Sliding Mode-Like Fuzzy Logic Control With Adaptive Boundary Layer for Multiple-Variable Discrete Nonlinear Systems

1 Introduction

The sliding mode control (SMC) attracted so many interests of scholars during the past several decades, thereby, spread research work could be referred for its main theory [5, 18, 19]. Owing to the robustness, fast response, invariance to system uncertainties, and compensation ability to the external disturbance, SMC has widely been applied to many industrial processes, such as flying craft control, motor drive and chemical engineering, etc.

In practice, the control scheme is always implemented by a computer or a DSP apparatus. Therefore, the discrete sliding mode control (DSMC) is with valuation of engineering. It is convenient for realization on digital devices, which forms a sampled data system. The field of DSMC has attracted numerous research works for a long time. Typically, classic reports were preferred in [8, 17], and recently, research works were preferred in [3, 4, 10, 12, 14–16].

However, SMC has the obvious shortcoming, chattering phenomenon. It impedes its practical application. In particular, for discrete systems, the chattering becomes more serious because of the sampling, digitization of analog signals, and complicated discrete modeling [13]. Herewith, Monsees and Scherpen presented a method of on-line tuning, the switching gain by adaptive law to reduce the chattering for the concerned discrete systems.

On the other hand, fuzzy logic control (FLC) is widely applied especially to practical control system, in which the precise mathematics model cannot be acquired easily. Fuzzy logic systems (FLS) are applied to sliding mode control systems in order to improve the performance of SMC, especially for the chattering elimination problem. Correspondingly, there are two combined methods: fuzzy sliding mode control (FSMC) [2, 7, 9, 20] and sliding mode fuzzy logic control (SMFC) [1, 6, 11, 22]. In the former, the unknown system dynamics are identified by FLS for computing the equivalent control of SMC. In the latter, FLC is designed based on the sliding mode, in order to be equivalent to the sliding mode control. In this field, a sliding mode controller and a FLC controller are amalgamated. The advantages of the SMC and the FLC controller can be combined, and their disadvantages can be removed. Allaoua proposed a new FLC based on sliding mode taking advantage of support vector machines (SVM) [1] and applied it to the electric vehicle propulsion system. Farhoud and Erfanian presented a FLC control method based on higher-order sliding mode [6], which owns good transient and steady-state responses.

Lhee et al. proposed a sliding mode-like fuzzy logic control (SMLFC) approach [11]. In their method, the SMC controller first is designed in which boundary layer thickness is tuned online by some appropriate adaptive rules. Then FLC controller is designed as equivalence to the predesigned SMC controller. In this way, the controller has no chattering. However in [11], the methods are only applicable to systems with second-order dynamics. Furthermore, as the dead zone parameter converges to zero, signal output of fuzzy controller also has a little chattering. Zhang et al. proposed an improved SMLFC method based on [11]. The method is extent, which has the application ability for high-order nonlinear systems, and the dead zone parameter does not converge to zero [22]. Furthermore, the method was improved for more general affine nonlinear systems with uncertainties [21]. For discrete nonlinear systems, there rarely is reported research works of the SMLFC with a boundary layer self-tuning so far.

Motivated by [11, 21, 22] and extending the results for continuous systems to discrete systems, in this paper, a SMLFC design issue to discrete nonlinear systems is mainly addressed. The SMLFC for discrete nonlinear systems is mainly considered in this paper. Based on the reported works, controller design is enhanced to be applicable to discrete nonlinear systems with nonlinear control gains. Dynamic fuzzy logic systems (DFLS) are also utilized for a more smoothing performance in which parameters can be regulated online to acquire better performance. Like [11, 21, 22], the FLC controller, which is equivalent to the predesigned SMC controller, is also obtained finally. The chattering phenomenon will be eliminated completely.

