Independent tracking and regulation adaptive control with forgetting factor

This paper considers a discrete-time adaptive control algorithm with a forgetting factor applicable to minimum phase plants. The tracking and regulation objectives are independently specified. The relevance of the eigenvalues of the gain matrix (F_k) used in the updating equation for the adaptive parameters ($\text{}\text{?}\text{gq(k)}$) is shown. It is proved that if the maximum eigenvalue of the inverse of the gain matrix F_k has an upper bound and a non-zero lower bound then the global convergence of the control algorithm is insured. The result of the design is a simple control scheme using a linear constant feedforward controller and a nonlinear feedback controller. The performance of the control structure in tracking and regulation are evaluated by simulations.     

Spectral portrait for non-hermitian large sparse matrices

The spectral portrait of a matrix is the picture of its ?-spectra forε ∈[ε ₁,ε ₂], where an ?-spectrum ofA is the union of all the eigenvalues of all the matricesA+Δ with ∥Δ∥₂≤ε∥A∥₂. The spectral portrait is, for example, useful to study the stability of various problems, or, as we illustrate in this paper, to visualize the condition number of an eigenvalue. Some methods to estimate the spectral portrait already exist, but only for small matrices. We propose here a new algorithm for non hermitian large sparse matrices.     

Recursive Least Squares Identification With Variable-Direction Forgetting via Oblique Projection Decomposition

In this paper, a new recursive least squares (RLS) identification algorithm with variable-direction forgetting (VDF) is proposed for multi-output systems. The objective is to enhance parameter estimation performance under non-persistent excitation. The proposed algorithm performs oblique projection decomposition of the information matrix, such that forgetting is applied only to directions where new information is received. Theoretical proofs show that even without persistent excitation, the information matrix remains lower and upper bounded, and the estimation error variance converges to be within a finite bound. Moreover, detailed analysis is made to compare with a recently reported VDF algorithm that exploits eigenvalue decomposition (VDF-ED). It is revealed that under non-persistent excitation, part of the forgotten subspace in the VDF-ED algorithm could discount old information without receiving new data, which could produce a more ill-conditioned information matrix than our proposed algorithm. Numerical simulation results demonstrate the efficacy and advantage of our proposed algorithm over this recent VDF-ED algorithm.      

A globally convergent adaptive pole placement algorithm without a persistency of excitation requirement

This paper presents an indirect adaptive control scheme for deterministic plants which are not necessarily minimum phase. Global convergence is established for the scheme in the sense that the closed-loop poles are asymptotically assigned for the given data sequence and the system input and output remain bounded for all time. A key feature of the scheme is that no persistency of excitation condition is required. The algorithm uses recursive least squares with variable forgetting factor, normalized regression vectors, and a matrix gain with constant trace.     

Faster Algorithms for All-Pairs Small Stretch Distances in Weighted Graphs

Let G=(V,E) be a weighted undirected graph, with non-negative edge weights. We consider the problem of efficiently computing approximate distances between all pairs of vertices in?G. While many efficient algorithms are known for this problem in unweighted graphs, not many results are known for this problem in weighted graphs. Zwick?(J. Assoc. Comput. Mach. 49:289–317, 2002) showed that for any fixed ε>0, stretch 1+ε distances (a path in G between u,v∈V is said to be of stretch t if its length is at most t times the distance between u and v in G) between all pairs of vertices in a weighted directed graph on n vertices can be computed in $\tilde{O}(n^{\omega})$ time, where ω<2.376 is the exponent of matrix multiplication and n is the number of vertices. It is known that finding distances of stretch less than 2 between all pairs of vertices in G is at least as hard as Boolean matrix multiplication of two n×n matrices. Here we show that all pairs stretch 2+ε distances for any fixed ε>0 in G can be computed in expected time O(n ^9/4). This algorithm uses a fast rectangular matrix multiplication subroutine. We also present a combinatorial algorithm (that is, it does not use fast matrix multiplication) with expected running time O(n ^9/4) for computing all-pairs stretch 5/2 distances in?G. This combinatorial algorithm will serve as a key step in our all-pairs stretch 2+ε distances algorithm.     

Robust self-tuning control based on discrete-time sliding mode for auto-regressive mathematical model in the presence of unmodelled dynamics

In this paper, we propose a new robust self-tuning control, called the generalized minimum variance αl-equivalent selftuning control (GMVSTC-αl) for the linear timevarying (LTV) systems, which can be described by the discrete-time auto-regressive exogenous (ARX) mathematical model in the presence of unmodelled dynamics. The estimation of the parameters contained in this mathematical model is made on the basis of the proposed modified recursive least squares (m-RLS) parametric estimation algorithm with dead zone and forgetting factor. The stability analysis of the proposed parametric estimation algorithm m-RLS is treated on the basis of a Lyapunov function. A numerical simulation example is used to prove the performances and the effectiveness of the explicit scheme of the proposed robust self-tuning control GMVSTC-αl.     

