The reaching phase drives the system towards a “**sliding** surface” and the **sliding** phase “slides” the states towards an equilibrium point. Lyapunov’s **method** is used to ensure asymptotic stability during the reaching phase. A discontinuous term is added to the **control** law to compensate for the system uncertainties and disturbances. However, the **method** requires knowledge of the mathematical **model** of the system and hence it is unique to each system which restricts its use. Therefore, there is a clear need to develop a **Model**-**Free** **control** **algorithm** **based** on the **Sliding** **Mode** **Control** (MFSMC) which derives the **control** law from previous **control** inputs, system measurements, **control** input gains and system’s order. An initial MFSMC **control** strategy was first proposed for Single-Input-Single-Output (SISO) **systems** [3] and then extended to Multi- Input-Multi-Output (MIMO) **systems** [4]. The goal of this research is to successfully integrate the proposed **control** **algorithm** on an **Unmanned** **Aircraft** System (UAS), e.g. quadrotor drone, for motion **control** and path tracking. In this work, a comparison in the performance of the proposed MFSMC **algorithm** to a traditional PID controller for tracking precision, power consumption and tuning time is presented.

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The study of fault tolerant **control** (FTC) has received much attention in the last decade. Many different schemes have been proposed, ranging from active to passive **control** methods [1], with **applications** running the gamut from large scale petrochemical plants to automotive and aerospace **systems** [2]. The work in [1], [2], [3], [4] provides an excellent literature review on the different schemes used for FTC including different **applications**; however, research into FTC has been significantly driven by problems encountered in the safety critical aerospace industry. This is due to the practical requirement for increasing safety as well as lowering operational costs. Many different FTC methods, specific to **aircraft** **applications**, have been proposed including linear approaches (e.g. H ∞ [5], LQG [6], **model**-following [7], multiple **model** [8], **model** predictive **control** [9]), and nonlinear approaches (e.g. nonlinear dynamic inversion [10], backstepping [11], neural networks [12], **sliding** **mode** [13]). Some of these methods have attracted the attention of industry and have been further tested and evaluated in order to assess their potential for future **applications** and implementations on **aircraft** (see for example the European study on state-of-the-art FTC by the GARTEUR AG-16 group [14] and the recent ADDSAFE project [15]). However many of these proposed FTC schemes are **based** on linear plant representations and are therefore only valid in the vicinity of the designed trim point. Therefore, one of the main challenges for practical implementation, especially for **aircraft**, is to ensure good performance for a wide range of operating conditions. Some of the linear **based** designs can be extended to handle variations in operating conditions (see for example, gain scheduling and linear parameter varying (LPV) schemes [16], [17]), but direct nonlinear methods such as nonlinear dynamic inversion (NDI) and backstepping provide equally viable alternatives – with many benefits compared to the extended linear cases. One obvious benefit is the direct exploitation of the well known **aircraft** equations of motion, which provide good and consistent performance throughout the flight envelope.

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these limitation of fuzzy logic **based** **control**. Several researchers [5-9] have also reported similarity between fuzzy logic **control** and **sliding** **mode** **control** and have utilized it to design fuzzy controllers **based** on **sliding** **mode** theory for a class of non-linear system. Most of the work reported in the design of **Sliding** **mode** fuzzy controller use representative functions like ( , σ σ . ) , where σ is the expression for **sliding** plane which results in the number of fuzzy rules being as high as 49. **Sliding** **mode** **based** Fuzzy **control** reported [5] uses 32 rule base. In this paper a new **Sliding** **mode** fuzzy **control** technique has been developed, which addresses all the above listed limitations of fuzzy **control** design. The proposed **control** technique is **based** on **model** following approach [7], where in a fuzzy controller minimizes error between plant vector and the **model** vector. The plant is forced to follow the response of an ideal mathematical **model** whose response has been tailored as par time domain specifications. In this way a fuzzy controller forces the plant to follow the response of a stable, ideal mathematical system and in a way fuzzy controller adapts itself to the specified time domain specifications and thus a fuzzy controller can be designed whose response can be controlled by varying the time domain specifications of the mathematical **model**. This technique enables one to do away with, defining complicated cost functions as the figure of merit of the system performance. The proposed fuzzy **algorithm** **based** on **Sliding** **mode** theory results in development of fuzzy rules in a very systematic way and the **control** law developed can be easily extended to higher order **systems**. The **algorithm** is **based** on driving the state vector to the **sliding** surface and simultaneously forcing it to the origin of state space. In this **method** two fuzzy inputs are suggested. The first input measures the magnitude of the distance of the state vector from the **sliding** plane and the second vector is the Euclidean norm of the state vector. Since, both the inputs are treated as positive quantity unlike the earlier cases [5], the proposed technique results in significant reduction of the rule base. The stability of proposed fuzzy **control** **based** on **sliding** **mode** theory is easier to prove [10].

