Improving existing “reaching law” for better discrete control of seismically-excited building structures

Abstract

In this paper, a novel “composite reaching law” was explained in details: 1) the equation of discrete motion for a control system; 2) the design of discrete-time variable structure control. In addition, the model of a three-storey shear-type building structure was used to verify the effectiveness of the discrete variable structure control method. The results of numerical example analysis of the model show that the control law can effectively reduce the peak value of seismic response of the building structure and the chattering effect of the control system.

Keywords

discrete-time variable structure control / composite reaching law / chattering effect / saturated control law

Introduction

The necessity of active control systems is due to the lack of adaptability of passive control systems against varying dynamic effects and base isolation systems that are generally capable of applying only for short buildings. In recent years, considerable research effort has been devoted to the development and application of active control to reduce dynamic responses and increase serviceability of civil engineering structures under environmental loadings such as wind and earthquake.

One of the popular control algorithms about active control is the so-called variable structure control (VSC). Variable structure control (VSC) method is well known due to its attractive features, such as fast response, insensitivity to parameter variations, and invariance to certain external disturbances [1-6]. However, due to the chattering problem (too many switches in the control bounds) of conventional VSC, the time histories of control forces can not be realized by the real controllers. So the excessive chattering effect is the major disadvantage of conventional variable structure controller [7].

To avoid excessive chattering effect, Zhao et al. [8] and Cai et al. [9] proposed an exponential reaching law method to be applied for variable structure control law. However, the undue chattering effect will appear when the “ϵ” parameter—the parameter mainly influences the control system performance to resist the outer disturbance— becomes a bigger value. In fact, in order to guarantee to exist a sliding mode, we must select a bigger value for the “ϵ” parameter. In this paper, a new discrete-time variable structure control method involving discrete-time composite reaching law has been applied for seismically-excited linear structure. The proposed control method can keep the chattering effect sufficiently low so as to ensure the system stable. At the end of this paper, a numerical example for a three-storey building model is carried out to demonstrate the feasibility of the control method.

Dynamic equation of structural systems

For an n-degree-of-freedom building structure subjected to the horizontal earthquake ground acceleration {\ddot{x}}_{\mathrm{g}}(t) and the active control force BoldItalic(t), the matrix equation of motion can be expressed as

(1)\mathit{M}\ddot{\mathit{x}}(\mathit{t})\mathit{+}\mathit{C}\dot{\mathit{x}}(\mathit{t})\mathit{+}\mathit{K}\mathit{x}(\mathit{t})\mathit{=}\mathit{D}\mathit{u}(\mathit{t})-\mathit{m}{\ddot{\mathit{x}}}_{\mathrm{g}}(t),

in which BoldItalic=[x₁, x₂,..., x_n]^T, is an n-vector with x_i(t) being the relative displacement of the ith storey unit with respect to the ground x_g(t); BoldItalic=diag[m₁, m₂,..., m_n] and BoldItalic=[m₁, m₂,..., m_n]^T are an (n×n) diagonal mass matrix and an n-dimensional mass vector with m_i being the mass of the ith storey unit; BoldItalic and BoldItalic are (n×n) stiffness and damping matrices, respectively; BoldItalic is an (n×r) location matrix; BoldItalic(t) is an r-vector consisting of r control forces.

In the state space, Eq. (1) can be written as

(2)\dot{\mathit{Z}}(t)=\mathit{A}\mathit{Z}(t)+\mathit{B}\mathit{u}(t)+\mathit{E}{\ddot{x}}_{\mathrm{g}}(t),

where BoldItalic(t) is a (2n×1) state vector; BoldItalic is a (2n×2n) system matrix; BoldItalic is a (2n×r) control matrix and BoldItalic is a (2n×1) excitation vector, respectively, given by

