A Simplified Neural Network for Linear Matrix Inequality Problems

A simplified neural network model is proposed to solve a class of linear matrix inequality problems. The stability and solvability of the proposed neural network are analyzed and discussed theoretically. In comparison with the previous neural network models (Lin and Huang, Neural Process Lett 11:153–169, 2000; Lin et al., IEEE Trans Neural Netw 11:1078–1092, 2000), the simplified one is composed of two layers rather than three layers, and the neuron array in each layer is triangular rather than square. The proposed approach can therefore reduce the complexity of the neural network architecture. In addition, the simplified neural network can also be extended to solve multiple linear matrix inequalities with specific constraints, which enlarges the application domain of the proposed approach. Finally, examples are given to illustrate the effectiveness and efficiency of the simplified neural network.     

Simultaneous stabilization for a collection of nonlinear control systems in canonical form

A controller design method is provided to simultaneously stabilize a collection of nonlinear control systems in canonical form. It is shown that, under a mild assumption, any collection of nonlinear systems in canonical form can be simultaneously stabilized by one continuous state feedback controller. A constructive universal formula is presented explicitly. An illustrative example is given to demonstrate the validity of the method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A new adaptive backstepping Coulomb friction compensator for servo control systems

A new Coulomb friction compensator is proposed for servo control systems in this paper. The novelty of the new approach lies in its capability of assigning the eigenvalues of the resulting closed loop system while attacking the problem. First, based on the standard backstepping methodology, an implicit Lyapunov function, with part of the components being only symbolically constructed at the very beginning, is utilized. To increase the robustness of the system against disturbance and model inaccuracy, an integral term is employed in the design. Using part of the variable gradient method, we are able to turn the implicit Lyapunov function into an explicit one, which is positive definite, and whose time‐derivative is negative definite. Second, it will be shown that the resulting closed loop error system is a switched linear system with two possible active modes that share the same set of eigenvalues, which is at our disposal. Unlike the common adaptive control design methods, such as the Control Lyapunov Function approach, in which the gains are typically positive but otherwise arbitrary, and are hence difficult to choose and have a lack of connection with the system's performance, our new scheme imposes two further constraints on the gains. It turns out that we can then match these gains with the coefficients of the desired characteristic equation of the closed loop system. In this respect, the gains are linked to the system's overall performance, which is a new and very appealing feature for such a scheme. Finally, a procedure of constructing a common Lyapunov function is provided to prove exponential stability of the aforementioned switched linear system. In addition, using the invariance principle, we will show the convergence of the estimated Coulomb friction coefficient to its real value. Numerical simulations are given to validate the effectiveness of the design and its robustness against friction time‐variations. Compared to existing results, the proposed scheme is much simpler, hence, much more advantageous computationally. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Lyapunov-based feedback design and experimental verification of IC engine speed control

