Global Stabilization of Triangular Systems with Time-Delayed Dynamic Input Perturbations

TL;DR

This paper presents a robust adaptive control method for a broad class of uncertain nonlinear systems with time delays, ensuring global stability despite complex uncertainties and unmodeled dynamics.

Contribution

It introduces a delay-independent adaptive output-feedback controller for strict-feedback-like systems with dynamic input uncertainties and time delays, expanding control design capabilities.

Findings

Achieves global stabilization under complex uncertainties.

Designs a delay-independent robust adaptive controller.

Handles unmeasured states and time-varying delays effectively.

Abstract

A control design approach is developed for a general class of uncertain strict-feedback-like nonlinear systems with dynamic uncertain input nonlinearities with time delays. The system structure considered in this paper includes a nominal uncertain strict-feedback-like subsystem, the input signal to which is generated by an uncertain nonlinear input unmodeled dynamics that is driven by the entire system state (including unmeasured state variables) and is also allowed to depend on time delayed versions of the system state variable and control input signals. The system also includes additive uncertain nonlinear functions, coupled nonlinear appended dynamics, and uncertain dynamic input nonlinearities with time-varying uncertain time delays. The proposed control design approach provides a globally stabilizing delay-independent robust adaptive output-feedback dynamic controller based on a dual dynamic high-gain scaling based structure.

Tables

	$R(x_1,\hat{\theta },\dot{\hat{\theta }})$	$=\max (\overline{R},16{\nu }_a{\overline{w}}_1(x_1,\hat{\theta },\dot{\hat{\theta }}),{\left(16{\nu }_a{\overline{w}}_2\left(x_1\right)\right)}^2,{\left(\frac{4{\overline{q}}_3(x_1,\hat{\theta }){\vartheta }_1\left(x_1\right)}{\hat{\theta }{\varphi }_{(1,2)}\left(x_1\right)}\right)}^2,\frac{4\left({\overline{q}}_4(x_1,\hat{\theta })+\hat{\theta }{\overline{q}}_5\left(x_1\right)\right)}{\hat{\theta }{\varphi }_{(1,2)}\left(x_1\right){\vartheta }_1\left(x_1\right)})$		(34)
	$\Omega (x_1,\hat{\theta },\dot{\hat{\theta }},r)$	$=\max (\overline{\Omega },2{\nu }_{b}r{\overline{w}}_1(x_1,\hat{\theta },\dot{\hat{\theta }}),2{\nu }_b{r}^{\frac{3}{2}}{\overline{w}}_2\left(x_1\right),2\left({\overline{q}}_3(x_1,\hat{\theta }){\vartheta }_1^2\left(x_1\right)r^{\frac{3}{2}}+r{\overline{q}}_4(x_1,\hat{\theta })+\hat{\theta }r{\overline{q}}_5\left(x_1\right)\right))$		(35)

	$R_u(y,u,r,\dot{r},\hat{\theta },\dot{\hat{\theta }},\varpi )$	$=\max ({\overline{R}}_u,\frac{4r^n{\overline{\Xi }}_{u1}(y,u,r,\dot{r},\hat{\theta },\dot{\hat{\theta }})}{c_u\underset{¯}{\mu }},\frac{8r^n\overline{\Pi }\left({\overline{\Xi }}_{u2}(y,u,r,\varpi )+{\nu }_u\right)\left(1+{\overline{u}}_d^2(y,u,r,\varpi )\right)}{c_u\underset{¯}{\mu }})$		(36)
	${\Omega }_u(y,u,r,\dot{r},\hat{\theta },\dot{\hat{\theta }},\varpi ,r_u)$	$=\max ({\overline{\Omega }}_u,\frac{2{\Pi }^2\left(r_u\right)r^n}{c_u{\Pi }^{\prime }\left(r_u\right)}\left(\frac{{\overline{\Xi }}_{u1}(y,u,r,\dot{r},\hat{\theta },\dot{\hat{\theta }})}{\Pi \left(r_u\right)}+{\overline{\Xi }}_{u2}(y,u,r,\varpi )+{\nu }_u\right))$		(37)

Equations

$${\dot{x}}_i$$

$${\dot{x}}_n$$

$$y$$

$$\displaystyle\left.\begin{array}[]{rcl}\dot{\hat{x}}_{i}&\!\!=&\!\!\phi_{(i,i+1)}(x_{1})[\hat{x}_{i+1}+r^{i}f_{i+1}(x_{1})]\\ &\!\!&\!\!-r^{i-1}g_{i}(x_{1})[\hat{x}_{2}+rf_{2}(x_{1})]\\ &\!\!&\!\!-(i-1)\dot{r}r^{i-2}f_{i}(x_{1})\,\,\,,\,\,\,2\leq i\leq n-1\\ \dot{\hat{x}}_{n}&\!\!=&\!\!\tilde{u}-r^{n-1}g_{n}(x_{1})[\hat{x}_{2}+rf_{2}(x_{1})]\\ &\!\!&\!\!-(n-1)\dot{r}r^{n-2}f_{n}(x_{1})\\ \hat{x}&\!\!=&\!\![\hat{x}_{2},\ldots,\hat{x}_{n}]^{T}\\ f_{i}(x_{1})&\!\!=&\!\!\int_{0}^{x_{1}}\frac{g_{i}(\pi)}{\phi_{(1,2)}(\pi)}d\pi\ ,\ i=2,\ldots,n\end{array}\right\}$$

$$\displaystyle\!\!\!\!\left.\begin{array}[]{rcl}\varpi_{2}&\!\!\!\!=&\!\!\!\!\frac{\hat{x}_{2}+rf_{2}(x_{1})+\vartheta(x_{1},\hat{\theta})}{r}\\ \varpi_{i}&\!\!\!\!=&\!\!\!\!\frac{\hat{x}_{i}+r^{i-1}f_{i}(x_{1})}{r^{i-1}}\,\,\,;\,\,\,i=3,\ldots,n\end{array}\!\!\!\right\}\ ;\ \varpi=[\varpi_{2},\ldots,\varpi_{n}]^{T}$$

$$\vartheta (x_1,\hat{\theta })=\hat{\theta }{x}_1{\vartheta }_1(x_1)$$

$$\tilde{u}$$

$$\dot{\zeta }$$

$$\dot{r}$$

$${\dot{r}}_u$$

$$\times {\Omega }_u(y,u,r,\dot{r},\hat{\theta },\dot{\hat{\theta }},\varpi ,r_u);r_u(0)\ge 1$$

$$\dot{\hat{\theta }}$$

$${\dot{x}}_1$$

$${\dot{x}}_2$$

$${\dot{x}}_3$$

$${\dot{x}}_4$$

$${\dot{\psi }}_1$$

$${\dot{\psi }}_2$$

$$P_o{A}_o(x_1)+A_o^T(x_1)P_o\le -{\nu }_o{I}_{n-1}-{\tilde{\nu }}_o{\varphi }_{(2,3)}(x_1)C^{T}C$$

