The problem of stabilizing the zero solution of a nonautonomous Hamiltonian system with guaranteed estimation of the quality of control is solved. This problem arises from the problem of optimal stabilization 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.
The problems of controlled motions of mechanical systems are relevant and attract the attention of many researchers. The problem of constructing and investigating 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 investigating 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 University 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.
2
The governed mechanical system is considered. Its motion is described by Hamilton’s equations: where q = (q1; : : : ; qn)T , n-dimensional vector of generalized coordinates in a real linear space Rn, with norm k k
q , p = (p1; : : : ; pn)T , n-dimensional vector of generalized linear momentum, p 2 Rn with norm kpk and H(t; q; p) is the Hamiltonian of the system, u(t; q; p) 2 U are forces of control action. Sign T means transpose operation.
It is assumed that system (
q_ = p_ = ; + u;
such control actions u = u0(t; q; p) of all u(t; q; p) 2 U . These effects ensure the
asymptotic stability of the unperturbed motion q = p = 0 of the system (
The problem of optimal stabilization is obtained from the previous problem
under the condition of a minimum of the quality criterion (
An equation satisfied for any other control action u = u (t; q; p) 2 U , ensuring asymptotical stability of solution q = p = 0:
I0 =
W (t; q0[t]; p0[t]; u0[t]) dt
W (t; q [t]; p [t]; u [t]) dt = I for t0 0, q0(t0) = q (t0) = q0, p0(t0) = p (t0) = p0.
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 (
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) 2 U . The control action should
ensure asymptotical stability of non-perturbed motion q = p = 0 of system (
Lyapunov’s function V = V (t; x) and Bellman function
B[V; t; x; u] = +
T
X(t; x; u) + W (t; x; u); 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 (
H = 1 pT Ap + U (t; q);
2
where A = A(t; q) is the symmetric coefficient matrix of non-negative quadratic
form with variables p. Equations of perturbed motion (
V_ =
T = p_ +
T q_ +
u0 =
Cq
T (q_ = Ap; p_ =
Derivation of Stabilizing Control
(
A0E
C
A1E; 0 < A0 < A1 const; where E is the identity matrix.
V (t; q; p) = 1 pT Ap + 1 qT Cq: 2 2
V_ = pT AT + = pT AT p
T Cq A 1Dp +
T p +
_ V pT
D
Basic Results
where D = D(t) is the symmetrical, positive-definite, n n matrix. If we will
add forces (
Cq
A 1Dp: Basic results about stabilization of the Hamilton systems with quality estimate of zero solution q = p = 0 are the following statements.
1. A and C do not depend on t. That is, the conditions @@At = 0, @@Ct = 0 are
satisfied;
2. B[V; t; p; q; u0(t; p; q)] 0:
Then control (
P (t0; q0; p0) = V (t0; q0; p0) = 12 p0T A(q0)p0 + 12 q0T Cq0: _ V
pT Dp
The function V is positive-definite on variables q, p and the derivative V_ ,
according to (13), is negative-definite only on variable p. Then the solution q =
(
1. @@At = 0, @@Ct < 0, 2. B(V; t; p; q; u0(t; p; q))
0.
Then the control (
Remark 1. The proof of statement 2 is like the proof of statement 1. The
difference is that derivative of the function (
0:
Then the control (
Statement 4. Let’s suppose the conditions are satisfied:
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 ! around a fixed vertical axis Oy (Fig. 1).
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 (
= sin t + 2; > :
= !0t:
We construct equations of controlled rod motion. We obtained the equations of perturbed motion by introducing deviations: 8>q_1 = p1 > >>>q_2 = p2; > > ><q_3 = kp32 ; >>p_1 = !2q1; >>>>p_2 = 0; > >:p_3 =
!2k2 sin q3 cos(q3 + 2!0t):
W (t; q; p) = d1p12 + d2p22: 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.
V = >>p_1 = >>>>p_2 = > >:p_3 = c1q1 c2q2 c3q3 d1p1; d2p2; k2d3p3:
We have asymptotically stable solution q = p = 0 with guaranteed quality estimate by statement 3: P (t0; q0; p0) = V (t0; q0; p0) =
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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regulator with bounded torques for robot manipulators. International Journal of
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