The brief introduction of Digital Lagrangian ver. A and ver. B
Generally Speaking, dynamical behaviour of a material or plate would be modelled by Partial Differential Equation (PDE) but control engineering require to be modelled by Oridinarlly Differential Equation (ODE). When about piezo was known at B3 of University of Tsukuba, the author thought that's it. It is suppoosed that vibration of material of bicycle could be controlled using piezo material. PDE? So what? It could be imaginzed dynamical behaviour with some continuous material in my brain. The author thought it could be done even for control. The rest is a hypothesis of physics and solid equations by mathematics based on that hypothesis. Then, how does Smart Material Systems should be modelled? The answer is Digital Lagrangian.
Until this part is a story at B.Eng course at University of Tsukuba, Japan. The following part is about issues at M.Phil/Ph.D course at Coventry University, England.
What is a Digital Lagrangian? The basic idea is very simple and which is related to Finite Element Method (FEM). Here, the author describes short summary of Digital Lagranian ver. A and ver. B.
There are no figure for this physics even it deployed modelling of material especially CFRP with piezo. The main reason is having no figure, making abstract physics, it is supposed to be extended this theory to other kind of systems such as weather systems, fluid dynamical systems or even for electro-magnetic dynamical systems.
iPad tuition material is expressive and wet. Ordinary text book for students are dry, no meaning? It would be said No. Making figure or moving material by python with matplotlib, C++/JavaScript with OpenGL, functional calculator with graph paper, etc, one may have deep understaning toward physics or mathematics. Similar thing could be said for Computer Science as well. Thus, it is encouraged for students to make figure of digital lagrangian by themselves.
It means that Swift Playgrounds is not good for education? What is a purpose of Swift Playgrounds and Smalltalk/Squeak? Using them, one could interact with program or object using programming language and environment by thinking themeselves. Although author has not been used those programming environment, it seems it would be better for programming, mathematics and physics especially for primary school students since there are one's thoughts.
The first approach of Digital Lagrangian of a Smart Material Systems, say Digital Lagrangian ver. A, requires energy of bending of a material. Why bending? That is it is supposed that vibration is target of control and dynamical behaviour would be modelled by the summation of all bending over the material with respect to x-y axis. Basic idea of ver. A is very simple. Make digitize lagrangian with respect to space both for energy and partial difference.
First, a material is modelled with host material which is Carbon Fiber Reinforced Plastic (CFRP) and attached material which Is piezo material. Then, micro mechanics is utilized to model macro behaviour of composite material.
Next, bending behaviour is considered, and then, energy is obtained over x-y axes. The trick is hear. Energy is function of x and y but what will be happened if which is obtained via digitized axes which is x_i and y_i over all i (i = 0, 1, 2 ... n)? Since energy is function of x_i and y_i (i = 0 ... n), partial differential property with x_i and y_i for each i can be obtained. If lagrangian is obtained via x-y axes and lagrangian is evaluated with x and y, PDE will be obtained. Since it is digitimzed lagrangian, newton's equations of motion as high order ODE can be obtained. Then, control can be done with high order ODE.
Thus, smart material systems will be target of control since it is high order ODEs with input of electricity to pieze materal and sensing of behaviour with pieazo material.
This is the bried introduction of Digital Lagrangian though original paper utilized 16 page by LaTeX.
It is interesting to think about digital lagrangian but difficult to understand for undergraduate student? It should be Master of Science candidate of Univ of Tokyo or Oxford? It would be No. If one really understand 1-2 year grade of undergraduate level of physics, with the support of some referenecs such as book of Composite Material, it is possible to understand about digital lagrangian. If one cannot understand, it would be evidence such that that person has high gradte for exam but not understand the essence of physics.
Finally, Digital Lagrangian ver. B is described here. The basic idea is very simple. Think about multi-jointed robot arm whose arm is rigid body and cannot be bended but each joint can be bent. Since it is multi-jointed robot arm, it can be modelled by ODE. Then, make length of arm L small enough, say delta_L. Since delta_L is close to 0, its behaviour would be close to plate. Which means that it is high order ODE but it models dynamical behaviour of plate or smart material systems.
That is fine for modelling of dynamical behaviour by Digital Lagrangian. But it is just an approximation using digitimized point (x_i, y_i) and there would be modelling error. What could said in that situation, then? The answer is robust control. This thesis is hybid of science and engineering. The author took the way of engineering in this topic. This is described in summary of this thesis. Check the detail for latter chapters for robust control.
The following is out of scope of this thesis but commented. What is PDE? What is ODE? Is there any relation? Deploying the idea of Digital Lagrangian, infinite order ODEs would be converged to PDE. That is a because of the fact that physically speacking, both are come from energy and derived by the very similar mathematical techniques. It is supposed that there are no distinct difference between PDE and set of ODEs.
Why author did not include article of digital lagrangian, which is 16 pages analysis by LaTeX to the original M.Phil thesis? That is because 1. major of my primary supervisor is pure mathematics and theoritical robust control; 2. major of my secondary supervisor is applications of control engineering and some physics. Digital Lagrangian and analysis of composite plate require more physics than ordinary control engineering person have. Thus, author did not included in my original M.Phil thesis. However, at now, author thinks it could be included it if it were appendix since it is just support main argument but not main argument.
The original 16 pages proposal is half math, half physics. Actually, the original core M.Phil thesis is started from that proposal but not included in thesis. By this part of thesis, now physics is complete. Here is the real start point of this real M.Phil thesis.
From the following chapters, discussion of adaptive robust control is given based on assumptions made by physical insight composed of analysis of smart material system by digital lagrangian. It could be said that without physical insight, appropriate assumptions and the structure of system matrix A of a nominal model for start point of mathematical analysis cannot be given. It had been heard at undergraduate study that mathematicians make toy by physics and then, play with mathematics. Here is toy by physics for theoritical adaptive robust control. This part is close to pure physics. Thus, it could be said this thesis can be started from physics, then, mathematics, applications of theory and finally philosophical dicussion is given.