This paper presents an **adaptive** **PI** **Hermite** **neural** **control** (APIHNC) system for multi-input multi-output (**MIMO**) **uncertain** **nonlinear** **systems**. The proposed APIHNC system is composed of a **neural** controller and a robust compensator. The **neural** controller uses a three-layer **Hermite** **neural** network (HNN) to online mimic an ideal controller and the robust compensator is designed to eliminate the effect of the approximation error introduced by the **neural** controller upon the system stability in the Lyapunov sense. Moreover, a proportional–integral learning algorithm is derived to speed up the convergence of the tracking error. Finally, the proposed APIHNC system is applied to an inverted double pendulums and a two-link robotic manipulator. Simulation results verify that the proposed APIHNC system can achieve high-precision tracking performance. It should be emphasized that the proposed APIHNC system is clearly and easily used for real-time applications.

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Abstract – Most practical **systems** have multiple inputs and multiple outputs, and the applicability of **neural** networks as practical **adaptive** identifiers and controllers will eventually be judged by their success in multivariable problems. In this paper, we design a model following **adaptive** controller for a class of a discrete time multivariable **nonlinear** **systems**. Radial Basis Function (RBF) **neural** network with Minimal Resource Allocation Network (MRAN) training algorithm is used for off-line stable identification. It implements a stable model following **adaptive** controller by utilizing the identification results. S imulation results demonstrate the proposed controller can drive unknown **MIMO** **nonlinear** **systems** to follow the desired trajectory very well.

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Finite-time **control** has received much attention because it can provide many beneﬁts such as strong robustness and better disturbance resistance capability [3, 4, 61]. The Lya- punov theory of ﬁnite-time stability for **nonlinear** **systems** has been clearly established by several authors [62, 63]. It is necessary to point out that the **nonlinear** functions in these **systems** all meet the linear growth condition. However, in practice, the **nonlinear** functions are often completely unknown for the constraints of the modeling method or unknown dynamic disturbances. In this case, the linear growth condition might not be satisﬁed. To eliminate this limitation, a new ﬁnite-time stability criterion was proposed in [64]. However, the controller proposed in [64] cannot be applied to the **nonlinear** system with unmodeled dynamics. In other words, there is still some room for improvement in making the ﬁnite-time **control** scheme implemented more eﬃciently. These facts moti- vate us to provide a new ﬁnite-time **adaptive** backstepping **control** scheme for **uncertain** **nonlinear** system with unmodeled dynamics. In contrast with the existing literature, the **control** scheme in this note oﬀers the following beneﬁts.

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The contents of the paper are as follows. In Section 2, we provide mathematical preliminaries on nonnegative dynamical **systems** that are necessary for developing the main results of this paper. In Section 3, we develop new Lyapunov-like theorems for partial boundedness and partial ultimate boundedness for **nonlinear** dynamical **systems** necessary for obtaining less conservative ultimate bounds for neuroadaptive controllers as compared to ultimate bounds derived using classical boundedness and ultimate boundedness notions. In Section 4, we present our main neuroadaptive **control** framework for **adaptive** set-point regulation of **nonlinear** **uncertain** nonnegative and compartmental **systems**. In Section 5, we extend the results of Section 4 to the case where **control** inputs are constrained to be nonnegative. Finally, in Section 6 we draw some conclusions.

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In this chapter, stable direct and indirect **adaptive** fuzzy controllers for a class of **MIMO** **nonlinear** **systems** with **uncertain** model dynamics are presented. In the direct scheme, fuzzy **systems** are used to construct adaptively an unknown ideal controller and their adjustable parameters are updated by using the gradient descent method in order to minimize the error between the unknown controller and the fuzzy controller. In the indirect scheme, the controller design is based on the approximation of the system’s unknown nonlinearities by using fuzzy **systems**. The free parameters of the used fuzzy **systems** in this case are updated using a gradient descent algorithm that is designed to minimize the identification error between the unknown nonlinearities and their **adaptive** fuzzy approximations. Both approaches do not require the knowledge of the mathematical model of the plant, guarantee the uniform boundedness of all the signals in the closed-loop system, and ensure the convergence of the tracking errors to a neighbourhood of the origin. Simulation results for direct **adaptive** **control** scheme performed on a two-link robot manipulator illustrate the method. Future works will focus on extension of the approach to more general **MIMO** **nonlinear** **systems** and its improvement by introducing a state observer to provide an estimate of the state vector.

