Adaptive Control of Time-Varying Parameter Systems with Asymptotic Tracking

Adaptive control of nonlinear dynamical systems with time-varying uncertain parameters is an open and practically relevant problem. It has been well established that traditional gradient-based update laws can compensate for constant unknown parameters yielding asymptotic convergence. Moreover, the development of robust modifications of such adaptive update laws result in uniformly ultimately bounded (UUB) results for slowly varying parametric uncertainty using a Lyapunov-based analysis, under the assumption of bounded parameters and their time-derivatives (cf. [1]).

More recent results focus on tracking and parameter estimation performance improvement, though still limited to a UUB result, using various adaptive control approaches for systems with unknown time-varying parameters. One such approach involves a fast adaptation law [2], where a matrix of time-varying learning rates is utilized to improve the tracking and estimation performance under a finite excitation condition. Another approach uses a set-theoretic control architecture [3, 4, 5] to reject the effects of parameter variation, while restricting the system error within a prescribed performance bound. While the aforementioned approaches can potentially yield improved transient response, the results still yield UUB error systems.

Motivation exists to obtain asymptotic convergence of the tracking error to zero, despite the time-varying nature of the uncertain parameters. Robust adaptive control approaches such as [6] yield asymptotic adaptive tracking for systems with time-varying uncertain parameters; however, such approaches exploit high-gain feedback based on worst-case uncertainty, rather than an adaptive control approach that scales to compensate for the uncertainty without using worst-case gains. In [7], the iterative learning control result in [6] is extended to yield asymptotic tracking for systems with periodic time-varying parameters with known periodicity.

To the best of our knowledge, an asymptotic tracking result has not been achieved for a generalized class of nonlinear systems with unknown time-varying parameters, where the parameters are not necessarily periodic. Asymptotic tracking is difficult to achieve for the time-varying parameter case because the time-derivative of the parameter acts like an unknown exogenous disturbance in the parameter estimation dynamics, which is difficult to cancel with an adaptive update law in a Lyapunov-based stability analysis.

To illustrate this problem, consider the scalar dynamical system¹¹1Note that the system $(\text{\ref{eq:illussystem}})$ is considered only for illustrative purpose. This paper presents result for a general system with a vector state and a linearly parameterizable uncertainty with time-varying parameters.

$$\dot{x}(t)=a(t)x(t)+u(t),$$

(1)

with the controller $u(t)=-kx(t)-\hat{a}(t)x(t)$, where $k$ is a positive constant gain, $a(t)$ is the unknown time-varying parameter, $\hat{a}(t)$ is the parameter estimate of $a(t)$ and the parameter estimation error $\tilde{a}(t)$ is defined as $\tilde{a}(t)\triangleq a(t)-\hat{a}(t)$. The traditional stability analysis approach for such problems is to consider the Lyapunov function candidate $V(x(t),\tilde{a}(t))=\frac{1}{2}x^{2}(t)+\frac{1}{2\gamma}\tilde{a}^{2}(t)$, where $\gamma$ is a positive constant gain. The given definitions and controller yield the following time-derivative of the candidate Lyapunov function: $\dot{V}(t)=-kx^{2}(t)+\tilde{a}(t)x^{2}(t)+\frac{\tilde{a}(t)}{\gamma}(\dot{a}(t)-\dot{\hat{a}}(t))$. For the constant parameter case, i.e., $\dot{a}(t)=0$, the well-known adaptive update law $\dot{\hat{a}}(t)=\gamma x^{2}(t)$ will cancel the cross term $\tilde{a}(t)x^{2}(t)$ in $\dot{V}(t)$. However, when the parameters are time-varying, it is unclear how to cancel or dominate $\dot{a}(t)$ via an update law such that $\dot{V}(t)$ becomes at least negative semi-definite. It would be desirable to have a sliding mode-like term based on $\tilde{a}(t)$ in the adaptation law, but $\tilde{a}(t)$ is unknown. Another approach could be to use a robust controller, e.g., $u(t)=-kx(t)-\bar{a}x(t)$, where $\bar{a}$ is a known constant upper bound on the norm of parameter $\left|a(t)\right|$, or an adaptive robust controller which involves certainty equivalence in terms of the unknown bound $\bar{a}$. Either of these approaches would yield an asymptotic tracking result (cf., [6]), but, as stated earlier, these approaches are based on a high-gain worst case scenario, rather than an adaptive control approach that scales to compensate for the uncertainty without using worst-case gains.