The convex-decomposable operator equation and its monotonic inclusive iteration

In this paper we discuss a class of nonlinear operators, that we name convex-decomposable operators, which have the nature:Fy?Fx≤(D ₁ (x, y)+D ₂ (x, y))(y?x) forx≤y, andD ₁ andD ₂ are isotone and antitone respectively. It is shown that it is a rather large class of operators, containing e.g. the operators whose second derivatives are bounded. For these operators we give a monotonic inclusive iterative algorithm, and show the convergence of this algorithm under some simple conditions.     

Optimal control of markov chains admitting strong and weak interactions

This paper is devoted to the study of an optimal control problem for a Markov chain with generator B + εA, where ε is a small parameter. It is shown that an approximate solution can be calculated by a policy improvement algorithm involving computations relative to an ‘aggregated’ problem (the dimension of which is given by N, the number of ergodic sets for the B matrix) together with a family of ‘decentralized’ problems (the dimensions of which are given by the number of elements in each ergodic set for the B matrix).     

<Emphasis Type="Italic">l</Emphasis><Subscript>∞</Subscript> Fuzzy filter design for nonlinear systems with missing measurements: Fuzzy basis-dependent Lyapunov function approach

In this paper, l _∞ fuzzy filtering problem is dealt for nonlinear systems with both persistent bounded disturbances and missing probabilistic sensor information. The Takagi–Sugeno (T–S) fuzzy model is adopted to represent a nonlinear dynamic system. The measurement output is assumed to contain randomly missing data, which is modeled by a Bernoulli distributed with a known conditional probability. To design the l _∞ fuzzy filter and guarantee tracking performance, the effect of the perturbation against persistent bounded disturbances is reduced by using the minimum l _∞ performance. By using the fuzzy basis-dependent Lyapunov function approach, a sufficient condition is established that ensure the mean square exponential stability of the filtering error. The proposed sufficient condition is represented as some linear matrix inequalities (LMIs), and the filter gain is obtained by the solution to a set of LMIs. Finally, the effectiveness of the proposed design method is shown via an example.     

Low complexity adaptive forgetting factor for online sequential extreme learning machine (OS-ELM) for application to nonstationary system estimations

Huang et al. (2004) has recently proposed an on-line sequential ELM (OS-ELM) that enables the extreme learning machine (ELM) to train data one-by-one as well as chunk-by-chunk. OS-ELM is based on recursive least squares-type algorithm that uses a constant forgetting factor. In OS-ELM, the parameters of the hidden nodes are randomly selected and the output weights are determined based on the sequentially arriving data. However, OS-ELM using a constant forgetting factor cannot provide satisfactory performance in time-varying or nonstationary environments. Therefore, we propose an algorithm for the OS-ELM with an adaptive forgetting factor that maintains good performance in time-varying or nonstationary environments. The proposed algorithm has the following advantages: (1) the proposed adaptive forgetting factor requires minimal additional complexity of O(N) where N is the number of hidden neurons, and (2) the proposed algorithm with the adaptive forgetting factor is comparable with the conventional OS-ELM with an optimal forgetting factor.     

Recursive identification of Hammerstein systems with dead-zone nonlinearity in the presence of bounded noise

The existing identification algorithms for Hammerstein systems with dead-zone nonlinearity are restricted by the noise-free condition or the stochastic noise assumption. Inspired by the practical bounded noise assumption, an improved recursive identification algorithm for Hammerstein systems with dead-zone nonlinearity is proposed. Based on the system parametric model, the algorithm is derived by minimising the feasible parameter membership set. The convergence conditions are analysed, and the adaptive weighting factor and the adaptive covariance matrix are introduced to improve the convergence. The validity of this algorithm is demonstrated by two numerical examples, including a practical DC motor case.     

Multivariable adaptive dynamic surface control based on norm estimation of unknown parameter matrices

In this article, a novel output-feedback adaptive dynamic surface control scheme is proposed for linear time-invariant multivariable plants based on the norm estimation of unknown parameter matrices. Besides avoiding the explosion of complexity problem in traditional multivariable backstepping design, the proposed scheme has the following features: (1) only one parameter needs to be updated on-line regardless of the plant order and input–output dimension, (2) only the Hurwitz condition is required for the high-frequency gain matrix and (3) the ?_∞ performance of the tracking error can be guaranteed. It is shown that all signals of the closed-loop system are semi-globally uniformly bounded. Simulation results are given to illustrate the effectiveness of the proposed scheme.     

