Optimal control of structures subjected to traveling load

Abstract

The problem of the optimal semi-active control of a structure subjected to a moving load is studied. The control is realized by a change of damping of the structure’s supports. The objective is to provide a smooth passage for vehicles and extend the time needed for the safety service of the carrying structures. In contrast to the previous works of the author, in this paper, the model used takes into account time-varying passage speeds, which allows a broader application, in particular, to robotics. The study of the optimal control problem produces a practical condition that justifies whether, for a given set of parameters, the controlled system can outperform its passively damped equivalent. For the optimization, an efficient method of parametrized switching times is developed and tested via a numerical example. The designed optimal control is examined on a real test stand. The experiments are carried out for three different passage scenarios. In terms of the assumed metrics the proposed method outperforms the passive case by over 40%.

Keywords

Optimal control structural control semi-active control vibration control moving load

1. Introduction

Problems of structures subjected to loads traveling with high velocity are of special interest to engineers. Numerous analytical and numerical solutions are being applied to solve the problems of transportation and robotic systems with single- or multi-point interactions, such as train–track, vehicle–bridge, or effector–guideway. The high vibration levels caused by the continually increasing speeds and load-carrying capacity requirements challenge scientists to seek new solutions. Constructing new railway tracks or bridges which would have a sufficiently higher load-carrying capacity or ability to resist dynamic strains frequently faces difficulties of an economic nature. Similarly, increasing the mass of a structure which results from a static strengthening is often challenging for technological reasons. To cope with undesired vibration effects, a variety of control systems acting on both the structures and the suspension of the traveling loads have been proposed and put into practice.

A common objective in structural control is to enhance the stability of a system subjected to impulsive or periodic excitation. The first group of control methods, referred to as active methods, is based on force actuators. An active control method to control the beam vibrations via linear force actuators is presented, for example, in Frischgesel et al. (1998). A controlled piezoelectric layer was considered in Baz (1997). An actively controlled beam subjected to a harmonic excitation was presented in Pietrzakowski (2001) whereas an actively controlled string system was studied in Tan and Ying (2000). Similarly, the control of the shape of the railway tracks can be listed among the active methods in structural control. Flont and Holnicki-Szulc (1997) developed an approach that uses active smart sleepers that enable the track to shift up and down. The objective was to minimize the deflection of the track. Stancioiu and Ouyang (2014) suggested an active control method to suppress the vibration of bridges. The controller was based on a linear quadratic regulator.

A recent trend is to replace force actuators with semi-active magneto-rheological dampers. This solution is usually less efficient. However, it has attracted the interest of engineers due to its significantly lower power consumption. It is also safer in the case of a control system failure. Unlike active systems, the semi-active ones based on controlled dampers can always be switched to a passive mode, providing system stability. One of the first concepts of semi-active control in mechanical systems was proposed in Karnopp et al. (1974), who presented the idea of stabilizing an oscillator with one degree of freedom moving upon uneven ground. The “Skyhook” algorithm developed by the authors is today one of the most widely used in suspension control systems for vehicles. The idea was initially designed to improve the comfort of the passengers. Later, a similar control method was adapted to an oscillator moving upon carrying structures. Extensive results were demonstrated in Chen et al. (2002). Controlled dampers are incorporated also for seismic isolation. Interesting results can be found in Ruangrassamee and Kawashima (2003). Fulin et al. (2002) proposed controling both the stiffness and the damping parameters. The control decision led to the maximum dissipation of energy.

The use of semi-active supports for a structure subjected to a moving load was first proposed in Bajer and Bogacz (2000), who demonstrated, by means of numerical simulations, that for a wide range of travel velocities, switching damping strategies outperform standard passive solutions. The idea was later extended in Pisarski and Bajer (2010) and Pisarski and Bajer (2011). By means of numerical experiments it was shown that the metric corresponding to the total deflection of the load trajectory from the desired straight line was reduced by up to 50%. The model used in those studies took into account neither a varying speed of passage nor the inertial forces associated with the moving mass, which are crucial to capturing the dynamics of systems used in robotic technologies.

This paper aims at designing a practical, semi-active control method dedicated to the applications to large-scale structures such as bridges and overpasses subjected to traveling trains as well as to robotic guideways subjected to effectors performing technological processes, for example, cutting, bonding or painting. The control will be realized by a change of damping of the structure’s supports. The control objective will be to provide a smooth passage for vehicles and extend the time needed for the safety service of the carrying structures. For this purpose, we will formulate and solve a finite-time bilinear optimal control problem. Initially, the total deflection along the traveling load trajectory will be assumed as the metric to be optimized. Three additional metrics corresponding to the structural vibration amplitudes will later be used to examine the performance of the designed method. With respect to the previous research, the present paper contributes to the following directions. The model takes into account a time-varying speed of the traveling load, which is essential to control robotic guideways where the effector is subjected to acceleration and deceleration. The study of the optimal control problem introduces a sufficient condition for the existence of optimally switched controls. This condition provides a simple test for optimality that is of particular importance here since semi-active controlled systems often fail to outperform the corresponding passive cases. For the optimization, an efficient method of parametrized switching times is developed and tested via numerical examples. Finally, for the first time, the designed control strategy is validated on an experimental test stand.

