Optimal vibration control of moving-mass beam systems with uncertainty

Abstract

A linear optimal regulator for uncertain system is designed through the application of the probability density evolution method to linear quadratic regulator controller. One important background of this work is bridge-vehicle/gun-projectile system. This type of optimal problem is currently transformed into a moving load problem. The developed optimal regulator can provide the law of probability densities of outputs varying with time. In order to make the advocated method reach an optimal performance, the beneficial weighting matrix pair (Q, R) is selected using a trade-off sense. The designed regulator is then applied to a coupled simply supported beam-moving mass system, choosing the mid-span deflection as an output response and considering stochastic system parameters. The numerical example shows that the robustness of the proposed optimal regulator cannot be overestimated in comparison with a deterministic linear quadratic regulator controller. Further, the proposed method can produce an efficient solution channel for modern optimal control theory, especially, when compared with different uncertain optimal control techniques.

Introduction

Resisting the respective elastic vibration of bridges and barrels caused by high-speed motions of vehicles and projectiles is essential to the design of bridges and firing accuracy of pills. Such bridge-vehicle system and gun-projectile system are generally taken as beam-moving mass systems (BMSs).^1–4 Recently, more and more attention has been attracted to optimal controller design of BMSs, such as time-varying optimal controller design in vehicle-bridge interaction, ⁵ constant gain feedback gain design in vehicle-railway system, ⁶ the contribution of displacement-velocity feedback controller on the BMSs, ⁷ the influence of Lyapunov-based boundary controller on statically indeterminate beam ⁸ and the usage of a proportional-integral controller on the magnetic wheels-guide rail system. ⁹ Due to the uncertainties of system parameters and/or input disturbances, an optimal controller should be designed as uncertain. Although the existing deterministic controller design methods, up to now, still play significant roles, these methods cannot accurately reflect the robustness of uncertain systems. Consequently, a series of research topics about uncertain optimal controller design have arisen. For BMSs, to name but a few, time-varying stochastic optimal problem ¹⁰ and robust problem of a stream of random moving loads ¹¹ have been researched in recent years. For other uncertain system, in Zhu and Ying ¹² an optimal feedback control scheme for nonlinear structural system excited by a non-Gaussian process was proposed based on the stochastic averaging method for quasi-Hamiltonian system and the stochastic dynamic programming principle. In Monti et al., ¹³ a new control approach for a power converter with parameter uncertainty was presented based on the Polynomial chaos Theory. In Fisher and Bhattacharya, ¹⁴ different control methods related to problems of linear quadratic control (linear quadratic regulator (LQR)) for systems with uncertain parameters was put forward by using the generalized Polynomial Chaos Theory. In Gallagher, ¹⁵ the finite-horizon optimal control as a method of robust control design for a stochastic system was developed through optimizing a polynomial chaos expansion (PCE). In Peng et al., ¹⁶ an optimal control was implemented for a system with stochastic earthquake excitation described using PCE for a nonlinear oscillator. The Karhunen-Loeve (KL) decomposition was used to describe the excitation. In Zhu and Huo, ¹⁷ a robust nonlinear control strategy was proposed for trajectory tracking of a model-scaled helicopter with parameter uncertainties, and this controller was based upon the back-stepping approach with nonlinear damping terms. In Song and Dyke, ¹⁸ an optimal control strategy of nonlinear stochastic system was developed through the application of pseudo-spectral (PS) method. In Jameson, ¹⁹ the sensitivities of robust optimal control was investigated by combing PCEs and adjoint theory.

It should be noted that the aforementioned studies mainly focus on attaining the regular distribution types and second-order moments of the responses of stochastic controlled systems. Although a PCE can obtain high-order statistical moments, this method cannot acquire the rule of probability densities of system outputs varying with time. To get over this difficulty, a generalized probability density evolution theory was developed based on the ideology of physical stochastic system.^20,21 Meanwhile, Li and Chen ²⁰ also pointed out that traditional stochastic expansion method was found not capable of adequately reflect the probabilistic characteristics of stochastic dynamic systems, and the probability density evolution method (PDEM) is able to capture the rule of probability densities varying with time rather than just the second-order moment of uncertain responses. Moreover, the density evolution theory was also applied to the dynamic analysis and dynamic reliability of structures. ²² Sequentially, the density evolution theory of importance measure was developed to discuss reliability sensitivity of truss structures. ²³ By using the method, the probability density function (PDF) is varying with time and the instantaneous PDF is sometimes quite irregular with multiple peaks, rather than regular distribution function types.^20,24–28

However, up to now, the vibration control problem of uncertain BMSs has been rarely researched, especially, the uncertain control strategy. To accurately reflect the law of probability densities of output responses of the controlled BMSs varying with time, an optimal regulator-based on density evolution theory is originally proposed to calculate the system’s outputs. This type of vibration control problems, which denotes the purport of the current research, is actually a procedure of uncertainty propagation. Therefore, a proposed linear optimal regulator-based density evolution theory is summarized as follows: (1) dynamic equation of controlled systems with uncertainties of inherent properties can be derived through designing a proposed linear optimal output regulator. For the sake of the optimal performance of the contrivable controller, the weighting matrices Q and R are chosen in a trade-off sense^29,30; (2) output responses of the controlled systems can be solved by using the probability density evolution, which can commendably reflect the probability densities evolution of the calculated responses rather than only the second-order statistics of response outputs. In this paper, it should be noted that the mid-span deflection is selected as output responses and the material parameters of the given BMSs are described as random variables. Sequentially, the robustness between the promoted regulator and the deterministic controller can be analyzed.

