Feedback control of nonlinear stochastic dynamic systems for accurately tracking a specified stationary probability density function

Abstract

Limited by the difficulty of obtaining the exact stationary solution of Fokker–Planck–Kolmogorov (FPK) equation, the general control strategy for tracking a specified stationary probability density function have not been obtained till now. However, the feedback control for tracking a pre-specified stationary probability density function plays an important role in engineering or industrial systems. In this paper, a feedback control strategy of nonlinear stochastic dynamic system for accurately tracking a specified stationary probability density function without the requirement of exact stationary solution of Fokker–Planck–Kolmogorov equation is proposed. According to the probability conservation form of the Fokker–Planck–Kolmogorov equation, the stationary Fokker–Planck–Kolmogorov equation is split into a probability circulation flow (PCF) equation and a probability potential flow (PPF) equation. The control force is divided into probability circulation flow part and probability potential flow part accordingly. The probability circulation flow part of the control force is determined to satisfy the probability circulation flow term of the controlled system constructed from the target stationary probability density function. The probability potential flow part of the control force is obtained by solving the probability potential flow equation. A two-dimensional nonlinear stochastic system is carried out as an example. The control force is designed to track different types of target stationary probability density functions. Numerical results show that the proposed control strategy can accurately track the stationary probability density functions without the requirement of the exact solution of Fokker–Planck–Kolmogorov equation. The control efficiency can be regulated by the control parameter C.

Keywords

Random excitation nonlinear system Fokker–Planck–Kolmogorov equation stationary probability density function feedback control

1. Introduction

Stochastic dynamics and motion/vibration control have been a significant subject in the field of engineering since engineering systems are usually subjected to random loadings, such as wind, ocean wave, and earthquake (Xu et al., 2011; Li and Chen, 2008; Wang et al., 2020). Among them, the probability density function (PDF) tracking control, which is designed to reshaping the PDF of key outputs of a stochastic dynamic system for tracking a pre-designed one attracts the attention from many researchers (Wang et al., 2008; Yue and Wang, 2003; Zhu and Zhu, 2019) due to its wide practical applications, such as flame shape control, papermaking process control, and chemical industry polymerization process control. However, limited by the difficulty of obtaining the exact stationary solution of Fokker–Planck–Kolmogorov (FPK) equation, the general analytical expression of the tracking control strategy for the multi-dimensional system, especially for the nonlinear system, have not been obtained till now.

For a Gaussian process, the statistical characteristics can be sufficiently described by its mean and variance value. Thus, for Gaussian outputs system, the target control is easy to be designed by tracking the mean and variance values (Lu and Skelton, 1998; Skelton et al., 1998; Sun, 2006; Wojtkiewicz and Bergman, 2001). However, due to the nonlinearity or non-Gaussian excitations, the output states of many practical systems usually appear in the form of non-Gaussian. For such systems, the tracking controls for targeting the mean and variance value are not suitable and the complete SPDF should be set as the target of tracking control design. Recently, Ren et al. made a very comprehensive summary on the feedback control of SPDF tracking, and divided the shaping control strategies into four categories (Ren et al., 2019). Zhu and Zhu (Zhu and Zhu, 2011, 2019) obtained the corresponding shaping control law according to the five classes of exact stationary solutions of the dissipative MDOF Hamiltonian system. Based on the limit average method (Huan et al., 2015), an effective procedure was proposed to reshape the PDF of stochastically excited quasi-integrable Hamiltonian systems against Markov jump disturbances (Xia et al., 2019). The above-mentioned shaping control methods are all designed under the condition that exact stationary solutions of FPK equations can be obtained. However, the exact stationary solutions of FPK are usually difficulty to be obtained, especially for nonlinear and multi-degree-of-freedom (MDOF) systems (Zhu and Yang, 1996; Zhu et al., 1990; Zhu, 2006). Some shaping control methods that do not rely on exact stationary solutions are proposed. Yin and Guo adopted the optimal recursive algorithm to realize the shaping control of the joint PDF by minimizing the distance between the output distribution and the expected distribution (Yin and Guo, 2012). Wang converted the PDF target control problem into neural network parameter control (Wang, 2012). Yin et al. realized the shaping control of the joint PDF through a data-driven method (Yin et al., 2015). In general, the advantage of the shaping control method based on the exact stationary solution is that the explicit tracking control law can be obtained, but the problem is that it is difficult to obtain the exact stationary solution of FPK equation. For other target control methods, the control force generally needs to be estimated online, which will result in a large online calculation load.

