=Paper=
{{Paper
|id=Vol-2018/paper-02
|storemode=property
|title=Equilibrium State Stabilization of the Hamilton Systems with Quality Estimate of Control
|pdfUrl=https://ceur-ws.org/Vol-2018/paper-02.pdf
|volume=Vol-2018
|authors=Aleksandr Andreev,Sergey Bezglasnyi,Michael Geraskin,Alena Mukhametzyanova
}}
==Equilibrium State Stabilization of the Hamilton Systems with Quality Estimate of Control==
https://ceur-ws.org/Vol-2018/paper-02.pdf
Equilibrium State Stabilization of the Hamilton
Systems with Quality Estimate of Control?
Aleksandr S. Andreev1[0000−0002−9408−0392] ,
Sergey P. Bezglasnyi2[0000−0001−8105−4921] , Michael I. Geraskin3 , and
Alena A. Mukhametzyanova4
1
Ulyanovsk State University, Ulyanovsk, Russia,
andreevas@ulsu.ru
2 3 4
Samara National Research University, Samara, Russia,
2
bezglasnsp@rambler.ru, 3 innovation@ssau.ru, 4 alain.20@mail.ru
Abstract. The problem of stabilizing the zero solution of a nonau-
tonomous Hamiltonian system with guaranteed estimation of the quality
of control is solved. This problem arises from the problem of optimal sta-
bilization by minimizing the requirements for the functional: instead of
minimizing it, it is necessary only that it does not exceed a predetermined
estimate. The solution is obtained by synthesizing the active program
control, applied to the system, and stabilizing the control based on the
feedback principle. The problem is solved analytically on the basis of the
Lyapunov direct method of stability theory using the Lyapunov function
with sign-constant derivatives. As examples, problems on the synthesis
and stabilization of program motions of a homogeneous variable-length
rod and a mathematical pendulum of variable length in a rotating plane
are solved.
Keywords: controlled mechanical system, equilibrium state, limiting
functions, stabilization, Lyapunov’s function, quality estimate, computer
modeling, math modeling
1 Introduction
The problems of controlled motions of mechanical systems are relevant and at-
tract the attention of many researchers. The problem of constructing and inves-
tigating the properties and stability conditions of such motions was considered in
the works of many scientists, for example [14, 16, 18, 20, 21]. As a rule, ensuring
the asymptotic stability of the solution of the problem reduces to investigat-
ing the zero solution of the non-autonomous system [6] and is carried out on
the basis of the direct Lyapunov method [17]. The method of limit systems [5]
and its modification [3] allow using the Lyapunov functions with constant-sign
derivatives to significantly expand the class of functions used to construct the
?
This work was supported in the framework of the national tasks of the Ministry of
Education and Science of the Russian Federation (project no. 9.5994.2017-БЧ).
desired controls in closed analytic form in the class of continuous functions. This
method has proved itself well in solving problems of constructing given motions
of mechanical systems, and with its help only at the Samara State Aerospace Uni-
versity a lot of problems were solved by a group of authors on the construction of
asymptotically stable program motions: a rigid body on a moving platform [8], a
double pendulum variable length with a movable suspension point [13], the arms
of the robot manipulator [9], a free gyrostat with variable moments of inertia,
depending on the time [7], spherical motions of the satellite a circular orbit [10]
fixed-rotor motions gyrostat with polostyu1 filled with a viscous liquid [12].
In many problems, in addition to stabilizing movements, it is important to
assess the quality of the transient process, given by some quality functional that
requires optimization [15]. A wide range of researchers have found the problems
of optimal stabilization, for example [1, 19]. In [4], a formulation was given and
sufficient conditions were obtained for the problem of stabilizing the zero solution
of non-autonomous systems with a guaranteed estimate of the quality of control
arising from the optimal problem.
This method can be used for mathematical and computer modeling of eco-
nomic systems.
2 Problem Formulation
The governed mechanical system is considered. Its motion is described by Hamil-
ton’s equations:
∂H(t, p, q)
q̇ = ,
∂p
(1)
∂H(t, p, q)
ṗ = − + u,
∂q
where q = (q1 , . . . , qn )T , n-dimensional vector of generalized coordinates in a
real linear space Rn , with norm kqk, p = (p1 , . . . , pn )T , n-dimensional vector
of generalized linear momentum, p ∈ Rn with norm kpk and H(t, q, p) is the
Hamiltonian of the system, u(t, q, p) ∈ U are forces of control action. Sign T
means transpose operation.
