Hybrid modelling and control of a synchronous DC-DC converter. Mahmood Mirzaei* and Ali A. Afzalian

Transcription

1 44 Int. J. Power Electronics, Vol., No. 4, 29 Hybrid modelling and control of a synchronous DC-DC converter Mahmood Mirzaei* and Ali A. Afzalian Department of Electrical Engineering, Shahid Abbaspour University of Technology, Tehran, Iran Mirzaei@stud.pwut.ac.ir Afzalian@pwut.ac.ir *Corresponding author Abstract: In this paper, the problem of hybrid modelling and control of a fixed frequency DC-DC converter, namely the Non-inverting Buck-Boost converter is addressed. The v-resolution method is used to capture the hybrid nature of the system. Two mixed logical dynamical (MLD) models are obtained for the converter using hybrid systems description language (HYSDEL). In the process of using v-resolution concept, the problem of non-linear terms in state update equations is solved using piecewise affine approximation. Three PWA models are obtained using least squares fitting over non-linear dynamics of the system. A suitable model in terms of accuracy and complexity is chosen for controller design. Model predictive control is used as the control strategy and explicit solution to the control problem is obtained. Finally, the simulation results are presented. The accuracy and complexity of the models are compared and the performance of the controller is evaluated over three different scenarios which are common in practice. Keywords: explicit solution; MLD model; model predictive control; Non-inverting Buck-Boost converter; PWA model. Reference to this paper should be made as follows: Mirzaei, M. and Afzalian, A.A. (29) Hybrid modelling and control of a synchronous DC-DC converter, Int. J. Power Electronics, Vol., No. 4, pp Biographical notes: Mahmood Mirzaei received his BSc in Control Engineering from the Department of Electrical Engineering, Shiraz University, Shiraz, Iran in 24. He obtained his MSc also in Control Engineering from Shahid Abbaspour University of Technology, Tehran, Iran in 28. His research interests include hybrid systems modelling, model predictive control of hybrid systems and applications (e.g., power electronics), network controlled systems (NCS) and fault tolerant control systems. Ali A. Afzalian received his BEng and MSc in Electrical Engineering from the University of Tehran, Tehran, Iran, in 988 and 99 respectively and his PhD in Control Engineering from the University of Sheffield, Sheffield, UK in 998. Since 99, he has been a Faculty Member in the Department of Electrical Engineering of Shahid Abbaspour University of Technology, Teheran, Iran. His research interests include supervisory control of discrete-event systems, fuzzy logic, neural networks, neurofuzzy systems, genetic algorithms and electrical power systems control and dynamics. Copyright 29 Inderscience Enterprises Ltd.

2 Hybrid modelling and control of a synchronous DC-DC converter 45 Introduction In the past decades, the growing demand for delivering electric power in various forms and with high performance and reliability has had a great impact on power electronics. Various areas of power electronics including power devices, circuit design and control methods have undergone rapid development. In this stream, various control strategies have been proposed to achieve high performance and at the same time, low cost power converters and a bridge between has been made automatic control and power electronics communities. DC-DC converters are extensively used in different applications ranging from space and medical applications to DC motor derives and many others because of their advantages, such as light weight, small size and high reliability. Because of the type of non-linearity in the mathematical model of DC-DC converters, the controller design has become an attractive topic in automatic control community. Various control techniques ranging from linear control based on linearised model (Leung et al., 99) to passivity based control (Sira-Ramirez et al., 997) and sliding mode (Sira-Ramirez, 23) have been proposed to deal with the problem. The reader is referred to Sira-Ramirez and Silva-Ortigoza (26) and Geyer (25) for more information on the various control methods. Most of the methods use simplified model of the converter which in turn reduce the validity of the controller for practical applications. The difficulty of controlling DC-DC converters stems from their hybrid nature. External signals change discrete variables of converters between two or more discrete states. The system has a specific continuous dynamics in each state. Thus, these systems can be categorised as hybrid systems with controlled switching (Branicky, 995). Considering the hybrid nature of the converters, Senesky et al. (23) proposed a controller based on a hybrid automation model of the Boost converter. A non-linear model predictive control (NMPC) was used based on the non-linear average model (Middlebrook and Cuk, 976) of the converter in Lazar and Keyser (24). Another problem in designing DC-DC converters rises from introducing constraints in the design process. These constraints can be hard, such as constraints on duty cycle value or soft such as security constraints imposed on the inductor current. Model predictive control of hybrid systems has proved its power in controlling systems with hybrid nature and subject to various constraints (Morari and Baric, 26). Online MPC has been successfully tested on Buck and Boost DC-DC converters in Geyer et al. (24) and Beccuti et al. (25). The main drawback of this approach is the problem of solving an optimisation problem in each step time. Because of the high rate of sampling and the demand for low cost converters, this approach is not realistic. Solution to the problem of online computational burden is explicit hybrid control (Borrelli, 22). Explicit hybrid control was tested on Buck and Boost converters in Papafotiou et al. (24) and Beccuti et al. (26). Control of DC-DC converters can be achieved only based on output voltage measurements, but some knowledge of inductor current can substantially improve the performance of the system. All methods of hybrid control use inductor current in feedback loop that in turn increase hardware complexity and cost of the converter. Two methods that use computations based on input and output voltage measurements to estimate inductor current were proposed in Mariethoz et al. (28). The validity and performance of the explicit method was proved by experimental results for Buck and Boost in Mariethoz et al. (28) and Beccuti et al. (27) respectively. This paper sketches the works that have been done on Buck and Boost DC-DC converters in Papafotiou et al. (24) and Beccuti et al. (26) to Non-inverting