In this paper, a feedback control strategy of the nonlinear stochastic dynamic system for accurately tracking a specified SPDF is proposed. This control law does not need the exact stationary solution form of FPK equation. It has an analytical expression which does not need to be calculated online, and it can track various forms of target SPDFs. This paper is organized as follows: In section 2, the controlled system model is given and the control problem is described; In section 3, the probability conservation equation of FPK equation is briefly reviewed, and the mathematical expressions of circulation and potential stationary conditions are given. The design process of the control law is given in section 4. In section 5, the effectiveness of the control law is verified by a stochastic time-delay ecological predator-prey system for tracking two different SPDFs. The conclusion of the paper is given in the last section.

2. Problem formulation

Consider a MDOF nonlinear stochastic system expressed by the following general stochastically excited, dissipated, and controlled Hamilton equations

{\dot{Q}}_{i} = \frac{\partial H'}{\partial P_{i}} + M_{Q_{i}}' + u_{Q_{i}} + σ_{2 i - 1, k} W_{k} (t)

(1)

{\dot{P}}_{i} = - \frac{\partial H^{'}}{\partial Q_{i}} + M_{P_{i}}' + u_{P_{i}} + σ_{2 i, k} W_{k} (t)

(2)

i = 1,2, \dots, n; k = 1,2, \dots, m

where

Q_{i}

and

P_{i}

are generalized displacements and generalized momenta of the ith degree of freedom, respectively;

H' = H' (Q, P)

is a Hamiltonian,

Q = [Q_{1}, Q_{2}, \dots, Q_{n}], P = [P_{1}, P_{2}, \dots, P_{n}]

M_{Q_{i}}' = M_{Q_{i}}' (Q, P), M_{P_{i}}' = M_{P_{i}}' (Q, P)

are the dissipated terms.

u_{Q_{i}}, u_{P_{i}}

are the control forces to be designed.

W_{k} (t)

are the Gaussian white noise with unit intensity.

σ_{j, k}

are the coefficients of the corresponding Gaussian white noise terms,

σ = [\begin{matrix} σ_{1,1} & σ_{1,2} \dots & σ_{1, m} \\ σ_{2,1} & σ_{2,2} \dots & σ_{2, m} \\ \begin{matrix} ⋮ \\ σ_{2 n, 1} \end{matrix} & \begin{matrix} ⋮ \\ σ_{2 n, 2} \dots \end{matrix} & \begin{matrix} ⋮ \\ σ_{2 n, m} \end{matrix} \end{matrix}]

System (1) is equivalent to the following Stratonovich stochastic differential equation

d Q_{i} = (\frac{\partial H'}{\partial P_{i}} + M_{Q_{i}}' + u_{Q_{i}}) d t + σ_{2 i - 1,} k^{o} d B_{k} (t)

(3)

d P_{i} = (- \frac{\partial H'}{\partial Q_{i}} + M_{P_{i}}' + u_{P_{i}}) d t + σ_{2 i,} k^{o} d B_{k} (t)

(4)

i = 1,2, \dots, n; k = 1,2, \dots, m

where ‘o’ denotes the Stratonovich differential.

According to the definition of Itô stochastic differential, the Stratonovich stochastic differential equations (3) and (4) can be converted to the following Itô stochastic differential equation by adding Wong-Zakai correction terms (Arnold, 1976; Wong and Zakai, 1965)

d Q_{i} = (\frac{\partial H^{'}}{\partial P_{i}} + M_{Q_{i}}' + u_{Q_{i}} + V_{i}) d t + σ_{2 i - 1, k} d B_{k} (t)

(5)

d P_{i} = (- \frac{\partial H^{'}}{\partial Q_{i}} + M_{P_{i}}' + u_{P_{i}} + Z_{i}) d t + σ_{2 i, k} d B_{k} (t)

(6)

X_{j} = {\begin{matrix} Q_{(j + 1) / 2}, j i s o d d \\ P_{j / 2}, j i s e v e n \end{matrix}

(7)

V_{i} = \frac{1}{2} \sum_{k = 1}^{m} \sum_{j = 1}^{2 n} σ_{j, k} \frac{\partial σ_{2 i - 1, k}}{\partial X_{j}}; Z_{i} = \frac{1}{2} \sum_{k = 1}^{m} \sum_{j = 1}^{2 n} σ_{j, k} \frac{\partial σ_{2 i, k}}{\partial X_{j}}

(8)

where

B_{k} (t)

are the standard Wiener processes,

V_{i} = V_{i} (Q, P)

Z_{i} = Z_{i} (Q, P)

are Wong–Zakai correction terms. Usually, Wong–Zakai correction terms can be expressed in the separated form, that is,

V_{i} = V_{i}' (P_{i}) + V_{i} ″ (Q_{i}, P_{i}), Z = Z_{i}' (Q_{i}) + Z_{i} ″ (Q_{i}, P_{i})

. Combined

V_{i}' (P_{i})

and

Z_{i}' (Q_{i})

