1928 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 8, AUGUST 2010 ITAE Optimal Sliding Modes for Third-Order Systems With Input Signal and State Constraints Andrzej Bartoszewicz and Aleksandra Nowacka-Leverton Abstract—In this note, the design of a time-varying switching plane for the sliding-mode control of the third-order system subject to velocity, ac- celeration and input signal constraints is considered. Initially, the switching plane passes through the system representative point (RP) in the error state space and then it moves with a constant velocity to the origin of the space. Having reached the origin the plane stops moving and remains fixed. The plane parameters are selected to minimize the integral of the time multi- plied by the absolute error (ITAE) without violating velocity, acceleration and input signal constraints. Furthermore, the switching plane is chosen in such a way that the reaching phase is eliminated, insensitivity of the system with reference to the external disturbances and the model uncertainty is guaranteed from the very beginning of the control action and monotonic tracking error convergence to zero is ensured. Index Terms—Sliding-mode control, switching plane design, time- varying switching planes, variable structure systems. I. INTRODUCTION Sliding-mode control (SMC) [1]–[4], provides an effective and ro- bust means of controlling nonlinear plants. The main advantage of this technique is that once the system state reaches a sliding surface, the system dynamics remain insensitive to a class of parameter variations and disturbances. However, robust tracking is assured only after the system state hits the sliding surface, i.e., the robustness is not guaranteed during the reaching phase. Provided a conventional time-invariant sliding plane is considered, the advantage of the SMC, namely the insensitivity of the system, is not obtained for some time from the beginning of its mo- tion. In order to overcome this problem the idea of the time-varying switching lines applied to the SMC of the second-order systems was introduced in [5] and further discussed in [6] and [7]. The control al- gorithms proposed in those papers eliminate the reaching phase and guarantee fast error convergence for the second-order uncertain sys- tems with arbitrary initial conditions. Further results on the application of the time-varying switching surfaces to the SMC of dynamic plants have been reported in [8]–[19]. In this note, we consider the third-order, nonlinear, time-varying system subject to the acceleration, velocity and input signal constraints. We introduce a time-varying switching plane which at the initial time passes through the RP, specified by the initial conditions of the system, in the error state space. Afterwards, the plane moves smoothly, with a constant velocity, to the origin of the space and having reached the origin remains fixed. Thus the proposed control algorithm eliminates the reaching phase and forces the RP of the system to always stay on the switching plane. As a consequence, our control is robust with respect to the external disturbances and parameter uncertainties from the very beginning of the system motion. Furthermore, in order to obtain good Manuscript received December 12, 2008; revised June 04, 2009; accepted March 31, 2010. First published May 03, 2010; current version published July 30, 2010. This work was supported by the Polish State budget in the years 2008–2010, under the research project “Design of the switching surfaces for the sliding mode control of dynamic plants” (N N514 300035) and by the Foun- dation for Polish Science (FNP). Recommended by Associate Editor A. Ferrara. The authors are with the Institute of Automatic Control, Technical Univer- sity of Lódz ´, Lódz ´ 90-924, Poland (e-mail: andrzej.bartoszewicz@p.lodz.pl; ola.nowacka@gmail.com). Digital Object Identifier 10.1109/TAC.2010.2049688 dynamic performance of the considered system, the switching plane is designed in such a way that the ITAE over the whole period of the control action is minimized and the constraints are satisfied. No matter which of the constraints are taken into account the proposed control algorithm ensures the tracking error convergence to zero without over- shoots or oscillations. This implies that the position constraint is always satisfied in the considered system. II. PROBLEM STATEMENT We consider the time-varying nonlinear third-order system (1) where , , and are the state variables of the system and is the state vector, denotes time, is the input signal, , – are a priori known, bounded functions of time and the system state, and are functions representing the system uncer- tainty and external disturbances, respectively. Further, in the note, it is assumed that there exists a strictly positive constant , which is the lower bound of , i.e., (2) Furthermore, functions and are unknown and bounded. There- fore, there exists a constant which for every pair satisfies the following condition . Initial condi- tions of the system are denoted as , , . System (1) is supposed to track the desired trajec- tory given as a function of time , where , and is a dif- ferentiable function of time. The tracking error is defined by the following vector . Hence, we have , , . In this note, it is assumed that at the initial time , the tracking error , where is an arbitrary real number different than zero. Furthermore, we assume that the first and second-order error derivatives satisfy the following relations: and . This condition is indeed satisfied in many practical applications such as for example point to point control of robot manipulators. Let us introduce a time-varying switching plane which originally moves with a constant velocity in the state space and then stops at the time instant . Consequently, for any (3) where , , , and are some constants. The selection of these con- stants will be considered further in the note. Since the considered plane stops at the time , then for any (4) First, constants , , , and should be chosen in such a way that the RP of the system at the initial time belongs to the switching plane. For that purpose, we require that (5) Notice that the input signal (6) where and is a strictly positive constant, ensures the stability of the sliding motion on switching plane (3). This input signal 0018-9286/$26.00 © 2010 IEEE