PDF Design of Optimal Sliding Mode Controller for Uncertain Systems

1] for uncertain linear system tracking 26 2.5 States obtained by applying the proposed disturbance observer-based OSMC and dis- . 46 3.5 Control input obtained by applying the compensator-based SOSMC proposed by Chang States obtained by applying the proposed OSOSMC.

Introduction

However, performance of an optimal controller is not guaranteed when the system is not in its nominal state. In the second step, a discontinuous switching control is used to keep the system on the sliding surface.

Optimal Sliding Mode Controller

The combination of optimal control with SMC led to optimal sliding mode control (OSMC). Hamilton's method was used to design the optimal regulation, which was combined with sliding mode regulation in [43, 62].

Chattering Mitigation

Motivation

Uncertainty that is not within the range of the system's input matrix is ​​known as the mismatched uncertainty. Recently, the disturbance observer [ 11 , 89 ] was proposed to estimate the mismatch uncertainty affecting a nonlinear system, and the estimate was used to design the SMC to stabilize the nonlinear system.

Contributions of this Thesis

First order optimal sliding mode controller for linear uncertain systems

Optimal second-order sliding mode controller for linear uncertain systems An optimal second-order sliding mode control (OSOSMC) method is proposed for linear uncertain-.

Optimal second order sliding mode controller for linear uncertain systems

State dependent Riccati equation (SDRE) based optimal second order sliding

The advantage of using a non-linear sliding surface, such as a finite sliding surface, is that it converges the proposed integral sliding variable in finite time. A proposed second-order sliding mode controller based on the Riccati equation (SDRE) is used to stabilize certain chaotic systems.

Control Lyapunov function (CLF) based optimal second order sliding mode

Thesis Organization

In the second part, an optimal first-order sliding mode controller is proposed for linear systems affected by the mismatch uncertainty. A second-order sliding mode control method is realized by designing an NTSMC based on an integral sliding variable.

First order optimal adaptive sliding mode controller for the linear uncertain system

In Section 2.3, a first-order optimal sliding state controller is proposed for the linear system affected by the mismatched uncertainty. In this section, a first-order optimal adaptive sliding mode controller (OASMC) is proposed for the linear system affected by the matched uncertainty whose upper bound is unknown.

Stabilization problem

The optimal controller is designed for the nominal linear system using the LQR technique. In Section 2.2, a first-order optimal adaptive sliding mode controller (OASMC) is developed for the linear system affected by matched uncertainty whose upper bound is unknown.

Problem statement

The objective is to design an optimal sliding mode controller for the above uncertain system to achieve stabilization and tracking at the expense of minimal control input. An adaptive tuning law is used to design a sliding state controller with unknown upper bound on the matched uncertainty.

Optimal controller design

Adaptive sliding mode controller design

Tracking problem

Here, an optimal adaptive sliding mode controller (OASMC) is designed to track the desired system trajectory at the expense of minimal control effort. So it will be taken carefully while designing the sliding mode controller by combining with the uncertainty.

Simulation Results

• Disturbance observer based first order optimal sliding mode controller

The states x1 and x2 obtained by using the proposed OASMC, the conventional SMC and the integral SMC proposed by Laghrouche et al. The performance of the proposed OASMC is then compared with that of the integrated SMC proposed by Laghrouche et al.

Figure 2.1: States obtained by applying the proposed OASMC, conventional SMC and integral SMC proposed by Laghrouche et al

Problem statement

A first-order optimal sliding-mode controller is proposed for the linear system affected by mismatched uncertainty. In the second part, the perturbative observer-based sliding mode controlleru2(t) is designed to tackle the mismatch uncertainty.

Optimal controller design

Sliding mode controller based on disturbance observer

Simulation Results

• Summary
• Stabilization of linear uncertain single input single output (SISO) system

The performance of the proposed disturbance observer-based OSMC is compared with that of a disturbance observer-based conventional SMC. In Section 3.2, the design procedure of the optimal second-order sliding mode controller (OSOSMC) for the stabilization of a single input single output (SISO) linear uncertain system is explained.

Figure 2.5: States obtained by applying the proposed disturbance observer based OSMC and disturbance observer based SMC

Sliding mode controller design

In order to bring the system to the sliding surface in finite time, a non-singular final sliding mode controller based on the proposed integral sliding variable s(t) is designed. Furthermore, the terminal slip variable based on the integral slip mode converges to zero at the finite time. The integral slip variable s(t) based on the non-singular finite slip surface σ(t) = 0 results in a second-order SMC.

Moreover, it can also be proved that the integral sliding variable s(t) converges to zero in finite time.

