Decentralized Sliding Mode Control (DSMC)

For decades, centralized control architecture, such as the LQR, the LQG, the SMC, and so on, has been used to design controllers for civil structural systems. One major disadvantage of the centralized control architecture is the installation of distributed sensors and control devices, the requirement of wiring for system connectivity, and the amount of data transmission between the distributed sensors, the control devices, and the central computer (or controller), especially in the practical implementation of full-scale structures. Although the innovative techniques of wireless communication have been developed in the field of civil and structural engineering, the time delay is still a challenging problem (Lynch, 2007; Wang et al., 2007). Likewise, the feedback state vector needs to be estimated through an estimator (or sometimes referred to observer or compensator) if the sensors in the strategic locations are hard to be implemented and connected. In that case, the design of an efficacious estimator for civil structural systems under uncertainties or nonlinear behavior is quite challenging. Moreover, the centralized control architecture is very sensitive to system failure.

When the sensors or the central computer is malfunction during the hazard episode, the operation of the whole control system will have a risk to shut down or to be disrupted.

On the other hand, decentralized control architecture deals with the difficulties came mainly from dimensionality, information structure constraints, uncertainties, and time delays, by decomposing the global system into a set of local sub-systems (Šiljak, 1996; Bakule, 2008). In the decentralized control architecture, the control force for a particular control device can be determined only according to the structural responses in the vicinity of the control device’s location (Lynch and Law, 2002). Moreover, the robustness may be compromised in the centralized control architecture since it depends on the accurate identification of structural systems as a whole. Theoretically, the decentralized control architecture is more robust because it need only the minimum requirement of wiring and the necessary communication between sensors as well as the partial system characteristics.

Although it is quite challenging to develop an effective decentralized control architecture, it still shows significant advantages in numerous studies (Loh and Chang, 2008; Lei et al., 2012). Hence, emphasis will be placed on the DSMC in this study and the development of the DSMC is described in the following.

DSMC based on Static Output Feedback (DSMC-SOF)

To achieve a decentralized control architecture, the controllers and their sliding surface can be designed based only on the measured information from a limited number of sensors installed at the strategic locations without an estimator. It has been shown that the SMC is stable for static (direct) output feedback with a limited number of sensors (Yang et al., 1995a; Yang et al., 1995b; Yang et

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al., 1996b). In other words, a sliding mode controller can be designed based only according to the structural responses in the vicinity of the control device’s location; one limitation is that the velocity sensors are required as a minimum.

Assuming that 𝑚 be the number of states measured from a limited number of sensors installed at the strategic locations, the constraint that 𝑟 ≤ 𝑚 < 2𝑛 are forced. Further, let 𝐳̅_𝑚(𝑡) be the modified state vector, consisting of 𝑚 measured outputs (observation) and some zero elements (for those states that are not measured). In other words, 𝐳̅_𝑚(𝑡) is obtained from 𝐳(𝑡) by replacing those states, which are not measured, by zero elements.

𝐳̅_𝑚(𝑡) = {𝐳^𝑖(𝑡) if the state can be measured

0 if the state cannot be measured (5.25)

Then, the sliding surface in Equation (5.12) is given by

𝐒(𝐳̅_𝑚) = 𝐏𝐳̅_𝑚(𝑡) = 0 (5.26)

where the sliding surface matrix, 𝐏, can also be determined by using the regular form, the method of pole assignment, the LQR, or the nonlinear optimization algorithms. A stable (static output) controller can be obtained from Equation (5.20) and Equation (5.21) by replacing 𝐳(𝑡) by 𝐳̅_𝑚(𝑡) as following

The controller given in Equations (5.29) is referred to as DSMC-SOF. Generally, there are two parameters to be predefined (or adjusted) to determine the control force for a particular control device, i.e. the corresponding elements in the sliding surface matrix, 𝐏, and switching parameter matrix, 𝛅.

Note that the ground acceleration needs to be measured for the DSMC-SOF, and the number of measured states, 𝑚, shall be larger than the number of control devices, 𝑟.

