Control design based on multi-index nonlinear control method

Firstly, a nonlinear control system based on the Riemannian manifold is defined from the perspective of holism, and the expression of the system’s state equation in the local coordinate system of the Riemannian manifold is given. The geometric structure of the Riemannian manifold is discussed. The influence of nonlinear systems is studied. The controllability and observability of nonlinear systems is studied. Secondly, using the properties of involute distribution and fully geodesic submanifolds, the local parts of nonlinear systems based on Riemannian manifolds are given. Controllable structured decomposition, local observable structure decomposition and Kalman decomposition. Thirdly, using the properties of orthogonal involute distribution family, incremental involute distribution family and fully geodesic submanifold family, the research is based on Riemannian manifolds. The parallel decoupling problem and cascade decoupling problem of nonlinear control systems on the above, as well as the local disturbance decoupling problem of affine nonlinear control systems.


Introduction.
The geometric theory of control systems is an important aspect of nonlinear control research. With the in-depth development of nonlinear control system research, people gradually realize that the geometric structure of the state space has an effect on nonlinear control. The precise control of the system has an important impact, such as a large-range aircraft system flying at a large angle of attack and a deep space detector system. In recent years, due to the wide application of differential geometry methods in the control theory of nonlinear systems, people can make in-depth discussions on the internal relationship between the geometric structure of the state space and the nonlinear control system. From the perspective of mechanical systems, related control problems, tracing problems of precise mechanical systems, and dynamics and control problems of non-precision mechanical systems are studied [1]. This discussion provides a clearer background to the actual problems and is easy to understand. However, due to mechanical Due to the limitation of the theorem, this research method is often trapped in a small local area to deal with problems, and the holistic thinking method of modern geometry is difficult to function in a systematic and in-depth manner. Some researchers have discussed nonlinear control systems based on general fiber bundles. The various characteristics of the system structure are discussed, but this method is not intuitive enough. Although the influence of the fiber space structure of the nonlinear control system is discussed, the influence of the geometric structure of the bottom manifold on the nonlinear control system is not studied.
After the 1990s, the geometric method research on nonlinear control systems has weakened. Although differential algebraic methods are involved, the progress has not been obvious. Some wellknown geometric cybernetics scholars have turned to the study of feedback stabilization and various quality designs of nonlinear systems. Some new control methods are involved, such as H ∞ control, adaptive control, etc, and many research results have appeared, but there is still a lack of deep understanding and understanding of the overall and global problems of nonlinear control systems. With the development of this theory, we found that the control systems studied by this theory seldom directly discuss the influence of the geometric structure of the state space on the system. This is because the state equation of the system does not introduce quantities related to the geometric structure of the state space. In this way, one On the one hand, the research on the control system is very inaccurate [2]. On the manifolds of different geometric structures, the equation of state of the system is the same; on the other hand, the powerful tools of geometry cannot fully play a role in the research of control systems. Therefore, we must The method establishes a direct connection between the geometric structure of the state space and the state equation of the control system.
In this paper, a smooth Riemannian manifold is selected as the state space of the system. In response to the above problems, according to the characteristics of the Riemannian manifold, we define a nonlinear control system based on the Riemannian manifold from the perspective of holism. Take the smooth tangent on the manifold. The vector field is the state vector field of the control system, and the Riemannian manifold is connected with the state vector field through the auxiliary curve. This has three advantages: First, the connection coefficient term related to the geometric structure of the Riemannian manifold is introduced in the state equation representation of the nonlinear system given in the local coordinate system of the manifold, and the flow can be directly studied through this connection. The influence of the geometric structure of the shape of the nonlinear control system [3]. Second, when the Riemannian manifold is a flat Euclidean space, the connection coefficients are all zero, and the nonlinear control system on the Riemannian manifold we define is different from the usual Euclidean space. The nonlinear control system is consistent. In addition, the Riemannian manifold has a metric structure, which can describe the orthogonality of the tangent vectored field, so that the reasoning, demonstration, and calculation of the control system can be simplified. Third, there are various fields in physics and mechanics, such as flow fields, Electric field, magnetic field, gravitational field, gauge field, etc, to study nonlinear systems with smooth tangent vector field as the state vector field, not only has important theoretical significance in mathematics, but also can provide for the in-depth study of corresponding problems in mechanics and physics. Precise language and powerful tools [4].
Below, for the nonlinear control system based on the Riemannian manifold, the relevant research results are introduced from three aspects: the local coordinate representation of the state equation, the local controllability, the decomposition of the observable structure, and various decoupling problems.