Methods

Probability density evolution equation of uncertain system

Because of the uncertainties of inherent system properties, a variety of uncertain control synthesis strategies needs to rise gradually. Although deterministic linear and nonlinear control strategies still have taken up some space, these schemes could not reasonably reflect the impacts of control performance on uncertain systems. Some literatures related to active control of uncertain systems, especially BMS systems, have been reported. For example, stochastic linear-quadratic-Gaussian (LQG) control and time-variant optimal polynomial chaos control were deeply researched in Stancioiu and Ouyang.^10,11 Meanwhile, there are also large amounts of stochastic control algorithms for other types of uncertain systems, such as PS control, ¹⁸ optimal stochastic averaging feedback control, ¹² and PCE control.^13–16 The aforementioned investigations mainly focus on implementing the second-order statistics of response outputs. Although a PCE is able to obtain high-order statistical moments, this method cannot exhibit the probability densities evolution of the calculated responses. To overcome the above problem, a physical stochastic optimal control strategy is developed by using a generalized PDEM.^20,21 This type of vibration control problems, which represents the purport of the current research, is actually a procedure of uncertainty propagation. Therefore, we originally concentrate on an optimal regulator-based on density evolution theory. First, the basic theory of the evolution was introduced as follows:

Without loss of generality, the equation of motion of uncertain system can be uniformly written as

$$\mathbf{M}(\mathbf{\Theta })\ddot{\mathbf{X}}(t)+\mathbf{C}(\mathbf{\Theta })\dot{\mathbf{X}}(t)+\mathbf{K}(\mathbf{\Theta })\mathbf{X}(t)=\mathbf{F}(t)$$

(1)

where $\mathbf{M}$ , $\mathbf{C}$ , $\mathbf{K}$ are $n\times n$ mass matrix, damping matrix and stiffness matrix, respectively; $\mathbf{F}$ is the deterministic or random external force; $\mathbf{\Theta }$ is c-dimensional random variable; $\ddot{\mathbf{X}}$ , $\dot{\mathbf{X}}$ , $\mathbf{X}$ are $n\times 1$ acceleration, velocity and displacement vectors, respectively.

According to the principle of preservation of probability, the conditional PDF of $\mathbf{X}$ excited by external forces can be described as follows^21,22

$$f_{\mathbf{X}|\Theta }\left(x,\theta ,t\right)=\delta \left(x-H\left(\theta ,t\right)\right)$$

(2)

where $\delta (\cdot )$ is Dirac function; $H(\theta ,t)$ is the physical solution of $\mathbf{X}$ .

The generalized probability density evolution equation (PDEE) can be obtained by taking the derivative of equation (2)

$$\frac{\partial f_{\mathbf{X}|\Theta }\left(x,\theta ,t\right)}{\partial t}+\dot{\mathbf{X}}\left(\mathbf{\theta },t\right)\frac{\partial f_{\mathbf{X}|\Theta }\left(x,\theta ,t\right)}{\partial x}=0$$

(3)

The initial condition for equation (3) is

$$\partial f_{\mathbf{X}|\Theta }{\left(x,\theta ,t\right)}_{t=0}=\delta \left(x-x_0\right)f_{\Theta }\left(\theta \right)$$

(4)

where $x_0=X(t_0)$ is the initial value for $\mathbf{X}$ .

The PDF of $\mathbf{X}$ , then, can be attained by taking integration with respect to equation (4)

$$f_{\mathbf{X}}\left(x,t\right)={\displaystyle \int f_{\mathbf{X}|\Theta }\left(x,\theta ,t\right)\text{d}}\theta$$

(5)

Sequentially, the law of probability densities of output responses $\mathbf{X}$ varying with time can be acquired by using equation (5). Consequently, a proposed linear optimal regulator-based density evolution theory is summarized as follows: (1) the dynamic equation of controlled systems with uncertainties of inherent properties can be derived through designing a proposed linear optimal output regulator; (2) the output responses of the controlled systems can be solved by using the probability density evolution, which can commendably reflect the probability densities evolution of the calculated responses rather than only the second-order statistics of response outputs. Next, the presented optimal regulator-based density evolution theory for uncertain BMSs can be elaborated.

Linear optimal regulator design-based on PDEM for uncertain BMSs

An Euler–Bernoulli beam with multiple active tendon control systems under the moving mass and simply supported boundary conditions is considered. The inclination angle of the ith tendon with respect to the support is denoted by ${\alpha }_i$ . Without regard to contact loss, the equation of the motion of the controlled BMS under parametric uncertainties is governed as

$$\mathit{EI}\frac{{\partial }^{4}w\left(y,t\right)}{\partial y^4}+\rho A\frac{{\partial }^{2}w\left(y,t\right)}{\partial t^2}=-m\left(\stackrel{\text{.}\text{.}}{w}\left(t,\mathit{vt}\right)\right)+g))\delta \left(y-\mathit{vt}\right)-{\displaystyle \sum _{i=1}^{n_L}4k_c\hspace{0.17em}\cos {\alpha }_i{u}_i}\left(t\right)\delta \left(y-y_i\right)$$

(6)

where $w(y,t)$ is the transverse deflection of beam; $\delta (\cdot )$ is the Dirac delta function; g is acceleration of gravity; $u_i(t)$ is the control force of the ith actuator; $y_i$ is the position of the ith actuator; $n_L$ is the number of independent active tendon control systems; $v$ is the constant travelling speed at the beam of length L; $k_c$ is the tendon stiffness; $E$ , I, $\rho$ , $A$ are Yong’s modulus, second moment of area, density per unit length and the cross-sectional area, respectively. These properties are assumed as random variables.

$$\text{Let}w\left(y,t\right)={\displaystyle \sum _{k=1}^N{\phi }_k}\left(y\right)q_k\left(t\right)$$

(7)

where ${\phi }_k(y)$ is the modal shape function; $q_k(t)$ is the modal coordinate.

