Direct adaptive neural control for affine nonlinear systems

doi:10.1016/j.asoc.2008.10.001

Applied Soft Computing

Volume 9, Issue 2, March 2009, Pages 756-764

https://doi.org/10.1016/j.asoc.2008.10.001 Get rights and content

Abstract

This paper presents a direct adaptive neural control scheme for a class of affine nonlinear systems which are exactly input–output linearizable by nonlinear state feedback. For single-input–single-output (SISO) systems of the form $\dot{x} = f (x) + g (x) u$ , the control problem is comprehensively solved when both $f (x)$ and $g (x)$ are unknown. In this case, the control input comprises two terms. One is an adaptive feedback linearization term and the other one is a sliding mode term. The weight update laws for two neural networks, which approximate $f (x)$ and $g (x)$ , have been derived to make the closed loop system Lyapunov stable. It is also shown that a similar control approach can be applied for a class of multi-input–multi-output (MIMO) systems whose structure is formulated in this paper. Simulation results for both SISO- and MIMO-type nonlinear systems have been presented to validate the theoretical formulations.

Introduction

The input–output behavior of a nonlinear system can be made linear under certain assumptions by means of nonlinear state feedback. The controller can be proposed in such a way that the closed loop error dynamics becomes linear as well as stable [1]. The main problem with this control scheme is that the cancellation of the nonlinear dynamics depends upon the exact knowledge of system nonlinearities. When the system nonlinearities are not known completely but some bounds on them are known, the nonlinearities can be approximated either by neural networks or by fuzzy systems. The controller then uses these estimates to linearize the system. This concept was used in several papers [2], [3], [4], [5], [6] dealing with the study of feedback linearizable systems. Sanner and Slotine [7] have studied the applications of Gaussian networks for direct adaptive control. Spooner and Passino [8] proposed both direct and indirect adaptive control schemes which use Takagi–Sugeno fuzzy systems to provide asymptotic tracking when the plant dynamics are poorly understood. Choi and Farrell [9] proposed a non-parametric stable adaptive control approach using a piecewise linear approximator. Lewis et al. [10] designed a direct adaptive control for robot manipulators using radial basis and multi-layer networks. Zang et al. [11] proposed a modified Lyapunov function to show the closed loop system stability. Wang and Hill [12] presented a new learning mechanism by which an adaptive neural controller is capable of learning the unknown nonlinear systems dynamics while controlling the system. In a recent paper [13], we have proposed variable gain controllers where the nonlinear systems have been approximated in the framework of T–S fuzzy model.

In this paper, we propose a direct adaptive control scheme to achieve output tracking of affine nonlinear systems. The main advantage of a direct adaptive control scheme over an indirect adaptive control scheme is that in a direct adaptive control scheme there is no need for explicit system identification. In indirect adaptive control scheme, the system is generally identified off-line from its input–output data and the controller is designed based on the identified system model. The identified model should be accurate enough for better performance of the controller. Moreover stability is a critical issue in indirect adaptive control. But in direct adaptive control, the controller is designed in such a way that the closed loop stability is maintained while the tracking error converges to 0 with time. The affine systems have a form $\dot{x} = f (x) + g (x) u$ where $f (x)$ and $g (x)$ are two nonlinear functions. For general applications, there may happen two distinct cases. First is when the system nonlinearity $f (x)$ is unknown but the input nonlinearity $g (x)$ is known. Second is when both $f (x)$ and $g (x)$ are unknown. The systems are assumed to have a relative degree of n, i.e. after n differentiation, the input will appear in the output. One should note that for a system to be feedback linearizable $| g (x) |$ should be greater than 0, i.e. $g (x)$ can be either positive or negative. The nonlinearities can be approximated using neural networks or fuzzy systems. Since the adaptive controller has a form $u = (1 / g (x)) [- f (x) + K e]$ , the control problem becomes difficult when the nonlinearity $g (x)$ is unknown because of the fact that the approximation of $g (x)$ can be 0 at times which makes the controller unbounded. This problem has been addressed in different manners by various researchers [14], [9], [11]. In [14], a second control component is added and a new design parameter is introduced to maintain the boundedness of the input. The region of operation is divided into two regions and the stability of the closed loop systems is analyzed separately for the two regions. Choi and Farrel [9] have modified the update law using a projection algorithm. A persistently exciting condition is imposed to show the stability of the closed loop system. In [11], a single approximator approximates f(x)/g(x) which is used in the control law. A modified Lyapunov function is used to prove the closed loop stability. However, it is also assumed that the |f(x)| is bounded by a known function and this information is used in a time varying design parameter. These modifications and assumptions make the design more complex and restricted. In this paper we have solved this problem by keeping $ĝ (x)$ away from 0 using a projection algorithm. The effect of the projection algorithm on the closed loop stability has been taken care of by introducing a sliding mode term in the controller. In many of the earlier works it is assumed that $g (x) > 0$ . In this paper, the assumption has been relaxed by considering $| g (x) | > 0$ . However it is assumed that $g (x)$ is lower bounded by a known constant $g_{l}$ . Two radial basis function (RBF) networks are used to approximate the nonlinearities $f (x)$ and $g (x)$ . The weight update laws of the RBF networks are derived in such a way that the closed loop system is Lyapunov stable and the output tracking error converges to 0 with time. When the approximation error of the RBF networks are taken into account, the weight update laws are modified in [10] for closed loop stability. But in this paper we have shown that the same update law can maintain the closed loop stability with a bounded tracking error. Though the proposed adaptive control scheme is implementable only for a class of nonlinear systems, however there are many practical systems like single link flexible joint manipulator, jet engine compression systems [1], MEMS devices [15] for which the proposed control scheme can be applied successfully. Even if the system dynamics are readily not available in affine strict feedback form, one can convert them into the specified form using some transformation. We have further extended the application of the proposed controller to multi-input–multi-output (MIMO) systems when $f (x)$ is unknown and $g (x)$ is known. Many of the mechatronics kits, manipulator systems are examples of affine MIMO systems for which the controller derived in this manuscript can be applied.

