Bang-Bang順滑控制在切換式電源轉換器之設計

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(1)國立交通大學電機與控制工程學系博士論文. Bang-Bang 順滑控制在切換式電源轉換器之設計. Bang-Bang Sliding Mode Control in Switching Power Converters. 研究生：蔡. 建. 峰. 指導教授：陳. 永. 平. 博士. 中華民國九十六年十月.

(2) Bang-Bang 順滑控制在切換式電源轉換器之設計 Bang-Bang Sliding Mode Control in Switching Power Converters. 研究生：蔡建峰. Student：Jian-Feng Tsai. 指導教授：陳永平教授. Advisor：Prof. Yon-Ping Chen. 國立交通大學電機與控制工程學系博士論文. A Dissertation Submitted to Department of Electrical and Control Engineering College of Electrical and Computer Engineering National Chiao-Tung University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Electrical and Control Engineering October 2007 Hsinchu, Taiwan, Republic of China. 中華民國九十六年十月.

(3) Bang-Bang Sliding Mode Control in Switching Power Converters Student: Jian-Feng Tsai Advisor: Yon-Ping Chen Department of Electrical and Control Engineering National Chaio-Tung University. Abstract In this dissertation, the bang-bang sliding mode control is proposed to design the switching controller for a buck DC-DC converter. Normally, the on and off states of the switching device is modeled as an input switching between 1 and 0, and the reaching-and-sliding (RAS) regions are crucially related to the parameters of the sliding function. As a result, the sliding motions may not globally exist and different switching behaviors may be generated. With the bang-bang sliding mode control, the system trajectories of a buck DC-DC converter will clearly depicted in the phase plane and experimental results will be given for verifications. Besides, some considerations about the practical implementation are also included and a periodic ramp signal is added into the controller to achieve constant switching frequency. Then, the bang-bang sliding mode control is also adopted to design the switching controller for a class of switched systems. Two sufficient conditions to guarantee the existence of stable sliding motions will be given and the complex switching behaviors resulting from two switching functions will be clearly described. Moreover, the robustness of this switching controller to model uncertainties will be discussed. Finally, numerical simulations are given for demonstrations.. i.

(4) Bang-Bang 順滑控制在切換式電源轉換器之設計研究生: 蔡建峰. 指導教授: 陳永平國立交通大學電機與控制工程學系. 中文摘要. 本論文利用 bang-bang 順滑控制來設計降壓式直流轉換電路之控制器，由於開關元件只操作在開和關兩個狀態，加上順滑迫近區間會因為選擇不同的順滑函數而改變，導致順滑行為無法全域地存在狀態空間中，並產生不同的切換行為。本論文會在相平面中詳盡地分析系統的狀態軌跡，同時經由實驗來驗證。此外，實務設計時所會面對的問題也將同時考量，並且加入週期性的斜坡函數來達到固定的切換頻率。最後，本論文將利用 bang-bang 順滑控制來設計某一類切換系統的控制器，同時提出存在穩定順滑行為的兩個條件，由於推導出的控制法則包含兩個切換函數，因此會產生複雜的切換行為，在本論文中也將清楚的描述這些切換行為，並更進一步探討該控制器對於系統中存在不確定模式的強健性，最後經由數值模擬來呈現。. ii.

(5) 致. 謝. 本論文的完成，最得感謝陳永平老師多年來的悉心指導，帶領我進入學術的殿堂與耐心的指導我論文寫作，並且讓我負責馬達組的研究，在此謹向老師表達最誠摯的感謝。同時，謝謝口試委員徐國鎧老師、鄭志強老師、楊谷洋老師、梁耀文老師與張浚林老師的寶貴意見與指正，使本論文更臻於完善。其次，謝謝世宏、倍榕與宗正在口試期間的幫忙，讓整個過程能夠順利進行。另外，謝謝馬達組的翰宏、欣達、思穎、胤宏與坤祐，由於你們的付出，讓研究能夠一點一滴的累積。對於可變結構實驗室的夥伴豐洲、桓展、子楊以及畢業的克聰學長、倉鴻、文魁和昌衢，也要說聲謝謝，因為有你們，研究生活才更加的多彩多姿。最後，要感謝最親愛的父母親與姊姊，由於你們的體諒與支持，讓我能專心於課業，堅持到畢業這一刻，我愛你們。. iii.

(6) Contents English Abstract ............................................................................................................. i Chinese Abstract ........................................................................................................... ii Acknowledgement ....................................................................................................... iii Contents ....................................................................................................................... iv List of Figure..................................................................................................................v List of Table ................................................................................................................ vii Chapter 1 Introduction ...................................................................................................1 1.1 Research Motivation ........................................................................................1 1.2 Sliding Mode in Switching Power Converters.................................................2 1.3 Stabilization Problems in Switched Systems ...................................................4 1.4 Organization of the Dissertation ......................................................................5 Chapter 2 Bang-Bang Sliding Mode Control.................................................................6 2.1 Preliminaries of General Sliding Mode Control ..............................................6 2.2 Properties of Bang-Bang Sliding Mode Control..............................................8 Chapter 3 Bang-Bang Sliding Mode Control in Switching Power Converters ...........10 3.1 Model Description of Buck DC-DC Converter ............................................. 11 3.2 Design Procedures and Phase Plane Analysis................................................13 3.2.1 Design Procedures of Bang-Bang Sliding Mode Control...................13 3.2.2 Phase Plane Analysis...........................................................................16 3.3 Experimental Results .....................................................................................21 3.4 Discussions ....................................................................................................28 3.4.1 Modifications of State Variables and Sliding Function ......................28 3.4.2 Constant Switching Frequency Strategy.............................................30 3.4.3 Numerical Simulation Results ............................................................38 Chapter 4 Bang-Bang Sliding Mode Control in Switched Systems ............................46 4.1 Problem Statement .........................................................................................47 4.2 Design Procedures of Bang-Bang Sliding Mode Control..............................49 4.3 Stabilization of the Switched Systems with Model Uncertainties .................54 4.4 Numerical Simulation Results .......................................................................58 Chapter 5 Conclusion and Future Work.......................................................................64 5.1 Conclusion .....................................................................................................64 5.2 Future Work ...................................................................................................65 Bibliography ................................................................................................................67. iv.

(7) List of Figure Fig. 2.1 Three modes of switching behaviors on s=0. .................................................8 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6. Buck DC-DC converter with resistive load. .................................................12 Six cases with l1, l2 and s in the phase plane. ...............................................15 System trajectories with different initial conditions. ....................................17 System trajectories in the cases of Type-I. ...................................................20 System trajectories in the cases of Type-II. ..................................................20 The bang-bang sliding mode controller for the buck DC-DC converter. .....21. Fig. 3.7 System trajectories with different λ in example 3.1. ....................................24 Fig. 3.8 Sliding functions with different λ in example 3.1. .......................................24 Fig. 3.9 Output voltage errors with different λ in example 3.1. ................................25 Fig. 3.10 System trajectories with different λ in example 3.2. ....................................25 Fig. 3.11 Sliding functions with different λ in example 3.2. .......................................26 Fig. 3.12 Output voltage errors with different λ in example 3.2. ................................26 Fig. 3.13 Output voltage errors by load variations. .....................................................27 Fig. 3.14 Bang-bang sliding motions inside the hysteresis region...............................32 Fig. 3.15 Hysteresis-type bang-bang sliding mode controller .....................................32 Fig. 3.16 Hysteresis-type bang-bang sliding mode controller at switching constant frequency.......................................................................................................35 Fig. 3.17 Waveforms of the signals inside the hysteresis region (d>0.5). ...................35 Fig. 3.18 Waveforms of the signals inside the hysteresis region (d<0.5). ...................36 Fig. 3.19 Waveforms of the signals inside the hysteresis region ( ε 0 >0). ....................37 Fig. 3.20 Waveforms of the signals inside the hysteresis region ( ε 0 <0). ....................37 Fig. 3.21 Hysteresis-type fixed-frequency bang-bang sliding mode controller with a PI-type compensator. ....................................................................................38 Fig. 3.22 Output voltage errors by percentage in four cases with different Vd. ...........40 Fig. 3.23 System trajectories in four cases with different Vd. ......................................40 Fig. 3.24 Waveforms of the case with Vd=5V. .............................................................41 Fig. 3.25 Waveforms of the case with Vd=10V. ...........................................................41 Fig. 3.26 Output voltage errors by percentage in four cases with different ε ramp. .....42 Fig. 3.27 Steady states of sliding functions in four cases with different ε ramp...........43 Fig. 3.28 Waveforms of the case (d) in example 3.5....................................................43 Fig. 3.29 Output voltage errors by percentage in three cases with different KI...........44 Fig. 3.30 Waveforms of the case (c) in example 3.6....................................................45 Fig. 4.1 Four regions in the state space separated by Ωs and ΩN...............................50 v.