The local representation of nonlinear control system on Riemannian manifold
The concepts and definitions of mathematics and cybernetics cited in this article, let (M, G) be a mdimensional smooth Riemannian manifold, G is the Riemannian metric on M, and D is the Levi-Civita connection of the metric G on M [5]. TM is the tangent bundle of M, and Γ (TM) is the set of smooth sections of the tangent bundle TM. Consider the nonlinear control system on M. 3 ( t)) is locally satisfied, that is, the state vector field X (t ) is the smooth tangent vector field X on the curve γ ( t ). The curve γ (t) is called the auxiliary curve of the state vector field X (t) in the coordinate neighborhood (U, xi). Further, if X (t) is the tangent along the auxiliary curve γ (t). The vector field, that is, X (t ) = d γ/dt, then the curve γ ( t ) is called the state curve of the state vectored field X (t ). After introducing the auxiliary curve γ (t ) of the state vector field X ( t ), Then dX (t ) /dt is the directional derivative of the state vector field X (t ) along the tangent direction d γ/dt of the auxiliary curve, and D is connected by Levi-Civita of M. In the local coordinate neighborhood (U, xi ), let : Then we can get by calculation: Therefore, in the local coordinate system (U, xi), the state equation of the nonlinear system (1) can be expressed as: Equations (3) and (4) are non-linear ordinary differential equations. According to the theory of ordinary differential equations, given the initial value, there will always be a unique solution that satisfies the initial value.
It is worth not that although the local coordinated neighborhood on the manifold is considered here, it is different from the local neighborhood in our usual sense. The local coordinated neighborhood here is relatively overall, and can actually be A relatively large area [7]. For example, for Euclidean space, it has only one coordinate neighborhood, which is itself. For a spherical surface, one of its local coordinate neighborhoods can be an open hemisphere.
It can be seen from the above that in the local coordinate system of the Riemannian manifold M, for the nonlinear system, the study on the state vector field X (t ) and the study on the state curve γ (t ) of X (t ). be equivalent. In fact, if the state vector field X (t ) is known, the integral curve of X (t ) is the state curve of X (t ). On the contrary, if the state curved γ ( t ) is known, then The tangent vector field along γ (t ) is the state vector field. Equations (3 ) and (4 ) contain the connection coefficients of Levi-Civita connection D on the Riemannian manifold M in the local coordinate system, and they will be based on the Riemann flow The metaphysical nonlinear control system is closely related to the internal geometric structure of the Riemannian manifold. On the one hand, the influence of the geometric structure of the state space on the nonlinear control system can be studied through the equations (3 ) and (4 ). On the other hand, the overall The powerful tools of differential geometry can work through this connection [8].
This article focuses on the study of the restriction and influence of the geometric structure of the state manifold on the nonlinear control system based on it. In order to further understood the nonlinear system based on the Riemannian manifold is related to the internal structure of the Riemannian manifold. The representation of the nonlinear control system based on three spatial forms : Euclidean space, unit surface, and hyperbolic space in the local coordinate system is given, and the effect of the curvature structure of the Riemannian manifold on the nonlinear system is shown [9]. The representation of the nonlinear control system built on the cylindrical surface and the cone surface in the local coordinated system is presented, and the influence of the special curved surface structured and the singular structure of the nonlinear control system is shown.

