Finite-Time Disturbance Observer of Nonlinear Systems

Most of the systems existing in nature are non-linear. However, especially in practical applications, for highly nonlinear systems, how to implement control tasks for dynamic systems with uncertain parameters is still a hot research issue. In recent years, the scientific community has developed multiple advanced control strategies for nonlinear systems. SMC became the most widely used and effective control method in complex nonlinear systems. In the field of nonlinear control, the sliding mode control (SMC) method is famous for its insensitivity to inaccurate parameter and uncertain disturbances [1,2,3]. Therefore, it is widely used in nonlinear system, hybrid system, cascade system, and constraint system [4,5]. However, these sliding mode methods have two unavoidable defects, including severe chattering and tardy convergence speed.

In traditional SMC, a large switching gain may bring higher control accuracy, but an excessive gain will obviously cause chattering problems, which will reduce system performance and even cause system failure. Intelligent SMC controllers have been discussed for nonlinear systems [6]. In general, it is feasible to eliminate chattering by adaptive adjusting switching gain. A dynamic sliding mode control (DSMC) is not only easy to implement, but also has a better property to eliminate the chattering [7]. This is because the dynamic sliding mode surface can transfer the discontinuous term to the higher order derivative of the designed controller, and a continuous DMSC law is obtained.

Another defect of traditional SMC is the low convergence speed in order to improve the convergence speed of the sliding mode. Mozayan et al. [8] proposed an enhanced exponential reaching law for the PMSG wind turbine system. Wang et al. [9] developed a new sliding mode reaching law that includes the state and the power term of the sliding function. These methods greatly improve the convergence rate of the dynamic system and have a certain effect on eliminating chattering. The terminal sliding mode controllers [10,11,12,13] are designed to ensure the tracking error of the system can finitely converge to zero. Therefore, in this paper, combining the advantages of DMSC and TSMC, a DTSMC method is proposed to increase the convergence speed and reduce chattering.

In practical applications, nonlinear systems often have unknown internal parameter perturbations and external disturbances, which makes the ideal sliding mode controller unable to be realized. Therefore, many scholars have proposed a series of new neural network approaches to approximate nonlinear functions, including nominal functions and parametric perturbations [14,15,16,17,18]. When the nonlinear function is unknown, these methods are effective, but the matched and mismatched disturbances of the system cannot be effectively resolved, and in order to avoid the influence of disturbances on the control performance, a large switching gain needs to be selected to decrease the influence of disturbances on the control performance, which may cause large chattering. At the same time, the FTDO is a simple way to estimate the matched and mismatched disturbances of the system, which means that only a small switching gain is needed to deal with the impact of the disturbance estimation error [19,20]. Wang et al. [21] designed a continuous fast nonsingular TSMC using finite-time exact observer for automotive electronic throttle systems. Considering the time-varying disturbances in the DC-DC boost converters, Wang et al. [22] proposed a continuous non-singular TSMC with a finite time disturbance observer. In [23,24], an improved nonsingular sliding mode algorithm is proposed for nonlinear systems with matched and mismatched disturbances. These methods fully demonstrate the superiority of the FTDO. Neural networks are widely used because of the good approximation ability [25,26,27,28,29] therefore they can be used to estimate the unknown parameters and system nonlinearities.

Inspired by the above methods, an adaptive DTSMC based on a FTDO is introduced. The FTDO is adopted to estimate the matched and mismatched disturbance of nonlinear system, then the disturbance compensation term is added to the DTSMC law to reduce the influence of the lumped disturbance. Although the existence of the disturbance observer can reduce the switching gain, how to choose the gain is still worth considering. A DHLRNN controller in [30] is introduced to obtain the optimal switching gain, instead of selecting it based on expert experience. Compared with the existing methods, the innovative points are listed as:

(1) By combining the advantages of DMSC and TSMC, dynamic terminal sliding mode (DTSM) surface is designed, which can not only weaken chattering but also ensure the tracking error can finitely converge to zero.

(2) A FTDO is proposed to remove the negative impact of matched and mismatched disturbances on the control performance. The disturbance compensation term is added to the DSMC law, so the switching gain only needs to be greater than the disturbance estimation error which indirectly weakens the system chattering to a certain extent.

(3) A DHLRNN is presented to approximate the optimal switching gain, which can effectively avoid the switching gain being selected too large or too small, which not only guarantees the superiority of the algorithm but also reduces the difficulty of algorithm application.

The rest of this paper is organized as follows. Section 2 explains a finite-time disturbance observer. A new double hidden layer recurrent neural network is introduced in Section 3. Section 4 designs an adaptive DTSM controller with a FTDO. To verify the effectiveness of the algorithm, simulation experiments are applied in Section 5. Finally, conclusions are summarized in Section 6.