$${\underline{\nu }}_o{I}_{n-1}\le P_o{\tilde{D}}_o+{\tilde{D}}_o{P}_o\le {\overline{\nu }}_o{I}_{n-1}$$

$$P_c{A}_c(x_1)+A_c^T(x_1)P_c\le -{\nu }_c{\varphi }_{(2,3)}(x_1)I_{n-1}$$

$${\underline{\nu }}_c{I}_{n-1}\le P_c{\tilde{D}}_c+{\tilde{D}}_c{P}_c\le {\overline{\nu }}_c{I}_{n-1}.$$

$$\dot{ϵ}$$

$${\dot{V}}_x$$

$$-x_1{\varphi }_{(1,2)}\vartheta -\frac{1}{2}\frac{\dot{r}}{r^2}{x}_1^2-\frac{2cr{ϵ}^T{P}_{o}B{u}_d}{r^{n-1}}$$

$$+r{w}_1(x_1,\hat{\theta },\dot{\hat{\theta }})[|ϵ{|}^2+|\varpi {|}^2]+q_1(x_1)x_1^2+{\theta }^{\ast }{q}_2(x_1)x_1^2$$

$$\dot{u}$$

$${\dot{u}}_d$$

$${\chi }_1$$

$$+\frac{\partial \mu (t,x,\psi ,u)}{\partial x_n}\tilde{u}+n{r}^{n-1}\dot{r}K(x_1)\varpi$$

$$+\sum _{i=2}^{n-1}\frac{\partial \mu (t,x,\psi ,u)}{\partial x_i}[{\varphi }_{(i,i+1)}(x_1)x_{i+1}+{\varphi }_i(t,x)]$$

$$+\frac{\partial \mu (t,x,\psi ,u)}{\partial \psi }{q}_{\psi }(t,x,\psi ,u,x_{\Delta },{\psi }_{\Delta },u_{\Delta })$$

$$+\frac{\partial \mu (t,x,\psi ,u)}{\partial t}+r^n\frac{\partial K}{\partial x_1}\varpi [{\varphi }_{(1,2)}(x_1)x_2+{\varphi }_1(t,x)]$$

$$+r^{n}K(x_1)\dot{\varpi }.$$

$${\beta }_1(y,u,r,\dot{r},\hat{\theta })$$

$$+\sum _{j=2}^{n-1}{\varphi }_{(i,i+1)}(x_1)r^i+r^{n-1}{n}^{\frac{3}{2}}\Gamma (x_1)\}$$

$$+n{r}^{n-1}\dot{r}|K(x_1)|+r^{n-1}\dot{r}|K(x_1)||D_c|$$

$$+r^n\left|\frac{\partial K(x_1)}{\partial x_1}\right||\hat{\theta }{\vartheta }_1(x_1)x_1|{\varphi }_{(1,2)}(x_1)$$

$$+r^{n+1}|K(x_1)||A_c(x_1)|+r^{n+1}|K(x_1)||G(x_1)|$$

$$+r^n|K(x_1)|\hat{\theta }|{\vartheta }_1(x_1)+{\vartheta }_1^{\prime }(x_1)x_1|{\varphi }_{(1,2)}(x_1),$$

$$V_u$$

$${\dot{V}}_u$$

$$\displaystyle\quad\!-\!\bigg{(}\frac{\Pi^{\prime}(r_{u})\dot{r}_{u}}{\Pi(r_{u})}+\frac{n\dot{r}}{r}\bigg{)}\frac{\ln(1+u_{d}^{2})}{2\Pi(r_{u})r^{n}}$$

$$+k_{\psi 1}\frac{{\alpha }_{\psi }(\psi )}{r^{2n-\frac{3}{2}}}+\frac{{\Xi }_{u1}(y,u,r,\dot{r},\hat{\theta },\dot{\hat{\theta }})}{\Pi (r_u)}\frac{u_d^2}{1+u_d^2}$$

$$+r{\tilde{w}}_1(x_1)[|ϵ{|}^2+|\varpi {|}^2]+\frac{1}{r}{\tilde{q}}_1(x_1,\hat{\theta })x_1^2+\frac{1}{r}{\theta }^{\ast }{\tilde{q}}_2(x_1)x_1^2$$

Full text

I Introduction

Consider the following class of uncertain nonlinear systems:

[TABLE]

Here, the strict-feedback-like subsystem with state $x=[x_{1},\ldots,x_{n}]^{T}$ represents a nominal system and the subsystem with state $\psi\in{\cal R}^{n_{\psi}}$ represents an appended input unmodeled dynamics. $u\in{\cal R}$ and $y\in{\cal R}^{2}$ are the control input and measured output, respectively. The subscript $\Delta$ is used to denote time delay, i.e., the notations $x_{\Delta}$, $\psi_{\Delta}$, and $u_{\Delta}$ refer to the time delayed versions of the signals $x$, $\psi$, and $u$, respectively, i.e., $x(t-\Delta)$, $\psi(t-\Delta)$, and $u(t-\Delta)$. Here, $\Delta$ is a (possibly time-varying) time delay111To simplify notation, the time argument is omitted when referring to signal values at time $t$, e.g., $u(t)$ is written simply as $u$; the time delayed signal values are written as $u_{\Delta}=u(t-\Delta)$, etc.. The functions $\phi_{(1,2)},\ldots,\phi_{(n-1,n)}$, are assumed to be known and continuous. $\phi_{1},\ldots,\phi_{n-1}$, $\mu$, and $q_{\psi}$ are uncertain continuous functions.

In (1), the $x$ subsystem can be regarded as a “nominal” system, which if the value of $\mu$ could be directly specified by the controller, would be of the form $\dot{x}_{i}=\phi_{(i,i+1)}(x_{1})x_{i+1}+\phi_{i}(t,x)\,,\,i=1,\ldots,n-1;\dot{x}_{n}=u$. In the actual system (1), the $x$ subsystem is driven by a nonlinear uncertain function $\mu$, representing an input perturbation which involves $\psi$, which is the unmeasured state of the input unmodeled dynamics, as well as time delays versions of $x$, $\psi$, and $u$. The control objective considered in this paper is to globally (i.e., starting from any initial condition) regulate the signals $x$ and $\psi$ in the system (1) asymptotically to zero under the various uncertainties described above and using measurement of the output $y$.