(8) Fig. 4.2 Three types of reaching modes. ....................................................................53 Fig. 4.3 Geometric representations of the directional derivatives for the reaching mode of type 2. .............................................................................................53 Fig. 4.4 Geometric view of lN and l Δ related to model uncertainties. .....................55 Fig. 4.5 System trajectories of example 4.1: reaching modes of type 1 and type 2...62 Fig. 4.6 System trajectories of example 4.2: reaching modes of type 1 and type 3...63 Fig. 4.7 System trajectories of example 4.3. ..............................................................63. vi.

(9) List of Table Table 3.1 The relations among l1, l2 and s....................................................................15 Table 3.2 Parameters of the buck DC-DC converter....................................................21 Table 3.3 Tune-up of λ in terms of the desired performance. ..................................27. vii.

(10) Chapter 1 Introduction. 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.

(11) 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. 2.

(12) 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.. 3.

(13) 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.. 4.

(14) 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.. 5.

(15) Chapter 2 Bang-Bang Sliding Mode Control. In this chapter, the basic sliding mode theory is first introduced in §2.1 as preliminaries. Then, the problem resulting from the fixed input and the existence condition of sliding motion will be presented. Finally, the features of the bang-bang sliding mode control are discussed and three modes of the switching behaviors near the switching surface are described in §2.2.. 2.1 Preliminaries of General Sliding Mode Control Generally, there are two fundamental steps to design a sliding mode control. First, choose an appropriate sliding surface or sliding manifold s to guarantee the system stability in the sliding mode s=0. Second, derive the control algorithm such that the system trajectories can reach the sliding surface in a finite time and then stay thereafter [1-6]. For example, consider a linear time invariant system expressed as x = Ax + Bu + Bh( x ,t ). (2.1). where A is the n×n system matrix, B is the n×m input matrix, x is the n×1 state vector, u is the m×1 input vector and h is the m×1 matched input disturbance. Without loss of generality, the pair (A,B) is assumed to be controllable and B is of full rank. In the first step, the sliding surface is selected as s = Cx. (2.2). where s=[s1,s2,…,sm]T and C is a m×n coefficient matrix. Note that the choice of C must guarantee the existence of stable sliding motions and several approaches have been proposed, e.g., the transformation matrix method [41], the eigenstructure assignment method [42] and the Lyapunov-based method [43]. Then, design the input u=[u1 u2 … um] by discontinuous feedback, where 6.

(16) ⎧u + ( x ), si ( x ) > 0 ui = ⎨ i− , i = 1,2,…, m ⎩ui ( x ), si ( x ) < 0. (2.3). With the control algorithm (2.3), the reaching-and-sliding (RAS) condition must be satisfied by si si < −σ i si ,. si ≠ 0. (2.4). where σi are positive. Once (2.4) is satisfied, the system trajectories could reach the sliding mode s=0 in a finite time and then stay thereafter [1]. Next, consider a single input bilinear system [11] expressed as x = f ( x ) + ug ( x ) + h( x ,t ). (2.5). where f and g are both n×1 vector functions and h is a n×1 external disturbance vector. While the sliding mode occurring, the following invariance conditions [12] are satisfied s = 0 and L f +ug + h s = 0. (2.6). where Lf+ug+hs denotes the directional derivative of the scalar function s with respect to the vector filed f+ug+h. Then, the equivalent control ueq can be defined as u eq ( x ) = −. L f +h s. (2.7). Lg s. Theoretically, the ideal sliding dynamics or equivalent system dynamics can be obtained by substituting ueq into (2.5). However, there exist two considerable problems in (2.7). First, ueq must satisfy the intermediate condition [14], i.e., ueq must be bounded by. (. ). (. min u + ,u − < u eq ( x ) < max u + ,u −. ). (2.8). Therefore, RAS-condition can only be satisfied in the region determined from (2.8), which is referred to as the reaching-and-sliding (RAS) region [13]. Second, the existence of sliding motion on s is guaranteed if the transversality condition [12] is satisfied by Lg s ≠ 0. (2.9). For the linear time invariant system (2.1), it is easy to satisfy the transverality. 7.

(17) condition by choosing a C such that the m×m square matrix CB is invertible, i.e., (CB)-1 exists. However, for bilinear systems, the state variables are involved in g(x) and thus, (2.9) may not be satisfied in some region, in which the stability can not be guaranteed.. 2.2 Properties of Bang-Bang Sliding Mode Control In this dissertation, the sliding mode theory is applied to the systems with the input only switching between two fixed values, which is referred to as bang-bang sliding mode control. With the constrained input, the RAS-condition may not be globally satisfied and thus the system trajectories will not slide along s=0 outside the RAS-region. Generally, the switching behaviors around s=0, as shown in Fig. 2.1, can be classified into three following modes [44]: 1. Refractive mode: the system trajectories are directed toward s=0 on one side and away from s=0 on the other side. 2. Attractive mode: the system trajectories are directed toward s=0 on both sides. 3. Rejective mode: the system trajectories are directed away from s=0 on both sides.. Fig. 2.1 Three modes of switching behaviors on s=0. (a) Refractive mode. (b) Attractive mode. (c) Rejective mode. 8.

(18) Mathematically, these switching modes can be represented by following equations: s ( x ) u =u + ⋅ s ( x ) u =u − > 0 on s ( x ) = 0. (2.10). s( x ) ⋅ s( x ) < 0. around s ( x ) = 0. (2.11). around s ( x ) = 0. (2.12). for the refractive mode,. for the attractive mode, and s( x ) ⋅ s( x ) > 0. for the rejective mode. Obviously, the attractive mode on s(x)=0 is the sliding mode. In switching power converters, if they can be stabilized by the sliding mode control, only refractive mode and attractive mode will occur on s(x)=0. Usually, they are designed to operate only in the attractive mode (i.e., the sliding mode), and slide on it thereafter. Sometimes, they may be designed to operate from refractive mode to attractive mode purposely for achieving faster system dynamics. In next chapter, it will show that there exist only refractive mode and attractive mode around the sliding function in the buck DC-DC converter. The system trajectories will be clearly depicted in the phase plane and verified by experimental results. In Chapter 4, the switching behaviors in switched systems will be more complicated since the derived switching control laws consist of two switching functions: one is the predetermined sliding function and the other one is the switching boundary resulting from the bilinear model of switched systems. It will show that only attractive mode will occur around the sliding function but all three switching modes may exist around the other switching boundary. Besides, there may exist unstable hyper-switching motion around the intersection of these two switching functions in high-order switched systems. This complicated switching phenomenon will be briefly discussed in Chapter 5 as our future research.. 9.

(19) Chapter 3 Bang-Bang Sliding Mode Control in Switching Power Converters. With the switching property, the sliding mode theory provides an intuitive way to control switching power converters [12,15]. Compared with the state-space average method [27,28], the sliding mode theory leads to large signal stability. Besides, it is robust to uncertainties and much easier for the implementation. For simplicity, a buck DC-DC converter is conventionally modeled as a linear system by neglecting the unknown parasitic resistance, which usually results in a small uncertain variation and then reduces the system precision. For improvement, the unknown parasitic resistance is taken into consideration, which makes the buck DC-DC converter modeled as an uncertain bilinear system. However, it is not easy to deal with such an uncertain bilinear model by conventional control technologies. Generally, most of the existing works are based on the assumption that the buck DC-DC converter operates only in the continuous conduction mode (CCM). However, if the components’ values are not appropriately selected, the buck DC-DC converter may operate in the discontinuous conduction mode (DCM). In this chapter, it will show that the buck DC-DC converter can be finally driven into the predetermined sliding mode regardless of the existence of DCM during the transient state. The remainder of this chapter is organized as follows. In §3.1, the model of a buck DC-DC converter with parasitic resistance is first introduced as an uncertain bilinear system. Then, it is modified as a linear system in the phase-variable control canonical form. In §3.2, the design procedures of bang-bang sliding mode control will be given and the system trajectories with different switching behaviors will be clearly depicted. 10.

