1. Introduction
At present, the theory of control of dynamical systems is a fairly well-developed scientific field with important engineering applications [
1,
2,
3]. Mathematical problems arising in this scientific direction are associated with various aspects of the analysis of dynamical systems and optimization [
4,
5]. Taking into account the influence of inevitably present random perturbations complicates the formulation of control problems and leads the development of adequate methods of the stochastic analysis and synthesis. A method of stochastic Lyapunov functions for the stabilization problems was developed in [
6]. The theory of random differential equations in control problems is presented in [
7]. Seminal works [
8,
9] were devoted to the optimal stochastic control. A systematic review of theory of stochastic processes in the context of the feedback control was given in [
10]. Methods of the control engineering design on the base of stochastic analysis was considered in [
11]. Here, recent studies can be also mentioned [
12,
13,
14,
15,
16].
Many mathematical models of control systems contain strong nonlinearities. It is well known that in nonlinear systems, even small random perturbations can dramatically change the deterministic dynamics and give rise to unexpected stochastic phenomena such as noise-induced transitions [
17,
18,
19], stochastic resonance [
20,
21], noise-induced chaos [
22,
23], and stochastic excitability [
24,
25].
Even the analysis of nonlinear stochastic systems is a complex mathematical problem associated with solving the Fokker-Planck-Kolmogorov equation [
26]. Here, various kinds of approximations are a useful tool [
27]. The constructive technique of stochastic sensitivity functions, which makes it possible to approximate distribution densities near attractors at low noise [
28,
29], has been successfully used in the analysis of various kinds of stochastic phenomena [
30,
31,
32,
33].
In solving problems of synthesis for stochastic nonlinear systems with both continuous and discrete time, the use of the stochastic sensitivity method is also a fruitful direction. It was shown that the appearance of unacceptable large-amplitude oscillations induced by noise can be explained by the high sensitivity of initial deterministic attractors to random disturbances. In [
34], it was shown how one can suppress such oscillations by the synthesis of the appropriate regulator that decreases the stochastic sensitivity. This approach has been developed through consistent studies [
35,
36,
37,
38]. Note that in the case of complete information, the construction of such a regulator was reduced to the solution of linear matrix equations.
However, in practice, the complete information on the current system state usually is not available: only some coordinates can be measured and, moreover, this data contains random errors [
39]. Therefore, to develop a theory of the stochastic sensitivity synthesis for nonlinear stochastic systems with incomplete information is of particular interest [
40,
41,
42].
The present paper aims to study the problem of the the stochastic sensitivity synthesis in control dynamical systems with random disturbances. We will focus on a class of static feedback regulators that use noisy data on some system state coordinates.
In
Section 2, we present results of the asymptotic analysis of deviations of random solutions from the equilibrium in the general nonlinear control system. We derive a quadratic matrix equation that connects the assigned stochastic sensitivity matrix of the equilibrium and the feedback matrix of the regulator. Here, an important aspect of attainability of the assigned stochastic sensitivity is reduced to the mathematical problem of solvability of the corresponding matrix equation. Mathematical results are summarized in Theorem 1, where explicit formulas for feedback regulators synthesizing the required stochastic sensitivity are presented. These general theoretical results are applied in
Section 3 to a class of nonlinear oscillators. The efficiency of the suggested approach and the elaborated mathematical technique is illustrated by the stabilization of a stochastically forced van der Pol oscillator.
2. Controlling Stochastic Sensitivity by a Regulator with Noisy Measurements
Consider a general stochastic system with control in the form:
Here, x is an n-dimensional state, u is an l-dimensional control input, is an n-vector-function, is an -matrix-function, is an q-dimensional white Gaussian noise with parameters , is a symmetric non-negative definite -matrix, and is a scalar parameter of the noise intensity. The -matrix-function characterizes a dependence of random forcing on the system state x.
Let the unforced and uncontrolled system (
1) (with
,
) have an equilibrium
. The stability of
is not supposed.
Let available information on the current state
of the system (
1) be incomplete. Here, we consider a case that only the measurement
m-vector
connected with the state
is known:
where
is
m-vector function,
is
-matrix-function,
is white Gaussian
p-vector noise, uncorrelated with
, satisfying
Here,
is a symmetric non-negative definite
-matrix.
At present, regulators of very different structures are used to control systems with incomplete information. In this paper, in the problem of stabilization of the equilibrium
, we focus on the case of the static regulator in the following feedback form:
Here, K is a constant -matrix.
Let us denote by
a set of matrices
K that provide an exponential stability of the equilibrium
in the corresponding closed-loop deterministic system
in some neighbourhood of the equilibrium
.