Applying modal superposition method ³¹ to equation (6) by multiplying both sides of the equation by ${\phi }_j(y)$ and integrating from 0 to L, then equation (6) can be transformed into

$$\begin{array}{c}\left(\mathbf{M}\left({\mathbf{\Theta }}_{\mathbf{M}}\right)+\overline{\mathbf{M}}\left(t\right)\right)\ddot{\mathbf{q}}\left(\mathbf{\Theta },t\right)+\overline{\mathbf{C}}\left(t\right)\dot{\mathbf{q}}\left(\mathbf{\Theta },t\right)+(\mathbf{K}\left({\mathbf{\Theta }}_{\mathbf{K}}\right)+\overline{K}\left(t\right))\mathbf{q}\left(\mathbf{\Theta },t\right)t\le t_f\\ =-\mathit{mg}\mathbf{\psi }\left(\mathit{vt}\right)-{\displaystyle \sum _{i=1}^{n_L}4k_c\hspace{0.17em}\cos {\alpha }_i\mathbf{\psi }\left(y_i\right){\mathbf{u}}_i}\left(\mathbf{\Theta },t\right)\end{array}$$

(8)

where $\mathbf{\psi }\left(\mathit{vt}\right)={\left[{\phi }_1(\mathit{vt}),{\phi }_2(\mathit{vt}),{\phi }_3(\mathit{vt}),\dots ,{\phi }_N\left(\mathit{vt}\right)\right]}^{\mathbf{T}}$ ; ${\mathbf{\Theta }}_{\mathbf{M}}$ , ${\mathbf{\Theta }}_{\mathbf{K}}$ are random parameters involved in matrices $\mathbf{M}$ and $\mathbf{K}$ ; $\mathbf{\Theta }=[{\mathbf{\Theta }}_{\mathbf{M}},{\mathbf{\Theta }}_{\mathbf{K}}]=[{\text{Θ}}_1,{\text{Θ}}_2,{\text{Θ}}_3,\dots ,{\text{Θ}}_z]$ , where $\text{z}$ is the number of random parameters; $\mathbf{M}({\mathbf{\Theta }}_{\mathbf{M}})$ , $\mathbf{K}({\mathbf{\Theta }}_{\mathbf{K}})$ are represented by

$$\mathbf{M}\left({\mathbf{\Theta }}_{\mathbf{M}}\right)=\rho \left({\text{Θ}}_1\right)A\left({\text{Θ}}_2\right)\text{diag}\left\{M_{\mathit{kj}}\right\}\text{;\hspace{1em}}\mathbf{K}\left({\mathbf{\Theta }}_{\mathbf{K}}\right)=E\left({\text{Θ}}_3\right)I\left({\text{Θ}}_4\right)\text{diag}\left\{K_{\mathit{kj}}\right\}$$

(9)

where ${\mathbf{M}}_{\mathit{kj}}$ = ${\displaystyle {\int }_0^L{\phi }_k\left(y\right)}\cdot {\phi }_j{\hspace{0.17em}}^{\mathbf{T}}\left(y\right)\text{d}y$ ; $K_{\mathit{kj}}={\displaystyle {\int }_0^L{\phi }_k(y)}\cdot {\phi }_j{\hspace{0.17em}}^{″\prime \prime }\left(y\right)\text{d}y$ ; $\Delta \mathbf{M}(t)$ , $\Delta \mathbf{C}(t)$ , $\Delta \mathbf{K}(t)$ are expressed as follows

$$\overline{\mathbf{M}}(t)=m\text{diag}\left\{{\overline{M}}_{\mathit{kj}}\right\}\text{;}\overline{\mathbf{C}}(t)=2mv\text{diag}\left\{C_{\mathit{kj}}\right\};\overline{\mathbf{K}}(t)=m{v}^2\text{diag}\left\{{\overline{K}}_{\mathit{kj}}\right\}$$

where ${\overline{M}}_{\mathit{kj}}={\phi }_k(\mathit{vt}){\phi }_j(\mathit{vt})$ ; $C_{\mathit{kj}}={\phi }_j(\mathit{vt}){\phi }_k{\hspace{0.17em}}^{\prime }(\mathit{vt})$ ; ${\overline{K}}_{\mathit{kj}}={\phi }_j(\mathit{vt}){\phi }_k{\hspace{0.17em}}^{\prime \prime }(\mathit{vt})$

After terminal time instant t_f, the beam vibrates freely and the stochastic controlled system is expressed as follows

$$\mathbf{M}\left({\mathbf{\Theta }}_{\mathbf{M}}\right)\ddot{\mathbf{q}}\left(\mathbf{\Theta },t\right)+\mathbf{K}\left({\mathbf{\Theta }}_{\mathbf{K}}\right)\mathbf{q}\left(\mathbf{\Theta },t\right)=-{\displaystyle \sum _{i=1}^{n_L}4k_c\mathrm{\hspace{0.17em}cos}{\alpha }_i\mathbf{\psi }(y_i){\mathbf{u}}_i}t>t_f$$

(10)

where ${\left[\mathbf{q}{\left(t_f\right)}^{\mathbf{T}}\dot{\mathbf{q}}{\left(t_f\right)}^{\mathbf{T}}\right]}^{\mathbf{T}}$ is used as initial conditions of system (11).