, respectively, with

\frac{\partial H^{'}}{\partial P_{i}}

and

- \frac{\partial H^{'}}{\partial Q_{i}}

equations (3) and (4) becomes

d Q_{i} = (\frac{\partial H}{\partial P_{i}} + M_{Q_{i}} + u_{Q_{i}}) d t + σ_{2 i - 1, k} d B_{k} (t)

(9)

d P_{i} = (- \frac{\partial H}{\partial Q_{i}} + M_{P_{i}} + u_{P_{i}}) d t + σ_{2 i, k} d B_{k} (t)

(10)

i = 1,2, \dots, n

where

\frac{\partial H}{\partial P_{i}} = \frac{\partial H^{'}}{\partial P_{i}} + V_{i}'

- \frac{\partial H}{\partial Q_{i}} = - \frac{\partial H^{'}}{\partial Q_{i}} + Z_{i}'

, and

M_{Q_{i}} = M_{Q_{i}}' + V_{i} ″, M_{P_{i}} = M_{P_{i}}' + Z_{i} ″

The objective of the control problem can be described as making the SPDF of the states Q, P of system (1) or (4) which become the following pre-specified target SPDF through the control design

ρ^{t a r} = ρ^{t a r} (Q, P)

(11)

3. Fokker–Planck–Kolmogorov equation

The stationary FPK equation associated with system (4) can be formulated as

\sum_{i = 1}^{n} (- \frac{\partial [(\frac{\partial H}{\partial P_{i}} + M_{Q_{i}} + u_{Q_{i}}) ρ]}{\partial Q_{i}} - \frac{\partial [(- \frac{\partial H}{\partial Q_{i}} + M_{P_{i}} + u_{P_{i}}) ρ]}{\partial P_{i}} + \frac{1}{2} \sum_{j = 1}^{2 n} (\frac{\partial^{2}}{\partial Q_{i} \partial X_{j}} (b_{2 i - 1, j} ρ) + \frac{1}{2} \frac{\partial^{2}}{\partial P_{i} \partial X_{j}} (b_{2 i, j} ρ))) = 0

(12)

where $ρ = ρ (Q, P)$ is the SPDF of system (4) and $b_{j, l} = {(σ σ^{T})}_{j, l}$ . Generally, it is very difficult to solve FPK equation (12) for obtaining the analytical expression of $ρ$ , especially for nonlinear or MDOF systems. Zhu and his co-workers obtained the exact stationary solutions of FPK equation for five classes of Hamiltonian systems when certain conditions are satisfied (Zhu and Zhu, 2011).

Rewriting FPK equation (12) in the following probability conservation form gives

\sum_{j = 1}^{2 n} \frac{\partial G_{j}}{\partial X_{j}} = 0

(13)

G_{j} = a_{j} ρ - \frac{1}{2} \sum_{l = 1}^{2 n} \frac{\partial}{\partial X_{l}} (b_{j, l} ρ)

(14)

a_{j} = {(- 1)}^{j + 1} \frac{\partial H}{\partial X_{j}^{+}} + M_{X_{j}} + u_{X_{j}}

(15)

X_{j}^{+} = {\begin{matrix} P_{(j + 1) / 2}, j is odd \\ Q_{j / 2}, j is even \end{matrix}

j = 1,2, \dots, 2 n

where

G_{j}

is the probability flow of the jth dimension.

When solving equation (13), it is generally considered that $G_{j} = 0$ on the boundary. For the vibration system, the boundary probability flow is zero means that the PCF is balanced and the PPF is zero at every point in the state space. For non-vibration systems, the zero-boundary probability flow means that the PCF and the PPF are zero at every point in the state space. Then, by splitting $a_{j}$ into probability circulation part $a_{j}^{(c)}$ and probability potential part $a_{j}^{(p)}$ , that is, $a_{j} = a_{j}^{(c)} + a_{j}^{(p)}$ , we have

\sum_{j = 1}^{2 n} \frac{\partial}{\partial X_{j}} (a_{j}^{(c)} ρ) = 0

(16)

a_{j}^{(p)} ρ - \frac{1}{2} \sum_{l = 1}^{2 n} \frac{\partial}{\partial X_{l}} (b_{j, l} ρ) = 0

(17)

j = 1,2, \dots, 2 n

The targeting control law obtained by using the exact stationary solution is essentially based on the exact solution of equation (12) without control $u_{X_{j}}$ (Zhu and Zhu, 2011, 2019). However, for many practical systems, exact stationary solutions are difficult to obtain, which limits the application of targeting control. In this paper, a novel procedure of design the targeting control law without the requirement of exact SPDF is proposed.

4. Control law

The targeting control law is designed by the following steps.

