| This paper is concerned with control strategies of the triangular nonlinear control systems. It is the fundamental idea of this paper that, for a class of nonlinear system, one may design controller by equivalently transforming this system to a triangular nonlinear systems and then using the technique of backstepping or forwarding to design the controllers.In chapter 1, we reviewed the history of control theory, especially the progress of the canonical form of nonlinear systems and the backstepping approach.Chapter 2 includes a brief introduction of differentiable manifold.In chapter 3, using differential-geometric control theory, we solved some problems about the equivalence of nonlinear systems which concern whether a complicated system may be, in some sense, equivalent to a simpler system. The major contributions of this chapter are as follows:(1) Two criteria for converting an ordinary differential equation to normal form via coordinate transformation were presented. Firstly, it was proved that ordinary differential equations can be converted to normal forms if and only if there exists a single-input control system which treats the vector field of the ordinary differential equation as drift vector field and can be fully linearized by state feedback. Secondly, near the non-singular point, there always exists some coordinate transformation to perform the converting;and near the singular point, the converting can be found if and only if, at this point, the eigenpolynomial of the Jacobian matrix of the vector field is equal to the minimal polynomial of the same matrix.(2) The problem of two affine nonlinear systems equivalent to each other via coordinate transformation was considered. The solution of the problem was given when the distribute spaned by the drift vector field and input vector field of either of the two systems have regular involutive closure.(3) We showed the conditions of nonlinear systems equivalent to, via linear coordinate transformation and feedback, upper-triangular systems, low-triangular systems and the systems which are both upper-triangular systems and low-triangular systems.(4) Using the theories of singular distributions, a necessary and sufficient condition under which an affine non-autonomous system is locally feedback equivalent to, via a change of coordinates and state feedback, a non-autonomous low-triangular system.(5) We presented two necessary and sufficient conditions under which a non-autonomous system is locally feedback equivalent to, via coordinate transformation and state feedback, a non-autonomous p-normal form.In chapter 4, we solved two problems about the equivalence of discrete-time nonlinear systems which concern whether a complicated problem may be, in some sense, equivalent to a simpler system. The major contributions of this chapter are as follows:(1) A necessary and sufficient condition was provided for transforming a discrete-time nonlinear system to a low-triangular nonlinear system via feeback and coordinate transformation. Moreover, a necessary and sufficient condition was provided for transforming a discrete-time nonlinear system to a low-triangular nonlinear system via feeback and coordinate transformation.(2) A necessary and sufficient condition was provided for transforming a discrete-time nonlinear system to an upper-triangular nonlinear system via feeback and coordinate transformation.In chapter 5, we study the backstepping approach of the low-triangular nonlinear systems. The major contributions of this chapter are as follows:(1) For non-affine systems with feedback form, we improve on the former result of bounded backstepping, so the technique can be used in more general cases. The advances of our new approach are as the follows. Firstly, there are fewer restrictions in the new design, especially we reject the assume that the virtual control is bounded. Secondly, the bound of feedback is compressed successfully, in some sense, this result is stronger with the less restriction. Thirdly, the proposed Lyapunov function is not including the saturation functions, which is technically different from all of the former results on bounded backstepping. Finally, the proposed method allows applying to a more general class of non-affine control systems.(2) V° bounded robust control, the purpose of which is to restrict the trajectory of system in a given region, is really practical in engineering. Using the nonsmooth analysis theory, the applications of L°° bounded robust backstepping were extended to nonsmooth control systems. The right sides of considered system equations are locally IT bounded, and the designed Lyapunov functions are regular and local Lipschitz.(3) Consider a hybrid system which is combined by a nonlinear control system and a continuous Petri net, and the latter can represented as a set of min-plus linear algebraic equations. We applied the backstepping approach to make the nonlinear system asymptotically stable and make the Petri net limited buffer capacity.In the chapter 6, we considered the observablity. First, we reviewed the definition of observablity and the necessary and sufficient conditions under which a nonlinear system is completely uniformly locally weakly observable, respectively, a single-output nonlinear system is completely uniformly locally weakly observable. Then, a sufficient condition under which a multi-output nonlinear system is completely uniformly locally weakly observable was given. We also gave a canonical form of these multi-output systems. The sufficient conditions become necessary and sufficient conditions when the system is single-output.In the chapter 7, we applied the nonlinear control theory to model and observe the avermectin fermentative process. According to the mechanism of fermentation, we constructed a mathematic model of the fermentative process. The parameter of the model is identified by genetic algorithm. Based on thetheory of nonlinear observer and the main theorem of the former chapter, discrete and continuous observers are designed for observing the concentration of mycelia respectively.
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