\mathit{Z}(t)\left[\begin{array}{c}\mathit{x}\left(t\right)\\ \dot{\mathit{x}}\left(t\right)\end{array}\right],\mathit{A}\mathit{=}\left[\begin{array}{cc}\mathbf{0}& {\mathit{I}}_n\\ \mathrm{-}{\mathit{M}}^{\mathrm{-}\mathbf{1}}\mathit{K}& \mathrm{-}{\mathit{M}}^{\mathrm{-}\mathbf{1}}\mathit{C}\end{array}\right],\mathit{B}=\left[\begin{array}{c}\mathbf{0}\\ {\mathit{M}}^{\mathbf{-}\mathbf{1}}\mathit{D}\end{array}\right],\mathit{E}=\left[\begin{array}{c}\mathbf{0}\\ -{\mathit{M}}^{\mathbf{-}\mathbf{1}}m\end{array}\right].

Since the control system described by Eq. (2) is linear and time-invariant, its analytical solution can be obtained as [10]

(3)\mathit{Z}(\mathit{t})\mathit{=}{\mathrm{e}}^{\mathit{A}(\mathit{t}\mathit{-}{\mathit{t}}_{\mathit{0}})}\mathit{Z}({\mathit{t}}_{\mathit{0}})\mathit{+}{\int }_{{\mathit{t}}_{\mathit{0}}}^{\mathit{t}}{\displaystyle {\mathrm{e}}^{\mathit{A}(t-\tau )}}\mathit{B}\mathit{u}(\tau )\mathit{d}\tau \mathit{+}{\displaystyle {\int }_{{\mathit{t}}_{\mathit{0}}}^{\mathit{t}}{\mathrm{e}}^{\mathit{A}(t-\tau )}}\mathit{E}{\ddot{x}}_{\mathrm{g}}(\tau )d\tau .

Suppose all information for the online calculation of control forces is sampled with period T. Between two consecutive sampling points, kT and (k+1)T, the best available information about the external excitations is {\ddot{x}}_g(kT). Therefore, the external excitations are sampled as zero-order hold and are thus assumed to be constant between two consecutive sampling instants. During real-time control, the control forces are calculated once every sampling period. Then the discrete-time control signals are converted into zero-order hold continuous-time signals and applied to the structure. The control forces are constants between two consecutive sampling instant. Let t₀=kT and t=(k+1)T, Eq. (3) becomes

(4)\mathit{Z}[(k+\mathrm{1})T]={\mathrm{e}}^{\mathit{A}\mathit{T}}\mathit{Z}(kT)+{\displaystyle {\int }_{kT}^{(k+\mathrm{1})T}{\mathrm{e}}^{\mathit{A}[(k+\mathrm{1})T-\tau ]}}\mathit{B}\mathit{u}(\mathit{k}\mathit{T})\mathit{d}\tau \mathit{+}{\displaystyle {\int }_{kT}^{(k+\mathrm{1})T}{\mathrm{e}}^{\mathit{A}[(k+\mathrm{1})T-\tau ]}}\mathit{E}{\ddot{x}}_{\mathrm{g}}(kT)d\tau .

Making variable substitution of η=(k+1)T-τ, Eq. (4) becomes

(5)\mathit{Z}[(\mathit{k}\mathit{+}\mathit{1})\mathit{T}]={\mathrm{e}}^{\mathit{A}T}\mathit{Z}(kT)+{\int }_{\mathrm{0}}^T{\displaystyle e^{\mathit{A}\eta }\mathit{B}\mathit{u}(kT)d\eta }+{\displaystyle {\int }_{\mathrm{0}}^T{\mathrm{e}}^{\mathit{A}\eta }\mathit{E}{\ddot{\mathit{x}}}_{\mathrm{g}}(kT)d\eta }.