Firstly, $a_{j}$ is divided into probability circulation part and the probability potential part as seen in equations (16) and (17). Write each part as the combination of the uncontrolled system and control force as the following

a_{j} = a_{j}^{(c)} + a_{j}^{(p)} = (a_{j}^{(c)}' + u_{X_{j}}^{(c)}) + (a_{j}^{(p)}' + u_{X_{j}}^{(p)})

(18)

where

a_{j}^{(c)}'

and

a_{j}^{(p)}'

represent the PCF term and the PPF term of the uncontrolled system, respectively.

u_{X_{j}}^{(c)}

and

u_{X_{j}}^{(p)}

are the PCF term and the PPF term of the control force, respectively.

The probability circulation term $a_{j}^{(c)} = a_{j}^{(c)}' + u_{X_{j}}^{(c)}$ in equation (18) satisfies equation (16). According to the Hamiltonian theory, the probability circulation term of the controlled system usually has the following form

a_{j}^{(c)} = a_{j}^{(c)}' + u_{X_{j}}^{(c)} = {(- 1)}^{j + 1} C \frac{\partial H^{t a r}}{\partial X_{j}^{+}}

(19)

where

H^{t a r}

is the Hamiltonian function of the controlled system, which is unknown to be constructed,

C

can be any positive constant,.

Substituting term equation (19) and the target SPDF $ρ^{t a r}$ into equation (16), we obtain

\sum_{j = 1}^{2 n} ({(- 1)}^{j + 1} C \frac{\partial^{2} H^{t a r}}{\partial X_{j} \partial X_{j}^{+}} ρ^{t a r} + {(- 1)}^{j + 1} C \frac{\partial ρ^{t a r}}{\partial X_{j}} \frac{\partial H^{t a r}}{\partial X_{j}^{+}}) = 0

(20)

According to the definition of $X_{j}$ and $X_{j}^{+}$ , the summation of the first term of equation (20) is zero. Equation (20) is reduced and further is written in the form of Poisson brackets as the following

\sum_{j = 1}^{2 n} ({(- 1)}^{j + 1} C \frac{\partial ρ^{t a r}}{\partial X_{j}} \frac{\partial H^{t a r}}{\partial X_{j}^{+}}) = C [H^{t a r}, ρ^{t a r}] = 0

(21)

where [•] is the Poisson bracket.

The Hamiltonian $H^{t a r}$ can be determined from equation (21). An easy way to construct $H^{t a r}$ is that the target SPDF $ρ^{t a r} (Q, P)$ can always be written in the form of $ρ^{t a r} (f (Q, P))$ , and then, according to the definition of Poisson bracket, $H^{t a r}$ can be set as $H^{t a r} = f (Q, P)$ to satisfy equation (21).

Substituting the determined $H^{t a r}$ into equation (19), the probability circulation part of the control force can be obtained as

u_{X_{j}}^{* (c)} = {(- 1)}^{j + 1} C \frac{\partial H^{t a r}}{\partial X_{j}^{+}} - a_{j}^{(c)}'

(22)

Secondly, substituting $a_{j}^{(p)} = a_{j}^{(p)}' + u_{X_{j}}^{(p)}$ in equation (18) and the target SPDF $ρ^{t a r}$ into equation (17), we obtain

(a_{j}^{(p)}' + u_{X_{j}}^{(p)}) ρ^{t a r} - \frac{1}{2} \sum_{l = 1}^{2 n} \frac{\partial}{\partial X_{l}} (b_{j, l} ρ^{t a r}) = 0

(23)

Solving equation (23), the probability potential part of the control force can be obtained

u_{X_{j}}^{* (p)} = \frac{\frac{1}{2} \sum_{l = 1}^{2 n} \frac{\partial}{\partial X_{l}} (b_{j, l} ρ^{t a r})}{ρ^{t a r}} - a_{j}^{(p)}'

(24)

The total control force is the sum of the circulation part (equation (22)) and the potential part (equation (24)) and given as

u_{X_{j}}^{*} = u_{X_{j}}^{* (c)} + u_{X_{j}}^{* (p)} = {(- 1)}^{j + 1} C \frac{\partial H^{t a r}}{\partial X_{j}^{+}} - a_{j}^{(c)}' + \frac{\frac{1}{2} \sum_{l = 1}^{2 n} \frac{\partial}{\partial X_{l}} (b_{j, l} ρ^{t a r})}{ρ^{t a r}} - a_{j}^{(p)}'

(25)

