Information Technology Journal

Year: 2014 | Volume: 13 | Issue: 4 | Page No.: 697-702
DOI: 10.3923/itj.2014.697.702

Direct Adaptive Fuzzy Control of Nonlinear Systems with Unknown Control Directions

Long Luo, Fei Luo and Yuge Xu

Abstract: For the Adaptive Fuzzy Control (AFC) of uncertain nonlinear systems, the unknown control direction brings about great difficulty in control design. This study presents a novel direct AFC approach for a class of perturbed uncertain affine nonlinear systems with unknown control directions. The overall control input contains a basic direct AFC term and an additional robust control term. A Lyapunov-based ideal control law is proposed to solve the control singularity problem and the Nussbaum gain technique is applied to solve the control direction problem. Using an e²-modification in adaptive laws, it not only obtains bounded adaptive parameters, but also achieves asymptotic convergence of tracking errors. Moreover, the proposed controller has more compact structure compared with the previous indirect approaches. Simulated studies have demonstrated the effectiveness of the proposed approach.

Keywords: unknown control gain sign, Adaptive fuzzy control, asymptotic tracking, e2-modification and Nussbaum-type function

INTRODUCTION

For uncertain nonlinear systems, indirect and direct Adaptive Fuzzy Control (AFC) approaches (Wang, 1994) have been intensively developed in the past decades. Generally speaking, for facilitating AFC design, the control direction is assumed to be known a priori (Pan et al., 2011, 2013). However, the problem of unknown control direction often appears in practical applications which brings about great difficulty in controller design (Wen and Ren, 2010). Without knowing the control direction, (Ge and Wang, 2002) firstly introduced the Nussbaum gain technique into the Adaptive Neural Control (ANC) design for a class of strict feedback nonlinear systems. A Nussbaum gain is a control-direction estimator that can swing the sign according to the control performance (Nussbaum, 1983). Next, the approach of (Ge and Wang, 2002) was extended to the pure-feedback nonlinear system (Ren et al., 2009), the low-triangular nonlinear system (Du et al., 2006), the block-triangular nonlinear system (Chen et al., 2009), the nonaffine nonlinear system (Liu et al., 2009) and the output-feedback nonlinear system (Liu and Li, 2010). Note that all aforementioned approaches are based on the indirect scheme and make tracking errors Uniformly Ultimately Bounded (UUB). The direct adaptive design can lead to more compact control structure (Pan and Er, 2013). Direct Backstepping AFC was developed for the strict-feedback nonlinear system (Wang et al., 2007). Direct composite AFC was proposed for a class of affine nonlinear systems (Labiod and Guerra, 2011). However, Wang et al. (2007), using the norm of parameter vectors as adaptive parameters is not appropriate since the turning would be monotonously increased at positive direction. The achieved asymptotic convergence of the tracing errors is based on the neglect of Fuzzy Approximation Errors (FAEs) Labiod and Guerra (2011). To achieve the asymptotic stability in the presence of the FAE, a robust control term with a boundary estimation law was applied into the ANC of the triangular-structured nonlinear system by Zhang and Ge (2009). But the robust control term is unbound since the boundary estimation law is an integral of an absolute value.

PROBLEM FORMULATION

Consider the following perturbed nth-order Single-Input Single-Output (SISO) nonlinear system in the controllable canonical form, i.e., the Brunovsky form:

where, x = [x₁, x₂,..., x_n]^T = [x, , ..., x^(n-1)]^Tεⁿ is the state vector, uεú and yεú are the control input and system output, respectively, f(x)ε, g(x)ε and d(t)ε are the unknown continuously differentiable nonlinear driving function, control gain function and external disturbance, respectively. Let y_dε denote a desired output, y_d = [y_d1, y_d2,..., y_dn]^T = [y_d, _d,..., y_d^(n-1)]^Tεⁿ and yd₁ = [y_d, y_d⁽ⁿ⁾]^Tεⁿ⁺¹. Suppose that the sign of g is unknown but either positive or negative. The following assumptions are made as by Zhang and Ge (2009).

Assumption 1: There exist unknown functions (x) and and constants such that .

Assumption 2: y_d has the (n+1)th-order derivative such that where M_d is a finite constant.

Define the output tracking error and its error vector . The objective of this study is to design a robust direct AFC for the system in Eq. 1 such that the closed-loop system achieves asymptotic stability in the sense that all involving variables are UUB and the tracking errors converge to zero.

LYAPUNOV-BASED CONTROL STRUCTURE

Choose a real vector k = [k_n,…, k₁]^T so that h(s) = sⁿ+k₁s^n-1+…+k_n is a Hurwitz polynomial, where s is a complex variable. Let:

Thus, there exists a unique positive definite symmetric matrix Pε^nxn for any given positive definite symmetric matrix Qε^nxn such that:

Design an ideal control law as follows:

Making u = u* Eq. 1, one obtains:

Choose the Lyapunov function candidate for Eq. 4 as follows:

Differentiating Eq. 5 with respect to time t yields:

Using |g|’ = sgn(g) and sgn() = |g|/g, one gets:

where, denotes the minimum eigenvalue of Q and λ_max(P) denotes the maximum eigenvalue of P. Above derivation implies that e is exponentially convergent, i.e., with t₁ε⁺ to be a finite time.

