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This paper discusses the consensus tracking problem for a class of nonlinear multi-agent systems (MASs) with output constraints, unmodeled dynamics, nonsymmetric input dead-zones and actuator failures under directed graphs, and proposes an adaptive cooperative neural dynamic surface control (DSC) strategy. Using the properties of invertible nonlinear mapping and Gaussian function, output constraints and non-strict feedback terms are separately dealt with. A measurable dynamic signal produced by an auxiliary first-order system is used to eliminate the influence of unmodeled dynamics on the system. Two input models of input dead-zone and actuator failure are linearized, each follower control signal is constructed via DSC. All the signals of the closed-loop system are proved to be cooperative semi-globally uniformly ultimately bounded, and all the followers can accomplish a desired consensus results. Finally, the simulation results are provided to illustrate the availability of the presented adaptive control approach.

The main purpose of this paper is to obtain some exact computational formulas or upper bounds for hybrid mean value involving Hardy sums $S_{3}(h, p)$ and general Kloosterman sums $K(r, l, \lambda; p)$. By applying the properties of Gauss sums and the mean value theorems of Dirichlet $L$-function, we derive some new identities. As the special cases, we also deduce some exact computational formulas for hybrid mean value involving $S_{3}(h, p)$ and classical Kloosterman sums $K(n, p)$.

In this paper, an adaptive quantized fuzzy dynamic surface control (DSC) stratergy is investigated for a class of nonlinear systems with input unmodeled dynamics and output constraints. A challenge lies in that the input-quantized actuator is considered to possess both unknown control gain and nonlinear input unmodeled dynamics. By quantized controller design and DSC technique, combining normalized signal, integral Lyapunov functions, Nussbaum functions and the adaptive laws, the obstacle caused by quantization and multiple uncertainties is effectively overcome. The designed novel quantizer has the advantages of both the existing uniform quantizer and hysteresis quantizer, which can avoid the chattering and reduce the quantization error no matter the control volume is large or small. By defining a group of transformation based on a logarithmic one to one mapping, time-varying output constraints is satisfied. It is shown that all the signals are bounded, and the output signal is constrained within the preset range.

This study proposes an adaptive quantised fuzzy control strategy for a class of non-linear systems with time-varying delay, state constraints, and input unmodelled dynamics. Design difficulties focus on the fact that the input-quantised actuator possesses both unknown control gain and non-linear input unmodelled dynamics, and all the states are required to satisfy state constraints. To remove these obstacles, integral barrier Lyapunov functions combined with Nussbaum functions are designed to deal with the unknown control gains and state constraints, and a quantised controller combined with the normalised signal is developed to tackle the quantised input and input unmodelled dynamics. To avoid the chattering and reduce the quantisation error in a wide scope of the control volume, a novel logarithmic uniform hysteresis quantiser is employed, which has both advantages of the existing uniform quantiser and hysteresis quantiser. Two simulation examples of practical control systems are conducted to demonstrate the effectiveness of the proposed control protocol.

In this paper, finite time adaptive neural network (NN) tracking control is investigated for a class of uncertain nonstrict feedback systems with unknown dead-zone and unmodeled dynamics as well as full state restrictions. Using the linearized representation of dead-zone, finite-time adaptive control is presented by the aid of dynamic surface control (DSC) technique and the property of Gaussian function. By introducing logarithmic function as invertible nonlinear mapping, the full state restriction problem is solved. The dynamic signal generated by the low-pass filter is used to handle unmodeled dynamics. Through finite-time stability theory and introducing the bounded closed set in the proof, all the signals in the closed-loop system are proved to be semi-globally practical finite time stable (SGPFS). The restriction conditions are not triggered for all the states. Finally, two numerical examples are used to illustrate the feasiability of the finite time adaptive control strategy.

In this paper, the issue of adaptive neural event-triggered control (ETC) is studied for uncertain block-structure multi-input multi-output (MIMO) constrained non-affine nonlinear systems with unmodeled dynamics. A dynamic signal produced by the auxiliary system based on the property of unmodeled dynamics is employed to solve the dynamical disturbances. The unknown continuous function obtained at each step of recursion is estimated by using radial basis function neural networks (RBFNNs). Utilizing logarithmic function as an invertible mapping, the uncertain constrained MIMO non-affine system is changed into a novel unconstrained block-structure MIMO nonaffine system. Using improved dynamic surface control (DSC) strategy, adaptive event-triggered control scheme is developed for the transformed non-affine system based on relative threshold mechanism. According to the Lyapunov method, all the signals in the closed-loop system are shown to be semi-globally uniformly ultimately bounded (SGUUB). Output constraint requirements are not triggered, and Zeno behavior is avoided. A constrained pure-feedback system and a kind of 2-DOF flexible manipulator system are used to illustrate the theoretical findings.

