Pseudo-State Sliding Mode Control of Fractional SISO Nonlinear Systems

Abstract

This paper deals with the problem of pseudo-state sliding mode control of fractional SISO nonlinear systems with model inaccuracies. Firstly, a stable fractional sliding mode surface is constructed based on the Routh-Hurwitz conditions for fractional differential equations. Secondly, a sliding mode control law is designed using the theory of Mittag-Leffler stability. Further, we utilize the control methodology to synchronize two fractional chaotic systems, which serves as an example of verifying the viability and effectiveness of the proposed technique.

1. Introduction

Fractional calculus has a long history of three hundred years, over which a firm theoretical foundation has been established. In the past few decades, with deep understanding of the power of fractional calculus and rapid development of computer technology, an enormous number of interesting and novel applications have emerged in physics, chemistry, engineering, finance, and other sciences [1–3]. In particular, engineers and scientists from various fields have developed plenty of fractional dynamical systems, that is, systems which are better characterized by noninteger order mathematics models. As has been stated in [3], a suitable way to more efficient control of fractional dynamical systems is to design fractional controllers.

Several pioneering attempts to develop fractional control methodologies have been made, such as TID controller [4], CRONE controller [5], fractional PID controller [6], and fractional lead-lag compensator [7]. Basic ideas and technical formulations of the above four fractional control schemes with comparative comments have been presented in [8].

Very recently, by applying fractional calculus to advanced nonlinear control theory, several fractional nonlinear control schemes have been proposed, such as fractional sliding mode control, fractional adaptive control, and fractional optimal control. Exactly, to design fractional sliding mode controls, various fractional sliding surfaces have been constructed in [9–21]. In particular, adaptive sliding mode controls have been proposed in [12, 19–21], fractional terminal sliding mode controls in [15, 16], and sliding mode controls for linear fractional systems with input and state delays in [17]. In [22], the authors have presented two ideas to extend the conventional model reference adaptive control (MRAC) by using fractional parameter adjustment rule and fractional reference model. In [23], a fractional model reference adaptive control algorithm for SISO plants has been proposed, which can guarantee the stability and ability to reject disturbances. In [24–26], adaptive controllers have been designed to control and synchronize fractional chaotic systems. In [27], Agrawal has proposed a general formulation for a class of fractional optimal control problems, which is specialized for a system with quadratic performance index subject to a fractional system dynamic constraint. In [28], the authors have generalized the optimality conditions of [27]. Besides the above fractional control methodologies, fractional optimal synergetic control has been presented in [29] and active disturbance rejection control in [30].

Motivated by the above contributions, this paper proposes a sliding mode control design for fractional SISO nonlinear systems in the presence of model inaccuracies. By constructing a stable fractional sliding mode surface on the basis of Routh-Hurwitz conditions, a sliding mode control law is designed. Further, stability analysis is performed using Mittag-Leffler stability theory. Comparing this with methods in the previous papers, we utilize the fractional derivative of the sliding mode surface instead of first-order derivative, to obtain the equivalent control law. Moreover, to carry out the stability analysis of the closed-loop fractional nonlinear system, we use the fractional derivative of the Lyapunov function candidate in terms of Theorem 2 in [18].

The rest of the paper is organized as follows. Section 2 reviews some basic definitions for fractional calculus. Section 3 proposes the sliding control design for fractional SISO nonlinear systems. Numerical simulations of synchronization of the fractional Genesio-Tesi system and the fractional Arneodo system are presented in Section 4. Finally, Section 5 concludes this paper with some remarks on future study.

2. Basic Definitions for Fractional Calculus

Fractional calculus is a generalization of integration and differentiation to noninteger order fundamental operator _𝑎𝐷^𝛼_𝑡, where 𝑎 and 𝑡 are the bounds of the operation and 𝑎∈𝑅. The continuous integrodifferential operator is defined as [1] _𝑎𝐷^𝛼_𝑡=𝑑^𝛼𝑑𝑡^𝛼∫𝛼>01𝛼=0_𝑡𝑎(𝑑𝜏)^𝛼𝛼<0. (1)

The three most frequently used definitions for fractional calculus are the Grünwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition [1–3].