Rest of the paper is organized as follows. A brief introduction about feedback linearization techniques for SISO systems followed by directive adaptive control scheme for SISO systems is presented in Section 2. Section 3 presents the directive adaptive control technique for MIMO systems when $f (x)$ is unknown but $g (x)$ is known. Simulation results for three nonlinear systems are presented in Section 4 with concluding remarks in Section 5.

Section snippets

Single-input–single-output affine systems

A large class of single-input–single-output nonlinear systems can be represented by the following affine system: $\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ ⋮ \\ {\dot{x}}_{n} = f (x) + g (x) u \\ y = x_{1} \end{matrix}$ where $x = {[x_{1} x_{2} \dots x_{n}]}^{T} \in R^{n}$ , $y \in R$ and $u \in R$ .

The control problem: Find u such that $x (t)$ follows a desired trajectory $x^{d} (t)$ .

If we choose control input $u = (1 / g (x)) [- f (x) + k_{v} r + λ_{1} e^{(n - 1)} + \dots + λ_{n - 1} e^{(1)} + {\dot{x}}_{n d}]$ where $e = y^{d} - y$ is the output tracking error and $r = e^{(n - 1)} + λ_{1} e^{(n - 2)} + \dots + λ_{n - 1} e$ (power denotes respective derivatives), the closed loop error dynamics becomes $\dot{r} = - k_{v} r$ which

Direct adaptive control of MIMO systems

In this section we have extended the application of the proposed controllers to MIMO systems. A general MIMO system can be written as $\begin{matrix} \dot{x} & = f (x) + g (x) u \\ y & = C x \end{matrix}$ where $x \in R^{n}$ , $f (x) \in R^{n}$ , $u \in R^{m}$ , $g (x) \in R^{m \times n}$ , $y \in R^{p}$ , $C \in R^{p \times n}$ . Some MIMO systems can also be written in the following form: $\begin{matrix} {\dot{x}}_{1} & = x_{2} \\ {\dot{x}}_{2} & = f_{1} (x) + g_{11} (x) u_{1} + \dots + g_{1 m} (x) u_{m} \\ {\dot{x}}_{3} & = x_{4} \\ {\dot{x}}_{4} & = f_{2} (x) + g_{21} (x) u_{1} + \dots + g_{2 m} (x) u_{m} \\ ⋮ \\ {\dot{x}}_{2 n - 1} & = x_{2 n} \\ {\dot{x}}_{2 n} & = f_{n} (x) + g_{n 1} (x) u_{1} + \dots + g_{n m} (x) u_{m} \\ y & = [x_{1} x_{3} \dots x_{2 n - 1}] \end{matrix}$ For such cases, the system equations can be re-written as: $\begin{matrix} {\dot{z}}_{1} & = z_{2} \\ {\dot{z}}_{2} & = f (z) + g (z) u \\ y & = z_{1} \end{matrix}$ where $z_{1} = [\begin{matrix} x_{1} & x_{3} & \dots & x_{2 n - 1} \end{matrix}]$

Simulation results

The performance of the proposed controller is demonstrated through simulation results. Two nonlinear systems have been taken for this purpose. The first example considered here is a SISO system while the second example is a MIMO one.

Conclusion

Feedback linearization techniques have been successfully used over the past decades to design controllers for affine nonlinear systems. The controller design becomes difficult when the system is not known completely. In this paper, both the system nonlinearity and the input nonlinearity have been assumed to be unknown and two RBF networks have been used to approximate those nonlinearities. A direct adaptive controller along with the weight update laws for the networks were derived to establish

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The proposed NN controller clearly has an advantage, i.e., it is implemented without a priori knowledge of the nonlinear systems. In contrast, in the literature (Belarbi & Chemachema, 2007; Chen et al., 2010; Kar & Behera, 2009; Li et al., 2011; Wen & Ren, 2011), very restrictive assumptions such as upper and lower bounds on system nonlinearities, control gain sign and approximation errors are required to implement NN controllers. Furthermore, the proposed controller is more suitable for real time implementation since it does not require any state measurement.
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Direct adaptive neural control for affine nonlinear systems

Abstract

Introduction

Section snippets

Single-input–single-output affine systems

Direct adaptive control of MIMO systems

Simulation results

Conclusion

Neural Netw.

Non Linear and Adaptive control design

Adaptive control of linearizable systems

IEEE Trans. Autom. Control

Stable adaptive tracking of uncertain systems using nonlinearly parameterized on-line approximators

Int. J. Control

A neural approach for control of nonlinear systems with feedback linearization

IEEE Trans. Neural Netw.

Stabilization of feedback linearizable systems using a radial basis function network and its zero-form

IEEE Trans. Autom. Control

Direct adaptive nn control of a class of nonlinear systems

IEEE Trans. Neural Netw.

Gaussian networks for direct adaptive control

IEEE Trans. Neural Netw.