Optimal Control Of Delayed Systems And Its Applications To Wheeled Inverted Pendulum

The problem of optimal control is to seek an optimum control strategy in all of the feasible control trategy,this is a classical optimization problem and has enduring vitality due to it's wide application background.In particular,the linear quadratic optimal control is more popular in practice since it has simple form of solution and can be easily implemented.However,the optimal control requires accurate system,but uncertainty factor and time delay are common in actual controlled systems.Therefore,the control effect would be reduced or even lead to unstability if uncertainty factor and time delay are neglected in designing optimal control.Input delay is a kinds of time delay,which represented as the time gap in the control path from the time when measuring signal of the control plant begins,to the time when the control takes effect to the closedloop,it is an inevitable infactor in controlled systems.Usually,input delay is neglected in designing controller when it is a small quantity.Hovever,the performance of optimal control is too sensitive,and a very small input delay would greatly increase the actual performance criterion value.The optimal control problem of delayed input systems is one of the main research objects of this paper.This paper consists of seven parts.In Chapter 1,we give some introductions about the research status of the follows: optimal control of time delay systems,robust optimal control of time delay systems and the two wheeled inverted pendulum robots.In Chapter 2,the effects of the input deay to the controlled systems are expounded.In Chapter 3,we investigate the optimal feedback control for linear systems with input delays.The linear systems with input delays are converted into linear systems without delays,and then all the design procedures are based on the delay-free linear systems.We reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only.Based on this fact,we calculate the delayed optimal control and find that the optimal delayed feedback gain can be represented by a simple formula.In addition,we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays by using dynamic programming method.In Chapter 4,we present a design method for the optimal trajectory tracking problem of linear systems with an input delay,an external disturbance.A new integral state transformation is introduced to convert the original error system with time delay into a delay-free one,and observe a key but simple relationship between the original state variables and the new state variables.Therefore,the original problem is converted into an optimal control for the delay-free system.Finally,we design the controller in two parts: one is the delayed optimal trajectory tracking controller that is used to implement the given control tasks,and the other is obtained by the disturbance observer that is used for compensating the external disturbance.Due to the delay effect,the current state in the controller is replaced with a predictor state.Simulation results show that the combined controller not only implements the actual control task well but also has a strong robustness against the disturbance.In Chapter 5 and 6,the control of Back-and-Forth motion and Lower-Raise-Head motion to avoid obstacle of the two wheeled inverted pendulum are investigated.In controlling the Back-and-Forth motion,the linear optimal control theory is used because the tilt angle of the pendulum is forced to be small enough by minimizing a quadratic performance criterion with large weight of tilt angle error,and the controller is represented by using predictor-based feedback.When the uncertainties are taken into account,the optimal state of the nominal error system is chosen as the integral sliding mode manifold,and the control task of the Back-and-Forth motion can be well implemented by using the integral sliding mode control.Numerical simulations show that the proposed controller not only works very well in implementing the Back-and Forth motion task,but also has strong robustness against uncertainties.In controlling the Lower-Raise-Head motion,linearized models do not work due to the strong nonlinearity caused by big tilt angle for avoiding obstacle.Without considering yaw movement,the wheeled inverted pendulum system is decoupled,the subsystem governing the state of the tilt angle is transformed into a simple linear system by using feedback linearization.With a properly chosen trajectory tracking target,the control of Lower-Raise-Head motion is solved on the basis of optimal trajectory tracking control for linear subsystems with an input delay.Finally,some conclusions are made in Chapter 7.