It can be seen that $a_{j}^{(c)}'$ and $a_{j}^{(p)}'$ appear in the form of the sum in the control law (equation (25)). $a_{j}^{(c)}'$ and $a_{j}^{(p)}'$ represent the PCF term and the PPF term of the uncontrolled system, respectively. Although $a_{j}^{(c)}'$ and $a_{j}^{(p)}'$ are unknown, their sum can be determined by equation (15) with vanishment of control force $u_{X_{j}}$ . Then, the total control force in equation (25) becomes

u_{X_{j}}^{*} = {(- 1)}^{j + 1} C \frac{\partial H^{t a r}}{\partial X_{j}^{+}} + \frac{\frac{1}{2} \sum_{l = 1}^{2 n} \frac{\partial}{\partial X_{l}} (b_{j, l} ρ^{t a r})}{ρ^{t a r}} - {(- 1)}^{j + 1} \frac{\partial H}{\partial X_{j}^{+}} - M_{X_{j}}

(26)

where C is the circulation flow control coefficient, and it can be any positive constant. In particular, when

C = 0

, the control law design will not need to construct

H^{t a r}

, which will make the control very convenient.

For a dissipative Hamiltonian system, Zhu et al. obtained the feedback tracking control law according to the exact stationary solutions of five types of Hamiltonian system (Zhu and Zhu, 2011, 2019). The target SPDF that can be shaped by this method is in exponential form, namely $ρ^{t a r} = C_{1} e^{(- λ (Q, P))}$ . However, the exact solution of FPK equation is usually difficult to be obtained, and the target SPDF is also limited in the exponential form. Inspired by the method proposed by Zhu et al., this paper proposes a more general control law. The shaping control law is solved according to the steady-state equilibrium condition of the FPK equation, which dominates the probability density evolution process. Therefore, the acquisition of the control law in this paper does not depend on the requirement of the exact stationary solution and target SPDF does not limit in the exponential form.

5. Example and analysis

To verify the generality and the effectiveness of the proposed procedure, this paper takes a approximate model of the stochastic time-delay ecological predator–prey system as an example to conduct the control law for tracking two sets of target SPDFs.

The motion of an ecosystem is governed by the following stochastic delayed model (Cai and Lin, 2007)

{\dot{X}}_{1} = X_{1} [a_{1} - s X_{1} - b X_{2} + u_{x 1} + d_{1} W_{1} (t)]

(27)

{\dot{X}}_{2} = X_{2} [- c + f \int_{0}^{\infty} F (τ) X_{1} (t - τ) d τ + + u_{x 2} + d_{2} W_{2} (t)]

(28)

where

X_{1} (t)

and

X_{2} (t)

are two stochastic processes of population densities of preys and predators, respectively.

a_{1}

s

b

c

, and

f

are positive constants.

a_{1}

and

- c

are the intrinsic growth rates of preys and predators, respectively.

b

and

f

represent interspecific interaction.

s

reflects the intraspecific interaction of preys.

d_{1} W_{1} (t)

and

d_{2} W_{2} (t)

represent the stochastic disturbances of prey birth rate and predator mortality, respectively.

W_{1} (t)

and

W_{2} (t)

are assumed to be two independent unit density Gaussian white noises with 0 mean value.

F (t)

is called the delay function. For a specific predator-prey system,

F (t)

is specified as

F (t) = \frac{1}{γ} e^{- t / γ}

(29)

where

γ

is a measure of the average delay time.

Expanding $X_{1} (t - τ)$ in equation (28) in Taylor series at $τ = 0$ and taking the first-order approximation (Cai and Lin, 2007; Xia et al., 2021), equations (27) and (28) becomes

{\dot{X}}_{1} = X_{1} [a_{1} - s X_{1} - b X_{2} + u_{x 1} + d_{1} W_{1} (t)]

(30)

{\dot{X}}_{2} = X_{2} [- c + f (1 - γ a_{1} X_{1}) + s f γ X_{1}^{2} + b f γ X_{1} X_{2} + u_{x 2} + d_{2} W_{2} (t)]

(31)

where

u_{x 1}

and

u_{x 2}

are feedback control forces, which are used to shape the probability densities of population of preys and predators to track a pre-designed target SPDF.

To convert system (20) to Hamiltonian form, the following transformation is introduced

Q = - \ln (X_{1}), P = \ln (X_{2})

(32)

Then, system (30) and (31) becomes

\dot{Q} = \frac{\partial H}{\partial P} + M_{Q} (Q, P) + u_{q} + d_{1} W_{1} (t)

(33)

\dot{P} = - \frac{\partial H}{\partial Q} + M_{P} (Q, P) + u_{p} + d_{2} W_{2} (t)

(34)

where

M_{Q} (Q, P) = \frac{s}{f} [- c + f e x p (- Q)], M_{P} (Q, P) = f γ \exp (- Q) [- a + b e x p (P)]

(35)

and the Hamiltonian function associated with the Hamiltonian system of equation (33) and (34) is

H = c q + c l n (\frac{c}{f}) + (f - c s γ) (\exp (- q) - \frac{c}{f}) + 0.5 f s γ (\exp (- 2 q) - \frac{c^{2}}{f^{2}}) - a (p - \ln (\frac{a}{b})) + b (\exp (p) - \frac{a}{b})

(36)

Two different target SPDFs of the population probability densities of the prey–predator system are set as

ρ_{1}^{t a r} (q, p) = \frac{(p^{2} + q^{2})}{0.01 π} exp (\frac{- {(p^{2} + q^{2})}^{2}}{0.02})

(37)

ρ_{2}^{t a r} (q, p) = \frac{0.0496}{{(p^{2} + q^{2})}^{2} + 0.01}

(38)

where the first SPDF is exponential and the second is non-exponential.

