On active disturbance rejection control for lower-triangular systems with mismatched nonlinear uncertainties and unknown time-varying control coefficients

Abstract

It is a challenging issue to achieve the normal operation of control systems despite mismatched nonlinear uncertainties and unknown time-varying control coefficients. Based on the signs of control coefficients rather than nominal values or approximative mathematical expressions, the paper proposes a new active disturbance rejection control to tackle mismatched nonlinear uncertainties and unknown values of time-varying control coefficients. The design procedure can be concluded by three steps: determining the equivalent integrators chain form, constructing the extended state observer to estimate total disturbance and designing a dynamical system to generate the input approaching the desired input signal. Then, under a mild assumption for mismatched nonlinear uncertainties and unknown time-varying control coefficients, the paper rigorously analyzes the bounds of tracking error, estimating error and the error between the actual and desired inputs. Based on the presented error bounds, the tracking error with respect to the desired trajectory can be close to zero during the whole time period by suitably enlarging the observer parameter. The theoretical results reveal the strong robustness of the proposed method to mismatched nonlinear uncertainties and unknown time-varying control coefficients. Finally, by constructing the relationship between the observer parameter and the parameter in dynamical input design, the adjustable controller parameters remain observer parameter and feedback gain, which is friendly to practitioners.

Appendix

1.1 Proof of Proposition 1

Part I: The analysis of the mapping $\varphi$. With the combination of the dynamics (3) and the explicit form of ${\tilde{x}}_i~(1\le i \le n)$ (7), it can be directly verified that ${\tilde{x}}_i = \dot{{\tilde{x}}}_{i-1}$ for $2\le i \le n$. Then, for the transformation (7), we will prove that the term

$$\begin{aligned}&{\tilde{\tau }}_i \triangleq \Sigma _{j=1}^{i-2} \frac{d^{j-1}}{dt^{j-1}} \left( \frac{d(\Pi _{k=1}^{i-j-1}\theta _k(t))}{dt} x_{i-j} \right) \nonumber \\&\quad \quad \quad + \Sigma _{j=1}^{i-1} \frac{d^{j-1}}{dt^{j-1}} \left( (\Pi _{k=0}^{i-j-1} \theta _k(t)) \phi _{i-j} \right) \end{aligned}$$

(63)

is a function dependent on $(x_1,\cdots , x_{i-1},t)$ for $2\le i\le n$ by mathematical induction in the following three steps.

Step 1 (Consider the case that $i=2$). It can be obtained that

$$\begin{aligned} {\tilde{\tau }}_2 = \theta _0(t) \phi _1(x_1,t) = \phi _1(x_1,t). \end{aligned}$$

(64)

Hence, ${\tilde{\tau }}_2$ is a function dependent on $(x_1,t)$.

Step 2 (Consider the case that $i=k~(2 \le k\le n-1)$). Suppose that ${\tilde{\tau }}_k~(2 \le k\le n-1)$ is a function dependent on $(x_1,\cdots , x_{k-1},t)$.

Step 3 (Consider the case that $i=k+1~(2 \le k\le n-1)$). Since ${\tilde{\tau }}_k$ is a function dependent on $(x_1,\cdots , x_{k-1},t)$, the state ${\tilde{x}}_k$ has the following form:

$$\begin{aligned} {\tilde{x}}_k(t) = (\Pi _{j=1}^{k-1} \theta _j(t)) x_i (t) + {\tilde{\tau }}_k(x_1,\cdots , x_{k-1},t).\nonumber \\ \end{aligned}$$

(65)

By taking the derivative of ${\tilde{x}}_k$ and utilizing the dynamics (3), there is

$$\begin{aligned} \begin{aligned} \dot{{\tilde{x}}}_k(t)&={\tilde{x}}_{k+1}(t) \\&= (\Pi _{j=1}^{k-1} \theta _j(t)) (\theta _k(t) x_{k+1}(t) + \phi _k(x_1,\cdots ,x_k,t)) \\&\quad + \frac{d(\Pi _{j=1}^{k-1} \theta _j(t))}{dt} x_k (t)\\&\quad + \frac{\partial {\tilde{\tau }}_k(x_1,\cdots , x_{k-1},t)}{\partial t}\\&\quad + \Sigma _{j=1}^{k-1} \frac{\partial {\tilde{\tau }}_k(x_1,\cdots , x_{k-1},t)}{\partial x_j}\\ {}&(\theta _j x_{j+1}+ \phi _j(x_1,\cdots ,x_j,t)). \end{aligned} \end{aligned}$$

(66)

By comparing the form (66) with (7), it can be obtained that

$$\begin{aligned} \begin{aligned} {\tilde{\tau }}_{k+1}&= (\Pi _{j=1}^{k-1} \theta _j(t)) \phi _k(x_1,\cdots ,x_k,t) \\&\quad + \frac{d(\Pi _{j=1}^{k-1} \theta _j(t))}{dt} x_k (t) + \frac{\partial {\tilde{\tau }}_k}{\partial t} \\&\quad + \Sigma _{j=1}^{k-1} \frac{\partial {\tilde{\tau }}_k}{\partial x_j} (\theta _j x_{j+1}+ \phi _j(x_1,\cdots ,x_j,t)). \end{aligned} \end{aligned}$$

(67)

Due to the supposition in Step 2, (67) illustrates that ${\tilde{\tau }}_{k+1}$ is a function dependent on $(x_1,\cdots , x_{k},t)$.