Control designs for various structures/classes of nonlinear dynamic systems including parametric and functional uncertainties, input nonlinearities and input unmodeled dynamics, time delays, etc., have been addressed in the literature (e.g., [36, 1, 2, 3, 4, 5, 12, 13, 37, 6, 38, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20] and references therein). Scaling based control designs for various types of triangular and non-triangular system structures have been addressed in [14, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

The application of the dynamic scaling technique to systems with uncertain input unmodeled dynamics such as in (1) was considered in [32, 33, 34, 35]. In [32, 33], nominal feedforward-like systems coupled with nonlinear input uncertainties were considered and a three-time-scale control design was developed that utilized two dynamic scaling parameters (one being essentially analogous to the scaling parameter that was utilized in our prior dual dynamic high-gain scaling based control designs [22] and the second scaling parameter being introduced specifically to handle the dynamic nonlinear input uncertainties). In [35], the three-time scale (i.e., utilizing two dynamic scaling parameters) control design approach was extended to nominal strict-feedback-like systems coupled with nonlinear input uncertainties. In [34], it was shown that (under certain structural conditions of the nominal system and nominal controller) a scaling based control redesign can be introduced to add robustness to input unmodeled dynamics to a general nonlinear system with a given nominal control design. Here, we consider the general class (1) of uncertain systems which includes time delays in the input unmodeled dynamics and we will show that the scaling-based design concept from [32, 35] can be applied to this uncertain system to provide global stabilization with robustness to uncertain input unmodeled dynamics. Specifically, while $q_{\psi}$ was required to be a function of $(t,x,\psi,u)$ at the current time $t$ in [35], the control design here addresses the system structure shown in (1) wherein $q_{\psi}$ involves time-delayed versions of both the state variables and the control input signals. The control design methodology developed in this paper is based on [35] and introduces refinements in the overall control design and the Lyapunov analysis to address the uncertain input unmodeled dynamics with time delays.

II Notations and Assumptions

Notations: With $k$ being any integer, the notation $I_{k}$ denotes an identity matrix of dimension $k\times k$. $|a|$ denotes Euclidean norm if $a$ is a column vector, absolute value if $a$ is a scalar, and Euclidean norm of the vector obtained by stacking all the columns of $a$ if $a$ is a (square or non-square) matrix. Given any symmetric positive-definite matrix $P$, the notations $\lambda_{max}(P)$ and $\lambda_{min}(P)$ denote its maximum and minimum eigenvalues, respectively, of the matrix. If $\alpha:[0,a)\rightarrow[0,\infty)$ is a strictly increasing continuous function with $\alpha(0)=0$, then it is said to belong to class ${\cal K}$. If, the class ${\cal K}$ definition holds with $a=\infty$ and $\alpha(r)\rightarrow\infty$ as $r\rightarrow\infty$, then $\alpha$ is said to belong to class ${\cal K}_{\infty}$.

Here, we consider the output-feedback stabilization problem, i.e., only $y$ is assumed to be measured in the system (1). It is assumed that the functions $\phi_{i},\phi_{(i,i+1)},q_{\psi},$ and $\mu$ appearing in the system dynamics satisfy sufficient conditions for local existence and uniqueness of solution trajectories for the system (e.g., local Lipschitz conditions). The control objective in this paper is to make $x(t)$ and $\psi(t)$ in the system (1) asymptotically converge to zero as $t\rightarrow\infty$ starting from any initial conditions $x(0)\in{\cal R}^{n}$ and $\psi(0)\in S_{\psi}$. Here, $S_{\psi}$ is some known subset (possibly non-compact or unbounded) of ${\cal R}^{n_{\psi}}$. Also, $\overline{S}_{\psi}$ denotes the set of all possible values of $\psi(t)$ over all time considering the set $S_{\psi}$ of possible initial values of $\psi$ and the dynamics of the $\psi$ state variables. Here, $\overline{S}_{\psi}$ could be simply ${\cal R}^{n_{\psi}}$ in general or could be a subset of ${\cal R}^{n_{\psi}}$. The assumptions on the system considered here are given below.

Assumption A1: (lower bound on magnitude of the upper diagonal terms $\phi_{(i,i+1)}$) A positive constant $\sigma$ exists such that $\phi_{(i,i+1)}(x_{1})\geq\sigma\,\,,\,\,1\leq i\leq n-1$ for all $x_{1}\in{\cal R}$.

Assumption A2: (inequality bound on the uncertain functions $\phi_{i}$) A known continuous function $\Gamma:{\cal R}\rightarrow{\cal R}^{+}$ and an unknown constant $\theta\geq 0$ exist such that for all $t\geq 0$, $x\in{\cal R}^{n}$, and $1\leq i\leq n$, the following inequalities hold: $|\phi_{i}(t,x)|\leq\Gamma(x_{1})[\theta|x_{1}|+\sum_{j=2}^{i}|x_{j}|]$.

Assumption A3: (cascading dominance of upper diagonal terms) The inequalities $\phi_{(i,i+1)}(x_{1})\geq\overline{\rho}_{i}\phi_{(i-1,i)}(x_{1})$ and $\phi_{(i,i+1)}(x_{1})\leq\underline{\rho}_{i}\phi_{(i-1,i)}(x_{1})$ are satisfied for all $x_{1}\in{\cal R}$ and $3\leq i\leq n-1$ with $\overline{\rho}_{i}$ and $\underline{\rho}_{i}$ being positive constants.

Assumption A4: (assumptions on the uncertain nonlinear “input perturbation” function $\mu$) Known non-negative continuous functions $\overline{\mu}(y,u)$, $\overline{\mu}_{1}(y,u)$, $\overline{\mu}_{1a}(x_{1})$, $\tilde{\mu}_{1}(y,u)$, and $\overline{\mu}_{2}(y,u)$, non-negative (possibly uncertain) functions $\overline{\mu}_{1\psi}(\psi)$, $\tilde{\mu}_{1\psi}(\psi)$, and $\overline{\mu}_{2\psi}(\psi)$, an uncertain constant $\theta$, and a known constant $\underline{\mu}$ exist such that for all time $t$, $x\in{\cal R}^{n}$, $\psi\in\overline{S}_{\psi}$, and $u\in{\cal R}$: (a) $\frac{\partial}{\partial u}\mu(t,x,\psi,u)\geq\underline{\mu}>0$; (b) $|\mu(t,x,\psi,u)|\leq\overline{\mu}(y,u)$; (c) |\frac{\partial\mu(t,x,\psi,u)}{\partial\psi}q_{\psi}(t,x,\psi,u,x_{\Delta},\psi_{\Delta},u_{\Delta})|\leq\overline{\mu}_{1}(y,u)\sum_{k=0}^{1}\Big{\{}\overline{\mu}_{1a}(x_{1}(t-k\Delta))[\theta|x_{1}(t-k\Delta)|+\sum_{j=2}^{n}|x_{j}(t-k\Delta)|]+\overline{\mu}_{1\psi}(\psi(t-k\Delta))\Big{\}}; (d) $|\frac{\partial\mu(t,x,\psi,u)}{\partial t}|\leq\tilde{\mu}_{1}(y,u)[\theta|x_{1}|+\sum_{j=2}^{n}|x_{j}|+\tilde{\mu}_{1\psi}(\psi)]$; (e) $|\frac{\partial\mu(t,x,\psi,u)}{\partial x}|\leq\overline{\mu}_{2}(y,u)+\overline{\mu}_{2\psi}(\psi)$. Also, for all $x\in{\cal R}^{n}$, $t\in{\cal R}$, and $\psi\in\overline{S}_{\psi}$, we have $\lim_{|u|\rightarrow\infty}|\mu(t,x,\psi,u)|=\infty$.