A popular approach to design adaptive controllers for the time-varying parameter case is to consider robust modification of the update law and assume upper bounds of $\left|a(t)\right|$ and $\dot{a}(t)$ to obtain a UUB result. For instance, consider a standard gradient update law with sigma-modification [8], $\dot{\hat{a}}(t)=\gamma x^{2}(t)-\gamma\sigma\hat{a}(t)$, which yields $\dot{V}(t)=-kx^{2}(t)-\sigma\tilde{a}^{2}(t)+\tilde{a}(t)(\frac{\dot{a}(t)}{\gamma}+\sigma a(t))$, implying a UUB result when the parameter $a(t)$ and its time-derivative $\dot{a}(t)$ are bounded. Moreover, the approaches developed in results such as [2] and [4] can be used to improve the transient response of the UUB error system.

The major challenge to achieve asymptotic stability is the derivative of the time-varying parameter term in the Lyapunov analysis, which is addressed in this paper with a Lyapunov-based design approach, that is inspired by the modular adaptive control approach in [9]. This approach includes higher order dynamics which appear after taking the time-derivative of (1). Since these higher order dynamics contain the time-derivative of the parameter estimate $\dot{\hat{a}}(t),$ it is possible to design $\dot{\hat{a}}(t)$ to facilitate the subsequent stability analysis. With this motivation, a continuous adaptive control algorithm is developed for nonlinear dynamical systems with linearly parameterized uncertainty involving time-varying parameters, where a semi-global asymptotic tracking result is achieved. A key feature of the proposed method is a robust integral of the sign of the error (RISE)-like (see [10, 11, 12, 9]) update law, i.e., the update law contains a signum function of the tracking error term multiplied by some desired regressor based terms. The update law also involves a projection algorithm to ensure that the parameter estimates stay within a bounded set. However, the projection algorithm introduces a potentially destabilizing term in the time-derivative of the Lyapunov function candidate, leading to an additional technical obstacle to obtain asymptotic tracking. This challenge is resolved by using an auxiliary term in the control input, which facilitates stability by providing a stabilizing term and canceling the aforementioned potentially destabilizing term in the time-derivative of the candidate Lyapunov function. With the proposed method, the closed-loop system dynamics have the same structure as previous RISE controllers [10, 11, 12, 9], for which the stability analysis tools are well established, yielding asymptotic convergence of the tracking error to zero, boundedness of the parameter estimation error, and boundedness of the closed-loop signals.

Consider a control affine system with the nonlinear dynamics

$$\dot{x}(t)=h(x(t),t)+d(t)+u(t),$$

(2)

where $x:[0,\infty)\to\mathbb{R}^{n}$ denotes the state, $h:\mathbb{R}^{n}\times[0,\infty)\to\mathbb{R}^{n}$ denotes a continuously differentiable function, $d:[0,\infty)\to\mathbb{R}^{n}$ represents an exogenous disturbance acting on the system, and $u:[0,\infty)\to\mathbb{R}^{n}$ represents the control input. The function $h(x(t),t)$ in $(\text{\ref{eq:dynamicsystem}})$ is assumed to be linearly parameterized as

$$h(x(t),t)\triangleq Y_{h}(x(t),t)\theta_{f}(t),$$

(3)

The disturbance parameter vector $d(t)$ can be appended to the $\theta_{f}(t)$ vector, yielding an augmented parameter vector $\theta:[0,\infty)\to\mathbb{R}^{n+m}$ as

$$\theta(t)\triangleq\left[\begin{array}[]{c}\theta_{f}(t)\\ d(t)\end{array}\right],$$

(4)

$$Y(x(t),t)\triangleq\left[\begin{array}[]{cc}Y_{h}(x(t),t)&I_{n}\end{array}\right].$$

(5)

$$\dot{x}(t)=Y(x(t),t)\theta(t)+u(t).$$

(6)

