Asymptotic stabilization with locally semiconcave control Lyapunov functions on general manifolds

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Abstract

Asymptotic stabilization on noncontractible manifolds is a difficult control problem. If a configuration space is not a contractible manifold, we need to design a time-varying or discontinuous state feedback control for asymptotic stabilization at the desired equilibrium.

For a system defined on Euclidean space, a discontinuous state feedback controller was proposed by Rifford with a semiconcave strict control Lyapunov function (CLF). However, it is difficult to apply Rifford’s controller to stabilization on general manifolds.

In this paper, we restrict the assumption of semiconcavity of the CLF to the “local” one, and introduce the disassembled differential of locally semiconcave functions as a generalized derivative of nonsmooth functions. Further, we propose a Rifford–Sontag-type discontinuous static state feedback controller for asymptotic stabilization with the disassembled differential of the locally semiconcave practical CLF (LS-PCLF) by means of sample stability. The controller does not need to calculate limiting subderivative of the LS-PCLF.

Moreover, we show that the LS-PCLF, obtained by the minimum projection method, has a special advantage with which one can easily design a controller in the case of the minimum projection method. Finally, we confirm the effectiveness of the proposed method through an example.

Introduction

Asymptotic stabilization on noncontractible manifolds, a difficult control problem  [1], [2], [3], has been studied by a few researchers [1], [2], [4], [5], [6], [7]. The main problem is that a noncontractible manifold, as a configuration space, never has a continuous asymptotically stabilizing static state feedback control at any desired equilibrium. Hence one needs to design a discontinuous or time-varying stabilizing controller  [3].

Control Lyapunov functions (CLFs) play an important role in feedback control design  [3], [8], [9]. In particular, semiconcave strict CLFs enable designing discontinuous asymptotic stabilizing controllers  [10]. Rifford proposed a discontinuous controller, defined on Euclidean space, based on semiconcave strict CLFs  [10]. However, Rifford’s controller cannot be directly applied to stabilization on manifolds or systems with unbounded inputs.

In this paper, we introduce the framework of a locally semiconcave practical CLF for stabilization on manifolds. We consider the disassembled differential instead of the limiting subderivative of a locally semiconcave function. Then, we show that the directional subderivative used in the definition of the practical CLF is replaced with the disassembled differential. Further, we propose a Rifford–Sontag-type discontinuous asymptotically stabilizing static state feedback controller with the disassembled differential of the locally semiconcave practical control Lyapunov function (LS-PCLF) by means of sample stability.

For general differentiable manifolds, we proposed the minimum projection method to design a locally semiconcave strict CLF [11], [12], but we did not show how to stabilize the origin of control systems defined on manifolds with the LS-PCLFs. In this paper, we show that the locally semiconcave CLF, obtained by the minimum projection method, is particularly advantageous for calculating the disassembled differential. Therefore, one can easily design a controller when the LS-PCLF is obtained by the minimum projection method.

Section snippets

Differentiable manifolds

A brief introduction of differentiable manifolds is necessary to discuss the control systems defined on manifolds  [13], [14]. In this paper, X denotes an n-dimensional smooth manifold, TxX a vector space called the tangent space to X at x, and an element of TxX a tangent vector at x. TxX denotes the dual space to TxX, called the cotangent space at x, and an element of TxX a cotangent vector (or a differential 1-form) at x. A subset MX is said to be precompact in X, if its closure in X is

Locally semiconcave practical control Lyapunov functions

Strict CLFs are commonly used for the development of an asymptotically stabilizing controller. For discontinuous control design, semiconcave strict CLF was introduced by Rifford  [10]. The locally semiconcave strict CLF is defined as follows.

Definition 4 Locally Semiconcave Strict CLF

A global locally semiconcave strict control Lyapunov function for system (5) is a locally semiconcave function V:XR such that the following properties hold:

  • (A1)

    V is proper; that is, the set {xX|V(x)L} is compact for every L>0.

  • (A2)

    V is positive definite; that is,

Sample stabilization with locally semiconcave control Lyapunov functions

In this section, we consider an asymptotically stabilizing controller design problem for the following control-affine system:ẋ=f(x)+g(x)uf(x)+i=1mgi(x)ui, where xX and uF(R,Rm);tu(t)Rm. Moreover, mappings f,gi:XTxX are assumed to be locally Lipschitz continuous with respect to x for all i{1,,m}, and satisfies f(0)=0.

We introduced the disassembled differential for locally semiconcave functions and comment on some properties of the disassembled differential. In this subsection, we

Practical stabilization with minimum projection method

According to the preceding discussion, we can design an asymptotic stabilizing controller with an LS-PCLF by means of sample stability. To design an LS-PCLF on manifolds, we can use the minimum projection method. In this section, we show how the LS-PCLF generated by the method has an advantage in controller design.

Example: obstacle avoidance of a mobile robot

In accordance with the preceding discussion, we can design a sample stabilizing controller for control system (21) defined on manifold X. In this section, we show an example of the proposed method.

We consider the following mobile robot system discussed in [11]: {ẋ=u1ẏ=u2, where (x,y)R2 is a state and (u1,u2)R2 is an input. Also considered is a disk-shaped obstacle with radius 1 centered at (−2,0).

The problem here is to design a sample stabilizing controller at the origin (x,y)=(0,0) taking

Conclusion

In this paper, we proposed a sample stabilizing controller for a control system defined on a general manifold with a locally semiconcave practical control Lyapunov function. We employed the disassembled differential in the proposed controller, which enabled us to easily design a sample stabilizing controller when the locally semiconcave control Lyapunov function was designed by the minimum projection method. Finally, we confirmed the effectiveness of the proposed method through an example.

We

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