Assumption A5: (input-to-state stability – ISS – assumptions on the input unmodeled dynamics, i.e., the $\psi$ subsystem) A Lyapunov function $V_{\psi}:{\cal R}^{n_{\psi}}\rightarrow{\cal R}^{+}$ exists such that for all $t\geq 0$, $x\in{\cal R}^{n}$, $u\in{\cal R}$, and $\psi\in\overline{S}_{\psi}$, the following inequality holds: \frac{\partial V_{\psi}}{\partial\psi}q_{\psi}(t,x,\psi,u,x_{\Delta},\psi_{\Delta},u_{\Delta})\leq-\alpha_{\psi}(|\psi|)+\sum_{k=0}^{1}\Big{\{}\Gamma_{2}(x_{1}(t-k\Delta))[\theta x_{1}^{2}(t-k\Delta)+\sum_{j=2}^{n}x_{j}^{2}(t-k\Delta)+\gamma_{s}(y(t-k\Delta),u(t-k\Delta))]\Big{\}}, with $\alpha_{\psi}$ being a known class ${\cal K}_{\infty}$ function, $\gamma_{s}$ and $\Gamma_{2}$ being known non-negative functions, and $\theta$ being an unknown non-negative constant. Also, $\gamma_{s}(y,u)\leq\overline{\gamma}_{s}(x_{1})|\mu(t,x,\psi,u)|^{2}$ with $\overline{\gamma}_{s}$ being a known non-negative function. Furthermore, $\alpha_{\psi}(|\psi|)\geq\underline{V}_{\psi}V_{\psi}(\psi)$ and $[\overline{\mu}_{1\psi}^{2}(\psi)+\tilde{\mu}_{1\psi}^{2}(\psi)+\overline{\mu}_{2\psi}^{2}(\psi)]\leq\overline{k}_{\psi}\alpha_{\psi}(|\psi|)$ for all $\psi\in{\cal R}^{n_{\psi}}$ with $\underline{V}_{\psi}$ and $\overline{k}_{\psi}$ being known positive constants and $\overline{\mu}_{1\psi}$, $\tilde{\mu}_{1\psi}$, and $\overline{\mu}_{2\psi}$ being functions of $\psi$ as in Assumption A4.

Assumption A6: (assumptions on the time delay $\Delta$) The unknown time-varying time delay $\Delta$ is uniformly bounded in time and satisfies, for all time, the inequality given by $|\dot{\Delta}|\leq\overline{\Delta}<1$ with $\dot{\Delta}$ denoting $\frac{d}{dt}\Delta$ and with $\overline{\Delta}$ being a known positive constant.

Theorem 1: Under the Assumptions A1-A6, a positive constant $a_{\theta}$ and continuous functions $g_{i},i=2,\ldots,n$, $k_{i},i=2,\ldots,n$, and nonnegative continuous functions $\vartheta_{1}$, $\lambda$, $R$, $\Omega$, $R_{u}$, $\Omega_{u}$, and $\overline{Q}_{\theta}$ can be found such that the dynamic output-feedback controller given below (with $[\hat{x}_{2},\ldots,\hat{x}_{n},\zeta,r,r_{u},\hat{\theta}]^{T}$ comprising the state of the designed dynamic controller)

[TABLE]

[TABLE]

[TABLE]

when put in closed loop with the system with dynamics shown in (1) guarantees that for any initial conditions $(x(0),\psi(0),\hat{x}(0)$, $\zeta(0),r(0),r_{u}(0),\hat{\theta}(0))\in{\cal R}^{n}\times S_{\psi}\times{\cal R}^{n-1}\times{\cal R}\times[1,\infty)\times[1,\infty)\times[a_{\theta},\infty)$, all the closed-loop signals ($x_{1},\ldots,x_{n}$, $\psi$, $\hat{x}_{2},\ldots,\hat{x}_{n}$, $\zeta$, $r$, $r_{u}$, $\hat{\theta}$, $u$) are uniformly bounded over the time interval $[0,\infty)$ and, furthermore, the signals $x_{1},\ldots,x_{n},\psi,\hat{x}_{2},\ldots,\hat{x}_{n}$ converge to zero asymptotically as the time $t$ goes to $\infty$.

Remark 1: Comparing assumptions A1 through A6 given above with the corresponding assumptions in our earlier work [35], it is seen that A1 through A5 are essentially analogous to the assumptions considered before. The additional element in the system structure here is the presence of time delays in the input unmodeled dynamics. The required assumption on this additional element is addressed by Assumption A6. Assumption A6 is equivalent to the standard assumption utilized in the literature on time delayed systems (e.g., [33]) that essentially requires that the time delay value does not change faster than “real-time” (i.e., $|\dot{\Delta}|<dt/dt=1$). The proposed dynamic controller design approach can be applied to systems that also have time delays in other parts of the system (e.g., in the $\phi_{i}$ terms) and multiple time delay values (e.g., state and input time delays in the nominal system dynamics, multiple possible time delay values instead of a single $\Delta$, etc.) by appropriately adding additional terms in the overall system Lyapunov function. However, these additional possible time delays are not considered here so as to focus on the basic control design approach to handle the time delay $\Delta$ in the input unmodeled dynamics. $\diamond$

Remark 2: The proposed control design given in Theorem 1 comprises of a reduced-order observer (with state vector $[\hat{x}_{2},\ldots,\hat{x}_{n}]^{T}$) with dynamics in (9), the nominal control law (14), a dynamic state extension $\zeta$ with dynamics as in (15), dynamic scaling parameters $r$ and $r_{u}$ with dynamics as in (16) and (17), respectively, and an adaptation parameter $\hat{\theta}$ with dynamics as in (18). Here, the dynamic scaling parameters $r$ and $r_{u}$ are initialized with $r(0)\geq 1$ and $r_{u}(0)\geq 1$ and the dynamic adaptation parameter $\hat{\theta}$ is initialized with $\hat{\theta}(0)\geq a_{\theta}>0$. From the dynamics of these controller state variables, we see that $\dot{r}(t)$, $\dot{r}_{u}(t)$, and $\dot{\hat{\theta}}(t)$ are non-negative at all time $t$.It is noteworthy that the overall dynamic controller structure in Theorem 1 is essentially as in [35] and it is seen in Section III that the time delays in the input unmodeled dynamics are handled through changes in the designs of the functions $R$, $\Omega$, $R_{u}$, $\Omega_{u}$, etc., while retaining the overall controller structure. This is indeed illustrative of the flexibility and generality of the dynamic scaling-based controller design approach that enables (as noted in, for example [26, 34]) control designs for a wide range of classes of systems and uncertainty structures within a unified framework. $\diamond$

Remark 3: Consider the following system with output $y=[x_{1},x_{4}]^{T}$:

[TABLE]

where $x_{i,\Delta}$ and $\psi_{i,\Delta}$ denote $x_{i}(t-\Delta)$ and $\psi_{i}(t-\Delta)$, respectively. $\theta_{i},i=1,2,3$, $b_{i},i=1,\ldots,7$, and $a_{i},i=1,2$, are unknown constants; we assume that upper and lower bounds are known for $a_{i},i=1,2$, as $\overline{a}_{i}$ and $\underline{a}_{i}>0$, respectively. and upper bounds $\overline{b}_{i},i=1,\ldots,7$ are known for $|b_{i}|$. Also, an upper bound $\overline{\Delta}<1$ on $|\dot{\Delta}|$ is considered as known. This sixth-order system can be seen to satisfy the Assumptions A1-A6. The value of $\sigma$ in Assumption A1 can be picked as $3/4$. The function $\Gamma$ in Assumption A2 can be chosen to be $\Gamma(x_{1})=\max(1,\overline{b}_{1})|x_{1}|+x_{1}^{2}+\overline{b}_{1}x_{1}^{3}$ and the uncertain parameter $\theta$ can be defined as $\theta_{a}=\max(1,|\theta_{1}|,|\theta_{2}|,|\theta_{3}|)$. With $\overline{\rho}_{3}=0.8$, and $\underline{\rho}_{3}=4$, we see that Assumption A3 is also satisfied. Also, with $\underline{\mu}=\frac{2}{3}\underline{a}_{1}$, $\overline{\mu}(y,u)=3\overline{a}_{1}|u|\!+\!\overline{a}_{1}|x_{1}|u^{2}\!+\!(\overline{a}_{1}+\overline{a}_{2})(3+x_{1}^{2}+x_{4}^{2})|u|^{3}$, $\overline{\mu}_{1}(y,u)=[\overline{a}_{1}|u|+(\overline{a}_{1}+\overline{a}_{2})|u|^{3}][1+\overline{b}_{2}+\overline{b}_{3}|x_{1}|+\overline{b}_{4}+\overline{b}_{5}|u|+\overline{b}_{6}|x_{1}|+\overline{b}_{7}|u|]$, $\overline{\mu}_{1a}(x_{1})=1+|x_{1}|+x_{1}^{2}$, $\overline{\mu}_{1\psi}(\psi)=|\psi_{1}|+3|\psi_{2}|$, $\tilde{\mu}_{1}(y,u)=\overline{a}_{1}u^{2}$, $\tilde{\mu}_{1\psi}(\psi)=0$, $\overline{\mu}_{2}(y,u)=\overline{a}_{1}u^{2}+(\overline{a}_{1}+\overline{a}_{2})(1+2|x_{1}|+2|x_{4}|)|u|^{3}$, $\overline{\mu}_{2\psi}(\psi)=0$, and with $\theta$ appearing in Assumption A4 defined as $\theta_{b}=1$, it is seen that Assumption A4 is satisfied. Defining $V_{\psi}=\frac{1}{2}(\psi_{1}^{2}+\psi_{2}^{2})$, Assumption A5 holds with $\alpha_{\psi}(|\psi|)=\frac{1}{4}|\psi|^{2}$, $\theta$ given as $\theta_{b}=1$, $\Gamma_{2}(x_{1})=4(\overline{b}_{2}^{2}+\overline{b}_{3}^{4}x_{1}^{2}+\overline{b}_{3}^{4}x_{1}^{6}+\overline{b}_{4}^{2}x_{1}^{4}+\overline{b}_{5}^{2})+2(\overline{b}_{6}^{4}x_{1}^{2}+\overline{b}_{7}^{2})$, $\gamma_{s}(y,u)=(1+x_{1}^{2})^{2}u^{2}$, $\overline{\gamma}_{s}(x_{1})=[4(1+x_{1}^{2})/(3\underline{a}_{1})]^{2}$, $\overline{k}_{\psi}=36$, and $\underline{V}_{\psi}=\frac{1}{2}$. The overall uncertain parameter $\theta$ appearing in Assumptions A2 and A5 can be defined as $\theta=\max(\theta_{a},\theta_{b})$. Therefore, as described above, the example system (19) satisfies the Assumptions A1-A6 and the proposed control design methodology can be applied to this example system. $\diamond$

III Control Design and Stability Analysis

Choosing $g_{i},k_{i},i=2,\ldots,n$: Define $D_{o}=D_{c}=\mbox{diag}(1,2,\ldots,n-1)$ and $\tilde{D}_{o}=\tilde{D}_{c}=D_{o}\!-\frac{1}{2}I_{n-1}$. Let $C$ be the $1\times(n-1)$ matrix given as $[1,0\ldots,0]$. Let $A_{o}(x_{1})$ and $A_{c}(x_{1})$ be the square matrices of dimension $(n-1)\times(n-1)$ with $(i,j)^{th}$ elements defined as $A_{o_{(i,i+1)}}=\phi_{(i+1,i+2)}\,,\,i=1,\ldots,n-2$, $A_{o_{(i,1)}}=-g_{i+1}\,,\,i=1,\ldots,n-1$, $A_{c_{(i,i+1)}}=\phi_{(i+1,i+2)}\,,\,i=1,\ldots,n-2$ and $A_{c_{(n,j)}}=-k_{j+1}\,,\,j=1,\ldots,n-1$, and with all other elements being zero. Given Assumptions A1 and A3, we know ([23, 39, 40]) that constant $(n-1)\times(n-1)$ matrices $P_{o}>0$ and $P_{c}>0$, positive constants $\nu_{o}$, $\tilde{\nu}_{o}$ $\underline{\nu}_{o}$, $\overline{\nu}_{o}$, $\nu_{c}$, $\underline{\nu}_{c}$, and $\overline{\nu}_{c}$, and functions $g_{2},\ldots,g_{n}$,$k_{2},\ldots,k_{n}$ can be constructed such that the following two pairs of coupled Lyapunov inequalities are satisfied for all $x_{1}\!\in\!{\cal R}$:

[TABLE]

[TABLE]

The first of these pairs (20) can be considered the observer-context coupled Lyapunov inequalities that relates to the choice of the functions (observer gain functions) $g_{2},\ldots,g_{n}$ while the second of these pairs (21) can be considered the controller-context coupled Lyapunov inequalities that relates to the choice of the functions (controller gain functions) $k_{2},\ldots,k_{n}$. Also, the functions $g_{2},\ldots,g_{n}$ can be chosen ([23, 39, 40]) such that $\sqrt{\sum_{i=2}^{n}g_{i}^{2}(x_{1})}\leq\overline{g}\phi_{(2,3)}(x_{1})$ for all $x_{1}\in{\cal R}$ with $\overline{g}$ being a positive constant.