In this paper, the speed control problem of internal combustion engines is investigated based on mean-value engine models. The dynamics of internal combustion engines is a complicated nonlinear system, and usually, it is difficult to know the exact values of the physical parameters. First, a Lyapunov-based design method is shown without requiring the full information of the physical parameters. Then, to improve transient performance, the design method is extended to several cases under different operation conditions. Numerical simulation results are presented for comparing the proposed design methods. Finally, experiments are conducted on an engine test bench and the results demonstrate the validity of the proposed design methods. Recommended by Editorial Board member Myotaeg Lim under the direction of Editor Hyun Seok Yang. The authors are grateful to Kai Zheng for his assistance of the model identification experiments. Jiangyan Zhang received the B.E. and M.E. degrees in Electrical Engineering, Yanshan University, China, in 2005 and 2008, respectively. Now, she is a Ph.D. candidate with the Department of Engineering and Applied Sciences, Sophia University, Tokyo, Japan. Her current research interests include nonlinear system control theory and applications to powertrain system control. Tielong Shen received the Ph.D. degree in Mechanical Engineering from Sophia University, Tokyo, Japan, March, 1992. From April 1992, he has been a faculty member of the Chair of Control Engineering in Department of Mechanical Engineering, Sophia University, where he currently serves as professor of the Department of Engineering and Applied Science. His research interests include control theory and application in mechanical systems, power systems, and automotive powertrain. Currently, he is an Associate Editor for the IEEE Control System Society Conference Editorial Board, and is serving as Associate Editor of Journal of Control Theory and Applications, and the Regional Editor Asia-Pacific for International Journal of Modeling, Identification and Control etc. Junichi Kako received the B.E. degree from Nagoya Institute of Technology, Nagoya, Japan. He joined Toyota Motor Corporation, Tokyo, Japan in 1989. He worked on various aspects of automotive powertrain control. From 1989 to 1994, he took part in the team for the development of Laboratory Automation (LA) system, Engineering Office Automation (EOD) system, and embedded system of powertrain control. During 1995–2001, he focused on the engine control systems in Powertrain Management Engineering Division. In 2002, he was with Future Project Division in which he was responsible for the R&D of model-based engine control system. Currently, he is developing engine control systems in the Powertrain Management Engineering Division, Toyota Motor Corporation. Shozo Yoshida received the M.S. degree in Engineering from Kyoto University, Kyoto, Japan. He joined Toyota Motor Corporation, Tokyo, Japan in 2000. From 2000 to 2004, he was with Future Project Division and worked on physical combustion modeling for Model-based Control Development. Since 2005, he has been with the Powertrain Management Engineering Division Toyota Motor Corporation, and is a member of the R&D of Model-based Engine Calibration.     

Global asymptotic stabilization of the attitude and the angular rates of an underactuated non-symmetric rigid body

The paper deals with the global stabilization of both the attitude and the angular velocities of an underactuated rigid body. First a stability theorem is proven for a class of systems; subsequently, the equations describing the physics of the rigid body are presented, showing that the rigid body belongs to the considered class of systems, and a sufficient condition for the application of the theorem to the stability of the rigid body equilibrium is pointed out. Finally, some simulation results are reported showing the effectiveness of the proposed methodology.     

Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions

Counterexamples are given which show that a linear switched system (with controlled switching) that can be stabilized by means of a suitable switching law does not necessarily admit a convex Lyapunov function. Both continuous- and discrete-time cases are considered. This fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched system. In particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a Lyapunov function.     

Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T_n of increasing dimensions n+1 and with a domain shape in the form of a “hyper-pyramid”, 0≤ξ_n≤ξ_n−1?≤ξ₁≤x≤1. We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L² and H¹ local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.     

Nonlinear gliding stability and control for vehicles with hydrodynamic forcing

This paper presents Lyapunov functions for proving the stability of steady gliding motions for vehicles with hydrodynamic or aerodynamic forces and moments. Because of lifting forces and moments, system energy cannot be used as a Lyapunov function candidate. A Lyapunov function is constructed using a conservation law discovered by Lanchester in his classical work on phugoid-mode dynamics of an airplane. The phugoid-mode dynamics, which are cast here as Hamiltonian dynamics, correspond to the slow dynamics in a multi-time-scale model of a hydro/aerodynamically-forced vehicle in the longitudinal plane. Singular perturbation theory is used in the proof of stability of gliding motions. As an intermediate step, the simplifying assumptions of Lanchester are made rigorous. It is further shown how to design stabilizing control laws for gliding motions using the derived function as a control Lyapunov function and how to compute the corresponding regions of attraction.     

Global robust output regulation of lower triangular systems with unknown control direction

This paper presents the solvability conditions for the global robust output regulation problem for lower triangular nonlinear systems assuming the control direction is unknown. The approach used is an integration of the robust stabilization technique and Nussbaum gain technique.     

Stability analysis of a class of digital filters utilizing single saturation nonlinearity

A novel criterion for the global asymptotic stability of a class of digital filters utilizing single saturation nonlinearity is presented. An example showing the effectiveness of the present criterion is given.