Based on the fact that ${\tilde{\tau }}_i$ is a function dependent on $(x_1,\cdots , x_{i-1},t)$ for $2\le i\le n$, (7) can be rewritten in the following form.

$$\begin{aligned} \left\{ \begin{aligned}&{\tilde{x}}_1 = x_1,\\&{\tilde{x}}_i = (\Pi _{j=1}^{i-1} \theta _j(t)) x_i + {\tilde{\tau }}_i(x_1,\cdots ,x_{i-1},t),\quad 2\le i \le n. \end{aligned} \right. \end{aligned}$$

(68)

According to Assumption 2, the term $\Pi _{j=1}^{i-1} \theta _j(t)$ is nonzero for $2\le i \le n$. Then, the mapping from $({\tilde{x}},t)$ to (x, t) can be determined by the following recursive way.

From (68), the functions $\varphi _1$ and $\varphi _2$ can be obtained as follows.

$$\begin{aligned}&\varphi _1({\tilde{x}},t) = {\tilde{x}}_1, \quad \nonumber \\&\varphi _2({\tilde{x}},t) = \frac{{\tilde{x}}_2 - {\tilde{\tau }}_2( x_1,t )}{\theta _1(t)} =\frac{{\tilde{x}}_2 - {\tilde{\tau }}_2( \varphi _1({\tilde{x}},t), t )}{\theta _1(t)}.\nonumber \\ \end{aligned}$$

(69)

Suppose that $\varphi _i({\tilde{x}},t)$ exists for $2\le i\le k \le n-1$. Owing to (68), we can acquire the function $\varphi _{k+1}({\tilde{x}},t)$ as follows.

$$\begin{aligned}&\varphi _{k+1}({\tilde{x}},t) = \frac{{\tilde{x}}_{k+1} - {\tilde{\tau }}_{k+1} (\varphi _1({\tilde{x}},t) ,\cdots , \varphi _k({\tilde{x}},t),t)}{\Pi _{j=1}^{k} \theta _j(t)},\quad \nonumber \\&\quad 2\le k\le n-1. \end{aligned}$$

(70)

Additionally, it is obvious that $\varphi _{n+1}({\tilde{x}},t) = t$. Moreover, the inverse of the mapping $\varphi$ can be directly obtained by (68).

Then, the bounds of $\varphi _i({\tilde{x}},t)~(1\le i\le n+1)$ are analyzed. Owing to the dynamics (3), the expression of ${\tilde{\tau }}_i$ (63) implies that ${\tilde{\tau }}_i$ is composed of the finite sums and products of $x_j~(1\le j \le i-1)$, $\phi _j~(1\le j \le i-1)$, the partial derivatives of $\phi _j~(1\le j \le i-1)$ up to $(i-j-1)$-th order and $\theta _j^{(p)}~(1\le j \le i-1,0\le p \le i-j-1)$. According to Assumption 2, there exist continuous functions ${\tilde{\psi }}_{\tau ,i}(x_1,\cdots ,x_{i-1})~(2\le i \le n)$ such that

$$\begin{aligned}&\sup _{t\ge t_0 } |{\tilde{\tau }}_i (x_1,\cdots ,x_{i-1},t)| \le {\tilde{\psi }}_{\tau ,i}(x_1,\cdots ,x_{i-1}),\quad \nonumber \\&\quad \forall [x_1~\cdots ~x_{i-1}] \in R^{i-1},~2\le i \le n. \end{aligned}$$

(71)

By introducing the non-decreasing continuous function

$$\begin{aligned}&\psi _{\tau ,i}(\varrho _x) \triangleq \sup _{\Vert [x_1~\cdots ~x_{i-1}]\Vert \le \varrho _x} {\tilde{\psi }}_{\tau ,i} (x_1,\cdots ,x_{i-1}) \end{aligned}$$

for $\varrho _x \ge 0$, (71) implies that

$$\begin{aligned}&\sup _{t\ge t_0, \Vert [x_1~\cdots ~x_{i-1}]\Vert \le \varrho _x} |{\tilde{\tau }}_i (x_1,\cdots ,x_{i-1},t)| \nonumber \\&\quad \le \psi _{\tau ,i}(\varrho _x),\quad \forall \varrho _x\ge 0,~2\le i \le n. \end{aligned}$$

(72)

Next, the bounds of $\varphi _i$ are presented by recursive method. Denote ${\tilde{\varrho }}_x$ as a nonnegative constant. Based on (69), the bounds of $\varphi _1$ and $\varphi _2$ are shown as follows.

$$\begin{aligned} \left\{ \begin{aligned} \sup _{t\ge t_0,\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} |\varphi _1({\tilde{x}},t)|&\le \sup _{\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} |{\tilde{x}}_1| \le M_{\varphi ,1}({\tilde{\varrho }}_x) \triangleq {\tilde{\varrho }}_x,\\ \sup _{t\ge t_0, \Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} |\varphi _2({\tilde{x}},t)|&\le \sup _{t\ge t_0,\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} \frac{|{\tilde{x}}_2| + |{\tilde{\tau }}_2 (\varphi _1({\tilde{x}},t),t)|}{ | \theta _1(t)|} \\&\le M_{\varphi ,2}({\tilde{\varrho }}_x) \triangleq \frac{{\tilde{\varrho }}_x + \psi _{\tau ,2}(M_{\varphi ,1}({\tilde{\varrho }}_x))}{{\underline{M}}_{\theta ,1}}. \end{aligned} \right. \nonumber \\ \end{aligned}$$