Scaled observer errors $\epsilon_{i}$ and their dynamics: As noted in Remark 2, the dynamics (9) can be viewed as a reduced-order observer with state variables $\hat{x}=[\hat{x}_{2},\ldots,\hat{x}_{n}]^{T}$. Define the observer error vector $e=[e_{2},\ldots,e_{n}]^{T}$ and the scaled observer error vector $\epsilon=[\epsilon_{2},\ldots,\epsilon_{n}]^{T}$ as $e_{i}=\hat{x}_{i}+r^{i-1}f_{i}(x_{1})-x_{i}\,\,,\,\,\,i=2,\ldots,n$ and $\epsilon_{i}=\frac{e_{i}}{r^{i-1}}\,\,\,,\,\,\,i=2,\ldots,n$. Then, we have the dynamics

[TABLE]

where $B\in{\cal R}^{n-1}=[0,\ldots,0,1]^{T}$, and $\overline{\Phi}=[\overline{\Phi}_{2},\ldots,\overline{\Phi}_{n}]^{T}$ with $\overline{\Phi}_{i}=-\frac{\phi_{i}}{r^{i-1}}+g_{i}\frac{\phi_{1}}{\phi_{(1,2)}}$. The quantities $\varpi_{i}$ in (12) can be viewed as scaled observer estimates with dynamics given by $\dot{\varpi}=rA_{c}\varpi-\frac{\dot{r}}{r}D_{c}\varpi-rG\epsilon_{2}+G\frac{\phi_{1}}{\phi_{(1,2)}}+B_{1}\Psi$ where $G=[g_{2},\ldots,g_{n}]^{T}$, $B_{1}\in{\cal R}^{n-1}$ is given as $[1,0,\ldots,0]^{T}$, and $\Psi=\frac{\partial\vartheta}{\partial x_{1}}\phi_{(1,2)}(x_{1})(\varpi_{2}-\epsilon_{2})+\frac{1}{r}\frac{\partial\vartheta}{\partial x_{1}}[\phi_{1}-\phi_{(1,2)}\vartheta]+\frac{1}{r}\frac{\partial\vartheta}{\partial\hat{\theta}}\dot{\hat{\theta}}$. From (13), we have $\frac{\partial\vartheta}{\partial\hat{\theta}}=x_{1}\vartheta_{1}(x_{1})$ and $\frac{\partial\vartheta}{\partial x_{1}}=\hat{\theta}(\vartheta_{1}(x_{1})+x_{1}\vartheta_{1}^{\prime}(x_{1}))$ where the notation $\vartheta_{1}^{\prime}(x_{1})$ is used to represent \frac{d\vartheta_{1}(\pi)}{d\pi}\Big{|}_{\pi=x_{1}}.

Observer Lyapunov function $V_{o}$, controller Lyapunov function $V_{c}$, and composite Lyapunov function $V_{x}$: The observer and controller Lyapunov functions are defined as $V_{o}=r\epsilon^{T}P_{o}\epsilon$ and $V_{c}=r\varpi^{T}P_{c}\varpi+\frac{1}{2}\left(1+\frac{1}{r}\right)x_{1}^{2}$ and a composite Lyapunov function is defined as the linear combination $V_{x}=cV_{o}+V_{c}$ where $c>0$ is any constant such that $c>[4\lambda_{max}^{2}(P_{c})\overline{g}^{2}/(\tilde{\nu}_{o}\nu_{c})]$. Using (20) and (21), it can be shown that

[TABLE]

where $\theta^{*}=\theta+\theta^{2}$, $q_{1}(x_{1})=\frac{4}{c\nu_{o}}\phi_{(1,2)}^{2}(x_{1})+\frac{8}{\nu_{c}\phi_{(2,3)}(x_{1})}\phi_{(1,2)}^{2}(x_{1})+\lambda_{max}^{2}(P_{c})+2$, q_{2}(x_{1})=2\Gamma(x_{1})+1+\frac{8\lambda_{max}^{2}(P_{c})\overline{g}^{2}\phi_{(2,3)}(x_{1})}{\nu_{c}}\frac{\Gamma^{2}(x_{1})}{\phi_{(1,2)}^{2}(x_{1})}+\frac{8n}{c\nu_{o}}\lambda_{max}^{2}(P_{o})\Gamma^{2}(x_{1})\Bigg{[}1+\overline{g}^{2}\phi_{(2,3)}^{2}(x_{1})/\phi_{(1,2)}^{2}(x_{1})\Bigg{]}, and

w_{1}(x_{1},\hat{\theta},\dot{\hat{\theta}})=3\lambda_{max}(P_{c})\Big{|}\frac{\partial\vartheta(x_{1},\hat{\theta})}{\partial x_{1}}\Big{|}\phi_{(1,2)}(x_{1})+\vartheta_{1}^{2}(x_{1})\dot{\hat{\theta}}^{2}+\lambda_{max}^{2}(P_{c})\hat{\theta}^{2}(\vartheta_{1}(x_{1})+\vartheta_{1}^{\prime}(x_{1})x_{1})^{2}(\Gamma^{2}(x_{1})+\phi_{(1,2)}^{2}(x_{1})\vartheta_{1}^{2}(x_{1}))+3\lambda_{max}(P_{o})(n+n^{2})\Gamma(x_{1})+n\lambda_{max}^{2}{P_{o}}\hat{\theta}^{2}\vartheta_{1}^{2}(x_{1}).

Remark 4: The nominal $x$ subsystem can be seen to be globally stabilized if the virtual “control input” entering into the subsystem could be made to be $\tilde{u}$ instead of $\mu(t,x,\psi,u)$. Hence, $u_{d}=[\mu(t,x,\psi,u)-\tilde{u}]$ can be viewed as a “mismatch” term due to the fact that the actual control signal entering into the $x$ subsystem is $\mu(t,x,\psi,u)$ instead of the desired virtual control input $\tilde{u}$. It will be seen in the analysis below that the control “redesign” given by the dynamic state extension $\zeta$ and control input ($u$) definition in (15) along with the designs of dynamics of the scaling parameters $r$ and $r_{u}$ will make $\mu(t,x,\psi,u)$ track the desired/nominal control input signal $\tilde{u}$, thus making the actual system (1) with the input unmodeled dynamics globally stabilized with asymptotic convergence of $x$ and $\psi$. $\diamond$

Mismatch term $u_{d}=[\mu(t,x,\psi,u)-\tilde{u}]$ and Lyapunov function component $V_{u}$: We see from (15) that:

[TABLE]

Hence,

[TABLE]

where

[TABLE]

Here, $\frac{\partial\mu(t,x,\psi,u)}{\partial x_{i}}$ denotes $\frac{\partial\mu(t,x,\psi,u)}{\partial x}\varrho_{i}$ where $\varrho_{i}$ is a vector of dimension $n\times 1$ having 1 as its $i^{th}$ element and having 0’s everywhere else. The quantity $\chi_{1}$ can be shown to satisfy the magnitude bound |\chi_{1}|\leq\beta_{1}(y,u,r,\dot{r},\hat{\theta})[|\epsilon|+|\varpi|]+\beta_{2}(y,u,r,\hat{\theta},\dot{\hat{\theta}})|x_{1}|+\theta\beta_{3}(y,u,r)|x_{1}|+\overline{\mu}_{2\psi}(\psi)\Big{\{}r^{n}\beta_{4}(x_{1})[|\epsilon|+|\varpi|]+\beta_{5}(x_{1},\hat{\theta})|x_{1}|+\theta\beta_{6}(x_{1})|x_{1}|\Big{\}}+r^{n+1}\beta_{7}(x_{1})[|\epsilon|^{2}+|\varpi|^{2}]+r^{n}\beta_{8}(x_{1})\theta|\varpi||x_{1}|+\left|\frac{\partial\mu(t,x,\psi,u)}{\partial\psi}q_{\psi}(t,x,\psi,u,x_{\Delta},\psi_{\Delta},u_{\Delta})\right|+\left|\frac{\partial\mu(t,x,\psi,u)}{\partial t}\right| with

