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In this article, a new discrete-time sliding mode predictive control (DSMPC) with a PID sliding function is proposed for synchronous DC–DC buck converter, and its stability analysis is reported. Th...

A novel adaptive control scheme is presented for a class of non-affine nonlinear systems with non-affine nonlinear function possibly being discontinuous. A discontinuous condition for non-affine nonlinear systems is present to guarantee the controllability of system. The non-affine nonlinear function is modelled appropriately by using piecewise functions. Based on Lyapunov analysis method, the basic idea of invariant set theory is constructively introduced to prove the boundedness of all the signals in the closed-loop system. Finally, simulation example is provided to demonstrate the effectiveness of the proposed approach.

The intrinsic properties of an ultrasonic motor (high torque for low speed, high static torque, compact in size, no electromagnetic interference, etc.) offer great advantages for industrial applications. However, when load torque is applied, a dead-zone occurs in control input. Therefore, a sliding-mode controller, which is a non-linear controller, is adopted for the ultrasonic motor. The state quantities, such as acceleration, speed, and position, are needed to apply the sliding-mode controller for position control. However, a rotary encoder causes quantization errors in the speed information. This article presents a robust position control method for an ultrasonic motor by using a variable structure system observer. The state variables for the sliding-mode controller are estimated by the variable structure system observer. A small, low-cost, and good response sliding-mode controller is designed in this work by using a microcomputer that is essential in an embedded system for the development of ...

This article is mainly concerned with uncertain systems subject to periodic disturbances with known periodicity, and the methodology of discrete variable structure repetitive control is applied to address the control problem. The notion of quasi-sliding mode (QSM) is reviewed in this article. Switching function dynamics obtained with the proposed repetitive control strategies are desirable to govern both reaching phase and sliding phase. For characterising the switching-function dynamics, detailed estimation results of both attractive layer bound and QSM band are given, with which the repetitive controllers can be designed to ensure closed-loop robust stability, and achieve perfect tracking in the presence of periodic perturbations. As more knowledge about the periodicity is known, partial-period repetitive control schemes are presented for reducing memory occupation. Numerical simulation and experiment are carried out to verify the theoretical results.

In this article the control of mechanical systems with elastic joints is addressed. This kind of system is characterised by a high relative degree. The mechanical control input must be designed to be continuous. Different reduced order observers are introduced and conditions are found to guarantee the exponential convergence of the observation errors. The error stability is ensured provided a certain matrix inequality has a solution. The proposed second-order sliding mode point-to-point controller based on the observed state is proven to stabilise the mechanical system with elastic joints about the desired reference position.

In this paper we present a stability theory for discontinuous dynamical systems (DDS): continuous-time systems whose motions are not necessarily continuous with respect to time. We show that this t...

A novel variable structure control strategy for a class of second-order nonlinear systems is presented based on the phase-plane analysis and computing prediction. In this strategy control is switched by online prediction of the trajectory behaviour of the closed-loop system; control switching does not occur continuously, so the chattering that occurs in conventional variable structure control methods can be avoided completely. It is proved that the states of the closed-loop system can be stabilized to the origin exponentially and the trajectory of the closed-loop system does not overshoot in general. As an application, the proposed predictive variable structure control strategy is used to stabilize nonholonomic systems in chained form. An example of a wheeled mobile robot is studied and associated simulation results demonstrate the effectiveness of the proposed control strategy.

In this article, the integration of vehicle vertical and lateral motion control is investigated via control coordination between active suspension and four wheel steering. With respect to the different treatments of nonlinear tyre vertical and lateral behaviour correlativity, two control approaches based on linear optimal H ∞ synthesis and nonlinear sliding mode control are proposed and studied, respectively. For the former approach, the integrated control benefits have been examined through the overall performance improvement under the influence of external disturbance, such as road unevenness. For the latter approach, the potential of using tyre vertical load influence on lateral behaviour to improve vehicle critical handling performance has been investigated. Simulations and analysis indicate that both the above two approaches exhibit promising effects of vehicle chassis control integration.

The object of the study is to apply the Pacejka magic formula tyre model on a study of a hydraulic anti-lock braking system, especially applied to a light motorcycle. A sliding mode PWM controller is designed and tested. Both simulation and experimental studies of an anti-lock braking system are undertaken. The paper presents an analytical approach for estimating the longitudinal adhesive coefficient between a tyre and the road through the magic formula tyre model, the parameters of which are identified by a genetic algorithm. A dynamic analysis of a light motorcycle is carried out in detail. The experimental results show that the antilock braking system designed in the study is effective to prevent wheels locking during emergency braking. The proposed simulation results match experimental data well.