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In the past decade one may observe augmented attention on under- actuated **systems** in research field. The under -actuated **systems** have fewer actuators than the degrees of freedom to be controlled. These **systems** have **applications** in **free**-flying Space robots, Underwater robots, Surface vessels, Manipulators with structural flexibility, etc. The inverted pendulum has been considered as benchmark example in nonlinear **control** studies as a under-actuated system[1].Dynamics of the inverted pendulum are quintessential for the balance in wheeled Mobile robots, Robo walk and Robo thrusters [2-6]. With single input and two outputs, inverted pendulum has remained an exacting **control** problem owing to its characteristics like instability,non- linearity. It has two equilibrium points; one being stable while other is unstable [7].Various **control** strategies can be found in literature to stabilize the pendulum around unstable equilibrium point [8]Researchers have analysed PID **based** **control** [9], neural network **control** [11],fuzzy **control** [12], optimisation tools like linear quadratic regulator [10, 13].**Sliding** **mode** **control** has emerged as promising **method** for **control** of inverted pendulum. [15-17]

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The ﬁrst part is devoted to system modelling. A dynamic **model** can be identi- ﬁed from data collected (input and output data from the plant). However, the data obtained is often aﬀected by noise. Hence, such algorithms for modelling the plant should be robust enough to accurately predict the dynamic behaviour of the sys- tem in the presence of noisy data. Taking this into account, this thesis focuses on subspace-**based** identiﬁcation methods, and proposes an eﬀective **algorithm** **based** on the Least-Square Support Vector Regression (LS-SVR). In the proposed algo- rithm, the system identiﬁcation is formulated as a regression problem to be solved by applying multi-output LS-SVR.

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Dehghani and Menhaj [23] derived an integrated controller and estimator for the problem of autonomous **aircraft** leader-following formation **control** using integral SMC and assuming no communication between **aircraft** (i.e. relative state measurements are obtained by observation of the leader only) thereby necessitating the use of a robust **control** **method** like integral SMC and adaptive parameter estimation to determine desired trajectories from uncertain measurements. Both Brezoescu, Lozano, and Casillo [10] and Yang, Kang, and Sukkarieh [8] presented adaptive robust controllers that did not rely on SMC. In [10] an adaptive backstepping Lyapunov-**based** tracking controller was designed to reject wind disturbances and achieve geo-referenced path following for the lateral **mode** of a fixed wing aerial vehicle. Simulation results showed that the proposed controller was able to compensate for a Dryden gust **model** wind disturbance for different heading angles relative to the wind. Heading tracking results all converged to desired values after step changes in desired headings, and waypoints were followed nearly perfectly even in the presence of varying wind and state measurement noise intended to replicate actual sensors. In [8] a **method** of adaptive nonlinear MPC for a fixed wing UAV was derived. The objective was only to **control** the decoupled lateral directional **mode**, with the assumption that another controller would maintain altitude and heading. The system **model** used did not account for the true aerodynamic nature of the system, with **control** input being defined as the time rate of change of the heading angle. A smooth and continuously differentiable path was obtained by fitting cubic Bezier curves to a set of waypoints. MPC is (typically) a discrete **control** **method** that involves solving a finite horizon optimal **control** problem at each time-step using a nominal plant **model** and then using the output as a feedforward **control**. The proposed **method** of MPC is adaptive because it dynamically changes the time horizon used to predict future system behavior in order to reduce computational complexity when possible and to provide higher fidelity results during tracking of more complex (higher curvature) trajectories.