[TABLE]

$\beta_{2}(y,u,r,\hat{\theta},\dot{\hat{\theta}})=\overline{\mu}_{2}(y,u)\hat{\theta}\vartheta_{1}(x_{1})\{\phi_{(1,2)}(x_{1})+n\Gamma(x_{1})\}+r^{n-1}|K(x_{1})|\hat{\theta}^{2}(|\vartheta_{1}(x_{1})+\vartheta_{1}^{\prime}(x_{1})x_{1}|)\phi_{(1,2)}(x_{1})|\vartheta_{1}(x_{1})|+r^{n-1}|K(x_{1})||\dot{\hat{\theta}}\vartheta_{1}(x_{1})|$, $\beta_{3}(y,u,r)=(n+1)\overline{\mu}_{2}(y,u)\Gamma(x_{1})+r^{n}\frac{|K(x_{1})||G(x_{1})|}{\phi_{(1,2)}(x_{1})}\Gamma(x_{1})+r^{n-1}|K(x_{1})|\hat{\theta}|\vartheta_{1}(x_{1})+\vartheta_{1}^{\prime}(x_{1})x_{1}|\Gamma(x_{1})$, $\beta_{4}(x_{1})=\phi_{(1,2)}(x_{1})+|K(x_{1})|+\sum_{j=2}^{n-1}\phi_{(i,i+1)}(x_{1})+n^{\frac{3}{2}}\Gamma(x_{1})$, $\beta_{5}(x_{1},\hat{\theta})=\phi_{(1,2)}(x_{1})\hat{\theta}\vartheta_{1}(x_{1})+n\Gamma(x_{1})\hat{\theta}\vartheta_{1}(x_{1})$, $\beta_{6}(x_{1})=(n+1)\Gamma(x_{1})$, $\beta_{7}(x_{1})=\frac{3}{2}\left|\frac{\partial K(x_{1})}{\partial x_{1}}\right|\phi_{(1,2)}(x_{1})$, and $\beta_{8}(x_{1})=\left|\frac{\partial K(x_{1})}{\partial x_{1}}\right|\Gamma(x_{1})$.

Define

[TABLE]

where $\ln$ denotes $log_{e}$; $\Pi$ is any function from ${\cal R}^{+}$ to ${\cal R}^{+}$ such that the following properties are satisfied: $\Pi(a)\geq 1$ whenever $a\geq 1$; $\Pi(a)$ is a continuously differentiable and monotonically increasing function over the interval $[1,\infty)$; a constant $\overline{\Pi}>0$ exists such that $\Pi(a)\leq\overline{\Pi}$ for all $a\in[1,\infty)$. We utilize $\Pi^{\prime}(r_{u})$ to denote $\frac{\partial\Pi(r_{u})}{\partial r_{u}}$. An example of a function that satisfies the above properties is $\Pi(a)=\tanh(ka)+1$ with any constant $k>0$. $V_{u}$ satisfies:

[TABLE]

where $r_{\Delta}$, $\hat{\theta}_{\Delta}$, $x_{1,\Delta}$, $\varpi_{j,\Delta}$, $\epsilon_{j,\Delta}$, and $\psi_{\Delta}$ denote the time-delayed versions (with time delay $\Delta$) of the corresponding signals (e.g., $x_{1,\Delta}(t)=x_{1}(t-\Delta)$, $\epsilon_{j,\Delta}(t)=\epsilon_{j}(t-\Delta)$). Also, $\tilde{w}_{1}(x_{1})=c_{1}(1+\overline{\mu}_{1a}^{2}(x_{1}))+\beta_{7}(x_{1})+\frac{\beta_{4}^{2}(x_{1})}{c_{\psi 2}}+\frac{\beta_{8}^{2}(x_{1})}{2c_{4}}$, $\tilde{q}_{1}(x_{1},\hat{\theta})=c_{2}(1+\hat{\theta}^{2}\vartheta_{1}^{2}(x_{1})\overline{\mu}_{1a}^{2}(x_{1}))+\frac{\beta_{5}^{2}(x_{1},\hat{\theta})}{2c_{\psi 2}}$, $\tilde{q}_{2}(x_{1})=c_{3}(1+\overline{\mu}_{1a}^{2}(x_{1}))+\frac{\beta_{6}^{2}(x_{1})}{2c_{\psi 2}}+\frac{c_{4}}{2}$, $k_{\psi 1}=0.5(c_{\psi 1}+3c_{\psi 2})\overline{k}_{\psi}$, and

[TABLE]

where $c_{1},\ldots,c_{4}$, $c_{\psi 1}$, and $c_{\psi 2}$ are any positive constants and $\tilde{\overline{\mu}}_{1}(y,u)$ is defined as $[\overline{\mu}_{1}(y,u)+\tilde{\mu}_{1}(y,u)]$. Since we have the upper bound $|u_{d}|\leq\overline{u}_{d}(y,u,r,\varpi)$ where $\overline{u}_{d}(y,u,r,\varpi)$ is defined as \overline{u}_{d}(y,u,r,\varpi)\buildrel\triangle\over{=}\Big{[}\overline{\mu}(y,u)+r^{n}|K(x_{1})||\varpi|\Big{]}, the following inequality can be written for the term involving $u_{d}$ in (23): $-\frac{2cr\epsilon^{T}P_{o}Bu_{d}}{r^{n-1}}\leq r\lambda_{max}^{2}(P_{o})c^{2}|\epsilon|^{2}+\frac{1}{r^{2n-3}}[1+\overline{u}_{d}^{2}(y,u,r,\varpi)]\ln(1+u_{d}^{2})$. Note that $\overline{u}_{d}(y,u,r,\varpi)$ is a completely known function and involves only available variables.

