This paper considers a model-free control approach to Euler-Lagrange equations and proposes a new quantitative performance measure with its Lyapunov-based computation method. More precisely, this paper aims to solve a trajectory tracking problem for uncertain Euler-Lagrange equations by using a model-free controller with a proportional-integral-derivative (PID) control form. The $L_{\mathrm{\infty }}$-gain is evaluated for the closed-loop systems obtained through the feedback connection between the Euler-Lagrange equation and the model-free controller. To this end, the input-to-state stability (ISS) for the closed-loop systems is first established by deriving an appropriate Lyapunov function. The study further extends these arguments to develop a computational approach to determine the $L_{\mathrm{\infty }}$-gain. Finally, the theoretical validity and effectiveness of the proposed quantitative performance measure are demonstrated through a simulation of a $2$-degree-of-freedom ( $2$-DOF) robot manipulator, which is one of the most representative examples of Euler-Lagrange equations.

This paper proposed a new optimal control method for uncertain Euler-Lagrange systems, focusing on estimating model uncertainties and improving tracking performance. More precisely, a linearization of the nonlinear equation was achieved through the inverse dynamic control (IDC) and an $H_{\mathrm{\infty }}$ optimal estimator was designed to address model uncertainties arising in this process. Subsequently, a generalized $H_2$ optimal tracking controller was obtained to minimize the effect of the estimation error on the tracking error in terms of the induced norm from $L_2$ to $L_{\mathrm{\infty }}$. Necessary and sufficient conditions for the existences of these two optimal estimator and controller were characterized through the linear matrix inequality (LMI) approach, and their synthesis procedures can be operated in an independent fashion. To put it another way, this developed approach allowed us to minimize not only the modeling error between the real Euler-Lagrange equations and their nominal models occurring from the IDC approach but also the maximum magnitude of the tracking error by solving some LMIs. Finally, the effectiveness of both the $H_{\mathrm{\infty }}$ optimal disturbance estimator and the generalized $H_2$ tracking controller were demonstrated through some comparative simulation and experiment results of a robot manipulator, which was one of the most representative examples of Euler-Lagrange equations.

In this paper, we consider the $L_1$-induced norm analysis for linear multivariable differential equations. Because such an analysis requires integrating the absolute value of the associated impulse response on the infinite-interval $[0,\mathrm{\infty })$, this interval was divided into $[0,H)$ and $[H,\mathrm{\infty })$, with the truncation parameter $H$. The former was divided into $M$ subintervals with an equal width, and the kernel function of the relevant input\slash output operator on each subinterval was approximated by a $p$th order polynomial with $p=0,1,2,3$. This derived to an upper bound and a lower bound on the $L_1$-induced norm for $[0,H)$, with the convergence rate of $1/M^{p+1}$. An upper bound on the $L_1$-induced norm for $[H,\mathrm{\infty })$ was also derived, with an exponential order of $H$. Combining these bounds led to an upper bound and a lower bound on the original $L_1$-induced norm on $[0,\mathrm{\infty })$, within the order of $1/M^{p+1}$. Furthermore, the $l_1$-induced norm of difference equations was tackled in a parallel fashion. Finally, numerical studies were given to demonstrate the overall arguments.

This paper is concerned with a delay-independent output-feedback controller synthesis suppressing the $L_{\mathrm{\infty }}$-gain of linear time-delay systems with time-varying delays. We first proposed a continuous-time version of the existing discrete-time ellipsoidal $\mathcal{D}$-invariant set and established its existence condition in terms of some linear matrix inequalities (LMIs). This existence condition was further extended to characterizing the $L_{\mathrm{\infty }}$-gain of linear time-delay systems with time-varying delays. Because of the delay-independent property of the proposed $\mathcal{D}$-invariant set, the $L_{\mathrm{\infty }}$-gain analysis does not depend on the choice of delays including their magnitudes and time derivatives. Based on this analysis method, we also constructed an output-feedback controller synthesis for ensuring the $L_{\mathrm{\infty }}$-gain of time-delay systems bounded by a performance level $\rho$. In an equivalent fashion to the $L_{\mathrm{\infty }}$-gain analysis method, this controller synthesis is independent of the delays in the sense that the obtained controller coefficients do not depend on the delay characteristics. Finally, numerical results were given to demonstrate the effectiveness and validity of the proposed results.