(73)

Notice that $M_{\varphi ,1}(\cdot )$ and $M_{\varphi ,2}(\cdot )$ are increasing functions. Suppose that

$$\begin{aligned} \sup _{t\ge t_0,\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} |\varphi _i ({\tilde{x}},t)| \le M_{\varphi ,i}({\tilde{\varrho }}_x) \end{aligned}$$

where $M_{\varphi ,i}$ is a continuous increasing function for $2\le i \le k$, $2\le k\le n-1$ and ${\tilde{\varrho }}_x\ge 0$. Then, the bound of $\varphi _{k+1} ({\tilde{x}},t)$ can be obtained as follows.

$$\begin{aligned} \begin{aligned}&\sup _{t\ge t_0,\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} |\varphi _{k+1} ({\tilde{x}},t)|\\&\quad \le \sup _{t\ge t_0,\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} \frac{ |{\tilde{x}}_{k+1}| + |{\tilde{\tau }}_{k+1} (\varphi _1({\tilde{x}},t) , \cdots , \varphi _k({\tilde{x}},t),t)|}{ |\Pi _{j=1}^{k} \theta _j(t)|}\\&\quad \le M_{\varphi ,k+1}({\tilde{\varrho }}_x) \\&\qquad \triangleq \frac{{\tilde{\varrho }}_x + \psi _{\tau ,k+1}\left( \sqrt{ \Sigma _{j=1}^{k} (M_{\varphi ,j}({\tilde{\varrho }}_x))^2 } \right) }{ \Pi _{j=1}^{k} {\underline{M}}_{\theta ,j}}. \end{aligned} \end{aligned}$$

(74)

Hence, it can be concluded that there exist continuous increasing functions $M_{\varphi ,i}~(1\le i \le n)$ such that $\sup _{t\ge t_0,\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} |\varphi _i ({\tilde{x}},t)| \le M_{\varphi ,i}({\tilde{\varrho }}_x)$ for ${\tilde{\varrho }}_x\ge 0$ and $1\le i \le n$.

By taking the partial derivatives of $\varphi _i$ along the dynamics (3), it can be deduced from the similar analysis (69)–(74) that there exist continuous functions $M_{\frac{\partial \varphi }{\partial {\tilde{x}}},i}$ and $M_{\frac{\partial \varphi }{\partial t},i}$ such that

$$\begin{aligned}&\sup _{t\ge t_0,\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} \left\| \frac{\partial \varphi _i ({\tilde{x}},t)}{\partial {\tilde{x}}} \right\| \nonumber \\&\quad \le M_{\frac{\partial \varphi }{\partial {\tilde{x}}},i}({\tilde{\varrho }}_x),~~ \nonumber \\&\sup _{t\ge t_0,\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} \left| \frac{\partial \varphi _i ({\tilde{x}},t)}{\partial t} \right| \nonumber \\&\quad \le M_{\frac{\partial \varphi }{\partial t},i}({\tilde{\varrho }}_x),~~1\le i\le n. \end{aligned}$$

(75)

for any ${\tilde{\varrho }}_x \ge 0$. Due to (73)–(75), (10) is proved by defining

$$\begin{aligned}&\psi _{\varphi }({\tilde{\varrho }}_x) \triangleq \max _{1\le i\le n} \left\{ M_{\varphi ,i}({\tilde{\varrho }}_x),M_{\frac{\partial \varphi }{\partial {\tilde{x}}},i}({\tilde{\varrho }}_x) M_{\frac{\partial \varphi }{\partial t},i}({\tilde{\varrho }}_x) \right\} . \end{aligned}$$

Part II: The analysis of the mapping $\gamma$. Based on the analysis of ${\tilde{\tau }}_i$, the mapping $\gamma$ can be directly determined by (68). Combined with the bounds of ${\tilde{\tau }}_i$ (71)–(72), (68) further implies that

$$\begin{aligned} \left\{ \begin{aligned} \sup _{t\ge t_0, \Vert x\Vert \le \varrho _x} |\gamma _1(x,t)| \le&|x_1| \le M_{\gamma ,1}(\varrho _x) \triangleq \varrho _x,\\ \sup _{t\ge t_0, \Vert x\Vert \le \varrho _x} |\gamma _i(x,t)| \le&\sup _{t\ge t_0, \Vert x\Vert \le \varrho _x} \left( |\Pi _{j=1}^{i-1} \theta _j(t)| |x_i| + |{\tilde{\tau }}_i| \right) \\ \le&M_{\gamma ,i}(\varrho _x) \triangleq \varrho _x\Pi _{j=1}{\bar{M}}_{\theta ,i} + \psi _{\tau ,i}(\varrho _x),\quad 2\le i\le n, \end{aligned} \right. \end{aligned}$$

(76)

for any given $\varrho _x\ge 0$. Notice that $M_{\gamma ,i}(\varrho _x)~(1\le i\le n)$ are continuous increasing functions.