This paper deals with the dynamic behaviour of different DC-to-DC switching electronic converters in photovoltaic applications. In particular, the ability of the DC-to-DC converters to accomplish regulation of certain circuit variables, which are related to the control objectives, is examined. This task is addressed from a variable structure system approach. With this aim, the feasibility of establishing stable sliding regimes on different state co-ordinate surfaces is investigated. Differences in the dynamic behaviour of the DC-to-DC switching converters between photovoltaic and conventional power-supply applications are stressed.

An active vibration control technique for a double pendulum system, the pivot of which is mounted on a motorized cart and vertically transferred, is developed using a VSS control scheme, in which the Lyapunov function is characterized by a switching function that is weighted by all of the pendulum’s state variables. The designed control system, including a linear observer for estimating the cart velocity and a nonlinear observer for estimating the angular velocities, is actively effective in suppressing vibrations of double pendulum, although the control effect is slightly better for the first link than for the second for both the in-phase and out-of-phase modes.

In this paper an adaptive cruise control (ACC) of a convoy consisting of two passenger cars is designed and tested. For the ACC only on board sensors in the following vehicle are used, communication within the convoy or between the controlled vehicle and electronic systems on the roadside is not assumed. A laser scanner is applied for range measurements, derived from the complete vision data of the area in front of the car. Since the scanner provides the range only, a Kalman Filter is used to estimate the velocity and acceleration of the car. For controller design the concept of flat outputs in connection with the exact state linearization is applied. Moreover, the exact state linearization is combined with a sliding mode control. The control parameters are obtained by an optimization algorithm using optimal tracking formulation. The optimization also guarantees individual vehicle stability as well as string stability of the convoy. It is shown how the convoy is responding to disturbances resulting from i...

A general design method for the variable structure control of distributed time-delay systems is studied in this paper. The proposed switching function and variable structure controller are of finite dimension. The basic idea is to construct a finite dimensional system which contains all the unstable poles of the original time-delay system and hence those unstable poles of the time delay system will be absorbed by the developed variable structure controller. The proposed method is applied to the stabilization of combustion in a liquid monopropellant rocket motor and satisfactory results are obtained by simulation study. Simulation results show that the developed variable structure controller can still stabilize the combustion when the two key parameters, pressure exponent and maximal time lag r, change from their nominal value down to 0.0 and 0.1 respectively. By this result, we can somewhat overestimate the two parameters to obtain a reliable controller for the rocket motor control systems.

Conventionally, the discrete variable structure control always suffers from the chattering phenomenon. Therefore, in this paper, we shall show that, for the discrete variable structure control, there always exist a limit cycle if the traditional nonlinearity of the sign function is used. This limit cycle behaviour is just the chattering phenomenon in the variable structure control theory. In particular, the frequency of this limit cycle is identical to the Nyquist frequency in the sampling theorem. In addition, a theorem related to the chattering-free design for the multiple-input multiple-output variable structure control is proposed. If the nonlinear function is memoryless and belongs to the sector [0,2), the chattering phenomenon can be eliminated. If the width of the boundary layer is enough, the reaching condition as well as chattering-free design can be attained; otherwise, it will result in the chattering phenomenon.

For the control of anti-lock brake system (ABS), a longitudinal four-wheel vehicle model with brake actuator is described and a sliding mode controller with pulse width modulation (PWM) method has been developed for passenger vehicles. In our research, we introduce actuator dynamics of solenoid-solenoid valve type in system equation and derive the sliding mode control input theoretically. We propose using PWM method to compensate for the discrete nature of actuator dynamics by duty control. The effectiveness of the proposed control algorithms was confirmed by vehicle test on an in-door test bench that was specially constructed for the purpose concerned.

The equilibrium manifold of a singularly perturbed system has a close relationship with the sliding surface of a variable structure system. The fast time and slow time responses has a similar behavior to the "reaching mode" and "sliding mode", respectively. This paper aims to equip the powerful composite control method with robustness through variable structure control design. The major bridge in between is a Lyapunov function. It is found that a singularly perturbed system in sliding mode may preserve two-time-scale attribute, in which a new equilibrium manifold exists on the sliding surface. Sliding motions being attracted to the manifold can therefore be referred to as "sliding mode in sliding mode".