The proposed control will be designed in an open-loop structure. An open-loop control system is, in general, less robust with respect to system parameters variation and external disturbances compared with a closed-loop control system (comprehensive comparison for the vibrating systems was presented in Casciati et al. (2006). Various methods can be employed to identify the mechanical parameters of the structure (see, for instance, Zhang and Jankowski, 2013). Nevertheless, precise estimation of the velocity of the moving load is not always feasible. Real-world applications of the proposed control may require the collection of several solutions for different profiles of passage velocities. In the cases where the speed profile cannot be predicted (such as for vehicles passing bridges), the developed method can be implemented in the model predictive controller allowing the adaptation of the optimal solutions to the measured parameters (see, for example, Bordons and Camacho, 1998). Thanks to the closed-loop structure, the model predictive controller can also improve the robustness with respect to external disturbances.

The rest of the paper is structured as follows. Section 2 introduces the mathematical model of the carrying structure subjected to the traveling load. In Section 3, the optimal control problem for straight-line load passage is formulated and the necessary optimality condition is given. A solution method based on parametrized switching times is introduced in Section 4. In Section 5, the performance of the designed control method is validated by means of experimental tests. Conclusions and future outlook are provided in Section 6.

2. Mathematical model

In this paper, we will consider a carrying structure supported with a set of controlled dampers, as illustrated in Figure 1. For the model of the span, we will use the Euler–Bernoulli beam equation, which is widely applied to thin elastic bodies subjected to small deflections (Gere and Timoshenko, 1997). A beam of total length l is characterized by its bending stiffness EI and density per unit length μ. Controlled supports are located at the positions a_i. For each of the supports, we assume a controlled damping u_i and/or a fixed stiffness k_i. The beam is subjected to a mass m traveling at a time-varying velocity v(t) that is assumed to be given. The position of the moving mass at time t is denoted by x_m(t) and computed by using

x m (t) = \int_{0}^{t} v (t) d t

(1)

The dimensions of the moving mass are assumed to be relatively small compared with the beam. As a result, at the time t, the interaction between the mass and the beam will be represented by a point contact and described by the Dirac delta function δ(x − x_m(t)). Similarly, we will use δ(x − a_i) to introduce the reaction of the support located at the position a_i.

Figure 1.

A span supported with a set of controlled dampers and subjected to a mass traveling at a time-varying speed v(t).

Let w(x, t) be the transverse deflection of the beam at the spatial coordinate x and time t. Then, the system is governed by

μ \frac{\partial 2 w (x, t)}{\partial t 2} + EI \frac{\partial 4 w (x, t)}{\partial x 4} = - \sum_{i = 1}^{m} (k i w (a i, t) + u i \frac{\partial w (a i, t)}{\partial t}) δ (x - a i) - m (g + \frac{\partial 2 w (x m (t), t)}{\partial t 2}) δ (x - x m (t))

(2)

The endpoint supports impose the following boundary conditions:

w (0, t) = 0, w (l, t) = 0, (\frac{\partial 2 w}{\partial x 2}) | x = 0 = 0, (\frac{\partial 2 w}{\partial x 2}) | x = l = 0

(3)

For the initial condition, we assume

w (x, 0) = 0, \overset{\cdot}{w} (x, 0) = 0

(4)

The left-hand side of (2) consists of the standard terms of the Euler–Bernoulli equation corresponding to the potential and inertial forces of the beam. The first two terms of the right-hand side stand for the reactions of the controlled supports. The last terms correspond to the excitation due to the moving load. For this excitation, we take into account both gravity and the inertial force. The latter is often ignored for large-scale structures. For systems where the masses of the span and the moving load are comparable (for instance, maglev trains, robotics), the inertial force of the moving mass plays a key role in the dynamics (see Dyniewicz and Bajer, 2010).

2.1. ODE representation

It can be verified (see Fryba, 1999) that for the assumed boundary conditions (3) the set of eigenfunctions for (2) is given by

θ \bar{j} (x) = \sin (\bar{j} π x / l), \bar{j} = 1, 2, \dots

(5)

By using the separation of space and time variables the solution to (2) can be represented by the following Fourier series:

w (x, t) = \frac{2}{l} \sum_{\bar{j} = 1}^{\infty} V \bar{j} (t) θ \bar{j} (x)

(6)

In this work, we will rely on the approximated solutions by taking $\bar{j}$ from 1 to n = 10. The assumed 10 modes are sufficient to render the dynamics of the structure while keeping the size of the system eligible for efficient optimization.

