Chapter 5 Design of a Digital Sliding Mode Controller

Transcription

1 Chapter 5 Design of a Digital Sliding Mode Controller In chapter 4 the linear controllers PID and RST are developed to reglate the Bck converter pt voltage. Frthermore based on the PID and RST control laws, a Tri-mode controller is proposed to redce power consmption inside. However SMPS converters are nonlinear system in natre de to their switching property. One of the most important featres of the DC-DC converter is related to Variable Strctre Systems (VSS) characterises. VSS are systems physically changed intentionally dring time with respect to the strctre control law. The instances at which the changing of the strctre occrs are determined by the crrent state of the system. From this point of view SMPS represents a particlar class of VSS since their strctre is periodically changed by the action of controlled switches and diodes. Ths to enhance the dynamic performance of SMPS, nonlinear control methods, especially those nonlinear controllers with VSS are essential. Sliding-Mode-Control (SMC), a form of the large grop of VSS controller was theoretically introdced a few decades ago. SMC for VSS offers an alternative way to implement a control action which eploits the inherent variable strctre natre of DC-DC converters [L, P, P]. In particlar the converter switches are driven as a fnction of the instantaneos vales of the state variables in sch a way so as to force the system trajectory to stay on a sitable selected srface on the phase space called the sliding srface. SMC has been improved sitable for DC-DC converters by several researchers in recent literatres [R7, R8, A5, S4, S5, S6]. However becase of the SMC principles, the operation rires virtally infinite switching frency that challenges the feasibility of SMC in low-power SMPS converters. In order to fi the very high and variable switching frency, a fied-frency PWM-based SMC which derived from Hysteresis Modlator (HM) based SMC [B, P4, M] has been recently proposed in [S6, G] bt as an analog control for medim power SMPS applications where the switching frency is in the range of hndreds khz. Unfortnately analog PWM-based SMC is difficlt to control high-frency low-power SMPS. It is sensitive to analog component variations, and it can be mied only with analog signals in the converter and not with signals inside a high-level digital power management nit. By contrast, digital DPWMbased SMC designed inside the controller is easier to implement, is insensitive to analog 77

2 component variations and offers more fleibility. Theore, this chapter presents an original implementation of a DPWM-based SMC for a high-switching frency SMPS. 5. Review of Sliding Mode Control Before SMC application for digitally controlled SMPS, it is necessary to briefly introdce the theory and operation of SMC. This section covers the theoretical aspects of the SMC. 5.. Sliding Mode Controller: An Ideal Controller in Theory Sliding Mode Control (SMC) was introdced initially for the robst control of Variable Strctre Systems (VSS) [C, W]. The basic principle of SMC is to employ a certain sliding srface as a erence path, sch that the state variables trajectory can be directed towards the desired ilibrim. Theoretically sch ideology of the SMC can be flly achieved only with the absolte compliances to certain conditions, namely the eistence conditions, and the condition that the system operates at infinite switching frency. In sch respect what is derived is an idealized controlled system, whereby no eternal distrbance or system ncertainties can affect the ideal control performance for zero reglation error and very fast dynamic response. These featres fndamentally describe the epectancy of an ideally controlled system. Hence in a certain sense, the SMC is actally a type of ideal controller for the class of VSS. 5.. Theory of Sliding Mode Controller SMC theory has been applied to many nonlinear VSS to improve the systems performance. Variable Strctre Controllers (VSC) act as a high-speed switched feedback control reslting in sliding mode. The prpose of the switching control law is to drive the nonlinear plant state trajectory onto a pre-specified (ser-chosen) srface in the state space and to maintain the plant state trajectory on this srface for sbsent time. The srface is called a switching srface. When the plant state trajectory is above the srface, a feedback path has a given gain and another gain when the trajectory drops below the srface. This srface defines the rle for proper switching. The srface is also called a sliding srface (sliding manifold). Ideally, once intercepted, the switched control maintains the plant state trajectory on the srface for all sbsent time and the plant state trajectory slides along this srface ntil the origin. For eample the srface S and the plant state trajectory of a SMC is shown in Fig. 78