(20) in the phase plane. Then, experimental results of the buck DC-DC converter subject to load variation and different choices of sliding functions will be demonstrated in §3.3. Finally, some considerations about the practical implementation and the method of operating the switching power converters at a constant switching frequency will be included in §3.4.. 3.1 Model Description of Buck DC-DC Converter A buck DC-DC converter with resistive load is illustrated in Fig. 3.1, where E is the DC voltage source, L is the inductor, C is the capacitor, iL is the inductor current, iC is the capacitor voltage, vo is the output voltage, D is the diode, S is the switching device, R is the resistive load and rd is the unknown parasitic resistance in the circuit. Note. that R=R0+ΔR, where R0 is the nominal value of resistive load and ΔR varies in the range of [δr1, δr2]. Clearly, R∈[Rmin,Rmax], where Rmin=R0+δr1 and Rmax=R0+δr2. Assume that the buck DC-DC converter operates in CCM. Then, the state-space equation can be expressed as ⎡ ⎡ iL ⎤ ⎢ 0 ⎢ ⎥=⎢1 ⎣vC ⎦ ⎢ ⎣C. 1 ⎤ ⎡ rd L ⎥ ⎡ iL ⎤ + ⎢− 1 ⎥ ⎢⎣vC ⎥⎦ ⎢ L ⎥ − ⎣ 0 RC ⎦ −. ⎤ ⎡E⎤ 0⎥ ⎡ i L ⎤ ⎢ ⎥ ⎢ ⎥u + L u 0⎥⎦ ⎣vC ⎦ ⎢⎣ 0 ⎥⎦. (3.1). where u represents the on state and off state of the switching device S by the values of 1 and 0. Note that (3.1) is linear in control and linear in state variables iL and vC, but not jointly linear in control and state variables. That means (3.1) is a bilinear system [11]. Now, let the desired output voltage be a constant Vd. Define vo−Vd and iC/C as the new state variables x1 and x2. Then, we have x1 = d (vo − Vd )/dt = dvo /dt = iC /C = x2. (3.2). Note that vo is equivalent to vC if the equivalent serial resistance (ESR) of C is. 11.

(21) neglected. From Fig. 3.1, it is easy to obtain that iL = iC + vo /R = Cx2 + ( x1 + Vd )/R. (3.3). Further differentiating (3.3) and using the first equation in (3.1), it leads to x2 = −. (3.4). r V 1 1 E x1 − x2 + u − d iL u − d LC RC LC LC LC. Let x be [x1 x2]T, (3.2) and (3.4) could be rewritten as. x = Ax + ( B + ΔB ( x ))u + h. (3.5). where 1 ⎤ ⎡ 0 1 ⎥ A=⎢ 1 ⎢⎣− LC − RC ⎥⎦ ⎡ 0 ⎤ B=⎢ E ⎥ ⎢⎣ LC ⎥⎦. (3.6). (3.7). 0 ⎤ ⎡ 0 ⎤ ⎡ rd Vd ⎥ rd ΔB ( x ) = ⎢ rd i L ⎥ = ⎢ rd ⎢⎣− LC ⎥⎦ ⎢⎣− RLC x1 − L x2 − RLC ⎥⎦. (3.8). ⎡ 0 ⎤ h = ⎢ Vd ⎥ ⎢⎣− LC ⎥⎦. (3.9). Significantly, the bilinear system (3.1) is changed into (3.5), which is a linear system with control input u and encounters the state-dependent uncertainty ΔB(x) and external input h. Since both ΔB(x)u and h are matched disturbances, it is suitably to design the controller with the sliding mode control technique.. S rd. E. iL D. L vL. iC vC. C vo. R. Fig. 3.1 Buck DC-DC converter with resistive load. 12.

(22) 3.2 Design Procedures and Phase Plane Analysis In this section, the design procedures of bang-bang sliding mode control for a buck DC-DC converter are introduced first. Then, the RAS-regions will be shown to crucially relate to the coefficients of sliding functions. Finally, the system trajectories with different sliding functions will be clearly depicted in the phase plane.. 3.2.1 Design Procedures of Bang-Bang Sliding Mode Control. In general, there are two fundamental steps to design a sliding mode control. First, choose an appropriate sliding function s to guarantee the system stability in the sliding mode s=0. Second, derive the control algorithm such that the system trajectories can reach the sliding surface in a finite time and then stay on s=0 thereafter. However, unlike the conventional sliding mode control, the control input u in (3.5) only switches between 1 and 0, which makes the controller design more restrictive. In the first step, let the sliding function or sliding line be chosen as s = x2 + λx1. (3.10). where λ is a positive constant. Obviously, it will guarantee that the system dynamics is stable in the sliding mode s=0 since the system is represented in the phase-variable control canonical form. For the second step, the bang-bang sliding mode control algorithm is purposely designed as u = 0.5(1 − sgn(s )). (3.11). which switches between 1 and 0 depending on the scalar sign of s. Most importantly, (3.11) must satisfy the following RAS-condition: ss < 0 , ∀s ≠ 0. (3.12). such that the system trajectories will reach the sliding line, but may be not in a finite time, and then stay thereafter. From (3.5) and (3.10), the derivative of the sliding function with respect to time is 13.

(23) s=−. V r 1 1 ⎞ E ⎛ u − d iL u − d x1 + ⎜ λ − ⎟ x2 + LC LC LC LC RC ⎠ ⎝. (3.13). Further substituting the control input (3.11) into (3.13), it leads to E − rd iL Vd ⎧ 1 1 ⎞ ⎛ ⎪⎪ − LC x1 + ⎜ λ − RC ⎟ x2 + LC − LC ⎝ ⎠ s=⎨ V 1 1 ⎛ ⎞ ⎪− x1 + ⎜ λ − ⎟ x2 − d ⎪⎩ LC RC ⎠ LC ⎝. for s < 0 for s > 0. (3.14). If the following inequality is satisfied Vd E − rd iL V 1 1 ⎞ ⎛ − <− x1 + ⎜ λ − ⎟ x2 < d LC LC LC RC ⎠ LC ⎝. (3.15). the RAS-condition (3.12) is guaranteed. The region described by the inequality (3.15) is the RAS-region. Replacing iL by (3.3), (3.15) can be rewritten into two inequalities as ⎧ ⎪⎪ ⎨ ⎪ ⎪⎩. (R + rd )Vd − RE < − (R + rd ) x. 1 rd ⎞ ⎛ +⎜λ− − ⎟ x2 RLC RLC RC L ⎠ ⎝ V 1 1 ⎞ ⎛ − x1 + ⎜ λ − ⎟ x2 < d LC RC ⎠ LC ⎝ 1. (3.16). Clearly, the RAS-region must be determined from s and two lines, l1 and l2, with the slopes being m1=(R+rd)/(RLCλ−L−RCrd) and m2=R/(RLCλ−L) respectively. Since m1 and m2 are related to the values of components in the buck DC-DC converter and λ, the relations among l1, l2 and s can be classified into six cases: “A, B, C, D, E, F” as listed in Table 3.1 and they are also plotted in Fig. 3.2. Note that P1((RVd+rdVd−RE)/(R+rd),0) and P2(−Vd,0) are the crossing points of l1 and l2 in. x1-axis.. 14.

