3386 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009 High-Gain Observers With Sliding Mode for State and Unknown Input Estimations Kalyana C. Veluvolu, Member, IEEE, and Yeng Chai Soh, Senior Member, IEEE Abstract—To handle the state estimation of a nonlinear system perturbed by a scalar disturbance distributed by a known nonlin- ear vector, a sliding-mode term is incorporated into the nonlinear high-gain observer (HGO) to realize a robust HGO. By imposing a structural assumption on the unknown input distribution vector, the observability of the disturbance with respect to the output is safeguarded, and the disturbance can be estimated from the sliding surface. Under a Lipschitz condition for the nonlinear part, the nonlinear observers are designed under the structural assumption that the system is observable with respect to any input. In the sliding mode, the disturbance under an equivalent control becomes an increment of Lipschitzian function, and the convergence of the estimation error dynamics can be proven similar to the analysis of HGOs. The proposed technique can be applied for fault detection and isolation. The simulation results for the bioreactor application demonstrate the effectiveness of the proposed method. Index Terms—High-gain observers (HGOs), nonlinear trans- formation, sliding-mode observers (SMOs), unknown input estimations. I. I NTRODUCTION T HE STATE estimation of nonlinear systems has been an active field of research in the last few decades. The works in [1]–[3] presented some fundamental results on state estima- tion of systems via state transformation and nonlinear observer. Through a nonlinear change of coordinates, linearization is achieved by output and output derivative injections. High-gain observers (HGOs) [1], [3] have been proposed for a general class of single output systems that is uniformly observable. The approach was generalized to a more general class of nonlinear systems in [4] and [5]. In [5], a constant gain observer is proposed for a special class of nonlinear systems that does not require the nonlinear transformation. Furthermore, [6] im- proved the existing design of the HGO by incorporating the nonlinearity of the system into the gain design strategy. The de- sign of other exponential observers such as the Gauthier–Kupka Observer [2] and the Kalman-Like Observer [3] also follows a similar framework. Khalil et al. [7]–[10] further explored the use of an HGO for feedback control, numerical differentiation, and sampled-data control. Manuscript received September 12, 2008; revised November 12, 2008. First published June 5, 2009; current version published August 12, 2009. K. C. Veluvolu is with the School of Electrical Engineering and Com- puter Science, Kyungpook National University, Daegu 702701, Korea (e-mail: veluvolu@ee.knu.ac.kr). Y. C. Soh is with the School of Electrical and Electronics Engineer- ing, Nanyang Technological University, Singapore 639798 (e-mail: eycsoh@ ntu.edu.sg). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2009.2023636 HGOs with sliding-mode control have been used in the design of output feedback controllers due to their ability to robustly estimate the unmeasured states and to asymptotically attenuate the disturbances [7], [9]. The works in [7] also proved a nonlinear separation principle for the stabilization of nonlinear systems employing the HGO. In [8], discrete- time implementation of HGOs was also explored. The work in [11] designed a discrete-time controller using HGO with the separation principle. However, HGOs are designed for the nominal systems and the performance degrades in the presence of uncertainty/disturbances. In this paper, a robust HGO is developed by appending a sliding-mode term to the HGO. Sliding-mode control is a well-established method for han- dling disturbances and modeling uncertainties through the con- cepts of sliding surface design and equivalent control [12]. Based on the same concept, sliding-mode observers (SMOs) have been developed to robustly estimate the system states [12]–[21]. The Lyapunov-based approach of Walcott and Zak [13] considered the problems of state observation in the pres- ence of bounded uncertainties/unknown inputs based on a matching condition. The approach in [16] and [22] extended the design of [12] to linear systems such that the states affected by the unknown inputs are dealt with by the switching terms. The reconstruction of unknown inputs/faults from equivalent control was also discussed in the same work. The work in [18] extended the SMO design of [13] to Lipschitz nonlinear systems based on matching conditions [12]. In [23] and [24], estimation in the presence of unknown inputs for mechani- cal systems has been studied using high-order sliding-mode techniques. In addition, there has been tremendous interest in the application of these techniques in real time for industrial applications in both continuous and discrete time [25]–[32]. The equivalent control-based SMO was first proposed by Utkin [12], and its extensions to nonlinear systems were ad- dressed in [14], [15], and [17]. These methods chose the output estimation error and its higher order derivatives as the sliding surfaces, which are, in general, not measurable. These meth- ods require structural assumptions for the design of switching terms for all the states, which are, in general, conservative. In practical applications, as only the output estimation error is available, and so the higher order derivatives have to be obtained through low-pass filtering [12]. Furthermore, most of the works to date are mainly limited to linear systems with constant disturbance distribution matrix and rely on either the structural conditions or matching assumptions such that the error system is stable and free from unknown inputs. Recently, SMOs [19], [20] are designed for a class of nonlinear uncertain continuous and discrete-time systems. The gain design depends 0278-0046/$26.00 © 2009 IEEE Authorized licensed use limited to: Kyungpook National University. Downloaded on August 31, 2009 at 09:18 from IEEE Xplore. Restrictions apply.