The target marginal SPDFs associated with equations (37) and (38) can be obtained as

ρ_{i}^{t a r} (q) = \int_{- \infty}^{\infty} ρ_{i}^{t a r} (q, p) d p

(39)

ρ_{i}^{t a r} (p) = \int_{- \infty}^{\infty} ρ_{i}^{t a r} (q, p) d q

(40)

i = 1,2

The feedback control forces for both target SPDFs in equations (37) and (38) can be determined by the previous proposed procedure. It can be seen from equations (37) and (38) that both $ρ_{1}^{t a r} (q, p)$ and $ρ_{2}^{t a r} (q, p)$ are functions of $p^{2} + q^{2}$ . Therefore, the target Hamiltonian functions $H_{1}^{t a r}$ and $H_{2}^{t a r}$ can be determined by using equation (21) as

H_{1}^{t a r} = H_{2}^{t a r} = (p^{2} + q^{2})

(41)

Substituting equation (41) into equation (26) with the specified forms of SPDFs in equations (37) and (38), the analytical expressions of the total feedback control forces for these two target can be obtained.

The control forces for first target $ρ_{1}^{t a r} (q, p)$ are

u_{1} = u_{q 1} + u_{q 1}

(42)

u_{q 1} = C * 2 p + \frac{q d_{1}^{2}}{(p^{2} + q^{2})} - q d_{1}^{2} \frac{(p^{2} + q^{2})}{0.01} + a - b e x p (p) + \frac{s}{f} [c - f e x p (- q)]

(43)

u_{p 1} = - C * 2 q + \frac{p d_{1}^{2}}{(p^{2} + q^{2})} - p d_{1}^{2} \frac{(p^{2} + q^{2})}{0.01} - [- c + f e x p (- q)] [1 + s γ \exp (- q)] -

f γ \exp (- q) [- a + b e x p (p)]

(44)

The control forces for second target $ρ_{2}^{t a r} (q, p)$ are

u_{2} = u_{q 2} + u_{q 2}

(45)

u_{q 2} = C * 2 p + \frac{- 2 q d_{1}^{2} (p^{2} + q^{2})}{{(p^{2} + q^{2})}^{2} + 0.01} + a - b e x p (p) + \frac{s}{f} [c - f e x p (- q)]

(46)

u_{p 2} = - C * 2 q + \frac{- 2 p d_{1}^{2} (p^{2} + q^{2})}{{(p^{2} + q^{2})}^{2} + 0.01} - [- c + f e x p (- q)] [1 + s γ \exp (- q)] - f γ \exp (- q) [- a + b e x p (p)]

(47)

where

C

is the circulation flow control coefficient and it can be any positive constant.

Some numerical results are obtained as shown in Figures 1–5 for system’s parameters

a_{1} = 0.9, b = 1, c = 0.5, f = 0.5, s = 0.1, γ = 0.1, a = a_{1} - \frac{s c}{f} = 0.8, d_{1} = 0.1, d_{2} = 0.1, C = 0.0

Figure 1.

The joint stationary probability density function of Q and P of the uncontrolled system.

Figure 2.

The control effectiveness for target $ρ_{1}^{t a r}$ . (a) The SPDFs of Q and P of target and controlled system; (b) the SPDFs of Q of the target, uncontrolled system, and controlled system; (c) the SPDFs of P of the target, uncontrolled system, and controlled system. SPDF: Stationary Probability Density Function.

Figure 3.

The control effectiveness for target $ρ_{2}^{t a r}$ . (a) The SPDFs of Q and P of the target and controlled system; (b) the SPDFs of Q of target, uncontrolled system, and controlled system; (c) the SPDFs of P of target, uncontrolled system, and controlled system. SPDF: Stationary Probability Density Function.

Figure 4.

Marginal PDF evolution of state Q with $ρ^{t a r}$ as the shaping target. (a) $ρ^{t a r} = ρ_{1}^{t a r}$ ; (b). $ρ^{t a r} = ρ_{2}^{t a r}$ .

Figure 5.