As a whole, it is more logical and more realistic for the control system to be modeled in a discrete-time fashion as

(6)\mathit{Z}(k+\mathrm{1})={\mathit{A}}_{\mathrm{d}}\mathit{Z}(k)+{\mathit{B}}_{\mathrm{d}}\mathit{u}(k)+{\mathit{E}}_{\mathrm{d}}{\ddot{x}}_{\mathrm{g}}(k),

where BoldItalic_d=e^BoldItalicT is a (2n×2n) discrete-time system matrix;

{\mathit{B}}_{\mathrm{d}}={\displaystyle {\int }_{\mathrm{0}}^T{\mathrm{e}}^{\mathit{A}\eta }\mathit{B}\mathrm{d}\eta }

is a (2n×r) discrete-time control matrix and

{\mathit{E}}_{\mathrm{d}}={\displaystyle {\int }_{\mathrm{0}}^T{\mathrm{e}}^{\mathit{A}\eta }\mathit{E}\mathrm{d}\eta }

is a (2n×1) discrete-time excitation vector.

Design of discrete-time variable structure control

In the above section, the continuous-time system has been discretized into the standard discrete form. The new discrete-time variable structure control (DVSC) method involving discrete-time composite reaching law for the system will be investigated in this section.

For the DVSC system, the state trajectory starting from any initial state seldom arrives at the switching surface exactly because the state of the system is a discrete-time sequence at interval of sampling period. In state space, the state trajectory shows a point range and not a continuous-time curve, which implies that switching seldom occurs on the switching surface. So the VSC for discrete-time systems is different from the traditional VSC for continuous-time systems. Gao [1] has made detailed research on the DVSC, and the quasi-sliding mode and the quasi-sliding mode band have been rigorously defined in his research. From Gao [1], the DVSC should have the following three attributes:

1) Starting from any initial state, the trajectory will move monotonically toward the switching surface and cross it in finite time.

2) Once the trajectory has crossed the switching surface the first time, it will cross the surface again in every successive sampling period, resulting in a zigzag motion about the switching surface.

3) The size of each successive zigzagging step is non-increasing and the trajectory stays within a specified band.

The motion of the DVSC system satisfying attributes 1) and 2) is called quasi-sliding mode, whereas the specified band which contains the quasi-sliding mode is called the quasi-sliding mode band which is defined by Ref. [1]:

(7)

where 2Δ is the width of the band. When Δ=0, the quasi-sliding mode becomes an ideal quasi-sliding mode. In a practical situation, the switching surface can be determined by using the ideal-quasi-sliding mode, and the controller can be obtained by using the discrete-time reaching condition. Designs of the switching surface and the controller are as follows.

Determination of discrete switching surface

The first step of using the theory of VSC system design is to determine the switching function. In the design of the switching surface, the external excitation {\ddot{x}}_{\mathrm{g}}(k) is neglected; however it is taken into account in the design of the controller. Consider a linear switching surface. For simplicity, let BoldItalic(k)=[S₁(k),S₂(k),...,S_r(k)]^T be an r-dimensional switching surface with r sliding variables, S₁(k),S₂(k),...,S_r(k), given by

(8)\mathit{S}(k)=\mathit{\Theta }\overline{\mathit{Z}}(k)=\mathrm{0.}

There are several approaches that can be used to determine the matrixBoldItalic, such as the pole assignment method, the linear quadratic regulator (LQR) method and the null assignment method [1].

Design of controller with discrete-time composite reaching law

In the theory of DVSC, the controllers are designed to drive the state trajectory into the quasi-sliding mode band. To achieve the above goal, Gao [1] proposed a discrete-time exponential reaching law to be used for controller design in early nineties and it can be expressed as

(9)\mathit{S}(k+\mathrm{1})=(\mathrm{1}-qT)\mathit{S}(k)-ϵT\mathrm{sgn}(\mathit{S}(k)),

where T is the sampling period; ϵ>0 and q>0 are both positive real constant, and q is restricted by qT<1.

For the DVSC based the discrete-time exponential reaching law, the dynamic quality of DVSC system in the reaching mode can be controlled by the choice of the parameters (ϵ and q) of the difference equation. Generally, the “q” parameter influences the reaching velocity at which the control system gets to the switching surface. The control system can arrive at the switching surface rapidly when the “q” parameter has acquired a bigger value, especially, when the “q” parameter is almost 1/T. In comparison with the “q” parameter, the “ϵ” parameter mainly influences the control system performance to resist the outer disturbance. The control system can resist the outer disturbance more efficiently when the “ϵ” parameter has acquired a bigger value, but it may render the control system unstable because of the chattering effect in this condition.

