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The main goal of this thesis is to develop robust sliding mode control strategies for uncertain systems. Due to the discontinuous control behavior in sliding mode controllers, jitter becomes an inherent undesirable phenomenon.

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List of Publications

Refereed Journals

Conference Proceedings

Introduction
Motivation and purpose
Contributions of this Thesis
Organization of the Thesis

The sliding mode control (SMC) is a specific type of variable structural control system (VSCS) that uses a discontinuous control input. The idea of sliding mode control (SMC) was not known to the control community in general until an article published by Utkin [15] and a book by Itkis [18].

Preliminary Concepts

Introduction
Variable Structure System and Sliding Mode
Stability of the Sliding mode
Relative Degree in Sliding Mode
Order of the sliding mode
Finite time stability
Chattering
Summary

The standard sliding mode can be implemented only if the relative degree of the sliding variable is 1, i.e. The standard sliding mode used in traditional VSS is first order (σ is continuous and ˙σ is discontinuous).

Figure 2.1: Illustration of Filippov method

Adaptive Integral Sliding Mode Controller

Introduction
Problem Deﬁnition
Design of adaptive integral sliding mode controller

Finite time stabilization of an integrator chain system
Design of integral sliding mode controller

Simulation Examples

Adaptive integral chattering free sliding mode controller for the triple inte- grator systemgrator system
Adaptive integral chattering free sliding mode controller for the single in- verted pendulumverted pendulum

Summary

The sliding mode control of the system (3.1) with respect to the sliding variable σ(x) can be expressed as [61]. In particular, prior knowledge about the upper bound of system uncertainty is not a necessary requirement.

Figure 3.1: State response with the proposed control law (3.22)

Adaptive Sliding Mode Controller for Multiple Input Multiple Output

Introduction

The design prerequisite of the sliding mode controller is complete knowledge about the state vector which is not available in this example. In Section 4.4, the effectiveness of this adaptive glide mode controller is validated by applying it for stabilization of the VTOL system.

Adaptive sliding mode controller

Stability during the sliding mode
Design of the control law
Design of the adaptive tuning law

It should be noted that c1 and c2 are designed such that the eigenvalues of axis lie in the left half of the s-plane leaving a positive definite matrix P [74], such that. Thus, a sliding mode controller for the above system can be designed to maintain the system trajectories on the sliding manifold using the control input ˙u=v.

The twin rotor MIMO System

TRMS Description
System Modeling
Design of sliding mode controller for the TRMS horizontal subsystem
Design of adaptive tuning law for the horizontal subsystem
Stability during the sliding mode
Design of sliding mode controller for the TRMS vertical subsystem
Design of adaptive tuning law for the vertical subsystem
Stability of the sliding surface
Control parameters for the horizontal subsystem
Control parameters for the vertical subsystem

For the proposed sliding mode, the first- and second-order time derivatives of the sliding surface are obtained as, . To reduce the displacement of the pitch angle, a proportional plus integral sliding surface is designed for the vertical subsystem. The values of the reaching mode coeﬃcientk1v and the adaptive tuning parameter γv are chosen as 1.3 and 1.25.

The desired and actual TRMS step responses along with the control inputs for the horizontal and vertical subsystems are shown in Figure 4.2. To study the robustness of the adaptive sliding mode controller, an external disturbance d(t) is applied to the TRMS as given below.

Figure 4.1: The twin rotor MIMO system (TRMS) [1]

The vertical take-oﬀ and landing (VTOL) aircraft

Adaptive sliding mode controller design with PI sliding surface
The adaptive PI sliding surface design
Eﬀectiveness
Simulation Results

However, from the above discussion it is clear that the design of the sliding surface requires that the limits of the uncertainties are known a priori [75], which is extremely difficult in practice. Therefore, the need arises to design the sliding surface in such a way that prior knowledge about the limits of the uncertainties is not required. The trajectory of the closed loop system (4.90) can be driven on the sliding manifold l(t) in a finite time using the controller of.

Remark 4.5. The parameterτ in the controller (4.122) is very crucial as it is one of the parameters responsible for determining the convergence rate of the sliding surface. For comparison purposes, the associated design parameters of the proposed adaptive sliding mode controller are selected as follows [3].

Figure 4.7: A typical sketch of a VTOL aircraft in the vertical plane [2].