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In recent years, many **control** schemes have been proposed to accommodate actuator failures; see, for example, [–]. By applying backstepping technique for the linear sys- tems, a systematic actuator failure compensation **control** was presented in []. Then, in [] the proposed **control** method was extended to **nonlinear** **systems** with actuator fail- ures; in [] the problem of accommodating actuator failures was investigated for a lass of **uncertain** **nonlinear** **systems** with hysteresis input as a follow-up extension. In practice, the failure pattern in an actuator may change repeatedly, which makes failure parameters suﬀer from an inﬁnite number of jumps. Consequently, the considered Lyapunov function would experience inﬁnite number of jumps. In [], this problem was addressed by apply- ing a new tuning function under the frame of **adaptive** **control**. However, the proposed **control** strategy can only apply to the strict-feedback **systems**.

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The traditional method for obtaining fault information for non-linear **systems** is the fault diagnosis approach, which includes the procedures of fault detection and fault isolation (FDI). Many FDI methods involve residual generator designs, as well as isolation filters for fault location [4]–[9]. In contrast to these classical methods fault estimation (FE) directly reconstructs the fault shape (the magnitude with respect to time) without any of the aforementioned complex designs. The FE signals are thus conveniently available for use in a compensation scheme to robustly compensate the fault effects within all **control** loops. Significant literature on the subject of FE design methods for Lipschitz non-linear FTC **systems** has been established, e.g., for the **adaptive** observer (AO) [10], [11], the sliding mode observer (SMO) [12], the extended state observer (ESO) [13], and the non-linear unknown input observer (NUIO) [14]–[16]. However, in the AO faults are estimated with zone convergence, and a proportional-integral (**PI**) structure with carefully chosen learning rate is implemented for time-varying fault estimation. The canonical form SMO proposed in [12] requires several state transformations as well as a priori knowledge of the fault upper bounds. The ESO reconstructs the faults in polynomial form with an assumption of their orders. The NUIO approach can obtain asymptotic state and fault estimations with a comparatively simple design. Nevertheless, the NUIOs proposed in [14] and [15] are designed with rank requirement on system coefficient matrices in order to decouple the disturbance completely, which limits the applicability to real **systems**. Although [16] releases this rank requirement by considering partially decoupled disturbance, the effect of system uncertainties are not taken into account. A novel NUIO without rank requirement for Lipschitz non-linear **systems** subject to faults and both disturbance and uncertainty is of great interest in this paper.

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[17-22], satellite communications [23,24], amplifier mod- eling [25], **control** of **nonlinear** **MIMO** **systems** [6], etc. Recently, a **neural** network approach has been proposed to adaptively identify the overall input – output transfer function of this class of **MIMO** **systems** and to characterize each component of the system (i.e., the memoryless nonlinearities and the linear combiner) [4]. The proposed NN model is composed of a set of mem- oryless NN blocks followed by an **adaptive** linear com- biner. Each part of the **adaptive** system aims at identifying the corresponding part in the unknown **MIMO** system. The algorithm has been successfully ap- plied to system modeling, channel tracking, and fault detection.

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Model Predictive **Control** (MPC) [1] has been widely and successfully applied in industrial process, especially the multi-input, multi-output (**MIMO**) **nonlinear** process. Several recent publications have provided a good introduction to theoretical and practical issues associated with MPC technology. In 1999, Allgower, Badgwell, Qin, Rawlings, and Wright [13] presented a more comprehensive overview of **nonlinear** MPC and moving horizon estimation, including a summary of recent theoretical developments and numerical solution techniques, Rawlings (2000)[25] provided an excellent introductory tutorial aimed at **control** practitioners. A comprehensive review of theoretical results on the closed-loopbe havior of MPC algorithms was provided by Rawlings, Rao, and Scokaert (2000). Notable past reviews of MPC theory include those of Garsíaa, Prett, and Morari (1989)[18] ; Ricker (1991)[21] ; Morari and

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Recently, the **control** problem of switched **uncertain** **nonlinear** **systems** has attracted much attention [1], [2]. For non-strict feedback **uncertain** switched **nonlinear** **systems**, an **adaptive** fuzzy output-feedback stabilization **control** method has been already studied in [1]. The **adaptive** fuzzy **control** problem of **nonlinear** switched stochastic pure feedback **systems** has been discussed in Yin et al. [2]. Also, a lot of decentralized controller design methods for switched **nonlinear** **systems** have been investigated and various successful **control** applications have been developed. Meanwhile, many research results of strict feedback form **systems** or non-strict feedback form **systems** have been proposed [3], [4]. In [3], an **adaptive** fuzzy controller has been presented for a class of switched **uncertain** **nonlinear** **systems** with strict-feedback form. Liu et al. [4] focus on backstepping-based **adaptive** **neural** **control** for switched **nonlinear** **systems** in nonstrict-feedback form.

