Hisakazu Nakamura

Global Inverse Optimal Control With Guaranteed Convergence Rates of Input Affine Nonlinear Systems

In earlier works, global inverse optimal controllers were proposed for input affine nonlinear systems with given control Lyapunov functions. However, these controllers do not provide information about the convergence rates. If the systems have local homogeneous approximations, we can employ homogeneous controllers, which locally asymptotically stabilize the origin and specify the convergence rates. However, homogeneous controllers generally do not attain global stability for nonhomogeneous systems. In this paper, we design global inverse optimal controllers with guaranteed local convergence rates by utilizing local homogeneity of input affine nonlinear systems. If we do not consider inverse optimality, we can liberally adjust the sector margins. We also clarify that local convergence rates and sector margins are invariant under coordinate transformations.

Minimum Projection Method for nonsmooth control Lyapunov function design on general manifolds

Abstract Asymptotic stabilization on noncontractible manifolds is known as a difficult control problem. On the other hand, an important fact is every control system that is globally asymptotically stabilizable at a desired equilibrium must have nonsmooth control Lyapunov functions. This paper considers the problem of construction of nonsmooth control Lyapunov functions on general manifolds, and we propose a nonsmooth control Lyapunov function design method called the ‘Minimum Projection Method’. The proposed method considers a simple-structured smooth manifold associated with the original manifold by a surjective immersion, and then a control Lyapunov function defined on the simple-structured manifold is projected to the original manifold. A function on the original manifold is thus obtained. In this paper, we prove that the control system on another manifold associated with a surjective immersion is determined uniquely, and the resulting function by the proposed method is a nonsmooth control Lyapunov function on the original manifold. The effectiveness of the proposed method is confirmed by examples.

Asymptotic stabilization with locally semiconcave control Lyapunov functions on general manifolds

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.

Homogeneous Stabilization for Input Affine Homogeneous Systems

Dilated homogeneous systems are local canonical forms of nonlinear control systems. For input affine homogeneous systems, existence of continuous homogeneous stabilizing controllers is equivalent to existence of differentiable homogeneous control Lyapunov functions. In this technical note, we propose an homogeneous stabilizing controller and an homogeneous inverse optimal controller for multi-input affine homogeneous systems with given homogeneous control Lyapunov functions.

Multilayer minimum projection method for nonsmooth strict control Lyapunov function design

Asymptotic stabilization on noncontractible manifolds is known as a difficult control problem. To address this problem, we had proposed the minimum projection method to design nonsmooth control Lyapunov functions. This method, however, has some problems: difficult etale-surjection design, undesirable resulting control Lyapunov functions, etc. In this paper, we propose a new nonsmooth control Lyapunov function design method called the ‘Multilayer minimum projection method’ for nonsmooth control Lyapunov function design on general manifolds. The method considers many simple-structured smooth manifolds associated with the original manifold by etale mappings, and then a function on the original manifold is obtained by projecting control Lyapunov functions defined on the simple-structured manifolds onto the original manifold. In this paper, we prove that the resulting function by the proposed method is a nonsmooth control Lyapunov function on the original manifold. Moreover, we prove that if all control Lyapunov functions defined on simple-structured manifolds are strict, the control Lyapunov function on the original manifold is a strict control Lyapunov function. Finally, the effectiveness of the proposed method and the advantage over the conventional minimum projection method are confirmed by an example.

Smooth Lyapunov functions for homogeneous differential inclusions

This paper provides a construction method of a smooth homogeneous Lyapunov function associated with a discontinuous homogeneous system, which is locally and asymptotically stable. First, we analyze two similar converse Lyapunov theorems for differential inclusions and unify them into a simple theorem. Next, we propose a new definition of homogeneous differential inclusion. Then, we construct a smooth, homogeneous Lyapunov function associated with the homogeneous differential inclusion. Finally, we show that the order of homogeneity of a homogeneous system indicates the speed of convergence.

Robust Finite-time Control of Robot Manipulators

Abstract This paper considers finite-time position control of robot manipulators. The robot manipulators are modeled by discontinuous differential equations. In this paper, we prove that the Nakamuras local homogeneous controller based on a control Lyapunov function is valid to the position control of the robot manipulators, and show the effectiveness of the controller by experiments. Moreover, we compare the controller with other nonlinear controllers and show advantages of the controller.

Control of Two-wheeled Mobile Robot via Homogeneous Semiconcave Control Lyapunov function

We proposed a control law of a homogeneous semiconcave control Lyapunov function for a two-wheeled mobile robot. However, we have not confirmed the effectiveness of the proposed method by an experiments. In this paper, we apply the proposed method to a Roomba, a two-wheeled mobile robot and confirm the effectiveness of the proposed method.

HOMOGENEOUS EIGENVALUE ANALYSIS OF HOMOGENEOUS SYSTEMS

Abstract This paper focuses on the problem of stability in homogeneous systems with dilation. First, we propose an ‘homogeneous eigenvalue’ for homogeneous systems. Next, we analyze the stability of homogeneous systems using homgeneous eigenvalues, and we show that the use of positive real homogeneous eigenvalues implies instability. Finally, we show the effectiveness of the proposed method through an example.

CONTROLLER FOR A NONLINEAR SYSTEM WITH AN INPUT CONSTRAINT BY USING A CONTROL LYAPUNOV FUNCTION I

Abstract In this paper, we generalize the Malisoffs controller for a nonlinear system with an input constraint. Malisoff and Sontag proposed a universal control formula for a nonlinear system such that the k -norm of inputs is less than one. However, k is limited to 1 k ≤ 2. We improve the Malisoffs formula so that it can be applied in any case of k ≥ 1. We also confirm the effectiveness of the improved controller by computer simulation.