The controlled system (8) can be written in state-space form

$$\begin{array}{l}\dot{\mathbf{Z}}(\mathbf{\Theta },t)=\mathbf{A}(\mathbf{\Theta },t)\mathbf{Z}(\mathbf{\Theta },t)+\mathbf{B}(\mathbf{\Theta },t)\mathbf{U}(\mathbf{\Theta },t)+\mathbf{F}(t)\\ \mathbf{Z}(\mathbf{\Theta },t_0)={\mathbf{z}}_0\\ \mathbf{Y}(\mathbf{\Theta },t)\text{=}\mathbf{C}(\mathit{vt})\mathbf{Z}(\mathbf{\Theta },t)\end{array}$$

(11)

where $\mathbf{Z}(\mathbf{\Theta },t_0)$ is the initial condition; $\mathbf{A}(\mathbf{\Theta },t)$ is a 2n×2n stochastic coefficient matrix; $\mathbf{B}(\mathbf{\Theta },t)$ is a 2n×r controller location matrix and $\mathbf{F}(t)$ is a 2n-dimensional excitation force vector, respectively; $\mathbf{Y}(\mathbf{\Theta },t)$ is the output. $\mathbf{Z}(\mathbf{\Theta },t)$ , $\mathbf{A}(\mathbf{\Theta },t)$ , $\mathbf{B}(\mathbf{\Theta },t)$ , $\mathbf{U}(\mathbf{\Theta },t)$ and $\mathbf{F}(t)$ are expressed as

$$\begin{array}{l}\mathbf{Z}\left(\mathbf{\Theta },t\right)=\left[\begin{array}{l}\mathbf{q}\left(\mathbf{\Theta },t\right)\\ \ddot{\mathbf{q}}\left(\mathbf{\Theta },t\right)\end{array}\right];\\ \mathbf{B}\left(\mathbf{\Theta },t\right)=\left[\begin{array}{l}0\hfill \\ -{\left(\mathbf{M}({\mathbf{\Theta }}_{\mathbf{M}})+\Delta \mathbf{M}(t)\right)}^{-1}\left[\mathbf{\psi }(y_1)4k_c\cos {\alpha }_1\dots \mathbf{\psi }(y_i)4k_c\cos {\alpha }_i\right]\hfill \end{array}\right]\\ \mathbf{A}\left(\mathbf{\Theta },t\right)=\left[\begin{array}{ll}0& \mathbf{I}\\ -{(\mathbf{M}({\mathbf{\Theta }}_{\mathbf{M}})+\Delta \mathbf{M}(t))}^{-1}(\mathbf{K}({\mathbf{\Theta }}_{\mathbf{K}})+\Delta K(t))& -{(\mathbf{M}({\mathbf{\Theta }}_{\mathbf{M}})+\Delta \mathbf{M}(t))}^{-1}\Delta C(t)\end{array}\right]\\ \mathbf{U}\left(\mathbf{\Theta },t\right)={\left[{\mathbf{u}}_1(t),{\mathbf{u}}_2(t),\dots ,{\mathbf{u}}_r(t)\right]}^{\prime };\mathbf{F}(t)={\left[0,\mathit{mg}\mathbf{\psi }(\mathit{vt})\right]}^{\prime }\end{array}$$

(12)

When the control problem of the stochastic system (12) needs to be solved, the control objective can be defined as the minimization of the mid-span deflection response of the beam. Then, the optimal problem is changed to a linear quadratic problem, which is obtained by a quadratic performance index

$$J(\mathbf{Z},\mathbf{U})=\frac{1}{2}{\mathbf{Z}}^{\mathbf{T}}(\mathbf{\Theta },t_f)\mathbf{SZ}(\mathbf{\Theta },t_f)+\frac{1}{2}{\displaystyle {\int }_0^{t_f}({\mathbf{Z}}^{\mathbf{T}}(\mathbf{\Theta },t)\mathbf{QZ}(\mathbf{\Theta },t)+{\mathbf{U}}^{\mathbf{T}}(\mathbf{\Theta },t)\mathbf{RU}(\mathbf{\Theta },t))}\text{d}t$$

(13)

where $\mathbf{Z}(\mathbf{\Theta },t_f)$ is the terminal state; $\mathbf{S}$ and $\mathbf{Q}(t)$ are nonnegative definite matrices; ${\mathbf{R}}_i(t)$ is a positive definite matrix. Therefore, the minimum of the standard quadratic performance index J gives rise to an issue of the conditional extreme value of cost function. For a closed-loop controlled system, the control force function of the optimal problem can be generated by the following expression

$$\mathbf{U}(\mathbf{\Theta },t)=-{\mathbf{R}}^{-1}{\mathbf{B}}^{\mathbf{T}}(\mathbf{\Theta },t)\mathbf{P}(\mathbf{\Theta },t)\mathbf{Z}(\mathbf{\Theta },t)=-{\mathbf{G}}_g(\mathbf{\Theta },t)\mathbf{Z}(\mathbf{\Theta },t)$$