Similarly, by taking the partial derivatives of $\gamma _i$ along the dynamics (3), it can be deduced from Assumption 2 that there exist continuous functions $M_{\frac{\partial \gamma }{\partial x},i}$ and $M_{\frac{\partial \gamma }{\partial t},i}$ such that

$$\begin{aligned}&\sup _{t\ge t_0,\Vert x\Vert \le \varrho _x} \left\| \frac{\partial \gamma _i (x,t)}{\partial x} \right\| \le M_{\frac{\partial \gamma }{\partial x},i}(\varrho _x),~~ \nonumber \\&\sup _{t\ge t_0,\Vert x\Vert \le \varrho _x} \left| \frac{\partial \gamma _i (x,t)}{\partial t} \right| \le M_{\frac{\partial \gamma }{\partial t},i}(\varrho _x),~ \forall ~1\le i\le n,\nonumber \\ \end{aligned}$$

(77)

for any $\varrho _x \ge 0$. Based on (76)–(77) and the notation

$$\begin{aligned} \psi _{\gamma }(\varrho _x) \triangleq \max _{1\le i\le n} \left\{ M_{\gamma ,i}(\varrho _x),M_{\frac{\partial \gamma }{\partial x},i}(\varrho _x) M_{\frac{\partial \gamma }{\partial t},i}(\varrho _x) \right\} , \end{aligned}$$

then (9) is proved.

1.2 Proof of Proposition 2

Firstly, the bounds of b(t), $(b(t))^{-1}$ and ${\dot{b}}(t)$ are analyzed. According to the bounds of $\theta _i(t)$, $(\theta _i(t))^{-1}$ and ${\dot{\theta }}_i(t)$ in Assumption 2, the expression of b(t) (12) implies that

$$\begin{aligned} \left\{ \begin{aligned}&|b(t)| \le \Pi _{i=1}^{n} {\bar{M}}_{\theta ,i},\\&|b^{-1}(t)| = |\Pi _{i=1}^{n} \theta _i^{-1}(t)| \le \Pi _{i=1}^{n} {\underline{M}}_{\theta ,i}^{-1},\\&|{\dot{b}}(t)| =\Sigma _{j=1}^{n} |{\dot{\theta }}_j(t)| \Pi _{1\le i\le n,i\ne j} |\theta _i(t)| \le \Pi _{i=1}^{n} {\bar{M}}_{\theta ,i}. \end{aligned} \right. \end{aligned}$$

(78)

By denoting $\psi _b = \max \{ \Pi _{i=1}^{n} {\bar{M}}_{\theta ,i}, \Pi _{i=1}^{n} {\underline{M}}_{\theta ,i}^{-1} \}$, (28) is proved.

Next, the bound of $f({\tilde{x}},t)$ is analyzed. Proposition 1 illustrates that $\varphi _{n+1-j}({\tilde{x}},t) = x_{n+1-j}$ for $1\le j\le n-1$. Additionally, ${\tilde{\phi }}_i({\tilde{x}},t) = \phi _i(x_1,\cdots ,x_i,t)$ for $1 \le i \le n$. From the expression of $f({\tilde{x}},t)$ (12), $f({\tilde{x}},t)$ can be regarded as a function of (x, t), denoted as ${\bar{f}}(x,t)$.

Owing to the dynamics (3) and the notation ${\bar{f}}(x,t)$, it can be verified that ${\bar{f}}(x,t)$ is composed of the finite sums and products of $x_j~(1\le j \le n)$, $\phi _j~(1\le j \le n)$, the partial derivatives of $\phi _j~(1\le j \le n)$ up to $(n-j)$-th order and $\theta _j^{(p)}~(1\le j \le n-1,0\le p \le n-j-1)$. Due to Assumption 2, there exists a continuous function $\psi _{f,1}(x)$ such that

$$\begin{aligned} \sup _{t\ge t_0} |{\bar{f}}(x,t)| \le \psi _{f,1}(x), \end{aligned}$$

(79)

which further implies that

$$\begin{aligned}&\sup _{t\ge t_0, \Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} |f({\tilde{x}},t)| = \sup _{t\ge t_0, \Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} \nonumber \\&\quad |{\bar{f}}(x,t)| \le \sup _{\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} \psi _{f,1}(x),\quad \forall {\tilde{\varrho }}_x \ge 0. \end{aligned}$$

(80)

Recalling the continuous increasing function $\psi _\gamma (\cdot )$ in Proposition 1, owing to inverse function theorem, the inverse function of $\psi _\gamma (\cdot )$ can be denoted as $\psi _\gamma ^{-1}(\cdot )$ which is a continuous function in $[0,\infty )$. According to (8)–(9) in Proposition 1, the following equation holds.

$$\begin{aligned} \begin{aligned} \sup _{\Vert x\Vert \le \psi _\gamma ^{-1}({\tilde{\varrho }}_x/\sqrt{n})} \Vert {\tilde{x}}\Vert&= \sup _{\Vert x\Vert \le \psi _\gamma ^{-1}({\tilde{\varrho }}_x/\sqrt{n})} \left\| \begin{bmatrix} \gamma _1(x,t)&\cdots&\gamma _n(x,t) \end{bmatrix} \right\| \\&\le \sqrt{n} \psi _\gamma (\psi _\gamma ^{-1}({\tilde{\varrho }}_x/\sqrt{n})) = {\tilde{\varrho }}_x. \end{aligned} \end{aligned}$$