Definition 1. The Grünwald-Letnikov derivative definition of order 𝛼 is described as _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=lim_ℎ→01ℎ^𝛼∞_𝑗=0(−1)^𝑗𝛼𝑗𝑓𝑡−𝑗ℎ. (2)

Definition 2. The Riemann-Liouville derivative definition of order 𝛼 is described as _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡_𝑎𝑓(𝜏)𝑑𝜏(𝑡−𝜏)^{𝛼−𝑛+1},𝑛−1<𝛼<𝑛. (3)

However, applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contains 𝑓(𝑎), 𝑓^′(𝑎), and so forth. Unfortunately, the Riemann-Liouville approach fails to meet this practical need. It is M.  Caputo who solved this conflict.

Definition 3. The Caputo definition of fractional derivative can be written as _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎𝑓^(𝑛)(𝜏)𝑑𝜏(𝑡−𝜏)^{𝛼−𝑛+1},𝑛−1<𝛼<𝑛. (4)

In the rest of the paper, we use the Caputo approach to describe the fractional systems and the Grünwald-Letnikov approach to propose numerical simulations. To simplify the notation, we denote the fractional-order derivative of order 𝛼 as 𝐷^𝛼 instead of ₀𝐷^𝛼_𝑡 in this paper.

3. Sliding Control of Fractional SISO Nonlinear Systems

Consider fractional SISO nonlinear systems 𝐷^𝛼𝑥_𝑖=𝑥_𝑖+1𝐷,𝑖=1,2,…,𝑛−1,_𝛼𝑥_𝑛=𝑓(𝐱)+𝑔(𝐱)𝑢,𝑦=𝑥₁, (5) where 𝛼∈(0,1] is the order of the dynamic system, 𝐱=(𝑥₁,𝑥₂…,𝑥_𝑛)^𝑇∈ℝ^𝑛 is denoted as state vector, and the scalar 𝑢,𝑦∈𝑅 are system’s input and output, respectively. The dynamic 𝑓(𝐱) (possibly nonlinear or time varying) is not exactly known, but estimated as 𝑓(𝐱). The control gain 𝑔(𝐱) (possibly time varying or state dependent) is an unknown function.

Assumption 4. The estimation error on 𝑓(𝐱) is assumed to be bounded by some known function 𝐹(𝐱): 𝑓(𝐱)−𝑓(𝐱)≤𝐹(𝐱). (6)

Assumption 5. 𝑔(𝐱) is assumed to be bounded by 0<𝑔_min(𝐱)≤𝑔(𝐱)≤𝑔_max(𝐱), (7) where both 𝑔_min(𝐱) and 𝑔_max(𝐱) are known functions.

Assumption 6. The desired tracking signal is denoted as 𝑦_𝑑(𝑡), and it is assumed that 𝑦_𝑑(𝑡)∈𝐶^𝑛[0,∞) and 𝑦_𝑑(𝑡),  𝐷^𝛼𝑦_𝑑(𝑡),…,𝐷^𝑛𝛼𝑦_𝑑(𝑡) are all bounded.

In this paper, our goal is to design a suitable fractional sliding mode controller to make the output 𝑦 of the system (5) track the desired signal 𝑦_𝑑; that is, lim_𝑡→∞𝑦−𝑦_𝑑=0. (8)

Now we are ready to give the design steps.

Firstly, the tracking error is defined as 𝑥=𝑦−𝑦_𝑑. (9)

Then, the tracking error vector is defined as 𝐱=𝑥−𝑦_𝑑,𝐷^𝛼𝑦_𝑑,…,𝐷^{(𝑛−1)𝛼}𝑦_𝑑^𝑇=𝑥,𝐷^𝛼𝑥,…,𝐷^{(𝑛−1)𝛼}𝑥^𝑇. (10)

A fractional sliding surface is proposed as 𝑠(𝑡)=𝑐₁𝑥+𝑐₂𝐷^𝛼𝑥+⋯+𝑐_𝑛−1𝐷^{(𝑛−2)𝛼}𝑥+𝐷^{(𝑛−1)𝛼}𝑥, (11) where the polynomial 𝑃(𝑝)=𝑐₁+𝑐₂𝑝+⋯+𝑐_𝑛−1𝑝^𝑛−2+𝑝^𝑛−1 satisfies Routh-Hurwitz conditions for fractional differential equations [31].