Consider deviations
of the system (
4) states
from the equilibrium
. For these deviations, the following first approximation system can be written:
As follows from (
5), the set
is determined as
Here,
are the eigenvalues of the matrix
. We suppose that the set
is not empty. Details of the algebraic analysis of this set can be found in [
43].
Let us consider now the corresponding closed-loop stochastic system. As follows from (
1)–(
3), this closed-loop stochastic system is written in the form:
Let
be a solution of the stochastic system (
6). For the asymptotics
of the deviation of the solution
of the controlled closed-loop system (
6) from the equilibrium
, one can write the following system
In the following mean-square analysis of the sensitivity of the equilibrium
to the random disturbances, a key role is played by the covariance matrix
. Due to uncorrelatedness of noises
and
, for
one can write the matrix equation:
For any
, an arbitrary solution
of this equation has a limit:
where
W is a unique stable stationary solution of the Equation (
8). The matrix
W satisfies the following quadratic algebraic equation:
The matrix
W is called a
stochastic sensitivity matrix of the equilibrium
. For weak noise, this matrix gives us a first approximation of the covariance matrix of the stationary distributed solutions
of the system (
6):
For any
, the regulator (
3) providing an exponential stability of the equilibrium
for the deterministic system (
4), forms the corresponding stochastic sensitivity matrix
of this equilibrium in the closed-loop stochastic system (
6).
Varying
, we can change the values of the stochastic sensitivity matrix
and thereby control the variance of the states
of the stochastic system (
1) near the equilibrium
. So, the control problem for the dispersion of random states around the equilibrium
can be reduced to the synthesis of the assigned stochastic sensitivity matrix
W by the appropriate regulator (
3).
Denote by
a set of symmetric and positive definite
-matrices. Note that not any matrix
can be synthesized. Indeed, multiplying the Equation (
9) by the projective matrix
P from the left and righ, we get for the matrix
W the following necessary condition of the attainability:
Here,
is the projective matrix, and the sign “+” means pseudoinversion. Emphasize that the condition (
10) does not depend on the measurement parameters.
Let us now formulate the corresponding problem of the stochastic synthesis in terms of solving the algebraic Equation (
9).
Problem of the stochastic sensitivity synthesis:
For the assigned matrix , it is necessary to find a matrix guaranteeing the equality where is a solution of Equation (9). In many cases, not any matrix is attainable.
Definition 1. The element is said to be attainable if there exists that the equality holds.
Definition 2. The set of all attainable elements,is called the attainability set for system (6). The notion of attainability in context of stochastic sensitivity synthesis was introduced and discussed in [
34].
Thus, as it follows from (
9), to synthesize the assigned stochastic sensitivity matrix
W, it is necessary to find the feedback gain
K satisfying the quadratic matrix equation
It what follows we suppose that noise in measurements is non-singular in the sense that .
Let us consider the substitution
For this new matrix variable
L, the Equation (
11) can be written as
It is convenient to rewrite the Equation (
12) in the form:
Let us denote
then the Equation (
13) can be rewritten as
The non-negative definiteness of the matrix
Q
is a necessary condition of the solvability of the Equation (
14). From the decomposition
, where
A is an
-matrix, a general solution of the quadratic Equation (
14) can be written as:
Here,
U is an arbitrary orthogonal
-matrix. Finally, for the feedback matrix
K, we get the matrix equation
So, the problem of the solution of the quadratic matrix Equation (
11) is reduced to the solution of the linear matrix Equation (
17).
If the matrix
B is quadratic (
) and has a full rank (
), then the Equation (
17) has a solution
If
, then it is necessary to take into account one more condition for the solvability of the Equation (
17):
Under these conditions, the feedback matrix has the form
Results of this Section are summarized in the following Theorem.
Theorem 1. Let noise in the system (2) be non-singular. - (a)
If , then the attainability set consists of the matrices satisfying inequality (15). - (b)
For any attainable matrix W, the Equation (11) has a solution (18). If then the attainability set consists of the matrices satisfying relations (15) and (19). Under these conditions, the Equation (11) has a solution (20).
This Theorem generalizes the results presented in [34,40]. Examples
Example 1. For , the Equation (11) is written as In this one-dimensional case, the attainability condition (15) is From this quadratic inequality, taking into account the non-negativeness of
W, we get for stochastic sensitivity
W the attainability condition
where
To synthesize the stochastic sensitivity
, one has to use the regulator with the feedback coefficient
K found from the formula (
18) with
:
In (
21),
is the minimum value of the stochastic sensitivity. This value is achieved by the regulator with
Let us fix
. For this case, the function
that determines the minimum of stochastic sensitivity is plotted in
Figure 1a for different values of
F. Plots of the feedback coefficients
of the corresponding optimal regulators providing such a minimum stochastic sensitivity are shown in
Figure 1b.
