1.1 Research Motivation
The sliding mode control (SMC) is well known for its robustness to parameter variations and bounded matched disturbances [1-6]. With this advantage, the sliding mode control has been extensively applied to various fields of engineering applications, e.g., robotic systems, electric drives and switching power converters [7-10]. The basic idea of the sliding mode control is enforcing the system trajectories to reach a predetermined sliding surface or sliding regime in a finite time and then stay on it thereafter. Once the sliding mode occurs, the controlled system will possess excellent robustness and invariance properties to the matched disturbance.
When the input only takes two possible values, it is called the bang-bang input, which originates from the optimal control and some examples can be found in [11].
While a bang-bang input is only available in the sliding mode control, it is referred to as the bang-bang sliding mode control. The major difference between it and the conventional sliding mode control is that the reaching-and-sliding (RAS) region may not be globally satisfied [12,13].
Switching power converters are the most significant systems with their control inputs only switching between 0 and 1. For the switched systems consisting of two subsystems, their inputs also take only two possible values of 1 and –1. Therefore, it is proper to design the switching controllers of these systems via bang-bang sliding mode control. This dissertation will mainly focus on the design of the switching controller for a buck DC-DC converter based on bang-bang sliding mode control and then apply it to stabilize the switched systems.
1.2 Sliding Mode in Switching Power Converters
A variety of SMC-based methods have been proposed for switching power converters [12,14-26]. Compared to the state-space average method [27,28], the sliding mode theory provides large signal stability and is more robust to uncertainties.
Earlier, Sira-Ramirez presented detailed analyses about bilinear switched-networks and showed that based on the sliding mode theory, the output voltage regulation of a buck DC-DC converter could be achieved via indirect control with the exactly known model [12]. Then, Carpita and Marchesoni presented a robust sliding mode controller for the power conditioning system with resistive load variation and input disturbance;
however, they didn’t consider how the system performances are affected by different choice of the coefficients in sliding functions [17]. During that time, Spiazzi et al.
proposed general purpose sliding mode control for DC-DC switching power converters [25,26]. More detailed analyses about the system stability were given and a simple method for switching power converters to operate at a constant switching frequency was proposed. Recently, Tan et al. give guidelines on the practical design of the sliding mode controller for buck DC-DC converters [19] and propose other SMC-based controllers to operate switching power converters at a constant switching frequency [20-22]. He and Luo also give another type of SMC-based controller to achieve constant switching frequency [23,24].
From above, there are mainly two purposes in these works. First, design a sliding function on which the system dynamics can be stabilized. For buck DC-DC converters, the simplest sliding function can be determined from the linear combination of state variables in the phase-variable control canonical form [17,19,25].
Other types of sliding functions are proposed by adding an integral term of the output
voltage error [20,24]. Second, modify the controllers such that switching power converters can operate at a constant switching frequency with the invariant property to input and load variations. Several possible methods have been proposed to achieve this objective [20,21,24,26]. Based on the equivalent control in the sliding mode, the duty cycle control signal of a PWM controller can be determined from the work of Tan et al. [20] and He and Luo [24]. An adjustable hysteresis band is proposed based on the adaptive feedforward and feedback control scheme in [21]. The adaptive feedforward loop is adopted to reduce the frequency deviation resulting from the input voltage variation. As for the adaptive feedback loop, it is used to adjust the parameter of the sliding function such that the frequency deviation resulting from the load variation can be eliminated. Another simple method is adding a periodic ramp signal into the hysteresis-type sliding mode controller and then using an additional PI-type compensator to reduce the steady-state error in the sliding mode [26]. Among these methods, the last one possesses the advantage of easier realization and preserves the original fast dynamic response in transient state. Therefore, this method will be adopted in this dissertation for the purpose of controlling the buck DC-DC converter at a constant switching frequency.
In switching power converters, their large-signal models depend on the different states of switching devices or diodes. Thus, large-signal multi-models should be adopted to describe the overall system dynamics and these switching power converters are treated as switched systems [29,30]. As a result, the stabilization problems in switched systems are very significant and worthy of research.
1.3 Stabilization Problems in Switched Systems
Switched systems are a special class of hybrid systems consisting of more than one subsystem [31]. Recently, more and more attentions have been paid to this filed and one of the most attractive problems is to stabilize the switched systems consisting of unstable subsystems (see, e.g., [31-35] and the references cited therein). In the work of Wicks et al. [33,34], they incorporated the sliding mode theory to design the switching controllers for the switched systems with stable convex combinations of two subsystem matrices. In the work of Xu and Antsaklis [36], they discussed three classes of switched systems with subsystems possessing unstable foci, unstable nodes and saddle points. Their main idea is to choose an active subsystem such that the distance of the state to the origin is minimized, where the switching criterions are based on the angles of the subsystem vector fields and the geometric properties of the phase plane. In [37], Bacciotti proposed another stabilizing switching rule based on the damping feedback originated from the work by Jurdjevic and Quinn [38]. In his later work with Ceragioli [39], a state-static-memoryless stabilizing feedback law was proposed to stabilize a different class of switched systems, in which one of the subsystems has a pair of conjugate imaginary eigenvalues. Recently, Lin and Antasklis consider a class of uncertain switched systems satisfying the assumptions that their subsystems contain stable auxiliary systems and there exist no unstable sliding motions [40]. Motivated by the works of Wicks et al. [33,34], this dissertation will extend the bang-bang sliding mode control to design the switching controller for another class of switched systems. Compared with their work, we will give more theoretical analyses about the sliding motions in the switched systems and provide two important assumptions to guarantee the existence of stable sliding motions.
1.4 Organization of the Dissertation
The remaining contents of this dissertation are organized as follows. In Chapter 2, the fundaments of the sliding mode theory are given as preliminaries and then the properties of the bang-bang sliding mode control are discussed. In Chapter 3, the switching controller of a buck DC-DC converter is designed based on the bang-bang sliding mode control. The switching behaviors will be analyzed in the phase plane and experiment results will be given for verifications. Some practical considerations will also be included and the original controller will be modified to achieve constant switching frequency by adding a periodic ramp signal. In Chapter 4, the bang-bang sliding mode control is extended to design the switching controller for a class of switched systems and two important assumptions to guarantee the existence of stable sliding motions will be given. Moreover, it will show that the switching control laws consist of two switching functions and the complex switching behaviors will be clearly described. Further, the robustness of this switching controller to the model uncertainties will be discussed. Finally, conclusions and suggestions for future research are given in Chapter 5.