(81)

Based on (81), it can be verified that

$$\begin{aligned}&\left\{ {\tilde{x}}|~\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x \right\} \nonumber \\&\quad \subset \left\{ {\tilde{x}}|~\Vert x\Vert \le \psi _\gamma ^{-1}({\tilde{\varrho }}_x/\sqrt{n}) \right\} ,\quad \nonumber \\&\quad \forall {\tilde{\varrho }}_x\ge 0. \end{aligned}$$

(82)

With the combination of (80) and (82), it can be obtained that

$$\begin{aligned}&\sup _{t\ge t_0, \Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} |f({\tilde{x}},t)| \le \sup _{\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x} \psi _{f,1}(x) \le {\bar{\psi }}_{f,1}({\tilde{\varrho }}_x)\nonumber \\&\quad \triangleq \sup _{\Vert x\Vert \le \psi _\gamma ^{-1}({\tilde{\varrho }}_x/\sqrt{n})} \psi _{f,1}(x),~~ \forall {\tilde{\varrho }}_x\ge 0. \end{aligned}$$

(83)

Due to the continuity of $\psi _{\gamma }^{-1}(\cdot )$ and $\psi _{f,1}(\cdot )$, it can be deduced that ${\bar{\psi }}_{f,1}({\tilde{\varrho }}_x)$ is a continuous function with respect to the variable ${\tilde{\varrho }}_x$.

From Proposition 1, $\varphi _{n+1}({\tilde{x}},t)=t$ and

$$\begin{aligned} x_i = \varphi _i ({\tilde{x}},t),\quad 1\le i\le n. \end{aligned}$$

(84)

Then, the partial derivatives of $f({\tilde{x}},t)$ can be expressed as follows.

$$\begin{aligned} \left\{ \begin{aligned} \frac{\partial f({\tilde{x}},t)}{\partial {\tilde{x}}_i}&= \frac{\partial {\bar{f}}(\varphi _1({\tilde{x}},t),\cdots ,\varphi _n({\tilde{x}},t),t)}{\partial {\tilde{x}}_i} \\&= \Sigma _{j=1}^{n} \frac{\partial {\bar{f}}(x_1,\cdots ,x_n,t)}{\partial x_j} \frac{\partial \varphi _j({\tilde{x}},t)}{\partial {\tilde{x}}_i},\quad 1\le i\le n,\\ \frac{\partial f({\tilde{x}},t)}{\partial t}&= \frac{\partial {\bar{f}}(x_1,\cdots ,x_n,t)}{\partial t}. \end{aligned} \right. \nonumber \\ \end{aligned}$$

(85)

Due to Assumption 2, with the similar derivation as (79), there exist continuous functions $\psi _{fx,j}$ and $\psi _{ft}$ such that

$$\begin{aligned}&\sup _{t\ge t_0} \left| \frac{\partial {\bar{f}}(x_1,\cdots ,x_n,t)}{\partial t} \right| \le \psi _{ft}(x),~~ \nonumber \\&\sup _{t\ge t_0} \left| \frac{\partial {\bar{f}}(x_1,\cdots ,x_n,t)}{\partial x_j} \right| \le \psi _{fx,j}(x),~~1\le j \le n.\nonumber \\ \end{aligned}$$

(86)

Based on the bounds of $\frac{\partial \varphi _j}{\partial x_i}$ (10) and with the similar procedure as (79)–(83), there exist continuous functions ${\bar{\psi }}_{f,2}(\cdot )$ and ${\bar{\psi }}_{f,3}(\cdot )$ such that

$$\begin{aligned}&\sup _{t\ge t_0, \Vert {\tilde{x}} \Vert \le {\tilde{\varrho }}_x} \left\| \frac{\partial f({\tilde{x}},t)}{\partial {\tilde{x}}}\right\| \le {\bar{\psi }}_{f,2}({\tilde{\varrho }}_x),\quad \nonumber \\&\sup _{t\ge t_0, \Vert {\tilde{x}} \Vert \le {\tilde{\varrho }}_x} \left| \frac{\partial f({\tilde{x}},t)}{\partial t}\right| \le {\bar{\psi }}_{f,3}({\tilde{\varrho }}_x),\quad \forall {\tilde{\varrho }}_x\ge 0. \end{aligned}$$

(87)

By denoting $\psi _f({\tilde{\varrho }}_x) \triangleq \max \{{\bar{\psi }}_{f,1}({\tilde{\varrho }}_x),{\bar{\psi }}_{f,2}({\tilde{\varrho }}_x),{\bar{\psi }}_{f,3}({\tilde{\varrho }}_x)\}$, (83) and (87) imply (29).