(24) Table 3.1 The relations among l1, l2 and s.. λ. Case. m1, m2. A. 0<λ≤1/RC−R/L. m1<0, m2<0, m1≤m2<−λ (L/C>R2). B. 1/RC−R/L<λ<1/RC. m1<0, m2<0, m2<m1<−λ. C. λ=1/RC 1/RC<λ<1/RC+rd/L λ=1/RC+rd/L 1/RC+rd/L<λ. m1<0, m2=∞, m1<−λ<m2. D E F. l1. l2. x2. m1<0, m2>0, m1<−λ<m2 m1=∞, m2>0, −λ<m2< m1 m1>0, m2>0, −λ<m2< m1. x2. l1. l2. l1. x2. l2 P1 P2. x1. P1. P2. Slidin g line. Sli din g. Case A l2. x2 l1 P1. P2. x1. Sl. lin e. x1 l2. e. e. Case F. Fig. 3.2 Six cases with l1, l2 and s in the phase plane.. 15. l in. P1. P2 in ing l Slid. l in. Case E. g. e. x2. l1. ne g li. in g. din Sli. d Sli. l2. id in. x1. Case C. P1. P2. P1. P2. Case B. x2. Case D. x1. l1. x1.

(25) 3.2.2 Phase Plane Analysis. In normal operation, the output voltage vo of a buck DC-DC converter is nonnegative, which implies x1 = vo − V d ≥ −Vd. (3.17). and then the system trajectories should be inherently in the right-half plane of x1=−Vd. Moreover, when a buck DC-DC converter operates in CCM, iL should be larger than zero and (3.3) can be consequently rewritten as iC (t ) > −. vo (t ) R. (3.18). While the components of the buck DC-DC converter are not well designed, it may operate in DCM with iL=0 and the system dynamics is governed by iC (t ) = −. vo (t ) R. (3.19). Since x1=vo−Vd and x2=iC/C, (3.18) and (3.19) can be rewritten as x1 + RCx2 > −Vd. (3.20). x1 + RCx2 = −Vd. (3.21). and. Note that the linear function (3.21) is referred to as the drifting line in this dissertation. According to (3.17), (3.20) and (3.21), the system trajectories of a buck DC-DC converter should be restricted to the region: x1 ≥ −Vd ⎧ Ω=⎨ ⎩ x1 + RCx2 ≥ −Vd. (3.22). as shown in Fig. 3.3, where the system trajectories with different initial conditions are also included for u=0(dashed line) and u=1 (solid line). The system trajectories depicted by dashed line and solid line will converge to Q1(−Vd,0) and Q2(E−Vd,0) respectively. All the system trajectories with u=0 move clockwise to reach the drifting line first and then approach Q1. As for the system trajectories with u=1, they move spirally clockwise to Q2. 16.

(26) x2. Ω. x1 = −Vd. x1. Q1 Dr if. Q2 tin g. lin e. Fig. 3.3 System trajectories with different initial conditions. According to Table 3.1, the RAS-regions could be mainly classified into two types:. λ≤1/RC as Type-I and λ>1/RC as Type-II. Clearly, Type-I consists of case A, case B and case C and Type-II consists of case D, case E and case F. For Type-I, Ω is the RAS-region and separated into Ω0 and Ω1 by s=0, as depicted in Fig. 3.4. From (3.16), Ω0 and Ω1 are bounded as s>0 ⎧ ⎪ l2 < 0 ⎪ , for u = 0 Ω0 = ⎨ > − x V 1 d ⎪ ⎪⎩ x1 + RCx2 > −Vd s<0 ⎧ ⎪ l1 > 0 ⎪ , for u = 1 Ω1 = ⎨ x1 > −Vd ⎪ ⎪⎩ x1 + RCx2 > −Vd. (3.23). (3.24). Two system trajectories related to J0 in Ω0 and J1 in Ω1 are also shown in Fig. 3.4, where J0 and J1 represent the initial conditions. Since all the system trajectories in Ω0 and Ω1 satisfy the RAS-condition (3.12), the system trajectories starting from J0 and 17.

(27) J1 will approach the sliding line and then generate the desired sliding mode s=0. From (3.10), the system dynamics will be exponentially stable in the sliding mode. Obviously, the RAS-condition is globally satisfied in Ω for the cases of Type-I and the system trajectories will never enter the drifting line. Thus, if the following inequality is satisfied. λ ≤ 1 / RC. (3.25). all the switching behaviors around s=0 are exactly of the attractive mode as discussed in §2.2, and the buck DC-DC converter will operate only in CCM. Typically, λ will be chosen as 1/RC such that the RAS-region is large enough [19]. However, the system dynamics with λ =1/RC may be too slow if faster response is required during the start-up or in the transient state. As for the cases of Type-II, the switching behaviors are more complicated than the cases of Type-I, due to the RAS-condition is only locally satisfied in Ω. From (3.16), four sub-regions, Ω0, Ω′0, Ω1 and Ω′1, as shown in Fig. 3.5, can be obtained and they are bounded as s>0 ⎧ ⎪ l2 < 0 ⎪ , for u = 0 Ω0 = ⎨ > − x V 1 d ⎪ ⎪⎩ x1 + RCx2 > −Vd ⎧ s>0 ⎪ Ω′0 = ⎨ l 2 > 0 , for u = 0 ⎪ x > −V d ⎩ 1 s<0 ⎧ ⎪ l1 > 0 ⎪ , for u = 1 Ω1 = ⎨ x1 > −Vd ⎪ ⎪⎩ x1 + RCx2 > −Vd s<0 ⎧ ⎪ Ω1′ = ⎨ l1 < 0 , for u = 1 ⎪ x + RCx > −V 2 d ⎩ 1. (3.26). (3.27). (3.28). (3.29). where Ω0 and Ω′0 are related to u=0 and Ω1 and Ω′1 are related to u=1. There are. 18.

(28) mainly five kinds of system trajectories shown in Fig. 3.5, corresponding to five different points J0, J1, J2, J3 and J4. For the system trajectories through J0 and J1, they are just like the cases of Type-I, which will reach the sliding line. For the system trajectory through J2, it may come from Ω1 or start from Ω′0. From Fig. 3.5, this system trajectory will enter the sub-region Ω0. For the system trajectory through J3, it may come from Ω0 or start from Ω′1. Also viewing from Fig. 3.5, this system trajectory will enter the sub-region Ω1. Finally, for the system trajectory through J4, it comes from Ω′0 and then moves along the drifting line with u=0. Once this system trajectory passes through the sliding line, it will move into the sub-region Ω′1, where u is changed from 0 to 1. According to the trends of these five system trajectories, all of them will be eventually reach the segment, between S1 and S2 of the sliding line and then slide along it to the origin. Compared with the cases of Type-I, the system trajectories of Type-II may include DCM in the transient state, but they can still be successfully driven into the sliding line and kept on s=0 with a larger convergent rate. Obviously, it corresponds to the discussion in §2.2 that the switching power converters will possess faster dynamic response while they are operated from refractive mode to attractive mode. From the above analyses, the bang-bang sliding mode control is shown to be an effective method to drive all the system trajectories to reach the sliding line and then converge to the origin, i.e., the output voltage regulation of the buck DC-DC converter is theoretically achieved. In next section, several experiments will be conducted to verify the proposed bang-bang sliding mode control.. 19.

(29) x2 Slid. P2. Ω. J0. ing. Ω0. line. x1. P1. J1. Ω1 Dr ift ing. l2 x1 = −Vd. l1 lin. e. Fig. 3.4 System trajectories in the cases of Type-I.. J2. x2. Ω. Ω0 /. li ing Slid. S2. P2. J0. ne. Ω1. P1. l2. Dr. x1 = −Vd. Ω0. J1. ift ing l. S1. ine. l1. J3. Ω1. J4. /. Fig. 3.5 System trajectories in the cases of Type-II.. 20. x1.