Throughout the rest of the paper, we will use the first-order ordinary differential equation (ODE) representation of the dynamics (2). First, we introduce the state vector $y \in ℝ 2 n$ defined as

y (t) = [V 1 (t), \overset{\cdot}{V} 1 (t), V 2 (t), \overset{\cdot}{V} 2 (t), \dots, V n (t), \overset{\cdot}{V} n (t)] T

(7)

Next, we introduce the following auxiliary matrices $\bar{A} (t)$ , $\bar{B}$ , $\bar{C} (t)$ and the vector $\bar{F} (t)$ :

Now let the matrices A(t), B(t), and the vector F(t) be defined by

A (t) = (I + \bar{C} (t)) - 1 \bar{A} (t), B (t) = (I + \bar{C} (t)) - 1 \bar{B}, F (t) = (I + \bar{C} (t)) - 1 \bar{F} (t)

(9)

Here $I$ stands for the identity matrix. Then, it can be verified (see the Appendix) that the system dynamics given by (2)–(4) can be represented by the following system of the ODEs:

\overset{\cdot}{y} = A (t) y + \sum_{i = 1}^{m} u i B i (t) y + F (t) y (0) = 0

(10)

For the controls, we assume that each one is bounded by two positive values, corresponding to the minimum and maximum admissible damping coefficients, i.e.

u \in U = [u min, u max] m \subset ℝ_{+}^{m}

(11)

3. The optimal control problem

Let T stand for the time of the moving load passage that is computed by using

l = \int_{0}^{T} v (t) d t

(12)

Throughout this paper, we will aim to find the controls that arrange that the passages of the load traveling over a structure are performed as close as possible to desired straight-line trajectory w_d(x_m(t),t) = 0 for all $t \in [0, T]$ . For that purpose, we shall now introduce an integral objective corresponding to the total vertical displacement of the moving load from the desired trajectory:

J = \int_{0}^{T} (w (x m (t), t) - w d (x m (t), t)) 2 d t = \int_{0}^{T} w 2 (x m (t), t) d t

(13)

Also of importance is the fact that the objective (13) captures the extremal amplitudes of the carrying structure, since we can expect that the maximum deflection lies on the trajectory w(x_m(t),t) (see Dyniewicz, 2012).

To rewrite our metric in state-space coordinates, let us define the vector

q (t) = \frac{2}{l} [θ 1 (x m (t)), 0, θ 2 (x m (t)), 0, \dots, θ n (x m (t)), 0] T

(14)

By introducing the matrix $Q (t) = q (t) q T (t)$ the objective (13) is represented by

J = \int_{0}^{T} y T Q (t) y d t

(15)

The problem of an optimal passage for the system (10) can be written as follows:

Find u * = argmin u \in U J = \int_{0}^{T} y T Q (t) y d t Subjectto \overset{\cdot}{y} = A (t) y + \sum_{i = 1}^{m} u i B i (t) y + F (t), y (0) = 0

(16)

3.1. First-order necessary optimality condition

To design an effective solution method for the problem (16), it is essential to first investigate the structure of the optimal controls. The problem is classified as a finite horizon bilinear optimal control problem. It has been studied extensively (see Mohler, 1973) that if the objective function is not explicitly dependent on the control and the set of admissible controls is bounded from above and below, then the solution to a bilinear optimal control problem is given by controls of the bang–bang type. This result follows directly from the Pontryagin maximum principle (Pontryagin et al., 1962).

By introducing the adjoint state p and the Hamiltonian

H (y, p, u) = p T (A (t) y + \sum_{i = 1}^{m} u i B i (t) y + F (t)) - y T Q (t) y

(17)

the Pontryagin maximum principle states that the optimal controls for the problem (16) satisfy

u * = argmax u \in U H (y, p, u)

(18)

Here the adjoint state is given by

\overset{\cdot}{p} = - \frac{\partial H}{\partial y}, p (T) = 0

(19)

From (18) we immediately obtain the optimal controls

u_{i}^{*} = {u max, p T B i (t) y \geq 0 u min, p T B i (t) y < 0

(20)

Naturally, to determine the trajectories of u^*, one has to solve the following two-point boundary value problem:

\overset{\cdot}{y} = A (t) y + \sum_{i = 1}^{m} u_{i}^{*} B i (t) y + F (t), y (0) = 0 \overset{\cdot}{p} = - A T (t) p - \sum_{i = 1}^{m} u_{i}^{*} B_{i}^{T} (t) p + 2 Q (t) y, p (T) = 0

(21)

3.2. Existence of switched optimal controls

The solution given by (20) defines the bang–bang structure of the optimal controls. However, it does not provide any information on the number of switches. It does not even justify whether we can expect any switches. Note that our optimal control problem is defined on a time horizon dependent on the speed of the moving load. In the case of a fast passage, this horizon becomes relatively short. There are many vibrating systems where the short time-horizon semi-active optimal policy is to set the damping to the maximum admissible value. The reader can consider a semi-active controlled oscillator subjected to a harmonic force as a good example. For a wide range of initial conditions, the minimization of energy-like functions leads to the constant maximum admissible damping, unless the time horizon is sufficiently long. Our goal now is to provide a condition that, for a given set of system parameters, justifies whether the optimal solution (20) exhibits a switched structure or is constant at the maximum admissible value. This condition is intended to serve as a test before performing the optimization procedures.

For simplicity, let us consider the system (10) with only one control, i.e.

\overset{\cdot}{y} = A (t) y + uB (t) y + F (t), y (0) = 0

(22)

The corresponding adjoint state is

\overset{\cdot}{p} = - A T (t) p - uB T (t) p + 2 Q (t) y, p (T) = 0

(23)

The proposition given below establishes a sufficient condition for the existence of a switched control u^*≠ u_max that results in a more beneficial value of the objective function (15).

