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Abstract
We suggest a method for solving the terminal control problem for multidimensional affine systems that are not linearizable by feedback. We prove a sufficient condition for the existence of a solution and present a numerical procedure for its construction.
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... For nonlinear dynamical systems, approaches to solve problems with fixed ends are known only for some classes of systems. For example, for a statically linearizable system, the system can be transformed to a special form called the regular canonical form [2,3], and then one can find the open-loop trajectory in the terms of time polynomials, the order of which is determined by the number of boundary conditions. For some dynamical systems, it is possible to transform the system to a quasi-canonical form [4], for which a similar approach is applied. ...
... one finds a solution x(t), u(t) of system(3).The solution thus found is a solution of the problem (3)-(6), because the function x(t) satisfies the initial conditions by construction and the final conditions by virtue of condition (B). Using the flatness of the system (3), one can find the open-loop control for the problem with fixed ends. ...
A solution of a nonlinear perturbed unconstrained point-to-point control problem, in which the unperturbed system is differentially flat, is considered in the paper. An admissible open-loop control in it is constructed using the covering method. The main part of the obtained admissible control correction in the limit problem is found by expanding the perturbed problem solution in series by the perturbation parameter. The first term of the expansion is determined by A.N. Tikhonov’s regularization of the Fredholm integral equation of the first kind. As shown by numerical experiments, the found structure of an admissible control allows one to find the final form of high precision point-to-point control based on the solution of an auxiliary variational problem in its neighborhood.
The paper presents the results of optimization of the flight program for a passenger aircraft. In mathematical modeling, real operational limitations and possible changes in atmospheric parameters were taken into account. The mathematical model is characterized by the description of the flight of an airplane as a material point and the description of a gas turbine engine by a model of the first level of accuracy. The following operational limitations were considered: the possibility of cruising at certain levels, the restriction on the vertical speed with a decrease, the possibility of transition to the next echelon with an available stock of thrust of 20%. The optimization of the flight program is based on the criterion of the minimum amount of fuel expended for the flight, at a given range. Calculations were conducted for several standards of air temperature change, (depending on the climatic zone). A comparative analysis of the obtained optimization results is carried out, the degree of influence of the change in atmospheric conditions is estimated. When comparing flights in different climatic zones, the fuel costs were compared for a flight program optimized for the given atmospheric conditions and for a flight program optimized for use in the mathematical model of the International Standard Atmosphere (ISA).
State-feedback linearization is widely used to solve various problems of the control theory. An affine system is said to be state-feedback linearizable if there are a smooth change of variables in the space of states and an invertible change of controls, which transform the system to the system of a regular canonical form. However if a system is not state-feedback linearizable it yet can be orbitally feedback linearized, i.e. the system can be transformed to a regular canonical form after a change of the independent variable.The article solves the following terminal problem for multi-dimensional stationary affine systems: for given two states, find controls and a time T such that the corresponding trajectory of the system joins these states for the time T. We make an integrable change of the independent variable depending on controls. As a result, we obtain a non-stationary affine system, its dimension being one less than dimension of the original system. The new terminal problem with the restriction on controls is formulated for the transformed system. We prove the relation between solutions of the original terminal problem and solutions of the terminal problem for the transformed system. It is shown that to solve the original terminal problem it is sufficient to solve terminal problem for the transformed system. Then, we check whether the transformed system can be state-feedback linearized. For this purpose, we check the necessary and sufficient conditions of state-feedback linearization for non-stationary affine systems. If the conditions are met then we transform the system to a regular canonical form for which the concept of inverse dynamics problems can be used to solve terminal problems. However, due to the restriction on controls an additional check is necessary whether the found controls meet the restriction.An example of the terminal problem for the five-dimensional affine system with two controls is given. We prove that the system in question is not state-feedback linearizable on any open subset of the state space. However, the system can be transformed to a regular canonical form after the change of the independent variable depending on controls. The proposed method allows us to solve the terminal problem.
The symmetry of Cauchy problem solution for linear matrix differential equations is under research in the present article. The coefficients of the equation in question are supposed to be analytical functions in some domain of the complex plane. We find a formula for high-order derivatives of an arbitrary solution of the equation. We prove the sufficient conditions for the symmetry of Cauchy problem solution for linear matrix differential equations on the basis of the devised formula. To check these conditions, we need to analyse the properties of the special matrix sequence. Since the sequence consists of the infinite number of elements, the check is difficult to implement. It is shown that if some requirements are met, then it is sufficient to check only first several elements of the sequence. The example of the linear matrix differential equation is given to illustrate how the proposed condition may be used in proving the solution symmetry. The obtained results may be used in solving various problems of the control theory.
The problem under consideration is trajectory planning of an aircraft in a vertical plane with variable state constraints. The time of maneuver is considered to be known. The main problem is to find the permissible trajectory which meets given requirements. The most developed methods for solving similar problems do not allow us to take into account the restrictions on the system condition. The approach applied in this work allows us to automatically take into account the current restrictions during the required trajectory construction, not using any iterative methods. Building a program trajectory is carried out in a certain class of functions. The paper proposes an optimization approach to choosing the trajectory. The program control implementing this trajectory is based on the concept of inverse dynamic problems. It enables us to synthesize a proper program control, to build the control stabilizing program trajectories and to choose numerical optimization of the trajectory by a definite criterion. We describe a nonlinear mathematical model of the plane movement as a material point in the trajectory reference frame. We illustrate the findings of the research with examples and show the results of numerical modeling.
