Adaptive Quasi-Sliding Mode Control Based on a Recursive Weighted Least Square Estimator for a DC Motor Juan D. Valladolid 1 , Juan P. Ortiz 1 , Member, IEEE and Luis I. Minchala, Member, IEEE Abstract— This paper presents the methodology of design of a discrete-time adaptive quasi-sliding mode controller (QSMC) based on a recursive weighted least square (RWLS) estimator for a dc motor. The proposed control scheme allows handling the classic problem of a QSMCs, which is the steady-state error due to the use of a saturation function instead of a switching function in the sliding mode control (SMC) algorithm. The use of linear and nonlinear signal references helps to show the closed-loop performance of the control system and its tracking capabilities. Experimental results show a better performance of the RWLS-QSMC algorithm applied on the speed control of a dc motor than a classic SMC. I. INTRODUCTION SMC is a powerful controller that offers good perfor- mances in systems, linear and nonlinear, with uncertainties in their models. Many researches of SMC applied to the energy field have been reported in literature [1]–[4] . Addi- tional applications of SMC for unmanned aerial vehicles are reported in [5], [6]. Further studies of SMC applied to an engine cooling system [7], congestion control in communica- tion networks [8], and the regulation of an electro-hydraulic system [9] are also analyzed. The implementation of SMC seems limited, sometimes, for the theoretical requirement of a switching control signal. The signal discontinuity due to such switching requirement causes the chattering phenomenon. This issue has been studied, and a solution where a saturation function is used instead of the switching function is proposed under the name of QSMC [10]–[12]. Although, the use of the saturation function typically causes steady-state error. Diminishing the steady-state error caused by the QSMC is typically done by the introduction of an integral term in the sliding surface [13]–[15]. This paper proposes a discrete QSMC based on a RWLS estimator. The RWLS estimator algorithm [16]–[18] allows the reduction of the steady state error by adapting the parameters of an autoregressive model with exogenous input (ARX) of the system to be controlled. The proposed controller (RWLS-QSMC) is tested in the speed regulation of a dc motor. The results show a good performance of the system in the regulation and tracking of linear and nonlinear references. One of the strengths of the This work was supported by the Transportation Engineering Research Group (GIIT) of the Universidad Politecnica Salesiana, Cuenca, Ecuador. 1 The authors are with the Department of Automative Engineering, Universidad Politecnica Salesiana, Cuenca, Ecuador jvalladolid@ups.edu.ec; jortizg@ups.edu.ec Luis I. Minchala is with the Department of Electrical, Electronic and Telecommunications Engineering, Universidad de Cuenca, Cuenca, Ecuador ismael.minchala@ucuenca.edu.ec RWLS-QSMC is the possibility to operate the dc motor in nonlinear zones of its dynamics, with good results. This paper is organized as follows: Section II presents the model of the dc motor. Section III presents the algorithm of the RWLS estimator. Section IV presents the methodology of design of the QSMC. Section V presents the details on the implementation and results of the RWLS-QSMC. Section VI presents the conclusions. II. DC MOTOR MODEL A time-variant ARX model is used to represent the motor dynamics by using the state-space approach: x (k + 1) = A (k) x (k)+ B (k) u (k) y (k)= C (k) x (k) (1) where x (k) ∈ℜ n is the state vector, y (k) ∈ℜ m is the output vector, u (k) ∈ℜ r is the input vector, A (k) ∈ℜ n×n is the state matrix, B (k) ∈ℜ n×r is the input matrix and C (k) ∈ℜ m×n is the output matrix. Without loss of generality, the dc motor is represented by a second order model where n =2, m =1 and r =1, which was adjusted from data measured from the system by applying a pseudorandom binary signal. III. RECURSIVE WEIGHTED LEAST SQUARE ESTIMATOR The ARX model to be used by the RWLS estimator is: y k = a 1 y k-1 + a 2 y k-2 + b 0 u k-1 + b 1 u k-2 (2) where y(k) is the system output, and u(k) is the control signal. The time delay of the system is one sample, d =1; therefore b 0 =0. The RWLS algorithm to be used in this work is presented in [18]-[19], and summarized here as follows: 1) Select initial values of the weighting factor (a), the forgetting factor (γ ), and the number of samples (N ) 2) Obtain the covariance matrix, Q N = ( Ψ T N W N Ψ N ) -1 where W N =        aγ N-nm 0 ... 0 0 0 aγ N-(nm+1) ... 0 0 . . . . . . . . . . . . . . . 0 0 ... aγ 0 0 0 ... 0 a        2016 IEEE Conference on Control Applications (CCA) Part of 2016 IEEE Multi-Conference on Systems and Control September 19-22, 2016. Buenos Aires, Argentina 978-1-5090-0755-4/16/$31.00 ©2016 IEEE 886