The spectral responses of Q of the controlled system with $ρ^{t a r}$ as the control target. (a) $ρ^{t a r} = ρ_{1}^{t a r}$ ; (b) $ρ^{t a r} = ρ_{2}^{t a r}$ .

5.1. Effectiveness of the control law

Figure 1 plots curved surfaces of the joint SPDF of Q and P of the uncontrolled system. Figure 2(a) plots the curved surfaces of the first target SPDF $ρ_{1}^{t a r} (q, p)$ in equation (37) and the SPDF of the controlled system. The good agreement of these two SPDFs can be observed. In Figures 2(b) and (c), the dashed lines are the integral projection of Figure 1 on the Q-axis and P-axis, respectively. The asterisk symbol and solid lines denote the marginal SPDFs of the controlled system and target, respectively. It is seen that the uncontrolled is quite different from the target one. However, after the proposed control, the reshaped SPDF agrees well with the target one, which verifies the effectiveness of the proposed control law.

In order to verify that the control law proposed is not restricted by the function form of the target SPDF, Figure 3(a) plots the curved surfaces of the second non-exponential target SPDF $ρ_{2}^{t a r} (q, p)$ and the associated SPDF of the controlled system. Figure 3(b) and (c) plot the marginal SPDFs of Q and P of the uncontrolled system (dashed line), controlled system (asterisk symbol), and target (solid line), respectively. The effectiveness of the control law for non-exponential target is also verified.

5.2. Efficiency regulation of the control law

The parameter C in the expression of control force is an arbitrary positive. That means any non-negative real value of C can achieve the purpose of tracking the same target SPDF. So what is the difference for different control parameter C? To answer this question, the effects of parameter C on the control effectiveness are investigated and shown in Figures 4 and 5.

Figure 4(a) shows the evolution of the marginal PDF of Q of the controlled system for different control parameter C = 0, 1, and 10, for the first target $ρ_{1}^{t a r}$ . At initial time t = 0 s, the marginal PDF of Q is a delta function, as shown by the vertical solid line at the abscissa 0.1. When t = 0.5 s, the marginal PDFs of the controlled system for C = 0, 1, and 10 are evolved and shown by the asterisk symbol, the circle symbol, and the cross symbol in Figure 4(a), respectively. The dashed line, dash-dotted line, and dotted line, respectively, represent the marginal PDFs of the controlled system when t = 5 s. It is seen that as time t increasing, all the controlled marginal PDFs for different C will converge to the target one; however, for larger parameter C, the controlled PDF has larger convergence rate. That means larger C will lead to better control efficiency.

Figure 4(b) plots the marginal PDF evolution of Q of the controlled system for different control parameter C = 0, 1, and 10, for the target $ρ_{2}^{t a r}$ . Similar observation with Figure 4(a) can be obtained.

Figure 5(a) and (b), respectively, show the spectral responses of Q of the controlled system with $ρ_{1}^{t a r}$ and $ρ_{2}^{t a r}$ in equations (37) and (38) as the control targets. The asterisk symbol, the circle symbol, and the cross symbol represent the related results of C = 0, 1, and 10, respectively. It can be seen from Figure 5 that the cross symbols (C = 10) are mainly distributed around 3 Hz, and the distribution of asterisk symbols (C = 0) and circle symbols (C = 1) are different, but they are basically located at relative low frequency range between 0 Hz and 1 Hz. This indicates that although different parameter C can converge the same target SPDF, the spectral responses for different C are quite different. Large C will increase the frequency of the response, which explains why PDF converges much more quickly for larger parameter C observed in the previous.

6. Conclusion

In this paper, a procedure for designing a feedback control strategy of nonlinear stochastic dynamic system for accurately tracking a specified SPDF without the requirement of exact stationary solution of FPK equation is proposed. The proposed procedure is applicable for vibrational and non-vibrational systems. The designed control force consists of a PCF part and a PPF part, which are determined from the PCF term and PPF term of the FPK equation, respectively. The advantages of the proposed control law include: the target control law is designed without the requirement of the exact solution of FPK equation; the target control law is in analytical expression and do not need real-time online calculation; the control law can track various forms of SPDFs, and the target SPDF do not limit to the non-exponential form; the control efficiency can be easily regulated by the control parameter C. The control effectiveness of the control law is validated by a 2-dimensional nonlinear stochastic system for tracking two different types of target SPDFs.

The proposed control strategy is designed for accurately tracking a specified SPDF, however, it also has the perspective of the applications in the structural vibration control if we design the target SPDF according to the objective of vibration control.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No.12172323 and No.12072312) and Zhejiang Science and Technology Project (No. 2019C03129). Zhejiang Provincial Natural Science Foundation of China (No. LZ22A020003).