Scaling of Lyapunov function for $\psi$ subsystem: Defining $\tilde{V}_{\psi}=\frac{V_{\psi}}{r^{2n-\frac{3}{2}}}$, we have

[TABLE]

Overall composite Lyapunov function $V$ including terms to handle time-delayed terms in $\dot{V}_{u}$ and $\dot{\tilde{V}}_{\psi}$: Define

[TABLE]

with $c_{u}$, $c_{\theta}$, and $c_{\psi}$ being any positive constants with furthermore $c_{\psi}$ additionally satisfying $c_{\psi}\geq c_{u}[2k_{\psi 1}+\tilde{\Delta}c_{\psi 1}\overline{k}_{\psi}]$. Hence, using (29) and (31), we obtain

[TABLE]

where $\overline{w}_{1}(x_{1},\hat{\theta},\dot{\hat{\theta}})=w_{1}(x_{1},\hat{\theta},\dot{\hat{\theta}})+\lambda_{max}^{2}(P_{o})c^{2}+c_{u}[\tilde{w}_{1}(x_{1})+c_{1}\tilde{\Delta}\overline{\mu}_{1a}^{2}(x_{1})]+3\tilde{c}_{\psi}\Gamma_{2}(x_{1})$, $\overline{w}_{2}(x_{1})=2\tilde{c}_{\psi}\overline{\gamma}_{s}(x_{1})\Gamma_{2}(x_{1})|K(x_{1})|^{2}$, $\overline{\Xi}_{u1}(y,u,r,\dot{r},\hat{\theta},\dot{\hat{\theta}})=c_{u}\Xi_{u1}(y,u,r,\dot{r},\hat{\theta},\dot{\hat{\theta}})$, \overline{\Xi}_{u2}(y,u,r,\varpi)=\Big{[}\frac{1}{r^{2n-3}}+2\tilde{c}_{\psi}\Gamma_{2}(x_{1})\frac{\overline{\gamma}_{s}(x_{1})}{r^{2n-\frac{3}{2}}}\Big{]}[1+\overline{u}_{d}^{2}(y,u,r,\varpi)], $\overline{q}_{1}(x_{1})=q_{1}(x_{1})$, $\overline{q}_{2}(x_{1})=q_{2}(x_{1})+\tilde{c}_{\psi}\Gamma_{2}(x_{1})$, $\overline{q}_{3}(x_{1},\hat{\theta})=3\tilde{c}_{\psi}\Gamma_{2}(x_{1})\hat{\theta}^{2}$, $\overline{q}_{4}(x_{1},\hat{\theta})=c_{u}[\tilde{q}_{1}(x_{1},\hat{\theta})+c_{2}\tilde{\Delta}\hat{\theta}^{2}\vartheta_{1}^{2}(x_{1})\overline{\mu}_{1a}^{2}(x_{1})]$, and $\overline{q}_{5}(x_{1})=c_{u}[\tilde{q}_{2}(x_{1})+c_{3}\tilde{\Delta}\overline{\mu}_{1a}^{2}(x_{1})]$, $\tilde{c}_{\psi}=c_{\psi}\frac{2-\overline{\Delta}}{1-\overline{\Delta}}$, and $\tilde{\Delta}=\frac{1}{1-\overline{\Delta}}$.

Design of function $\vartheta_{1}$ appearing in (13): The function $\vartheta_{1}(x_{1})$ is chosen as \vartheta_{1}(x_{1})=\frac{4}{\phi_{(1,2)}(x_{1})}\Big{(}\frac{\overline{q}_{1}(x_{1})+\vartheta_{1}^{*}}{a_{\theta}}+\overline{q}_{2}(x_{1})\Big{)} with $\vartheta_{1}^{*}$ being any positive constant.

Design of function $\overline{Q}_{\theta}$ appearing in dynamics of $\hat{\theta}$ in (18): \overline{Q}_{\theta}(x_{1})=c_{\theta}\Big{(}\overline{q}_{2}(x_{1})+\frac{\overline{q}_{5}(x_{1})}{r}\Big{)}x_{1}^{2}.

Design of functions $\lambda$, $R$, $\Omega$, $R_{u}$, and $\Omega_{u}$ appearing in dynamics of scaling parameters in (16) and (17): $\lambda:{\cal R}\rightarrow{\cal R}^{+}$ is chosen to be any continuous function such that $\lambda(s)=1$ for any $s>0$ and $\lambda(s)=0$ for any $s<-\epsilon_{r}$ with $\epsilon_{r}>0$ being any constant. The functions $R$ and $\Omega$ are chosen as shown in (34) and (35), respectively, with $\overline{R}$ and $\overline{\Omega}$ being any nonnegative constants and with \nu_{a}=\max\Big{(}\frac{1}{c\nu_{o}},\frac{1}{\nu_{c}\sigma}\Big{)} and \nu_{b}=\max\Big{(}\frac{1}{c\underline{\nu}_{o}},\frac{1}{\underline{\nu}_{c}}\Big{)}. $R_{u}$ and $\Omega_{u}$ are chosen as shown in (36) and (37), respectively, where $\overline{R}_{u}$ and $\overline{\Omega}_{u}$ are any nonnegative constants and $\nu_{u}$ is any positive constant.

Analysis of closed-loop stability and asymptotic convergence: There are two possible cases: (i) $r\geq R(x_{1},\hat{\theta},\dot{\hat{\theta}})$; (ii) $r<R(x_{1},\hat{\theta},\dot{\hat{\theta}})$. In case (i), we have $r\geq R(x_{1},\hat{\theta},\dot{\hat{\theta}})$, and in case (ii), we have $\dot{r}=\Omega(x_{1},\hat{\theta},\dot{\hat{\theta}},r)$. In either of these cases, i.e., at all time instants $t$, we have

[TABLE]

Now, similarly, there are two possible cases for $r_{u}$: (A) $r_{u}\geq R_{u}$; (B) $r_{u}<R_{u}$. In case (A), we have $r_{u}\geq R_{u}$, and in case (B), we have $\dot{r}_{u}=\Omega_{u}$. It can be shown that in either of cases (A) and (B), i.e., at all time instants $t$,

[TABLE]

From (39) and the system dynamics, the closed-loop stability and asymptotic convergence properties can be inferred. Firstly, to show that solutions exist for all time, consider the maximal interval of existence of solutions of the closed-loop dynamic system to be $[0,t_{f})$ with some $t_{f}>0$. Then, from (39), it can be seen that $V$ is bounded on $[0,t_{f})$ and that therefore, from the definition of $V$ and the dynamics of the closed-loop system, a process of signal chasing can be used to show that all closed-loop signals remain bounded on $[0,t_{f})$. Therefore, solutions exist for all time, i.e., $t_{f}=\infty$. Also, from (39), $\epsilon$, $\varpi$, $x_{1}$, $\psi$, and $u_{d}$, and therefore, $x_{1},\ldots,x_{n},\psi,\hat{x}_{2},\ldots,\hat{x}_{n}$ go to zero asymptotically as $t\rightarrow\infty$.

IV Conclusion

We considered a general class of uncertain nonlinear systems with input unmodeled dynamics with time-varying time delays and showed that a dynamic scaling based robust adaptive output-feedback controller can be designed to globally stabilize this uncertain nonlinear system. While the input unmodeled dynamics subsystem involves uncertain (and time-varying) time delays both on the state and control input signals entering into this subsystem, the developed control design methodology is itself delay-independent in two important senses: firstly, the controller does not utilize any delayed versions of the measured output, input, or controller internal variable signals; secondly, the control design does not require knowledge of the actual time delay magnitudes (specifically, only requires an upper bound on rate of change of the time delay magnitude). In further research, applicability of the proposed techniques to more general classes of nonlinear systems (e.g., nontriangular systems, more general structures of appended dynamics, etc.) are being considered.