(30) 3.3 Experimental Results A lab-prototype of buck DC-DC converter for theoretically verifying is fabricated as shown in Fig. 3.6, in which sg denotes the gate driver signal, iC is the sensed capacitor current, vo is the sensed output voltage, and a PC-based controller is used to implement the bang-bang sliding mode control. The maximum sampling rate is limited at 20kHz and the parameters of the buck DC-DC converter are listed in Table 3.2. In the following examples, it will show the performances of the bang-bang sliding mode control in output voltage regulation subject to different sliding functions and load variations.. S. L vL. iL. IRF530. E. io iC. D. C vo. vC. 1N5822. R. sg 1/C. s. +. +. +. λ. _ Vd. Bang-Bang Sliding Mode Controller. Fig. 3.6 The bang-bang sliding mode controller for the buck DC-DC converter. Table 3.2 Parameters of the buck DC-DC converter. Parameter E. L. C. R. Vd. Value. 2.47(mH). 470(μF). 15.35(Ω). 8(V). 12.28(V). 21.

(31) Example 3.1:. The experimental results shown in Fig. 3.7, Fig. 3.8 and Fig. 3.9 are obtained by choosing three different sliding functions with λ=100, λ=500 and λ=1000. All the system trajectories are all successfully driven to the corresponding sliding lines and then move toward the origin, as shown in Fig. 3.7. For the system trajectory with. λ=100, it corresponds to the case of Type-I that goes through J1. As for the system trajectories with λ=500 and 1000, they correspond to the case of Type-II that goes through J1. Note that 1/RC=141.8 and thus the case with λ=100 is of Type-I, and the cases with λ=500 and 1000 are of Type-II. From Fig. 3.8, it is easy to find that the system trajectories with smaller λ will reach the sliding line faster than those with larger λ. However, from Fig. 3.9, their convergent rates of the output voltage errors will be slower than those with larger λ.. Remark 3.1. From Fig. 3.9, there are steady-state errors in the output voltages, which can be reduced if λ is increased. The steady-state errors result from the use of phase-variable control canonical form in the design of sliding function, which is equivalent to a PD-type feedback control [19,45]. Another reason of the existence of steady-state errors is that the averaging value of the sliding function may not be ideally zero [26].. Example 3.2:. Let λ=100, 3500 and 20000. The experimental results are shown in Fig. 3.10, Fig. 3.11 and Fig. 3.12. Fig. 3.10 shows three system trajectories with λ=100, 3500 and 20000. The system trajectory with λ=100 has been described in example 3.1. As for the system trajectories with λ=3500 and 20000, they correspond to the cases of Type-II that goes through J2 and J4 respectively. Note that there exists DCM during the 22.

(32) transient state in the case with λ=20000. From Fig. 3.11, it can also be found that the sliding functions with λ=3500 and λ=20000 do not always converge since the RAS-condition for the cases of Type-II is not globally satisfied. However, these system trajectories will eventually enter the RAS-region and reach the sliding line after a short time. Unlike example 3.1, Fig. 3.12 shows that larger λ will lead to larger overshoot, which may not be desired in some applications. In this example, it shows that the experimental results actually confirm the switching behaviors described in §3.3 and the bang-bang sliding mode control can successfully drive these system trajectories into the corresponding sliding lines.. Example 3.3:. Let λ=1000, 3300 and 4000 and connect the buck DC-DC converter to a variable resistive load with the value of 20.5Ω and 6.9Ω. Fig. 3.13 shows the output voltage errors of experimental results, where the sub-figures on the left side demonstrate the transient state from light load (3W) to heavy load (9W) and the sub-figures on the right side demonstrate the transient state from heavy load to light load. Obviously, if λ is chosen to be larger than 1000, the buck DC-DC converter will possess the robustness to load variations with less than 0.3V deviation, i.e., less than 3.75% of Vd. ■ From above examples, we can choose a suitable λ to attain the desired performances, as summarized in Table 3.3. If the overshoot is larger than the desired specification, it can be improved by decreasing the value of λ according to Fig. 3.12. If the steady-state error of the output voltage is larger than the acceptable value, it can be reduced by increasing λ according to Fig. 3.9. While the convergent rate is too slow, it can be accelerated by increasing λ also viewing from Fig. 3.9.. 23.

(33) 7000 6000 5000 4000. λ =1000. x2. 3000. λ =500. 2000 1000. λ =100. 0 -1000 -10. -8. -6. -4 x1. -2. 0. 2. Fig. 3.7 System trajectories with different λ in example 3.1.. 4000. Sliding Function. 2000 0. λ =100. -2000 -4000 λ =500 -6000 -8000 λ =1000. -10000 0. 0.001. 0.002 0.003 Time(s). 0.004. 0.005. Fig. 3.8 Sliding functions with different λ in example 3.1.. 24.

(34) Output Voltage Error(V). 2 0. λ =1000. -2 -4. λ =500 λ =100. -6 -8 -10 0. 0.01. 0.02 0.03 Time(s). 0.04. 0.05. Fig. 3.9 Output voltage errors with different λ in example 3.1.. 10000 8000 λ =3500. x2. 6000. λ =20000. 4000 2000 λ =100. 0 Drifting Line. -2000 -10. -8. -6. -4 x1. -2. 0. 2. Fig. 3.10 System trajectories with different λ in example 3.2.. 25.

(35) 10000 7500 Sliding Function. 5000. λ =100. 2500 0. -2500. λ =3500. -5000 -7500 -10000 0. λ =20000. 0.002. 0.004 0.006 Time(s). 0.008. 0.01. Fig. 3.11 Sliding functions with different λ in example 3.2.. Output Voltage Error(V). 2. λ =20000. 0 λ =3500. -2 λ =100. -4 -6 -8 -10 0. 0.01. 0.02 0.03 Time(s). 0.04. 0.05. Fig. 3.12 Output voltage errors with different λ in example 3.2.. 26.

(36) Output Voltage Error (V). 0 -0.25 -0.5 -0.75 2.5. 5. 7.5 10 12.5 15 17.5 20 22.5 25 Time(ms) λ=1000. Output Voltage Error (V). 0 0.5 0.25 0 -0.25 -0.5 -0.75 0. 5. 0.5. 10 Time(ms). 15. λ=3300. 20. 25. Output Voltage Error (V). Output Voltage Error (V) Output Voltage Error (V) Output Voltage Error (V). 0.5 0.25. 0.25 0 -0.25 -0.5 -0.75 0. 5. 10 Time(ms). 15. λ=4000. 20. 25. 0.5 0.25 0 -0.25 -0.5 -0.75 0. 5. 10 Time(ms). λ=1000. 5. 10 Time(ms). λ=3300. 5. 10 Time(ms). λ=4000. 0.5. 15. 20. 25. 15. 20. 25. 15. 20. 25. 0.25 0 -0.25 -0.5 -0.75 0 0.5 0.25 0 -0.25 -0.5 -0.75 0. Fig. 3.13 Output voltage errors by load variations.. Table 3.3 Tune-up of λ in terms of the desired performance. Desired performance. λ. Reduce overshoot. Decreased. Reduce output error. Increased. Increase convergent rate. Increased. 27.

(37) 3.4 Discussions Theoretical analyses of the bang-bang sliding mode control for a buck DC-DC converter are given in §3.2, and the hardware to realize the control algorithm in the PC-based environment is demonstrated in §3.3. It shows that the experimental results agree with the theoretical analyses. However, there are still some gaps between the PC-based controller and the practical controller by analog circuits such that the original control algorithm could not be directly implemented. Thus, some modifications for practical design will be included in this section. Besides, there exists unavoidable switching frequency deviation in the bang-bang sliding mode controller, which results from different sliding functions, input voltage, output voltage and load variations. The deviation in switching frequency is not desired since it makes the input and output filters design more difficult. Based on the method proposed in [25], a periodic ramp signal will be adopted in this section to achieve the constant switching frequency. More detailed discussions about the bang-bang sliding motions in the steady state will be given. Finally, numerical simulations will demonstrate the effectiveness by these modifications and some conclusions will be given.. 3.4.1 Modifications of State Variables and Sliding Function. Usually, there is only one external reference voltage Vref available in analog circuits. The sensed output voltage vo cannot be directly compared with Vref but should be scaled with a predetermined scalar α. Therefore, the original state variable x1 will be redefined by a new variable z1 as Vref ⎛ z1 = αvo − Vref = α ⎜⎜ vo − α ⎝. 28. ⎞ ⎟⎟ = α (vo − Vd ) = αx1 ⎠. (3.30).