The fractional sliding surface (11) can be rewritten as 𝑠(𝑡)=𝐷^{(𝑛−1)𝛼}𝑥+^𝑛−1_𝑖=1𝑐_𝑖𝐷^{(𝑖−1)𝛼}𝑥=𝐷^{(𝑛−1)𝛼}𝑦−𝐷^{(𝑛−1)𝛼}𝑦_𝑑+^𝑛−1_𝑖=1𝑐_𝑖𝐷^{(𝑖−1)𝛼}𝑥=𝐷^{(𝑛−1)𝛼}𝑥₁−𝐷^{(𝑛−1)𝛼}𝑦_𝑑+^𝑛−1_𝑖=1𝑐_𝑖𝐷^{(𝑖−1)𝛼}𝑥=𝑥_𝑛−𝐷^{(𝑛−1)𝛼}𝑦_𝑑+^𝑛−1_𝑖=1𝑐_𝑖𝐷^{(𝑖−1)𝛼}𝑥. (12)

Taking its 𝛼th order derivative with respect to time yields 𝐷^𝛼𝑠(𝑡)=𝐷^𝛼𝑥_𝑛−𝐷^𝑛𝛼𝑦_𝑑+^𝑛−1_𝑖=1𝑐_𝑖𝐷^𝑖𝛼𝑥. (13)

Substituting the second equation of system  (5) into  (13), one has 𝐷^𝛼𝑠(𝑡)=𝑓(𝑥)+𝑔(𝑥)𝑢−𝐷^𝑛𝛼𝑦_𝑑+^𝑛−1_𝑖=1𝑐_𝑖𝐷^𝑖𝛼𝑥. (14)

In terms of Assumption 5, one of the estimated values of 𝑔(𝐱) can be chosen as 𝑔(𝐱)=𝑔_min(𝐱)𝑔_max(𝐱). (15)

Bounds of 𝑔(𝑥) are written as 𝛽⁻¹≤𝑔(𝐱)𝑔(𝐱)≤𝛽, (16) where 𝛽=𝑔_max/𝑔_min.

Design the control law as 𝑢(𝑡)=1𝑔(𝐱)−𝑓(𝐱)+𝐷^𝑛𝛼𝑦_𝑑−^𝑛−1_𝑖=1𝑐_𝑖𝐷^𝑖𝛼𝑥(𝐱)(𝑠)=𝑢−𝑘⋅sign−𝜎𝑠_𝑒𝑞(𝑡)+𝑢_𝑟(𝑡), (17) where 𝑢_𝑒𝑞(𝑡)=(1/𝑔(𝐱))[−𝑓(𝐱)+𝐷^𝑛𝛼𝑦_𝑑−∑^𝑛−1_𝑖=1𝑐_𝑖𝐷^𝑖𝛼𝑥], 𝑢_𝑟(𝑡)=(1/𝑔(𝐱))[−𝑘(𝐱)⋅sign(𝑠)−𝜎𝑠], 𝑘(𝐱)≥𝛽(𝐹(𝐱)+𝜂)+𝛽(𝛽−1)|𝑢_𝑒𝑞|, and 𝜂 is a positive constant.

To ensure the stability of the fractional system (5), we have the following theorem.

Theorem 7. Under Assumptions 4–6, the fractional SISO nonlinear system (5) can be controlled using the sliding mode control law (17) with the fractional sliding surface   (11).

