The refined summary
This thesis was born on the boundary of the theory of control and a particular application, namely a smart material system. More specifically, the primary motivation of this study comes from the following questions. Is it really possible to use smart material system in a practical situation? What is the problem from the view point of the theory of control? In this work, it is assumed that one of the most important problem is that of robustness of the controlled system, and it is supposed that one of the solution would be estimate uncertainty and disturbance without a priori knowledge in order to cancel their effect on system behaviour. This study provides a method, including its theoritical foundation, to estimate and cancel out any bounded disturbance and/or uncertainty.
So, why smart material systems? What is the aim of this research? The following is the triple of the reason:
- Aim: To reduce energy consumption of mechanical systems;
- Objective: To make adaptive robust controller for a class of smart material systems;
- Methodology: analysis, proof and hypothesis driven numerical experiment. Real experiment is out of scope.
Smart material system is a material composed from Carbon Fiber Reinforced Plastic (CFRP) and piezo material. Generally speaking, CFRP is light material. Is it possible to CFRP more light? If it are more light, it is possible to reduce energy used for vehicle, train and aeroplane. Smart material systems can be used to implement body of vehicle, train and aeroplane. Using that material for those systems, those systems will get more light but not bend or vibrate. That material will be ideal material for those industry applications.
How to obtain or supply oil is the problem over the world but especially for Japan. Since in Japan, there is not source of oil. There is EV. So oil is not matter? What is electricity in Japan? There was 3.11 which is massive earth quake in east Japan and nuclear energy system had been collapsed. It is too dangerous to replace all electric source to nuclear power. That means, Japan need to utilize to obtain electricity.
Author suppose that The World War2 is an economical war. Why Japan had been required to fight against US? Is it really required to do so? Author suppose it isn't. Why Japan went to Asia during that war? That is because of oil. If Japan was declared that we are not fight against any country, do some economic trade with some countries and make some fair connection with Arabics, war is not required. Strategically speaking, Japan had failed. Actually, atomic bombs were applied to Japan. This weak point is still remain in Japan.
Therefore, it is supposed that more light material is really required over the world especially for Japan.
The world is fragile. We all people in the world suffer to energy problem and inflation. Why inflation despite the fact that there is only war between two countries? There is statical value called core core CPI in Japan. In core core CPI, energy and foodstuffs are not included in the value of price to measure inflation. But that assumption may not be correct. It might be because of the fact that energy is used in transportation and fee of transportation is included in the PL of companies. Mitigation plans of this phenomenon would be derivative, restructure of global supply chain and engineering/research. Thus, even for global economics, energy and light material are important for all over the world.
Private motivation is come from making the ultimately the new bicycle. If CFRP is used for frame of bicycle, especially for road bike, its stiffness is high, it is light, and it absorb vibration. However, if CFRP bike is used for serious situation, it is not enough. A better frame is required. Thus, author thought to control frame with piezo material. That is the reason to go to the world of material and control.
Yet another question is why I chose bicycle for my application. That is because author had been involving in sport bicycle club both for road and track race when he was high school student. Author had a right to go inter-high school race which is nation wide race in Japan. If it were team pursuit, its speed is around 50km/h. Author was serious for bicycle. There are a lot of interesting physical phenomena in bicycle. If baton wheel and disk wheel are used, it is better especially for time trial. Why? Fluid dynamics? If it were team pursuit, front person is changed time by time to balance effect of wind pressure. It is obvious it is about fluid dynamcis. But what is flow of wind? Is it possible to analyze? If aluminum is used for frame, its shape is differnet from that of CFRP or other alloy. Why? If stem is long, it bend slightly. If special alloy is used for long stem, it does not bend. What is alloy? The question is how to analyze behaviour of bicycle? Or even ultimately new bicycle could be made if the power of engineering is applied? Therefore, author went to the world of mechanical engineering.
One of the most important problems in control systems is the robustness of the controlled system.
Why robustness? The primary question is what is modelling error and how to control provided by the fact that there is always modelling error. In CFRP research lab of University of Tsukuba, it was shown that even model gets complex, there is always modelling error by master's student. In that time, is it possible to control? That is the reason author chose robustness for the topic of postgraduate study.
It is known that almost all physical systems, such as mechanical or structural system, contain some form of uncertainty. Even smart material systems cannot escape from this problem. Such system consist of host materials, sensing and actuating layers, which are attached or embedded to the host materials. The modelling of smart material systems gives rise to infinite dimention models if it were modelled by ordinary differential equations (ODE). In practice, however, a model is obtained by using the Digital Lagrangian, and hence, a high order finite ODE is obtained. Such approximate models will inevitably generate uncertainty, representing unmodelled dynamics. In addition, such system models would suffer from parametric uncertainty and external disturbances. Thus, this type of system model would include many type of uncertainties.
In past decades, much research has been done using a deterministic approach for the robust control problem. The majority of this work assumes a known upper bound to uncertainty and disturbance and robust controllers are determined deterministically. However, for smart material systems, it may be difficult or even impossible to obtain such a “priori” knowledge of any disturbance. If this is the case, what can be said about robustness of controlled system without a “priori” knowledge of any disturbance? Part of the answer to this question can be found in this thesis.
This thesis considers a linear uncertain system in which the uncertainty and/or disturbance is known to be bounded, but its bound is unknown. The main contribution is that an adaptive feedback control law is designed to estimate the bounded disturbance. The design of the adaptive control algorithm is novel and the adaptive control algorithm is easy to implement. This information can then be used cancel the effect of the disturbance in the system. This has the advantage that, if further design objective are to be realized, the control can be designed based on the information from the known nominal model only and not on the model with uncertainty.
This thesis is organized as follow. In first chapter, Digital Lagrangian for modelling of smart material systems is introduced. In next chapter, the problem of adaptive robust control is defined and proof and analysis is given for arbitrary N dimensional system with assumptions derived from smart material system model obtained from Digital Lagrangian. At next chapter, applications of robust control is shown. At final chapter, what is a modelling? What is a control? and relation between Mori-Tanaka is discussed. Finally, conclusion remarks and further work is shown.