The influence of the input data and a Hurwitz matrix on the stability of the system in general is investigated. The factors that influence the behavior and timing of the transient process are analyzed. An expression is obtained for finding the time of the transient process as well as its dependence on the parameters of the Hurwitz matrix.
Functional stability theory originated in the 1980s [
The dynamic object (Fig. 1) is written in the form of linear difference equations:
Q( q )y( n ) qk Pu ( q )U( n ) P ( q ) ( n ) , where y(n) – initial value;
Q(q), Pu(q), Pξ(q) – polynomials of q; an, bn, cn; q; an, bn, cn – coefficients of polynomials; q – delay operator: qmχ(n) = χ(n–m) (sometimes the delay operator q–1 is used in the literature, so q–mχ(n) = χ(n–m)).
The control device (Fig. 2) is described as:
R( q )U( n ) Pr ( q )r( n ) Py ( q )y( n ) , where r(n) – input influence.
G(q) y(n) qk 1Pu (q)Pr (q)r(n) P (q)R(q)(n)
In functionally stable systems, vector of reducing filter parameters (Fig. 4) must change so as to ensure the optimality of the whole system over time.
To solve this problem we will use the method of inverse dynamics problems (Fig. 5).
Description of the mathematical model
The mathematical description is based on the matrix equations of the form [
X (t) F[ X (t),U (t), Z (t)] P(t);
X (t0 ) X 0, t [t0, tT ], (1) where U(t) is a vector of control; Z(t) is a vector of the object parameters; P(t) is a vector of disturbances acting on the object; X0 is the initial conditions.
The presented theory is intended to provide a given programmatic movement (state) of the object of control and may provide for solving the problems of stabilization, terminal control, and adaptive tracking (Fig. 6). An object that can be described by the system (1) must be on a given trajectory (correspond to a certain state) and provide a minimum of the functionality of quality:
J[Un0 (), X n0 ()] min J[Un (), X n ()] ,
(2)
where X n0 (t) , U n0 (t) are optimal vectors of state and control.
In Refs. [
The system of matrix differential Eq. 1 can be represented as a structural diagram (Fig. 7).
To solve the problem, we assume that the programmatic movement is asymptotically stable.
The following conditions are imposed on this class of controlled dynamic systems. 1. There is a subspace R Rm × Rn and a unique function U : R × Ξ → Rm such that for any pair of points {X, Z} R and the point {ξ} Ξ, the identity is satisfied Z F[ X , U (x, z, ), ] , (3) i.e., Eq. 1 can be solved with respect to control on the subspace R.
The condition (3) guarantees the existence and uniqueness of an ideal programmatic control law of the form Un = Un(Xn, Ẋn, Ξ), which ensures the accurate implementation of programmatic movement provided Xn(t0) = X0; P(t) = 0; (Xn(t), Ẋn(t)) R t [t0 ,tT ] .
2. The following restrictions are imposed on the initial and constant external
disturbances [
X (t0 ) X n (t0) 0;
P(t) C p , (4) where δ0, Cp are positive parameters.
3. There is a Hurwitz matrix Г with prime eigenvalues γi where X(t0) = X0 is the determined initial vector of state;
X X , (5) (6) (7) (8) (9) γi are the roots of the characteristic equation of the system, i 1, n , for any X Rn, the condition holds:
( X , X n (t) ( X X n (t))) R , t t0 .
U (t, X ) U[ X , X n ( X X n ), ] .
For any disturbances that satisfy the condition (4), the movement X(t) (X(t0) R) asymptotically approaches a determined programmatic movement Xn(t), i.e., it provides the asymptotic stability of the programmatic movement in general.
From the expressions (1), (3), (6), we find
F[ X (t),U (t), ] X X n (t) ( X X n (t)) .
X X n (t) ( X X 0 ) .
Due to the selection of the Hurwitz matrix Г, the required transient process is
provided [
According to the condition (6), the matrix is chosen with prime eigenvalues, then there are positive numbers C and γ such that
Re i , X (t) X n (t) C X (t0 ) X n (t0 ) e (t t0 ) , t t0. (10)
Thus, the programmatic movement in general will be asymptotically stable. 4
Results and Discussions Let us give an estimate of the maximum time of the transient process
Case 1. external influences are absent π(t) = 0.
Let the law of control that guarantees for any ξ Ξ, π(t) Qπ (ε is the closeness of
real and programmatic movements [
The expression (11), given the expression (10), can be represented in the form: hence
X (t) X n (t) , t tn .
C X (t0 ) X n (t) e (tn t0) ,
C (tn t0 ) ln
X (t0 ) X n (t0 ) .
Tn 1 ln
C X (t0 ) X n (t0 ) .
(11) (12) (13) (14) Denote as Tn = tn – t0 the time of the transient process in the system.
The time Tn of the transient process can be estimated using the expression (13) Case 2. External disturbances occur π(t)≠ 0.
Let the controlled object be X (t) F[ X (t),U (t), ] (t), X(t0) =X0, X n (t) ( X X n ) QX .
described t [t0, tn].
by
the
X QX,
The control U is of the chosen form: U (t, X ) U[ X , X n ( X X n ), ] , t [t0, ∞).
The equations of the X (t) X n (t) [ X (t) X n (t)] (t) .
Suppose that Xn(t) and Ẋn(t) both lie on sets QX and QẊ with stocks δ1 and δ2, respectively (Fig. 8), when closed-loop system have the form: 1 C X0 X n (t0 ) , 2 C X0 X n (t0 ) , or given 0.
1 C X 0 X n (t0 ) 0 , 2 1 C X 0 X n (t0 ) 0 . (15) (16) (17)
X min[1 C X 0 X n (t0 ) , 2 1 C X 0 X n (t0 ) ] ,
Hence, given (20), it follows: C X 0 X n (t0 ) e (tt0 ) C 1C . Solving the obtained inequality with respect to t = tn, we find C 1C C X 0 X n (t0 ) e (tn t0) . (20) (21) (22) (23) (24)
tn t0 1 ln
C X 0 X n (t0 ) C 1C
.
Tn 1 ln C X0 X n (t0) .
C 1C
Comparing the expression (23) with the expression (14), we can conclude that in the presence of external disturbances, the time of transient process increases Tn (C 0) Tn (C 0) 1 ln
C 1C .
With great disturbances (increase in Cπ), the time of transient process increases significantly.
To reduce the time of the transient process in the system, it is necessary to increase γ by means of an appropriate choice of control. 5
Conclusions For a multidimensional controlled object, satisfying det C ≠ 0, the structure of the control algorithm does not explicitly contain the equation of motion of the object. The proposed approach to the construction of control algorithms allows obtaining algorithms without using detailed equations of the controlled process. Moreover, it is sufficient to use as a mathematical model the generalized equations that reflect the fundamental laws of motion.
To synthesize the aircraft control algorithms, complete nonlinear equations of motion can be used without linearizing them. The resulting algorithms are also nonlinear. Their structure is adequate to the structure of mathematical models of controlled processes, and the parameters of these algorithms are determined by the parameters of mathematical models of assigned motion trajectories.
In essence, the construction of aircraft motion control algorithms along the assigned trajectory has two aspects. The first one is related to the direct formation of the vector of the required control force f*[x], and the other is related to the calculation of the values of the elements of the vector of the control function Un(t) that creates the necessary force f*. The calculation relations according to which f* and Un(t) are calculated form the contents of the motion control algorithm.
The coefficients of the control system are determined by the basic parameters of the motion of the object, as well as by the parameters of the assigned program trajectory. This allows changing the parameters of the programmatic trajectory in the object movement.