This paper presents an improved adaptive Kalman filter (AKF) to obtain state and fault estimation for linear time-varying (LTV) discrete stochastic systems. In the stochastic framework, the fault is modeled as parameter change, then the improved AKF is introduced to estimate state and fault. The exponential convergency of the improved AKF is proved. Numerical simulation is used to verify the proposed approach.

In this article, the problem of backstepping-based adaptive neural control is discussed for nonlinear systems in strict-feedback form with full state restrictions and dynamical uncertainties. The constrained affine nonlinear system is transformed into a nonaffine nonlinear system without state restrictions by constructing invertible asymmetric nonlinear mapping. The uncertainties are disposed of by the aid of first-order auxiliary dynamic system. Based on modified backstepping design and using the property of Gaussian function, an improved adaptive backstepping control scheme is proposed for the novel unconstrained nonaffine nonlinear systems. The computation complex generated by the conventional backstepping design and mean value theorem is avoided. All signals of the whole system are proved to be semi-globally uniformly ultimately boundedness (SGUUB).

This article focuses on the issue of model reference adaptive control (MRAC) for first-order linear time-invariant systems (LTIS) with output constraints and unknown gain sign. The completely unknown control coefficient is handled based on Nussbaum function. The output constraints are effectively dealt with by introducing barrier Lyapunov function (BLF). With the help of the Lyapunov synthesis approach, the updating laws of unknown parameters are determined. With the aid of Babalat’s lemma and the property of BLF, the tracking error is proved to converge asymptotically to zero, and output restrictions are not triggered. Simulation findings are used to verify the effectiveness of the proposed MRAC algorithm.

A novel event-triggered anti-disturbance PI control strategy for nonlinear systems is studied in this essay. Firstly, the general method of event-triggered control is extended, and the event-triggered mechanism is leaded into the control system. Secondly, the designed observer is used to estimate the external disturbances. Then, by combining the state feedback with disturbance estimation, a PI-type dynamic tracking controller is introduced to ensure that the output tracking error approaches zero. What’s more, the Lyapunov function model is established, and the relevant theorem is proved. In addition, by solving the lower bound of the minimum trigger time interval to be greater than zero, it is guaranteed that the Zeno phenomenon in the event-triggered mechanism can be avoided. Finally, the A4D model is simulated, which proves the algorithm is effective.

In this paper, the problem of adaptive neural control is discussed for multi-input multi-output (MIMO) strict feedback nonlinear systems with time-varying output constraints and unmodeled dynamics. Using the composite function of hyperbolic tangent function as a nonlinear mapping, the original system is transformed into a novel unconstrained MIMO strict-feedback system with uncertain terms. A auxiliary dynamic signal is designed to estimate the dynamical uncertainties in the novel system. Based on the transformed system, adaptive neural controller is constructed by using modified dynamic surface control(DSC) technique. Using the defined compact set in stability analysis and Lyapunov function method, all signals in the closed-loop system are proved to be the semi-global uniform ultimate boundedness (SGUUB). Each element of the output vector can strictly abide by the time-varying constraint conditions.

Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums K ( r , l , &lambda, p ) and certain Hardy sums S 1 ( h , q ) &sum, m = 1 p - 1 &sum, s = 1 p - 1 K ( m , n , &lambda, p ) K ( s , t , &lambda, p ) S 1 ( 2 m s &macr, p ) , &sum, s = 1 p - 1 | K ( m , n , &lambda, p ) | 2 | K ( s , t , &lambda, p ) | 2 S 1 ( 2 m s &macr, p ) . Our results not only cover the previous results, but also contain something quite new. Actually the previous authors just consider the case of the principal character &lambda, modulo p, while we consider all the cases.

