Construction and Researching Aircraft High Potential of Robust Stability Control System in the Form of Single-parameter Structurally Stable Mapping

Corresponding Author: М. Beisenbi, Department of System Analysis and Control, L.N. Gumilyov Eurasian National University, Astana, Republic of Kazakhstan This work is licensed under a Creative Commons Attribution 4.0 International License (URL: http://creativecommons.org/licenses/by/4.0/). 599 Research Article Construction and Researching Aircraft High Potential of Robust Stability Control System in the Form of Single-parameter Structurally Stable Mapping


INTRODUCTION
Control system design is one of the main tasks in automation of all branches of industry, including machine manufacturing, energy sector, electronics, chemical and biological, metallurgical, textile, transportation, robotics, aviation, space systems, highprecision military systems, etc.In these systems, the uncertainty can be caused by the presence of uncontrolled disturbances acting on an object control and ignorance of the true values of the parameters of control objects and unpredictable change them in time.The main goal in modern control system design is, in some sense, to provide the best protection against uncertainty in the knowledge of the system.The ability of a control system to keep stability in the conditions of parametrical or nonparametric uncertainty is realized as robust stability of system (Polyak and Sherbakov, 2002).Research of system robust stability consists in the indication of restrictions on control system parameters change (Polyak and Sherbakov, 2002;Dorato and Yedavalli, 1990).
The many papers research a problem of robust stability (Polyak and Sherbakov, 2002;Dorato and Yedavalli, 1990;Kuntsevich, 2006;Liao and Yu, 2008).In these works investigated the robust stability of polynomials, matrixes, within the linear principle of stability of continuous and discrete control systems, in works (Kuntsevich, 2006, Liao andYu, 2008) are solving the problems of absolute robust stability.In the practical tasks, connected with development and creation of control systems in technology, economy, biology and other spheres, in the conditions of essential parametrical uncertainty, the increase in potential of robust stability is one of the key factors, which guaranteeing to a control system protection from entry in regime of determined chaos and strange attractors.And guarantees applicability of models and reliability of the designed control systems work.
At present it is conventional that, real control objects are nonlinear and one of the main properties of nonlinear dynamic systems is functioning in the mode of the determined chaotic traffic (Andrievsky and Fradkov, 1999;Nicols and Prigogine, 1989;Loskutov and Mikhaylov, 2007).In linear dynamic systems it is appear in the form of control system's zero steady state stability loss (Beisеnbi, 2011b;Beisеnbi and Erzhanov, 2002).
In this regard, in the conditions of uncertainty, there was a need for development of models and methods of design of control system with rather wide area of robust stability, which called control systems with the increased potential of robust stability (Beisеnbi, 2011b;Beisеnbi and Erzhanov, 2002).The concept of creation of a control system with the increased potential of robust stability is based on results of the catastrophes theory (Gilmore, 1984;Poston and Stewart, 2001), where the main structural-steady maps are received.
This study is devoted to design of control systems with increased potential of robust stability by dynamical objects with uncertain parameters in a class of the single-parameter structurally steady maps (Beisеnbi, 1998(Beisеnbi, , 2011a(Beisеnbi, , 2011b;;Beisеnbi and Erzhanov, 2002;Ashimov and Beisenbi, 2000).
Researches of the recent years showed, that the method of Lyapunov functions can be (Barbashin, 1967;Krasovsky, 1959;Malkin, 1966) successfully used to analyse the robust stability of linear and nonlinear control systems.Usage of Lyapunov's functions method for the solution of a set of practical linear or nonlinear tasks is constrained by the lack of a general method for selecting or constructing Lyapunov functions and difficulties with their algorithmic representation (Barbashin, 1967;Malkin, 1966).An inappropriate choice of a Lyapunov function or the inability to construct one does not indicate instability of the system, only that a proper Lyapunov function has not been found.
The method of design of Lyapunov vector function (Voronov and Matrosov, 1987), on the basis of geometrical interpretation of asymptotic stability theorem and concepts of stability is offered.Therefore, the origin corresponds to a predetermined condition of the system, the unperturbed state and the equations of the state are formed concerning perturbations, i.e., in deviations of the perturbed motion from unperturbed (Malkin, 1966).Consequently, the state equations express the speed of change of a perturbations vector (deviations) and for steady system is directed toward the origin.And the gradient vector from required Lyapunov function, for stable system, will be always directed to the opposite side.It allows to present Lyapunov function in the form of a potential surface (Gilmore, 1984).Research of robust stability of the control system with uncertain parameters are based on ideas of a Lyapunov direct method.