It is assumed that system (1) has trivial solution q = p = 0 for u ≡ 0.
Let control quality estimation of this system be equal to
Z∞
I= W (t, x[t], u[t]) dt. (2)
t0
This is done for transferring process with control u[t] and trajectory x[t] of
system (1).
Integrated function W [t, x, u], defining quality control estimation (2) of pro-
cess x[t] is continuous non-negative function defined on whole motion area in
general.
We will give three diverse types of system stabilization problem.
Problem of stabilization. The solution of the stabilization problem is to find
such control actions u = u0 (t, q, p) of all u(t, q, p) ∈ U . These effects ensure the
asymptotic stability of the unperturbed motion q = p = 0 of the system (1).
The problem of optimal stabilization is obtained from the previous problem
under the condition of a minimum of the quality criterion (2) and consists in find-
ing such control actionsu = u0 (t, q, p) of all u(t, q, p) ∈ U . This control ensures
asymptotical stability of non-perturbed motion q = p = 0 of the system (1).
An equation satisfied for any other control action u = u∗ (t, q, p) ∈ U , ensuring
asymptotical stability of solution q = p = 0:
Z∞ Z∞
I = W (t, q [t], p [t], u [t]) dt ≤ W (t, q ∗ [t], p∗ [t], u∗ [t]) dt = I ∗
0 0 0 0
t0 t0
for t0 ≥ 0, q (t0 ) = q (t0 ) = q0 , p (t0 ) = p∗ (t0 ) = p0 .
0 ∗ 0
The problems of stabilization and optimal stabilization were posed by
N. N. Krasovsky in work [15].
The problem of optimal stabilization of motion of a controlled system on
an infinite time interval reduces to finding the optimal Lyapunov function and
optimal control actions satisfying the Bellman partial differential equation. This
equation must be solved taking into account the additional inequality. The result
is a rather difficult task. In this paper, we study a variable formulation of the
problem of stabilization of motion-stabilization with a guaranteed estimate of
the quality of control. It arises from the problem of optimal stabilization when
the requirement for the functional is weakened: it is not necessary to minimize
it, it is only necessary that it does not exceed some estimate [4].
Definition. Control action u = u0 (t, q, p) is called stabilizing with guaranteed
estimate of the control quality P (t, q, p) if it ensures asymptotical stability of
non-perturbed motion q = p = 0 of system (1). The following condition satisfied
for any controlled motions q 0 [t], p0 [t], q 0 (t0 ) = q0 , p0 (t0 ) = p0 :
Z∞
I= W (t, q 0 [t], p0 [t], u0 [t])dt ≤ P (t0 , q0 , p0 ). (3)
t0
According to this definition, we place the problem of stabilization with a
guaranteed estimate of the quality of control of Hamiltonian systems: to find
control action u = u0 (t, q, p) among all u(t, q, p) ∈ U . The control action should
ensure asymptotical stability of non-perturbed motion q = p = 0 of system (1).
The inequality (3) inequality is satisfied for any controlled motion.
The above statement of the problem due to the weakening of the requirement
of minimization of the functional (2) makes it possible to simplify the problem
[15] and substantially extend the class of solvable problems in comparison with
the problem of optimal stabilization.
Lyapunov’s function V = V (t, x) and Bellman function
T
∂V ∂V
B[V, t, x, u] = + X(t, x, u) + W (t, x, u), (4)
∂t ∂x
where x = (q, p)T is still important for getting solution of this problem.
Variables q, p can be considered as deviations from trivial solution q = p = 0
because we are investigating stabilization problem of zero solution of system (1).
Equations system (1) is equations of perturbed motion. This will allow us to
apply to methods of stabilization with a guaranteed estimation of the quality
of control of Hamiltonian systems methods and results developed to study the
stability and stabilization of the zero equilibrium position of non-autonomous
systems [4].
We will consider classical systems with Hamiltonian:
1 T
H= p Ap + U (t, q), (5)
2
where A = A(t, q) is the symmetric coefficient matrix of non-negative quadratic
form with variables p. Equations of perturbed motion (1) will be written, con-
sidering the Hamiltonian (5):
(
q̇ = Ap,
(6)
ṗ = − 21 pT ∂A ∂U
∂q p − ∂q + u.
3 Derivation of Stabilizing Control
We assume that C is the positive-definite, symmetric, not disappearing, limited
matrix:
A0 E ≤ C ≤ A1 E, 0 < A0 < A1 − const,
where E is the identity matrix.