Simulation results

• Tracking of linear uncertain single input single output (SISO) system
• Stabilization of linear uncertain multi input multi output (MIMO) system
• Summary
• Optimal second order sliding mode controller for nonlinear uncertain systems

The proposed objective is to trace the output of the system (y(t) = x1(t)) along the desired trajectory xd(t). The proposed OSOSMC is applied to the suspension control of the magnetic levitation (maglev) vehicle model [5]. Also, the control input of the proposed OSOSMC is significantly smoother than that of Shieh et al.

TV) and the 2-norm of the control input for the proposed OSOSMC and the SMC by Chang [6].

Figure 3.1: Inverted pendulum system

Extended linearization

Simulation results obtained using the proposed optimal second-order sliding mode controller (OSOSMC) are compared with results from the adaptive sliding mode controller (ASMC) proposed by Roopaei et al. Simulation results determine the effectiveness of the proposed SDRE-based optimal second-order sliding mode controller (OSOSMC). The proposed optimal second order sliding mode controller (OSOSMC) is applied to stabilize the system (5.27).

The proposed disturbance observer-based second-order optimal sliding mode controller (DOB- OSOSMC) is compared with the disturbance observer-based sliding mode controller (DOB-SMC) proposed by Yang et al.

System defined in error domain

Optimal control

Optimal feedback control u1(t) is designed in such a way that it minimizes the cost function (4.9) subject to the nonlinear differential constraint (4.8) and also converges the system (4.8) to zero. The basic approach to design an optimal controller for nonlinear systems using the SDRE method is by using extended linearization. The control action is achieved by solving the linear quadratic optimal control problem as state-dependent coefficient matrices that are considered constant.

To design the optimal controller based on SDRE method, HJB equation does not need to be solved.

Stability analysis

Minimization of the performance index

Sliding mode control

However, non-singular terminal sliding mode [77,79] can achieve finite time convergence of the system dynamics. Moreover, it can be proved that integral sliding variable s(t) converges to zero in a finite time. The proposed OSOSMC is designed for trajectory tracking of the nonlinear uncertain system, but it can also be used for stabilization of the nonlinear uncertain system.

To demonstrate the effectiveness of the proposed SDRE-based OSOSMC, it is applied for both trajectory tracking and stabilization of the nonlinear uncertain system.

Tracking of the nonlinear uncertain system

The output and desired states of the proposed SDRE-based OSOSMC and terminal sliding mode controller (TSMC) proposed by Chen et al. The control inputs required by these methods to track the desired state xd= 2 sin(2πt) are shown in Figure 4.4. b) TSMC proposed by Chen et al. a) Control inputs for proposed SDRE-based OSOSMC. The control inputs required by these controllers to track the desired state xd= cost+ 2 sin(2πt) are shown in Figure 4.6. b) TSMC proposed by Chen et al. a) Control inputs for proposed SDRE-based OSOSMC.

It is observed from Table 4.3 that the proposed OSOSMC methodology based on SDRE produces a smoother control input than that of TMSC proposed by Chen et al.

Figure 4.1: Output tracking in Van der Pol circuit [7] for desired state x d = 2

Stabilization of the nonlinear uncertain system

Optimal second order sliding mode controller for chaotic systems

Certain chaotic systems can be represented in a linear structure with state-dependent coefficient matrices. Table 4.5 shows that these chaotic systems can be represented as linear-like structures containing a state-dependent coefficient matrix (SDC). To demonstrate the effectiveness of the proposed optimal sliding mode controller, it is applied for the stabilization of a chaotic system and simulation studies are carried out on different types of chaotic systems.

As such, simulation results are discussed for the case when chaotic systems are affected by uncertainties.

Table 4.5: Examples of chaotic systems

Lorenz system

The states x1(t), x2(t) and x3(t) obtained by applying the proposed OSOSMC are compared with those obtained by using the ASMC proposed by Roopaei et al. The control inputs obtained by using of the proposed OSOSMC and ASMC proposed by Roopaei et al. Moreover, in the case of the proposed OSOSMC, the control input is smooth and without any chatter.

It is clear from Table 4.6 that the proposed OSOSMC is able to produce a smoother control input with significant reduction in control effort than the ASMC proposed by Roopaei et al.

Figure 4.9: State space trajectories of Lorenz system

Liu system

In Section 5.2, a second-order optimal sliding mode controller is designed for a nonlinear system affected by coordinated uncertainty. In Section 5.3, a second-order optimal sliding mode controller is designed for a nonlinear system affected by mismatched uncertainty. A second-order sliding mode controller is implemented using a non-singular finite sliding surface based on an integral sliding variable.

The states and control inputs obtained using the proposed second-order optimal sliding mode control (OSOSMC) and adaptive sliding mode control (ASMC) proposed by Kuo et al.

