Discrete Regulator Equations Research Articles

Experimental Output Regulation of Linear Motor Driven Inverted Pendulum With Friction Compensation

Over the past few decades, the nonlinear output regulation (NOR) theory has attracted extensive attentions in control society. However, few experimental results have been reported about the NOR theory. This article first presents experimental results on discrete-time NOR problem for a linear motor driven inverted pendulum (LMDIP) system. To solve this problem, it is essential to find the solution of the complicated discrete regulator equations (DREs). Moreover, the friction force commonly exists in mechanical systems and cannot be neglected for control systems requiring high precision tracking performance. We present a novel discrete-time controller by combining neural network (NN) approach and friction-feedforward compensation mechanism. Finally, we verify the proposed control algorithm by experiment and make some comparisons with the linear controller.

Discrete output regulator design for linear distributed parameter systems

In this manuscript, we address discrete-time state and error feedback output regulator designs for a class of linear distributed parameter systems (DPS) with bounded input and output operators. By utilising the Cayley–Tustin bilinear transform, a linear infinite-dimensional discrete-time system is obtained without model spatial approximation or model order reduction. Based on the internal model principle, discrete state and error feedback regulators are designed. In particular, discrete Sylvester regulator equations are formulated, and their solvability is proved and linked to the continuous counterparts. To ensure the stability of the closed-loop system, the design of stabilising feedback gain and its dual problem of stabilising output injection gain design are provided in the discrete-time setting. Finally, three simulation examples including a first-order hyperbolic partial differential equation model and a 1-D heat equation with considerations of step-like, ramp-like and harmonic exogenous signals are shown to demonstrate the effectiveness of the proposed method.

Position control of spherical inverted pendulum via improved discrete-time neural network approach

Much recently, a discrete-time neural network (NN) approach from the output regulation theory was adopted to solve the position tracking problem of the spherical inverted pendulum (SIP) system. The key of this approach is to find the approximate solution of the corresponding discrete regulator equations (DREs) of the SIP system, which are composed of 10 nonlinear algebraic functional equations. However, the procedure for calculating the approximate solution of the DREs is quite tedious and is dependent on the system parameters. In this paper, an improved discrete-time NN control algorithm is proposed, which relies on the NN approximation of the feedforward function. Since the feedforward function is two-dimensional, the improved NN approach is much simpler compared with the existing NN approach. Moreover, a distinct advantage of our approach is that it allows certain robustness to the system parameters when every state is available. Simulation results demonstrate that our approach leads to much smaller tracking errors than the existing NN approach.

Discrete-Time Neural Network Approach for Tracking Control of Spherical Inverted Pendulum

In recent years, the tracking problem of spherical inverted pendulum (SIP) system with multi-input, multi-output nature and unstable zero dynamics has been well addressed based on continuous-time nonlinear output regulation (NOR) theory. For the convenience of digital implementation, this paper further investigates the approximate NOR problem of the SIP system in discrete-time framework. The key for solving the discrete-time NOR problem lies in how to solve a set of algebraic functional equations known as discrete regulator equations (DRE). Since the equations are very complicated, the accurate solution of the DRE can not be obtained. In this paper, we first show that the DRE associated with the SIP system are solvable by center manifold theorem and then use neural network approach to tackle with the tracking problem. Finally, we compare our method with polynomial approximation method.

Neural-Network-Based Approximate Output Regulation of Discrete-Time Nonlinear Systems

The existing approaches to the discrete-time nonlinear output regulation problem rely on the offline solution of a set of mixed nonlinear functional equations known as discrete regulator equations. For complex nonlinear systems, it is difficult to solve the discrete regulator equations even approximately. Moreover, for systems with uncertainty, these approaches cannot offer a reliable solution. By combining the approximation capability of the feedforward neural networks (NNs) with an online parameter optimization mechanism, we develop an approach to solving the discrete nonlinear output regulation problem without solving the discrete regulator equations explicitly. The approach of this paper can be viewed as a discrete counterpart of our previous paper on approximately solving the continuous-time nonlinear output regulation problem.

On the discrete-time robust nonlinear servomechanism problem

The paper addresses the robust servomechanism problem for nonlinear discrete-time systems. The solvability conditions for the discrete-time k th -order robust servomechanism prob- lem is first established by using the internal model principle. Then it is further shown that, under an additional assumption on the solution of the discrete regulator equations, the solvability of the discrete-time k th -order robust servomechanism problem leads to the solvability of the discrete-time robust servomechanism problem. The results of this paper give a discrete-time counterpart of the results on the continuous-time robust servomechanism problem.

A neural network-based approximation method for discrete-time nonlinear servomechanism problem

A feedback controller that solves the discrete-time nonlinear servomechanism problem relies on the solution of a set of nonlinear functional equations known as the discrete regulator equations. The exact solution of the discrete regulator equations is usually unavailable due to the nonlinearity of the system. The paper proposes to approximately solve the discrete regulator equations using a feedforward neural network. This approach leads to an effective way to practically solve the discrete nonlinear servomechanism problem. The approach has been illustrated using the well-known inverted pendulum on a cart system. The simulation shows that the control law designed by the proposed approach performs much better than the conventional linear control law.