3 46 M. Mirzaei and A.A. Afzalian Buck-Boost converter. This converter can provide an output voltage below and above the source voltage. Its main advantage is that the provided output voltage has the same polarity of the source because of its specific topology utilising four switches (Figure ). The switches are dependent and make two distinct topologies. Thus, the converter can be categorised as a mono-variable converter (Sira-Ramirez and Silva-Ortigoza, 26). The main control objective here is to control the semiconductor switches (Figure ) such that the DC component of the output voltage reaches a specific reference value. This must be done in the presence of changes of the source voltage and the load resistance. The rest of the paper is organised as follows. Firstly, the physical set-up of the converter is shown and hybrid automaton model of the system is obtained in Section 2.. In Section 2.2, v-resolution method is explained and piecewise affine (PWA) approximation is explained in Section 2.3 as a method of solving the problem of non-linear terms in state update equations of the model. Mixed logical dynamical (MLD) framework is used in Section 2.4 to integrate all the three dynamics of the system that will be explained in one model and it is explained how hybrid systems description language (HYSDEL) is used to obtain matrices of the model. PWA model is another modelling framework which is used to model the converter and it is explained in Section 2.5. Control objectives and strategy are explained in Section 3.. Finally, simulation results are presented to check the validity of the models and the performance of the controlled system. In the first part of this section, simulation results are presented for the obtained models. Performance of the controlled system is evaluated under three different scenarios namely start-up, change in the load resistance and change in the source voltage. 2 Modelling for controller design purposes 2. Physical set-up and hybrid automata model The physical set-up for the synchronous Non-inverting Buck-Boost converter is shown in Figure. It is assumed that the converter works in the continuous conduction mode (CCM). Figure Physical set-up for the Non-inverting Buck-Boost In the set-up R, L and C denote load resistance, inductance and capacitance, respectively. r L and r C are parasitic elements of the inductor and capacitor and v s is the input voltage. The converter including its switches has two distinct dynamical modes. The duty cycle which is a variable bounded between zero and one determines how long each of the dynamics is in charge. At the beginning of the first interval ( kts t < ( k + d( k)) Ts ), the switches are in u = position (Figure 2) which means S and S3 are ON and S2 and S4