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for the measurement. There exist certain **systems** where it is feasible to measure some of the states while the remaining ones are not measurable. Such **systems** need an observer design to estimate only unmeasured states so as to have high system efficiency and accuracy. For such **systems**, second category of observers is introduced, named as reduced order observer [15]-[18]. Designing of an observer for **systems** having unknown dynamics is an active area of research. Such observers, referred as **adaptive** observers, mainly emphasizes on simultaneously estimating the unknown states and uncertainties of a class of **nonlinear** **systems**. An **adaptive** observer performs the role of state estimation as well as parameter identification. It comprises two coupled algorithms for the tasks. The state estimation algorithm works under unknown parameters, where updated parameters are used for estimating state variables. The parameter identification algorithm is also based on measured outputs and estimated states. Various **adaptive** observer methods have been introduced for **nonlinear** **systems** with unknown parameters. The conventional design approach for **adaptive** observer mainly emphasizes on the designing of the observers for the **systems** where uncertainties follow Lipschitz condition, however it results in a conservative observer design and is applicable to limited class of **systems**. Designing of **adaptive** observers which uses approximation tools like **Neural** Networks (NN) or Wavelet **Neural** Networks (WNN) for system identification is new domain of research in the field of observer design. Use of these system identification tools relaxes the Lipschitz restriction and hence it enhances the class of **uncertain** **nonlinear** **systems** under consideration. Owing to the universal approximation property of these identification tools, the results provided by these observers are highly accurate in comparison to conventionally designed **adaptive** observers.

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In this paper, consensus tracking problem for high order **MIMO** multi-agent **systems** with **nonlinear** dynamics have been studied. The proposed protocol was distributed NN robust **adaptive** method under undirected connected topologies. The proposed **control** method was constructed based on filtered error which obtained using relative state error. To estimate unknown nonlinearities of the controller, RBFNNs were employed and approximation error and effect of **uncertain** disturbances was compensated for by additional robust term in the controller. Update laws of unknown parameters of **neural** networks were determined from Lyapunov stability analysis. Lyapunov stability analysis was applied to guarantee overall system stability and convergence of unknown parameters. Simulation results presented to confirm the validity of the proposed controllers.

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Abstract: This paper proposes an **adaptive** approximation-based controller for **uncertain** strict-feedback **nonlinear** **systems** with unknown dead-zone nonlinearity. Dead-zone constraint is represented as a combination of a linear system with a disturbance-like term. This work invokes **neural** networks (NNs) as a linear-in-parameter approximator to model **uncertain** **nonlinear** functions that appear in virtual and actual **control** laws. Minimal learning parameter (MLP) algorithm is proposed to decrease the computational load, the number of adjustable parameters, and to avoid the “explosion of learning parameters” problem. An **adaptive** TSK-type fuzzy system is proposed to estimate the disturbance-like term in the dead-zone description which further will be used to compensate the effect of the dead-zone, and it does not require the availability of the dead-zone input. Then, the proposed method based on the dynamic surface **control** (DSC) method is designed which avoids the “explosion of complexity” problem. Proposed scheme deals with dead-zone nonlinearity and **uncertain** dynamics without requiring the availability of any knowledge about them, and it develops a **control** input without singularity concern. Stability analysis shows that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded and the tracking error converges to the vicinity of the origin. Simulation and comparison results verify the acceptable performance of the presented controller.

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particular, **adaptive** backstepping approach has played an important role in the **control** of strict-feedback **nonlinear** sys- tems. Generally, **adaptive** backstepping provides a systematic **control** approach to solve the tracking or regulation **control** problems of **uncertain** **nonlinear** **systems**, in which the classic **adaptive** **control** is applied to deal with the unknown parameter and backstepping technique is used to construct controller. The main feature of **adaptive** backstepping **control** is that it can handle the **control** problems of **nonlinear** **systems** without the requirement of matching condition. **Adaptive**

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The main contributions of this note can be summarized as follows: 1) In the proposed scheme, the problems of “explosion of complexity” and “curse of dimensionality” are solved from the root causes, different from DSC and MLP. The number of online learning NNs is reduced to only n, which is equal to the number of the **systems** outs and independent of the system orders. The intermediate controls would not appear in the **control** scheme. That will lead to a much simpler controller with less computational burden. 2) The **adaptive** law proposed in this note is merely dependent on the state variables, the reference signals and their mth order derivatives. With the special property and structure of our algorithm, the potential controller singularity problem existing in may **adaptive** **control** algorithm is avoided.

