Controller Design for a Second-Order Plant with Uncertain Parameters and Disturbance: Application to a DC Motor

and Applied Analysis 3 The effect of the reconstructed error is tackled by means of an updated parameter provided by an additional updating law.The Lyapunov function involves a quadratic form for the difference between the reconstructed error and its updated parameter. The drawback is that the time derivative of the reconstructed error is neglected in the time derivative of the Lyapunov function, which is not realistic and could degrade the robustness of the controller. In summary, the main drawbacks of the above control schemes are the following: (D4) upper and lower bounds of plant model parameters and lumped plant model terms are required to be known; (D5) the upper bound of the transient behavior of the tracking error depends on integral terms that involve Nussbaum functions; (D6) the time derivative of identification error is neglected in the time derivative of the Lyapunov function. In the present work, an adaptive controller is developed for a permanent magnet DC motor. The state adaptive backstepping (SAB) of [4] is used as the basic framework for the controller design. In order to handle the unknown varying model parameters, significant modifications are introduced in the approach, on the basis of the modifications appearing in [32]. The main modifications are as follows: (i) use a truncated version of the quadratic form that depends on the backstepping states, and (ii) develop a convergence analysis based on the truncated version of the quadratic form. Using the scheme proposed in this paper, the following benefits are obtained: (RC1) the resulting upper bound of the transient behavior of the tracking error is constant and does not depend on integral terms involving Nussbaum functions, so that the transient behavior of the tracking error can be rendered small by properly choosing the controller parameters; (RC2) none of the exact values of the plantmodel parameters are required to be known; (RC3) none of the upper bounds of the plant model parameters are required to be known; (RC4) the tracking error converges to a residual set whose size is user defined, despite the lack of knowledge on both the exact values and the upper bounds of the plant model parameters; (RC5) discontinuous signals are avoided in the control and update laws; (RC6) the time derivative of the Lyapunov function does not neglect the time derivative of any varying parameter. The controller was applied to a permanent magnet DC motor whose voltage input is supplied by a buck power converter. With the aim to obtain a good agreement between simulations and experimental set-up, the numerical simulation includes realistic characteristics such as internal resistances, discretization, and time delay. The controller was implemented in a digital platform. The control design procedure and the stability analysis indicate that the drawbacks (D1), (D2), (D3), (D4), (D5), and (D6) are overcome, as the benefits (RC1) to (RC6) of the control scheme in [32] are achieved in the presentwork. In addition, the bounded nature of all the closed loop signals is guaranteed. This paper is organized as follows. In Section 2 the plant model used to design the controller and the goal of the control are presented. In Section 3 the design of the controller is developed. In Section 4 the bounded nature of the closed loop signals and the convergence of the tracking error are proven. In Section 5 numerical and experimental results are presented, and finally, Section 6 is devoted to conclusions. 2. Plant Model and Control Goal The linear model corresponding to a DC permanent magnet motor is given by