4 Hybrid modelling and control of a synchronous DC-DC converter 47 are OFF (Figure ). At the end of the first interval, all the switches in Figure toggle (switches in Figure 2 change to u = ) and the dynamic of the system changes. In the second interval, the inequality ( k + d( k)) Ts t < ( k + ) Ts holds. By defining x() t = [ i () ()] T L t vc t as the state vector, where i L (t) is the inductor is current and v C (t) is the capacitor voltage, the dynamics of the system can be defined by the following affine continuous time state space equations: Fx fv s kts t ( k d( k)) Ts xt & + < + () = F2x+ f2vs ( k + d( k)) Ts t < ( k + ) T () s Where matrices F i and f i can be found by Kirchhoff s laws and simple mathematical operations. rl x l F =, f = x l, xc R r + c Rrc R rl + xl R+ rc xl R+ rc F2 =, f2 =. R xc R+ rc xc R+ rc (2) Figure 2 Schematic circuit of the converter with parasitic elements The first dynamic of the system is active at the beginning of each period. At the end of the first interval ( kts t < ( k + d( k)) Ts ), a transition occurs and the second dynamic becomes active. The hybrid automaton model for the converter can be easily found. This model of the system has two locations (Senesky et al., 23) (Figure 3). Figure 3 Hybrid automaton model of the system q G q2 x& ( t) = f q ( x( t)) x& ( t) = f ( x( t)) q2 G2 In the hybrid automaton model, G and G 2 are the guards of the model which are:

5 48 M. Mirzaei and A.A. Afzalian G = [ kt t < ( k+ d( k)) T ] G = ( k+ d( k)) T t < ( k + ) T 2 s s s s (3) q and q 2 are discrete states and: f ( xt ( )) = Fxt ( ) + fv fq2( xt ( )) = Fxt 2 ( ) + fv 2 q s s (4) In this paper, time scale and state variable transformation are normalised. Normalisation has several advantages (Sira-Ramirez and Silva-Ortigoza, 26). For example, it simplifies the mathematical description of the system by eliminating superfluous parameters. Only those parameters which are responsible for important qualitative changes in system behaviour will remain in the mathematical model. Simulation runs on normalised models facilitates mathematical operations for the processor. It eliminates the stiffness that is present in most power electronic devices models which are caused by small capacitance and inductance values. Returning from the normalised model to non-normalised model is simple and can be achieved by simple multiplications. It also eases the implication of analytic results of the system model and shows only relevant parameters that have effect on results such as equilibrium point, steady state behaviour and control amplitude restrictions. DC-DC converters are somewhat difficult to simulate with their un-normalised parameter values. Small values of inductance and capacitance multiply the left hand side of the differential equation which defines the dynamic of the system and produces large right hand side. This can make the model numerically stiff for simulations. For more information about normalisation, the authors refer the reader to Sira-Ramirez and Silva-Ortigoza (26). To normalise parameters of the model, v s, R and T s are used as reference values and all other parameters are normalised. As a result, the state space equation can be reformulated as follows: x& () t ( ( )) ( ( )) ( ) F x + f k t< k + d k = F x + f k+ d k t< k (5) 2.2 v-resolution model v-resolution model was first introduced in Geyer et al. (24) as a modelling strategy to capture the hybrid nature of the synchronous DC-DC converters. This modelling method provides a discrete time model which can be easily implemented in control design strategies. It also gives intermediate values of the state variables which can be used for imposing constraints. This is particularly important for inductor current whose value can change drastically inside a period. Measuring inductor currents at the beginning and the end of the period cannot provide enough information for applying constraints. In v-resolution, the period of T s is divided into v sub-periods of duration τ s = T s /v with v N and v > 2. States which are sampled in the period are denoted by ξ(n). The v-resolution model gets x(k) and assigns it to ξ(). Then, a dynamic should be chosen to determine the transition to ξ() which ξ () = f ( ξ (), d). This procedure continues until determining the value of x( k + ) = ξ ( v) (Figure 4).

6 Hybrid modelling and control of a synchronous DC-DC converter 49 Figure 4 Evolution of the state variable in one period for v = 3 Three dynamics can be used for the function f in ξ ( n+ ) = f( ξ ( n), d) which are denoted by D, D2 and D3. D and D2 are discrete dynamics of the system with sampling time T s /3, when switches are in u = and u = respectively. The method of averaging (Middlebrook and Cuk, 976) is used to find the third dynamic, therefore, the third dynamic can be written as: n n+ ξ( n+ ) = [ F ( d( k) ) + F2 ( d( k))] ξ( n) v v n n+ + [ f ( d( k) ) + f2 ( d( k))] v v This equation is non-linear because of the multiplicative terms of state and duty cycle. In order to determine the dynamic which is in charge of state evolution, we need to define some binary variables first: n σ n = true d( k) for n =, L, v (7) v Figure 5 shows the binary variables for v = 3. (6) Figure 5 Binary variables values in a period (see online version for colours) Based on obtained binary variables, state update equations can be written as: F ξ( n) + f v f σ σ ξ( n + ) = F2 ξ( n) + f2 vs if σn F ( d( k)) ξ( n) + f ( d( k)) v if σ σ s n n+ 3 s n n+ (8) 2.3 PWA approximation As mentioned in Section 2.2, the average model of the converter contains non-linear terms which need to be linearised before using in the HYSDEL codes. The method of PWA approximation (Rodrigues and How, 2) is used to solve the problem of