Proposition 1

Let $y u max (t)$ and $p u max (t)$ be the solutions of the state (22) and the adjoint state (23) when the following constant control function is given: $u (t) = u max$ , for all $t \in [0, T]$ . If there exists an interval $[t 1, t 2] \subseteq [0, T]$ such that for all $t \in [t 1, t 2]$ we have $p_{u max}^{T} B (t) y u max < 0$ , then there also exists a control $u * \in U$ , $u * \neq u max$ such that $J (u *) < J (u max)$ .

Proof

Let u* = u_max + δu. Then, we can write the differential of the objective function as

J (u max + δ u) - J (u max) = δ J (u max) (δ u) + r J (u max, δ u)

(24)

where δJ(u_max)(δu) is the first variation of J(u_max) and r_J(u_max,δu) = o(δu), i.e. r_J(u_max,δu)/∥δu∥→ 0 as ∥δu∥→ 0. For a sufficiently small δu the sign of the right-hand side of (24) depends on the sign of the variation. Therefore, we need to prove that δJ(u_max)(δu) < 0.

By introducing the Hamiltonian

H (y, p, u) = p T (A (t) y + uB (t) y + F (t)) - y T Q (t) y

(25)

the objective function (15) can be represented by

J = \int_{0}^{T} (p T \overset{\cdot}{y} - H) d t

(26)

An infinitesimal change δu causes variations of the functions δy, δ $\overset{\cdot}{y}$ , δp. This results in the following variation of the objective function:

δ J (u) (δ u) = \int_{0}^{T} (- \frac{\partial H}{\partial u} δ u - (\frac{\partial H}{\partial y}) T δ y) d t + \int_{0}^{T} (p T δ \overset{\cdot}{y} + (\overset{\cdot}{y} - \frac{\partial H}{\partial p}) T δ p) d t

(27)

Here the last term vanishes, since

\overset{\cdot}{y} = \frac{\partial H}{\partial p}

(28)

Under the assumption

δ \overset{\cdot}{y} = \frac{d}{dt} (δ y)

(29)

an integration by parts yields

δ J (u) (δ u) = \int_{0}^{T} - \frac{\partial H}{\partial u} δ u d t - \int_{0}^{T} (\overset{\cdot}{p} + \frac{\partial H}{\partial y}) T δ y d t + [p T δ y]_{0}^{T}

(30)

Here the second and the last term vanish, since we have

\overset{\cdot}{p} = - \frac{\partial H}{\partial y}, p (T) = 0, δ y (0) = 0

(31)

Thus,

δ J (u) (δ u) = - \int_{0}^{T} p T B (t) y δ u d t

(32)

Now, let the variation of the control be given by

Then $u * \in U$ . For such a control, we conclude that

forall t \in [t 1, t 2] p_{u max}^{T} B (t) y u max < 0 \Rightarrow δ J (u max) (δ u) = - \int_{t 1}^{t 2} p_{u max}^{T} B (t) y u max ε d t < 0

(34)

which completes the proof.

Proposition 1 can be easily generalized to a system with multiple inputs.

4. Solution method: optimal switching times approach

As demonstrated in the previous section, the optimality condition for the problem (16) consists of a two-point boundary value problem. To solve this problem, shooting (Stoer and Bulirsch, 1980) and relaxation (Press et al., 1992) methods can be applied. In the shooting method, the idea would be to find such an initial condition for the adjoint state that has its zero terminal condition fulfilled. The relaxation method implements another approach. The time domain is represented as a set of points creating a mesh. The system’s dynamics is represented by finite difference equations. An iterative procedure is to adjust all the state and adjoint state values on the mesh to bring them into successively closer agreement with the finite-difference equations together with the boundary conditions. In many cases, shooting and relaxation methods are combined. Both methods exhibit good performance in the case of low-dimensional problems, except when the solutions are highly oscillatory or not smooth.

In the case of more complex systems, we can expect difficulties in obtaining accurate, switching-shaped, numerical solutions. Increasing the precision of the calculations is associated with a higher dimensional optimization problem, which results in the rapid extension of the time required for the computations. There is a need to use a more efficient numerical algorithm for computing the optimal switching controls. For this purpose, it is very intuitive to parametrize the switching times and reformulate the problem (16). The objective function will be optimized with respect to new parameters: the switching times. This approach transforms the optimal control problem to a nonlinear programming problem, where gradient-based methods can be applied. To obtain the gradient, we will use fundamental facts from the calculus of variations as well as the properties of the Dirac delta function. In what follows, a complete computational algorithm will be given. A numerical example will examine the convergence and the computational burden of the algorithm.