Neural network-based variable structure control is proposed for the design of nonlinear discrete systems. Sliding mode control is used to provide good stability and robustness performance for nonlinear systems. An affine nonlinear neural predictor is introduced to predict the outputs of the nonlinear process, and to make the variable structure control algorithm simple and easy to implement. When the predictor model is inaccurate, variable structure control with sliding modes is used to improve the stability of the system. A recursive weight learning algorithm for the neural networks based affine nonlinear predictor is also developed, and the convergence of both the weights and the estimation error is analysed.

The problem of model-referenced control for interconnected systems with series non-linearities is investigated. By applying variable structure control theory, we propose a controller which is able to drive the error subsystem between the model state and the plant state to zero. The presented variable structure controller ensures the global reaching condition of the sliding mode of the error system. It shows that the uncertain nonlinear dynamic system also possesses the property of insensitivity to uncertainties and disturbances as a linear system does. Furthermore, the sliding mode can be designed to converge within a specified exponential speed.

This paper considers the problem of stabilizing large scale dynamic systems (LSS) via decentralized controllers based on Variable Structure Systems (VSS), using the principle of dominant subsystems. The principle and applications for LSS are described and simulation results on a two-machine power system with a linear and nonlinear model are presented.

This paper gives a systematic approach in designing VS(Variable structure)stabilizer for excitation control problem of SMIB(Single Machine Infinite Bus) system. By observing the behaviour of variation of eigenvalues with respect to change of operating points,using participation factors analysis a tentative pole placement has been given such that switching plane obtained at a nominal operating point is valid over a wide range of operating points. This paper gives a computational procedure for determination of gains using the condition to be satisfied inorder that sliding takes place. The gains are determined in such a way that they guarantee sliding mode over a wide range of operating conditions. The performance of VS stabilizer is compared with Constant Gain Power system Stabilizer(CGPS) using feedback from all state variables of the system and conventional Power System Stabilizer(PSS) in the form of lead/lag blocks and reset block.

Based on variable structure systems (VSS) theory, this paper presents a new method of estimating the disturbances (or system nonlinearities and any model uncertainties) for continuous-time multi-input multi-output (MIMO) minimum phase (with respect to the relation between the disturbance and the output) dynamical systems. The designed robust method requires only the input and output measurements of the system. Even for MIMO systems under the assumption that the partial states directly affected by the disturbances do not directly appear in the outputs (when it is considered in the state space), the disturbances can still be estimated by applying the proposed formulation. The estimated disturbances are then employed to construct a robust state observer. Further, the estimated disturbances and the state observer are applied to synthesize a controller to place the desired stable poles and to cancel the disturbances. A design example and simulation results are presented to show the practicality and effectivene...

This paper proposes a design technique for the variable structure control (VSC) to stabilize the discrete-time model-following systems. That is, for a giving model with ideal response characteristics, a VSC is designed so that the error between the plant's state and the model's state is forced to exhibit like a sliding mode behavior, then the output of the plant faithfully follows that of the model. This VSC includes two portions, one is the “sliding control” which depends on the determination of the matrix G for the sliding surface such that the sliding mode is stable; the other is the reaching control which is designed to force the error state approching the sliding surface. It is also shown that in this controlled system the sliding surface is extended to a sliding region which is determined by a diagonalizing transformation matrix. Furthermore, the designed reaching control gain is a simple vector with the same elements. This may make the practical controller be easily implemented.

Based on the VSS theory, this paper presents a new method of estimating the disturbance for minimum-phase (with respect to the relation between the disturbance and output ) continuous-time dynamical systems with arbitrary relative degrees. By using the transfer function method, first, the higher order filter of the disturbance is estimated, then, the lower order filter of the disturbance is estimated inductively. Eventually, the disturbance is estimated. The estimated disturbance is employed to construct a VSS-type state observer. Further, the estimated disturbance and the state observer are applied to a controller to place the desired stable poles and to cancel the disturbance. A design example and simulation results are presented to show the practicality and effectiveness of the proposed algorithm.

This paper is on the analysis and design of sampled-data high-gain control and sliding-mode control systems. In continuous high-gain control, it is usually true that the higher the gains, the more effective the system uncertainties may be suppressed. When implemented digitally, however, there are limits to the feedback gains without causing instability.This paper analyses high-gain,sampled-data,systems utilizing the results of the singular perturbation theory. The system can be decomposed into fast and slow subsystems under high gain feedback. It is shown how the fast subsystem goes unstable when the gains get high enough due to the sampling effect. With reasonable approximations, a set of simple stability criteria is derived relating the feedback gains to the sampling period and other factors. It is also shown that control chattering, which is typical in variable structure systems with sliding mode, also happens in high-gain control systems when the gains exceed or are close to the critical values. Moreo...