7 42 M. Mirzaei and A.A. Afzalian non-linearity. The first step in PWA approximation is to determine the variation range of the variables of non-linear terms. Non-linear terms are multiplications of the duty cycle which are state variables. The variation range of the state variables can be found using the hybrid automaton model of the system. After applying a sequence of duty cycles to the hybrid automaton of the system, the variable variations are observed to be il [ 6,2] and vc [ 2,4]. The next step in PWA approximation is to partition the domain space of the non-linear terms to sufficient number of polytopes (Figure 6) and then to find the inequalities that define the partitions in the form of: H x + G d K (9) i i i Figure 6 Partitions of the first third of duty cycle for v = 3 and their defining half spaces Inductor Current Duty Cycle Here, each domain is partitioned into 8 (3 6) and 24 (3 8) partitions for models MLD and MLD2, respectively. The last step in PWA approximation is to find the PWA equation which can best fit the non-linear terms in its corresponding partition and to determine its equation. Figure 7 shows the non-linear term and its PWA approximations in the first third of the period. Figure 7 PWA approximation of the non-linear term I L d (see online version for colours)

8 Hybrid modelling and control of a synchronous DC-DC converter MLD framework An appropriate modelling framework is now needed to describe the three explained dynamics for the system which were explained in Section 2.2 and finally derived in (8) in one model, a model which is capable to be used in the control design process. MLD framework was introduced by Bemporad and Morari (999) which has been considered as a foundation for addressing different problems in hybrid systems ranging from control design to verification and state estimation. In MLD framework, the evolution of the system is governed by discrete time mixed integer linear dynamics subject to mixed integer linear inequalities. The idea of using mixed integer programming to control dynamical systems has first appeared in Tyler and Morari (999) and this idea is used in MLD framework to solve the optimal control problem of hybrid systems. MLD framework eases the problem by providing a device to transform optimal control problem of the hybrid systems to mixed integer optimisation problems. Switching between dynamics and jumping in state values are phenomena that occur in hybrid systems and can be modelled in MLD framework. Generally speaking, MLD models can be shown with the following equalities and inequalities: x( t+ ) = Ax t ( t) + Btu( t) + B2tδ ( t) + B3tz( t) yt () = Cxt t () + Dut t () + D2tδ () t + D3tzt () () E δ () t + E z() t E u() t + E x() t + E 2t 3t t 4t 5t Where t Z is time label and x, y and u are states, outputs and inputs respectively which have the form: [ c l] [ c l] [ c l] n n c l pc R, {, }, c R, Pl m m c l {, }, R, {,} T T T x = x x, y = y y, u = u u, x xl y c y u u l c l As it is described, MLD framework provides a model of hybrid system which is well-suited for optimal control design. HYSDEL programming language was developed for convenient modelling of systems in MLD framework (Torrisi and Bemporad, 24). This programming language gets dynamical description of the system as a high level language and provides matrices A, B i, C, D i and E i of the MLD model. HYSDEL is used here to produce MLD model of the converter. 2.5 PWA model Another method of DC-DC converter modelling (the other methods that were used are hybrid automata and MLD modelling) which is used here results in PWA model of the system. This method was first introduced in Beccuti et al. (26) for hybrid modelling of a Boost converter to be used in an optimal control problem. A discrete time model of the system is obtained by employing a sampling interval equal to the switching period T s. The least square fitting (LSF) over several regions of the exact system update equation is used to obtain PWA description of the non-linear dynamic. The system equation was obtained by integrating () from t = k to t = k + : ()