3 5-, where the SMC incldes three state variables, and. The main idea of SMC is to bring and keep the error on a sliding srface sch that the system is insensitive to the distrbances and parameter changes. Theore the SMC is very robst. The SMC design approach consists of two components: <> The first step is to bild a sliding srface with the different forms of control objectives. <> The second step is to determine the eistence of sliding motion and ensre the stability. Srface S= Converging to the origin Plant state trajectory O Fig. 5- Graphical representation of the plant state trajectory behavior in SMC process Considering a general SISO (single-inpt-single-pt) atonomos nonlinear system, the sliding-mode controlled system can be modelled by: f ( ) g( ) S( ) T K (5.) Where n is the variable vector of SMC, f and g are state vectors defined on n, S is the T sliding srface,,..., T K K K is the sliding gains parameters and is the switch position which makes the system discontinos: n, if S( ), if S( ) (5.) where and are the switching signals respectively in the range of S ( ) and S ( ). In order that the sliding motion can eist, the state variable phase trajectory mst be directed towards the sliding srface S ( ) to obtain a stable soltion of the system as shown in Fig. 5-. Ths it is necessary to determine the sliding motion range for the SMC, i.e. to determine the eistence conditions [F, J7, V] and ensre the srface eist. This task can be performed sing the Lyapnov s second method [J6], where the Lyapnov s fnction V is generally defined as V S. The eistence conditions mst be satisfied: Ths the eistence conditions of SMC can be written as: S( ) S( ) (5.) 79

4 S( ) if S( ) S( ) if S( ) (5.4) Sbstitting ation (5.) and (5.) into (5.4) then the eistence conditions of SMC can be rewritten as: S( ) T K f ( ) g( ) if S S( ) T K f ( ) g( ) if S (5.5) In that way the eistence conditions for the SMC are ensred. 5. Sliding Mode Control for DC-DC SMPS DC-DC converters, a type of VSS, are non-linear in natre. Moreover the parameters of SMPS change with the load variation. One of the important featres of the SMC in the VSS is its robstness, which makes the system insensitive to the parameters variation. From this point of view the SMC is particlarly sitable for the application of SMPS converter. 5.. Qasi-Sliding-Mode (QSM) Controller In order to acqire high performance for VSS, the ideal SMC mst be operated at high enogh switching frency on the sliding srface. Ths the natre of the sliding mode controller is to ideally operate at an infinite switching frency sch that the controlled variables can track a certain erence path to achieve the desired dynamic response and steady-state operation [V]. This rirement for operation at infinite switching frency, however, challenges the feasibility of applying SMC in DC-DC power converters. Hence for SMC to be applicable in power converters, their switching frencies mst be constricted within a practical range. In order to limit the switching frency, different methods sch as Hysteresis-Mode, Constant-on-Time and Limited-Maimm-Frency are proposed [B]. Among these conventional limiting-frency methods, the most poplar way is the Hysteresis-Mode (HMbased) SMC that will be introdced in net sb-section. Nevertheless this restriction of the sliding mode controller switching frency transforms the sliding-mode controller into a type of qasi-sliding-mode (QSM) controller, which operates as an approimation of the ideal SMC. Since all SMC in practical power converters 8

5 are frency-limited, they are, strictly speaking, a type of qasi-sliding-mode (QSM) controllers. 5.. Conventional HM-based SMC A review of the literatre [L, B, M, P4, V] shows that most of the previosly proposed SMC for switching power converters are Hysteresis-Modlation (HM) based, which rires a bang bang type controller to perform the switching control shown in Fig. 5-. S O Hysteresis-base SMC O S Fig. 5- Hysteresis Modlation-based SMC S Switch on S Switch off Since there are only two available choices ON ( ) and OFF ( ) for switch action in SMPS, then this method is easily accomplished as shown the ation (5.): where ' ON ' if S( ) ' OFF ' if S( ) previos state is an arbitrarily small vale arond zero. otherwise The introdction of a hysteresis band with the bondary conditions S ( ) S t (5.6) S ( ) provides a form to limit the infinite high switching frency. However de to the lack of systematic design methods and implementation criteria, the implementation of HMbased SMC for SMPS still relies on the trial-and-error tning of the and magnitde to achieve the desired switching frency for a particlar operating condition. The performance of HMbased SMC depends on the eperience of designer and engineer. 5.. The Rirement of Fied-Frency SMC Clearly from ation (5.6) that the infinite high switching frency is limited by the hysteresis band vale magnitde, bt the SMPS operation frencies yet rely on the bang-bang, i.e. HM-based SMC switching frency is still variable, which inherits the typical disadvantages of having variable switching frency operation and being highly control-sensitive to noise. Obviosly, when switching frency is variable, designing the 8