The FTDO is essentially a high-order sliding mode differentiator, which was first proposed by Levant [19]. Consider an n-order SISO system with n-order differentiable bounded uncertain disturbance $g\left(t\right)$ as follows:

For the matching disturbance term contained in system (1), a FTDO is proposed, and the differential equation is defined as: where $v_0,v_1,\cdots ,v_n$ are the internal state variables of the observer, ${\lambda }_0,{\lambda }_1,\cdots ,{\lambda }_n,K$ are the observer gains which are greater than zero, and $K$ is a known Lipshitz constant that satisfies the condition $\left|g^{\left(n\right)}(t)\right|<K$.

Levant [19] and Shtessel [20] prove Theorem 1 and Theorem 2 in detail.

Suppose take the order of the observer as $n=2$. Finally, a FTDO is obtained as follow: where ${\lambda }_0,{\lambda }_1,{\lambda }_2,K$ are the observer gains which are greater than zero. The output meaning of the observer is defined as:

Symbol ^ represents the estimation of the state $x_2$, unknown disturbance $g\left(t\right)$, and its various derivatives. Therefore, define the estimation error as:

Find the first derivative of Equation (7) and bring the observer (5) into it. The dynamic equation of the estimation error can be obtained as follows: where $L$ is a known Lipshitz constant satisfying condition $\left|\ddot{g}(t)\right|<L$.

A DHLRNN is designed in [30] to approximate the equivalent controller term, which has better dynamic fitting capability. This paper will adopt the DHLRNN to obtain the optimal sliding mode switching gain. Figure 1 is the proposed network structure, which is a 4-layer structure including input, first hidden, second hidden and output layers, with a symmetry structure.

The signal propagation and the results of each layer are explained as follows:

Layer 1—Input layer: The number of nodes in the input layer and the number of input signals $x_i\left(i=1,2,\cdots ,m\right)$ are the same. Usually, the tracking error and the derivative of the error are selected as the input signal. Each input node is related to the output $exY$ at the previous moment, and the corresponding feedback weight is defined as $W_{ri}\left(k=1,2,\cdots ,m\right)$. Therefore, the i-th node output in the input layer is computed as: where $X={\left[x_1,x_2,\cdots ,x_m\right]}^T\in R^{m\times 1}$ is the input of neural network, the feedback weight is $W_r={[W_{r1},W_{r2},\cdots ,W_{rm}]}^T\in R^{m\times 1}$, the output of the input layer is $\theta ={[{\theta }_1,{\theta }_2,\cdots ,{\theta }_m]}^T\in R^{m\times 1}$.

Layer 2—First hidden layer: The proposed network has two hidden layers. In this layer, the Gaussian activation function is used to initially extract the input feature information. Therefore, the Gaussian function output of j-th node is calculated as: where the center vector and base width vector are defined as $C_1={\left[c_{11},c_{12},\cdots ,c_{1n}\right]}^T\in R^{n\times 1}$ and $B_1={\left[b_{11},b_{12},\cdots ,b_{1n}\right]}^T\in R^{n\times 1}$. Finally, output vector of the first hidden layer is ${\Phi }_1={\left[{\varphi }_{11},{\varphi }_{12},\cdots ,{\phi }_{1n}\right]}^T\in R^{n\times 1}$.

Layer 3—Second hidden layer: The purpose of the second hidden layer is to realize the nonlinear mapping from low latitude to high latitude. We also choose the Gaussian function as the activation function. In addition, there is no special requirement for the number of nodes in the first and second hidden layers. Finally, the second hidden layer output of the k-th node is designed as: where $C_2={\left[c_{21},c_{22},\cdots ,c_{2l}\right]}^T\in R^{l\times 1}$ is a center, base width is $B_2={\left[b_{21},b_{22},\cdots ,b_{2l}\right]}^T\in R^{l\times 1}$, output of the second hidden layer is ${\Phi }_2={\left[{\varphi }_{21},{\varphi }_{22},\cdots ,{\phi }_{2l}\right]}^T\in R^{l\times 1}$.

Layer 4—Output layer: It is a linear map. Each node in the second hidden layer is linked to the output layer by a weight $W_k\left(k=1,2,\cdots ,l\right)$. The DHL-RNN output is expressed as: where $W={\left[W_1,W_2,\cdots ,W_l\right]}^T\in R^{l\times 1}$ is the weights of output layer.

We first introduce a dynamic terminal sliding mode surface that not only ensures fast error convergence but also reduces chattering. Then the sliding mode control law is derived, and additionally, a FTDO is added to the control law to estimate the lumped unknown disturbance. Finally, the neural network proposed in section IV is introduced to approximate the switching gain in the sliding mode control law.

The purpose of the SMC law is to realize the compensation current $x_1=i_c$ tracking its reference current $r=i_c^{*}$. In order to design the control law, we first define the tracking error $e=x_1-r$ and its derivative $\dot{e}=x_2-\dot{r}$, then the error is $E={\left[\begin{array}{cc}e& \dot{e}\end{array}\right]}^T$.

