150Danial Senejohnny, Mohammadreza Faieghi and Hadi Delavari

1.Introduction

Among existing control methodologies, the sliding mode control method, first proposed in the early fifties, is one of the control design methods to dominate the uncertainties and disturbances (i.e., the so-called 'matched' disturbances) acting on the systems. The method has gained significant research attention since early sixties in the former USSR and in the modern world since the late seventies and has been widely applied in a variety of applications [1, 2].

In first order sliding mode controller design, the sliding surface is selected such that it has relative degree one with respect to the control input. That means the control input acts on the first derivative of the sliding surface. Higher-order sliding mode is the generalization of the first-order sliding mode and the control input is performed so that it acts on higher derivatives of the sliding surface. The chattering is due to the inclusion of the sign function in the switching term and it can cause the control input to start oscillating around the zero sliding surface, resulting in unwanted wear and tear of the actuators. In general two approaches have been proposed in the literature to solve the problem. The first is to smoothen the switching term as the sliding surface gets closer to zero (soft switching) by using the continuous approximations of the discontinuous sign function, and the second is to generate "higherorder sliding modes'', first introduced by Levant in 1987 [3, 4].

The second-order and higher-order sliding mode approaches have been actively developed over the last two decades for chattering attenuation and robust control of uncertain systems in variety of applications with relative degree two and higher respectively [3-12]. The second-order sliding mode control compared to first-order SMC has the advantage of providing a smooth control and better performance in the control implementation yielding less chattering and better convergence accuracy while preserving the robustness properties.

In recent years, numerous studies and applications of fractional-order systems in many areas of science and engineering have presented [13-16]. Fractional calculus, as old as the ordinary differential calculus, goes back to times when Leibniz and Newton invented differential calculus. Emergence of effective methods in differentiation and integration of non-integer order equations make fractional-order systems and controllers more and more attractive to the control engineers.

In most cases, researchers consider the fractional order controllers applied to the integer or fractional order plants to enhance the system control performance [17-25]. In [17] a new tuning method for fractional order proportional and derivative controller (FO-PD) is proposed for a class of typical second-order plants, existence of fractional adaptive controller based on high gain output feedback for linear, time-invariant, minimum phase, and single input single output systems of relative degree one is investigated in [18], several alternative methods for the control of power electronic buck converters applying fractional order control (FOC) [19], a fractional order disturbance observer for robust vibration suppression in [20], fractional order reference models in model-reference adaptive control investigated in [21], and various kind of fractional sliding mode control strategies with different fractional sliding surfaces in [2225].

In this paper, the objective is to construct a second-order sliding mode control based on a fractional PIDDα sliding surface with independent gain coefficients. Although the theme of 2- SMC is inherently proposed for high frequency chattering suppression, but an adaptive algorithm for of parameter estimation in switching surface is proposed to further suppress this phenomenon. As a way of illustrating the control scheme, simulations are carried out on interconnected twin tank system.

Complimentary Contributor Copy

The rest of this paper is organized as follows, basic mathematical definitions of fractional calculus and problem statement are presented in Section 2. Second-order fractional sliding mode control strategy with fractional PIDDα sliding surface, widely investigated in section 3. Adaptive algorithm for estimating switching surface parameters is given in Section 4. In Section 5, simulation results are given to support the theoretical analysis of the proposed second order fractional sliding mode control and validate its usefulness. A summary of the present research is given in Section 6.

2. Preliminaries

2.1. Basic Definition of Fractional-Order Calculus

Fractional-order integration and differentiation are the generalization of the integer-order ones. Efforts to extend the specific definitions of the traditional integer-order to the more general arbitrary order context led to different definitions for fractional derivatives [13-16]. One of the most commonly used definitions is Caputo definition. In this paper, authors have

where t0 and t are the limits of operation, and is the Gamma function.