Proof: Let $\mathcal{D}\subset\mathbb{R}^{2n+1}$ be an open connected set containing $y(t)=0$, where $y:[0,\infty)\to\mathbb{R}^{2n+1}$ is defined as

$$y(t)\triangleq\left[\begin{array}[]{cc}z^{T}(t)&\sqrt{P(t)}\end{array}\right]^{T}.$$

$$V_{L}(y(t),t)\triangleq\frac{1}{2}r^{T}r+\frac{1}{2}e^{T}e+P,$$

$$\text{$\dot{P}$}(t)\triangleq-L(t),\,$$

(24)

	$\displaystyle\dot{\widetilde{V}}_{L}(y,t)$	$\displaystyle=$	$\displaystyle\underset{\xi\in\partial V_{L}(y,t)}{\bigcap}\xi^{T}K\left[\begin{array}[]{ccc}\dot{e}^{T}&\dot{r}^{T}&\frac{1}{2}P^{-\frac{1}{2}}\dot{P}\end{array}\right]^{T}$		(30)
		$\displaystyle=$	$\displaystyle\nabla V_{L}^{T}K\left[\begin{array}[]{ccc}\dot{e}^{T}&\dot{r}^{T}&\frac{1}{2}P^{-\frac{1}{2}}\dot{P}\end{array}\right]^{T}$		(32)
		$\displaystyle\subset$	$\displaystyle\left[\begin{array}[]{ccc}e^{T}&r^{T}&2P^{\frac{1}{2}}\end{array}\right]\times$		(36)
			$\displaystyle K\left[\begin{array}[]{ccc}\dot{e}^{T}&\dot{r}^{T}&\frac{1}{2}P^{-\frac{1}{2}}\dot{P}\end{array}\right]^{T},$

	$\displaystyle\dot{\widetilde{V}}_{L}$	$\displaystyle\overset{a.e.}{\subset}r^{T}(\widetilde{N}+N_{B}-\beta\,\textrm{sgn}(e)-Kr-e)$
		$\displaystyle+e^{T}(r-\alpha e)-r^{T}(N_{B}-\beta\textrm{sgn}(e))$		(37)

$$\textrm{SGN}(e_{i})=\begin{cases}\left\{1\right\},&e_{i}>0\\ \left[-1,1\right],&e_{i}=0\\ \left\{-1\right\},&e_{i}<0.\end{cases}$$

$$\dot{\widetilde{V}}_{L}\overset{a.e.}{\leq}\rho(\left\|z\right\|)\left\|z\right\|\left\|r\right\|-K\left\|r\right\|^{2}-\alpha e^{2}.$$

Using Young’s Inequality on $\rho(\left\|z\right\|)\left\|z\right\|\left\|r\right\|$ yields $\rho(\left\|z\right\|)\left\|z\right\|\left\|r\right\|\leq\frac{\rho^{2}(\left\|z\right\|)\left\|z\right\|^{2}}{2}+\frac{1}{2}\left\|r\right\|^{2}$. Therefore,

	$\displaystyle\dot{\widetilde{V}}_{L}$	$\displaystyle\overset{a.e.}{\leq}$	$\displaystyle\frac{\rho^{2}(\left\|z\right\|)\left\|z\right\|^{2}}{2}-(K-\frac{1}{2})\left\|r\right\|^{2}-\alpha e^{2}$		(38)
		$\displaystyle\overset{a.e.}{\leq}$	$\displaystyle-\left(\lambda_{3}-\frac{\rho^{2}(\left\|z\right\|)}{2}\right)\left\|z\right\|^{2},$

$$\dot{V}_{L}\overset{a.e.}{\leq}-c\left\|z\right\|^{2}\,\forall\,y\in\mathcal{D},$$

(39)

$$\mathcal{D}\triangleq\left\{y\in\mathbb{R}^{2n+1}|\left\|y\right\|\leq\rho^{-1}\left(\sqrt{2\lambda_{3}}\right)\right\}.$$