6 filters nder a worst-case (lowest) frency condition will reslt in oversized filters. Moreover, the variation of the switching frency also deteriorates the reglation performance of the converters. In order to keep the switching frency fied, two basic approaches have been proposed in the implementation of conventional HM-based SMC. One approach is to incorporate a constant ramp or timing fnction directly into the controller [P, B, L]. However this method comes at an epense of additional hardware circits, as well as deteriorated transient response in the system performance cased by the sperposition of the ramp fnction pon the SMC switching fnction. Another approach is to inclde some forms of adaptive control into the HM-based SMC to contain the switching frency variation [V, S7]. However the architectre of this adaptive sliding mode controller is relatively more comple, and increase the implementation cost of the controller. On the other hand, fied switching frency SMC can also be obtained by employing plse-width modlation (PWM) instead of HM [S4, V4, S8]. In practice, this is similar to classical PWM control schemes in which the control signal is compared to the ramp waveform to generate a discrete gate plse signal [D5]. The advantages of PWM-based SMC are that it does not need additional hardware circitries since the switching fnction is performed by the PWM modlator, which can be implemented inside the digital controller. However in order to preserve the original sliding mode control laws, the practical implementation of PWM-based SMC is nontrivial, especially when both crrent and voltage state variables are involved. Hence this approach is not always implementable for some conventional HM-based SMC types PWM-Based SMC Conventional HM-based SMC application is based on the ivalent control [V, C, W], bt PWM-based SMC application is based on the averaged dty control [S4, V4, S8, G]. Becase of the shortage of variable frency operation in HM-based SMC, PWM-based SMC recently has been sed to replace the HM-based SMC. However there is seldom theoretical analysis for the establishment of their relationship. In order to better nderstand how the PWM can replace the HM-based sliding mode control, it is necessary to briefly stdy ivalent control and averaged dty control. 8

7 A. Eqivalent Control As discssed previosly, to achieve an ideal SMC operation, the system mst be operated at an infinite switching frency so that the state variables trajectory is oriented precisely on the sliding srface. However in the practical case of finite switching frency, the trajectory will oscillate in the vicinity of the sliding srface while moving towards the origin (see Fig. 5- with three variables). It is possible to identify the movement of the trajectory as a composition of two isolate components: a fast-moving (high-frency) component and a slow-moving (low-frency) component shown in Fig. 5-. Sliding srface Actal trajectory Converging to the origin O Sliding srface = Converging to the origin slow moving (low-frency component) O Sliding srface + Converging to the origin +Ve fast-moving (high-frency component) -Ve O Fig. 5- High and low frency components of the state trajectory on sliding srface where sliding srface S and S, ve and ve are the directions respectively below and above the srface S. It can be seen that the high-frency component is actally a discontinos trajectory that alternates between ve and ve direction, whereas the lowfrency component is actally a continos trajectory that moves along the sliding plane. Since the movement of the trajectory is an effective consence of the inpt switches action, it is theore possible to relate the corresponding low-frency and high-frency components of the trajectory to a low-frency continos switching action low low, and a high-frency discontinos switching action high low high. Under sch assmptions, the switching action of high, where, and that prodces only the highfrency trajectory component, and the switching action of low prodces only the lowfrency trajectory component. Becase the high-frency component is often filtered by the SMPS converter filter, it is reasonable to consider the effect of the high-frency discontinos switching action high, and only the low-frency continos switching action low acts as the desired switching action that will prodce a trajectory that is a near ivalence to an ideal SMC operation trajectory. This is the so-called ivalent control. The ivalent control signal, i.e.,, is actally the low-frency continos switching action low described above: low Then the sliding-mode controlled system in ation (5.) can be rewritten as low (5.7) 8