(14)

where matrix $\mathbf{P}(\mathbf{\Theta },t)$ is calculated by solving backward the differential Riccati equation.

$$\begin{array}{l}\dot{\mathbf{P}}(\mathbf{\Theta },t)+\mathbf{A}(\mathbf{\Theta },t)\mathbf{P}(\mathbf{\Theta },t)+{\mathbf{A}}^{\mathbf{T}}(\mathbf{\Theta },t)\mathbf{P}(\mathbf{\Theta },t)-\mathbf{P}(\mathbf{\Theta },t)\mathbf{B}(\mathbf{\Theta },t){\mathbf{R}}^{-1}{\mathbf{B}}^{\mathbf{T}}(\mathbf{\Theta },t)\mathbf{P}(\mathbf{\Theta },t)+\mathbf{Q}=0\\ \mathbf{P}(\mathbf{\Theta },t_f)=\mathbf{S}\end{array}$$

(15)

where $\mathbf{P}(\mathbf{\Theta },t_f)$ is final condition and can be obtained using Lyapunov equation. The control effectiveness of the optimal control depends on the specified control strategy, which is related to the objective performance of systems.

The most important advantage of the state-space control technique is that it can consider the control to be expressed as a state-feedback function (11) of the manuscript and can transform the system from an open-loop control system into a closed-loop control system. By employing this method, the control function can be synthesized based on dynamic equations of the system regardless of the effect of any force acting at the system input. Then, a simple change of variable can be introduced which can consider the forcing term F(t). Sequentially, the time-varying modal force $\mathit{mg}\mathbf{\psi }(\mathit{vt})$ can be considered during the design process. Therefore, the modal force can be augmented to the matrix B, which can be termed as the augmented B. The state-space equation is transformed into the augmented system, whose advantage is that the required control action is able to directly consider the excitation force rather than only considering it as a disturbance. It should be stressed that the matrix B depends on the location of the actuators, and it not only affects the design process but also affects largely the controllability of the system. Moreover, the control problem of BMSs is that not only it is a time-varying process but also the control action requires time-varying weighting matrices. Therefore, to take better advantage of control effort, the weighting matrices originated from the state-space equation are allowed to change with the position of the mass on the beam.

Figure 1 depicts the system construction drawing of uncertain optimal output regulator in a finite time. From this figure, it can be deduced that the propositional regulator gives a novel controller alternative to a deterministic controller.

Figure 1.

System construction drawing of uncertain optimal output regulator in a finite time. PDEM: probability density evolution method; BMS: beam-moving mass system; LQR: linear quadratic regulator.

According to equations (13) and (14), one realizes that the critical procedure of LQR problem relies on the selection of weight matrices Q and R. The beneficial weighting matrices can be obtained in a trade-off sense.^29,30

$$\mathbf{Q}=a\left[\begin{array}{l}\mathbf{I}0\hfill \\ 0\mathbf{I}\hfill \end{array}\right],\mathbf{R}=b\mathbf{I}$$

(16)

where a, b are coefficients of state matrices and control force matrices, respectively, whose ratio indicates the effectiveness and economy of the optimal control. The optimal values a and b can be obtained using the system second-order statistics assessment.

Likewise, equation (11) can be also written in state-space form

$$\begin{array}{l}\dot{\mathbf{Z}}(\mathbf{\Theta },t)=\mathbf{A}(\mathbf{\Theta })\mathbf{Z}(\mathbf{\Theta },t)+\mathbf{B}(\mathbf{\Theta })\mathbf{U}(\mathbf{\Theta },t)\hfill \\ \mathbf{Z}(\mathbf{\Theta },t_f)={\mathbf{z}}_{f}t>t_f\hfill \\ \mathbf{Y}(\mathbf{\Theta },t)\text{=}\mathbf{C}(\mathit{vt})\mathbf{Z}(\mathbf{\Theta },t)\hfill \end{array}$$

(17)

where $\mathbf{Z}(\mathbf{\Theta },t_f)$ is the initial condition for system (11); $\mathbf{A}(\mathbf{\Theta })$ is a 2n×2n time-invariant system matrix and $\mathbf{B}(\mathbf{\Theta })$ is a 2n×r controller location matrix, respectively. $\mathbf{A}(\mathbf{\Theta })$ and $\mathbf{B}({\mathbf{\Theta }}_{\mathbf{M}})$ are

$$\begin{array}{l}\mathbf{A}\left(\mathbf{\Theta }\right)=\left[\begin{array}{ll}0& \mathbf{I}\\ -\mathbf{M}{({\mathbf{\Theta }}_{\mathbf{M}})}^{-1}\mathbf{K}({\mathbf{\Theta }}_{\mathbf{K}})& 0\end{array}\right]\\ \mathbf{B}\left({\mathbf{\Theta }}_{\mathbf{M}}\right)=\left[\begin{array}{l}0\\ -\mathbf{M}{({\mathbf{\Theta }}_{\mathbf{M}})}^{-1}\left[\mathbf{\psi }(y_i)4k_c\cos {\alpha }_1.\text{.}\text{.}\mathbf{\psi }(y_i)4k_c\cos {\alpha }_i\right]\end{array}\right]\end{array}$$