Positive-definite on variables p and q Lyapunov’s function is considered [11].
It admits of an infinitesimal upper bound.
1 T 1
V (t, q, p) = p Ap + q T Cq. (7)
2 2
The total derivative on time of Lyapunov’s function by system (6) will take
the form:
T T
∂V ∂V ∂V
V̇ = ṗ + q̇ + =
∂p ∂q ∂t
T T
∂V 1 T ∂A ∂U ∂V ∂V
= · − p p− +u + · (Ap) + . (8)
∂p 2 ∂q ∂q ∂q ∂t
We choose stabilizing control in form:
∂U
u0 = − Cq − A−1 Dp, (9)
∂q
where D = D(t) is the symmetrical, positive-definite, n × n matrix. If we will
add forces (9) to system (6), we have equations of controlled motion:
(
q̇ = Ap,
−1
(10)
ṗ = − 21 pT ∂A
∂q p − Cq − A Dp.
Wherein the total derivative on time (8) of Lyapunov’s function by system
(10) will take the form:
1 ∂A ∂U ∂U
V̇ = pT AT − pT p− + − Cq − A−1 Dp +
2 ∂q ∂q ∂q
T !
1 T ∂A 1 ∂A 1 ∂C
+ p p − q T C T Ap + pT p + qT q=
2 ∂q 2 ∂t 2 ∂t
T
T T 1 T ∂A 1 T ∂A
=p A − p p + p p Ap + pT AT (−Cq) −
2 ∂q 2 ∂q
1 ∂A 1 ∂C
− pT AT A−1 Dp + q T C T Ap + pT p + qT q
2 ∂t 2 ∂t
We discard the terms of the third order on variables p:
T 1 ∂A 1 ∂C
V̇ ≈ −p D− p + qT q (11)
2 ∂t 2 ∂t
4 Basic Results
Basic results about stabilization of the Hamilton systems with quality estimate
of zero solution q = p = 0 are the following statements.
Statement 1. Let’s suppose:
1. A and C do not depend on t. That is, the conditions ∂A ∂C
∂t = 0, ∂t = 0 are
satisfied;
2. B[V, t, p, q, u0 (t, p, q)] ≤ 0.
Then control (9) is stabilizing for motion q = p = 0 of system (6) with the
guaranteed quality estimate:
1 T 1
P (t0 , q0 , p0 ) = V (t0 , q0 , p0 ) = p A(q0 )p0 + q0T Cq0 . (12)
2 0 2
Proof. We chose control in the form (9) for function V (t, q, p) (7) according to
conditions ∂A/∂t = 0, ∂C/∂t = 0 and (11), then:
V̇ ≈ −pT Dp (13)
The function V is positive-definite on variables q, p and the derivative V̇ ,
according to (13), is negative-definite only on variable p. Then the solution q =
p = 0 is asymptotically stable by the theorem on asymptotic stability zero
solution of non-autonomous system [3]. The matrixes A, C are limited. p →
0, q → 0 are satisfied under t → ∞. Then we have estimate for Lyapunov’s
function V :
1 1
V (t, q(t), p(t)) = pT Ap + q T Cq → 0.
2 2
According to (4), we have:
B = V̇ + W (14)
We have V̇ +W ≤ 0 or W ≤ −V̇ , according to 2. We integrate this expression
into the interval [t0 , t]:
ZT
W dt ≤ V (t0 , q0 , p0 ) − V (t, q, p) (15)
t0
Further, we take the limit by t → ∞:
Z∞
W dt ≤ V (t0 , q0 , p0 ) = P (t0 , q0 , p0 ).
t0
The statement is proved. t
u
Statement 2. Let’s suppose:
1. ∂A ∂C
∂t = 0, ∂t < 0,
2. B(V, t, p, q, u0 (t, p, q)) ≤ 0.
Then the control (9) is stabilizing for motion q = p = 0 of system (6) with the
guaranteed quality estimate (12).
Remark 1. The proof of statement 2 is like the proof of statement 1. The dif-
ference is that derivative of the function (7), according to (11) and conditions
∂C
∂t < 0 is negative-definite on variables q, p. Then the solution q = p = 0 is
asymptotically stable by the Lyapunov’s theorem on asymptotic stability.