The implementation problems by a digital computer for a well-designed linear continuous-time variable structure control system are considered. By using the zero-order-hold of the designed continuous-time VSS controller, a corresponding discrete-time controlled system is obtained. For the discrete-time system a new definition, the discrete-time weak-pseudo-sliding mode, is given to describing the zigzag behaviour corresponding to the continuous-time VSS sliding mode. An upper bound of the sampling period is given to guarantee the existence of the weak-pseudo-sliding mode along the prescribed hyperplane for the discrete-time system. If the chosen sampling period is not greater than this upper bound, the zero-order-hold of the continuous-time VSS controller can still stabilize sampled data system without any modification. Finally, simulation results are presented to show the effectiveness of the decision of the upper bound.

The paper presents a novel, simple, variable structure, self adjusting, squashable error-driven, adjustable gain power system stabilizer (PSS) design. The elastic error-driven and error-scaled PSS design falls in between the categories of bang-bang and linear error scaled variable gain. The digital simulation studies were conducted on a single machine infinite bus system. From the digital simulation results, the proposed PSS is shown to be very effective in damping the oscillations and it is robust as it tolerates any system structural and parametric variations. In addition, the proposed PSS can be easily implemented. The design was simulated and tested using MATLAB simulation package software and compared to conventional analog gain/lead/lag compensator PSS design.

A redundant dynamical sliding mode control scheme is proposed for the asymptotic stabilization of a rigid spacecraft. The input-dependent nonlinear sliding manifolds are derived from a smooth feedback control law, based on the use of nonlinear characteristic values and characteristic vectors. A scheme tolerating sudden perturbation failure is presented. Simulations studies are provided.

SUMMARY This study deals with dynamics and control of a rider-motorcycle system model in Hamiltonian form. The model is established to account for not only the motorcycle dynamics but also changes in rider's postures. Tyre forces are treated as external forces and rider's control actions contain three actuation torques. The Hamiltonian system is subject to dissipation effects due to tyre forces and rider's control torques. In addition to the total Hamiltonian, a generalized free energy is employed in formulation. Lyapunov's second method provides sufficient conditions for stability. Variable structure system (VSS) control is used to account for stabilization control exerted by the rider. Simulation results validate the proposed method.

A design method is presented for fuzzy control to stabilize a system subject to non-holonomic constraints. Although a time-varying feedback law has been proposed for stabilizing non-holonomic systems, it results in slow convergence. By contrast, the current study employs fuzzy control designed with sliding modes to stabilize non-holonomic systems. The non-holonomic system becomes input-output linearizable via a proper choice of output function. Fuzzy control is formulated to become a class of variable structure system (VSS) control. Moreover, it is designed with the aid of sliding modes to ensure stability. A rider-motorcycle system, assuming rolling contact between tyres and ground, can be treated as a non-holonomic system. Owing to the motorcycle's static natural instability, stabilization control becomes an important task while riding. Using the proposed method, both hands-off and hands-on rider control are investigated. The effects of stabilization control exerted by the rider are examined according to simulation results.

The design of a model reference adaptive controller with terminal sliding modes, while only input and output measurements are available, is discussed in this paper. In terminal sliding modes, the equilibrium point is reached in a finite time. By employing terminal sliding modes, the model reference adaptive controller enables the error dynamics to reach zero in a finite time. State variable filters are used to obtain differentiator-free controllers. Simulation results are presented to confirm the discussion.

A controller/observer pair is presented, on the basis of sliding mode ideas, which provides robust output tracking of a reference signal using only measured output information. Closed-loop analysis indicates that asymptotic tracking of a constant reference signal will be achieved despite the presence of a class of matched uncertainty. Furthermore, a form of ‘separation principle’ is shown to hold for this class of controller and observer, in the sense that they can be designed independently apart from a scalar function of the uncertainty bounds.

The paper presents a new approach to the design of variable structure current and VAR regulators for improving the dynamic stability of parallel AC-DC systems. The variable structure regulators are based on the concept of sliding mode control which constrains the system motion to a state trajectory. The sliding mode control provides greater robustness and insensitivity to plant parameter variations and is feasible for practical implementation. The transient simulations for a typical parallel AC-DC system connected to a weak AC system are performed to demonstrate the effectiveness of the variable structure control.