Case study on 1 degree of freedom (DOF) vertical take-oﬀ and landing (VTOL) aircraft system

Linear extended state observer (LESO) design
Experimental Results

This counterweight allows the position of the weight to be changed, which in turn affects the system dynamics. The initial condition for the VTOL system is chosen as [−0.45 0]T and the initial condition for the observer is also chosen as [−0.45 0]T. The desired path xd(t) to be followed is chosen as xd(t) = 0, i.e. the VTOL system will have to position itself relative to the horizontal plane.

It is clearly seen from Figure 4.18 that the proposed control law is smooth and vibration free. In order to verify the robustness of the proposed adaptive sliding mode controller, the device parameter was

Figure 4.14: QNET VTOL trainer on ELVIS II

Summary

Real-time experiments performed on a 1 DOF VTOL system using its QNET VTOL laboratory prototype are described. The proposed adaptive glide mode controller is applied to a 1 DOF VTOL system to track the desired reference trajectory and the experimental performance is studied. In the experimental setup, the VTOL pitch rate is an unavailable condition that is required for controller design.

So a linear extended mode observer (LESO) is combined with the adaptive sliding mode controller to estimate pitch velocity. The obtained experimental results confirm that the proposed controller is successful in achieving faithful trajectory tracking for 1 DOF VTOL.

Adaptive Terminal Sliding Mode Controller

Introduction

To tackle the problems of global asymptotic stabilization, a terminal sliding mode (TSM) control scheme is developed to achieve time-bounded stabilization. Unfortunately, the terminal sliding mode control has the same drawback of chat [30] as in the case of conventional sliding mode control. In [98], a second-order sliding state controller was developed for multivariable linear systems using the non-singular terminal sliding manifold.

In this chapter, a chatter-free adaptive terminal-sliding-mode (TSM) controller is proposed to achieve fast and finite-time convergence. In Section 5.2, the proposed chattering-free adaptive terminal slide mode (TSM) control strategy is derived.

Design of chattering free adaptive terminal sliding mode con- trollertroller

The number of adaptive rules r is determined by the designer in accordance with the knowledge of the relative order of disturbance that the system may encounter. For example, if r = 0, then the nature of the disturbance is periodic and it is well represented by a known constant value. It will be proved in Theorem 5.1 that since the derivative of the control input contains the discontinuous term, the actual control signal that will be obtained after the integration operation will not contain any high-frequency switching component.

In addition, the controller does not need prior knowledge of the upper limit of the disturbance. In practice, a compromise must be made between response speed and control input.

Stabilization of a triple integrator system

Although the torsion control law reduces the chatter, the control signal is still not smooth as observed in Fig. The state trajectory and control inputs obtained by using the proposed adaptive TSM controller are shown in Fig.

Fig. 5.1 shows the system trajectories and the control input obtained by using the third order sliding mode controller proposed by Defoort et al

Tracking control of a robotic manipulator

Eﬀectiveness
Simulation Studies

It is clear that the implementation of the NTSM controller is very complicated as it involves the computation of the nominal model of the robotic manipulator accompanied by the difficult obligation to know ||F(q,q,˙ q)¨|| A priori. From these figures it is observed that although the tracking performance is satisfactory, the main drawback of NTSM. 4] ensures rapid convergence of the states with the reference, the transient response is highly oscillatory at first.

Output Performance: To evaluate the output performance, the integrated absolute error (IAE) of the output is calculated. The output and input performance of the proposed adaptive TSM controller as well as the controls designed by Feng et al.

Figure 5.4: Conﬁguration of a two-link robotic manipulator

Summary

Nonlinear Sliding Surface based Adaptive Sliding Mode Controller

Introduction

Initially, the damping ratio is chosen as a very low value to ensure fast response, and as the output approaches the reference, the damping ratio is increased to reduce the overshoot. In the CNF method was used to design a non-linear sliding surface which increased the damping ratio from its initial low value as the output approached the set point. This chapter focuses on improving the transient performance of an uncertain system by developing a chatter-free sliding state controller which uses a nonlinear sliding surface. Here, a proportional plus constant reaching law based chatter free gliding mode controller is proposed using a non-linear sliding surface to improve the transient performance.

Furthermore, a discrete integral sliding mode controller based on a nonlinear sliding surface is designed to investigate the operation in the discrete domain. The proposed adaptive sliding mode (SM) controller using a nonlinear sliding surface is discussed in Section 6.2.