The problems of controllability and observ-ability decomposition for nonlinear control systems
3.1. The local controllability de composition of structure Below, we use the local orthogonality of the Riemannian metric on the Riemannian manifold and the properties of the involute distribution and the fully geodesic submanifolds to prove the local controllable structure decomposition theorem of the nonlinear system based on the Riemannian manifold. For convenience, The specified index change range is as follows : A, B, C =1, 2,., m ; i, j, k =1, 2,., d ; α, β, σ=d +1, d +2,., m.
We noticed that if the Riemannian manifold M is regarded as a configuration manifold, γ ( t ) = (γ1 ( t ), γ2 ( t ),., γm ( t )) ∈ M is regarded as a configuration, then the system of equations ( 5 ). Describe a class of inexact mechanical systems. The equations (5 ) can be analyzed and discussed using relevant research methods of the literature. Inexact mechanical systems are often encountered in practice, such as flexible robot control systems, satellites Attitude control system, redundant mechanical control system, etc.
The local observability decomposition of structure and local Kalman decompositions Let Δc be a non-singular involute distribution on the Riemannian manifold M, and the nonlinear system (2.1) for any allowable control u(t), has f(X(t), u(t)) ∈ Δc. Δu is the largest non-singular involute distribution contained in Ker(h) on M, and the nonlinear system is invariant to any allowable control u(t), Δu is f(X(t), u(t)).
4. The decoupling problems of nonlinear control systems on Riemannian manifold For many years, the various decoupling problems of the system have been one of the important contents of the geometric theory research of linear and nonlinear control systems. This is not only because the method and design of the research system decoupling problem allow us to study the control system Turn the complex about simple, and make it easier. More importantly, the decoupling problem is the link between the structure of the control system and the state space structure of the system. The study of the decoupling problem reveals the essential structure of the control system. This section studies the foundation The decoupling problem of nonlinear systems on Riemannian manifolds. Using the properties of orthogonal involute distribution families and fully geodesic submanifold families, the nonfeedback parallel solutions to the state equations of nonlinear control systems based on Riemannian manifolds are studied [10]. The coupling problem and the state feedback parallel decoupling problem of the nonlinear control system. Using the properties of the increasing involute distribution family and the fully geodesic submanifold family, the non-feedback cascade of the state equations of the nonlinear control system based on Riemannian manifolds is studied The decoupling problem and the state feedback cascade decoupling problem of the nonlinear control system. As an application of the state feedback decoupling problem, the local disturbance decoupling problem of the affine nonlinear control system based on the Riemannian manifold is studied.

The cascade decompo-sition problem of state equation
Using the properties of the increasing involutive distribution family and the increasing full geodesic submanifold family, the problem of non-feedback cascade decoupling of the state equation of the nonlinear control system based on Riemannian manifolds is studied [11].
4.2. The state feedback decoupling problem of nonlinear control system By discussing the decoupling problem of nonlinear systems. If the nonlinear system has parallel (cascade) decoupling, then the nonlinear system has state feedback parallel (cascade) decoupling [12]. It can be proved that the nonlinear system state feedback parallel decoupling theorem.

The local disturbance decoupling problem of affine nonlinear control system
As an application of state feedback decoupling of nonlinear systems based on Riemannian manifolds, we will discuss the problem of local disturbance decoupling of affine nonlinear control systems based on Riemannian manifolds.

Conclusion
This paper studies the geometric theory of nonlinear control systems based on Riemannian manifolds. In order to study the influence of the geometric structure of Riemannian manifolds on nonlinear control systems, we define a kind of nonlinear control system based on Riemannian manifolds from the point of view of integration. For nonlinear control system, the expression of the state equation of the nonlinear control system in the local coordinated system of the Riemannian manifold is given. This establishes a direct connection between the state equation of the nonlinear control system and the geometric structure of the Riemannian manifold. Starting from this, we have studied the Riemannian manifold structure and connection structure of the state space from several aspects such as the controllability, observability, controllability, observable structure decomposition of the nonlinear system, and various decoupling problems of the control system. The influence of, curvature structure and submanifold structure on nonlinear control system.
These theoretical studies enable us to have a deeper understanding and understanding of the overall nonlinear control system. We know that general finite-dimensional differential manifolds are always established on the background of Euclidean space. The famous Whitney theorem tells us that a differential manifold can always be embedded in a sufficiently high-dimensional Euclidean space as a submanifold. This kind of differential manifold that starts in Euclidean space and converges on Euclidean space are the overall differential geometry and large-scale analytical research provides a wide range of markets. Therefore, for the theoretical framework of nonlinear control systems based on general differential manifolds, in-depth research on various algorithms and designs will be an important aspect of the development of modern control theory one.