The remainder of the paper is as follows. Section 2 presents the problem formulation and general SMC design. SMC with boundary layer thickness self-tuning is approached in Section 3, and the reachability of the sliding mode will be verified by the Lyapunov stability theory. Section 4 shows the FLC controller design, which is equivalent to the predesigned SMC. In order to validate the scheme, numerical examples are given, and their simulation results are proposed in Section 5. Section 6 summarizes the conclusions.

2 Problem Formulation

Consider a class of multiple-variable discrete nonlinear systems with the form of

where y_k=[y_1,k, y_2,k, …, y_m,k]^T∈R^m denotes the output vector, yk(r) = [y1,k(r1), y2,k(r2), …, ym,k(rm)]T denotes its r_i-order difference (i=1,2, … m) with r=[r₁, r₂, …, r_m]^T defined as the system relative degree and fulfilled ∑i = 1m ri = n, and u_k=[u_1,k, u_2,k, …, u_m,k]^T denotes the control input vector and the form of

xk = [x1,k, x2,k, …, xn,k]T = [y1,kT, y2,kT, …, ym,kT]T

denotes the state vector with yi,k = [yi,k, yi,k+1, …, yi,k+ri−1]T, i = 1, 2, …, m.

Additionally, f(x_k, k)=[f₁(x_k, k), f₂(x_k, k), …, f_m(x_k, k)]^T denotes the nonlinear dynamic function vector, and g(x_k, k)=diag{g₁(x_k, k), g₂(x_k, k), …, g_m(x_k, k)} denotes the control gain matrix.

It is assumed that y_d,k=[y_d1,k, y_d2,k, …, y_dm,k]^T∈R^m is the corresponding desired trajectory vector. Correspondingly, the trajectory state vector is

xd,k = [xd1,k, xd2,k, …, xdn,k]T = [yd1,kT, yd2,kT, …, ydm,kT]T

where ydi,k = [ydi,k, ydi,k+1, …, ydi,k+ri−1]T, i = 1, 2, …, m. Without loss of generality, suppose that y_di,k is bounded for all time interval k (k∈[0, +∞]). The tracked model

is given, and r_i(k) is the reference input signal.

The control problem is to construct the sliding mode control law u_i,k such that the error vectors

converge to the tolerable range, where s_i(k) is the sliding mode surfaces

and the vectors c_i for i=1, 2, …, m are coefficients of Hurwitz polynomials.

As follows, only one subsystem of Eq. (1) is analyzed as for i=1, 2, …, m every subsystem fulfills the same principles.

The following assumptions underlie the proposed method.

Assumption 1: The nonlinear dynamic function f_i,k [i.e. f_i(x_k, k)] can be estimated as f^i,k, and the nonlinear control gain g_i,k [i.e. g_i(x_k, k)] can be estimated as g^i by designers where g^i is a constant scalar.

It follows that in the above Assumption 1, the nonlinear uncertainty Δf_i and Δg_i are defined as

Assumption 2: Without loss of generality, suppose that g_i(x_k, k)>0, g^i > 0 and Δg̅_i are the upper boundary of Δg_i. Then, g^i−1Δgi > −1 holds.

For convenience, define that

Then, δ_i>0, 0<γ_i≤γ̅_i with γ¯i = 1 + g^i−1Δg¯i. Here, u_eq,i is the equivalent control of the sliding mode control u_i,k.

Assumption 3: The uncertainties Δf_i and Δg_i are all bounded.

According to the basic sliding mode theory in [5] and parts of the content in [18] if the SMC control input u_i,k is constructed as

where

and

is the thickness of the boundary layer [11] with η_i>0 and ε_i>0 as positive scalars that can be optionally small. k_i,1 is given by

Then, the reaching condition

can be satisfied, where Δ[·]:z(k)→z(k+1)–z(k), z∈R. Then, the system tracking error vector e_i,k is bounded.