This paper is concerned with adaptive output feedback tracking control for a class of uncertain nonlinear systems with quantized input and unmodeled dynamics as well as output constraints. Quantized reduced-order K-filters are designed to observe parts of unmeasured states. The considered system is of two types of uncertainties. To deal with these uncertainties, we adopts global exponential stability technique of the state unmodeled dynamics with a Lyapunov description and the fuzzy approximation approach to those uncertain functions. To avoid the chattering of the quantized control signal, a hysteretic quantizer is introduced, and by designing a novel control law based on dynamic surface control (DSC) method, many assumptions of the quantized system in early literatures are removed. Combining quantized DSC with an one to one mapping from output error to the first dynamic surface, the robust quantized adaptive controller is constructed to guarantee that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB), and the tracking error is restricted within the prescribed output constraints. A numerical simulation example shows the effectiveness of the proposed approach.

In this paper, adaptive neural dynamic surface control(DSC) is developed for a class of constrained strict-feedback nonlinear systems with input unmodeled dynamics. By introducing a one to one nonlinear mapping, the output constrained strict-feedback system in the presence of unmodeled dynamics is transformed into a novel unconstrained strict-feedback system. Neural networks (NNs) are employed to approximate unknown nonlinear continuous functions. A normalization signal and an updating parameter are used to handle the uncertain term which input unmodeled dynamics brings about in the design final step. By adding the normalization signal to the whole Lyapunov function and using the defined compact set in stability analysis, all the signals in the closed-loop system are proved to be semi-globally uniformly ultimately bounded (SGUUB), and output constraint is not violated. Two numerical examples are used to illustrate the effectiveness of the proposed adaptive DSC method in handling input unmodeled dynamics.

This paper is concerned with the adaptive control problem of a class of output feedback nonlinear systems with unmodeled dynamics and output constraint. Two dynamic surface control design approaches based on integral barrier Lyapunov function are proposed to design controller ensuring both desired tracking performance and constraint satisfaction. The radial basis function neural networks are utilized to approximate unknown nonlinear continuous functions. K-filters and dynamic signal are introduced to estimate the unmeasured states and deal with the dynamic uncertainties, respectively. By theoretical analysis, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded, while the output constraint is never violated. Simulation results demonstrate the effectiveness of the proposed approaches.

In this paper, the problem of adaptive neural network (NN) dynamic surface control (DSC) is discussed for a class of strict-feedback nonlinear systems with full state constraints and unmodeled dynamics. By introducing a one to one nonlinear mapping, the strict-feedback system with full state constraints is transformed into a novel pure-feedback system without state constraints. Radial basis function (RBF) neural networks (NNs) are used to approximate unknown nonlinear continuous functions. Unmodeled dynamics is dealt with by introducing a dynamical signal. Using modified DSC and introducing integral-type Lyapunov function, adaptive NN DSC is developed. Using Young’s inequality, only one parameter is adjusted at each recursive step in the design. It is shown that all the signals in the closed-loop system are semi-global uniform ultimate boundedness (SGUUB), and the full state constraints are not violated. Simulation results are provided to verify the effectiveness of the proposed approach.

By using the properties of Kloosterman sum and Dirichlet character, an optimal upper bound estimate of incomplete Cochrane sum is given.

In this paper, a model reference adaptive control scheme is proposed by using Nussbaum function for first order linear system with unknown gain sign. Only one self-tuning parameter is designed. Based on Lyapunov method, the Nussbaum parameter is determined. Using Nussbaum function, unknown gain sign is handled. According to Babalat’s lemma, the tracking error is proved to converge to zero. A numerical example is provided to demonstrate the effectiveness of the proposed method.

Let q ≥ 2 $q\geq2$ be a fixed integer, A = A ( q ) ≤ q $A=A(q)\leq q$ , B = B ( q ) ≤ q $B=B(q)\leq q$ , and H = H ( q ) ≤ q $H=H(q)\leq q$ . Define ħ ( A , B , H ) = { a ∈ Z ∣ ( a , q ) = 1 , a b ≡ 1 ( mod q ) , 1 ≤ a ≤ A , 1 ≤ b ≤ B , | a − b | ≤ H } . $$\hbar(A,B,H)= \bigl\{ a\in\mathbb{Z}\mid (a,q)=1, ab\equiv1\pmod{q}, 1\leq a\leq A, 1\leq b\leq B, \vert a-b\vert \leq H \bigr\} . $$ With the aid of the estimates for the general Kloosterman sums and the properties of trigonometric sums, we obtain an upper bound of the general partial Gaussian sums over the number set ħ ( A , B , H ) $\hbar(A,B,H)$ .