9 422 M. Mirzaei and A.A. Afzalian F ( d( k)) Fd ( k) x k + = e e x k 2 ( ) ( ) Equation (2) can be written as: + + d( k) F ( d( k)) F ( d( k) s) d( k) F2 ( d( k) s) e f2 ds 2 e e f ds x ( k + ) =Φ ( d( k)) x ( k) +Γ ( d( k)) (3) Where Φ ( dk ( )) and Γ ( dk ( )) are matrices that depend non-linearly on d(k). In PWA model of the system, expression (2) is approximated by determining Ai, B i and f i in the following formula: x ( k + ) = Ax i ( k) + Bd i ( k) + fi if d( k) Di i =, L, v (4) dk ( ) To determine Ai, B i and f i, the following expression is minimised over grided series of points of x ( k) in the state space [ il min, il max ] [ vc min, vc max ]. D i s are the v intervals [, /v],, [v /v, ] and il min, i L max, v C min and v C max are determined by checking state variable deviations of the hybrid automaton model of the system which was obtained in Section 2. under changes of duty cycle. (2) 3 The control problem 3. Control objectives and strategy The main control objective of the converter is to derive DC component of the output voltage to its reference value as fast and with as little overshoot as possible. The control must be done in the presence of source voltage and load resistance changes. These are control objectives for the transient response of the controlled system. It must also have an acceptable error in steady state. Also, the controller should produce constant value for duty cycle in the steady state to prevent chattering (subharmonic oscillations). Basically, two control methods can be used for controlling DC-DC converters: voltage mode control and current mode control. The capacitor voltage of the Non-inverting Buck-Boost converter has two possible solutions for a specific reference value of output voltage which results in two optimal points for the cost function. In one of these optimal points, system works with high value of inductor current which is undesirable and also decreases performance of the converter. Also, v C exhibits non-minimum phase behaviour. Thus, current mode control is used to bypass these problems. The controller is designed in order to derive inductor current to its reference value. Reference value of inductor current can be computed based on known values of capacitor voltage reference, source voltage and load resistance. This strategy also makes the use of feed forward loops possible to better alleviate disturbance effects.

10 Hybrid modelling and control of a synchronous DC-DC converter Model predictive control MPC is the only advanced control strategy that has had a significant impact on industry and its simplicity made it so popular. MPC has been widely accepted as a control tool because of its ability to deal with constraint and multivariable systems. MPC uses the model of the system to predict future behaviour of the plant to use in a cost function to predict future performance. This strategy computes a sequence of control signals by solving an optimisation problem based on predicted performance and some measurements on the plant in each step. Thus, given the model of the plant, one only needs to formulate an adjective function based on control objectives. Each sampling time ideally, a constraint infinite time optimal control (CITOC) should be solved as the optimisation problem. Because in general, a simple closed form solution for CITOC problem does not exist, it can be simplified by defining a finite horizon and approximate CITOC problem with a constraint finite time optimal control (CFTOC) problem. 3.3 CFTOC problem and the online solution The control objective is to regulate DC component of the output voltage to its reference value which is equivalent to regulate inductor current to its calculated reference value. In practice, this reference value can be achieved by incorporating a PI controller. Thus, il, err = il il, ref is in the objective function. To prevent unwanted chattering, a term also is added to objective function that penalises the change of duty cycle in two consecutive steps Δ dk ( ) = dk ( ) dk ( ). Thus, the error vector can be defined as: T lerr,, (5) ( k) = i ( k) Δd( k) ε Thus, the objective function can be defined as follows: L J ( D( k), x( k), d( k ) ) = Qε ( k + l k) (6) l= in which the penalty matrix is Q = diag(q, q2). Q penalises ε(k + l, k) over finite horizon L. The online method involves repeated solution of the optimisation problem in each period. Thus, this method is confined to those problems that the solution of the optimisation problem can be obtained in one period. 3.4 CFTOC problem and the explicit solution The main drawback of online solution to the problem of CFTOC is its computational burden which limits the method to plants whose optimisation problem can be solved in one step. The online solution is particularly not applicable for DC-DC converters which mostly work with sampling times around micro seconds. Also, using a processor to solve the optimisation problem is not cost effective. This problem is solved by carrying the computational burden of the optimisation problem offline and using explicit solution. Recently, it was shown that the problem of CFTOC of PWA systems can be reformulated

