Fixed-time integral sliding mode tracking control of a wheeled mobile robot

Fixed-time disturbance observer

Firstly, for the attitude error subsystem (7), define an auxiliary variable as

where $$ \varpi_1 $$ satisfies

The parameters $$ l_{11} $$ and $$ l_{12} $$ are positive constants with $$ l_{11}>k_{1m}/J $$. Let variable exponential coefficient $$ \gamma_1 = \frac{\lambda_0\varsigma_1^2}{1+\mu_0\varsigma_1^2} $$ with $$ \lambda_0 $$ and $$ \mu_0 $$ satisfying $$ 0<\mu_0<1 $$ and $$ 1+\mu_0<\lambda_0 $$.

Theorem 1For the second-order subsystem (7), if the disturbance observer is constructed as

then it can estimate $$ d_1 $$ accurately in a fixed time. That is to say, the observation error $$ \tilde{d}_1 = d_1-\hat{d}_1 $$ can converge into a small region within a fixed time.

Proof of Theorem 1 Select a Lyapunov function as $$ V_2 = \varsigma^2_1 $$, differentiating it, one has

where $$ \epsilon_1 $$ is a positive constant.

Case 1 When $$ V_2 > 1 $$ and $$ |\varsigma_1|> 1 $$, one has $$ \operatorname{erf}(|\varsigma_1|)>\operatorname{erf}(1) $$ and $$ \frac{\lambda_0 \varsigma_1^2}{1+\mu_0 \varsigma_1^2}\geq \frac{\lambda_0}{1+\mu_0}>1 $$. Then (12) can be rewritten as

As $$ l_{12}\operatorname{erf}(1)-l_{11}\epsilon_1>0 $$ and $$ \bar{\gamma}_1 = \frac{\lambda_0+\mu_0+1}{2(1+\mu_0)}>1 $$, then all the solutions of $$ \{V_2> 1\} $$ will reach the set $$ \{V_2 \leq 1\} $$ within a fixed time $$ t_{f1} \leq \frac{1}{2(l_{12}\operatorname{erf}(1)-l_{11}\epsilon_1)(\bar{\gamma}_1-1)} $$.

Case 2 In the converse case $$ V_2 \leq 1 $$, one has

As $$ 1+\mu_0 \varsigma^2_1 \geq 1 $$ and $$ |\varsigma_1|\leq 1 $$, it can be obtained that $$ \min\big(|\varsigma_1|^{\gamma_1}\big) \geq \min\big(|\varsigma_1|^{\lambda_0 \varsigma^2_1}\big) = e^{\frac{-\lambda_0}{2e}} $$. Considering the Lemma Lemma3, then (14) is converted into the following form

with $$ b_1 = 2l_{12}e^{\frac{-\lambda_0}{2e}} $$, and $$ \tilde{\epsilon} = 2l_{11}\epsilon_1+2l_{12}\epsilon_2 $$. When $$ 0<l_{13}<1 $$, and $$ \tilde{\epsilon}-(1-l_{13})b_1V^{\frac{1}{2}}_2 \geq 0 $$, (15) can be simplified as $$ \dot{V}_2 \leq -l_{13} b_1V^{\frac{1}{2}}_2 $$. Then, the solution of $$ V_2 $$ will reach a small set $$ \Delta_1 $$, which is defined as $$ \Delta_1 = \big\{\varsigma_1|V_1(\varsigma_1)\leq(\frac{\tilde{\epsilon}}{b_1(1-l_{13})})^2\big\} $$ within a settling time $$ t_{f2}\leq \frac{2}{b_1l_{13}} $$.

In view of the above two cases, the auxiliary variable $$ \varsigma_1 $$ will converge into a small set $$ \Delta_1 = \big\{\varsigma_1|V_1(\varsigma_1)\leq (\frac{\tilde{\epsilon}}{b_1(1-l_{13})})^2\big\} $$ within settling time $$ t_f = t_{f1}+t_{f2} $$.

Then, the disturbance observation error

The disturbance $$ d_1 $$ is bounded according to Assumption 1. Thus, the disturbance observer (11) can estimate $$ d_1 $$ accurately, and the observation error $$ \tilde{d}_1 $$ can remain in a small set

3.1.2. Fixed-time sliding mode controller

For the subsystem (7), define $$ \omega_{e} = \omega-\omega_{r} $$. A fixed-time integral sliding mode surface is introduced as follows ^[22]

with $$ 0<p_i<1 $$, $$ q_i>1 $$, and $$ (i = 1, 2) $$. For any $$ x \in \rm{R} $$, $$ \alpha\in \rm{R}^+ $$, the notation is defined as $$ \lceil{x}\rfloor^{\alpha} = |x|^{\alpha}{\rm{sign}}(x) $$. Based on the sliding mode surface as (17), the fixed-time controller is designed as follows:

where $$ \alpha_i, k_{1i}, (i = 1, 2) $$ are positive constants, $$ \alpha_3 $$ satisfies $$ \alpha_3\geq\frac{k_{1m}}{J} $$. In addition, $$ p_i, q_i, (i = 1, 2, 3) $$ are all positive odd integers with $$ 0<p_i<1, q_i>1 $$.