Theorem 1: For every error subsystem of systems (1)–(3) and its sliding mode (4), the reaching condition (11) can be satisfied, and the switching area B_i={e_i,k||s_i(k)|≤φ_i} is also reached and stable, if the control is designed as Eqs. (7), (8). Furthermore, s_i(k) can be driven to S_i={e_i,k|s_i(k)=0} if Δf_i=Δg_i=0.

Proof: For every subsystem of Eq. (1), the Lyapunov function Vi(k) = si2(k) is selected, and then, its difference

ΔVi(k) = Vi(k + 1) − Vi(k) = Δsi(k)[Δsi (k) + 2si(k)]

is derived, whereas Eq. (11) equals

According to Eqs. (1)–(6),

is obtained certainly.

If s_i(k)>φ_i, let s_i(k)=φ_i+ξ_i,1(k), where ξ_i,1(k)>0, then Eq. (12) becomes

[Δfi + Δgiui,eq − δi − ηi]︸negative[Δfi + Δgiui,eq − δi − ηi + 2φi + 2ξi,1(k) − ηi] < 0.

Using Eqs. (6) and (9), the above becomes

[Δfi + Δgiui,eq − δi − ηi]︸negative[Δfi + Δgiui,eq + δi + 2εi + 2ξi,1(k)]︸positive < 0,

which implies that inequality (12) is satisfied.

If s_i(k)<–φ_i, let s_i(k)=–φ_i–ξ_i,1(k), where ξ_i,1(k)<0, then Eq. (12) becomes

[Δfi + Δgiui,eq + δi + ηi][Δfi + Δgiui,eq + δi + ηi − 2φi − 2ξi,1(k) + ηi] < 0.

Using Eqs. (6) and (9), the above becomes

[Δfi + Δgiui,eq + δi + ηi]︸positive[Δfi + Δgiui,eq − δi − 2εi − 2ξi,1(k)]︸negative < 0,

which implies that inequality (12) is again satisfied. Thus inequality (11) holds, and the switching area B_i is reachable and stable according to the Lyapunov stability theory.

If Δf_i=Δg_i=0, from Eq. (13), one has si(k + 1) = [1−(δi + ηi)φi]si(k), and it is obvious from Eq. (9) that −1 < 1 − (δi + ηi)φi < 1, which implies that s_i(k) is a stable first-order filter that asymptotically approaches S_i={e_i,k|s_i(k)=0}.■

However, parameters γ_i and δ_i are all unknown. Therefore, k_i,1 and φ_i cannot be determined, which makes control (7) be not implemented. In the next section, we will find a way to estimate them and consider the chattering elimination at the same time.

3 SMC With Boundary Layer Self-Tuning

Suppose that k_i,1 is a constant scalar. If k_i,1 is replaced by k_i,1–Δφ_i(k)–|s_i(k)| in Eq. (7), then Eq. (11) changes to

Theorem 2: For every error subsystem of systems (1)–(3) and its sliding mode (4), the reaching condition (14) can be satisfied, and the switching area B′i = {ei,k||si(k)| ≤ φi(k)} is also reached and stable, if the control is designed as Eqs. (7), (8). Furthermore, the sliding mode s_i(k) can be driven to S′i = {ei,k|si(k) = 0} if Δf_i=Δg_i=0.

Proof: According to the proof of Theorem 1, the Lyapunov function Vi(k) = si2(k) is also used here, and φ_i is replaced with φ_i(k), η_i is replaced with η_i–Δφ_i(k)–|s_i(k)|, then Eq. (11) equals

Δsi(k){Δsi(k) + 2si(k) + [ηi − Δφi(k) − |si(k)|]sgn[Δsi(k)]} < 0.

If s_i(k)>φ_i, let s_i(k)=φ_i+ξ_i,1(k), where ξ_i,1(k)>0, one has

[Δfi + Δgiui,eq − γiki,1]︸negative[Δfi + Δgiui,eq − γiki,1 + 2φi + 2ξi,1(k) − ηi + Δφi(k) + |si(k)|] < 0.

