Abstract

Control of Structures with Cubic and Quadratic Non-Linearities with Time Delay Consideration

O. C. Pinto

Technical School

Fundação Universidade Federal do Rio Grande

96201-900 Rio Grande, RS. Brazil ctiosval@super.furg.br

P. B. Gonçalves

Departamento de Engenharia Civil

Pontifícia Universidade Católica - PUC-Rio

22453-900 Rio de Janeiro, RJ. Brazil

paulo@civ.puc-rio.br

Introduction

In recent years, considerable attention has been paid to active structural control research. Over the last two decades, remarkable progress has been made in research on using active and hybrid systems as a means of structural protection against wind, earthquakes and other hazards (Soong, 1990, Spencer Jr 1996). Non-linear structural control has a more recent history and some interesting strategies have been proposed lately (Yang et al. 1988 and 1994). The investigation of time delayed actively controlled structural systems is also relatively recent and the importance of considering its effects in the design has been analyzed by Roorda (1980), Abdel-Rohman (1987) and Agrawal and Yang (1997), among others.

In a paper presented in COBEM97, the authors of this paper presented a methodology for the non-linear active control of flexible structures, in order to limit the amplitude of oscillations within safe allowable bounds (Pinto and Gonçalves, 1997). The adopted control method was based on non-linear optimal control, using an indicial formulation and state feedback control (Tomasula et al., 1996, Pinto and Gonçalves, 1999), assuming an idealized system, without any delay between the moment that the control force is assumed to be applied to the structure and the moment that it is actually applied. But in real systems there is an unavoidable delay. Actually, time delay is one of the main issues concerning to real time active structural control and it emerges due to the time needed to data acquisition and conditioning, computing the required control forces, generating and transmitting the signal to the actuators, and applying control forces to the structure (reaction time of actuators). The magnitude of the time delay is expected to decrease as more advanced control system software and hardware become available, but, even with the advances of technology, time delay can not be eliminated, only minimized. Therefore, it is an intrinsic parameter and should be considered in the project of active control systems. Time delay induces a phase shift, which may degrade the performance of the control system and, if it is not handled properly, can not only render it ineffective but also cause instability of the controlled system. In this paper, the effect of time delay on the controlled system is studied. Numerical studies for a single degree of freedom system with quadratic and cubic non-linearities under state feedback control and different kind of loading are presented and analyzed. Results emphasize the importance of considering time delay in the project of active structural control systems.

Nomenclature

A = coefficients related to system's properties

= coefficient that relates u^j with xⁱ

F₀ = force amplitude

I = moment of inertia

J = performance index

K = gain

Q = weighting tensors related to system response

R = weighting tensors related to control effort

t = time

u = control force

V = Taylor series expansion of the performance index

x = displacement

= velocity

= acceleration

x^I = state variable

Greek Symbols

a = quadratic nonlinear coefficient

b = cubic nonlinear coefficient

g = coefficient of the control force

f = small rotation around an equilibrium position

m = damping coefficient

Subscripts and Superscripts

0 = initial

f = final

a, b, i, j, k, l, m = tensor indices

T = transpose

Problem Formulation - Control Strategy

Using an indicial formulation of tensor algebra (Suhardjo et al. 1993, Pinto & Gonçalves, 1998) the state equations of a certain class of nonlinear autonomous controlled systems can be expressed in the polynomial form

where xⁱ is the i-th state variable, are coefficients related to system's properties, u^j is the j-th control force and is a coefficient that relates u^j with xⁱ. Here we define x^ij = xⁱx^j, x^ijk = xⁱx^jx^k, and so on.

Assuming state feedback, the control forces can be expressed as

where are the i-th gains of control order 1, 2, 3... respectively.

The adopted performance index has the general form

where t₀ is the initial time, usually set to zero, t_f is the final time, Q_ij, Q_ijk, Q_ijkl, ... are symmetric positive semi-definite tensors with order 2, 3, 4... respectively, and R_ij is a symmetric positive definite tensor of second order. Q and R are weighting tensors, with elements chosen depending on the relative importance attributed to state variable bounds and to control forces. High values of Q_ij, Q_ijk, ... stress the reduction of system response, while high values of R_ij result in less control effort (less energy consume). Choosing these values appropriately one can get a control as efficient as possible without great energy consumption.