Theorem 2For the second-order system (7), if the fixed-time controller is constructed in the form of (18), then the real sliding mode variable will converge into a small set within a fixed time.

Proof of Theorem 2 Choose a Lyapunov function as $$ V_3 = \frac{1}{2}s^2_1 $$ and refer to Lemma 2 to Lemma 4, the time derivative of $$ V_3 $$ is

where $$ \bar{p}_3 = \frac{p_3+1}{2}, \bar{q}_3 = \frac{q_3+1}{2}, \bar{\vartheta}_1 = \alpha_3\vartheta_1 $$ with $$ \vartheta_1 $$ being a positive constant. By using Lemma Lemma4, the second-order system (7) is fixed-time stable. The sliding mode surface $$ s_1 $$ will converge into a small region $$ \Delta_3 = \Big\{s_1|V(s_1)\leq {\rm{min}}\big\{\big(\frac{c_2}{\alpha_1 2^{\bar{p}_3}}\big)^{\frac{1}{\bar{p}_3}}, \big(\frac{c_2}{\alpha_2 2^{\bar{q}_3}}\big)^{\frac{1}{\bar{p}_3}}\big\}\Big\} $$ around the origin in a fixed time $$ t_{s_1} $$, which is determined by $$ t_{s_1}\leq \frac{1}{\alpha_12^{\bar{p}_3}\phi_1(1-\bar{p}_3)}+\frac{1}{\alpha_2 2^{\bar{q}_3}\phi_1(\bar{q}_3-1)} $$. Then, one can obtain that variables $$ \theta_{e} $$ and $$ \omega_{e} $$ converge to zero along the real sliding mode in a fixed time^[23].

3.2.1. Fixed-time disturbance observer

Introduce the following auxiliary variable for the simplified third-order subsystem (20)

where $$ \varpi_2 $$ satisfies

where $$ \gamma_2 = \frac{\lambda_3\varsigma_1^2}{1+\mu_3\varsigma_1^2} $$, and $$ \lambda_3 $$ and $$ \mu_3 $$ are integers satisfying the constraints: $$ 0<\mu_3<1 $$, $$ 1+\mu_3<\lambda_3 $$. The parameters $$ l_{21} $$ and $$ l_{22} $$ are positive constants with $$ l_{21}>k_{2m} $$ and $$ l_{22}>0 $$.

Theorem 3For the simplified third-order subsystem (20), a fixed-time disturbance observer is developed in the form of

then it can estimate $$ d_2 $$ in a fixed time, and the observation error $$ \tilde{d}_2 = d_2-\hat{d}_2 $$ can converge into a small region around the origin within a fixed time $$ t_{d_2} $$.

Proof of Theorem 3 Similar to the proof of Theorem 1.

3.2.2. Fixed-time sliding mode controller

For the third-order subsystem (20), introduce the following auxiliary variable:

where $$ \lambda_1 $$$$ \lambda_2 $$, $$ k_1 $$, and $$ \epsilon_3 $$ are positive constants, $$ \lambda_2\leq\lambda_1\operatorname{erf}(1) $$. Let $$ \gamma_3 = \frac{\lambda_4 {x_e}^{2}}{1+\mu_4{x_e}^{2}} $$, with $$ \lambda_4 $$ and $$ \mu_4 $$ being integers and $$ 0<\mu_4<1 $$, $$ 1+\mu_4<\lambda_4 $$.

Select a fixed-time sliding mode surface as:

where $$ k_{21} $$ and $$ k_{22} $$ are positive constants, $$ 0<p_i<1 $$ and $$ q_i>1 $$, $$ i = 4, 5 $$ are positive odd integers.

Theorem 4For the third-order subsystem (20), if the fixed-time sliding mode surface is chosen as (25) and the fixed-time controller is designed as (26),

then the sliding mode surface $$ s_2 $$ is fixed-time stable, which will converge into a small region of origin within settling time $$ t_{s_2}\leq \frac{1}{\alpha_4 2^{\bar{p}_6}\phi_2(1-\bar{p}_6)}+\frac{1}{\alpha_5 2^{\bar{q}_6}\phi_2(\bar{q}_6-1)} $$.

Proof of Theorem 4 The proof process will be conducted in 3 steps: (1) After the angular error $$ \theta_{e} $$ converges to zero according to Theorem 2, $$ s_2 $$ and the auxiliary variable $$ \xi $$ can converge into a small region around the origin within a fixed time; (2) The error variables $$ x_e, y_e $$ can converge into a small region around the origin within a fixed time; (3) It should be proved that $$ x_e $$ and $$ y_e $$ do not escape to infinity before the angular error $$ \theta_{e} $$ converges to zero.