Figure 4.16: State space trajectories of Liu system after applying the proposed SDRE based OSOSMC

Lorenz-Stenflo

Summary

To reduce the noise, a second-order sliding mode methodology is proposed by designing a non-singular terminal sliding mode controller based on the sliding integral variable. In this chapter, a second-order optimal sliding mode controller (OSOSMC) is proposed to control uncertain nonlinear systems affected by matched and mismatched types of uncertainties. The optimal controller is designed by defining a Lyapunov control function (CLF) and the second-order sliding mode controller is realized by designing a terminal non-singular sliding mode controller based on a sliding integral variable.

A disturbance observer is used to estimate the mismatch uncertainty and the second-order sliding mode controller is designed using an integral sliding variable based non-singular terminal sliding surface.

Optimal second order sliding mode controller for nonlinear systems affected by matched

The optimal controller is designed based on the CLF defined for the nominal nonlinear system. System uncertainties ∆f(x), ∆g(x) and external disturbance d(t) are assumed to satisfy the matching condition which means that they are in the range space of the input matrix. The purpose of the proposed control scheme is to design a chatter-free optimal sliding mode controller for the nonlinear uncertain system (5.3).

The design of the optimal sliding mode controller is followed in two steps viz. i) design the optimal controller for the nominal nonlinear system and (ii) design a sliding mode controller to tackle the uncertainty affecting the system.

Optimal control for the nominal system

If there exists a continuously differentiable, positively determined solution of the HJB equation (5.12), then the optimal controller is defined as [22]. The optimal controller u∗(t) is exactly the same as the controller u1(t) found using Sontag's formula (5.10). To apply the CLF-based optimal controller, exact knowledge of the system under consideration is a necessary condition.

But if the system is affected by uncertainty near the equilibrium state, the performance of the optimal controller degrades and may even fail.

Second order sliding mode control

• Optimal second order sliding mode controller for nonlinear systems affected by mis-
• Summary
• Scope for future work

The proposed optimal second-order sliding mode controller (OSOSMC) is applied for stabilization and tracking problems involving nonlinear systems affected by matching uncertainties. Now, an optimal second-order sliding mode controller is proposed for the nonlinear system affected by mismatch type uncertainty. In order to overcome this inherent difficulty of the OSMC, an optimal second order sliding mode controller (OSOSMC) is proposed.

A second-order optimal sliding mode controller (OSOSMC) using the state-dependent Riccati equation (SDRE) is proposed to control nonlinear uncertain systems affected by coordinated uncertainty.

Figure 5.1: States obtained by applying proposed OSOSMC and ASMC proposed by Kuo et al

Dynamic compensator-based second-order sliding mode controller design for mechanical

A novel higher order sliding mode control scheme proposed by Defoort et al

Adaptive integral higher order sliding mode controller for uncertain systems proposed

Robust output tracking control of an uncertain linear system via a modified optimal

Dynamic sliding mode controller design for chattering reduction proposed by Chang . 146

Nonsingular terminal sliding mode control of nonlinear systems proposed by Feng et al. 148

Sliding mode control with self-tuning law for uncertain nonlinear systems proposed by

Sliding mode control for systems with mismatched uncertainties via a disturbance ob-

• States obtained by applying proposed OSOSMC and ASMC proposed by Kuo et al. [10] 120
• States obtained by applying the proposed CLF based OSOSMC
• States obtained by applying the integral SMC [4]
• Control input obtained by applying the proposed CLF based OSOSMC
• Control input obtained by applying the integral SMC [4]
• States x 1 (t) and x 2 (t) obtained by applying the proposed DOB-OSOSMC
• States x 1 (t) and x 2 (t) obtained by applying the DOB-SMC proposed by Yang et al. [11] 133
• Comparison of control energy of conventional SMC, integral SMC proposed by Laghrouche
• Comparison of control indices of the proposed disturbance observer based OSMC and
• Comparison of Control Indices for Inverted Pendulum
• Comparison of Performance Indices for Inverted Pendulum
• Comparison of Control Indices for Triple Integrator System
• Comparison of Performance Indices for Triple Integrator System
• Comparison of Control Indices for Maglev System
• Comparison of Performance Indices for Maglev System
• Comparison of Control Indices for MIMO System in Example 1
• Comparison of Performance Indices for MIMO System in Example I
• Comparison of control indices to track the output according to x d (t) = 2
• Comparison of control indices to track the output according to x d (t) = 2 sin(2πt)
• Comparison of control indices to track the output according to x d (t) = cos t + 2 sin(2πt) 90
• Examples of chaotic systems
• Comparison of Control Indices
• Comparison of Control Indices
• Comparison of Control Indices

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