As can be seen, a decrease of noise intensity in measurements implies the decrease of the stochastic sensitivity.
Example 2. Consider the following cubic systemwith the measurementhere, and are uncorrelated white Gaussian noises. The deterministic uncontrolled system () possesses three equilibria: unstable , and stable Our control aims to stabilize the equilibrium using the regulator that minimizes the stochastic sensitivity.
In this example,
and
. As follows from (
21) and (
22), the minimum value of the stochastic sensitivity is
The feedback coefficient of the optimal regulator is written as
Let us fix
. In the system without control, a random trajectory starting from the unstable equilibrium
transits to the vicinity of the stable equilibrium (see the red trajectory in
Figure 2). In numerical simulations of random trajectories, we used a standard Euler-Maruyama scheme with the time step 0.001.
For
, according to the Formula (
23), the minimum value of the stochastic sensitivity is
The corresponding optimal regulator providing this stochastic sensitivity, has the feedback coefficient
(see Formula (
24)). Results of the application of this regulator are shown in
Figure 2 by blue. As one can see, a sample random trajectory of the controlled system exhibits small-amplitude oscillations near
3. Synthesis of Stochastic Sensitivity in Nonlinear Oscillators
In this Section, we consider a nonlinear oscillator with random disturbances and control input:
Here, f and are scalar functions, is white Gaussian noise with parameters , and is the noise intensity.
The Equation (
25) can be rewritten as a 2D-system:
Assume that the only coordinate
is measured with random errors:
where
is white Gaussian noise with parameters
. We suppose that
and
are uncorrelated.
Let
be an equilibrium of the uncontrolled deterministic system (
26) (
). Stability of this equilibrium is not supposed. To stabilize the equilibrium
, we will use the following feedback regulator:
where
K is a scalar feedback coefficient.
Eigenvalues of the matrix
satisfy the equation
We suppose that the regulator (
28) stabilizes the equilibrium
in the deterministic system (
26) and (
27) for
, so it holds that
In what follows, we consider the case . So, the set of feedback coefficients of stabilizing regulators is .
Now, let us turn to the problem of the stochastic sensitivity synthesis. Let be an assigned positive definite stochastic sensitivity matrix.
For the system (
26), we have the following form of the projective matrix:
So, as it follows from (
10), the element
, and therefore the attainable stochastic sensitivity matrix
W for system (
26) has a be diagonal:
.
The next step in attainability analysis is related to the matrix
Q. In considered case,
The condition (
15) of non-negative definiteness of the matrix
Q can be written as
Taking into account
, we have the restriction for the element
:
Further, for
Q, we have to find the decomposition
where
A is
-matrix. So, for elements
of the matrix
, the following system can be written:
This system of three equations for two unknown
and
is solvable if and only if
and has a solution
So, the restrictions
give us necessary and sufficient conditions of attainability for elements
of the stochastic sensitivity matrix
W. Note that these conditions determine some curve in the plain
.
In considered case, from the general Formulas (
19) and (
20), it follows that
Thus, the regulator (
28) with such a coefficient
synthesizes in the system (
26) with measurements (
27) a diagonal stochastic sensitivity matrix
where elements
and
are connected by the attainability conditions (
29).
Example 3. Stabilization of the stochastic van der Pol oscillator.
Let us consider how general results presented above can be applied to the solution of the stabilization problem of the equilibrium in the randomly forced van der Pol oscillatorwith measurementsand feedback control . In this model, we restrict ourselves by additive random disturbances where and are uncorrelated white Gaussian noises with parameters . The equilibrium
in the deterministic system without control (
) is exponentially stable for
. In presence of noise, the stochastic sensitivity matrix
of this equilibrium has equal elements
For small negative
, the stochastic sensitivity is high. This can cause large-amplitude stochastic oscillations near the equilibrium. Such oscillations are shown by blue in
Figure 3b for
and
. For these parameters,
.
Consider abilities of the regulator
in conditions of the measurement (
31) to change parameters
and
of the stochastic sensitivity. For the system (
30) and (
31) with
, the attainability conditions (
29) give us
As can be seen, in the considered example, the attainability set is a curve in the plain
determined by relations (
32). For fixed
, these curves are shown in
Figure 3a for different
. As can be seen, our regulator allows one essentially decrease the value
.
For the fixed values
and
, the pair
and
is attainable. The regulator that synthesizes such sensitivity has the feedback coefficient
. Results of direct numerical simulation of the system (
30) and (
31) with such a regulator are shown by red in
Figure 3b for
. As one can see, dispersion of random solutions is essentially reduced.