Obviously, η_i is replaced with η_i–Δφ_i(k)–|s_i(k)| in Eqs. (9), (10); the above becomes

[Δfi + Δgiui,eq − γiki,1]︸negative[Δfi + Δgiui,eq + δi + 2εi + 2ξi,1(k)]︸positive < 0,

which implies that inequality (12) is satisfied.

If s_i(k)<–φ_i, let s_i(k)=–φ_i–ξ_i,1(k), where ξ_i,1(k)<0, then Eq. (14) becomes

[Δfi + Δgiui,eq + γiki,1]︸positive[Δfi + Δgiui,eq + γiki,1 − 2φi − 2ξi,1(k) + ηi − Δφi(k) − |si(k)|] < 0.

Using Eqs. (6), (9), and η is replaced with η–Δφ(k)–|s(k)|, the above becomes

[Δfi + Δgiui,eq + γiki,1]︸positive[Δfi + Δgiui,eq − δi − 2εi − 2ξi,1(k)]︸negative < 0,

which implies that inequality (12) is satisfied.

Then, s_i(k) converges to B′i = {ei,k||si(k)| ≤ φi(k)}, where φ_i(k)=δ_i+η_i–Δφ_i(k)–|s_i(k)|+ε_i. Namely,

|si(k)| ≤ δi + ηi − Δφi(k) − |si(k)| + εi ⇔ 0.5|si(k)| ≤ δi + ηi − Δφi(k) + εi,

and furthermore,

φi(k) = δi + ηi − Δφi(k) + εi ⇔ φi(k + 1) = δi + ηi + εi,

thus, |s_i(k)|≤φ_i(k) is reachable.■

Now the reaching condition (14) is modified as

where λ_i,1, λ_i,2>1 are optional positive scalars. Then, the following theorem is referred.

Theorem 3: For every error subsystem of systems (1)–(3) and its sliding mode (4), the reaching condition (15) can be satisfied, and the switching area B′i = {ei,k||si(k)| ≤ φi(k)} is also reached and stable, if the control is designed as Eqs. (7), (8). Furthermore, the sliding mode s_i(k) can be driven to S′i = {ei,k|si(k) = 0} if Δf_i=Δg_i=0.

Proof: According to the proof of Theorem 2, if φ_i is replaced with φ_i(k) and η_i with η_i–λ_i,1Δφ_i(k)–λ_i,2|s(k)|, it is referred that B′i = {ei,k||si(k)| ≤ φi(k)} is reachable and φ_i(k)=δ_i+η_i–λ_i,1Δφ_i(k)–λ_i,2|s_i(k)|+ε_i. Namely,

|si(k)| ≤ δi + ηi − λi,1Δφi(k) − λi,2|si(k)| + εi ⇔ (1 + λi,2)|si(k)| ≤ δi + ηi − λi,1Δφi(k) + εi

Holds, and then, φ_i(k)=δ_i+η_i–λ_i,1Δφ_i(k)+ε_i, i.e.

Obviously, |s_i(k)|≤φ_i(k), where φi(k) = 1λi,1(δi + ηi + εi).■

However k_i,1 or k_i,1–Δφ_i(k) is unknown in practice. Definingk^i,1, it is the estimating value of k_i,1–Δφ_i(k), and k˜i,1 = k^i,1 − [ki,1 − Δφi(k)] is the error of the estimation; then, condition (11) should be satisfied. Choose the following adaptive laws

where parameters satisfy

βi > 1 + γ¯i, αi < −4(1+γ¯i), τi > 1.

Suppose that k^i,1(0) > 0,φ_i(0)>0, we get the following theorem.

Theorem 4: For every error subsystem of systems (1)–(3) and its sliding mode (4) with the control law (7) and the adaptive law (17), (18), the sliding mode s_i(k) reaches to B′i = {ei,k||si(k)| ≤ φi(k)}, and the closed-loop system is asymptotically stable if the control gain k_i,1 in Eq. (7) is replaced with k^i,1 by being adaptively tuned online.