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There are some inevitable uncertainties in actual system which will cause instability and difficulties in dealing with system. Therefore, the study of **uncertain** **nonlinear** system is of vital importance. **Control** of **uncertain** **nonlinear** dynamic **systems** is still a challenging problem though it attracted many researchers in **control** community during the past few decades led to development of fruitful methods based on **adaptive** **control** concepts. Alternatively, in recent years, **adaptive** **neural** network [3, 7, 8, 9, 10, 12, 19, 20, 22, 25] and fuzzy logic **control** [1, 2, 5, 6, 13, 14, 15, 16, 17, 18, 21, 23] become an active research area. These methodologies become especially more helpful if **control** of highly **uncertain**, **nonlinear** and complex **systems** is the design issue. The main philosophy that is exploited heavily in system theory applications is the universal function approximation property of **neural** networks or fuzzy logic. Benefits of using **neural** networks or fuzzy logic for **control** applications include its ability to effectively **control** **nonlinear** plants while adapting to unmodeled dynamics. In general, a two-step procedure is taken. First, based on implicit function theorem an ideal controller developed to

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The simulation result for the output is shown in Fig.1, the node changes are shown in Fig.2, and the **control** input signal is shown in Fig.3.Fig.4 shows the evolution of the Euclidian norm of the parameter estimates It can be seen that the actual trajectories converge rapidly to the desired ones. The **control** signal and the estimated parameters are bounded. These simulation results demonstrate the tracking capability of the proposed controlled and its effectiveness for **control** tracking of **uncertain** **nonlinear** **systems**.

design **control** capable of handling uncertainties is of practical interest and is challenging. To achieve the desired system performance, **adaptive** **control** is a valid methodology, which supplies adaptation mechanisms to regulate controllers for **systems** with some uncertainties, such as parametric, structural, and environmental uncertainties [6-7]. For non-switched **nonlinear** **systems** using fuzzy logic **systems** or **neural** networks to parameterise the unknown non-linearities, **adaptive** **control** of **uncertain** **nonlinear** **systems** has attracted much attention [8-9]. In recent years, **adaptive** fuzzy or **neural** backstepping approaches for strict-feedback form **systems** [10-11] provide some systematic methods to achieve good tracking performance.

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2016 ), back-stepping method (Kwan and Lewis 2000 ), terminal sliding mode **control** method (Van et al. 2018 ), etc. The **adaptive** observer method for fault tolerant **control** usually uses an on-line estimator to estimate a fault, where the fault estimator can be implemented using different components and are adapted with different learning algo- rithms. A radial basis function (RBF) network was used in Trunov and Polycarpou ( 1999 ) and Polycarpou and Trunov ( 2000 ) as the on-line estimator, and a projection-based learning algorithm was developed to tune the weights of the network. As the reported work was in an early stage, simulations showed that the tuning of the estimator is very difficult and the convergence of the estimation is slow. Rather than directly estimate disturbance, **neural** networks have also been used to estimate unknown parameters in a **nonlinear** **uncertain** system without combining with a **nonlinear** state observer. In the air-to-fuel ratio (AFR) **control** of air path in a spark ignition (SI) engine using a sliding mode method (Wang and Yu 2008 ), an RBF net- work was used to estimate two unknown parameters, the partial derivative of air passed the throttle with respect to the air manifold pressure and that w.r.t. crankshaft speed. The **adaptive** law of the network estimator was derived so that the states out of sliding mode will be guaranteed to converge to the sliding mode in finite time. Moreover, a RBF network was used to estimate the optimal sliding gain in Wang and Yu ( 2007 ) to achieve an optimal robust per- formance in AFR **control** against model uncertainty and measurement noise.

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In Section 4, **adaptive** fuzzy tracking **control** has been developed for the **uncertain** **MIMO** nonlin- ear system based on disturbance observer. However, **control** gain matrix G(x) has been required nonsingular. Since G(x) depends on the system state x, there exists the singular feasibility at a special moment in the practical system, i.e, |G(x)| = 0 . On the other hand, input saturation has not been considered. In fact, input saturation always exists due to actuator output constraint. If input saturation is ignored in the **control** design, the closed-loop **control** system performance may be degraded. Therefore, **adaptive** fuzzy tracking **control** design will be presented for the **uncertain** **MIMO** **nonlinear** **systems** with **control** singularity and unknown input saturation in this section. Considering the **uncertain** **MIMO** **nonlinear** system (3) with unknown non-symmetric input satura- tion, the **control** input u = [u 1 , u 2 , . . . , u m ] T is given by

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