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that requires no **model** for the underlying system. The data-driven **control** is such a **control** procedure which uses only the measurements from input and output, thus the unmodeled time varying uncertain- ties are not required known. The data driven concept was proposed firstly in the field of computer science, and has been introduced gradually into the **control** field. For instance, this concept has application in the **control** area of fault diagnosis and tolerance [40, 41]. Data-driven **control** methods have been devel- oped into many types, which include virtual reference feedback tuning [6], iterative learning **control** [21], dynamic programming approach [45], neural net- work **method** [19], **model** **free** **control** [1, 13] and oth- ers [22, 33].

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Reliable manufacturing processes and reasonable structural design methods are the basis for precise lifting of a polar crane. Literature [1] took the physical structure of the rope-sheave system of the polar crane into account and proposed a systematic **method** for analyzing the dynamics of the system **based** on the virtual power principle. Literature [2] presented a quasi-static **method** to analyze the equilibrium path and trajectory deflections for the load of the polar crane, which is of great significance for improving the structural layout of the lifting system. Literature [3] proposed a static balanced approach to analyz- ing the straightness deviation of the hook block during the lifting process. Literature [4] analyzed the influence of bridge deformation on the locating accuracy under different working conditions by the finite element **method**. Literature [5] presented an effective **control** **method** for the welding deformation of corbel by improving the welding process and welding sequence. Literature [6] analyzed the rigid-flexible coupling **model** of a polar

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Then, the BIC system [19] is applied to the chaos syn- chronization again. The sensory input of the BIC system is chosen as es ¼ e _ þ e and the controller parameter that is initiated from zero can be online tuned in the sense of the BELM. The simulation results of the BIC system are shown in Figs. 9 and 10 for scenario 1 and scenario 2, respectively. The responses of states ðx; yÞ are shown in Figs. 9a and 10a; the responses of states ð x; _ yÞ _ are shown in Figs. 9b and 10b; and the **control** inputs are shown in Figs. 9c and 10c. All of the tracking errors for both of the test scenarios would converge to zero after controller parameter learning, and it means that the chaos synchro- nization is stable using the BIC system. Although the simulated results show the effectiveness of the bio-inspired emotional learning approach, the convergence speed of tracking error is slow. In addition, the BELM cannot **model** the qualitative aspects of human knowledge.

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A chattering-**free** **sliding**-**mode** **control** has been mainly proposed in the discrete-time dynamical **systems** with considering of the bounded input. For this purpose, a **sliding**-**mode** **control** law is analytically derived in a typical second order discrete-time system. It is shown that the system states are quickly moved to a predefined **sliding** surface in some finite sampling times. Then the system states are guided to the origin. The effect of the system uncertainty is also investigated in order to design a robust SMC. Two numerical examples are provided to show the effectiveness of the proposed **method** in comparing with the existed results.