Expressing the performance index by the Taylor series

where V_ij, V_ijk, V_ijkl, etc are symmetric related to their indices, one can obtain by a minimization procedure the following Hamilton-Jacobi-Bellman equation

where J is given by Eq. 4 and

Manipulating Eq. 1 and 4 to 7, the first order control gain and the corresponding control equation, known as the Riccati equation, can be obtained:

where sym is the symmetry operator, defined in a way that when acting on a tensor T represents its symmetric form with respect to the free indices, i.e.

Once V_ij is obtained solving Eq. 9, one can get the first order control gains from Eq. 8. So the control forces can be computed with Eq. 2, resulting in the classical linear optimal control formulation. Similarly, we can get the equations for the second, third, fourth and fifth orders control, as shown by the following equations:

a) second order:

b) third order:

c) fourth order:

d) fifth order:

and so on, up to the desired order. These equations have a well-defined pattern, in a way that one can get higher order control equations without difficulty.

In order to get the tensors V, one has to solve numerically a set of differential equations, with a certain computational cost. However, in most structural problems, t_f is much longer than the natural period and setting it to ¥ doesn't change considerably the results and simplify significantly the problem, converting the differential equations into algebraic ones. In this work we derived equations for control up to the fifth order using this methodology.

Single Degree of Freedom Nonlinear Systems

In this section, the application of the strategy presented in the previous section for single degree of freedom (sdof) systems with quadratic and/or cubic nonlinearities is studied. These models are capable of representing approximately, at least in a qualitative way, most of the elements usually used in civil and mechanical engineering structures, such as beams, plates, shells and arches. The equation of motion of such sdof nonlinear controlled autonomous systems can be expressed as

where x is the displacement, is the velocity, is the acceleration, m is the damping coefficient, w is the natural frequency of the system, a and b are the quadratic and cubic nonlinear coefficients, respectively, and g is the coefficient of the control force u.

Using the formulation presented in the previous section, the state equations can be expressed as in Eq. 1 with the state variables x¹ = x and x² = , and the control force u¹= u, here a scalar. In this case, the only non-zero coefficients are = -w² , = 1 , = -2m , = -a , = -b and = -g .

Assuming the performance index

the first order control gains for displacement and velocity are given by

that are the same obtained using the algebraic Riccati equations.

Using these results, the second order control gain for displacement takes the form

Control gains of higher order can be obtained in a similar way. Here we developed a computer code capable of generating the gains up to the fifth order.

It is interesting to note that when a is null we have the well known Duffing equation, object of study of a number of nonlinear control works (Hackl et al. 1993, Cheng et al. 1993, Cui et al. 1997, Yabuno, 1997). In such case, the second order and forth order gains are null, and the third order gain for displacement is given by

It is interesting to notice in equations (21) to (22) that, more important than the individual values of Q_ij and R_ij in the weighting matrices, are the quotients of (Q_ij /R_ij).

Numerical Example

In this section the control strategy presented in the previous sections is applied to the problem of a very shallow pressure loaded spherical cap, firstly for the system without any delay and afterwards considering a time delay for the application of the control force. Unit values for g and R₁₁ were adopted, and in Q only Q₁₁ is not null.

For such a problem, the first mode is dominant and a simplified one-degree-of-freedom model is capable of describing, with a reasonable degree of accuracy, the nonlinear behavior of the cap (Gonçalves, 1994). The same pressure loaded thin-walled spherical shell presented in Gonçalves (1994) is considered. The dimensionless sdof equation of motion modeling the vertical displacement w is given by

This system has a two-well potential function, with two stable equilibrium states at w = 0 (the reference state) and at w = 1.92. The final state of the systems depends on the initial conditions and load characteristics, as can be observed in Fig. 1, where the free vibration response of the beam is shown for two different sets of initial conditions, w(0) = 0.70 and w(0) = 0.75, both with (0) = 0. While one response converges to a pre-buckling configuration, the other converges to a post-buckling configuration.

For a harmonic excitation F(t) = F₀ sin(12t), depending on the force amplitude F₀, the response will be attracted to one of the two existent potential wells, corresponding to pre and post-buckling equilibrium positions. For the system without control, the escape load is 10.76, as can be seen in Fig. 2. For a step load F(t) = F₀, the cap jumps to the second potential well when F₀ reaches the escape load, here 26.87, as can be seen in Fig. 3.

