Model Based Control

Typical MBCs include H∞ based control, linear quadratic regulator (LQR) control and nonlinear Lyapunov-based control theory. They have shown great potential but require some levels of knowledge about the system to be controlled. Here, we will have a short review of LQR control and Lyapunov-based control.

3.2.1.1 Linear quadratic regulator (LQR) control

The LQR control is a typical linear control and has been widely used. The control force u(t) is formulated as a dynamic function of negative linear feedback:

u(t) = −gdx(t) − gvx(t)˙

= G_LQRX(t)

(3.5)

where G_LQR= −[ gd gv ] is the LQR control gain vector with displacement and velocity gains gd, gv. The LQR control is concerned with controlling the dynamic response of SDOF system over a specified time period. The objective is to minimize a performance index J which consider a 2 norm of the error of the actual response X and the desired response Xdas well as the cost associated with the control force:

J= 1

where Qd, Qv and R are control weights for the displacement, velocity and control force. The mini-mization of J (Eq.3.6) will lead to the solution of control gain GLQR:

G_LQR= R⁻¹B^T_uH (3.7)

where H is determined by the Riccati equation:

A^TH + HA − HBuR⁻¹B^T_uH = Q (3.8)

The LQR control requires knowledge of displacement and velocity observations, as well as system parameters A and Bu. It has been widely used in structural control. Loh et al. [77] utilized LQR control theory with acceleration measurements to control a 20-story structure response under seismic excita-tions. Amini et al. [5] used active tuned mass dampers associated with LQR controller for mitigation of a 10-story structure response under near-field earthquake.

However, the LQR method has its own implementation challenges. For example, the weight func-tions of LQR is pre-designed and it is difficult to select appropriate weight funcfunc-tions to achieve the optimal performance [5]. Also, it is unrealistic to obtain full-state feedback given the large size of structure. Furthermore, some sensors are likely to fail given the long service time of control systems.

3.2.1.2 Lyapunov-based control

Lyapunov-based control is a nonlinear control strategy and especially useful for structures equipped with semi-active or hybrid control system with complex nonlinear dynamic behaviors. As its name suggest, the principle of Lyapunov-based control is based on Lyapunov stability theory. The theory states that the system is stable if , for any ε > 0, there exists a ball of radius δ > 0 such that if the initial condition of system is inside the radius δ and it will never leave ε at any time [114]. The mathematical form of Lyapunov stability theory is

∀ε > 0, ∃δ > 0, k X(0) k< δ ⇒ ∀t ≥ 0, k X(t) k< ε (3.9) This concept also can be demonstrated by using of a Lyapunov function V (t) > 0 that involves the potential function of system energy. If the Lyapunov function V (t) → 0 as t → ∞, the system is stable.

This requires

V˙(t) < 0 (3.10)

To further illustrate this approach, we use a SDOF system as example and focus on free vibration response ( f = 0). For typical passive damping systems, such as viscous and friction dampers, the control force u is the dynamic function of state feedback, taken as u = −GX. Taking the Lyapunov function V which contains the total energy of system as:

V =1

2X^TPX (3.11)

where P is a given symmetric positive definite matrix. Taking its time derivative ˙V yields

V˙ =1

2( ˙X^TPX + X^TP ˙X)

=1

2[X^T(PA + A^TP)X − X^T(G^TB^T_uPBuG)X] (3.12)

Eq.4.26demonstrate that any G with all positive elements will satisfy the Lyapunov stability. There-fore, passive damping systems will never destabilize the system since they add stiffness or damping into the system. Also, the dynamic of semi-active damper is equivalent to positive semi-definite gain G and it will always stabilizes the system [19].

Sliding mode control

The principle of sliding mode control is based on Lyapunov stability and it aims at directing a controlled system onto a desired state on which surface error will converge exponentially to 0. The surface error e of a state x can be expressed as e = x − xd where xd represent the desired state and is usually taken as xd = 0 for civil structure. We use the same SDOF system as previous subsection to describe the application of sliding mode control in civil structure. Define the sliding surface s as:

s= ˙e+ λ e = ˙x + λ x = ΛX (3.13)

where Λ = [ 1 λ ] with velocity weight λ . The system is stable if the surface s will converge to 0.

Therefore, consider the following Lyapunov candidate:

V=1

2s² (3.14)

Its derivative yields:

V˙ = s ˙s

= sΛ[AX + Buu+ Bff

= X^TΛ^TΛAX + sΛ Buu+ sΛBf f

(3.15)

To ensure stability, Eq. (4.26) needs to be negative definite. The first term of ˙V is negative definite, and the excitation f is considered as unmeasurable. A strategy is to select the control force u to make

the second term as negative as possible, such that sΛ Buu= −ηs² with pre-defined control parameter η . Therefore, the sliding mode control force u is given by:

u= −ηs(Λ Bu)⁻¹ (3.16)

Lyapunov-based control has been widely used in structural control. Fan et al. [34] used sliding model control algorithm to reduce the vibration of a 3-story steel frame equipped with base isolation and MR damper. Yang et al. [135] applied sliding mode control method in hybrid protective systems for applications to seismic-excited bridge structures. Lee et al. [73] utilized sliding mode control method for enhancing the seismic performance on irregular isolated bridges.

Although Lyapunov-based control is an effective and useful control method, especially in designing control algorithm for nonlinear damping device, there are several challenges for its application. For ex-ample, the first derivative of Lyapunov function (Eq.4.26) is required to be negative definite. However, it is not feasible to predict (e.g. earthquake) and measure (e.g. wind) external excitations and ensure the stability of the system. Also, it is difficult to pre-tune the control weighting parameters for providing the best performance under unkown excitations.