11 424 M. Mirzaei and A.A. Afzalian as a multi-parametric program by treating the state vector as parameters (Baotic et al., 23; Borrelli, 23). The optimal state feedback law can be expressed in PWA format (Borrelli, 23) as follows: d * ( k) = F x ( k) + g j If H x ( k) K j =, L, N j j j (7) H j and K j define polyhedral partitions of state space. F j and g j are the values of the PWA control law. 3.5 The state feedback law Multi-parametric toolbox (MPT) (Kvasnica et al., 24) is used to find the explicit solution of the MPC problem. MPT is also used to merge regions which contain the same expressions of the control law. This results in a controller with the same laws over reduced number of partitions. Because normalisation is used in the authors approach, the dependence of control law on v s is removed, thus, the partitions can be viewed in three-dimensional state space. Simplified partition of the explicit solution is shown in Figure 8. Implementing this state feedback in real applications is straightforward. This can be simply done by determining the partition of measured state and finding its corresponding affine control law from a look-up table with a micro-control based hardware. Figure 8 Partitions of the simplified explicit controller (see online version for colours) The control law is originally of the form dk ( ) = fi ( L, vc, dk ( )) and can be drawn in a four-dimensional space. Figure 9 visualises the state feedback law for a specific value of d(k ). This is done to reduce the dimension of the PWA function by removing one of its parameters.

12 Hybrid modelling and control of a synchronous DC-DC converter 425 Figure 9 Visualised state feedback for d(k ) =.6 (see online version for colours) 4 Simulation results 4. Simulation results for modelling In this section, based on explained methods, different hybrid models are obtained and a suitable model in terms of accuracy and complexity is chosen. Obtained hybrid models of the converter will be used in control design. Simulations over static behaviour of different obtained models will be used to choose the model. In this process, a prominent factor is the complexity of the models. Two MLD and three PWA models are obtained which are different in complexity and accuracy because of the number of regions used in PWA approximation. Inductance and capacitance are chosen to be p.u. and 5 p.u. respectively. Parasitic elements are.5 p.u. and.8 p.u. for capacitor equivalent series resistor (ESR) and inductor resistance. Source voltage and load resistance are reference values and therefore are p.u. finally switching is chosen to be 2 KHz. 4.2 Obtained models HYSDEL is used to obtain two MLD models with different numbers of PWA approximation regions which in turn cause models with different accuracy and complexity. MLD is obtained by using 8 regions (six regions in each of three subperiods) for approximating non-linear terms i L (d n/v) and v C (d n/v) of the average model. MLD2 is obtained by using 24 (3 8) regions. Table shows the comparison between complexities of the two MLD models. Three PWA models are also obtained using LSF of PWA models on non-linear exact mapping model. v (number of PWA dynamics) is used as a parameter to produce models with different levels of accuracy and complexity. Table 2 shows a comparison between the complexities of the PWA models.

13 426 M. Mirzaei and A.A. Afzalian Table Complexity of MLD models Model Number of PWA regions Number of mixed integer linear inequalities Number of PWA dynamics MLD 8 (3 6) 4,53 MLD2 24 (3 8) 59 4,29 Table 2 Complexity of PWA models Model Value of the v in the model Number of PWA dynamics PWA 5 5 PWA2 PWA Static behaviour of the models In this section, static behaviour of the models are compared. To find static behaviour, different values of duty cycle have been applied to the models and state variables have been saved in steady state. In Figure, static behaviour of capacitor voltage of MLD models are compared. Although MLD2 is a model with good accuracy (Figure ), as it is seen in Table, it is too complex to be used in control design process. In Figure, static behaviours of capacitor voltage of three PWA models are compared. The models MLD and PWA3 can be considered as good choices in terms of accuracy and complexity. As it is seen in Figure, PWA models are mostly different for duty cycle higher than the pick point around.8. For a specific reference voltage, two duty cycles can be found which one of them has duty cycle value larger than the pick point with large values of inductor current which results in degraded performance of the system, thus, duty cycles lower than pick point are used and models should have good accuracy in this interval. As it is seen in Figure, three PWA models are almost the same in the interval between zero and.8. Therefore, a good choice is the model with the least complexity. Figure Comparison between accuracy of two MLD models (see online version for colours) Hybrid Automaton Model MLD MLD2 V C Values (Different Models) Duty Cycle Values