1.3 Proof of Proposition 3

For $t\in [t_0,t_u)$, the control input satisfies that $u(t)=0$. Hence, the dynamics of ${\tilde{x}}$ in (11) can be rewritten as follows.

$$\begin{aligned} \dot{{\tilde{x}}}(t) = A {{\tilde{x}}}(t) + B f({\tilde{x}},t). \end{aligned}$$

(88)

Combined with the dynamics of ${\tilde{x}}^*(t)$ (17), the dynamics of e(t) for $t\in [t_0,t_u)$ is shown as follows.

$$\begin{aligned} {\dot{e}}(t) = A e(t) + B \Delta _{e0}(e,t),\quad t\in [t_0,t_u), \end{aligned}$$

(89)

where

$$\begin{aligned} \Delta _{e0}(e,t) = f({\tilde{x}}^*+e,t)+K^T({\tilde{x}}^*(t)-{\bar{r}}(t))-r^{(n)}(t).\nonumber \\ \end{aligned}$$

(90)

Then, consider the case that $t\in [t_u,\infty )$. With the notation (1) and the desired input (18), the dynamics of ${\tilde{x}}$ in (11) can be reformulated as follows.

$$\begin{aligned} \begin{aligned} \dot{{\tilde{x}}}(t)&= A {\tilde{x}}(t) + B (b(t)u(t)+f({\tilde{x}},t) -b(xt) u^*(t)+b(t)u^*(t))\\&= A {\tilde{x}}(t)-BK^T({\tilde{x}}(t)-{\bar{r}}(t))+Br^{(n)}(t)+Bb(t)\delta _u(t)\\&= A_K e(t) +Bb(t)\delta _u(t)+\dot{{\tilde{x}}}^*(t). \end{aligned}\nonumber \\ \end{aligned}$$

(91)

Owing to the dynamics of ${\tilde{x}}^*(t)$ (17), the dynamics of e(t) for $t\in [t_u,\infty )$ is obtained as follows.

$$\begin{aligned} {\dot{e}}(t) = A_K e(t) + Bb(t) \delta _u(t),\quad t\in [t_u,\infty ). \end{aligned}$$

(92)

Before analyzing the closed-loop form of $\zeta (t)$ and $\delta _u(t)$, the derivative of $u^*(t)$ is calculated due to (18).

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{u}}^*(t) = -\frac{{\dot{b}}(t)(-f({\tilde{x}},t) - K^T ({\tilde{x}}-{\bar{r}})+r^{(n)})}{b^2(t)} \\&\quad \quad \quad \quad \, +\frac{-\frac{\partial f}{\partial {\tilde{x}}} (A(e+{\tilde{x}}^*)+ B f(e+{\tilde{x}}^*,t) ) - \frac{\partial f}{\partial t} }{b(t)}\\&\quad \quad \quad \quad \, +\frac{- K^T ( A(e+{\tilde{x}}^*) + B f(e+{\tilde{x}}^*,t) -\dot{{\bar{r}}})+r^{(n+1)}}{b(t)} ,\\&\quad \quad \qquad \qquad t\in (t_0,t_u),\\&{\dot{u}}^*(t) = -\frac{{\dot{b}}(t)(-f({\tilde{x}},t) - K^T ({\tilde{x}}-{\bar{r}})+r^{(n)})}{b^2(t)} \\&\quad \quad \quad \quad \, + \frac{ -\frac{\partial f}{\partial {\tilde{x}}}( A_K e +Bb\delta _u+\dot{{\tilde{x}}}^*) - \frac{\partial f}{\partial t} }{b(t)} \\&\quad \quad \quad \quad \, + \frac{ -K^T ( A_K e +Bb\delta _u+\dot{{\tilde{x}}}^*-\dot{{\bar{r}}} ) +r^{(n+1)}}{b(t)},\\&\quad \quad \qquad \quad \quad t\in (t_u,\infty ). \end{aligned} \right. \nonumber \\ \end{aligned}$$

(93)

With the help of Remark 8, the derivative of u(t) is shown as follows.

$$\begin{aligned} {\dot{u}}(t)= & {} -sgn(b(t)) \kappa (\omega _o) ({\hat{f}}_t(t) + K^T (\hat{{\tilde{x}}}(t) \nonumber \\&\quad \,\, -{\bar{r}}(t))-r^{(n)}(t)) \nonumber \\&= -\kappa |b(t)| \frac{f_t({\tilde{x}},u,t) \!+\! K^T ( {\tilde{x}}(t)\!-\!{\bar{r}}(t)) \!-\!r^{(n)}(t)\!-\!K_e^T T_1 \zeta }{b(t)} \nonumber \\&= -\kappa |b(t)|\delta _u(t) + sgn(b(t)) \kappa K_e^T T_1 \zeta (t), \end{aligned}$$

(94)

where $K_e = [K^T~1]^T$. Since $T_1^{-1} A_L T_1 = \omega _o A_\varsigma$, the dynamics of $\zeta$ for $t\in [t_0,t_u)$ can be obtained from (11) and (15):

$$\begin{aligned} {\dot{\zeta }}(t) = \omega _o A_\varsigma \zeta (t) +B_f \Delta _{\zeta 0}(e,t),\quad t\in [t_0,t_u), \end{aligned}$$

(95)

where

$$\begin{aligned} \Delta _{\zeta 0}(e,t)= & {} \frac{\hbox {d} f({\tilde{x}},t)}{\hbox {d}t}|_{\text {along}~(88)} = \frac{\partial f}{\partial {\tilde{x}}} (A(e+{\tilde{x}}^*) \nonumber \\&+ B f(e+{\tilde{x}}^*,t) ) + \frac{\partial f}{\partial t}. \end{aligned}$$

(96)

Owing to (94), the dynamics of $\zeta (t)$ for $t\in [t_u,\infty )$ is presented as follows.