Hybrid adaptive robust control of static var compensator in power systems

To improve the transient response of an electric power transmission system, a hybrid adaptive robust control method is proposed in this paper for the static var compensator by incorporating the immersion and invariance adaptive (I&I adaptive) and L₂‐gain control. In contrast to the standard I&I adaptive control algorithm, establishing a target system is not required in constructing the robust control law with the proposed method. Thus, the procedure of solving PDEs to satisfy the immersion condition can be avoided. In addition, both parametric and non‐parametric uncertainties, which commonly exist in electric power transmission systems, are considered. The parametric uncertainty induced by the damping coefficient of the system is estimated by the designed adaptive law, which is constructed by ensuring the estimation error converges to zero. The non‐parametric uncertainty is caused by external disturbances and approximation errors in modeling the uncertain structure. By assuming that the L₂‐gain of the system to the non‐parametric uncertainties satisfies a dissipation inequality, we found that the robustness of the controller can be guaranteed. It is proved that all the system states are globally bounded and converge to a new stable equilibrium. Simulation results are also presented to show the effectiveness of the proposed control method in improving the transient response of the system and the convergence speed of the system states. Copyright © 2013 John Wiley & Sons, Ltd.     

State estimation in the presence of bounded disturbances

This contribution proposes a robust recursive algorithm for the state estimation of linear models with unknown but bounded disturbances corrupting both the state and measurement vectors. A novel approach based on state bounding techniques is presented. The proposed algorithm can be decomposed into two steps: time updating and observation updating that uses a switching estimation Kalman-like gain matrix. Particular emphasis will be given to the design of a weighting factor that ensures the stability of the estimation error.     

Bias-compensation-based least-squares estimation with a forgetting factor for output error models with white noise

In this paper, the bias-compensation-based recursive least-squares (LS) estimation algorithm with a forgetting factor is proposed for output error models. First, for the unknown white noise, the so-called weighted average variance is introduced. With this weighted average variance, a bias-compensation term is first formulated to achieve the bias-eliminated estimates of the system parameters. Then, the weighted average variance is estimated. Finally, the final estimation algorithm is obtained by combining the estimation of the weighted average variance and the recursive LS estimation algorithm with a forgetting factor. The effectiveness of the proposed identification algorithm is verified by a numerical example.     

Robust switching-type H∞ filter design for linear uncertain systems with time-varying delay

This paper is concerned with the problem of robust H_∞ filter design for a class of linear uncertain systems with time-varying delay. The uncertainty parameters are supposed to be time-varying, unknown, but bounded, which appear affinely in the matrices of the considered system model. Based on linear matrix inequality (LMI) method and switching laws, a new switching-type filter is designed to guarantee the asymptotic stability and H_∞ performance level of the filtering error systems. The key feature is that the new proposed filter parameters are switching between certain fixed gains automatically via the designed switching law. It is shown that the new filter design method is less conservative than the traditional fixed gain filter design method. An example is given to illustrate the validity of the proposed design.     

Observer design methodology for stochastic and deterministic robustness

A unified observer with stochastic and deterministic robustness is developed in this paper so that an observer is less sensitive to both stochastic and deterministic uncertainties. For stochastic robustness, the norm of the observer gain and the lower bound of the observer decay rate are shown to be design factors which can minimize the upper bound of the estimation error variance. For deterministic robustness, the L ₂ norm-based condition number of the observer eigenvector matrix is utilized to address robust estimation performance against deterministic uncertainties. In order to justify the proposed method, a graphical approach is first introduced, and then a multi-objective optimization problem including linear matrix inequality constraints is formulated to provide the unified robustness.     

On mitigating hotspots to maximize network lifetime in multi-hop wireless sensor network with guaranteed transport delay and reliability

In wireless sensor networks (WSNs), sensor nodes close to the sink consume more energy than others because they are burdened with heavier relay traffic destined for the sink and trend to die early, forming hotspots or energy holes in WSNs. It has a serious impact on network lifetime. In this paper, three optimization algorithms are proposed to mitigate hotspots and prolong network lifetime for adaptive Mary Phase Shift Keying (MPSK) based wireless sensor networks while transport delay and reliability can be still guaranteed. Based on the insight gained into the relationship between nodal data load and energy consumption in different regions, the first algorithm (GlobalSame) can extend considerably the network lifetime by selecting the optimal nodal transmission radius r, bit error rate ε and transmission rate allocations in bits per symbol (BPS) τ. The second algorithm (RingSame) can further improve network lifetime by comparison to the GlobalSame algorithm, which by selecting different ε _i and τ _i for nodes in different regions under constraints of total BER and transport delay . While the third algorithm (NodeDiff) can further improve the network lifetime by adopting different BER ε and BPS τ parameters of the same node for data packets received according to its distance to the sink. Extensive simulation studies show that our algorithms do considerably prolong the network lifetime.     

On a word avoiding near repeats

In this paper we construct an infinite binary word w with the following property: the minimal distance among two occurrences of a same factor of length n cannot be polynomially upperbounded. In particular, for all positive ε the number of distinct factors of w with exponent larger than 1+ε is finite.