Adaptive chattering free sliding mode (SM) controller using nonlinear sliding surface

Stability in sliding mode
Choice of nonlinear function Υ(r, y)

Time response of second order process with time delay
Comparison with adaptive SM controller using linear sliding surface

Stabilization of an uncertain system
Performance comparison with third order sliding mode controller

The performance of the proposed adaptive SM controller using nonlinear sliding surface is compared with that obtained by using adaptive SM controller with different linear sliding surfaces. The simulation results obtained by using the proposed adaptive SM controller are plotted in Fig. Convergence of the slip surface, the slip manifold, and the adaptive gain obtained by using the proposed method is shown in Fig.

The proposed adaptive SM controller using a nonlinear sliding surface is now applied to the triple integrator system. Convergence of the adaptive gain, slip manifold and slip surface using the proposed controller is shown in Fig.

Figure 6.1: Output response of adaptive SM controller with diﬀerent sliding surfaces

Composite nonlinear feedback based discrete integral sliding mode controller

Discrete ISM controller for linear system with matched uncertainty
Composite nonlinear feedback (CNF) based controller design

Single input single output (SISO) system
Comparison of the proposed discrete CNF-ISM controller with diﬀerent discrete ISM controllers
Multiple-input multiple-output (MIMO) system

In this section, two examples are presented to demonstrate the performance of the proposed CNF-ISM discrete controller. show the states x2 and x3 obtained using the proposed discrete CNF-ISM controller as well as the discrete ISM controller (6.57). The proposed CNF-ISM discrete controller consumes 0.5% more power compared to the ISM discrete controller.

show control inputs su1,u2 obtained by using the proposed discrete CNF-ISM controller and those obtained by using the discrete ISM controller proposed by Abidi et al. The proposed discrete CNF-ISM controller consumes 0.7% more energy compared to the controller proposed by Abidi et al.

Fig. 6.10 shows the block diagram of the proposed composite nonlinear feedback (CNF) based discrete integral sliding mode controller (ISM)

Summary

A major advantage offered by the proposed controller is the reduction of chatter in the control input. Simulation results show that the proposed adaptive SM controller provides smooth control action, can converge quickly, and has low sensitivity to parameter variations. The proposed controller uses a state feedback law, which has a low value of the damping ratio at the start, which ensures a fast response.

The nonlinear control law used in the controller increases the damping ratio as the system response approaches the reference input. Illustrative examples demonstrate the effectiveness of the proposed controller for both insecure single-input single-output (SISO) and insecure multiple-output multiple-input (MIMO) systems.

Conclusions and Scope for Future Work

Conclusions

This thesis attempts to design sliding mode controllers for uncertain systems having both matched and mismatched types of uncertainty. For conventional first-order sliding mode controllers, jitter in the control input is a major drawback. This thesis attempts to design free-sliding (SM) controllers to overcome the shortcomings of conventional first-order sliding-mode controllers.

An adaptive sliding mode integral controller using the no-slip technique is proposed in the thesis. An adaptive no-slip control scheme is proposed for a class of dynamic systems with matched and unmatched disturbances.

Scope for future work

An adaptive terminal-sliding mode (TSM) controller is proposed where the non-singular terminal-sliding manifold guarantees fast and finite time convergence. A nonlinear sliding surface-based chatter-free adaptive sliding mode controller is proposed to improve the transient performance of an uncertain system. To improve transient performance in discrete time-uncertain systems, an integral sliding mode with compound nonlinear feedback (CNF) is used.

Another possible extension of this work could be the incorporation of an optimal sliding mode controller to optimize the control effort. Nonlinear sliding surface based adaptive sliding mode controller can be extended to improve the transient performance of general references such as sinusoidal and other periodic signals.

Appendix

Modeling of 1 DOF VTOL Aircraft System

Perruquetti, "En tredje-ordens glidende-mode-controller til en stepmotor,"IEEE Transactions on Industrial Electronics, vol. Usai, "Chattering avoidance by second-orders sliding mode control,"IEEE Transactions on Automatic Control, vol. Utkin, "Om multi-input chatter-fri anden-ordens glidende tilstand kontrol,"IEEE Transactions on Automatic Control, vol.

Yu, “Terminal sliding mode control of linear MIMO systems,” IEEE Transactions on Circuits and Systems -I, vol. Sabanovic, “Sliding mode control for high-precision piezostatic motion,” IEEE Transactions on Industrial Electronics, vol.