This book, in fact, has been read before, blindly following the book and deducing formulas, but I always feel that I can't grasp the essence.
Therefore, this time it is a new look and a new foundation.
First of all, the representation of state space is essentially the theory of vector space, which is a kind of filling to the structure of vector space. This piece is very common in physics. The phase space of the system is a typical example. The displacement and velocity of all particles in the system determine the state of the system.
The state space can be regarded as the generalized phase space of the system, and the generalized displacement and generalized velocity are used to replace the usual mechanical velocity and displacement, which are called state parameters. Independence is the concept of independent degree of freedom, which corresponds to vector space and is linear independence. The largest linear independent group constitutes a set of bases, so the largest independent state parameter set constitutes the base of state space, and all state vectors can be represented by them.
This representation is also quite common, because it is natural to use vector space to represent the whole representation set of independent parameters.
There is nothing to say in this area. It can be considered that there is an isomorphism between state space representation and vector space representation, so all concepts have their isomorphism concepts.
Then there is the key part. The purpose of control theory is to find a suitable parameter set in the state space, so that this parameter set can naturally be kept in an optimal state.
It can be understood that taking two dimensions as an example, the state space is a plane, the optimal state is a point, and the control system is a curve on the plane, because the control system itself is a differential equation. According to the qualitative theory of ordinary differential equation, the solution of differential equation is a family of solution curves in phase space, and the initial conditions are selected to become a solution curve.
The classical control system is directly two-dimensional, so the phase diagram of the system can always be drawn. So it is the theory of differential equations. If we learn the basis of control system and look at various analytical solutions of differential equations, it will certainly be of great benefit.
For modern control theory, it is actually a complex differential equation with high order, variable coefficient and type mutation. These things are difficult to deal with, and many questions have no answers. Therefore, people need to study it.
Therefore, once the words "modern" and "advanced" are added to these courses, most of them are theories to be learned, and some basic and recognized methods are often given. For special and difficult courses, we can't find them. This is called cutting-edge research.
Therefore, the content given in the modern control theory book is still a solvable problem, but it uses more complicated mathematical theory, that is, the solution of differential equations, which is also the eigenvalue problem that people often say in physics. The solution of differential equations naturally needs the language of matrix and vector, that is, the method learned from matrix theory. Therefore, to some extent, modern control can be regarded as an application of matrix theory.
In fact, the relationship between matrix theory and functional analysis is also very close, one is finite dimension and the other is infinite dimension, both of which are used to solve the problem of parameter representation set.
Sequence convergence and curve convergence can also be regarded as point convergence and function convergence.
Then there are some random questions. Without parameters, it is a common probability problem. With parameters, it becomes a stochastic process problem. Stochastic analysis and stochastic differential equations are more complicated problems.
These things are too difficult, unreliable and difficult to apply.
The other part is the synthesis of control system, which is a more practical part. PID control in classical control theory, using specific principles to realize the designed control system, is also a very technical test. Just like the realization of the circuit, it will face many practical problems, such as interference error and delay, which can not be grasped only in theory.
Modern control no longer uses this simple control element. For complex situations, complex systems, computers, networks and special data processing chips are naturally needed. Control has become a hodgepodge, which is also a feature of the information age. Computing and network have penetrated into various fields.
It can be said that the idea of machine learning has subverted all fields, and the optimization theory, the core of machine learning, is sweeping all primary labor. Just as the popularity of calculators makes arithmetic meaningless, universal data processing and calculation units can make simple theoretical application very easy. According to these parameters, you can get the result. What book are you reading? Those simple knowledge have already been written into the chip.
So now the control system has become a systematic project, and it is difficult for someone to grasp all the details. Of course not. This is also the productivity progress brought by cooperative labor.
Although many people are encouraged by such scientific and technological progress, people's basic concepts have not changed, and they are still producing for their desires, overexploiting natural resources and producing many things beyond their needs. It is a naked waste in the name of high quality and high enjoyment. This kind of progress is hardly a good thing, which will only make people further squeeze the limits of natural resources and cause more unknown damage. Ignorance is a very dangerous thing, and extreme weather is already a sign. Harmony between man and nature is the long-term way.