# Single link manipulator with flexible joints

## Model description:

A single link manipulator with flexible joints and negligible damping can be represented by

\begin{align*} I\ddot{q}_1 + MgL\sin{q_1} + k(q_1 - q_2) &= 0 \\ J\ddot{q}_2-k(q_1-q_2) &=u, \end{align*}

where $q_1$ and $q_2$ are the angular positions, and $u$ is a torque input. The physical parameters $g, I, J, k, L,$ and $M$ are all positive. Taking $y=q_1$ as the output, it can be verified that $y$ satisfies the fourth-order differential equation

$$y^{(4)}=\dfrac{gLM}{I}(\dot{y}^2\sin{y}-\ddot{y}\cos{y})- \left(\dfrac{k}{I}+\dfrac{k}{J}\right)\ddot{y}-\dfrac{gkLM}{IJ}\sin{y}+\dfrac{k}{IJ}u.$$

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## Publication details:

 Title Adaptive output feedback control of nonlinear systems represented by input-output models Publication Type Journal Article Year of Publication 1996 Authors Khalil, H.K. Journal IEEE Transactions on Automatic Control Volume 41 Issue 2 Start Page 177 Pagination 177-188 Date Published 02/1996 ISSN 0018-9286 Accession Number 5202146 Keywords adaptive control, linearisation techniques, nonlinear control systems, state feedback Abstract We consider a single-input-single-output nonlinear system which can be represented globally by an input-output model. The system is input-output linearizable by feedback and is required to satisfy a minimum phase condition. The nonlinearities are not required to satisfy any global growth condition. The model depends linearly on unknown parameters which belong to a known compact convex set. We design a semiglobal adaptive output feedback controller which ensures that the output of the system tracks any given reference signal which is bounded and has bounded derivatives up to the nth order, where n is the order of the system. The reference signal and its derivatives are assumed to belong to a known compact set. It is also assumed to be sufficiently rich to satisfy a persistence of excitation condition. The design process is simple. First we assume that the output and its derivatives are available for feedback and design the adaptive controller as a state feedback controller in appropriate coordinates. Then we saturate the controller outside a domain of interest and use a high-gain observer to estimate the derivatives of the output. We prove, via asymptotic analysis, that when the speed of the high-gain observer is sufficiently high, the adaptive output feedback controller recovers the performance achieved under the state feedback one DOI 10.1109/9.481517