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LMI for adaptive barrier global sliding mode control (ABGSMC)

Uncertain systems with faulty actuators can be simplified and modeled via an LMI for adaptive barrier global sliding mode control (ABGSMC). The LMI can be derived by using a global nonlinear sliding surface in order to achieve system stability and the occurrence of sliding when actuator faults are present within the system. This LMI is important for continued research into LMI-based actuator improvements as modern systems are heavily prone to faults (including actuator faults).

The System
The following system is assumed:

$$ \begin{align} \dot{x}(t) &= A x(t) + B L u(t) + D \\ D &= \Delta \  B u(t) + b f(x) + d \\ \end{align} $$

where $$ x(t) = [x1, x2]^{\text{T}} \in R^2 $$ notes the system’s state vector, f(𝑥) is a nonlinear system function, $$ \Delta f $$ and $$ \Delta g $$ are parametric uncertainties and d represents the external disturbances. L is the post-fault coefficient which represents the actuator's effectiveness. We can assume that: $$ \begin{align} \begin{bmatrix} A \end{bmatrix} = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \end{align} $$

$$ \Delta B =  [0, \Delta B_2 ]^{\text{T}} $$

$$ B  =  [0, \Delta B_2^{\text{T}}]^{\text{T}} $$

and $$ L  =  [0,  L_2]^{\text{T}} $$

The Data
The system can then be described as:

$$ \begin{align} \dot{x_1}(t) &= A_{11} x_1(t) + A_{12} x_2(t) \\ \dot{x_2}(t) &= A_{21} x_1(t) + A_{22} x_2(t) + (B_2 L_2) u(t) + D \\ \end{align}

$$

With the state space form described above.

Assumptions
1. Pair (A, B) is totally controllable. Using this assumption for the system, the controllability of $$ (A_{11}, A_{12}) $$ can be obtained by the controllability of (A, B)

2. The parametric uncertainty and external disturbance term Dx is bounded and is less than $$ \eta_k $$

The LMI
Using the Lyapunov function, Schur Complement, the adaptive control law, and the assumption of sliding surfaces an LMI can be brought together to minimize faults via actuators

$$ \begin{align} &\text{min} \quad \eta_k < Dx \\ &\begin{bmatrix} (B_{1}W)^{T} + A+B_{1}W & B_{2} & (D_{1}W)^{T} \\ B_{2}^{T} & I & D^{T} \\ L_2W + D W & D & -L_2 \end{bmatrix} < 0 \end{align}$$

Conclusion:
As mentioned, the goal of this LMI is to mitigate the effects of external disturbances, parametric uncertainties, and actuator faults within the system. Minimizing the parametric uncertainty and disturbance term allows the system to mitigate faults within the actuators as well as determine a proper solution on how to minimize the external disturbance Dx so it falls below the parametric uncertainty.