8 f ( ) g( ) S( ) T K (5.8) B. Averaged Dty Control In conventional PWM control which is also known as the averaged dty control, the control inpt is switched between and once per switching cycle for a fied small dration. The time instance where the switching occrs is determined by the sampled vale of the state variables at the beginning of each switching cycle. Dty ratio d is then the fraction of the switching cycle in which the control holds the vale. It is normally a smooth fnction of the state vector, and it is denoted by d, where d. Then for each switching cycle interval dring the time,, the control inpt can be written as if t d if d t It allows that a system f ( ) g( ) can be epressed as d (5.9) dt f +g dt f dt (5.) The ideal average model of the PWM-controlled system response is obtained by allowing the dty ratio frency to tend to infinity, i.e., the above ation becomes i.e., lim lim lim d dt d d to approach zero. Under sch consideration, d d d f dt f dt g dt f dt g dt f g d (5.) (5.) which is erred as the averaged PWM-controlled system. Theore, it is shown that as the dty frency tends to infinity ( ), the ideal average behavior of the PWM-controlled system is represented by the smooth response of the system constitted by the dty ratio d. It shold also be noted that the dty ratio d replaces the discrete fnction in the same manner as the ivalent control of the SMC scheme to obtain (5.8). Hence the relationship between ivalent control and averaged dty control can be established d (5.) 84

9 C. PWM-Based SMC Replace HM-Based SMC Hence from the above review of ivalent control for HM-based SMC and averaged dty control for PWM-based SMC, it is very interesting that the PWM-based SMC can be sed to replace the HM-based SMC. The PWM-based SMC can not only limit infinite high switching frency bt also can fi the variable switching frency for SMPS application. The analysis of ivalent control and averaged dty control has proven that their relationship can be established. First in sliding mode control, the discrete control inpt (gate signal) can be theoretically replaced by a smooth fnction known as the ivalent control signal [V, C, W, S7]. Second at high switching frency, the ivalent control is effectively an averaged dty control signal d [S4, V4, S8, G]. Since averaged dty ratio is basically also a smooth analytic fnction of the discrete control plses in PWM, we can obtain a PWM-based SMC by mapping the ivalent control fnction into the averaged dty control in the plse-width modlator, i.e., SMC for HM-based SMC can be shown in Fig d. Finally the sbstittion of PWM-based PWM Sliding Control S Eqivalent control O Sliding Control S Dty ratio control d HM-based SMC PWM-based SMC clock t on t off Fig. 5-4 PWM-based SMC is sed to replace HM-based SMC 5. Design of a DPWM-Based SMC Most of the eisting SMC applied for SMPS converters are designed in analog control operating at low to medim frency range (hndreds of khz). This is not adate to meet the rirements of small-size and high-frency for today s low-cost high-performance SMPS. These eplain why the application of SMC in DC DC converters has only been of academic/research interest bt of little practical application. Fortnately as the theoretical grondwork of SMC is fairly matred, PWM techniqe has been applied to SMC and digital CMOS technology has been rapidly developed, it is time to direct more research efforts towards developing practical high-frency digital PWM-based SMC for SMPS converters. In this section we present a practical design of DPWM-based SMC application to a bck converter. Or focs in this design is the application of SMC to converter operating in continos crrent mode (CCM), and the discontinos crrent mode 85