In this region, $\lambda_{3}>\frac{\rho^{2}(\left\|z\right\|)}{2}$, so a constant $c$ satisfies (39), and larger values of $\lambda_{3}$ expand the size of $\mathcal{D}.$ Furthermore, the relationship in $(\ref{eq:VLdot})$ implies that $V_{L}(y(t),t)\in\mathcal{L}_{\infty},$ hence $e(t),r(t)$, $P(t)$ $\in\mathcal{L}_{\infty}$. These facts along with the expression in $(\text{\ref{eq:rsub2}})$, indicate that $\mu(t)\in\mathcal{L}_{\infty}$. The parameter estimate $\hat{\theta}(t)\in\mathcal{L}_{\infty}$ due to the projection operation. The state and its time-derivative, i.e., $x(t),\dot{x}(t)\in\mathcal{L}_{\infty}$, because $e(t),r(t),x_{d}(t),\dot{x}_{d}(t)\in\mathcal{L}_{\infty}.$ Further the regression matrix $Y(x(t),t)\in\mathcal{L}_{\infty}$ since its a bounded function for a bounded argument $x(t)$. Similarly, $Y_{d}(t)\in\mathcal{L}_{\infty}$, hence $\dot{\hat{\theta}}\in\mathcal{L}_{\infty}$ by Corollary 1. From the expression in (11), since $\hat{\theta}(t),e(t),\dot{x}_{d}(t),\mu(t)\in\mathcal{L}_{\infty}$, $u(t)\in\mathcal{L}_{\infty}$. Hence all the closed-loop signals are bounded.

Consider $\lambda_{1}\left\|y\right\|^{2}\leq V_{L}\leq\lambda_{2}\left\|y\right\|^{2},$ where $\lambda_{1},\lambda_{2}\in\mathbb{R}_{>0}$. To ensure $\left\|z\right\|\leq\rho^{-1}(\sqrt{2\lambda_{3}})$, it is sufficient to obtain the result from $\left\|y\right\|\leq\rho^{-1}(\sqrt{2\lambda_{3}})$. Since $\sqrt{\frac{V_{L}}{\lambda_{2}}}\leq\left\|y\right\|$, then $\sqrt{\frac{V_{L}}{\lambda_{2}}}\leq\rho^{-1}(\sqrt{2\lambda_{3}})$, and $V_{L}$ is non-increasing, so $V_{L}(t)\leq V_{L}(0)$. Hence it sufficient to show that $\sqrt{\frac{V_{L}(0)}{\lambda_{2}}}\leq\rho^{-1}(\sqrt{2\lambda_{3}})$ to ensure that $\sqrt{\frac{V_{L}}{\lambda_{2}}}\leq\rho^{-1}(\sqrt{2\lambda_{3}})$. Since $\lambda_{1}\left\|y(0)\right\|^{2}\leq V_{L}(0)$ implies $\left\|y(0)\right\|\leq\sqrt{\frac{V_{L}(0)}{\lambda_{1}}}\leq\sqrt{\frac{\lambda_{2}}{\lambda_{1}}}\rho^{-1}(\sqrt{2\lambda_{3}}),$ so $y\in\mathcal{S}\triangleq\left\{y(t)\in\mathcal{D}|y(t)\leq\sqrt{\frac{\lambda_{2}}{\lambda_{1}}}\rho^{-1}(\sqrt{2\lambda_{3}})\right\}$ is the region where $y(0)$ should lie for guaranteed asymptotic stability. The gain condition $\lambda_{3}=\min\{\alpha,K-\frac{1}{2}\}\geq\frac{\rho^{2}\left(\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}\left\|y(0)\right\|\right)}{2}$ needs to be satisfied according to the initial condition for asymptotic stability and the region of attraction can be made arbitrarily large to include any initial condition by increasing the gains $\alpha$ and $K$ accordingly. By the extension of LaSalle-Yoshizawa theorem for non-smooth systems in [14] and [15], $c\left\|z(t)\right\|^{2}\to 0$ and hence $\left\|e\right\|\to 0$ as $t\to\infty\,\,$$\forall\,\,y(0)\in\mathcal{S}$ , so the closed-loop error system is semi-globally asymptotically stable.

$\blacksquare$

A continuous adaptive control design was presented to achieve semi-global asymptotic tracking for linearly parameterizable nonlinear systems with time-varying uncertain parameters. The key feature of this design is a RISE-like parameter update law along with a projection algorithm, which allows the system to compensate for potentially destabilizing terms in the closed-loop error system, arising due to the time-varying nature of parameters. Semi-global asymptotic tracking for the error system is guaranteed via a Lyapunov-based stability analysis. Future work will involve improvement of the parameter estimation performance of time-varying parameter systems and its extension to the system identification problem.