(18)

This optimal problem of system (11) is obtained by using the following quadratic performance index

$$J\left(\mathbf{Z},\mathbf{U}\right)=\frac{1}{2}{\displaystyle {\int }_{t_f}^{+\infty }\left({\mathbf{Z}}^{\mathbf{T}}\left(\mathbf{\Theta },t\right)\mathbf{Q}\mathbf{Z}\left(\mathbf{\Theta },t\right)+{\mathbf{U}}^{\mathbf{T}}\left(\mathbf{\Theta },t\right)\mathbf{R}\mathbf{U}\left(\mathbf{\Theta },t\right)\right)}\text{d}t$$

(19)

and the matrix $\mathbf{P}(\mathbf{\Theta })$ can be solved by the algebraic Riccati equation

$$-\mathbf{P}(\mathbf{\Theta })\mathbf{A}(\mathbf{\Theta })-{\mathbf{A}}^{\mathbf{T}}(\mathbf{\Theta })\mathbf{P}(\mathbf{\Theta })+\mathbf{P}(\mathbf{\Theta })\mathbf{B}({\mathbf{\Theta }}_{\mathbf{M}}){\mathbf{R}}^{-1}{\mathbf{B}}^{\mathbf{T}}({\mathbf{\Theta }}_{\mathbf{M}})\mathbf{P}(\mathbf{\Theta })-\mathbf{Q}=0$$

(20)

where the matrix $\mathbf{P}(\mathbf{\Theta })$ is time-independent. Hence, the control force for system (11) can be given as follows

$$\mathbf{U}(\mathbf{\Theta },t)=-{\mathbf{R}}^{-1}{\mathbf{B}}^{\mathbf{T}}({\mathbf{\Theta }}_{\mathbf{M}})\mathbf{P}(\mathbf{\Theta })\mathbf{Z}(\mathbf{\Theta },t)=-{\mathbf{G}}_g(\mathbf{\Theta },t)\mathbf{Z}(\mathbf{\Theta },t)$$

(21)

Analogously, the system structure graphic of uncertain optimal output regulator for infinite time is described in Figure 2. For this type of the regulator, terminal state must be equal to zero, namely, $\mathbf{Z}(\mathbf{\Theta },\infty )=0$ . Otherwise, performance index will be infinity and then system (17) cannot be solved. Because $\mathbf{Z}(\mathbf{\Theta },\infty )=0$ , the terminal cost is not required in index (19).

Figure 2.

System construction drawing of uncertain optimal output regulator for infinite time. PDEM: probability density evolution method; BMS: beam-moving mass system; LQR: linear quadratic regulator.

Based on the probability density evolution theory, the PDEE with respect to $q$ is described as follows

$$\frac{\partial f_{\mathbf{q}|\Theta }(q,\theta ,t)}{\partial t}+\dot{\mathbf{q}}(\mathbf{\theta },t)\frac{\partial f_{\mathbf{q}|\Theta }(q,\theta ,t)}{\partial q}=0$$

(22)

The initial condition for equation (22) is

$$\partial f_{\mathbf{q}|\Theta }{(q,\theta ,t)}_{t=0}=\delta (q-q_0)f_{\Theta }(\theta )$$

(23)

where $q_0=q(t_0)$ is the initial value for $\mathbf{q}$ .

The PDF of the mid-span deflection $\mathbf{w}(\mathbf{y},t)$ , then, can be attained as follows

$$f_{\mathbf{w}}(w,t)={\displaystyle \int {\mathbf{\psi }}^{\mathbf{T}}(\mathbf{y})f_{\mathbf{q}|\Theta }(q,\theta ,t)\text{d}}\theta$$

(24)

where $\mathbf{w}(\mathbf{y},t)={\mathbf{\psi }}^{\mathbf{T}}(\mathbf{y})\mathbf{q}(t)$

Similarly, the PDEE with respect to $\mathbf{U}$ is expressed as follows

$$\frac{\partial f_{\mathbf{U}|\Theta }(q,\theta ,t)}{\partial t}+\dot{\mathbf{U}}(\mathbf{\theta },t)\frac{\partial f_{\mathbf{U}|\Theta }(q,\theta ,t)}{\partial U}=0$$

(25)

The initial condition for equation (25) is

$$\partial f_{\mathbf{U}|\Theta }{(U,\theta ,t)}_{t=0}=\delta (U-U_0)f_{\Theta }(\theta )$$

(26)

where $U_0=U(t_0)$ is the initial value for $\mathbf{U}$ .

The PDF of the control force $\mathbf{U}(\mathbf{\Theta },t)$ is

$$f_{\mathbf{U}}(U,t)={\displaystyle \int f_{\mathbf{U}|\Theta }(U,\theta ,t)\text{d}}\theta$$

(27)

Procedures of designing and solving the optimal output regulator

The specific procedures of designing and solving the optimal output regulator are depicted in Figure 3 and can be summarized as follows:

Figure 3.

The flowchart of designing and solving the optimal output regulator. PDEM: probability density evolution method; BMS: beam-moving mass system; LQR: linear quadratic regulator; PDF: probability density function.