The initial conditions for the fractional order differential equations with the Caputo’s derivatives are in the same form as for the integer-order differential equations. The formula for

transform of the Caputo derivative allows utilization of initial values of classical integer order derivatives. For zero initial conditions, Laplace transform of any fractional derivatives reduces to:

2.2. Problem Statement

Suppose a generalized nonlinear dynamic model can be described by a coupled secondorder nonlinear system of the form

Complimentary Contributor Copy

The tracking error can be defined as:

Replacing (4) in (7) yields:

The control objective is to steer the system’s states x(t) to the desired trajectory xd (t) . In

next section the theory of second order sliding mode control with fractional PIDDα sliding surface is studied.

3. Second Order Sliding Mode Control Strategy

In first order sliding mode controller design a desirable sliding surface, namely S , is selected and the feedback control law, acting discontinuously on S , aims to fulfill the constraint S 0 in finite time.

Provided that S, S, , S(r 1) are continuous functions and the rth order sliding set

S S S(r 1) 0 , the rth order sliding mode control objective is to steer to zero not only the sliding surface S in finite time, but also its (r-1) first successive time derivatives by defining a suitable discontinuous control function acting on the rth time derivative of S [9, 12].

In the particular case of the second-order sliding mode control, the control acting on the second derivative of the sliding variable, namely S , aims to steer to zero not only the sliding surface S , but also its first-order time derivative as S S 0 [11].

Complimentary Contributor Copy

The idea of the proposed second-order sliding mode, designed for the special class of nonlinear systems (4), is based on the controller proposed in [10]. The selected sliding surface in this research is a fractional PIDDα function of error (section 3.1) and the switching controller parameters are estimated according to an adaptation law (section 4). The block diagram of the proposed control scheme is depicted in Figure 1

Figure 1. Block diagram of second-order sliding mode controller with fractional sliding surface.

3.1. Second Order Fractional Slidng Surface Synthesis

The problem of interest in the present case is to generate a second-order sliding mode control on a chosen sliding surface S(t) . In the literature, some different sliding functions

were used in the derivation of sliding mode controllers such as, Terminal and fractional sliding surface [24, 26], fractional integral sliding surface [27], PD and fractional PDα sliding surface [25, 28]. In order to utilize the robustness properties of fractional order controller and

reduce the sliding surface settling time, we used the term c D e (t ) beside the term PID where

unanimously construct a robust fractional sliding surface PIDDα [22, 29].

The PIDDα fractional sliding surface with constant coefficients can be introduced as:

Complimentary Contributor Copy

Taking time derivative from sliding surface given in (10) one gets

S (t) S (t) k Pe (t) k D e (t) xd (t) xd (t) k FD D k e (t) k I e (t) (11)

k k k k 2k 1 k 2k 2k 1 2k k 2k 1 k 2k 1

The control input can be given as uk ukeq uksw , where ukeq and uksw are the equivalent control and the switching control, respectively. The equivalent control, ukeq , proposed by Utkin [30] is based on the nominal (estimated) plant parameters (i.e. model uncertainty are not considered) and provides the main control action, while the switching control, uksw ,

ensures the discontinuity of the control law across sliding surface, supplying additional control to account for the presence of model uncertainty.

By substituting (8) into (11) and then equating Sk (t) to zero, the overall control signal become as:

Substitution of (12) into (11), results in:

The switching controllers (13) involve a discontinuous control action owing to the term sgn(Sk (t)) . There arises the practical issue of chattering due to imperfections in switching

devices and delays. In addition, the chattering causes oscillations of the control input around the zero sliding surface, resulting in unwanted wear and tear of actuators. Recently, researchers have utilized the adaptive techniques together with the sliding mode control for many engineering systems to smooth the output from a sliding mode controller and alleviate the chattering in the pure sliding mode control [31]. Therefore, an adaptation law is derived in

section 4 to estimate switching control gains ks , kks . Stability and Lyapunov analysis are carried out in the next subsection.

Complimentary Contributor Copy