Consider a terminal sliding mode surface: where $C=\left[\begin{array}{cc}c& 1\end{array}\right]$, $c$ is a positive constant to ensure Hurwitz stability, and $P\left(t\right)={\left[\begin{array}{cc}p\left(t\right)& \dot{p}\left(t\right)\end{array}\right]}^T$.

According to Assumption 1, the following terminal function $p\left(t\right)$ is constructed as:

Due to the inherent chattering problem of sliding mode, we construct a new switching function $\sigma$ by differentiating the traditional switching function $s$. Therefore, a dynamic terminal sliding mode surface is designed as: where $\lambda$ is a strictly positive constant.

Ignoring the lumped unknown disturbance $g\left(t\right)$, the first derivative of Equation (19) can be obtained:

Defining $\dot{\sigma }=0$, we can get the ideal equivalent control term: where $\left\{\begin{array}{l}{I}_1=b\left(c+\lambda \right)\\ I_2=c\left[f\left(x\right)-\ddot{r}-\ddot{p}\right]+\dot{f}\left(x\right)-\stackrel{⃛}{r}-\stackrel{⃛}{p}\\ I_3=\lambda c\left(\dot{e}-\dot{p}\right)+\lambda \left[f\left(x\right)-\ddot{r}-\ddot{p}\right]\end{array}\right.$.

After considering the lumped uncertainty $g\left(t\right)$, a FTDO is introduced to compensate for the disturbance:

A switching control item is designed to ensure the robustness as:

Finally, the designed dynamic terminal sliding mode control law based on disturbance compensation is derived: where $\left\{\begin{array}{l}{I}_1=b\left(c+\lambda \right)\\ I_2=c\left[f\left(x\right)-\ddot{r}-\ddot{p}\right]+\dot{f}\left(x\right)-\stackrel{⃛}{r}-\stackrel{⃛}{p}\\ I_3=\lambda c\left(\dot{e}-\dot{p}\right)+\lambda \left[f\left(x\right)-\ddot{r}-\ddot{p}\right]\end{array}\right.$.

In order to find the adaptive law later, we will derive the approximation error in the neural network.

A DHLRNN is designed to approximate the switching control term, the following equation is true: where $\epsilon$ is the minimum error between the ideal and actual values.

The optimal parameter is expressed as:

Considering that the optimal parameters cannot be obtained, we use the output of DHLRNN as the estimated value of the switching term. The following equation can be derived: where ${\widehat{W}}_r,{\widehat{C}}_1,{\widehat{B}}_1,{\widehat{C}}_2,{\widehat{B}}_2,\widehat{W}$ is the estimated value of the optimal parameter $W_r^{*},C_1^{*},B_1^{*},C_2^{*},B_2^{*},W^{*}$.

Therefore, the approaching switching control term is:

Approximation error between actual and estimated value is obtained as: where ${\epsilon }_0={\tilde{W}}^T{\tilde{\Phi }}_2+\epsilon$.

The Taylor series expansion of ${\Phi }_2^{*}\left(W_r^{*},C_1^{*},B_1^{*},C_2^{*},B_2^{*}\right)$ at $W_r^{*}={\widehat{W}}_r$, $C_1^{*}={\widehat{C}}_1$, $B_1^{*}={\widehat{B}}_1$, $C_2^{*}={\widehat{C}}_2$, $B_2^{*}={\widehat{B}}_2$ is derived: where $O_h$ is high-order term, ${\Phi }_{2C_1},{\Phi }_{2B_1},{\Phi }_{2C_2},{\Phi }_{2B_2},{\Phi }_{2W_r}$ is the derivative of $W_r^{*},C_1^{*},B_1^{*},C_2^{*},B_2^{*}$ to ${\Phi }_2$ respectively. The above partial derivative is the Jacobian matrix determinant, like:

Taking the above derivative results (32)–(36) into the approximation error (29), we can obtain the following equation: where ${\Delta }_0={\widehat{W}}^T{O}_h+{\epsilon }_0={\widehat{W}}^T{O}_h+{\tilde{W}}^T{\tilde{\Phi }}_2+\epsilon$ is a lumped higher-order approximation error term, assumed to be bounded by $\left|{\Delta }_0\right|\le {\overline{\Delta }}_d$.

Figure 2 is a block diagram of the proposed adaptive DTSMC using the FTDO. From the Figure 2, the designed controller consists of an equivalent control item, an approximate switching control item, and a compensated disturbance item.

The detailed information is developed as: where $\left\{\begin{array}{l}{I}_1=b\left(c+\lambda \right)\\ I_2=c\left[f\left(x\right)-\ddot{r}-\ddot{p}\right]+\dot{f}\left(x\right)-\stackrel{⃛}{r}-\stackrel{⃛}{p}\\ I_3=\lambda c\left(\dot{e}-\dot{p}\right)+\lambda \left[f\left(x\right)-\ddot{r}-\ddot{p}\right]\end{array}\right.$.