Step 1 Select a positive Lyapunov function $$ V_4 = \frac{1}{2} {s}_2^2 $$, differentiating it and substituting (24)-(26) yields to

where $$ \bar{p}_6 = \frac{p_6+1}{2}, \bar{q}_6 = \frac{q_6+1}{2} $$, $$ \bar{\vartheta}_2 = \beta_3 \vartheta_2 $$ with $$ \vartheta_2 $$ being a positive constant. Using the Lemma Lemma4, the third-order subsystem (14) is fixed-time stable, and $$ s_2 $$ will converge into a small set $$ \Delta_4 = \Big\{s_2|V(s_2)\leq {\rm{min}}\big\{\big(\frac{c_3}{\alpha_1 2^{\bar{p}_6}}\big)^{\frac{1}{\bar{p}_6}}, \big(\frac{c_3}{\alpha_2 2^{\bar{q}_6}}\big)^{\frac{1}{\bar{p}_6}}\big\}\Big\} $$ around zero in the fixed time $$ t_{s_2} $$, which is determined by

Then, $$ s_2 $$ will hold in a small region of origin, which guarantees a real sliding mode surface ^[23]. Therefore, the auxiliary variable $$ \xi $$ and its derivative $$ \dot{\xi} $$ will also converge into the origin along the sliding mode surface ^[24].

Step 2 According to (25), when the auxiliary $$ \dot{\xi} = 0 $$, one has

Choose a Lyapunov function as $$ V_5 = x^{2}_e $$, the time derivative of $$ V_5 $$ is

where $$ \epsilon_4 $$ is a positive constant. The rest of the proof is similar to the proof of Theorem 2. There exists a constant $$ 0<\vartheta_3<1 $$ such that the variable $$ x_e $$ will reach and keep in a small region $$ \Delta_5 $$ around the origin within a fixed time $$ T_2 $$:

Then, it can be obtained that the $$ \dot{x}_e $$ is a uniformly continuous form (29). Employ Barbalat Lemma ^[25] to prove $$ \dot{x}_e \rightarrow 0 $$ as $$ t \rightarrow \infty $$, then $$ \dot{x}_e $$ is bounded after the variable $$ x_e $$ converges. Hence, there exists a small region $$ \Delta_5 $$ around the origin that $$ y_e $$ can converge into $$ \Delta_5 $$.

Step 3 Before the angular error $$ \theta_{e} $$ converges to zero, $$ \theta_e\neq0 $$, such that subsystem (13) cannot be simplified as (19). It should be proved that system state variables $$ x_e, y_e, $$ and $$ v $$ are bounded before the angular error $$ \theta_{e} $$ converges to zero.

Consider the following bounded function:

The time derivative of $$ V_6 $$ is

Let $$ \eta_1 = \sqrt{x^{2}_e+y^{2}_e+|v|}\geq\eta>1 $$, then one has the following inequalities: $$ |x_e|\leq\eta_1, |y_e|\leq\eta_1, |v|\leq \eta_1 $$. Furthermore, there exist positive constants $$ a_i(i = 3, 4), b_l, c_l, (l = 4, 5, ..., 9) $$, which satisfy $$ |x_e|^{\frac{\lambda_4}{\mu_4}}\leq a_3 \eta_1 $$, $$ |x_e|^{\frac{\lambda_4}{\mu_4}-1}\leq a_4 \eta_1 $$, $$ |s_2|^{p_6}<b_4+c_4\eta_1 $$, $$ |s_2|^{q_6}<b_5+c_5\eta_1 $$, $$ |\xi|^{p_4}<b_6+c_6\eta_1 $$, $$ |\xi|^{q_4}<b_7+c_7\eta_1 $$, $$ |\dot{\xi}|^{p_5}<b_8+c_8\eta_1 $$, $$ |\dot{\xi}|^{q_5}<b_9+c_9\eta_1 $$. According to Theorem 3, the state variable $$ \theta_e, \omega $$ will converge into the origin within a fixed time $$ t_{s_1} $$, then one has $$ \theta_{e}\leq \theta_m $$, $$ |\omega|\leq \omega_m $$, $$ |\tilde{d}_2|<k_d $$. Further $$ |\dot{y}_e|<|\dot{x}_e|\leq v_{1_{\rm{max}}}+\omega_m \eta_1 $$ can be obtained. Then, (33) can be simplified as

where $$ K $$ and $$ \varrho_1 $$ satisfy the following constraints:

On the contrary, if $$ \eta_1>1 $$, there exists a positive constant $$ \varrho_2 $$, which satisfies $$ \dot{V}_6\leq\varrho_2 $$. One has $$ \dot{V}_6\leq KV_6+\varrho_3 $$ for the state variable $$ x_e, y_e, v $$. Further, before the angular error $$ \theta_{e} $$ converges to zero, one can obtain

Remark 1 The auxiliary variable $$ \xi $$ in (24) can reduce the order of the third-order subsystem, which simplifies the process of the controller design. In addition, the controller developed in this paper can guarantee that the system state variables converge in a fixed time and the chattering problem is solved by using the error function erf($$ \cdot $$). Furthermore, utilizing the variable exponent coefficient in (24) avoids the common singularity problem.