ORCID iD

Ronghua Huan

References

Arnold

(1976) Stochastic Differential Equations: Theory and Applications. New Jersey: Wiley.

Cai

Lin

(2007) Stochastic analysis of time-delayed ecosystems. Physical Review E 76(4): 041913. DOI: 10.1103/PhysRevE.76.041913.

Huan

Zhu

, et al. (2015) Stationary response of a class of nonlinear stochastic systems undergoing Markovian jumps. Journal of Applied Mechanics 82(5): 051008. DOI: 10.1115/1.4029954.

Skelton

(1998) Covariance control using closed-loop modelling for structures. Earthquake Engineering and Structural Dynamics 27(11): 1367–1383. DOI: 10.1002/(SICI)1096-9845.

Chen

(2008) The principle of preservation of probability and the generalized density evolution equation. Structural Safety 30(1): 65–77. DOI: 10.1016/j.strusafe.2006.08.001.

Ren

Zhang

(2019) An introductory survey of probability density function control. Systems Science & Control Engineering 7(1): 158–170. DOI: 10.1080/21642583.2019.1588804.

Skelton

Iwasaki

Grigoriadis

(1998) A Unified Algebraic Approach to Linear Control Design. London, UK: Taylor & Francis.

Sun

(2006) Stochastic Dynamics and Control. London, UK: Elsevier.

Wang

Guo

Wang

(2008). Advances in stochastic distribution control. In 2008 10th International Conference on Control, Automation, Robotics and Vision. Hanoi, Vietnam. 17–20 Dec 2008. pp. 1479–1483. Piscataway, NY: IEEE Publishing. DOI: 10.1109/ICARCV.2008.4795742.

10.

Wang

(2012) Bounded Dynamic Stochastic Systems: Modelling and Control. Berlin, Germany: Springer-Verlag.

11.

Wang

Cherstvy

Liu

, et al. (2020) Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise. Physical Review E 102(1): 012146. DOI: 10.1103/PhysRevE.102.012146.

12.

Wojtkiewicz

Bergman

(2001) A moment specification algorithm for control of nonlinear systems driven by Gaussian white noise. Nonlinear Dynamics 24(1): 17–30. DOI: 10.1023/A:1026575320113.

13.

Wong

Zakai

(1965) On the relation between ordinary and stochastic equations. International Journal of Engineering Science 3(2): 213–229. DOI: 10.1016/0020-7225(65)90045-5.

14.

Xia

Huan

Zhu

, et al. (2019) Reshaping of the probability density function of nonlinear stochastic systems against abrupt changes. Journal of Vibration and Control 26(7–8): 532–539. DOI: 10.1177/1077546319889845.

15.

Xia

Sun

Ying

, et al. (2021) Dynamics and response reshaping of nonlinear predator–prey system undergoing random abrupt disturbances. Applied Mathematics and Mechanics 42(8): 1123–1134. DOI: 10.1007/s10483-021-2755-8.

16.

Duan

(2011) An averaging principle for stochastic dynamical systems with Lévy noise. Physica D: Nonlinear Phenomena 240(17): 1395–1401. DOI: 10.1016/j.physd.2011.06.001.

17.

Yin

Guo

(2012) Joint stochastic distribution tracking control for multivariate descriptor systems with non-Gaussian variables. International Journal of Systems Science 43(1): 192–200. DOI: 10.1080/00207721.2010.488754.

18.

Yin

Zhang

Guo

(2015) Data driven output joint probability density function control for multivariate non-linear non-Gaussian systems. IET Control Theory and Applications 9(18): 2697–2703. DOI: 10.1049/iet-cta.2015.0451.

19.

Yue

Wang

(2003) Recent developments in stochastic distribution control – A review. Measurement and Control 36(7): 209–215. DOI: 10.1177/002029400303600703.

20.

Zhu

(2011) Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density. Automatica 47(3): 539–544. DOI: 10.1016/j.automatica.2010.10.044.

21.

Zhu

(2019) Control of quasi non-integrable Hamiltonian systems for targeting a specified stationary probability density. International Journal of Control 92(5): 1117–1122. DOI: 10.1080/00207179.2017.1383631.

22.

Zhu

(2006) Nonlinear stochastic dynamics and control in Hamiltonian formulation. Applied Mechanics Reviews 59(1–6): 230–248. DOI: 10.1115/1.2193137.

23.

Zhu

Cai

Lin

(1990) On exact stationary solutions of stochastically perturbed Hamiltonian systems. Probabilistic Engineering Mechanics 5(2): 84–87. DOI: 10.1016/0266-8920(90)90011-8.

24.

Zhu

Yang

(1996) Exact stationary solutions of stochastically excited and dissipated integrable hamiltonian systems. Journal of Applied Mechanics 63(2): 493–500. DOI: 10.1115/1.2788895.