Text By using the properties of Gauss sums and the mean value theorem of the Dirichlet L-function, a hybrid mean value problem involving certain Hardy sum and Kloosterman sum is studied. Two exact computational formulae are given, through which the cancellation phenomenon is revealed. Video For a video summary of this paper, please visit https://youtu.be/d-X81xErU7Q .

In this technical note, $H_{\infty}$ control problem of continuous Markovian jump systems is investigated. A linear feedback control scheme, combined with a state observer design, is proposed in the form of linear matrix inequalities, which can ensure the systems' mean-square stability with $H_{\infty}$ performance. Then, a multi-step state transition conditional probability function is introduced for the continuous Markovian process, which is used to define the system's mean-square stability probability. Furthermore, the formulas for calculating the mean-square stability probability are derived for situations where the control force may not be strong enough to ensure the full stability. Simulation results are presented to show the effectiveness of the theoretical results.

In this paper, an adaptive dynamic surface output feedback control scheme is proposed for a class of nonlinear systems with unmodeled dynamics and output constraint. A description based on Lyapunov function for unmodeled dynamics is proposed to handle dynamic uncertainties. An one to one nonlinear mapping is introduced to solve output constrained problem. K-filters is introduced to estimate the unmeasured states. The radial basis function neural networks are utilized to approximate unknown nonlinear continuous function. It is shown that all the signals in the closed-loop system are bounded, and the output constraint is not violated. Simulation results demonstrate the effectiveness of the proposed approach.

Let k be a positive integer and n a nonnegative integer, 0 < λ 1, ..., λ k+1 ≤ 1 be real numbers and w = (λ 1, λ 2, ..., λ k+1). Let $$q \geqslant \max \left\{ {\left[ {\frac{1} {{\lambda _i }}} \right]:1 \leqslant i \leqslant k + 1} \right\}$$ be a positive integer, and a an integer coprime to q. Denote by N(a, k,w, q,n) the 2n-th moment of (b 1 ... b k − c) with b 1 ...b k c ≡ a (mod q), 1 ≤ b i ≤ λ i q (i = 1, ..., k), 1 ≤ c ≤ λ k+1q and 2 ∤ (b 1 + ... + b k + c). We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N(a, k,w, q, n), which generalized the result of Zhang. Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N(a, k,w, q, n) in the case of $$w = \left( {\tfrac{1} {2},\tfrac{1} {2}, \ldots ,\tfrac{1} {2}} \right)$$ and n = 0. In order to show our result is close to the best possible, the mean-square value of $$N\left( {a,k,q} \right) - \tfrac{{\varphi ^k (q)}} {{2^{k + 2} }}$$ and the mean value weighted by the high-dimensional Cochrane sum are studied too.

The main purpose of this paper is using the analytic methods to study the upper bound of the hyper Cochrane sum, and obtain an interesting result which generalized the main theorem of Liu [2].

By using the analytic methods, the mean value of the generalquadratic Gauss sums weighted by the first power mean of character sumsover a short interval is investigated. Several sharp asymptotic formulaeare obtained, which show that these sums enjoy good distributive prop-erties. Moreover, interesting connections among them are established. 1. Introduction and main resultsFor any integer n, the general quadratic Gauss sums G(n,χ;q) is defined asG(n,χ;q) =X qa=1 χ(a)e na 2 q ,where e(y) = e 2πiy . This summation is very important, since it is the general-ization of the classical quadratic Gauss sums. But we still know little aboutthe properties of G(n,χ;q), we do not even know how large G(n,χ;q) is. Sincethe value of |G(n,χ;q)| is irregular as χ varies, one can only get some upperbound estimates. For example, for any integer n with (n,q) = 1, from thegeneral result of Cochrane and Zheng [1] we can deduce that|G(n,χ;q)| ≤ 2 ω(q) q 12 ,where ω(q) denotes the number of distinct prime divisors of q. The case thatq is prime is due to A. Weil [2].However, weighted sums [4] involving G(n,χ;q) enjoys many good valuedistribution properties, through which interesting connections among them areestablished. Now we shall use analytic methods to study the mean value of thegeneral quadratic Gauss sums weighted by the first power mean of character

Using dynamic surface control method, an adaptive output feedback prescribed performance tracking control scheme is proposed for a class of uncertain nonlinear systems with input unmodeled dynamics and dead-zone in this paper. Reduced-order K-filters are designed to observe unmeasured states. Input dead-zone characteristic is considered on the basis of the control input having unmodeled dynamics. A normalization signal is employed to deal with the negative impact of nonlinear input unmodeled dynamics, and a transformation of tracking error is designed to implement constrained performance tracking control. The state unmodeled dynamics is described by Lyapunov function based on global exponential stability. By the theoretical analysis, the transient tracking performance is guaranteed, and the signals in the closed-loop system are proved to be semi-globally uniformly ultimately bounded. A numerical simulation example shows the effectiveness of the proposed approach.