For simplicity, let us first consider the system (10) driven by only one control consisting of one switch from the minimum to the maximum admissible value

u = u min + (u max - u min) U (t - τ), τ \in [0, T]

(35)

Here, $U$ stands for the unit step function. The dynamics is now represented by

\overset{\cdot}{y} = A (t) y + (u min + (u max - u min) U (t - τ)) B (t) y + F (t), y (0) = 0

(36)

By defining the Hamiltonian

H (y, p, τ) = p T (A (t) y + F (t)) + p T (u min + (u max - u min) \times U (t - τ)) B (t) y - y T Q (t) y

(37)

the objective function (15) takes the form

J = \int_{0}^{T} (p T \overset{\cdot}{y} - H) d t

(38)

An infinitesimal change dτ causes variations of δy, δ $\overset{\cdot}{y}$ , δp. This results in the following variation of the objective function:

δ J (u) (δ τ) = \int_{0}^{T} (- \frac{\partial H}{\partial τ} δ τ - (\frac{\partial H}{\partial y}) T δ y) d t + \int_{0}^{T} (p T δ \overset{\cdot}{y} + (\overset{\cdot}{y} - \frac{\partial H}{\partial p}) T δ p) d t

(39)

By implementing steps in analogy to (28)–(31), the variation (39) is represented by

δ J (u) (δ τ) = - \int_{0}^{T} \frac{\partial H}{\partial τ} d τ d t

(40)

This implies that the derivative of the objective function with respect to the switching time is

\frac{\partial J}{\partial τ} = - \int_{0}^{T} \frac{\partial H}{\partial τ} d t

(41)

Insertion of the Hamiltonian (37) results in

\frac{\partial J}{\partial τ} = - (u max - u min) \int_{0}^{T} p T B (t) y \frac{\partial U (t - τ)}{\partial τ} d t

(42)

Finally, we get

\frac{\partial J}{\partial τ} = (u max - u min) \int_{0}^{T} p T B (t) y δ (t - τ) d t = (u max - u min) p T (τ) B (τ) y (τ)

(43)

For a control with the reversed switching action, i.e.

u = u max + (u min - u max) U (t - \bar{τ}), τ \in [0, T]

(44)

the reader can verify that the gradient of the objective function with respect to the switching time

\bar{τ}

is given by

\frac{\partial J}{\partial \bar{τ}} = (u min - u max) p T (\bar{τ}) B (\bar{τ}) y (\bar{τ})

(45)

Before we develop the computational algorithm, let us first introduce the relevant settings. For each of the controls, we assume z switching actions from the minimum to the maximum admissible value [u_min] → [u_max] and z switching actions from the maximum to the minimum admissible value [u_max] → [u_min]. The switching times will be collected into two matrices: $τ = [τ i, j] m \times z$ , $\bar{τ} = [\bar{τ} i, j] m \times z$ . Here τ and $\bar{τ}$ consist of the times of the actions [u_min] → [u_max] and [u_max] → [u_min], respectively. ${τ i, j}$ and ${\bar{τ} i, j}$ are increasing sequences with respect to j, where for every pair (i,j) we have τ_i,j ∈ [0,T) and $\bar{τ}$ _i,j ∈ (0,T]. Moreover, we assume that τ_i,j < $\bar{τ}$ _i,j for all i,j. In the algorithm, we will use the following dynamics for the state:

\overset{\cdot}{y} = A (t) y + u min \sum_{i = 1}^{m} B i (t) y + (u max - u min) \times \sum_{i = 1}^{m} \sum_{j = 1}^{z} (U (t - τ i, j) - U (t - \bar{τ} i, j)) B i (t) y + F (t), y (0) = 0

(46)

and for the adjoint state,

\overset{\cdot}{p} = - A T (t) p - u min \sum_{i = 1}^{m} B_{i}^{T} (t) p - (u max - u min) \times \sum_{i = 1}^{m} \sum_{j = 1}^{z} (U (t - τ i, j) - U (t - \bar{τ} i, j)) B_{i}^{T} (t) p + 2 Q (t) y, p (T) = 0

(47)

The computational algorithm based on the steepest descent gradient method consists of the following steps.

Step 1. Set α to be a small positive number and guess the initial matrices [τ_i,j] and [ $\bar{τ}$ _i,j ].

Step 2. Solve the state equation (46) by substituting [τ_i,j] and [ $\bar{τ}$ _i,j ].

Step 3. By backward integration solve the adjoint state equation (47) by substituting [τ_i,j], [ $\bar{τ}$ _i,j ] and y.

Step 4. By using (43) and (45) compute the derivatives for all components of the switching time matrices.

Step 5. Update the switching times by using the formulas

τ_{i, j}^{+} = τ i, j - α \frac{\partial J}{\partial τ i, j}, {\bar{τ}}_{i, j}^{+} = \bar{τ} i, j - α \frac{\partial J}{\partial \bar{τ} i, j}

Optionally, a line search method can be implemented here to provide the optimal decrease of the value of the objective function.

Step 6. Check whether the switching times τ_i,j or $\bar{τ}$ _i,j extend their limited values 0 or T, respectively. If so, then set these times to the appropriate infinium or supremum of the set [0,T] and then go to Step 2.

Step 7. Check whether the length of any of the intervals [τ_i,j, $\bar{τ}$ _i,j] approaches zero. If so, discard those switching times, re-size the matrices [τ_i,j], [ $\bar{τ}$ _i,j ], and go back to Step 2.

Step 8. Repeat Steps 2–7 until a terminal condition (based, for example, on the norm of the gradient) is fulfilled.

4.1. Numerical example

Before we proceed to the experimental validation of the control method designed, let us first briefly investigate the computational aspects of the algorithm developed. Namely, we are interested in the performance and the convergence of the gradient-based procedure. The optimization will be executed for the system parameters that correspond to the test stand presented in the next section.

