# An Improved Data-Driven Integral Sliding-Mode Control and Its Automation Application
The aim of this study is to propose a novel model-free adaptive ISMC scheme to address the issues existing in the water-level-control system of circulating fluidized bed boilers. The proposed scheme equivalently linearizes the nonlinear system into an FFDL data model by using the dynamic linearization method. Then, a discrete-time double integral sliding-mode control law and a new reaching law are designed to improve the control-precision and disturbance-rejection performance. This study is practical and advanced, and it is of great significance for improving the performance of the water-level-control system in circulating fluidized bed boilers in industrial production. It provides useful references for the technological development and application in related fields. Section 2 provides the flow of the traditional MFAC method. Section 3 proposes the ISMC method based on the FFDL data model and rigorously proves its stability. Section 4 shows the numerical and boiler water-level-system simulation experiments. **Lemma 1** ([28]). : _Consider the following scalar dynamical system:_  
\[x(i+1)=x(i)-c_{1}\mathrm{sig}^{a}x(i)-c_{2}x(i)+l(i) \tag{1}\]_If \(|l(i)|<R\), then \(x(i)\) is bounded, and the following equation always holds:_  
\[|x(i)|\leq\left(1+\alpha^{\frac{\alpha}{1-\alpha}}-\alpha^{\frac{1}{1-\alpha}} \right)\cdot\max\left\{\left(\frac{R}{c_{1}}\right)^{\frac{1}{2}},\left(\frac{ c_{1}}{1-c_{2}}\right)^{\frac{1}{1-\alpha}}\right\}\forall i\geq i* \tag{2}\]  
**2. Traditional Model-Free Adaptive Controller**  
_2.1. Dynamic Linearization_  
Consider a class of discrete-time nonlinear systems that can be represented as follows:  
\[I(i+1)=f(I(i),\ \cdots,\ I(i-n_{I}),O(i),\ \cdots,\ O(i-n_{O})) \tag{3}\]  
where \(O(i)\) and \(I(i)\) denote the input and output, respectively. \(n_{O}\) and \(n_{I}\) are the orders. \(f(\cdot)\) is a nonlinear function. Define \(\mathbf{L}(i)\) as follows:  
\[\mathbf{L}(i) =\left[\Delta I(i),\cdots,\ \Delta I(i-L_{I}+1),\Delta O(i), \cdots,\Delta O(i-L_{O}+1)\right]^{\mathrm{T}} \tag{4}\]  
where \(L_{I}\) and \(L_{O}\) are the linearization length constants, also known as pseudo-orders. **Assumption 1** ([29]). : _The partial derivatives of \(f(\cdot)\) for all variables of the system exist._  
**Assumption 2** ([29]). : _For any \(i_{1}\
eq i_{2}\geq 0\) and \(\mathbf{L}(i_{1})\
eq\mathbf{L}(i_{2})\), one has_  
\[|I(i_{1}+1)-I(i_{2}+1)|\leq p\|\mathbf{L}(i_{1})-\mathbf{L}(i_{2})\| \tag{5}\]  
_where \(p>0\) is a constant._  
**Theorem 1** ([29]). : _For a time-varying system (1) satisfying Assumptions 1 and 2, when \(\|\Delta\mathbf{L}(i)\|\
eq 0\), then there must exist a parameter vector \(\mathbf{H}(i)\)[29], which makes the following hold:_  
\[I(i+1)=I(i)+\mathbf{H}^{\mathrm{T}}(i)\Delta\mathbf{L}(i) \tag{6}\]  
_where the time-varying parameter vector is defined as \(\mathbf{H}(i)=[h_{1}(i),\cdots,h_{L_{I}+L_{O}}(i)]^{\mathrm{T}}\)._  
**Remark 1**. : _Theorem 1 has been proven [29]. Choosing different linearization length constants can result in different FFDL data models. The reasonable selection of the time-varying parameter vector and length constant can improve the accuracy of the data model in describing the system._