10 (DCM) is not discssed here. The system modeling, derivation of eistence condition and selection of parameters for SMC are detailed in following sections. 5.. System Modelling for Sliding-Mode Controller The first step to the design of a SMC is to determine the state variables in terms of the desired the sliding model controller. Fig. 5-5 shows the schematic diagrams of the proposed PID-type SMC voltage controller for a bck converter. The sliding mode controller involves the pt voltage error and its integral and differential portions. Unlike most eisting SMC voltage controllers, besides the differential portion, it also takes into accont an additional voltage error integral term to redce the steady-state dc error of the pt voltage. Vin Q Q L V L C i L i C V C i V R PWM PWM ct () Hybrid DPWM PWM S( ) if S( ) S S( ) if S( ) SMC K + d X + K + dt dt K V X - + V Fig. 5-5 The schematic diagram of a PID-type SMC for a digitally controlled bck converter Where the pt voltage V is the SMC control objective, K K, K, K is the sliding parameter of SMC and the control variable can be epressed as: V d V V dt V V V dt T (5.4) where V represents the erence voltage,, and are respectively the voltage error and its differential and integral portions. Etracting the time differentiation of ation (5.4) leads to d V V dt d d V V dt dt d V V dt dt (5.5) 86

11 For easy discssion, assming that V is constant and capacitor ESR is zero, then d dv dvc ic V V dt dt dt C From ation (5.5) and (5.6), comes as: d dt Considering the CCM operation, i C d dt C Sbstitting ation (5.8) into (5.7) reslts in d i i dil di dt C C dt dt (5.6) (5.7) ic il i (5.8) L Under the sitation of CCM and averaged dty control, VL dil L dt V V V Since Then L in i V R di dv dt R dt Sbstitting ations (5.) and (5.) into (5.9), we can obtain Vin V dv C L R dt Becase of V V, ation (5.) can be rewritten as V V dv LC LC RC dt Sbstitting ation (5.6) into (5.4), then Eqation (5.7) finally leads to: in Vin V LC LC RC Vin V LC LC RC Sbstitting ation (5.6) into ation (5.5), then V V in LC RC LC LC (5.9) (5.) (5.) (5.) (5.) (5.4) (5.5) (5.6) (5.7) 87

12 Eqation (5.7) can be rewritten in a standard form of state-space description: Confronting with (5.8), it is clear that A B H (5.8) f g f A H, g B T S K K K K (5.9) T T S K K A B H 5.. Derivation of DPWM-Based SMC As mentioned previosly, since the relationship between ivalent control and averaged dty control has been established and the ivalent control signal has been proven ivalent to PWM dty ratio d for SMC, then the derivation of the DPWM-based SMC can be achieved by the ivalent control on the sliding srface S. Combining the preceding ations (5.4), (5.7) and (5.8) with (5.9), we can derivate the ivalent control inpt signal sing the invariance conditions by setting S, i.e., T T S K K A B H (5.) Now solving for ivalent control fnction (5.) yields T T K K K B K A H LC LC V V K RC K LC Sbstitting V V and in dv dt into ation (5.), then K dv K V LC LC V V Vin K RC dt K LC (5.) (5.) where is continos and als to PWM dty ratio d, and d, parameters K K and K K are to be determined which corresponds to the desired SMC dynamics that will be discssed in net section. A close inspection of ation (5.) reveals that the control signal involves the time differentiation of pt voltage, dv dt ic C, which reslts in the need of measrements (sch as crrent sensor or crrent-to-voltage sampling circit) for capacitor crrent i C and ths increase the size and cost of the digital controller. Since the bck converter has the pt voltage feedback V and the SMC controller is implemented in fll digital form, it is feasible to design a software observer or a mathematic fnction for V to eliminate the need for hardware measrement. Unfortnately sophisticated software observer needs spplement 88

13 resorces. The observer design isse is not discssed here. A simple nmeric derivation is adopted to calclate the rate of pt voltage changing in the FPGA implementation: dv V n V n V n dt T where V n and V n are the pt voltage in s th n and n th (5.) cycle respectively, T s is PWM switching period. Link ations (5.) and (5.), we can obtain the dty ratio of PWM-based SMC controller K K V LC V LC V V n Vin K RC K LC (5.4) 5.. Determination of SMC Parameters Eqation (5.4) gives the complete information for the dty ratio signal of DPWMbased SMC, where two sliding parameters K K and K K still need to be determined corresponding to the desired dynamics. For this prpose we employ the Ackermann s Formla [J7] to design the SMC also called as PID-type SMC to select the sliding parameters in this case. The selection of sliding parameters is based on the desired second-order dynamic properties. In this way, the sliding motion ensres that the state trajectory of the system nder SMC operation will always reach a stable ilibrim point. As stated in Eqations (5.4) and (5.4), the srface of the PID-type SMC at the stable ilibrim point, S can be detailed as: d dt T S K K K K K K K dt (5.5) According to Ackermann s Formla [J7] for a standard second-order system dynamics, the ation (5.5) can be transformed into Eqation (5.4) can be rewritten as K d K d dt K dt K n n (5.6) (5.7) where n K/ K is the ndamped natral frency and K KK is the damping ratio. Ths the sliding parameters now depend on the desired dynamic performance that can be chosen by ser. 89