The optimization algorithm was implemented in the MATLAB programming language. For the integration of the state and adjoint state dynamics, the fourth-order Runge–Kutta scheme was employed. At each of the optimization loops, the parameter α (see Step 5 in the algorithm) was taken so that the updated switching times could move only one sample time forward or backward. The procedure was terminated when each of the switching times started oscillating between two neighboring samples.

In the example, we consider four controls associated with four controlled dampers. For each control we assume two switching actions. The first action, denoted by τ_i,₁ (i stands here for the damper i), switches the damping from u_min to u_max. The second, denoted by $\bar{τ}$ _i, ₁, switches from u_max to u_min. The time interval [0,T] was discretized into 2000 time samples where samples no. 1 and no. 2000 correspond to t = 0 and t = T, respectively. The optimization was initiated with the following values: sample no. 200 for the switches τ_i,₁, and sample no. 1800 for the switches $\bar{τ}$ _i, ₁. After 931 iterations (126 seconds performed with the use of an Intel Quad-Core processor) the optimization was terminated. The evolution of the switching times during the optimization process is demonstrated in Figure 2 (see also Figure 3 for the evolution of the value of the objective function). The last switching time that met the terminal condition was τ₄ _, ₁. The optimal switching times are denoted by bullets (the numerical values are summarized in Table 1). The structures of the optimal controls are illustrated in the top of Figure 2. The practical interpretation of the results is that damper no. 1 is supposed to operate permanently at its maximum value. The remaining dampers are supposed to be switched twice within the considered period.

Figure 2.

Evolution of the switching times during the optimization process. The optimal values are indicated by bullets. The structures of the optimal controls are given at the top.

Figure 3.

Evolution of the value of the objective function during the optimization process.

Table 1.

The optimal switching times for the numerical example.

Damper	Switch from u_min to u_max [Time sample]	Switch from u_max to u_min [Time sample]
1	0	2000
2	240	1808
3	630	1678
4	966	1958

The increased number of switching actions resulted in a minor improvement of the value of the objective function. The problem was also solved assuming four and six switches. The results are summarized in Table 2. We observe a 2% improvement of the objective value while the number of required iterations doubles. It is worth noting that in the case of six switches, some of them were canceled when performing the algorithm. Aiming at practical simplicity, the experimental test will be performed under the assumption of two switches for each of the controls.

Table 2.

The values of the objective function and the number of iterations performed. The objective values are normalized to the case of two switches.

Number of switches	Objective value	Number of iterations
2	1.0000	931
4	0.9824	1546
6	0.9798	2012

5. Experimental validation

In this section, the designed control method will be examined by means of experimental results. We will investigate the performance for different passage speeds. For that purpose, we will evaluate four different metrics, corresponding to the state of the carrying structure and the trajectory of the moving load. In terms of these metrics, we will compare the system driven by optimal switched controls with one that operates permanently at the maximum admissible damping.

5.1. The test stand

A real view of the test stand is shown in Figure 4. The supporting structure was made of an aluminum truss frame. The carrying structure is a guideway supported with two springs that reduce the static deflection. For the traveling load we use a carriage powered by an electric motor. For the controlled devices we use four magneto-rheological rotary dampers equipped with encoders. The locations for the dampers are a₁ = 1/5l, a₂ = 2/5l, a₃ = 3/5l and a₄ = 4/5l. During the passage, the carriage first rapidly accelerates from zero to a given constant speed v_max (see Figure 5). Then, this speed remains constant until the last stage of the passage (corresponding to approximately 4/5 of the length of the guideway), when the motor starts its braking process. For both the acceleration and the deceleration, we use the maximum admissible values of approximately 6 m/s². The assumed speed trajectory is typical for many manufacturing processes engaging robotic technology.

Figure 4.

Real view of the test stand (on the left) and the rotary controlled damper supporting the guideway (on the right).

Figure 5.

The shape of the velocity of the moving load used in the experiments.

In order to design the parameters for the test stand, a set of numerical simulations were first performed. Custom software simulates the dynamic interactions between the traveling carriage and the whole structure. Various combinations of the support rails and guideways were considered. For the notable deflection of the beam, the pivotally mounted guideway without the supporting rail was selected. This results in the maximum transverse displacement being within the range of ±30 mm for a mass of 5 kg traveling at a speed of 3 m/s. The carriage is attached to the guideway with a special type of conical rollers, which protect it from jamming in the case of traveling over large deflections. The carriage is powered by a stepper motor via a toothed belt girded over a gearbox. Our setup enables accelerating a mass of 6 kg to a speed of 5 m/s and stopping it within 4 m. The major parameters of the test stand are given in Table 3.

Table 3.

Parameters of the test stand.

Guideway length (l)	4 m
Guideway stiffness (EI)	801 Nm²
Guideway density per unit length (μ)	2.3 kg/m
Damping range (u_min–u_max)	200–2000 Ns/m
Spring stiffness (k₁, k₂)	1000 N/m
Mass of carriage (m)	0.7–10 kg
Maximum motor torque	21 Nm
Maximum speed of carriage	6 m/s
Maximum acceleration of carriage	6 m/s²

Table 4.