14 Hybrid modelling and control of a synchronous DC-DC converter 427 Figure Comparison between accuracy of three PWA models (see online version for colours) PWA Model with ν=5 PWA Model with ν= PWA Model with ν=5 Hybrid Automaton Model V C (Capacitor Voltage) Duty Cycle In Figure 2 and Figure 3, comparison between static behaviour of inductor current and capacitor voltage of the hybrid automaton, the selected MLD and the selected PWA models in steady state, are presented. As complexity of the controller depends exponentially on the complexity of the model and because the change of parameters of the converter during the operation is inevitable, the accuracy of the model is of relative importance. Therefore, PWA model makes a good choice for the controller design. Figure 2 Comparison between accuracy of steady state values of inductor currents values of the obtained models (see online version for colours) 4 2 Hybrid Automaton Model MLD Model PWA Model I L Inductor Current Duty Cycle

15 428 M. Mirzaei and A.A. Afzalian Figure 3 Comparison between accuracy of steady state values of capacitor voltage of the obtained models (see online version for colours).6.4 Hybrid Automaton Model MLD Model PWA Model.2 V C (Capacitor Voltage) Duty Cycle 4.4 Simulation results for the controller design The non-linear model is approximated by 3 PWA dynamics (v = 3) on grided state space of [ 6, 2] [ 2, 4]. The bounds of state space were obtained by running hybrid automaton model of the converter with a sequence of duty cycle to incorporate state variations for different scenarios of duty cycle change. Explicit MPC controller was obtained which resulted in 29 regions for PWA feedback and 7 regions after simplification (Figure 8). 4.5 Simulation scenarios Three scenarios namely start-up, load resistance change and source voltage change which are common in practice are chosen (Beccuti et al., 25). In start-up scenario, the system starts with zero values for inductor current and capacitor voltage. Disturbances that can affect the system are changed in the load resistance and source voltage. The responses of the controller to these disturbances are also shown. In all the figures, the quantities are in per unit and one can use simple multiplication to find real values. Also, in all the figures, Figure A shows the result of state evolution for different scenarios in which the solid line is capacitor voltage, the dashed line is v C,ref and the dotted line is inductor current and Figure B shows duty cycle value.

16 Hybrid modelling and control of a synchronous DC-DC converter 429 Figure 4 Start-up scenario in the Boost operation mode (see online version for colours) 5 Figure A State Variables V CRef =.2 p.u Figure B Duty Cycle Figure 5 Start-up scenario in the Buck operation mode (see online version for colours) 2.5 Figure A State Variables V CRef =.8 p.u Figure B Duty Cycle

17 43 M. Mirzaei and A.A. Afzalian Figure 6 5% change in the source voltage; it changes from p.u. to.5 p.u., Boost operation mode (see online version for colours) 5 Figure A Duty Cycle State Variables V CRef =.2 p.u. Figure B Figure 7 5% change in the source voltage; it changes from p.u. to.5 p.u., Buck operation mode (see online version for colours) 2.5 Figure A Duty Cycle State Variables V CRef =.8 p.u. Figure B

18 Hybrid modelling and control of a synchronous DC-DC converter 43 Figure 8 % change in the load resistance; it changes from p.u. to 2 p.u., Boost operation mode (see online version for colours) 5 Figure A State Variables V CRef =.2 p.u Figure B.8 Duty Cycle Figure 9 % change in the load resistance; it changes from p.u. to 2 p.u., Buck operation mode (see online version for colours) 2.5 Figure A 2 State Variables.5.5 V CRef =.8 p.u Figure B.8 Duty Cycle