Statement 3. Let’s suppose the conditions are satisfied:
1. ∂A ∂C
∂t 6= 0, ∂t = 0,
1 ∂A
2. D > 2 ∂t ,
3. B(V, t, p, q, u0 (t, p, q)) ≤ 0.
Then the control (9) is stabilizing for motion q = p = 0 of system (6) with the
guaranteed quality estimate (12).
Statement 4. Let’s suppose the conditions are satisfied:
1. ∂A ∂C
∂t 6= 0, ∂t < 0,
2. D > 12 ∂A
∂t ,
3. B(V, t, p, q, u0 (t, p, q)) ≤ 0.
Then the control (9) is stabilizing for motion q = p = 0 of system (6) with the
guaranteed quality estimate (12).
Remark 2. The proofs of statements 3, 4 are like the proofs of statements 1, 2
respectively.
5 Stabilization of Homogeneous Rod’s Motion
We consider a homogeneous heavy rod. The rod length is variable. A rod mass is
m = 1. The rod moves without friction in the plane Oxy . The plane Oxy rotates
with a constant angular velocity ω around a fixed vertical axis Oy (Fig. 1).
Fig. 1. The rod in a rotate plane
Let’s suppose the ε, η are center mass coordinates of rod. The angle θ is
the deviation angle of the rod from the horizontal. Suppose that the rod length
changes according to law k = k(t) = a + b cos t, a = const > b = const > 0. The
system has three degrees of freedom.
The Hamiltonian of system takes form:
1 2
H(ε, η, θ) = (ε̇ + η̇ 2 + k 2 (t)θ̇2 + ω 2 k 2 (t) cos2 θ + ω 2 ε2 ) − gη
2
We chose program motion. Suppose that center mass of system moves on the
circle with radius R = 1. The center of a circle lies in the point O (2, 2). The rod
rotates in the plane Oxy around the center mass with constant angular velocity
ω0 = const:
∗
ε = cos t + 2,
η ∗ = sin t + 2,
∗
θ = ω0 t.
We construct equations of controlled rod motion. We obtained the equations
of perturbed motion by introducing deviations:
q̇1 = p1
q̇2 = p2 ,
q̇ = p3 ,
3 k2
ṗ1 = ω 2 q1 ,
ṗ2 = 0,
ṗ3 = −ω 2 k 2 sin q3 cos(q3 + 2ω0 t).
Suppose that quality estimate of transition process is given by the func-
tional (2) with integrand
W (t, q, p) = d1 p21 + d2 p22 .
It satisfies condition 3 of statement 3. We set the problem of stabilization zero
solution q = p = 0 of disturbed motion set of equations with guaranteed quality
estimate of the control for the proposed motion.
We chose the Lyapunov’s function:
1 1 2
V = p21 + p22 + 2 2 2
p + A1 q 1 + A2 q 2 + A3 q 3 .
2 k 2 (t) 3
The control is chosen in form (9):
2
cu1 = −ω q1 − c1 q1 − d1 p1 ,
u2 = −c2 q2 − d2 p2 ,
u3 = ω 2 k 2 sin q3 cos(q3 + 2ω2 t) − c3 q3 − k 2 d3 p3 .
Equations of motion of the stabilized system are obtained:
lq̇1 = p1 ,
q̇2 = p2 ,
q̇ = p3 ,
3 k2
ṗ 1 = −c 1 q1 − d1 p1 ,
ṗ2 = −c2 q2 − d2 p2 ,
ṗ3 = −c3 q3 − k 2 d3 p3 .
We have asymptotically stable solution q = p = 0 with guaranteed quality
estimate by statement 3:
1 1
P (t0 , q0 , p0 ) = V (t0 , q0 , p0 ) = p210 + p220 + p2
+ A 1 q 2
+ A 2 q 2
+ A 3 q 2
30 .
2 k 2 (t0 ) 30 10 20
6 Conclusion
In this paper, the problem of determining the control stabilizing the motion of a
mechanical system described by Hamilton’s equations is formulated and solved
with the additional condition of finding an assured estimate of the quality of
control. The solution of the problem is reduced to the investigation of the zero
solution of the nonautonomous system and was carried out on the basis of the
direct Lyapunov method with the use of the method of limit systems, which
made it possible to use the Lyapunov functions with sign-constant derivatives to
construct the desired control in a closed analytic form in the class of continuous
functions. Four statements that solve the problem are formulated and proved.
Based on the results obtained, two illustrative examples are solved. The results
of the work develop and generalize the corresponding results of [2, 4, 6, 11].
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