14 5..4 Derivation of Eistence Conditions To achieve the design specifications, the SMC mst maintain the variables state trajectory on the srface for all sbsent time and the state trajectory slides along this srface. Ths it is necessary to determine the sliding motion range for the SMC, i.e., to ensre the srface eist and to determine the eistence conditions for sliding srface. As stated previosly, this task is performed sing the Lyapnov s second method [J6]. Recalling of the eistence conditions in ation (5.5), the eistence conditions mst be satisfied: S( ) T K f ( ) g( ) if S S( ) T K f ( ) g( ) if S Sbstitting ation (5.) and (5.9) into (5.), T S( ) K A B D if S S T S( ) K A B D if S Detailing ation (5.9) leads to S (5.8) (5.9) Case : S,, then S : Eqation (5.9) can be detailed: K K K S K S K V V RC LC LC in Case : S,, then S : Eqation (5.9) can be detailed: K K V S K K K S RC LC LC (5.4) (5.4) It can be seen that, and S constrct the sliding srface in a -D space, which is the eistence range of sliding motion for SMC. In order to simplify the analysis of the eistence, the -D space can be mapped onto the -phase-plane ( ). Assming K is positive, the eistence range of srface for the PID-type SMC, ation (5.4) and (5.4), can be represented with the two lines in phase-plane: K K V V K RC K LC LC K K V K RC K LC LC in (5.4) Ths the two lines and respectively determine the two bondaries of sliding srface eistence range. From ation (5.4), it can be seen the two lines has the 9

15 same slope in the phase plane, where line passes throgh the point A: Vin V K K LC LC, and point B:, Vin V K K RC LC, line passes throgh the points point C: V K K LC LC, and point D:, V K K RC LC, and the ais V V is limited with the minimm vale point E: V V, and the maimm vale point V, in. For different vales of the parameters K K and K K, the slope of the two parallel lines is variable. Theore the eistence conditions can be illstrated in two sitations: () positive slope region and () negative slope region for each line. As stated above, the sliding parameters K K and K K are variable with different dynamical response frency f cr, ths the points A, B, C and D will be located at ais in positive or negative region. The regions of eistence for SMC is shown in Fig. 5-6 with for possibilities: (a) K K RC and K K LC, (b) K K RC and K K LC, (c) K K RC and K K LC, and (d) K K RC and K K LC, where (a) and (d) belong to sitation () with negative slope, (b) and (c) belong to sitation () with positive slope. According to ation (5.4), when the bck converter operates at 4MHz, the parameters are K K RC and K K LC. Ths the eistence region for the PID-type SMC in this case can be represented in Fig. 5-6 (c). The eistence conditions show the region of eistence of the SMC, which provides a range of employable sliding area that will ensre the state trajectory keep sliding along the srface S ( ) ntil reaching the stable operation at the origin O. S= E A C F O λ = S= λ = D B (a) D (c) λ = E C F O A B λ = S= λ = E A F O C S= λ = (b) (d) B D B λ = λ = E C A F O D Fig. 5-6 region of eistence for the sliding mode mapped in the phase plane with for possibilities (a, b, c, d) 9