• Govern the equation of motion of the controlled BMSs;

• Determine the distribution types $\mathbf{\Theta }$ of BMSs’ parameters;

• Discretize the representative points of BMSs’ parameters;

• Design the optimal output regulator-based on PDEM;

• Transform the solution of an optimal regulator problem into a problem of First-order partial differential equation sets; and

• Fit instantaneous PDFs of output responses and analyze the evolution process of PDFs.

Simulation

A simply supported Euler–Bernoulli beam is used to demonstrate the efficiency of the proposed design method, as shown in Figure 4. A moving mass with constant speed travels on the beam. In this example, the performance of the stochastic controlled system with multiple actuators is observed as a solution for suppressing the mid-span deflection response of the beam. All numerical parameters of the BMS are listed in Table 1. ⁵ The constant speed of 2 m/s is selected in the BMSs. The control force of the ith actuator is denoted by ${\mathbf{u}}_i(t)$ . The ith mode of the simply supported beam can be expressed as follows: ${\phi }_i=\sin (i\pi y/L)$ . The control effect of the beam response is selected at the position y = L/2. Figure 5 shows the rationality of the selection of the first two modes of the beam. From the figure, it can be seen that increasing the number of modes over two does not improve the accuracy of the mid-span deflection.

Figure 4.

BMSs with two active tendons.

Table 1.

Numerical parameters of the example system.

System parameters	Unit	Distribution type	Mean	COV
Yong’s modulus, E	N $\cdot$ m−2	Normal	2.1e8	0.1
Density per unit length, $\rho$	kg $\cdot$ m−3	Normal	7800.00	0.08
Cross-section width, b	m	Normal	3.28e−04	0.09
Cross-section height, h	m	Normal	0.0860	0.1
Mass of moving load	kg		0.5
Length of beam	m		1
Position of the first actuator	m		0.10
Position of the second actuator	m		0.60

COV: coefficient of variation.

Figure 5.

Mid-span deflection of the beam at the uncontrolled condition.

The gain control matrix ${\mathbf{G}}_g(\mathbf{\Theta },t)$ involved in the promoted optimal regulator must be calculated and then used to suppress the beam response from $t_0=0$ until $t=2L/v$ . Table 1 gives the position of two actuators. Next, by varying the coefficient b in a trade-off sense, the beneficial coefficients a = 1000 and b = 0.1 can be obtained. The use of the presented method with the beneficial weight matrices Q = 1000×diag([1 1 1 1]) and R = 0.1 ${\mathbf{I}}_{2\times 2}$ results in a time history of the mean of the mid-span deflection of the controlled system. It is clear from Figure 6 that the deflection response process of the controlled system is reduced significantly compared with that of the uncontrolled system. In this figure, it should be figured out that the optimal design control-based density evolution (ODEC) represents the proposed optimal controller scheme-based PDEM, which can accurately reflect the impacts of control actions on uncertain BMSs. Actually, the proposed ODEC is a non–deterministic controller scheme. Meanwhile, the deflection in the time with stronger variability, i.e. the time interval from 0.2 to 1 s, obviously reduces, which indicates the effectiveness of the novel design controller. It is noteworthy that the deflection response of free vibration system suppressed by the designed method has an amplitude fading. This is due to the fact that the optimal output regulator system for infinite time depends on the term ${\mathbf{G}}_{g1}\stackrel{.}{\mathbf{q}}(t)$ but the uncontrolled free vibration phase does not include the term.

Figure 6.

Time history of the mean of the mid-span deflection of BMSs with/without control. ODEC: optimal design control-based density evolution.

Meanwhile, the time history of the mean of the mid-span deflection of BMSs is compared with ODEC and Monte Carlo simulation, as shown in Figure 7. It can be derived that the response of uncertain BMSs with ODEC is close to that with Monte Carlo simulation. It should be noted that the ODEC strategy can be verified by using Monte Carlo simulation because the ODEC scheme involves the system’s uncertainty. These results indicate that the proposed ODEC scheme behaves well in precision. Here, the respective discrepancies of time histories of the mean of mid-span deflection and control forces between the coefficient b = 0.1and b = 0.9 are presented in Figure 8, where a equals 1000. It is seen that: (1) as b = 0.9, the control effect for the time history of the mean of mid-span deflection is underestimated; (2) however, the time history of the mean of control forces ${\mathbf{u}}_1(t)$ and ${\mathbf{u}}_2(t)$ will be reduced as the coefficient b increase. Moreover, the weight matrix R has a direct influence on the amplitude attenuation of the free vibration system. The results illustrate the influence of the choice of the weight matrix R on robust performance of the given optimal output regulator system.

Figure 7.

Time history of the mean of the mid-span deflection of BMSs: comparison between ODEC and Monte Carlo. ODEC: optimal design control-based density evolution.

Figure 8.

Time histories of the mean of mid-span deflection and control forces between the coefficient b = 0.1and b = 0.9. (a) Mid-span deflection of BMSs, (b) u₁(t) and (c) u₂(t). ODEC: optimal design control-based density evolution.

