Robust iterative learning control for uncertain continuous‐time system with input delay and random iteration‐varying uncertainties

Abstract

This study deals with the problem of robust iterative learning control (ILC) for linear continuous-time systems with input delay subject to uncertainties in input delay, plant dynamic, reference trajectory, initial conditions and disturbances. Using the internal model control (IMC) structure in the frequency domain, an ILC scheme is proposed in which the IMC structure is responsible for coping with uncertainties in both delay time and plant dynamic. Sufficient conditions are derived to ensure that the tracking error expectation is bounded and converges monotonically to a small neighbourhood of zero (in the L 2 -norm sense) when uncertainties in reference trajectory, initial conditions and disturbances vary randomly from trial to trial. It is shown that the derived conditions are still valid to guarantee both boundedness and monotonic convergence of the tracking error variance (in the L 2 -norm sense). Illustrative examples are provided to demonstrate the effectiveness of the proposed method.

Robust convergence conditions of iterative learning control for time‐delay systems under random non‐repetitive uncertainties

“…To accommodate both the characteristics of randomness and non‐repetitiveness in both factors of input‐delay and plant dynamic for the system studied in [35], the description of the system output in the frequency domain can be considered as follows (see [31, 32] for time‐delay systems, and also [45] for delay‐free systems): $$$$\begin{equation}{Y}_k\left( s \right) = {P}_k\left( s \right){U}_k\left( s \right) + {\Psi }_k\left( s \right){\rm{,\ }}\forall k \in \mathbb{Z}\end{equation}$$$$ $$$$\begin{equation}{P}_k\left( s \right) = {G}_k\left( s \right){e}^{ - {\theta }_ks}{\rm{,\ }}\forall k \in \mathbb{Z}\end{equation}$$$$where $${U}_k( s ) = \ell [ {{u}_k( t )} ]$$ and $${Y}_k( s ) = \ell [ {{y}_k( t )} ]$$ represent the control input and system output at iteration k , respectively; $${\Psi }_k( s ) = \ell [ {{\psi }_k( t )} ]$$ is a random nonrepetitive exogenous signal; $${G}_k( s )$$…”

“…Now, to obtain our results in this work, we first need to make fundamental assumptions on the system described by (1) and (2) and then revisit the definition given in [35] as follows: Assumption Like [31, 32], it is assumed that $${P}_k( s )$$ takes the multiplicative uncertainty form of: $$$$\begin{equation}{P}_k\left( s \right) = \left[ {1 + {\Delta }_k\left( s \right){W}_a\left( s \right)} \right]\hat{P}\left( s \right)\end{equation}$$$$…”

“…where the elements of the set $$\{ {{\Delta }_k( s ):k \in \mathbb{Z}} \}$$ are considered to be random stable transfer functions with the property of $${\| {{\Delta }_k( s )} \|}_\infty \le 1$$; $${W}_a( s )$$ represents a fixed and stable transfer function as the uncertainties weighting function; and $$\hat{P}( s )$$ is given by: $$$$\begin{equation}\hat{P}\left( s \right) = \hat{G}\left( s \right){e}^{ - \hat{\theta }s}\end{equation}$$$$where $$\hat{\theta }$$ and the stable transfer function $$\hat{G}( s )$$ are the estimate (or the nominal model) of $${\theta }_k$$ and $${G}_k( s )$$, respectively (see also [8, 28, 35] for addressing the repetitive uncertainties in both factors of plant dynamic and input‐delay). Remark From the viewpoint of practical application, the designer does not have access to the exact values ​​of both $${G}_k( s )$$ and …”

“…Furthermore, we assume that $${\Delta }_i( s )$$, $${\Psi }_j( s )$$, and $${R}_m( s )$$ are independent of each other, $$\forall i,j,m \in \mathbb{Z}$$. Consequently, we have (for a similar discussion, see [31, 32, 35]): $$$$\begin{equation}\mathbb{E}\left[ {{\Delta }_k\left( s \right)} \right] = {\Delta }_e\left( s \right),\forall k \in \mathbb{Z}\end{equation}$$$$ $$$$\begin{equation}\mathbb{E}\left[ {{\Psi }_k\left( s \right)} \right] = {\Psi }_e\left( s \right),\forall k \in \mathbb{Z}\end{equation}$$$$ $$$$\begin{equation}\mathbb{E}\left[ {{R}_k\left( s \right)} \right] = {R}_e\left( s \right),\forall k \in \mathbb{Z}\end{equation}$$$$ …”

“…In comparison with [31], we relax the restrictive assumption of the repetitive reference trajectory at each iteration. In comparison with our previous work [35], at each iteration, the restrictive assumptions of repetitive uncertainties in both factors of plant dynamic and input‐delay are relaxed. In contrast with the robust control methods presented in [23–25], the proposed method can deal with the problem of robust tracking for a continuous‐time system by operating the system over‐and‐over, no matter if (a) there exists time‐delay mismatch between the system and its model and (b) the reference trajectory is not a step‐function. Based on the frequency domain analysis, it is shown that both monotonic convergence and boundedness of the expected tracking error can be guaranteed (in the sense of L 2 ‐norm) when all of the existing uncertainties in the controlled system (i.e.…”