16 5..5 Stability for SMC Besides the eistence conditions, the sliding mode controller shold also comply with the stability conditions. The stability is an important item which is to ensre that the sliding srface will direct the state trajectory toward the stable ilibrim points in eistence regions. In this work the analysis of dynamic response and stability for the PID-type SMC can be started the srface S and S. Since the state variables,, are in phase canonical form, the S can be rewritten in Laplace form as: K K KX s KsX s KX s s s s K K Firstly sing the Rh s stability criterion to this second order linear polynomial determine the stability conditions, s K K s K K s K K K (5.4) to (5.44) The condition for the stability mst meet, which means all the coefficients K, K, K mst be with the same sign (positive or negative), i.e., K,, or K,,. This can ensre all roots have negative real parts. Secondly Etracting the time differential, the S can be rearranged into a stand second-order system form: s s (5.45) n n which is identical to ation (5.7), where the vales of damping ratio frency n and two eigenvales s, are s n, K K, K K K n j n, ndamped natral (5.46) Since then K mst be positive. Recalling that K, K, K mst be with the same sign, ths finally all the coefficients shold be positive for stable operation Matlab Simlation of SMC for a Bck Converter The time domain behavior of the proposed digital DPWM-based SMC is verified on a bck converter sing Matlab/ simlink shown in Fig. 5-6, where the bck converter circit elements are: L 4.7µH, C µf, R 5, V.V, V.5V and switching frency fs 4MHz. in 9

17 PWM -bit In Ot V -bit d D[N] C(t) Ramp Soft start V sampling V SMC Vin Bck S-fnction Bck FiPit t -bit R Fied-Point Setting Ot In clock V sampling Fig. 5-6 The modelling of the PID-type SMC for a digitally controlled bck converter To comply with the design ations regarding with the stability and the transient response, the choice for system dynamic performance shold be considered with the trade-off between n and, where large of simlations have been carried. Finally we set the bandwidth of the SMC response f at one-fifteenth of the switching frency f s, i.e., f 5, and choose the damping ratio. Ths the sliding parameters are determined as: K K 4 f 5 and K K 4 f 5 (5.47) s s The modelling of the bck converter is modelled in a hybrid model [S] sing Matlab s- fnction and all the calclations are compted in fied-point comptation with -bit fraction (see section 4..). To meet the condition of non-limit cycle NDPWM N ADC [A7], the analogto-digital model has a -bit resoltion and DPWM model has an -bit resoltion. The simlation reslts and their analysis are shown as below. Initially the pt voltage V follows the erence V by a slope fnction ntil steady state. Then the load sddenly changes from.a to.46a (R: 5Ω.Ω), Fig. 5-7 shows the transient response of the SMC at switching frency 4MHz: pt voltage (a), PWM dty ratio (b) indctance crrent (c). It can be seen that the dynamic response of SMC is satisfying, where the transient response is very fast and the ndershot is small. To illstrate the details of sliding trajectory in the SMC operation, Fig. 5-8 shows the sliding trajectory in - -S() -D space (a) and in - two phase plane (b). Dring the operation of the slope fnction, the pt voltage V always tracks the V and the sliding srface S directs toward the ilibrim point ntil steady state V V. In steady state the sliding trajectory is on the stable srface near the origin, and in dynamic state the ilibrim is broken and the sliding trajectory will directs toward the new ilibrim point again. Consently as shown in Fig. 5-8 the sliding trajectory changes twice in two circles. In order to clearly eplain the change of the sliding trajectory dring the SMC operation, the sliding srface S is also illstrated in the time-domain. Ths the Fig. 5-9 shows the sliding trajectory in t- -S() -D space (a) and in t-s() two phase plane (b). It can be seen when load changes the sliding trajectory can qickly direct to the stable ilibrim. s n 9

18 PWM dty d Unit: A / div pt Voltage: V / div transient pt voltage at 4MHz Time: s / div -4 (a) Otpt voltage V transient indctance crrent at 4MHz Time: S / div (b) Indctance crrent I L transient dty ratio at 4MHz Time: s /div -4 (c) PWM dty ratio d Fig. 5-7 Dynamic response of SMC when load changes from.a to.46a (R: 5Ω.Ω): pt voltage V (a), indctance crrent I L (b) and PWM dty ratio d (c) at switching frency 4MHz 94