MATHEMATICAL MODEL FORMULATION
Area of system stability: System can be written in expanded form: -the elements of the control object.The control law is described by a vector function in the form of single-parameter structurally steady maps (Gilmore, 1984;Poston and Stewart, 2001): 1) is defined by the solution of the equations: From (3) we receive a steady state of system: Other stationary states will be defined by solutions of the equation: Great number of solutions of the Eq. ( 5) can be written as: Here the system of the nonlinear algebraic Eq. ( 3) has the trivial decision (4) and uncommon decisions (6) when the Eq. ( 5) has imaginary decisions that can't correspond to any physically possible situation (Nicols and Prigogine, 1989).These decisions are joined with (4) when and branch off from it when states (6) also will be asymptotically steady, in other words, branches appears as a result of bifurcation while the state (4) loses stability and these branches are steady.
Verification of these statements is made on the basis of Lyapunov vector functions ideas (Voronov and Matrosov, 1987).
If Lyapunov function V(x) is set in the form of vector function V(V 1 (x), …., V n (x)), then components of speed vector will be equal (Beisеnbi and Uskenbayeva, 2014a;Beisenbi and Yermekbayeva, 2013a;Beisenbi et al., 2015): In the Eq. ( 7), substituting values of components of a vector of speed, we will get: Full derivative on time from Lyapunov vector function V (x) taking into account the equation of a state (1), we can define as product of the gradient from Lyapunov vector function on a vector of speed (Beisenbi and Uskenbayeva, 2014b;Beisenbi andYermekbayeva, 2013a, 2013b;Beisenbi et al., 2015), i.e.: From here (10) follows that full derivate on time from Lyapunov function will be negative function.
From ( 9) for components of Lyapunov vector function we will get: We can present Lyapunov function in a scalar form in the view: 10) is Lyapunov function and conditions of positive definiteness are defined by inequalities: Thus, the area of system stability (1) for the established state ( 4) is defined by system of inequalities (11).

Research of stationary states (6) stability:
The equations of a state (3) in deviations in relative steady state (7) can be written as (Beisenbi et al., 2015;Beisеnbi, 2011aBeisеnbi, , 2011b;;Beisеnbi and Erzhanov, 2002): The full derivative from Lyapunov function V (x) taking into account the state equations in deviations (12) relative to the stationary state ( 6) is defined as: Function ( 13) is negative function.We can find components of the gradient vector of Lyapunov function: From here we receive Lyapunov function in a scalar form: Function ( 14) on the beginning of coordinates addresses in zero, is continuous differentiable function and has the form of variables with odd degrees.Therefore on the basis of the Morse lemma (Gilmore, 1984;Poston and Stewart, 2001) function ( 14) around the steady state (6) can be represented as a quadratic form: From here positive definiteness of Lyapunov function will be defined by an inequality: Let investigate stability of a steady state (6): The equation of a state (3) in deviations in relative steady state (6) can be written as (Beisenbi et al., 2015;Beisеnbi, 2011aBeisеnbi, , 2011b;;Beisеnbi and Erzhanov, 2002): Omitting formal actions for research of stability of stationary states of ( 6), similar for a steady state x S (6) we will receive Lyapunov function in a scalar form: On the Morse lemma we will lead (Gilmore, 1984;Poston and Stewart, 2001) Lyapunov function, by means of stability matrix, to a quadratic form (Beisenbi et al., 2015): Stability conditions of a steady state (6) it will be expressed by system of inequalities: Thus, the control system constructed in a class of one-parametrical structural steady maps will be steady in indefinitely wide limits of change of uncertain parameters of the control object.The steady state (4) exists and is stable at change of uncertain parameters of object in area (11) and stationary states and (6) appear at loss of stability of a state (4) and they are not simultaneously exist.Stationary states and (6) are stable when performing system of inequalities ( 15) and ( 17).
Thus, the control system constructed in a class of one-parametric structural stable maps will be stable in indefinitely wide limits of change of uncertain parameters of the control object.The steady state x s 1 (4) exists and is stable at change of uncertain parameters of object in area (11).And stationary states x s 2 and x s 3 (6) appear at loss of stability of a state x s 1 (4) and they are not simultaneously exist.Stationary states x s 2 and x s 3 (6) are stable when performing system of inequalities ( 15) and ( 17).