Proof. Consider the following candidate Lyapunov function: 𝑉(𝑡)=[𝑠(𝑡)]², (18) where 𝑠(𝑡) is the fractional sliding surface (11) constructed previously.
Taking its 𝛼th order derivative with respect to time along with the fractional sliding surface (11), one has 𝐷^𝛼𝑉=𝑠⋅𝐷^𝛼𝑠+^∞_𝑗=1Γ1+𝑞ΓΓ𝑠1+𝑗1−𝑗+𝑞^(𝑗)𝐷^𝑞−𝑗𝑠≤𝑠⋅𝐷^𝛼𝑠+𝜌, (19) where 𝜌 is assumed to be an arbitrarily large positive constant which is a bound on the series of (19).
Substituting (14) into (19) yields 𝐷^𝛼𝑉≤𝑠𝑓(𝐱)+𝑔(𝐱)𝑢−𝐷^𝑛𝛼𝑦_𝑑+^𝑛−1_𝑖=1𝑐_𝑖𝐷^𝑖𝛼𝑥+𝜌. (20)
Substituting the control law (17) into (20) gives 𝐷^𝛼𝑉≤𝑠𝑓−𝑔𝑔⁻¹𝑓+1−𝑔𝑔⁻¹−𝐷^𝑛𝛼𝑦_𝑑+^𝑛−1_𝑖=1𝑐_𝑖𝐷^𝑖𝛼𝑥−𝑔𝑔⁻¹𝑘sign(𝑠)−𝜎𝑠²𝑓+𝑔+𝜌=𝑠𝑓−𝑓−𝑔⁻¹𝑓+1−𝑔𝑔⁻¹×−𝐷^𝑛𝛼𝑦_𝑑+^𝑛−1_𝑖=1𝑐_𝑖𝐷^𝑖𝛼𝑥−𝑔𝑔⁻¹𝑘sign(𝑠)−𝜎𝑠²𝑓+𝑔𝑔+𝜌=𝑠𝑓−⁻¹−1𝑢_𝑒𝑞−𝑔𝑔⁻¹𝑘sign(𝑠)−𝜎𝑠²+𝜌. (21)
Substituting the estimation error (6), the equivalent control law 𝑢_𝑒𝑞, and control gain 𝑘 of (17) into (21), one derives 𝐷^𝛼𝑉≤−𝜂|𝑠|−𝜎𝑠²+𝜌<−𝜎𝑉+𝜌, (22) which implies that the sliding mode dynamic is globally stable and the tracking error vector (10) converges to zero according to Theorem 2 in [18]. This proves that the fractional SISO nonlinear system (5) can be controlled using the sliding mode control law (17) with the fractional sliding surface (11).

4. Numerical Simulations

In this section, we apply the fractional sliding mode control method proposed in Section 3 to deal with the problem of synchronization between the fractional Arneodo system and the fractional Genesio-Tesi system. To carry out numerical simulations, we utilize the algorithm for numerical calculation of fractional derivatives introduced in [1]. This method is derived from the Grnwald-Letnikov Definition 1 based on the fact that the three Definitions 1, 2, and 3 are equivalent for a wide class of functions.

The fractional Arneodo system is represented as 𝐷^𝛼𝑦₁=𝑦₂,𝐷^𝛼𝑦₂=𝑦₃,𝐷^𝛼𝑦₃=5.5𝑦₁−3.5𝑦₂−𝑦₃−𝑦³₁ (23) with the desired tracking signal 𝑦_𝑑=𝑦₁.

The fractional Genesio-Tesi system is described as 𝐷^𝛼𝑥₁=𝑥₂,𝐷^𝛼𝑥₂=𝑥₃,𝐷^𝛼𝑥₃=𝑎(𝑡)𝑥₁+𝑏(𝑡)𝑥₂+𝑐(𝑡)𝑥₃+𝑑(𝑡)𝑥²₁+𝑢(𝑡),𝑦=𝑥₁, (24) where 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), and 𝑑(𝑡) are assumed to be unknown parameters and satisfy |𝑎(𝑡)+6|≤0.1,  |𝑏(𝑡)+2.92|≤0.1, |𝑐(𝑡)+1.2|≤0.1, and |𝑑(𝑡)−1|≤0.1, from which one has 𝑓(𝐱)=𝑎(𝑡)𝑥₁+𝑏(𝑡)𝑥₂+𝑐(𝑡)𝑥₃+𝑑(𝑡)𝑥²₁,𝑓(𝐱)=−6𝑥₁−2.92𝑥₂−1.2𝑥₃+𝑥²₁,𝐹(𝐱)=0.1𝑥₁+𝑥₂+𝑥₃+𝑥²₁. (25)

Both of the fractional Arneodo system (23) and the fractional Genesio-Tesi system   (24) exhibit chaos with certain values of fractional order 𝛼. In the sequel we investigate synchronization between the two fractional chaotic systems using the technique proposed previously. The former system is taken as drive system, while the latter one is considered as response system.