Figure 9 shows probabilistic characteristics of the mid-span deflection with/without control, including typical PDFs at different instants of time. From this figure, it can be seen that the mid-span deflection is obviously reduced. These observations illustrate that the recommended controller is able to efficiently suppress the vibration of uncertain BMSs. Furthermore, the results show that PDF of the deflection is varying with time and quite different from the regular distribution. The probabilistic characteristics of the control inputs ${\mathbf{u}}_1(t)$ and ${\mathbf{u}}_2(t)$ are shown in Figure 10. It can be observed that the shapes of typical PDFs of the control forces are different from those of the mid-span deflection. The cause is that the control forces lies on the modal displacement and modal velocity, while the mid-span deflection hinges on the modal shape vector $\mathbf{\psi }(\mathbf{y})$ .

Figure 9.

Probabilistic characteristics of the mid-span deflection with/without control: (a)–(c) Typical PDFs of the mid-span deflection with control varying with time. (d)–(f) Typical PDFs of the mid-span deflection without control varying with time. PDF: probability density function.

Figure 10.

Probabilistic characteristics of the optimal control forces: (a)–(c) Typical PDFs of u₁(t) varying with time; (d)–(f) Typical PDFs of u₂(t) varying with time. PDF: probability density function.

Although the deterministic control (e.g. deterministic LQR, deterministic LQG) might be applied to suppress the responses of stochastic systems, the control force only depends on the constant gain matrix, rather than the stochastic gain matrix. Figure 11 shows the discrepancy of time history of mid-span deflection between the deterministic LQR controller (DLC) and the optimal controller scheme-based on PDEM (ODEC). Actually, ODEC represents the nondeterministic controller and DLC denotes the deterministic controller. From the figure, it is concluded that as the beneficial coefficients a = 1000 and b = 0.1, the control effect of the output response and the robustness of the advocated controller are overvalued using the DLC. The discrepancy of the control forces between the two methods is seen by analyzing Figure 12 where the control action is plotted for the time history. It can be found that the absolute control input ${\mathbf{u}}_1(t)$ produced by the DLC is obviously undervalued when the time interval is 0.48 s. Meanwhile, the control input ${\mathbf{u}}_2(t)$ generated by the DLC is obviously underestimated in the time interval from 0.3 to 1 s. Results indicate that the second actuator plays more important role in the control process and the ODEC can let the robust performance of the designed regulator system more accurately. Moreover, combining Figures 9 and 11, it can be derived that although both the ODEC and the DLC methods can reduce the deflection of the beam, the ODEC strategy behaves a higher control effect. This indicates that the ODEC is more suitable for dealing with uncertain control problems. Meanwhile, the results of Figures 9 and 11 demonstrate that the instantaneous PDF and its evolution of system responses can be obtained by the ODEC method, while the DLC method cannot acquire the above phenomena and also is found not capable of dealing with uncertain controlled systems.

Figure 11.

Time history of mid-span deflection between the ODEC method and DLC method. ODEC: optimal design control-based density evolution; DLC: deterministic LQR controller.

Figure 12.

Time history of the mean of the control forces between the ODEC method and DLC method: (a) u₁(t) and (b) u₂(t). ODEC: optimal design control-based density evolution; DLC: deterministic LQR controller.

Figure 13 shows the time history of mid-span deflection with various control strategies. It can be observed that although all the control schemes can be employed to suppress the vibration of the beams, both the ODEC method and the augmented time-varying method (Augmented matrix) exhibit a higher control performance. Time-invariant control method only depends on the time-invariant system matrix A and the controller location matrix B, expressed in equation (18). Time-invariant control method is employed to suppress the beam subjected to moving mass and is also applied to suppress the free vibration of the beam. However, it can be seen that the performance of the time-invariant control method is not significant. Moreover, the advantage of the augmented time-variant control strategy is that the required control action is able to directly consider the excitation force rather than only considering it as a disturbance. It can be observed that the augmented time-variant strategy behaves a bit poor in control performance when the moving mass acts on the beam. However, the control effect of the augmented time-variant scheme is better than that of the ODEC method when the mass leaves the beam. Therefore, both the ODEC and the augmented time-variant methods have a superior performance when the vibration of the beam should be suppressed.

Figure 13.

Time history of mid-span deflection with various control Strategies: ODEC-proposed control method; Augmented matrix-the modal force is augmented to the matrix B, whose advantage is that the required control action can directly consider the excitation force rather than only considering it as a disturbance; Time-invariant control-the matrix A and B is time-invariant.³² ODEC: optimal design control-based density evolution.

Conclusions

This paper devises an optimal regulator-based on PDEM for an uncertain BMSs. In order to obtain the law of probability densities of control inputs and output responses, the designed controller is then applied to the BMSs. Meanwhile, a quadratic performance index related to the optimal problem is defined. The beneficial weighting matrices Q and R are selected in a trade-off sense. It is found through numerical example that the weight matrix R has a significant impact on the robust performance of the optimal output regulator system. Further, the ODEC extends the deterministic control, such as the DLC, of which the control gain is treated as the deterministic quantity. The instantaneous PDFs of the deflection response and the control force can be obtained to characterize the controller’s transient performance. The comparative study between the ODEC and DLC schemes illustrates that the robustness of the developed optimal controller cannot be overestimated in comparison with a deterministic LQR controller. Further, the instantaneous feature engendered by the proposed method can provide an efficient solution channel for modern optimal control theory, especially, when compared with different uncertain optimal control techniques. Moreover, the advocated method provides a computationally efficient while comparing with the Monte Carlo simulation. Finally, both the ODEC and the augmented time-variant methods behave a superior performance when the moving mass acts on the beam and the mass leaves the beam.