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DSMC based on On-off Control (DSMC-OC)

On-off control (also known as bang-bang control) which leads to on/off switching command is the simplest example of a feedback control algorithm. Using the sliding surface in Equation (5.12), the Lyapunov function in Equation (5.13), the state-space equation in Equation (5.2), and assuming static output feedback, one obtains

𝐕̇ = 𝐒^𝑇(𝐳̅_𝑚)𝛌𝐮(𝑡) + 𝐒^𝑇(𝐳̅_𝑚)𝐒̇_𝑠(𝐳̅_𝑚) ≤ 0 (5.30)

where 𝐒̇_𝑠(𝐳̅_𝑚) = 𝐏[𝐀𝐳̅_𝑚(𝑡) + 𝐄𝐪̈_g(𝑡)] is actually used to design the equivalent controller.

Note that the second term in Equation (5.30) is the derivative of the Lyapunov function for the structural system without control. Since the structural system without control is inherently stable, it means that 𝐒^𝑇(𝐳̅_𝑚)𝐒̇_𝑠(𝐳̅_𝑚) ≤ 0. Therefore, to design the control force, it is necessary to guarantee that 𝐒^𝑇(𝐳̅_𝑚)𝛌𝐮(𝑡) ≤ 0. With this premise, one possible solution is obtained by minimizing the first term in Equation (5.30). For this concept and assuming 𝛋(𝐳̅_𝑚) = 𝐒^𝑇(𝐳̅_𝑚)𝛌 = 𝐳̅_𝑚^𝑇𝐏^𝑇𝐏𝐁 ≠ 𝟎, it can easily be shown that

𝐕̇ = 𝛋(𝐳̅_𝑚)𝐮(𝑡) = ∑^𝑟_𝑖=1𝜅_𝑖(𝑡)𝑢_𝑖(𝑡)≤ 0 (5.31) where 𝛋(𝐳̅_𝑚) ∈ ℝ^1×𝑟 is a state-weighted vector for the switching controller, 𝜅_𝑖(𝑡) is the ith element of this vector, and 𝑢_𝑖(𝑡) is the control force from the ith control device.

Hence, the minimization of Equation (5.31) depends on the signs of 𝜅_𝑖(𝑡) and 𝑢_𝑖(𝑡). Here, for the semi-active control devices like MR dampers, the control force can be regulated by different command voltages. If the value of 𝜅_𝑖(𝑡) is positive, a minimum command voltage, such as 0, will be selected for a positive velocity response so as to ensure the minimum control force. On the contrary, if the value of 𝜅_𝑖(𝑡) is negative, then a maximum command voltage needs to be selected for a positive velocity response to obtain a maximum control force. According to this criterion, the controller is:

𝑣_𝑖 = 𝑣_{𝑖,𝑚𝑎𝑥}𝐻[−𝜅_𝑖(𝑡)𝑞̇_𝑗(𝑡)] (5.32)

where 𝐻[∙] is Heaviside step function; 𝑣_𝑖 is the command voltage for the ith control device which is installed at the jth DOF (which also represented the DOF of the ith equipment isolation system), 𝑞̇_𝑗(𝑡) is the relative velocity responses of the jth DOF. Based on the DSMC-OC described above, only the relative velocity and displacement responses between the vibration-sensitive equipment and the floor are required.

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DSMC based on On-off Control and Constraint (DSMC-OCC)

Theoretically, the DSMC-OC is straightforward and robust. The minimization of Equation (5.31) is expected to weight on the reduction of relative displacement responses; therefore, the reduction of absolute acceleration responses may be compromised for the DSMC-OC, which is bad for the acceleration-sensitive equipment. Fortunately, the acceleration responses of equipment can usually be suppressed once the control force is limited. For the cases in which the acceleration responses are important, such as the vibration-sensitive equipment isolated by the CSS systems or the other isolators, the control force can be limited by decreasing the damper capacity. Thus, to lower the absolute acceleration responses, instead of using Equation (5.32) directly, the maximum command voltage can be modified as follows:

0 ≤ 𝑣_𝑖 ≤ 𝜐𝑣_{𝑖,𝑚𝑎𝑥} (5.33)

Consequently, the constraint is that 0 < 𝜐 < 1 and 𝜐 can be determined through the numerical simulation or optimization to reduce the acceleration responses of equipment.