In terms of (11), the sliding surface is constructed as 𝑠=𝑐₁𝑥₁−𝑦₁+𝑐₂𝑥₂−𝑦₂+𝑥₃−𝑦₃, (26) where 𝑐₁=25 and 𝑐₂=10 satisfy the Routh-Hurwitz conditions [31].

The control law is designed as 𝑢=6𝑥₁+2.92𝑥₂+1.2𝑥₃−𝑥²₁+5.5𝑦₁−3.5𝑦₂−𝑦₃−𝑦³₁−25𝑥₂−𝑦₂−10𝑥₃−𝑦₃(𝑠).−𝜎𝑠−𝜎𝑠−𝑘⋅sign (27)

Initial conditions for the above two systems are, respectively, chosen as 𝑥₀=(−0.1,0.5,0.2) and 𝑦₀=(−0.2,0.5,0.2), while parameters in the control law (27) are selected as 𝜂=0.1 and 𝜎=0.5. Numerical simulations of synchronization between the fractional chaotic systems (23) and (24) are presented in Figures 1, 2, 3, and 4, with the simulation time 𝑇_sim=10 and time step ℎ=0.0005. For interpretations of the references to color in these figure legends, the reader is referred to the online version of this paper.

(a)

(b)

(c)

Figure 1 

Synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system with the control input (27) (red line represents the trajectories of the drive system, while blue line represents the trajectories of the response system).

Figure 2 

Synchronization errors with the control input (27) (red line represents the first error state 𝑒1, blue line represents the second error state 𝑒2, and black line represents the third error state 𝑒3).

Figure 3 

Time history of fractional sliding mode with the control input (27).

Figure 4 

Time history of the control input (27).

From the simulation results, we see that synchronization performance is excellent but is obtained at the price of high control chattering. It can be eliminated by replacing the discontinuous switching control law sign(⋅) by the smooth function tanh(⋅) in (27); that is, 𝑢=6𝑥₁+2.92𝑥₂+1.2𝑥₃−𝑥²₁+5.5𝑦₁−3.5𝑦₂−𝑦₃−𝑦³₁−25𝑥₂−𝑦₂−10𝑥₃−𝑦₃−𝜎𝑠−𝑘⋅tanh(𝑠). (28)

Numerical simulations with the modified control law (28) are presented in Figures 5, 6, 7, and 8. For interpretations of the references to color in these figure legends, the reader is referred to the online version of this paper.

(a)

(b)

(c)

Figure 5 

Synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system with the control input (28) (red line represents the trajectories of the drive system, while blue line represents the trajectories of the response system).

Figure 6 

Synchronization errors with the control input (28) (red line represents the first error state 𝑒1, blue line represents the second error state 𝑒2, and black line represents the third error state 𝑒3).

Figure 7 

Time history of fractional sliding mode with the control input (28).

Figure 8 

Time history of the control input (28).

From the above simulation results, one can easily see that the fractional Arneodo system and the fractional Genesio-Tesi system can be effectively synchronized via the proposed sliding mode control technique. Furthermore, the control chattering caused by the discontinuous control law (17) is successfully eliminated by the modification of (28).

5. Concluding Remarks

In this paper, we have investigated the pseudo-state sliding control design for fractional SISO nonlinear systems with model inaccuracies. A stable fractional sliding mode surface has been constructed based on the Routh-Hurwitz conditions for fractional differential equations. Then, a sliding mode control law is designed using the Mittag-Leffler stability theorem. Finally, numerical simulations of synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system have been performed to demonstrate the effectiveness of the proposed control technique.

As for the future perspectives, our research activities will be on,(i)designing   adaptive sliding control to deal with parametric uncertainties in 𝑓(⋅),(ii)generalizing  the method to fractional MIMO nonlinear systems,(iii)generalizing  the method to incommensurate nonlinear systems.

Conflict of Interests

The authors do not have any possible conflict of interests.