DIRECT ADAPTIVE FUZZY CONTROL

Certain control scheme: Since, f and g are unknown and d ≠ 0 Eq. 1, some terms in Eq. 3 cannot be determined. For facilitating derivation, let:

Thus, one has Then, one employs a FLS in Eq. 1:

to approximate is a input vector, is an adjusting parameter vector and is a fuzzy baisc function vector. The elements of ξ(z) are given by:

in which, are the membership functions (MFs) of , j = l₁...l_n, l_i = 1, m, i = 1,..., 2n+1, j = 1,..., M and m is the number of fuzzy partitions.

Define compact sets and , where M_x, M_θεR⁺ are finite constants. Let . The optimal FAE is defined as:

where, θ* is an optimal parameter vector given by:

From assumptions 1-2 and the derivation in section 3, these exists a bounded u* Eq. 3 such that Thus, according to the fuzzy approximate theorem by Wang (1994), one can suppose that wεL_∞. Define a lump uncertain term as follows:

From the boundness of w, d and g, one can make the following assumption.

Assumption 3: The lump uncertain term w_L Eq. 10 is bounded, i.e., there exists a finite positive constant such that

To cope with the unknown sign of g, the Nussbaum gain technique is applied. A function N(ζ) is called a Nussbaum-type function if it has the following properties (Ryan, 1991):

Lemma 1: (Zhang and Ge, 2009): Let V(.), ζ(.) be smooth functions defined on [0, t_f) with V(t)≥0, ∀tε[0, t_f) and N(ζ) be an even smooth Nussbaum-type function. If:

where, c₀ε is a suitable finite constant and g₀ is a nonzero constant, then V(t), ζ(t) and:

must be bounded on [0, t_f) with t_f = ∞.

Accordingly, one can design the certain controller as:

in which u_r is the robust control term defined as:

where, is the estimation of and εε⁺ is chosen such that:

(Sun et al., 2011).

Adaptive law derivation: From Eq. 1 and 3, one gets:

Choose the Lyapunov function candidate as:

where, are learning rates.

Theorem 1: For the system Eq. 1, select Eq. 14 that is equipped with Eq. 7 and 15 as the controller and design the parameter adaptive laws as:

where, σε⁺ is a user-defined small constant subjected to:

in which λ_min(Q) is the minimal eigenvalue of Q. Then, the closed-loop system achieves asymptotic stability in the sense that all involving variables are UUB and

Proof: Differentiating Eq. 17 with respect to time t, one obtains:

Noting u* = sgn(g)u*0, one gets:

Using Eq. 9-10 and sgn(g) = |g|/g, one obtains:

Substituting Eq. 18 and 19 into above expression leads to:

in which g₀ = -sgn(g). Since, g is either positive or negative, g₀ is a constant equaling to 1 or -1. Using Eq. 14 and 20, one gets:

Adding and subtracting ζ on the right side of Eq. 22 and using Eq. 7, one has:

Applying Eq. 15 to above expression and noting:

as by Phan and Gale (2007), one obtains:

Accordingly, one deduces:

where, Noting Eq. 21, one gets λ_Qε⁺. Integrating above expression over [0, t] and using:

lead to:

Then, one directly obtains Eq. 13. From Lemma 1, one concludes that V(t), ζ(t):

on tε[0, ∞). Thus, one has e, x, From Eq. 7, 14 and 15, one directly gets uεL_∞. Consequently, all involving variables are UUB. Thus, all terms on the right side of Eq. 16 are bounded, i.e., From Eq. 23, one also has:

which implies eεL₂. Now, we have From the Barbalat lemma (Wang, 1994), one gets END.

AN ILLUSTRATIVE EXAMPLE

Consider an inverted pendulum model in the form of Eq. 1 [30] with n = 2 and:

and d(t) = 3cos(2t)+2sin((0.09t+1)t), where x₁ is the angular position of the pendulum, x₂ is the angular velocity, g_v = 9.8 m sec^-2 is the gravitational acceleration, m_c is the mass of the cart, m₁ is the mass of the pendulum and l_p is the half-length of the pendulum. For simulation, Select m_c = 1 kg, m₁ = 0.1 kg, l_p = 0.5 m, x(0) = [π/6, 0] and y_d = sin(t).

To construct the controller, select (x_i) = exp(-0.5 (x_i-π(l_i-3)/3)/0.25)²) and with l_i = 1,…, 5 and i = 1,…, 5, let k₁ = 8, k₂ = 16 and Q = diag(100,100) and choose N(ζ) = ζ²cos(ζ), γ_θ = 100, γ_w = 20 and σ = 0.1.

Simulation results are shown in Fig. 1-4.

Both the angular position and the angular velocity successfully track their corresponding desired trajectories with very small tracking errors and smooth control inputs.

CONCLUSION

A novel robust direct AFC for a class of perturbed uncertain nonlinear systems with unknown control directions has been successfully developed in this study. The novelties of this study are as follows: (1) A Lyapunov-based ideal control law was presented to solve the control singularity problem and (2) an e²-modification of adaptive laws was applied to obtain both the boundedness of the adaptive parameters and the asymptotic convergence of the tracking errors. Compared with the previous approaches, the proposed approach not only obtains more compact control structure, but also guarantees better. Simulated application has demonstrated the effectiveness of the proposed approach.

ACKNOWLEDGMENTS

This study is supported by the Natural Science Foundation of Guangdong Province under Grant No. 10151064101000075, S2011010001153 and the Pearl New Star Science and Technology Foundation of Guangzhou City under Grant No. 2011J2200084. We also get a helping hand from ‘Fund of Innovation Creation Academy Group’ established by the Guangzhou Education Bureau.

REFERENCES