In this note, an adaptive control scheme is developed for the time-delayed nonlinear Markovian jump systems. The explosion of complexity in traditional backstepping design is avoided by utilizing dynamic surface control. Filters are constructed to provide the proper auxiliary signals. Neural networks are employed to estimate the unknown continuous functions. A novel Lyapunov function is proposed, which contains the stochastically jumping parameters. New features of neural networks approximation in Markovian jump systems are revealed. It is proved that the closed-loop system is bounded in probability. The theoretic result is illustrated through a simulation example.

This paper deals with the problem of tracking control for a class of high order nonlinear systems with input delay. The unknown continuous functions of the system are estimated by fuzzy logic systems (FLS). A state conversion method is introduced to eliminate the delayed input item. By means of the backstepping algorithm, the property of semi-globally uniformly ultimately bounded (SGUUB) of the closed-loop system is achieved. The stability of the closed-loop system is proved according to Lyapunov second theorem on stability. The tracking error is proved to be bounded which ultimately converges to an adequately small compact set. Finally, a computer simulation example of high order nonlinear systems is presented, which illustrates the effectiveness of the control scheme.

The tracking control problem is studied for a class of uncertain non-affine systems. Based on the principle of sliding mode control (SMC), using the neural networks (NNs) and the property of the basis function, a novel adaptive design scheme is proposed. A novel Lyapunov function, which depends on both system states and control input variable, is used for the development of the control law and the adaptive law. The approach overcomes the drawback in the literature. In addition, the lumped disturbances are taken in account. By theoretical analysis, it is proved that tracking errors asymptotically converge to zero. Finally, simulation results demonstrate the effectiveness of the proposed approach.

In this paper, adaptive variable structure neural control is presented for a class of uncertain multi-input multi-output (MIMO) nonlinear systems with state time-varying delays and unknown nonlinear dead-zones. The unknown time-varying delay uncertainties are compensated for using appropriate Lyapunov-Krasovskii functionals in the design. The approach removes the assumption of linear function outside the deadband without necessarily constructing a dead-zone inverse as an added contribution. By utilizing the integral-type Lyapunov function and introducing an adaptive compensation term for the upper bound of the residual and optimal approximation error as well as the dead-zone disturbance, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded. In addition, a modified adaptive control algorithm is given in order to avoid the high-frequency chattering phenomenon. Simulation results demonstrate the effectiveness of the approach.

The main purpose of this paper is using the analytic methods to study the hybrid mean value involving the character sums, general quadratic Gauss sums and general Kloosterman sums, and give several interesting mean value formulae.

The main purpose of this paper is to use the properties of the Gauss sums, primitive characters and the mean value of Dirichlet L-functions to study the hybrid mean value of the error term E(n, 1, c, q) and the hyper-Kloosterman sums K(h, n+1, q), the asymptotic property of the mean square value Σ =1 E 2(n, 1, c, p), and give two interesting mean value formulae.

In this paper, adaptive neural control is proposed for a class of uncertain multi-input multi-output (MIMO) nonlinear state time-varying delay systems in a triangular control structure with unknown nonlinear dead-zones and gain signs. The design is based on the principle of sliding mode control and the use of Nussbaum-type functions in solving the problem of the completely unknown control directions. The unknown time-varying delays are compensated for using appropriate Lyapunov-Krasovskii functionals in the design. The approach removes the assumption of linear functions outside the deadband as an added contribution. By utilizing the integral Lyapunov function and introducing an adaptive compensation term for the upper bound of the residual and optimal approximation error as well as the dead-zone disturbance, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded. Simulation results demonstrate the effectiveness of the approach.