Proof: Choose a Lyapunov function as

and seek the time derivate of V_i along the trajectory of the closed-loop system. One can obtain

ΔVi(k) = si2(k + 1) − si2(k) + k˜i,12(k + 1) − k˜i,12(k),

and the inequality (15) consequently becomes

According to Eq. (13), one has

and from Eqs. (17), (18), one has Δk˜i,1(k) = Δk^i,1(k) + Δ2φi(k) = Δsi(k). Substitute the above into Eq. (20),

and let λ_i,1=4, λ_i,2=8 for convenience, then,

and by Eq. (16),

In the following, s_i(k)>φ_i(k) and s_i(k)<–φ_i(k) is considered correspondingly.

1. When s_i(k)>φ_i(k).

   1. If Δs_i(k)>0, by Eqs. (10), (18), and (24), the inequality (23) becomes

      Δsi(k){Δfi + Δgiui,eq − γiki,1︸negative + (1 − γi − βi)k^i,1 + 0.5ηi +(γi − 1)(δi + ηi) + αi4(δi + ηi + εi) − 4|si(k)|} < 0.

      Because β_i>1+γ̅_i and α_i<–4(1+γ̅_i),

      Δsi(k){Δfi + Δgiui,eq − γiki,1︸negative+ (1 − γi − βi)k^i,1︸negative−4|si(k)|︸negative +(γi − 1)δi + (γi − 0.5)ηi + αi(δi + ηi + εi)/4}︸negative < 0.

      It is implied that Eq. (23) holds.

   2. If Δs_i(k)<0, by Eqs. (10), (18), and (24), the proof is similar to (a), and inequality (23) becomes

      Δsi(k){Δfi + Δgiui,eq − γiki,1 + (γi − 1 − 3αi4)δi︸positive + (1 − γi + 3βi)k^i,1︸positive+(γi − 1.5 − 3αi4)ηi︸positive−3αi4εi︸positive + 4si(k) + 4|si(k)|}︸positive < 0,

      which implies that Eq. (23) holds.

2. When s_i(k)<–φ_i(k), the proof is similar to (A) and then omitted.

   From above all, B′i = {ei,k||si(k)| ≤ φi(k)} is reachable and stable as the time difference of Eq. (19) is negative for |s_i(k)|>φ_i(k). Simultaneously, ∀ei,k ∈ Rri,V_i(k)→∞, when ||e_i,k||→∞. Thereby, according to the Lyapunov stability theory, s is globally asymptotically stable and converges to B′i.■

4 Sliding Mode-Like Fuzzy Logic Control

The following DFLS

is considered, which is used to replace the control input u_i,v. The output of the DFLS u̅_i,v, has one-order continuous dynamic. In Eq. (25), ω_i,1>0, ω_i,2>0 are filtering parameters to be designed, ξ=[ξ₁ … ξ_M]^T is the support points vector of the fuzzy rule base, p=[p₁ … p_M]^T is the fuzzy rule base function vector, which element is determined by

where M is the rule number, and N is the rule number of the input variables. The input variables of DFLS are selected as the sliding mode s_i and its difference Δs_i. The DFLS design in detail refers to [23].

The membership functions of the input variables are all triangular, which are shown in Figure 1A, B, where N=2 is the number of input variables. The universe field partitions and the triangular membership function of the output variable u̅_i,v, are shown in Figure 1C. We adopt the product inferring method results in the following IF…THEN…sentences:

Figure 1:

The Fuzzy Sets of the Input and Output Variables in the DFLS.

IF s_i is N and Δs_i is N, THEN u̅_i,v is PB;

IF s_i is Z and Δs_i is N, THEN u̅_i,v is PL;

…

IF s_i is N and Δs_i is Z, THEN u̅_i,v is PL;

…

IF s_i is P and Δs_i is P, THEN u̅_i,v is NB.

These rules are shown in Table 1.