Two different object-oriented modeling approaches, DEVS and EMSY, constitute the background to explore the area of variable structure modeling. Realizations of various kinds of structural changes are discussed in both approaches. Against the background of their prime application domains, both approaches deal with the problem of structural change differently. While DEVS emphasizes intelligent control of structural change, EMSY stresses the autonomous character of the system. Like autonomy and control, holism and reductionism play different roles in both approaches and affect the realization of structural changes. However, unlike the former which tend to transcend each other, the reductionistic and holistic view realized in the two modeling approaches prove to set a rigorous framework for variable structure modeling

A new discrete-time control algorithm has been developed by combining variable structure systems (VSS) and Lyapunov design. It possesses all the good properties of the sliding mode control and avoids the unnecessary discontinuity of the control input, thus eliminating chattering which has been considered as a serious obstacle for applications of VSS. A unified control approach for the stator currents and torque control based on the discrete-time sliding mode for application in indirect vector control of an IM drive is developed. The sliding mode approach can be applied to the control of an IM servodrive due to the replacement of the hysteresis controller with the widely used PWM technique. All the theoretical issues are verified by experiments. The experimental system consists of a transputer T805 and a microcontroller, which enables parallel processing. >

The problem of model-reference control is considered for a large-scale variable-structure adaptive control system in the presence of interconnected terms, system parameter variations and external disturbances. Based on the technique of parameter estimations and the theory of the variable-structure systems, this paper presents a new and feasible design algorithm to synthesize a decentralized variable-structure adaptive controller which can easily tackle the model-reference control problem of linear time-varying large-scale systems subjected to interconnected terms, system parameter variations and bounded disturbances. In contrast with previous results, the restrictive condition that the exact value of the parameter about the input gain must be known can be relaxed. A series of computer simulations are also included to illustrate the proposed design algorithm.

The problem of output tracking in SISO uncertain non-autonomous nonlinear systems is treated in this study. It is shown that a nonlinear system experiencing non-matching disturbances and uncertainties can be transformed to a form similar to the normal form. Then, a novel variable structure controller with disturbance estimation and parameter adaptation is presented. This controller, while preserving the robustness properties of sliding control, avoids a sliding mode and generates a continuous control signal, which results in close output tracking.

The stable region of dynamic systems is discussed in this paper. A new approach to estimate the stable region is developed based on the decomposition of the system. The stable region of a variable structure system (VSS) is studied in detail, since the structure of a variable structure system is changed according to the states of the system. Furthermore, a useful method is proposed to estimate the stable region when the control signals are bounded owing to a physical constraint. A numerical example is given to illustrate the result obtained in the paper.

A new algorithm for decentralized adaptive control is proposed in this paper. This algorithm consists of an ordinary local adaptive controller and a variable structure adaptive controller. The adaptive variable structure component of this algorithm is used to compensate for uncertain interconnections among the subsystems and to ensure global stability of the overall system. Simulation results are also presented to demonstrate the performance of the closed-loop control system

In this paper, based on the bounds of parametric uncertainties the necessary and sufficient conditions for the existence of a sliging regime, and the occurrence of hitting in a variable structure system, are presented. A new variable structure control algorithm is proposed to solve the load-frequency control problem for an interconnected hydrothermal power system. The simulation results demonstrate that using the proposed controller the transient responses of the system are greatly improved and the dynamic performance of the system is better than that of the conventional control scheme. Furthermore, the proposed approach is robust to the plant parameters' variations, and the characterization of the system is as good as the full state switching when the reducing switching scheme is adopted.

A design method is proposed for actual finite-dimensional VS-MRAC (variable structure model reference adaptive control) of asymptotically stable distributed systems with bounded disturbances, bound...

This paper presents a novel and systematic approach to constructing a self-organizing and self-learning multivariable controller. The proposed controller is built on a hybrid neural network consisting of a variable-structure competitive network and a standard back-propagation neural network (BNN). We develop the corresponding self-organizing and learning algorithms. The controller shares certain features with traditional fuzzy controllers in terms of using error-based input forms and emphasizing the rule-based paradigms. However, knowledge representation and reasoning are here carried out by the network structure and computing, instead of logical inference. On the other hand, compared with ordinary BNN-based control paradigms the proposed controller possesses a distinctive characteristic; that is, while having the advantages of computational efficiency and trainable capability of the network paradigm, it maintains the clarity of the rule-based paradigm by self-organizing a rule-base, thereby providing an ...