This paper discusses the fault diagnosis problem for a class of non-Gaussian stochastic processes with unknown fault. By using the spline function and T-S model simultaneously, the probability density function (PDF) control problem can be transformed into the control problem of T-S fuzzy weight dynamics. In this framework, an adaptive fuzzy filter based on output PDF is designed to estimate the size of system fault. Meanwhile, in order to solve the bounded problem of the fault, an adaptive projection algorithm is applied into adjust the estimated value for the fault. As a result, the satisfactory stability and fault diagnosis ability can be guaranteed by rigorous theoretical proof.

In this paper, an adaptive output feedback neural control strategy is investigated for a class of nonlinear systems with states and input unmodeled dynamics based on dynamics surface control method. States unmodeled dynamics is described by introducing a kind of Lyapunov function, and the nonlinear input unmodeled dynamics is dealt with by using a normalization signal. The unknown control gain sign is solved with the help of Nussbaum function. By the theoretical analysis, all the signals in the closed-loop system are proved to be semi-globally uniformly ultimately bounded. A numerical example is provided to illustrate the effectiveness of the proposed approach.

An adaptive neural output feedback control scheme is investigated for a class of stochastic nonlinear systems with unmodeled dynamics and unmeasured states. The unmeasured states are estimated by K-filters, and unmodeled dynamics is dealt with by introducing a novel description based on Lyapunov function. The neural networks weight vector used to approximate the black box function is adjusted online. The unknown nonlinear system functions are handled together with some functions resulting from theoretical deduction, and such method effectively reduces the number of adaptive tuning parameters. Using dynamic surface control (DSC) technique, Itô formula, and Chebyshev’s inequality, the designed controller can guarantee that all the signals in the closed-loop system are bounded in probability, and the error signals are semiglobally uniformly ultimately bounded in mean square or the sense of four-moment. Simulation results are provided to verify the effectiveness of the proposed approach.

The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the r-th hyper-Kloosterman sums Kl(h;k+1;r;q) and the hyper Cochrane sums C(h;q;m;k); and give an interesting mean value formula.

This paper is concerned with the problem of adaptive neural tracking control for a class of uncertain pure-feedback nonlinear systems. Using the implicit function theorem and backstepping technique, a practical robust adaptive neural control scheme is proposed to guarantee that the tracking error converges to an adjusted neighborhood of the origin by choosing appropriate design parameters. In contrast to conventional Lyapunov-based design techniques, an alternative Lyapunov function is constructed for the development of control law and learning algorithms. Differing from the existing results in the literature, the control scheme does not need to compute the derivatives of virtual control signals at each step in backstepping design procedures. Furthermore, the scheme requires the desired trajectory and its first derivative rather than its first n derivatives. In addition, the useful property of the basis function of the radial basis function, which will be used in control design, is explored. Simulation results illustrate the effectiveness of the proposed techniques. Refereed/Peer-reviewed

The high-dimensional D. H. Lehmer problem over quarter intervals is studied. By using the properties of character sum and the estimates of DirichletL-function, the previous result is improved to be the best possible in the case ofq = p, an odd prime withp≡1(mod 4), which is shown by studying the mean square value of the error term.

This paper investigates the adaptive control problem for a class of nonlinear systems using Lyapunov's. stability theory. The proposed method is designed based on a modified Lyauunov function and removes the possible controller singularity problem in some of the existing adaptive control schemes through feedback linearization techniques. The resulting closed-loop system is proved to be globally stable and the tracking error converges to a small neighbourhood of zero. An example is given to illustrate the effectiveness of the proposed controller.

A decentralized adaptive fuzzy control scheme for a class of large-scale nonlinear systems is proposed in this paper. The design is based on the principle of sliding mode control and the approximation capability of fuzzy systems. The control architecture employs decentralized fuzzy systems to adaptively compensate for plant uncertainties. By using the Lyapunov function method, the control algorithm is proved to be globally stable, with tracking errors converging to a neighborhood of zero.

Based on the integral-type Lyapunov function and the approximation capability of neural networks, an adaptive dynamic surface control scheme is proposed for a class of stochastic strict-feedback nonlinear systems in this paper. The design makes the approach of dynamic surface control be extended to the stochastic nonlinear systems, and relaxes the extent of application of the dynamic surface control approach. By introducing the first order filter, the explosion of complexity caused by the repeated differentiations of certain nonlinear functions such as virtual controls in traditional backstepping design is avoided. Compared with the existing literature, the proposed approach reduces the number of adjustable parameters effectively. By theoretical analysis, it is shown that all signals in the closed-loop system are bounded in probability.