(38) Obviously, α is determined from the ratio of Vref to Vd and z1 is x1 multiplied by α. Similar to the procedures in §3.1, define another state variable z2 as the derivative of z1 with respective to time, i.e., z2 =. dz1 dx = α 1 = αx 2 dt dt. (3.31). From (3.30) and (3.31), the system dynamics is represented as ⎡ 0 z=⎢ 1 ⎢⎣− LC. 1 ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ 1 ⎥ z + ⎢ αE ⎥u + ⎢ Vref ⎥ − − RC ⎥⎦ ⎢⎣ LC ⎥⎦ ⎢⎣ LC ⎥⎦. (3.32). where z is [z1 z2]T. Note that rd is neglected in this section for simplification. Compared with (3.5), z1 and z2 are the original state variables multiplied by α, the input vector is B multiplied by α and the external input vector is h scaled down by α. The system matrix A is unchanged and thus the system dynamics of (3.32) is still the same as (3.5). As for the RAS-region, it can also be derived from the similar procedures described in §3.2. With the state variables z1 and z2, the sliding function becomes s = z 2 + λz1 = α ( x2 + λx1 ). (3.33). Note that, in (3.25), to make RAS-region large enough, λ is usually chosen as β/C, where β=1/R and R is supposed to be known. Then, substituting the feedback signals into (3.33), it results in s=. α C. iC +. αβ C. (vo − Vd ). (3.34). In switching power converters, the value of capacitance C is typically in the range of microfarad. Therefore, the coefficients in (3.34) will be too large for practical implementation, e.g., in analog circuits, saturations may occur in operational amplifiers. A straightforward idea to solve this problem is multiplying the sliding function with a scalar γC. Then, the sliding function eventually becomes s = γC ( z 2 + λz1 ) = γ (αiC + β (αvo − Vref )) 29. (3.35).

(39) where γ is an adjustable positive number. Note that multiplying the sliding function with γC will not affect the overall system dynamics and the RAS-region, and it makes the sliding function independent of capacitance value. Now, there are two undetermined parameters β and γ in (3.35). For β, it is equivalent to λ and can be adjusted according to Table 3.3. As for γ, it is used to scale the original sliding function in a reasonable range.. 3.4.2 Constant Switching Frequency Strategy. Practically, the switching power converters cannot operate at an infinite switching frequency and a hysteresis band is usually used by the SMC-based controllers. However, the switching frequency of the hysteresis-type SMC-based controller is affected by line and load variations and thus, several methods as introduced in Chapter 1 have been proposed to modify the SMC-based controllers to control switching power converters at a constant switching frequency. In order to achieve the same purpose, the bang-bang sliding motions inside the hysteresis region will first be discussed. Then, a brief review of existing methods for SMC-based controllers will be given. Finally, the simplest method of adding a periodic ramp signal to control bang-bang sliding motions at a constant frequency will be discussed.. A) Bang-Bang Sliding Motions Inside the Hysteresis Region. In the sliding mode theory, the input is assumed to be capable of switching at an infinite frequency such that the controlled system can be theoretically in the sliding mode. However, for the switching power converters, the switching devices cannot be operated with an infinite switching frequency and thus a hysteresis band is usually. 30.

(40) introduced to avoid this unachievable requirement. While a hysteresis band is introduced around the s=0, the control law in (3.11) will become ⎧ 0, s > ε u=⎨ ⎩ 1, s < −ε. (3.36). where ε is the width of the hysteresis band. In the steady-state operation, the bang-bang sliding motions inside the hysteresis region are illustrated in Fig. 3.14, in which f(z,u−) and f(z,u+) denote the vectors of state variable velocity with u−=1 and u+=0 respectively. The time taken for f(z,u−) to move from point P to point Q and the time taken for f(z,u−) to move from point Q to point R are denoted as Δt1 and Δt2, which can be calculated by 2ε grad (s ) ⋅ f z ,u − − 2ε Δt 2 = grad (s ) ⋅ f z ,u + Δt1 =. (. ). (. ). (3.37). As shown in [19], if λ is ideally chosen as 1/RC, the switching frequency fsw can be approximately obtained as. f sw. ⎛ v ⎞ vC ⎜ 1 − C ⎟ 1 1 E⎠ = = = ⎝ 2εL T Δt1 + Δt 2. (3.38). From (3.38), fsw is mainly affected by the variations of input voltage and output voltage. Besides, if the resistive load is not exactly known (i.e., λ cannot be exactly set as 1/RC), it will also affect fsw. With the hysteresis band being used, the control scheme of the hysteresis-type bang-bang sliding mode controller is shown in Fig. 3.15.. 31.

(41) grad(s) Q. s (z ) = ε. f(z,u-). f(z,u+). s(z)>0. s (z ) = 0. s(z)<0. P. s (z ) = −ε. R. t Δt1. Δt 2. Fig. 3.14 Bang-bang sliding motions inside the hysteresis region.. S. iL. E. L vL. io iC. D. C vo. vC. R. sg. α s. γ. +. +. α β. +. _ Vref. Hysteresis-Type Bang-Bang Sliding Mode Controller. Fig. 3.15 Hysteresis-type bang-bang sliding mode controller. 32.

(42) B) Existing methods for fixed frequency sliding mode controller. As discussed in previous paragraph, fsw is highly dependent on the input voltage, output voltage and the resistive load. To control the switching power converters at a constant fsw, several methods have been briefly introduced in Chapter 1. In [20], an extra state defined as the integral of output voltage error is used to formulate the specific sliding function, which results in a PID-type PWM controller. Then, the equivalent control ueq can be derived from the invariance conditions in the sliding mode. However, R is still involved in the representation of ueq such that ueq cannot be perfectly realized. Compared with [20], the sliding function in [24] is designed as the function of inductor current, output voltage and the integral of output voltage error. With this sliding function, the system is initially in the sliding mode, i.e., s(t=0)=0. As a result, ueq can be theoretically obtained for all time. Significantly, an adaptive law is proposed to eliminate the effect of the un-modeled parasitic resistance in the circuit. In [21], the idea of adopting a variable hysteresis band adjusted by adaptive feedforward and feedback control is proposed. The adaptive feedforward loop is used to reduce the deviation of fsw resulting from the input voltage variation. As for the adaptive feedback loop, it is used to adjust the coefficients of the sliding function such that the deviation of fsw resulting from the load variation can also be reduced. Actually, the measuring of input voltage is needed for the adaptive feedforward control scheme to directly change the width of hysteresis band ε, and the measuring of the output load current io is also needed for the adaptive feedback control scheme to compute the resistive load R. The simplest method is that proposed in [26], which is implemented by adding a periodic ramp signal and a PI-type compensator into the hysteresis-type sliding mode controller. This method possesses the advantages that fsw is constant under all operation conditions and it can be directly controlled by varying the frequency of the periodic ramp signal. Therefore, it will be used in this dissertation 33.

(43) and some design considerations will be given in the next.. C) Hysteresis-Type Bang-Bang Sliding Mode Controller at fixed-frequency. First, define a periodic ramp signal sramp with the amplitude, frequency and period being denoted as εramp, framp and Tramp respectively. Then, add sramp into the original control scheme as the block diagram shown in Fig. 3.16, in which sc is the summation of s and sramp, sg is the gate driver signal and εs is the hysteresis band of s. The ideal steady-state waveforms of s, sramp and sc are given Fig. 3.17. From Fig. 3.17, it can be found that these signals should satisfy the following equations: Δton ⎞ ⎛ ⎞ ⎛ 2ε ⎜ s + s dt ⎟ + ⎜ − ε ramp + ramp Δt on ⎟ = sc ∫ 0 1 t = u = ⎜ ⎟ ⎜ ⎟ Tramp 0 ⎝ ⎠ ⎝ ⎠. t = Δton. =ε. (3.39). and Tramp ⎛ ⎞ ⎜s ⎟ + (− ε ) = s + s dt ramp c ⎜ t = Δton Δ∫t u =0 ⎟ on ⎝ ⎠. t =Tramp. < −ε. (3.40). Note that (3.39) and (3.40) are used to guarantee that S will switch off during each cycle and it can immediately switch on at the beginning of each cycle. To satisfy (3.40), εramp is typically chosen to be larger than ε. While a converter operates at a constant switching frequency, Δton/Tramp is equivalent to the duty cycle d and (3.39) can be expressed as. ε s + (2d − 1)ε ramp = ε. (3.41). − ε s − ε ramp < −ε. (3.42). Besides, (3.40) can be rewritten. Obviously, when fsw is synchronizing to framp, the bang-bang sliding motions inside the hysteresis region will be governed by (3.41) and (3.42). From (3.41), if the averaging value of s is zero, εs will be smaller than ε when d>0.5, and εs will be larger than ε when d<0.5. The waveforms with d>0.5 have been shown in Fig. 3.17 and the waveforms with d<0.5 are given in Fig. 3.18 as a comparison. 34.

