data. However, most existing methods face a number of challenges primarily due to their high computational complexity. In light of the ever-increasing size of data, these approaches tend to be inefficient in dealing with large-scale data sets. We propose a novel approach, called Fast Unsupervised Feature Selection (FUFS), to efficiently tackle this problem. Firstly, an anchor graph is constructed by means of a parameter-free adaptive neighbor assignment strategy. Meanwhile, an approximate nearest neighbor search technique is introduced to speed up the anchor graph construction. The ?_2,1-norm regularization is then performed to select more valuable features. Experiments on several large-scale data sets demonstrate the effectiveness and efficiency of the proposed method.     

Model reference robust adaptive <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> controller design

In this paper, a novel model reference robust adaptive H _∞ controller is designed, which not only guarantees asymptotically stability, but also optimizes adaptive H _∞ performance by estimating the optimal unknown control parameters. Furthermore, a novel state feedback framework is introduced, which depends on the optimal unknown parameter estimations, to guarantee that the defined cost function is minimized. At the same time, the minimal adaptive H _∞ performance index is obtained.     

Design of discrete ℋ<Subscript>2</Subscript>-optimal reduced-order controllers

Consideration was given to the design of discrete dynamic reduced-order controllers minimizing the ℋ₂-norm of the transfer matrix of a closed-loop system. The problem of reducing the controller order is related to the solution of the singular problem of filtration (no measurement noise) and control (no control at the controlled output). Using the well-known structures of controllers based on the corresponding minimum-order observers, these problems were shown to be reducible to the solution of two Riccati equations of which one is of a reduced order. Peculiarities of solution that are characteristic of the digital controllers and caused by the allowance for astatism and presence of control delays were examined. An example of an ℋ₂-optimal reduced-order controller was presented to illustrate the results obtained.     

Small deviations for two classes of Gaussian stationary processes and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-functionals, 0 < <Emphasis Type="Italic">p</Emphasis> ≤ ∞

Let w(t) be a standard Wiener process, w(0) = 0, and let η _a (t) = w(t + a) − w(t), t ≥ 0, be increments of the Wiener process, a > 0. Let Z _a (t), t ∈ [0, 2a], be a zeromean Gaussian stationary a.s. continuous process with a covariance function of the form E Z _a (t)Z _a (s) = 1/2[a − |t − s|], t, s ∈ [0, 2a]. For 0 < p < ∞, we prove results on sharp asymptotics as ɛ → 0 of the probabilities

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control Design Using Dynamic Neural Networks

In this paper, a new approach is investigated for adaptive dynamic neural network-based H _∞ control, which is designed for a class of non-linear systems with unknown uncertainties. Currently, non-linear systems with unknown uncertainties are commonly used to efficiently and accurately express the real practical control process. Therefore, it is of critical importance but a great challenge and still at its early age to design a stable and robust controller for such a process. In the proposed research, dynamic neural networks were constructed to precisely approximate the non-linear system with unknown uncertainties first, a non-linear state feedback H _∞ control law was designed next, then an adaptive weighting adjustment mechanism for dynamic neural networks was developed to achieve H _∞ regulation performance, and last a recurrent neural network was employed as a neuro-solver to efficiently and numerically solve the standard LMI problem so as to obtain the appropriate control gains. Finally, case studies further verify the feasibility and efficiency of the proposed research.     

The <Emphasis Type="Italic">k</Emphasis>-Homotopic Thinning and a Torus-Like Digital Image in <Emphasis Type="Bold">Z</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In order to discuss digital topological properties of a digital image (X,k), many recent papers have used the digital fundamental group and several digital topological invariants such as the k-linking number, the k-topological number, and so forth. Owing to some difficulties of an establishment of the multiplicative property of the digital fundamental group, a k-homotopic thinning method can be essentially used in calculating the digital fundamental group of a digital product with k-adjacency. More precisely, let be a simple closed k _i -curve with l _i elements in . For some k-adjacency of the digital product which is a torus-like set, proceeding with the k-homotopic thinning of , we obtain its k-homotopic thinning set denoted by DT _k . Writing an algorithm for calculating the digital fundamental group of , we investigate the k-fundamental group of by the use of various properties of a digital covering (Z×Z,p ₁×p ₂,DT _k ), a strong k-deformation retract, and algebraic topological tools. Finally, we find the pseudo-multiplicative property (contrary to the multiplicative property) of the digital fundamental group. This property can be used in classifying digital images from the view points of both digital k-homotopy theory and mathematical morphology.