The discrete-time exponential reaching law has itself disadvantages: when the control system moves on the switching band, the control system cannot reach to the zero point and only can form a chattering near the zero point, which may induce the system unstable.

Analyzing the disadvantages of the discrete-time exponential reaching law, a discrete-time composite reaching law is considered and expressed as [10]

(10)\mathit{S}(k+\mathrm{1})=(\mathrm{1}-qT)\mathit{S}(k)-ϵT\arctan (\Vert \mathit{Z}(k)\Vert )\mathrm{sgn}(\mathit{S}(k)).

The discrete-time composite reaching law can solve the above problems. In order to further eliminate the chatting effect, one substitutes the “sat” function for the “sgn” function and Eq. (10) becomes

(11)\mathit{S}(k+\mathrm{1})=(\mathrm{1}-T)\mathit{S}(k)-ϵT\arctan (\Vert \mathit{Z}(k)\Vert )\mathrm{s}\mathrm{a}\mathrm{t}(\mathit{S}(k)),

in which

The left-hand term in Eq. (11) can be further expressed as

(12)\mathit{S}(k+\mathrm{1})-\mathit{S}(k)=\mathrm{\Theta }\mathit{Z}(k+\mathrm{1})-\mathrm{\Theta }\mathit{Z}(k)=\mathrm{\Theta }\left[{\mathit{{\rm A}}}_{\mathrm{d}}\mathit{Z}(k)+{\mathit{B}}_{\mathrm{d}}\mathit{u}(k)+{\mathit{E}}_{\mathrm{d}}{\ddot{x}}_{\mathrm{g}}(k)\right]-\mathrm{\Theta }\mathrm{{\rm Z}}\mathit{k}\mathit{.}

Hence the controller can be obtained from Eqs. (11) and (12), and expressed as

(13)\mathit{u}(k)=-{(\mathit{\Theta }{\mathit{B}}_{\mathrm{d}})}^{-\mathrm{1}}\left\{\mathit{\Theta }{\mathit{A}}_{\mathrm{d}}-\mathit{I}+qT\mathit{I}\mathbf{Z}\mathit{k}+\mathit{\Theta }{\mathit{E}}_{\mathrm{d}}{\ddot{x}}_{\mathrm{g}}(t)+ϵT\arctan (\Vert \mathit{Z}(k)\Vert )\mathrm{s}\mathrm{a}\mathrm{t}(\mathit{\Theta }\mathit{Z}\mathit{k})\right\}.

Design of saturated controller

We design the saturated control law to be used when the control force needed according to Eq. (13) exceeds the maximum control force of an actuator:

(14){\widehat{u}}_i(t)=\{\begin{array}{cc}{u}_i\left(t\right),& \left|u_i\left(t\right)\right|\le {\overline{u}}_i,\\ {\overline{u}}_i\mathrm{sgn}\left(u_i\right(t\left)\right),& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s},\end{array}\begin{array}{ccc}& & \\ & i=1,2,\mathrm{...},r,& \\ & & \end{array}

where u_i(t) is the computed control force of the ith actuator; {\overline{u}}_i is the maximum control force of the ith actuator and {\overline{u}}_i(t) is the real control force of the i th actuator applied for the building structure.