(44) S. iL. E. L vL. io iC. D. C vo. vC. R. α sg. s. γ. +. +. α +. β. _ Vref. sc. +. sramp +. Hysteresis-Type Bang-Bang Sliding Mode Controller at Constant Frequency. Fig. 3.16 Hysteresis-type bang-bang sliding mode controller at switching constant frequency.. ＋ε ramp +ε. s(z ) = ε s s(z ) = 0. s. s(z ) = −ε s. −ε. sramp. − ε ramp. sc Δt on. Δt off. t. Fig. 3.17 Waveforms of the signals inside the hysteresis region (d>0.5). 35.

(45) s (z ) = ε s ＋ε ramp +ε. s (z ) = 0. −ε. sramp. − ε ramp. s. s (z ) = −ε s. sc t Δt on. Δt off. Δt on. Fig. 3.18 Waveforms of the signals inside the hysteresis region (d<0.5). Ideally, sramp should not affect the averaging value of s. But, it may deteriorate s such that the averaging value of s will be a nonzero value ε0. In this situation, the waveforms are non-ideal as given in Fig. 3.19, where ε0=0.5(εs1−εs2), and εs1 and εs2 are not necessary to be positive. The equations of (3.41) and (3.42) will become. ε s1 + (2d − 1)ε ramp = ε. (3.43). − ε s 2 − ε ramp < −ε. (3.44). and. The waveforms with positive ε0 have been shown in Fig. 3.19 and the waveforms with negative ε0 are also given in Fig. 3.20 as a comparison. Compared with Fig. 3.18, d is smaller when ε0 is positive, and d is larger when ε0 is negative. Thus, a nonzero ε0 may lead to steady-state error in the output voltage. In order to reduce the steady-state error, a PI-type compensator will be further incorporated into the control scheme and the block diagram is shown in Fig. 3.21. In next section, numerical simulations based on the control scheme in Fig. 3.16 will 36.

(46) first be given to show the effectiveness of adding an additional periodic ramp signal for achieving constant switching frequency. Then, a PI-type compensator will further be incorporated into the controller and numerical simulations will demonstrate the reduction of the steady-state error in the output voltage.. ＋ε ramp +ε s(z ) = ε s1 s(z ) = ε 0 s(z ) = 0 s(z ) = −ε s 2. s. −ε. sramp. − ε ramp. sc Δt on. t. Δt off. Fig. 3.19 Waveforms of the signals inside the hysteresis region ( ε 0 >0).. ＋ε ramp +ε. s(z ) = ε s1 s(z ) = 0 s (z ) = ε 0. s. s(z ) = −ε s 2 −ε. sramp. − ε ramp. sc Δt on. Δt off. t. Fig. 3.20 Waveforms of the signals inside the hysteresis region ( ε 0 <0).. 37.

(47) S. iL. E. L vL. io iC. D. C vo. vC. R. α sg. s. γ. +. +. α +. β. _ Vref. PI Compensator. sc. +. sramp +. Hysteresis-Type Fixed-Frequency Bang-Bang Sliding Mode Controller. Fig. 3.21 Hysteresis-type fixed-frequency bang-bang sliding mode controller with a PI-type compensator.. 3.4.3 Numerical Simulation Results Consider a buck DC-DC converter with the same parameters listed in Table 3.2 and set ε as 0.1. The parameters β and γ of the sliding function in (3.35) are chosen as 0.5 and 1. In the following examples, it will first show the simulation results with only the periodic ramp signal sramp being added, as the control scheme in Fig. 3.16. Then, the bang-bang sliding motions inside the hysteresis region with different Vd and different. εramp will be discussed. Finally, the control scheme in Fig. 3.21 will be employed to shown the effectiveness of reducing steady-state errors by incorporating the PI-type. 38.

(48) compensator.. Example 3.4 Consider four cases with the desired output voltage Vd being set as (a) 3V, (b) 5V, (c) 7V and (d) 10V. When the reference voltage Vref is provided as 3.3V, the corresponding α are calculated as the values of 1.1, 0.66, 0.47 and 0.33. Besides, framp is set as 20kHz and εramp is set as 0.5. The simulation results of the output voltage errors by percentage in four cases are presented in Fig. 3.22 and the system trajectories are shown in Fig. 3.23, in which the x-axis and y-axis are defined in the output voltage and capacitor current. Viewing from Fig. 3.22, there exist un-avoidable steady-state errors in the output voltage and the minimum error will occur in the situation when Vd is about half of input voltage, i.e., d is around 0.5. For the case (a) and case (d), the steady-state errors even exceed 10%. The waveforms of case (b) and case (d) are represented in Fig. 3.24 and Fig. 3.25. Obviously, ε0 is positive in Fig. 3.24 and negative in Fig. 3.25, which correspond to the situations in Fig. 3.19 and Fig. 3.20 respectively. Besides, the gate signal sg will synchronize to sramp if εramp is properly chosen to be enough larger than ε. In this example, εramp is 5 times the value of ε.. 39.

(49) Output Voltage Error (%). 20. (a) (b) (c) (d). 0. -20. (a) (b) (c) (d). -40. V =3 d V =5 d V =7 d V =10 d. -60. -80. -100 0. 5. 10. 15. 20. 25. 30. Time (ms) Fig. 3.22 Output voltage errors by percentage in four cases with different Vd.. 3.5. Capacitor Current(A). 3 2.5 2 1.5 1 0.5 0 -0.5 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. Output Voltage (V) Fig. 3.23 System trajectories in four cases with different Vd.. 40. 10.

(50) 1. s s ramp s. 0.5. c. 0 -0.5 -1 29. 29.05. 29.1. 29.15. 29.2. 29.25. 29.3. 29.35. 29.4. 29.45. 29.5. Gate Signal. Time (ms) s. g. 1. 0 29. 29.05. 29.1. 29.15. 29.2. 29.25. 29.3. 29.35. 29.4. 29.45. 29.5. Time (ms) Fig. 3.24 Waveforms of the case with Vd=5V.. 1. s s ramp s. 0.5. c. 0 -0.5 -1 29. 29.05. 29.1. 29.15. 29.2. 29.25. 29.3. 29.35. 29.4. 29.45. 29.5. Gate Signal. Time (ms) s. g. 1. 0 29. 29.05. 29.1. 29.15. 29.2. 29.25. 29.3. 29.35. 29.4. Time (ms) Fig. 3.25 Waveforms of the case with Vd=10V.. 41. 29.45. 29.5.

(51) Example 3.5 Consider four cases with εramp being set as (a) 0.15, (b) 0.25, (c) 1 and (d) 5. Let Vd be 5V and then α is calculated as 0.66 for these cases. The simulation results of the output voltage errors by percentage in four cases are presented in Fig. 3.26. From Fig. 3.26, with the same Vd, the steady-state error will be increased when εramp is increased. It results from that ε0 is larger for the cases with larger εramp, as shown in Fig. 3.27. Although the best output performance exists in case (a), sg will not synchronize to. sramp and the waveforms of this case are represented in Fig. 3.28. In this example, it shows that εramp should be chosen larger enough to guarantee that sg can synchronize to sramp. However, larger εramp may make the steady-state error beyond the acceptable range. In next example, a PI-type compensator will be incorporated into the control scheme to reduce the steady-state errors.. Output Voltage Error (%). 60. (d). (a) (b) (c) (d). 40 20 0. (c). εramp=0.15 εramp=0.25 εramp=1 εramp=5. (b) (a). -20 -40 -60 -80 -100 0. 5. 10. 15. 20. 25. 30. Time (ms) Fig. 3.26 Output voltage errors by percentage in four cases with different ε ramp.. 42.