19 *C S() sliding trajectory at fs=4mhz X =V -V (a) *C...4 =V -V sliding trajectory mapped onto - phase plane at fs=4mhz = V -V (b) Fig. 5-8 SMC Sliding trajectory in - -S() -D space (a) and mapped in - two-phase plane (b) at switching frency 4MHz 95

20 S() S() sliding trajectory at fs=4mhz 5 5 steady state dynamic state steady state X =V rev -V Initially, SMC operates in a slope fnction, 6 5 (a) 4 Time: S Initially, SMC operates in a slope fnction, Time: t (s) -4-4 sliding trajectory in time-domain at fs=4mhz steady state dynamic state steady state (b) Fig. 5-9 SMC Sliding trajectory in time-domain: t- -S() -D space (a) and mapped in t-s() two-phase plane (b) at4mhz switching frency 96

21 Unit: A / div Unit: A / div Unit: A/ div transient pt voltage Time: S (a) Otpt voltage V transient indctance crrent Time: S (b) Indctance crrent I L transient dty ratio Time: S (c) PWM dty ratio d Fig. 5- Dynamic response of SMC when load changes from 5mA to.46a (R: Ω.Ω): pt voltage V (a), indctance crrent I L (b) and PWM dty ratio d (c) at switching frency 4MHz -4 97

22 S() PWM dty (d) pt Voltage V / div pt voltage at MHz pt voltage at MHz pt voltage at 4MHz Time: S (a) Otpt voltage V dty ration at fs=mhz dty ration at fs=mhz dty ration at fs=4mhz Time: (s) -4 (b) PWM dty ratio d sliding trajectory in time-domain at fs=mhz sliding trajectory in time-domain at fs=mhz sliding trajectory in time-domain at fs=4mhz -.5 Initially, SMC operates steady dynamic steady -.75 in a slope fnction, state state state Time: t (s) -4 (c) Sliding trajectory S Fig. 5- Comparison reslts of V, dty ratio d and sliding trajectory S for SMC when load changes from 5mA to.46a (R: Ω.Ω) at MHz, MHz and 4MHz respectively 98

23 In order to validate the distrbance rejection in larger range of variation, Fig. 5- shows the dynamic reslts when the load changes from 5mA to.46a (R: Ω.Ω). It can be seen that the SMC can well reglate the pt voltage even in large range of load variation. In order to compare the SMC operation performance at different switching frency, Fig. 5- shows the comparison reslts of V, dty ratio d and sliding trajectory S for SMC when load changes from 5mA to.46a (R: Ω.Ω) at MHz, MHz and 4MHz respectively. It can be seen that the dynamic response of SMC is better with increasing switching frency. As stated previosly that higher switching frency performs in higher dynamic performance, and an infinite frency directs toward an ideal SMC. 5.4 Smmary In this chapter a nonlinear DPWM-based sliding mode controller which is derived from the conventional analog HM-based SMC, is proposed for the digitally controlled high-frency low-power SMPS application. Firstly a brief review of sliding mode control is given in details, which contains the fndamental theory and eistence conditions for SMC. Sbsently the SMC is introdced to the SMPS application domain. The ideal controller in theory is redefined to meet practical limitations. Hence the Qasi-Sliding-Mode (QSM) control is introdced to redce the infinite high switching. For the sake of completeness, we present the conventional HM-based SMC and reveal its shortage in practical design. To meet the rirement of fied-frency, a PWM modlator is introdced to solve the problem of frency variation. The relationship between the ivalent control and averaged dty control is derived, so that the PWM-based SMC is adopted to replace HM-based SMC. Most of the conventional HM-based SMC and recent PWM-based SMC are designed as analog controllers in low to medim power level, where the switching frency and control performance are not adate for todays SMPS. Ths we present an original implementation of a DPWM-based SMC for a high-switching low-power frency SMPS. An eample design of DPWM-based SMC for a bck converter is detailed. The practical design involves system modelling, derivation of DPWM-based SMC, selection of sliding parameters, derivation of eistence and stability analysis. The time domain behavior of the proposed digital DPWM-based SMC is verified on a bck converter sing Matlab/ simlink. The simlation reslts verify the performance of proposed SMC. The FPGA implementation will be presented in chapter 6. 99