$$\begin{aligned} {\dot{\zeta }}(t)= & {} \omega _o A_\varsigma \zeta (t)\nonumber \\&+B_f \Delta _{\zeta 1}(e,\zeta ,\delta _u,\omega _o,\kappa ,t),\quad t\in [t_0,t_u), \end{aligned}$$

(97)

where

$$\begin{aligned} \begin{aligned} \Delta _{\zeta 1}(e,\zeta ,\delta _u,\omega _o,\kappa ,t)&= \frac{\hbox {d}(b(t)u(t)+f({\tilde{x}},t))}{\hbox {d}t}|_{\text {along }(91)} \\&= {\dot{b}} \delta _u + {\dot{b}}(-f(e+{\tilde{x}}^*,t)\\&\quad -K^T(e+{\tilde{x}}^*-{\bar{r}})+r^{(n)})/b \\&\quad +b(-\kappa |b|\delta _u + sgn(b) \kappa K_e^T T_1 \zeta )\\&\quad +\frac{\partial f}{\partial {\tilde{x}}}( A_K e +Bb\delta _u+\dot{{\tilde{x}}}^*) + \frac{\partial f}{\partial t}. \end{aligned} \end{aligned}$$

(98)

With the combination of (93) and (94), the derivative of $\delta _u(t)$ is shown as follows.

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{\delta }}_u(t) = -|b(t)| \kappa \delta _u(t) + \Delta _{\delta _u 0}(e,\zeta ,\omega _o,\kappa ,t), \quad t\in [t_0,t_u),\\&{\dot{\delta }}_u(t) = -|b(t)| \kappa \delta _u(t) + \Delta _{\delta _u 1}(e,\zeta ,\delta _u,\omega _o,\kappa ,t),\quad t\in [t_u,\infty ), \end{aligned} \right. \nonumber \\ \end{aligned}$$

(99)

where

$$\begin{aligned} \left\{ \begin{aligned}&\Delta _{\delta _u 0}(e,\zeta ,\omega _o,\kappa ,t) = \hbox {sgn}(b(t)) \kappa K_e^T T_1 \zeta +\frac{{\dot{b}}(t)(-f({\tilde{x}},t) - K^T (e+{\tilde{x}}^*-{\bar{r}})+r^{(n)})}{b^2(t)} \\&\qquad \qquad \qquad \qquad \quad -\frac{-\frac{\partial f}{\partial {\tilde{x}}} (A(e+{\tilde{x}}^*)+ B f(e+{\tilde{x}}^*,t) ) - \frac{\partial f}{\partial t} - K^T ( A(e+{\tilde{x}}^*) + B f(e+{\tilde{x}}^*,t) -\dot{{\bar{r}}})+r^{(n+1)}}{b(t)},\\&\Delta _{\delta _u 1}(e,\zeta ,\delta _u,\omega _o,\kappa ,t) = \hbox {sgn}(b(t)) \kappa K_e^T T_1 \zeta +\frac{{\dot{b}}(t)(-f({\tilde{x}},t) - K^T (e+{\tilde{x}}^*-{\bar{r}})+r^{(n)})}{b^2(t)} \\&\qquad \qquad \qquad \qquad \qquad \quad \qquad - \frac{ -\frac{\partial f}{\partial {\tilde{x}}}( A_K e +Bb\delta _u+\dot{{\tilde{x}}}^*) - \frac{\partial f}{\partial t} -K^T ( A_K e +Bb\delta _u+\dot{{\tilde{x}}}^*-\dot{{\bar{r}}} ) +r^{(n+1)}}{b(t)}. \end{aligned} \right. \end{aligned}$$

(100)

As for the dynamics of z(t), it can be directly obtained from (11) that

$$\begin{aligned} {\dot{z}}(t) = g(z,\varphi _1({\tilde{x}},t),\cdots ,\varphi _n({\tilde{x}},t),t), \quad t\ge t_0. \end{aligned}$$

(101)

Based on the dynamics of e, $\zeta$, $\delta _u$ and z, i.e., (89), (92), (95), (97), (99) and (101), the closed-loop form can be written as (31)–(32).

Next, we will prove that the bounds of $\Delta _{e0}$, $\Delta _{\zeta 0}$, $\Delta _{\delta _u 0}$, $\Delta _{e1}$, $\Delta _{\zeta 1}$ and $\Delta _{\delta _u 1}$ satisfy (33) for $e\in \{e|~\Vert e\Vert \le \varrho _e \}$, $\zeta \in \{\zeta |~\Vert \zeta \Vert \le \varrho _\zeta \}$, $\delta _u \in \{\delta _u|~|\delta _u|\le \varrho _u \}$ and $\omega _o\in \{\omega _o|~\omega _o\ge \omega _o^* \}$ with any given positives $\varrho _e$, $\varrho _\zeta$, $\varrho _u$ and $\omega _o^*$. Hence, we only need to present the detailed expressions of the functions $\pi _{e0}(\cdot )$, $\pi _{\zeta 0}(\cdot )$, $\pi _{\zeta 1}(\cdot )$, $\pi _{\delta _u}(\cdot )$, $\pi _{\delta _u 0}(\cdot )$ and $\pi _{\delta _u 1}(\cdot )$.