In this paper, a stable variable structure control (VSC) scheme is studied. With the existence of a sliding regime and the occurrence of hitting, a new approach to determine the switching gains of a variable structure stabilizer with uncertain parameter is presented. With the assignment of an eigenstructure when designing the switching plane, and the incorporation of appropriate switching gains to form the sliding regime, the system dynamic performance can be improved. Based on the bounds of uncertainties the proposed algorithm is a systematic procedure. Simulation results show that with the proposed stabilizer the dynamic performance of a synchronous generator system can be greatly improved and the dynamic response is better than the variable structure stabilizer with trial and error switching gains and the conventional proportional‐integral stabilizer. It is insensitive to the variations of parameters, and is more stable. Furthermore, if the variations of parameters are within an indicated rang...

A framework is proposed for the development and analysis of robust observers for uncertain dynamical systems. A variable structure systems approach is used. Emphasis is placed upon the numerical tractability of the associated synthesis procedure. A selected numerical example is used to illustrate the proposed algorithm.

Performance characteristics of a sliding mode controller depend essentially on a selected sliding manifold. Customarily, the variable-structure controllers operate with linear sliding surfaces. However, using nonlinear or time-varying manifolds may provide definite advantages; in particular, they may provide faster convergence. In this paper, a control algorithm is presented which guarantees the occurrence of the sliding mode on nonlinear or time-varying manifolds. The bounds needed for calculating the switching gains are determined in terms of parameter variations rather than bounds on the system matrices. The parameter bounds are usually evident and can be easily established or estimated, while matrix bounds can rarely be justified in practical problems. In this paper, we also present an adaptive version of the controller, which provides an automatic estimation of the parameter bounds. The robustness of the controller with respect to a class of state-dependent uncertainties is analysed by the Lyapunov m...

Most of the switching surfaces proposed so far for a variable structure control system (VSCS) have been determined independently of initial conditions. The VSCS with these typical switching surfaces may be sensitive to parameter variations and extraneous disturbances during the reaching phase. To overcome this drawback, we propose a new switching surface which is initially designed to pass arbitrary initial conditions, and subsequently move towards a predetermined switching surface by rotating or/and shifting. We call it a moving switching surface (MSS). Using the MSS a low sensitivity system is obtained through shortening the reaching phase. Furthermore, the system robustness is guaranteed during whole intervals of control action by eliminating the reaching phase. To illustrate the advantages of the proposed method, a simple second-order linear system subjected to external disturbance is considered as a preliminary example followed by a two-link manipulator.

Modern microprocessor capabilities permit the control designer to consider using relatively complicated nonlinear control algorithms, which would have been considered impractical in the past. This paper presents the results of a study of the variable structure control technique for the design of excitation and governor controllers for a power system. Control laws for rotor angle and field flux are derived. The closed loop system is shown to be asymptotically stable. The system can be transferred to a new operating condition corresponding to any desired terminal voltage Vt, and tie-line power Ptie.

We propose a design method for finite-dimensional decentralized VS-MRAC of a class of large-scale non-linear interconnected distributed parameter systems with bounded input and output disturbances. Each equivalent control is approximated by each average control and the norm of the output error vector can be made asymptotically arbitrarily small without the ℳ-matrix condition, independently of the reference inputs and in spite of the presence of non-linear interconnections, unmodelled dynamics and disturbances. Furthermore, it is assured that all signals in the closed-loop system are globally ultimately uniformly bounded. Finally, we show the effectiveness of the proposed method by using a numerical example.

To make input-output feedback linearization a practical and systematic design methodology for single-input nonlinear systems, two problems need to be addressed. One is to handle systematically the difficulties associated with the internal dynamics or zero-dynamics when the relative degree is less than the system order. The other is to account for the effect of model uncertainties in the successive differentiations of the output of interest. While the first problem has recently received considerable attention, the second has been largely unexplored. This paper represents a preliminary study of a systematic methodology to account robustly for parametric uncertainties in the original system model. The approach is based on combining sliding control ideas with the recursive construction of a closed-loop Lyapunov function, and is illustrated with a simple example.

The problem is considered of sliding mode synthesis along discontinuity surfaces. The design approach is based on the unit control method under uncertainty conditions and the maximization of control efficiency. The results obtained concern both continuous- and discrete-time systems.