Controlled System without Time Delay

The introduction of an adequate control system can avoid the escape. Figure 4 shows the response of the structure without any control and with controls of order two and three (Q₁₁ = 500), respectively, for a load amplitude of 15, about 40% higher than the escape load without control. One can see that only a third order control (or higher) can avoid the failure of the structure. Higher order algorithms can be more efficient, resulting in smaller displacements without increasing the energy consumption. The peak control force for the third order algorithm is 7.41, very small if compared to the static critical load.

Table 1 shows how the escape load is modified by the order of the control system adopted, compared with the escape load of the system without any control (F₀ = 10.76).

One can see that a higher order control system results in higher escape loads, enlarging the safe working capacity of the structure without demanding great control forces.

The same system was also subjected to step loading of infinite duration, F(t) = F₀, resulting in an escape load of 26.87. Figures 5 and 6 show, respectively, the time response and control forces demanded for a step load of 35, more then 30% bigger than the escape load for the structure without control. Here Q₁₁ = 3000.

For this load level, a control algorithm of order less than three is not able to avoid the escape of the cap. Here, one can see that higher order control algorithms are more efficient in reducing the response of the structure with control forces of the same order of magnitude. It is important to note that the maximum control force required is only about 5% of the static pressure load used in this example and 4% of the static critical load.

As in the case of the harmonic loading, the control system brings an increase of the escape load, enlarging the safe working capacity of the structure. For the fifth order control algorithm, for example, we have an escape load 40.12% higher than that without control, as can be observed in Table 2.

Controlled System with Time Delay

In this section a time delay for the application of the control force is included. The system was integrated using a Runge-Kutta algorithm of fifth order and the application of the control force was delayed increasingly. The influence of time delay on the free and forced oscillations (cap under harmonic and step loading) were studied. For each loading case, both the amounts of time delay that made the control system inefficient (b_i) and caused an escape (b_e) were computed, for control algorithms of order one to five. Here the control system is considered to be inefficient when the rms or the peak value of the controlled response for displacement, velocity or acceleration is greater than that obtained for the system without control.

For the case of free vibration one considered the system starting in a position with the displacement w = 0.6 and velocity equal to zero and computed the amounts of time delay that made the control system inefficient (b_i). Table 3 shows the results obtained.

For an harmonic excitation F(t) = 10 sin(12t), the critical values of time delay obtained are shown in Table 4, and Table 5 shows the results for the system subjected to the step loading F(t) = F_0.

One can observe that the system is less sensitive to time delay for the case of harmonic excitation. For step loading and free vibration the results are very similar concerning to the amounts of time delay that made the control system inefficient. It can be also observed that the critical values of time delay do not change significantly with the order of the control algorithm, except for the values that cause escape in the case of step loading, which decrease with the control order.

For the parameters used in this example, no case of instability was found in the simulations for delays from 0 to 100% of the natural period of the structure.

Conclusions

The numerical results here obtained show that the performance of a control system can be improved using a nonlinear algorithm instead of a linear one, chiefly in the case of strong nonlinearities such as those in the equation for the spherical cap studied in this paper. The control algorithm presented in this work has shown to be capable of great reductions of the dynamic response and of increasing the escape load (snap-through buckling), enlarging the working capacity of the structure without demanding great control forces. Results indicate that the control algorithm can be efficient for non-linear systems, but increasing time delay reduces the efficiency of the control system. This emphasizes the importance of considering time delay in the project of active structural control systems.

It is important to note that before applying this strategy for real structures, beside the time delay issue, a number of practical considerations like spillover effects, control-structure iteration, etc. should be studied, since the inadequate application of control forces to a structure could not only render the control ineffective but also cause instability. This is particularly important in the analysis of non-linear structural systems. So, one can conclude that this strategy of nonlinear control is attractive, has a good potential and can be used as a base for the study of more complex structures and for the design of control systems.

Presented at COBEM 99 - 15th Brazilian Congress of Mechanical Engineering, 22-26 November 1999, São Paulo. SP. Brazil. Technical Editor: José Roberto F. Arruda.

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