Table 1

The Fuzzy Inferring Rules of the DFLS.

Weighted averaging defuzzier was adopted. Therefore, the fuzzy rule base (26) was got; p_l is the l-th rule membership value of the output variable u̅_i,v, l=1, 2, …, 9 is the number of rules, i.e. M=9.

According to the fuzzy rules and the output variable partitions in Figure 1C, the support points vector of the fuzzy rule base is obtained as ξ = [θ5 θ4 … θ4 … θ1]T. Then, θ_q, q=1, 2, … 5 is the partition parameter of the defuzzier, which is set up by designer.

In order to approximate the variable structure control input u_i,v, the output of the DFLS u̅_i,v must satisfy the relation between u̅_i,v and k^i (k^i is the variable k^i,1 for convenience). So k^i can be got by

according to Eq. (7). Then, φ_i can be self-tuning by the adaptive law (18), while the adaptive law (17) is not used. Because φ_i is also the partition parameter of the DFLS input variables, the fuzzy inferring with self-tuning is implemented.

5 Numerical Simulation

Consider the numerical example system described as

where x(k)=[x₁(k), …, x₄(k)]^T is the state vector and u(k) = [u1(k) u2(k)]T is the control input vector. The sampling time T_s is 0.01 s.

Suppose that the tracked models are

yd1(k + 2) = −a1,1yd1(k) − a1,2yd1(k + 1) + r1(k)yd2(k + 2) = − a2,1yd2(k) − a2,2yd2(k + 1) + r2(k),

where y_d1(k) and y_d2(k) are tracked trajectories, the reference r₁(k)=5e^–t sin πt, r₂(k)=3e^–2t cos 2πt.

For the system (27), we estimated the nonlinearity f^ = [x2−x1x4x32]T and the controller gain g^ = [0101]T. It is difficult for the general method to design a stable controller obviously. By using the design method of the paper, the controller design becomes relevant. The sliding modes were designed as s₁=0.1x₁+x₂ and s₂=0.5x₃+x₄. The parameters of the adaptive law are preferred as α₁=α₂=–5, β₁=β₂=10, and ω_1,1=ω_2,1=0.01 by the system uncertainty characteristics by Eq. (6). The DFLS parameters ω_1,2=ω_2,2=1. The two DFLSs are all designed as that proposed in Section 4, and their fuzzy reasoning rules are, respectively, shown in Table 1. The parameters θ₁=0.1, θ₂=50.

When the initial state x(0) is set at [00.500]T, the simulation results of the example are shown in Figures 2–4. From Figure 2, it is seen that the closed-loop system is stable, and the trajectories are tracked successfully. Furthermore, the sliding modes step into the switching band shown in Figure 3. The control signals are smooth enough, shown in Figure 4. However, we do not require the boundaries of uncertainty during the control design.

Figure 2:

The Simulation Results: The Trajectories Traced and the System Outputs.

Figure 3:

The Simulation Results: the Sliding Mode Values.

Figure 4:

The Simulation Results: The System Control Inputs.

To reveal the application effect of the final DFLS in the SMLFC, we calculate the control parameters k^i by Eq. (26). Figure 5 indicates the value of k^i, whose corresponding variable k_i design is very difficult in general DSMC because the parameters γ_i and δ_i are all unknown. However, in the method of the paper, the approximation of k_i is implemented by the self-tuning DFLS, and therefore, an equivalent FLC is acquired. At the same time, the boundary layer is self-tuned as well, which intensifies the elimination of chattering.

Figure 5:

The Simulation Results: The System Parameters k^1, k^2.

6 Conclusions

A sliding mode-like fuzzy logic control (SMLFC) is proposed for a class of discrete nonlinear systems. The dynamic fuzzy logic systems in which parameters are tuned on-line to approximate the sliding mode control are used, and the overall system stability is analyzed by the Lyapunov method. The reachabilities of the sliding modes are verified, and the simulation results validate the proposed control method.