Description of dynamics of the aircraft angular motion:
We investigate a task of traffic control of the aircraft by the pitch.Let consider that aircraft have constants, aprioristic-uncertain parameters, which values are located in the set area.We will notice that the similar situation can take place when aircraft flying on various modes, when height, the speed and loading of aircraft changes slowly in comparison with rate of angular motion.For the description of dynamics of the aircraft angular motion we use the following linearized equations (Andrievsky and Fradkov, 1999;Bukov, 1987): Their values depend on the factors stated above and can change over a wide range depending on height and the speed of flight.Exact values of parameters a priori not defined.Also we assume, that dynamics of executive body it is possible to neglect and consider that control is the deviation of rudder Then the equation of the aircraft angular motion will assume the form: As the control law we will choose: Thus, the system (18) with the control law (19) will assume the form: From the Eq. ( 20) we define the established conditions: The system (21) has the following stationary states: And other stationary conditions of system ( 21) are defined by the solution of the equations: This equation has nonzero solutions in the form: We investigate stability of system (20) in stationary points by the method of Lyapunov functions.Lyapunov function V (x) is set in the form of a vector function V(V 1 (x), …., V n (x)), then from geometrical interpretation of the theorem of asymptotic stability we will get (Barbashin, 1967;Malkin, 1966) The full derivative on the time from Lyapunov vector function V(x) taking into account the equation of a state (20), is represented as product of the gradient vector from Lyapunov vector function on a vector of speed i.e.: From the expressions (24) follows, that the full derivative on time from Lyapunov functions is always negative function.
On the basis of the Morse lemma we will present Lyapunov function in a scalar form in the following view: The conditions of ( 20) system stability in a steady state ( 22), we obtain, taking into account the negative definiteness of the functions (24) in the form of a system of inequalities: Research of stationary states (23) stability: The equations of system state (20) with respect to deviations of the stationary state ( 23) is written: Full-time derivative of the Lyapunov function V (x) with the equation of state (26) with respect to the stationary state ( 23) is defined as: From the expressions (27) follows, that the full derivative on time from Lyapunov function will be a negative function.We find the gradient vector components from Lyapunov vector function: On a gradient we will construct Lyapunov's function: By the Morse lemma from the catastrophe theory we can replace Lyapunov function ( 28) with a quadratic form: The condition of positive definiteness of Lyapunov function ( 28) or (29) we will get in a view: Hence a necessary and sufficient condition for the stability of the stationary state ( 23) of ( 20) system is performance of an inequality (30).

SIMULATION RESULTS
Control law is designed for linearized model ( 18) and, we find sufficient conditions for the stability of the stationary state and positive definiteness of Lyapunov function.
For the equations of dynamics of the aircraft angular motion: The matrices of coefficients are defined as follows: .Figure 1 show the results of the simulation system with the parameters from Table 1.
Figure 2 show the results of the simulation system with the parameters from Table 2.

CONCLUSION
Thus, the control system of aircraft motion with the increased potential of robust stability constructed in a class of single-parameter structurally steady maps provides stability for changes of uncertain parameters of the system.
It appears, the steady state ( 22) is globally asymptotically steady when performing conditions (25) and unstable at violation of conditions (25) and stability of a steady state (23) requires performance of conditions (30).When In other words, branches (23) appear as a result of bifurcation while the steady state ( 22) loses stability and these branches are steady.Stationary states ( 22) and ( 23) at the same time don't exist.It allows to increase the potential of robust stability of system in the conditions of uncertainty of parameters.

B
According to the conditions of positive definiteness of Lyapunov function (29) we get gain of the system.From (30) we define ,

Fig. 1 :Fig. 2 :
Fig. 1: Coefficient k 3 in the governing limits is a branching and there are new steady branches.

Table 1 :
System parameters the varying gain parameter k 3 .