Numerical simulations

The application of the control method presented in this paper is illustrated in this section. An idealized three Degree-of-Freedom (DOF) structure model proposed by Yang et al. [5] in Fig. 1 is used to study thoroughly the effect of the time-delay control. The 3DOF structure is counteracted by an active brace system (ABS) implemented on the first storey unit. The mass, stiffness and damping coefficient of each storey unit are m_i=1 metric ton, k_i=980 kN/m, and c_i=1.407 kN·s/m, respectively(i=1,2,3).The N-S component of the 1940 EI Centro earthquake record scaled to a maximum ground acceleration of 0.35 g is used as the input excitation, and its episode is taken to 8 s. The sampling period is given by T=0.02 s. The coefficient Δ in Eq. (11) is chosen as Δ=0.05. In this paper, the DLQR method is used for determining the coefficient matrix BoldItalic of the switching surface.

To verify that it can efficiently remove the chattering effect to use the discrete-time composite reaching law, the comparison of the control time histories for DVSC system between using the discrete-time composite reaching law and using the discrete-time exponential reaching law as the value of the “ϵ” parameter is changed is presented in Fig. 2. In Fig. 2, the control time histories for DVSC system using the discrete-time composite reaching law and using the discrete-time exponential reaching law are denoted by “Composite RL” and “Exponential RL”, respectively. For simplify, let the coefficient q in Eqs. (9) and (13) are chosen as q=40, and vary the values of the “ϵ” parameter from 0.1 to 0.6.

From Fig. 2, we can see clearly that it can reduce the chattering effect more efficiently to use the discrete-time composite reaching law when the “ϵ” parameter becomes a larger value.

When the two coefficientsϵand q in Eq. (13) are chosen asϵ=0.1 and q=40, the maximum control force u_max, the maximum interstorey drifts x_i and the maximum absolute acceleration {\ddot{x}}_i of each floor are presented in the forth column and the fifth column of Table 1 denoted by “Composite RL”. When the structure system is uncontrolled, the maximum interstorey drifts x_i and the maximum absolute acceleration {\ddot{x}}_iof each floor are presented in the second column and the third column of Table 1 denoted by “uncontrolled”. As observed from Table 1, the maximum control force u_max using the discrete-time composite reaching law is 12.749 kN. The maximum interstorey drift and the maximum absolute acceleration of the first storey unit are reduced respectively by nearly 90 percent and 68 percent, them of the second storey unit respectively by nearly 93 percent and 60 percent, and them of the third storey unit respectively by nearly 60 percent and 51 percent. When the maximum control force u_max of an actuator is 10 kN, the maximum interstorey drifts x_i and the maximum absolute acceleration {\ddot{x}}_i of each floor are presented in the sixth column and the seventh column of Table 1; when the maximum control force u_max of an actuator is 5 kN, the maximum interstorey drifts x_i and the maximum absolute acceleration {\ddot{x}}_i of each floor are presented in the eighth column and the ninth column of Table 1. From Table 1, one can observe that the saturated controller is quite effective in reducing the structural responses, particularly during the peak response period.

To show more detailed control performances, the comparisons of the interstorey drift and the absolute acceleration time histories of the first storey unit between DVSC using the discrete-time composite reaching law and uncontrolled are presented in Fig. 3.

From Fig. 3, one can observe that the control effectiveness by using the control method presented in this paper is remarkable. In fact, this case possesses the best control results but it is not realistic since time-delay is unavoidable.

Conclusions

In this paper, the discrete-time variable structure control (DVSC) with discrete-time composite reaching law is developed to solve this practical problem for seismically-excited linear structure. The control system is first formulated in discrete-time form. It is well known that the excessive chattering effect is the major disadvantage of conventional discrete-time variable structure controller. In this paper, it can keep the chattering effect sufficiently low by introducing DVSC with a composite reaching law.

Performance and robustness of the DVSC method proposed in this paper are both demonstrated by numerical simulations. The simulation results prove that the presented discrete-time variable structure control method is quite effective, which not only can reduce the peak-response of the ground motion, but also can reduce the undue chattering effect of the control system so as to ensure the system stable when the “ϵ” parameter becomes a larger value. Theoretical development and numerical verification show that the proposed control algorithm is feasible. Such simple on-line calculation makes the proposed control algorithm favorable to real implementation. So it is an attractive control strategy for the application of vibration control of building structures.

References

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