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condition of 0 < m < 1 limits the application of the fractional-order integrator. Obviously, the usual observers or differentiators [9, 10, 11, 12] only can estimate the derivatives of the signal. Recent years, Kalman filter is used to handle the separation of probabilistic noise and to estimate signal integral [13, 14]. However, for Kalman filter, the process noise covariance and measurement noise covariance are assumed to be zero-mean Gaussian distributed, and the process noise covariance is uncorrelated to the estimation error. These as- sumptions are different from the real noise in signal. The inaccurate noise information in sensed angular velocity may lead to the estimate drifts of attitude angle. In [15], a nonlinear double-integral observer with the abilities of noise rejection and drift correction was presented to estimate synchronously the onefold and double integrals of a signal. In [16], a general- ized multiple integrator was designed to estimate the multiple integrals for a signal. In [17], a nonlinear integral-derivative observer was proposed to estimate synchronously the integral and derivative of a signal. However, these observer cannot be use to estimate the uncertainties in the flight dynamics directly. On the other hand, some sensors provide usually the position-related information. Representative designs are: GPS positioning **systems** [18, 19]; GPS/INS **systems** [20, 21, 22]; ultrasonic rangers [23]; GPS module when outdoors and infrared rangers when indoors [24]; carrier phase differential GPS [25]; laser rangefinder [26]; vision system [27, 28, 29]; indoor motion capture system [30, 31]; laser rangefinder and vision system [32]. However, these strategies are dependent on the accurate **model**, and all the states are required to be known.

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The results for adaptive gain SMC is shown in figure 2. The responses for, both, the controlled out- put and input are smooth. However, a delay in evo- lution of substrate concentration is observed. This can be seen more clearly from the graph showing the evolution of the **sliding** function. It is clear that the controller is taking more than 0.5 h before driv- ing the state back to the **sliding** surface, hence re- sulting in the delay. From the above results, it is ob- vious that the gains calculated are too small to force the state back to the **sliding** surface when a pertur- bation in the set point occurs. But once it reaches the **sliding** surface the **control** action is not over stressed as it can be seen from the plot of du/dt vs. time. Unlike in the fixed gain approach, the adap- tive gain approach showed very little variation in du/dt with time. From these two studies it can be seen that high fixed gain approach should be used to drive the state quickly towards the **sliding** sur- face, soon after the perturbation in set point occurs. But once the state is near the **sliding** surface, adap- tive gain approach should be used to minimise over stressing of **control**. We combined these features in the **control** law for the hybrid approach.

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The main aim of this project is to revise and define the automatic steering con- trol of passenger cars for general lane-following maneuver. A 2-DOF controller **based** on H loop-shaping methodology is used by lateral vehicle **control** system is success- fully designed [5]. The 2-DOF controller supplies good lane-keeping and lane-change abilities on both curved and straight road segments. Moreover, it provides a com- putationally efficient **algorithm** and does not require explicit knowledge of the vehicle uncertainty. But, the test results show that the higher the vehicle’s speed, the less stable the vehicle system.

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In this paper, a new adaptive **control** **method** is proposed for direct matrix converters. The proposed **method** uses interval type-2 fuzzy logic integrated with **sliding** **mode** **control**. Employing the **sliding** **mode** **control** in matrix converters leads to an efficient choice of switching combinations and a reliable reference tracking. The main problem of the **sliding** **mode** **control** is the chattering phenomenon that degrades the controller performance through injecting high-frequency variations in the controller variables. The proposed **method** incorporates the interval type-2 fuzzy with the **sliding** **mode** **control** to mitigate the chattering problem. The **sliding** **mode** switch surface can be adjusted adaptively according to the system state and the proposed fuzzy compensation **based** on the Lyapunov stability theorem, so that the **control** system has the characteristics of low chattering effect and appropriate operation quality. Comprehensive evaluations of the waveforms are conducted for the new matrix converter through various simulations. Simulation results verify the effectiveness of the proposed adaptive **control** **method** for matrix converter in various conditions, and its superiority in chattering suppression in comparison to the conventional **sliding** **mode** **control** and the boundary layer **method**.

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ABSTRACT This paper proposes a novel result on optimal sliding mode control OSMC approach for general nth order nonlinear systems with matched and mismatched uncertainties based on exten[r]

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This paper presents a design **method** for attitude **control** of an autonomous underwater vehicle(X4-AUV) **based** **sliding** **mode** **control**. We are interested in the dynamic modeling of X4-AUV because of its complexity. The dynamic **model** is used to design a stable and accurate controller to perform the best tracking and attitude results. To stabilize the overall **systems**, each **sliding** **mode** controller is designed **based** on the Lyapunov stability theory. The advantage of **sliding** **mode** **control** is it’s not being sensitive to **model** errors, parametric uncertainties and other disturbances. Lastly, we show that the **control** law has a good robust and good stability through simulation.