The values of the metric J₁ normalized to the optimal passive cases.

v_max (m/s)	Optimal passive strategy	Optimal switched strategy
1	1.0000	0.8771
3	1.0000	0.7756
5	1.0000	0.5909

The control system is based on a PC equipped with an I/O data acquisition cards and rotary magneto-rheological dampers. The computed optimal switched control signals are first magnified by an amplifier to provide an operating current of 0.1–1 A (the nominal current range for the employed dampers: LORD TFD steering units). These signals sent to the dampers result in a damping coefficient of approximately 200–2000 Ns/m. The control process is triggered by the motor encoder’s indicating the start of the passage. The control signals are sent to the dampers every 5 ms. All switches are shifted backwards in time by 30 ms to compensate for the response time of the dampers. During the process, the encoders incorporated into the dampers measure the local transverse deflections, which are then collected by the acquisition card.

Figure 6.

Moving load trajectories in the case of v_max = 1 m/s. Comparison of the model-based and experimental results.

Figure 7.

Moving load trajectories in the case of v_max = 3 m/s. Comparison of the model-based and experimental results.

To validate the parameters of the model used in the optimization (see the model given by (58)), some numerical simulations were compared with the experimental results. The comparative data was obtained for of the speed range of v_max = 1–5 m/s, where a constant damping of 2000 Ns/m was assumed. To reconstruct the experimental trajectories, the first four terms of the series (6) were taken into account. The limitation to four terms is due to the number of available local measurements w(a_i,t) provided by the encoders incorporated in the dampers. To investigate how this truncation may impact the shapes of the moving load trajectories, some numerical simulations were performed taking into account four, six, eight and ten modes of the series (6). It was observed that for the assumed structural parameters and the traveling speed of v_max = 1–5 m/s, the dynamics is smooth and the moving load trajectories can be fairly represented by the first four modes. An increased number of terms in the series did not produce any significant changes. For a high traveling speed (v_max > 10 m/s), the trajectories started to involve angular shapes, in particular, at the points where the moving load was passing by a damper performing a switching action. For such cases, one should consider using the first seven to ten modes in the numerical model and the corresponding number of sensors for the experimental study. The designed optimization procedure was implemented with the first ten modes in the numerical model so that it may be used for a wide range of traveling speeds. The same number of modes was assumed for the comparative results presented below.

The experimental and numerical shapes of the moving load trajectories are compared in Figures 6–8. For each of the cases, the experimental result coincides with its corresponding model-based simulation. In each case, the relative error for the maximum deflection is less than 5%.

Figure 8.

Moving load trajectories in the case of v_max = 5 m/s. Comparison of the model-based and experimental results.

5.2. Experimental results

The experiments were conducted using the following parameters: the mass of the carriage was m = 5 kg; the different passage speeds had maximum values of v_max = 1 m/s, v_max = 3 m/s and v_max = 5 m/s. The performance of the control method will be examined by the following metrics:

J 1 = \int_{0}^{T} w 2 (x m (t), t) d t, J 2 = max_{t \in [0, T]} | w (x m (t), t) |, J 3 = \int_{0}^{l} \int_{0}^{T} w 2 (x, t) d x d t, J 4 = max_{x \in [0, l], t \in [0, T]} | w (x, t) |

(48)

Here J₁ is the optimized objective (13). All implemented optimal controls are based on this metric. For comparative results, we will verify the values of J₂, denoting the maximum deflection over the trajectory of the carriage, as well as of J₃ and J₄, corresponding, respectively, to the total and the maximum deflection of the guideway. Despite the fact that the experiments exhibited good replicability (with a maximum difference of 4% for the assumed metrics), each of the passages was repeated five times, and the values of the metrics were averaged. For each passage, we applied two damping strategies. The first, referred to as the optimal passive strategy, employed constant damping, where all controls were set to the maximum admissible value u_max. We shall demonstrate that among the admissible set of damping coefficients [u_min,u_max], the value u_max provides the best performance of passive damping. The second strategy, referred to as the optimal switched strategy, was based on the optimal switched solutions.

The typical shape of the deflected beam subjected to a moving load is shown in Figure 9. In this example, the passage at speed v_max = 3 m/s was performed under the passive strategy. In this case, the maximum deflection takes place at t ≈ 0.8 T . By increasing the traveling speed, we can observe that the maximum deflection shifts toward the end of the passage.

Figure 9.

Guideway deflection in space and time in the case of v_max = 3 m/s under the passive strategy.

The first set of experiments was performed to establish the optimal passive strategy to serve as a comparative result to justify the effectiveness of the designed optimal strategy. In Figure 10, the moving load trajectories are presented in the case of v_max = 3 m/s and constant damping of u = 200 Ns/m, u = 500 Ns/m, u = 1000 Ns/m, u = 1500 Ns/m and u = 2000 Ns/m. We can clearly observe that for the straight line passage the best performance is exhibited in the case of the maximum admissible value u_max = 2000 Ns/m, and this case was taken for the optimal passive strategy.

Figure 10.

Moving load trajectories in the case of passive damping and v_max = 3 m/s. The best performance is provided by the maximum admissible damping u_max = 2000 Ns/m.

Table 5.

The values of the metric J₂ normalized to the optimal passive cases.

v_max (m/s)	Optimal passive strategy	Optimal switched strategy
1	1.0000	0.9512
3	1.0000	0.9123
5	1.0000	0.8056

Table 6.