(52) 0.4. (a) (b) (c) (d). 0.35. Sliding Function. 0.3 0.25. εramp=0.15 εramp=0.25 εramp=1 εramp=5. (d). 0.2 0.15. (c). 0.1 (b). 0.05 0. (a). -0.05 -0.1 29. 29.05. 29.1. 29.15. 29.2. 29.25. 29.3. 29.35. 29.4. 29.45. 29.5. Time (ms) Fig. 3.27 Steady states of sliding functions in four cases with different ε ramp.. 0.2. s s ramp s. 0.1. c. 0 -0.1 -0.2 29. 29.05. 29.1. 29.15. 29.2. 29.25. 29.3. 29.35. 29.4. 29.45. 29.5. Gate Signal. Time (ms) s. g. 1. 0 29. 29.05. 29.1. 29.15. 29.2. 29.25. 29.3. 29.35. 29.4. 29.45. Time (ms) Fig. 3.28 Waveforms of the case (d) in example 3.5. 43. 29.5.

(53) Example 3.6 Let εramp=1 and Vd=5 and introduce a PI-type compensator into the control scheme as shown in Fig. 3.21, where the proportional constant and integral constant are defined as KP and KI. Then, consider three cases with KP being set as the same value of 1 and KI being set as (a) 50, (b) 100 and (c) 200 respectively. To show the effectiveness of adding a PI-type compensator, the simulation results are represented in Fig. 3.29. Obviously, the steady-state errors can be effectively eliminated and the buck DC-DC converter can operate at a desired frequency framp by the modified bang-bang sliding mode control scheme given in Fig. 3.21. The waveforms of case (c) are given in Fig. 3.30, in which sramp and sc are scaled down and the magnitude of s is too small such that sramp and sc are almost the same. Note that there will exist a very small deviation in s for compensating the steady-state error.. Output Voltage Error (%). 16. (a) K =50 I (b) K =100 I (c) K =200. 14. I. 12 10 8 (a). 6. (b) 4 (c). 2 0 -2 0. 5. 10. 15. 20. 25. 30. Time (ms) Fig. 3.29 Output voltage errors by percentage in three cases with different KI.. 44.

(54) 0.05. s 0.05×s ramp 0.05×s. 0.025. c. 0 -0.025 -0.05 29. 29.05. 29.1. 29.15. 29.2. 29.25. 29.3. 29.35. 29.4. 29.45. 29.5. Gate Signal. Time (ms) s. g. 1. 0 29. 29.05. 29.1. 29.15. 29.2. 29.25. 29.3. 29.35. 29.4. 29.45. 29.5. Time (ms) Fig. 3.30 Waveforms of the case (c) in example 3.6.. Some considerations for practical implementation by analog circuits are first given in §3.4.1. Then, the original controller is modified to operate at a constant switching frequency in §3.4.2 and the steady-state bang-bang sliding motions inside the hysteresis region are clearly discussed. In §3.4.3, the simulation results are given to demonstrate the effectiveness of the proposed control scheme. Altogether, the purposes of scaling the original sliding function into a reasonable range and controlling the buck DC-DC converter at a constant switching frequency are both achieved.. 45.

(55) Chapter 4 Bang-Bang Sliding Mode Control in Switched Systems. In this chapter, a switching controller based on the bang-bang sliding mode control will be proposed for a class of switched systems consisting of two second-order unstable subsystems. In §4.1, the stabilization problem in these switched systems is first briefly introduced and two important assumptions will be given for the bang-bang sliding mode control to guarantee the existence of stable sliding motions. Then, the design procedures and the system stabilities related to three types of reaching modes are provided in §4.2. Another class of switched systems with model uncertainties is considered in §4.3 and the robustness of the bang-bang sliding mode control to model uncertainties will be discussed. Finally, several numerical simulation results are demonstrated in §4.4.. 46.

(56) 4.1 Problem Statement Consider a switched system consisting of two second-order subsystems, given by. x = Aσ (t ) x. (4.1). where σ(t ) : [0, ∞ ) → {1,2} , x ∈ R2 and both A1 and A2 are 2×2 constant non-Hurwitz matrices. The main purpose in this chapter is to determine the switching conditions for these two subsystems such that the overall system dynamics is stabilized. Interestingly, (4.1) can be rewritten into a second-order homogeneous bilinear system, similar to the procedure in [37], controlled by a switching input as x = Ax + uNx. (4.2). where A=0.5(A1+A2), N=0.5(A1−A2) and the switching input u∈{−1, 1}. Clearly, the stabilization problem in (4.1) is equivalent to designing the switching input u to stabilize (4.2). To deal with the stabilization problem in (4.2), first define λi and ci as the i-th eigenvaule and its corresponding left eigenvector of A+u0N, where u0 is a real number. Then, the bang-bang sliding mode control is proposed to stabilize the system (4.2) under the following assumptions:. Assumption 4.1. There exists a u0 in the range of (−1, 1) such that the eigenvalues, λ1 and λ2, of A+u0N are stable and real.. Assumption 4.2. There exists at least one left eigenvector ci corresponding to λi that satisfies rank([ci ;ciN])=2.. 47.

(57) Similar to several literatures [32-34], the existence of stable matrices combination. A+u0N is also required in this dissertation. Besides, λ1 and λ2 are further assumed to be real, as declared in Assumption 4.1, such that their left eigenvectors c1 and c2 are real and could be used as the coefficient vector of a sliding function [13,46]. As for Assumption 4.2, it is required to guarantee the existence of stable sliding motions, which will be explained later. In next section, with both assumptions being satisfied, a switching controller based on the bang-bang sliding mode control will be proposed for the homogenous bilinear system (4.2). It will show that the switching input must be determined from two switching functions rather than a single one as the general sliding controls. The stability of the switched system will be proven and different switching behaviors resulting from these two switching functions will be clearly described.. 48.

(58) 4.2 Design Procedures of Bang-Bang Sliding Mode Control Under Assumption 4.1, system (4.2) can be further rewritten as (4.3). x = AS x + uS Nx. where AS=A+u0N, uS=−u0+u and u∈{−1, 1}. Obviously, system (4.3) is still a homogeneous bilinear system but possesses a stable system matrix AS with real eigenvalues and a new input uS. Let λ be one of the eigenvalues of AS and c be the corresponding left eigenvectors, i.e., (4.4). cAS = λ c Then, the sliding function s(x) can be defined as s ( x ) = cx. (4.5). Note that c must satisfy the condition of rank([c;cN])=2. From (4.2)−(4.5), the derivative of s(x) with respective to time is (4.6). s( x ) = c ( A + u 0 N )x + uS cNx = λ s ( x ) + uS cNx. Since u∈{−1, 1}, choose u=−sgn(ρ(x))·sgn(s(x)), where ρ(x)=cNx, and then uS will be (4.7). uS = −u 0 + u = −u 0 − sgn(ρ ( x )) ⋅ sgn(s ( x )) From (4.5)−(4.7), we have s ( x )s ( x ) = λ s ( x ) + [− u 0 − sgn(ρ ( x )) ⋅ sgn(s ( x ))]⋅ ρ ( x ) ⋅ s ( x ) 2. = λ s ( x ) − u 0 ⋅ ρ ( x ) ⋅ s ( x ) − sgn(ρ ( x )) ⋅ sgn(s ( x )) ⋅ ρ ( x ) ⋅ s ( x ) 2. = λ s( x ) − [u 0 ⋅ sgn(ρ ( x )) ⋅ sgn(s( x )) + 1]⋅ ρ ( x ) ⋅ s ( x ) 2. = λ s( x ) − α ⋅ ρ ( x ) ⋅ s( x ) 2. ≤ −α ⋅ ρ ( x ) ⋅ s ( x ). (4.8). 49.