In this paper, the shaping tracking problem based on output probability density functions (PDFs) is studied for a class of discrete-time non-Gaussian stochastic processes. Following B-spline approximation for the output PDFs, the discrete-time PI control input is constructed to solve the tracking control problem for complex nonlinear weight dynamics with multiple unknown uncertainties and state delays. Moreover, by designing the new convex linear matrix inequalities (LMIs) based on delay-dependent, the satisfactory delay-dependent stability, robustness, dynamic tracking performance as well as state constraint can be achieved simultaneously. Finally, a simulation example is presented to illustrate the effectiveness of the proposed approach.

Based on backstepping design, a novel adaptive tracking control scheme is proposed for a class of strict feedback nonlinear systems with unmodeled dynamics and completely unknown function control gain in this paper. An available dynamic signal is used to dominate the unmodeled dynamics. The unknown virtual control gain signs are dealt with using the property of Nussbaum function. The controller singularity problem is avoided using integral Lyapunov function. By theoretical analysis, the closed-loop systems is proved to be semi-global uniformly ultimately bounded.

In this paper, the novel probability density function (PDF) shaping control problem is studied for a class of non-Gaussian stochastic system with time-delay and multi-uncertainties. After the B-spline expansion is applied to approach the output PDF, the PDF shaping control problem is transformed to the control of dynamical weights vector. Furthermore, the PI tracking control algorithm is proposed for weighting systems with time delay and multi-uncertainties, such that the dynamic system can satisfy multi-objective delay-dependent control demands including stability, robustness, tracking performance and state constraint. Compared with previous works, the method weakens the conservative performance and improves the robustness of the system based on L 1 performance index. A simulation example is presented to illustrate the effectiveness of the proposed approach.

Learning is an important capability for an individual robot, which provides an effective way for understanding, planning, and decision-making in a complex environment. For robot motion control, a local weighted k-nearest neighbors states selection method based on environment information and task information is presented. Based on this method, TD reinforcement learning algorithm is combined to reduce the misclassified probability of kNN-TD method, which is finally verified by the simulations.

Based on the approximation capability of radial basis neural networks and the integral-type Lyapunov function, adaptive dynamic surface control(DSC) is investigated for a class of strict-feedback nonlinear systems with unknown virtual control gain functions. The main advantages of the proposed scheme are that only one parameter is adjusted in the whole backstepping design by using Young's inequality and dynamic surface control, and the computational burden is effectively alleviated. By theoretical analysis, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded, with arbitrary small tracking error by appropriately choosing design constants. Simulation results demonstrate the effectiveness of the proposed method.

In this paper, a novel adaptive neural network (NN) dynamic surface control(DSC) is developed for a class of strict-feedback nonlinear systems with unknown virtual control gain functions. The explosion of complexity in traditional backstepping design is avoided by utilizing dynamic surface control and introducing integral-type Lyapunov function. Using Young's inequality, only one parameter is adjusted at each recursive step in the backstepping design. It is shown that the proposed design method is able to guarantee semi-global uniform ultimate boundedness of all signals in the closed-loop system, with arbitrary small tracking error by appropriately choosing design constants. Simulation results verify the effectiveness of the approach.

Based on the principle of sliding mode control, an adaptive control scheme is proposed for a class of multi-input multi-output (MIMO) nonlinear time-varying delay systems with unknown dis urbances. Neural networks are used to approximate the unknown nonlinear functions. By choosing appropriate Lyapunov-Krasovskii functionals, unknown time-varying delay uncertainties are compensated for. By utilizing the integral-type Lyapunov function and Young's inequality, the restrictions of the control gain and the assumptions of time-varying delay uncertainties are relaxed. By theoretical analysis, it is shown that all the signals in the closed-loop system are uniformly ultimately bounded. Finally, simulation results are provided to illustrate the effectiveness of the proposed approach.

In this paper, robust adaptive control is proposed for a class of time delay nonlinear systems with unknown constant control gain and control direction. The design is based on the principle of sliding mode control and the use of Nussbaum-type function in solving the problem of the completely unknown control gain. The unknown function and the upper bound of the unknown time delay term are assumed to be in the form of nonlinear functions with unknown coefficients. The uncertain time delay terms are compensated for using appropriate Lyapunov- Krasovskii functional and Young's inequality in the design. The closed-loop control system is proved to be semi-globally uniformly ultimately bounded. Simulation results demonstrate the effectiveness of the approach.