The values of the metric J₃ normalized to the optimal passive cases.

v_max (m/s)	Optimal passive strategy	Optimal switched strategy
1	1.0000	0.8522
3	1.0000	0.7689
5	1.0000	0.5630

The comparison of the moving load trajectories under the optimal passive and the optimal switched strategies in the case of v_max = 3 m/s is shown in Figure 11 (for the optimal controls in this case see Figure 12). Under the optimal passive damping, the maximum deflection exceeds 28 mm. Under the optimal switched control, this deflection is reduced by 9%. For comparison of the passive strategies in the case of v_max = 5 m/s see Figure 13. As in the previous case, for the optimal passive strategy we assume u_max = 2000 Ns/m. The moving load trajectories under the optimal passive and the optimal switched strategies are depicted in Figure 14. The corresponding optimal controls are depicted in Figure 15. Again, the optimal switched strategy outperforms the best passive case. In terms of the maximum deflection, the improvement is now 20%. For the passages at speed v_max < 3 m/s, we can observe a tendency of a loss of efficiency of the designed control method. This is due to the fact that for low travel velocity, the system operates quasi-statically, and the effect of switched damping becomes negligible. The trajectories of the passive and optimally controlled systems at the speed v_max = 1 m/s are almost identical. On the other hand, in the case of fast passages, we can notice a very high efficiency. The experimental stand does not enable reaching v_max > 5 m/s, but the numerical simulation shows that at the speed v_max = 10 m/s, by using the optimal switched strategy, the maximum deflection can be reduced by over 45% and the total deflection by over 70%.

Figure 11.

Comparison of the moving load trajectories under optimal passive and optimal switched strategies in the case of v_max = 3 m/s.

Figure 12.

The optimal controls in the case of v_max = 3 m/s.

Figure 13.

Moving load trajectories in the case of passive damping and v_max = 5 m/s. The best performance is provided by the maximum admissible damping u_max = 2000 Ns/m.

Figure 14.

Comparison of the moving load trajectories under optimal passive and optimal switched strategies in the case of v_max = 5 m/s.

Figure 15.

The optimal controls in the case of v_max = 5 m/s.

The set of values for the metrics defined in (48) is presented in Tables 4–7. We can clearly observe the correlation between all metrics. In particular, the maximum deflection along the trajectory of the traveling carriage J₂ is almost identical with the maximum deflection of the guideway J₄. The total deflections represented by J₁ and J₃ also coincide. Thus, the selection of (13) to be optimized is justified.

Table 7.

The values of the metric J₄ normalized to the optimal passive cases.

v_max (m/s)	Optimal passive strategy	Optimal switched strategy
1	1.0000	0.9479
3	1.0000	0.9108
5	1.0000	0.7974

Regarding the structures of the optimal controls (see Figures 12 and 15) one can notice two practical features. The first is that in both cases the damper placed on the left-hand side operated at the maximum damping for almost all the time. A similar conclusion was made based on the numerical simulations. Thus, in the practical design, one can consider replacing a semi-active damper located on the side where the load starts its travel, with a passive one. The second feature of the structures of the optimal controls is the pattern that each damper needs to be switched to the maximum admissible value just before being approached by the load, while the switch back to the minimum value is supposed to be done after that passage. In other words, the optimal trajectory of the traveling load is provided when the maximum damping follows the load. The last statement need to be treated roughly since each traveling speed requires slightly different timing for dampers’ activation and deactivation. Nevertheless, the second feature suggests the possibility of approximating the optimal policy by using a simple and practical state-feedback control law. To do so, one can consider introducing the threshold values for vertical deflection (or velocity). Based on the fact that the deflection (or velocity) reaches the highest absolute values at the position under the traveling load, the switching policy employing the threshold values could activate and deactivate the dampers at required time instants. The key to reaching the desired performance would not only be in the selection of the threshold values but also in the design of a structure of the state-feedback control. The presented idea is out of the scope of this paper and is dedicated for future works.

6. Conclusion and future extensions

The problem of the optimal control of a structure subjected to a moving load has been studied. For the assumed objective of a straight-line passage, the optimal controls are of the bang–bang type. A simple test based on the integration of the state and adjoint dynamics justifies whether, for a given set of parameters, a switched control strategy can outperform a passively damped system. The developed method of parametrized switching times can efficiently handle the posed problem. A numerical study has shown that even under two switches per control, the optimal solutions exhibit high performance. The validation of the optimal controls was investigated on a real test stand. The experiments were performed for three different passage speeds. In each case, the controlled system outperformed its passively damped equivalent. With respect to the optimized metric, the improvement reached over 40%. The observation made in Pisarski and Bajer (2010) and Pisarski and Bajer (2011) that an increased speed of travel results in a better efficiency of the switched control strategy has also been verified. The experimental test stand does not allow for passages with speeds over 5 m/s; however, numerical simulations demonstrated that for a moving load traveling at 10 m/s, the value of the objective function can be reduced by over 70%.

The on-going research of the present author is focused on the problem of the real-time adaptation of the optimal controls according to the measured passage speed. This problem is of special importance for transport where the precise velocity profiles of the vehicles cannot be predicted.

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: The research leading to these results has received funding from the Foundation for Polish Science (grant agreement HOMING PLUS/2013-8/11).

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