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Under the assumption of linearity of the process **model**, the basic structure of the estimator is always the same, but its realization will depend on the chosen context: continuous or discrete, deterministic or stochastic. In the case where this **model** is a deterministic **model**, the state reconstructor will be called observer. In the case of noisy **systems** where random phenomena occur, it is called filter. The use of an observer can be seen in monitoring, diagnosis, and **control**.

photovoltaic (PV) system by **sliding** **mode** **control**(SMC). Here, open circuit voltage MPPT technique is used to track maximum power point. There is a difficulty in tracking the maximum power point of the photovoltaic system due to nonlinearity of the I-V characteristics which is dependent of the temperature and irradiation conditions. The system involves a PV panel, dc/dc boost converter, a load and a **control** that generates PWM signal that goes to the boost converter. The open circuit voltage **based** MPPT uses open circuit voltage to calculate maximum power output voltage. The input to the **sliding** **mode** controller is the change in reference voltage and PV voltage and the output of the SMC is the change in duty ratio. The SMC is used to track the maximum power point by changing the duty cycle of the boost converter. Using this **method**, the output power of PV array directly controls the dc/dc converter, hence reduces the complexity of the system. The advantages of this **method** are high efficiency, best accuracy, good convergence speed, and is robust to weather condition changes. The effectiveness of proposed **sliding** **mode** **control** can be validated using simulation

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shows the time required for 100 meters by the car with different **control** **method**. The x-axis label indicates the cases of different road condition and mass, for example, DA1000 says that the car with the mass 1000 (kg) is driving on the dry asphalt, WA1200 shows that the case with the mass 1200 (kg) on the wet asphalt and IR1400 is the case with mass 1400 (kg) on the ice road. As shown in the bar graph, it takes the minimum time for the car with the proposed **method** for the 100 meters in every case. So we can see that the car with proposed SMC have gained the best acceleration. In other words, the results also indicate the car with the proposed SMC decreases the loss of driving force mostly. Moreover, the time re- quired is long on th

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The **control** of nonlinear **systems** is still a challenging area in the literature of **control** **systems** theory and some efforts have been made to study this subject [1]. However, most of them can only be applied to a certain class of nonlinear **systems**. For example, feedback linearization is only applicable to a class of nonlinear **systems** meeting the involutivity condition and can be transformed into the companion form [1]. Many other methods have some limitations. For example, chattering is the most important problem in **sliding** **mode** **control** (SMC) [2]. It has been shown that **sliding** **mode** controller (SMC) is a powerful tool in facing uncertainties, disturbances, and noises that always produce difficulties in the realization of the designed controller for real **systems** [3]. This is due to the invariance property (insensivity against disturbances), which is stronger than robustness [1-4]. The invariance property motivates researchers to use SMC for various **applications** [5- 8] especially the precise **systems** [9]. The greatest limitation of SMC is the chattering, i.e. the high (but finite) frequency oscillations with small amplitude, which produces heat losses in electrical power circuits and wear mechanical parts [3, 4]. Four design methodologies have been proposed to overcome this problem: boundary layer, adaptive boundary layer, higher order SMC (HOSMC) and DSMC [2, 3]. Boundary layer and adaptive boundary layer methods cannot preserve the invariance property of SMC [1-3]. HOSMC is proposed to reliably prevent chattering [4, 10]. In higher order SMC, the effect of switching is totally eliminated by moving the switching to the higher order derivatives of desired output [10]. Many algorithms are proposed for implementation of second or higher order SMC [11]. However, the main drawback is that the controller design generally requires the Corresponding author; Email: karami@hsu.ac.ir

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