Due to Assumption 1, there is

$$\begin{aligned}&\sup _{t\ge t_0}\Vert {\bar{r}}(t)\Vert \le nM_r,\quad \nonumber \\&\sup _{t\ge t_0}\{ |r^{(n)}(t)|, |r^{(n+1)}(t)|\} \le M_r. \end{aligned}$$

(102)

Combined with the dynamics of ${\tilde{x}}^*$ (17), there exists a positive constant $M_{x^*}$ such that

$$\begin{aligned} \sup _{t\ge t_0}\{ \Vert {\tilde{x}}^*(t)\Vert , \Vert \dot{{\tilde{x}}}^*(t)\Vert \} \le M_{x^*}. \end{aligned}$$

(103)

According to the bounds of b, ${\dot{b}}$, $b^{-1}$, f, $\frac{\partial f}{\partial {\tilde{x}}}$ and $\frac{\partial f}{\partial t}$ shown in Proposition 2, (90) implies that

$$\begin{aligned}&\sup _{\Vert e\Vert \le \varrho _e}|\Delta _{e0}(e,t)| \le \pi _{e0}(\varrho _e) \nonumber \\&\quad \triangleq \psi _f(\varrho _e+M_{x^*}) + \Vert K\Vert (M_{x^*}+n M_r) + M_r,\nonumber \\&\qquad \forall \varrho _e \ge 0. \end{aligned}$$

(104)

Due to (92), it can be directly obtained that

$$\begin{aligned} |\Delta _{e1}(\delta _u,t)| = |b(t) \delta _u(t)| \le \psi _b |\delta _u|. \end{aligned}$$

(105)

Owing to (96) and (98), the bounds of $\Delta _{\zeta 0}$ and $\Delta _{\zeta 1}$ are provided as follows.

$$\begin{aligned}&\sup _{\Vert e\Vert \le \varrho _e}|\Delta _{\zeta 0}(e,t)| \le \pi _{\zeta 0}(\varrho _e)\\&\quad \triangleq \psi _f(\varrho _e+M_{x^*}) \left( 1+ \Vert A\Vert (\varrho _e+ M_{x^*}) + \psi _f(\varrho _e+M_{x^*}) \right) ,\\&\quad \sup _{\Vert e\Vert \le \varrho _e, |\delta _u| \le \varrho _u, \omega _o \ge \omega _o^*}|\Delta _{\zeta 1}(e,\zeta ,\delta _u,\omega _o,\kappa ,t)| \le \pi _{\zeta 1}(\varrho _e) \\&\quad + \pi _{ \omega }(\omega _o^*) \kappa \Vert \zeta \Vert + (\kappa +1) \pi _{\delta _u}(\varrho _e,\varrho _u), \end{aligned}$$

for any $\varrho _e\ge 0$, $\varrho _u\ge 0$ and $\omega _o^*\ge 0$, where $\pi _{\zeta 1}(\varrho _e) =\psi _b^2( \psi _f(\varrho _e+M_{x^*})+\Vert K\Vert (\varrho _e+M_{x^*}+nM_r)+M_r ) + \psi _f(\varrho _e+M_{x^*}) ( 1+ \Vert A_K\Vert \varrho _e + M_{x^*} )$, $\pi _\omega (\omega _o^*)=\max \{1,\psi _b\}\Vert K_e\Vert \cdot$ $\Vert T_1(\omega _o^*)\Vert$ and $\pi _{\delta _u}(\varrho _e,\varrho _u)=\max \{\psi _b^2 \varrho _u, \psi _b \varrho _u (1+\psi _f(\varrho _e+M_{x^*}))\}$. With the help of Proposition 2, the following bounds of $\Delta _{\delta _u 1}$ and $\Delta _{\delta _u 2}$ are obtained from (100).

$$\begin{aligned} \left\{ \begin{aligned}&\sup _{\Vert e\Vert \le \varrho _e}|\Delta _{\delta _u 0}| \le \pi _{\delta _u 0}(\varrho _e)+\pi _{ \omega }(\omega _o^*)\kappa \Vert \zeta \Vert ,\\&\sup _{\Vert e\Vert \le \varrho _e,|\delta _u|\le \varrho _u}|\Delta _{\delta _u 1}| \le \pi _{\delta _u 1}(\varrho _e , \varrho _u) + \pi _{ \omega }(\omega _o^*)\kappa \Vert \zeta \Vert , \end{aligned} \right. \quad \forall \varrho _e\ge 0,~\varrho _u\ge 0,\nonumber \\ \end{aligned}$$

(106)

where $\pi _{\delta _u 0}(\varrho _e) =\max \{\psi _b, \psi _b^3\} ( \psi _f(\varrho _e+M_{x^*})( 2+\Vert A\Vert (\varrho _e+M_{x^*})+\psi _f(\varrho _e+M_{x^*}) ) + \Vert K\Vert (1+\Vert A\Vert )(\varrho _e+M_{x^*}+nM_r + \psi _f(\varrho _e+M_{x^*})) + 2M_r )$ and $\pi _{\delta _u 0}(\varrho _e,\varrho _u) = \pi _{\delta _u 0}(\varrho _e) + \psi _b ( \psi _f(\varrho _e+M_{x^*}) +\Vert K\Vert ) ( \Vert A_K\Vert \varrho _e+ \psi _b \varrho _u + M_{x^*} + nM_r )$.

Based on (104)–(106), (33) is proved.