DSMC based on SDOF Equipment Isolation System (DSMC-SEIS)

For the DSMC-SOF, the interaction between the structural systems and the equipment isolation systems has been taken into account so the controller also depends on the earthquake excitations, indicating that the ground acceleration needs to be measured in the practical implementation. Since each equipment and its isolation system are firmly installed on a specific floor, the structure-equipment interaction can be considered through the structural responses of this specific floor. Hence, a single DOF (SDOF) equipment isolation system under the external excitation represented by the absolute acceleration responses can be used to design the controller directly. For example, the sliding surface and the derivative of the Lyapunov function can be expressed as

𝑉̇ = 𝑆(𝐳)𝑆̇(𝐳) = [𝑃𝑞_𝑗(𝑡) + 𝑞̇_𝑗(𝑡)][𝑃𝑞̇_𝑗(𝑡) + 𝑞̈_𝑗(𝑡)] ≤ 0 (5.34) where 𝑞̈_𝑗(𝑡) is the relative acceleration responses of the jth DOF where the control device is installed, and it can be obtained from the equation of motion derived by considering only this equipment isolation system.

𝑚_𝑗𝑞̈_𝑎,𝑗(𝑡) + 𝑘_𝑗𝑞_𝑗(𝑡) = 𝑢_𝑖(𝑡) (5.35) where the subscript 𝑎 denotes the absolute responses and 𝑢_𝑖(𝑡) denotes the control forces of the ith control device. The friction of the equipment isolation system is ignored in Equation (5.35); however, it can be replaced by 𝑚_𝑗𝑞̈_𝑎,𝑗(𝑡) + 𝑘_𝑗𝑞_𝑗(𝑡) + 𝑚_𝑗𝜇_𝑗sgn[𝑞̇_𝑗(𝑡)] = 𝑢_𝑖(𝑡) to consider the friction

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coefficient, 𝜇_𝑗, of the ith equipment isolation system. Moreover, the relative acceleration responses of the jth DOF can be obtained from Equation (5.35) by substituting 𝑞̈_𝑎,𝑗(𝑡) = 𝑞̈_𝑗(𝑡) + 𝑞̈_𝑎,𝑘(𝑡).

𝑞̈_𝑗(𝑡) =^{𝑢𝑖(𝑡)−𝑘}_𝑚^{𝑗𝑞𝑗(𝑡)}

𝑗 − 𝑞̈_𝑎,𝑘(𝑡) (5.36)

After that, substituting Equation (5.36) into Equation (5.34), the control force can be obtained by assuming a linear state-feedback controller

𝑢_𝑖(𝑡) = −𝑃𝑚_𝑗𝑞̇_𝑗(𝑡) + 𝑘_𝑗𝑞_𝑗(𝑡) + 𝑚_𝑗𝑞̈_𝑎,𝑘(𝑡) −^{𝑃𝑞𝑗(𝑡)+𝑞̇}_𝑚 ^𝑗(𝑡)

𝑗 𝛿 (5.37)

where 𝑞̈_𝑎,𝑘(𝑡) is the absolute acceleration responses of the kth floor where the equipment isolation system and the control device are located.

Unlike the DSMC-SOF, the absolute acceleration responses of the specific floor need to be measured in order to calculate the control force in Equation (5.37). Again, there are two parameters to be predefined (or adjusted) to determine the control force for one control device, 𝑃 and 𝛿. Since the SMC frequently favors the reduction of relative displacement responses rather than absolute acceleration responses, Equation (5.37) can be modified as

𝑢_𝑖(𝑡) = −𝑃𝑚_𝑗𝑞̇_𝑗(𝑡) + 𝑘_𝑗𝑞_𝑗(𝑡) + 𝜂𝑚_𝑗𝑞̈_𝑎,𝑘(𝑡) −^{𝑃𝑞𝑗(𝑡)+𝑞̇}_𝑚 ^𝑗(𝑡)

𝑗 𝛿 (5.38)

where 𝜂 is another parameter to be predefined (0 ≤ 𝜂 ≤ 1) and can be determined through the numerical simulation or optimization, too.

Here, because the equipment isolation system without control is inherently stable, the derivative of the Lyapunov function is always smaller than or equal zero. This important result guarantees that the stability of the controller is independent of the external excitation and hence Equation (5.38) can be secured (Lu et al., 2008). In the case that 𝜂 = 0, the structural responses of the specific floor are unnecessary, and the structure-equipment interaction is ignored.