The paper considers a problem of the time-specified control terminal for the second order system with restrictions on the state variables.Most developed methods for solving problems of the terminal [1, 2, 3, 4, 5] do not allow us to take into account the restrictions on the system condition. To solve such problems are widely used methods based on the concept of inverse dynamics problems [6, 7, 8, 9, 10], with one step of which being to specify a kinematic object trajectory. Some methods use an iterative [11] process of finding a desired program trajectory.This work is based on the results presented in [12]. It is shown that the solution of the original problem is equivalent to finding the terminal phase of the trajectory that satisfies the restrictions imposed on the state variables, as well as the certain additional conditions. It is assumed that the restrictions imposed on the state variables can be represented as functions for which, in a certain class of functions, special approximations are built. A desired phase trajectory is built as a linear combination of obtained functions-approximations. Thus constructed phase trajectory is a solution to the original terminal problem. The presented formulas are true for both the upper and lower half-plane of the phase space. The paper proposes an optimization approach to a choice of the trajectory as well as the options to extend the set in which the phase trajectories are sought. It gives the numerical simulation results, a presented in [12] algorithm, and also the results of numerical solution to the optimization problem.This approach can be used to solve the terminal problems of vector-controlled mechanical systems with restrictions on the state variables.
One of the approaches to solving terminal control problems for affine dynamical systems is based on the use of polynomials of degree 2n − 1, where n is the order of the system in question. In this paper, we investigate the terminal control problem for which the final state of the system coincides with the origin in the phase space. We seek a set of initial states such that the solution of the terminal control problem can be constructed by using a polynomial of degree 2n − 2.Note that solution of the terminal control problem in question can be used to solve the problem of stabilizing the zero equilibrium in a finite time.For the second-order systems we prove the necessary and sufficient conditions for existence of the polynomial of the second degree which determines the solution of the terminal problem. The solutions of the terminal control problem based on the polynomials of second and third degree are given. As an example, the terminal control problem is considered for the simple pendulum.We also discuss solution of the terminal problem for affine systems of the third order, based on the use of the fourth and fifth degree polynomials. The necessary and sufficient conditions for existence of the fourth-degree polynomial such that its phase graph connects an arbitrary initial state of the system and the origin are obtained.For systems of arbitrary order n we obtain the necessary and sufficient conditions for existence of a solution of the terminal problem using the polynomial of degree 2n − 2. We also give the solution of the problem by means of the polynomial of degree 2n − 1.Further research can be focused on extending the results obtained in this note to terminal control problems where the desired final state of the system is not necessarily the origin.One of the potential application areas for the obtained theoretical results is automatic control of technical plants like unmanned aerial vehicles and mobile robots.
We consider the terminal problem for affine systems in quasicanonical form that are not feedback linearizable but admit orbital linearization. We suggest a method for solving terminal problems for this class of systems. We present an example in which the solution of a terminal problem is constructed by our method for a fourth-order system in quasicanonical form.
A new method for solving the terminal control problem for dynamical systems is formulated. This problem is to determine a program trajectory and a program control that takes the system from a given initial state to a given final state. The method is based on the addition of equations with control derivative to the source system and reformulation of the problem in the boundary value problem for the augmented system E. Additional equations must be chosen so as to satisfy the following conditions. There is a surjective map (covering) from the phase space E to the phase space of some dynamical system Y. The covering takes solutions of E to solutions of Y. Boundary conditions in the final moment are mapped to the boundary conditions on the solutions of Y. Any solution of Y satisfies the boundary conditions in the initial moment. Then the solution of the terminal control problem is as the solution of two Cauchy problems for dynamical systems E and Y. Augmented system E satisfying mentioned properties is called r-closure of the terminal control problem. It is shown that this approach generalizes the well-known method for solving the terminal control problem for flat systems. A flat system is a system whose solutions are uniquely determined by a certain set of functions of time (flat output). The mentioned well-known method is based on polynomial dependence of flat output of time and do not take into account constraints on the system.It is proved that for an arbitrary flat system r-closure can be chosen any determined system of ordinary differential equations of the corresponding order. It is showed how to construct a covering with the above-mentioned properties using the general solution of this system. The properties of the covering are proved only locally, i.e. when the initial time is close to the final time, and the initial conditions are close the final conditions. But this covering may be applicable to other terminal problems with the same final conditions. This result can be used to solve the terminal control problem for flat systems with constraints. In addition, an example demonstrates the possibility of applying this method to non-flat systems.
Nelineinye sistemy: geometricheskie metody analiza i sinteza (Nonlinear Systems: Geometric Methods of Analysis and Synthesis
Jan 2005
V I Krasnoshchechenko
A P Krishchenko
VI Krasnoshchechenko
Krasnoshchechenko, V.I. and Krishchenko, A.P., Nelineinye sistemy: geometricheskie metody analiza
i sinteza (Nonlinear Systems: Geometric Methods of Analysis and Synthesis), Moscow, 2005.
Investigation of the Controllability of Regular Systems of a Quasicanonical Form
Jan 2006
12
D A Fetisov
DA Fetisov
Fetisov, D.A., Investigation of the Controllability of Regular Systems of a Quasicanonical Form, Vestn.
Moskov. Gos. Tekhn. Univ. Estestv. Nauki, 2006, no. 3, pp. 12-30.
Transformation of Affine Systems with Control and the Stabilization Problem
Jan 1992
1945
A P Krishchenko
M G Klinkovskii
AP Krishchenko
Krishchenko, A.P. and Klinkovskii, M.G., Transformation of Affine Systems with Control and the Stabilization Problem, Differ. Uravn., 1992, vol. 28, no. 11, pp. 1945-1952.
Investigation of the Controllability of Affine Systems
Jan 2013
15-18
S V Emel'yanov
A P Krishchenko
D A Fetisov
Emel'yanov, S.V., Krishchenko, A.P., and Fetisov, D.A., Investigation of the Controllability of Affine
Systems, Dokl. Akad. Nauk, 2013, vol. 449, no. 1, pp. 15-18.