19 432 M. Mirzaei and A.A. Afzalian 5 Conclusions and future works PWA model of the Non-inverting Buck-Boost converter was obtained. The current mode control was employed and a set of non-linear equations was solved to find i L,ref based on known values of source voltage, load resistance and v C,ref. Explicit model predictive controller was designed to regulate i L to its reference value i L,ref which is equivalent to steering v C to v C,ref. Performance of the controller was evaluated with three different scenarios for Buck and Boost operating modes. In this paper, current mode control was used to bypass non-minimum phase behaviour of the capacitor voltage. This problem also can be solved by choosing long prediction horizon (N = 3) and the Move Blocking strategy can be used to reduce complexity of the controller (Beccuti et al., 27). It is also desirable to eliminate inductor current sensor for some reasons (Mariethoz et al., 28), thus, sensorless control method which was recently introduced (Mariethoz et al., 28) can be a topic for future research. Also, implementing the obtained controller using MPT code generator on hardware can be a topic of future works. References Baotic, M., Christophersen, F.J. et al. (23) Infinite time optimal control of hybrid systems with a linear performance index, CDC. Beccuti, A.G., Papafotiou, G. et al. (25) Optimal control of the boost dc-dc converter, Proceedings of the CDC-ECC 25. Beccuti, A.G., Papafotiou, G. et al. (26) Explicit model predictive control of the boost Dc-Dc converter, 2nd IFAC Conf. on Analysis and Design of Hybrid Systems, Alghero, Italy. Beccuti, A.G., Papafotiou, G. et al. (27) Explicit hybrid model predictive control of the dc-dc boost converter, IEEE PESC, Orlando, Florida, USA. Bemporad, A. and Morari, M. (999) Control of systems integrating logic, dynamics, and constraints, Automatica, Vol. 35, No. 3, pp Borrelli, F. (22) Discrete Time Constrained Optimal Control, Swiss Federal Institute of Technology, Zurich. Borrelli, F. (23) Constrained optimal control of linear and hybrid systems, Lecture Notes in Control and Information Sciences, Springer 29. Branicky, M.S. (995) Studies in Hybrid Systems: Modeling, Analysis, and Control, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. Geyer, T. (25) Low Complexity Model Predictive Control in Power Electronics and Power Systems, Zurich, ETH: 3. Geyer, T., Papafotiou, G. et al. (24) On the optimal control of switch-mode DC-DC converters, in R. Alur and G. Pappas (Eds.): Hybrid Systems: Computation and Control, Vol. 2993, pp Kvasnica, M., Grieder, P. et al. (24) Multi parametric toolbox (MPT). Hybrid Systems: Computation and Control, Springer Verlag, Pennsylvania, Philadelphia, USA. Lazar, M. and Keyser, R.D. (24) Non-linear predictive control of a DC-to-DC converter, Symposium on Power Electronics, Electrical Drives, Automation and Motion, (SPEEDAM), Capri, Italy. Leung, F.H.F., Tam, P.K.S. et al. (99) The control of switching DC-DC converters a general LQR problem, IEEE Transactions on Industrial Electronics, Vol. 38, No., pp Mariethoz, S., Beccuti, A.G. et al. (28) Sensorless explicit model predictive control of the DC-DC buck converter with inductor current limitation, IEEE Applied Power Electronics Conference.

20 Hybrid modelling and control of a synchronous DC-DC converter 433 Middlebrook, R.D. and Cuk, S. (976) A general unified approach to modeling switching power converter stages, IEEE Power Electronics Specialists Conference Records, pp Morari, M. and Baric, M. (26) Recent developments in the control of constrained hybrid systems, Computers & Chemical Engineering, Vol. 3, Nos. 2, pp Papafotiou, G., Geyer, T. et al. (24) Hybrid modeling and optimal control of switch-mode DC-DC converters, in IEEE Workshop on Computers in Power Electronics (COMPEL), Champaing, IL, USA. Rodrigues, L. and How, J.P. (2) Automated control design for a piecewise-affine approximation of a class of nonlinear systems, American Control Conference, Arlington. Senesky, M., Eirea, G. et al. (23) Hybrid modeling and control of power electronics, in Proceedings of the HSCC 23 6th International Workshop, Springer-Verlag. Sira-Ramirez, H. (23) On the generalized PI sliding mode control of DC-to-DC power converters: a tutorial, International Journal of Control, Vol. 76, Nos. 9, pp Sira-Ramirez, H. and Silva-Ortigoza, R. (26) Control Design Techniques in Power Electronics Devices, Springer-Verlag. Sira-Ramirez, H., Perez-Moreno, R.A. et al. (997) Passivity-based controllers for the stabilization of DC-to-DC power converters, Automatica, Vol. 33, pp Torrisi, F.D. and Bemporad, A. (24) HYSDEL a tool for generating computational hybrid models, IEEE Trans. Contr. Systems Technology, Vol. 2, No. 2, pp Tyler, M.L. and Morari, M. (999) Propositional logic in control and monitoring problems, Automatica.