Hybrid Predictive and Input-output feedback linearization controllers design for Half Bridge Multicellular Inverter: Comparative study

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1 The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria Hybrid Predictive and Input-output feedback linearization controllers design for Half Bridge Multicellular Inverter: Comparative study K. Mokhtari *, M. Benmiloud and A. Benalia 3 * LACoSERE Laboratory, Amar Telidji University of Laghouat, Algeria Keywords «Half-bridge multicellular inverter», «hybrid system», «hybrid predictive controller», «inputoutput feedback linearization». Abstract Power converters knows important technological developments. However, accuracy and simplicity of their mathematical models are essential to obtain powerful controllers and thus enhance system performances. Hybrid models are qualified suitable for the representation of power converters due to the coexistence of continuous variables and discrete variables. This paper proposes hybrid model for an interesting application in high level voltage conversion which is half bridge multicellular inverter. Hybrid predictive controller is developed based on the proposed hybrid model. A comparative study between the proposed controller and input-output feedback linearization controller based on average model has been investigated. Simulation results clearly bring out the advantages and the effectiveness of the hybrid predictive controller compared to the input-output feedback linearization controller under variations of the load current and input voltage. I. Introduction The last decade has witnessed increasing developments on multilevel converters to rectify problem of high power processing via switch mode converters. The fruit of these developments is expressed by many innovative topologies. One of them is multicellular converter shown in fig. associated to R-L load []. It attracts the attention of many teams of research, due to its interesting characteristics and high dynamic performances such as: low switching ripple, few conduction losses, small and modularity of the topologies []. Multicellular converters have the particularity to be constructed by floating capacitors where their voltages must be caully controlled since the survival of the converter depends on it []. For this purposes, many studies have been proposed in the literature based on various mathematical models of the converter (instantaneous, averaging and harmonic model) [, 3]. In [, 4], input output feedback linearization have been developed based on the average model. Fuzzy logic controller based harmonic model of multicellular converter has been analyzed in []. Recently, other emerging control strategies based on the instantaneous model have been studied. For example: sliding mode control [5], hybrid control [3], predictive control [6], optimal control [6], controlled Lyapunov function [7], practical stabilization [8], and quadratic stabilization [9]. In this paper, two controllers for two cells half bridge inverter (C-HBI) have been investigated: i) Hybrid predictive controller is developed based on hybrid model and ii) Input-Output feedback ICPEA 5 Paper ID 8 URL:

2 The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria linearization controller is synthesized based on average model. Comparative study has been done to show the robustness and the effectiveness of the two controllers. This paper is organized as follows: in Section, description and different models of half bridge inverter will be presented. Section 3 will be devoted to the input-output feedback linearization controller design for the C-HBI. In section 4, the hybrid predictive controller will be presented. The obtained results through simulations will be shown in the last section. II. Half Bridge Multicellular Inverter Modelling Fig. shows the topology of two cells half bridge inverter associated to a passive R-L load. It is based on the association of two commutation cells with floating capacitor inserted between them. Each commutation cell is composed of pairs of complementary switches. For the k th cell: u k ( ) means that the upper switch is closed (open) and the lower switch is open (closed). u k Different mathematical models describing the dynamics of power converters can be found in the literature. The main used models of multicellular converter are as follows [3], Direct instantaneous model; Average model; Harmonic model []; Hybrid model. Fig.. Topology of two cells half bridge inverter. In order to synthesize an input-output feedback linearization controller, a continuous model is needed. Thus, an average model is used for this purpose. In the other hand, the hybrid predictive controller is designed using a hybrid dynamical model. First of all, we recall different models of multicellular converter. II.. Direct instantaneous model of C-HBI This nonlinear model represents the dynamics of the converter exactly where all the harmonic phenomena are present. The C-HBI can be described by the following affine differential equation [], With f ( x( ) E L R L x ( f ( x( ) g( x( ) u( y( h( x( ), and x x g( x( ) c x L x c ( x E ) c () ICPEA 5 Paper ID 8 URL:

3 The state vector x x T i L x The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria x represents the capacitor voltage and the vector u u T u contains the cells signals (inputs). II.. Average Model v c x and the load current The basic idea is to replace each variable by its average value over a period T. This model is continuous, which allows the use of nonlinear and linear systems theory for the control design. However, a PWM must be added to transform the continuous inputs (generally corresponds to the duty cycle) to boolean inputs. We consider the following transformation to obtain the average model of the C-HBI, We have, And f ( ) E R L L t ( x( ) d T tt x( t U( u( ) d T u( Y h( ) tt f ( ) g( ) U, g( ) c L c ( E ) c (3) (4) Where the state vector T contains the average value of the capacitor voltage vc and the average value of the load current duty cycles that correspond to the average values of the inputs. i L and the vector U U U T II.3. Hybrid Model represents the From (), we can remark the coexistence of continuous and discrete variables. Thus, a hybrid dynamical model can be established to benefit from the powerful proposed controllers in this area of research. The model () can be written as follows: x ( A ( x( B ( y( x( Where ( is a right continuous piecewise function, called the switching signal. It takes its values in a finite set of discrete modes Q q, q, q 3, q 4. Each discrete mode corresponds to a specific configuration of the inverter obtained by the cells states. Table. I resumes different discrete modes and their dynamics. (5) ICPEA 5 Paper ID 8 URL:

4 The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria Discrete Mode Mode Mode Mode Mode q q q 3 q 4 Table I: Different discrete modes Cells signals Dynamics A qi B qi u, u A q B q R / L E / L / C u, u A q B q / L R / L E / L / C u, u A q B 3 q 3 / L R / L E / L u, u A q B 4 q 4 R / L E / L II.4. Control requirements The control objective is to design a state dependent switching control law selects the active mode at each instant for the following requirements: Stabilization of the capacitor voltage around vc E / Stabilization of the load current around a given sinusoidal erence The first aim allows the share of the input voltage constraint across the two cells. ; I ( x( ): R Q that III. Input-Output Feedback Linearization controller (I-O FLC) The input output feedback linearization is an approach for the controllers design for nonlinear systems. Its aim is to perform an algebraic transformation of the nonlinear dynamic system in order to obtain a totally or partially linear one, which permit the decoupling between input and output variables by applying feedback linearization. The goal of feedback linearization is to control separately voltage and current of C-HBI. III.. Recall of the transformation The state feedback described in [] is expressed by, With U ( ) ( ) ( ) v (6) ( ) ( ) ( ) ( ) Where present the new input, ( ) and ( ) presents decoupling matrix and vector respectively and they are shown as follows, v ( ). r r L g L ( ) ( ) f h Lg L h m f ( ) r m rm Lg L h ( ) L L h ( ) f m gm f m, r L f h ( ) ( ) (7) rm L f hm ( ) Such as L f h j ( ) is called Lie derivative of h j with respect f or along f [] and m represent the number of inputs. III.. Application to C-HBI According to the equations (3) and (4) and the use of (7) we obtain after calculation the decoupling matrices given as below, ICPEA 5 Paper ID 8 URL:

5 The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria R C( E ) L E E R E ( ), ( ) (8) C L E E E By applying the state feedback to the system, the linearized system obtained is define as follows, y x v y x v Fig.. The decoupled system after input-output linearization. III... Closed Loop Controller After the input-output feedback linearization control, we obtain two decoupled sub-systems represented by integrators. We will elaborate proportional correctors for the voltage and current regulation, which the control loop equation is given by, E Where x vc, x I applied to the decoupled system. v k K p k x x, k, and K k k p k are the erence states. Fig.3 represents the linear loop control + _ + _ V β () + + U System α ) Fig. 3: Control loop with proportional correctors. The choice of the proportional gains can be performed by comparing the first order closed loop system by a erence one. III... Generation of Boolean signals (PWM) The objective is to generate the real signals to be applied to the switches for the different cells. It is demonstrated in [] that the control orders must be out of phase with each other in the general case by / p ( p present the cells numbers). To ensure this constraint for the C-HBI ( p ) ), it is sufficient to generate two-phase carrier regularly out of phase by, each carrier takes values between and. The function that can generate the carrier is giver by, p k arcsinsin Fd t k (9) p ICPEA 5 Paper ID 8 URL:

6 Where F d / T d two cells signals ( The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria is the switching frequency and u and u )., IV. Hybrid Predictive Control represents the phase shift between the In this section another control approach is presented, this control is also called hybrid control []. The hybrid control approach includes four fundamental steps [] in which we will present them in the next section by applying these steps to the C-HBI. Simulation results will be compared between the hybrid predictive control and the presented input-output feedback linearization control. IV. Control steps The algorithm below resumes the hybrid predictive steps for the C-HBI control. Measuring the capacitor voltage x ( kt ) e q Prediction of the state vector x k T i ) order prediction model, x q i and the load current ( k ) T x( kt ) T ( A x( kt ) B ), i,..., 4 e e e q i e x ( kt ) e (, for different discrete modes, using the below first e q i Choice of the discrete mode minimizing the weighted Euclidean distance between the state at time and the erence as follows, k ( x( ) argmin qiq dist, q T i qi With dist x ( k ) x Px ( k ) x qi qi Applying the configuration that corresponds to the selected discrete mode in the next sampling period. Fig.4 illustrates the proposed control strategy for the C-HBI. This sequence of actions is given by figure 4 as below, measure. i= calculate calculate Calculate the euclidean distance Yes No choose the configuration min Apply the configuration at the moment k+ Fig. 4: Hybrid predictive control algorithm for the C-HBI. ICPEA 5 Paper ID 8 URL:

7 The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria V. Simulation Results The presented two controllers are validated through Matlab simulations for two cells inverter with the parameters in Table II. Table II: The Inverter Parameters Element Values Unit Supply (E) 8 V Resistor (R) Ω Capacitor (C) 4 µf Inductor (L) 5 mh To compare the performance of hybrid predictive control and input-output linearization feedback control, two robustness tests were performed: i) variation of the load current and ii) variation of the input voltage. The parameters of the two controllers are presented in table III and table IV. Parameters Table III: Input-Output Feedback Linearization Parameters Values Proportional action Proportional action K K p p Switching frequency F d= KHz Table IV: Hybrid Predictive Control Parameters. Parameters Values Sampling time T e=µs Weight matrix P =diag (,) The following subsections resume the obtained results under the two robustness tests. V.. Test : Robustness with respect the variation of the load current, In this first test, a variation has been imposed on the erence load current (frequency and amplitude) as follows, A, f I Asin( f A 4, f A, f 5Hz 5Hz Hz t,. t.4,.6, t.,.4 Fig. 5 shows the evolution of the capacitor voltage. One can remark the stabilization of the capacitor voltage around its erence value c E / 4V without overshoot for the two controllers. A v fast response time is guaranteed by the input-output feedback linearization controller (ms) compared to the hybrid predictive controller (.5ms). Fig. 6 illustrates the robustness of the two controllers in load current stabilization. An overshoot in the load current is present in the case of input-output feedback linearization controller, which justifies the fast convergence of the capacitor voltage. ICPEA 5 Paper ID 8 URL:

8 The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria 4 v c (I-O FLC) v c (Hybrid Predictive Controller) v =E/ 4 i L (I-O FLC) i L (Hybrid Predictive Controller) I Capacitor Voltage (V) x Fig. 5: Capacitor voltages evolution for a variable erence current Load Current (A) Fig.6: Load current evolution for a variable erence current Fig. 8. a and Fig. 8. b represent the harmonics analysis of the load current using FFT tool of Matlab. The THD in the load current of the I-O FLC is equal to.54%, which is twice bigger than the hybrid predictive controller. Thus, one can conclude that the hybrid predictive controller is more convenient for the C-HBI control, which is confirmed by the analysis of the harmonic content in load current. These results are clear in the output voltage waveform given by Fig. 7 for the two cases. 4 I-O FLC Hybrid Predictive Controller 4 Output Voltage (V) - Selected signal: 3 cycles. FFT window (in red): cycles - Selected signal: 3 cycles. FFT window (in red): cycles Fig.7: Output voltage evolution of the C-HBI Fundamental (5Hz) = 39.97, THD=.54% Fundamental (5Hz) = 39.96, THD=.5%.5. Mag (% of Fundamental)..5. Mag (% of Fundamental) Harmonic order Harmonic order -a- -b- Fig.8: FFT analysis of C-HBI load current for a variable load current: a) I-O FLC (Input-Output feedback linearization controller), b) Hybrid predictive controller. ICPEA 5 Paper ID 8 URL:

9 The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria V.. Test : Robustness with respect the variation of the voltage input For this test, a sinusoidal perturbation was imposed on the input voltage as below, E 8 Asin( f, A V, f 3Hz The erence load current is equal to sin(. Fig. 9 and Fig. show that the two controllers ensure the same performance compared to test. I Fig. illustrates that the TDH in the load current is very small in the case of hybrid predictive controller compared to the Input-Output feedback Linearization Controller. One can conclude that the hybrid predictive control guarantees a high performance (good tracking without overshoot, good perturbation rejection and less harmonics in the output waveforms) with respect to the input-output feedback linearization control in all variations phases i L (I-O FLC) i L (Hybrid Predictive Controller) I Capacitor Voltage (V) Load Current (A) - v c (I-O FLC) v c (Hybrid Predictive Controller) - v =E/ Selected signal: cycles. FFT window (in red): cycles Selected signal: cycles. FFT window (in red): cycles Fig. 9: Capacitor voltages evolution for a - variable input voltage Fig.: Load current evolution for a variable - input voltage Mag (% of Fundamental) Fundamental (5Hz) = 9.8, THD= 9.89% Mag (% of Fundamental) x Fundamental (5Hz) =, THD=.6% Harmonic order Harmonic order -a- -b- Fig.: FFT analysis of C-HBI load current for a variable input voltage: a) I-O FLC, b) Hybrid predictive controller. VI. Conclusion In this paper, two control approaches were presented for the control of two cells half bridge inverter associated to RL load. Different dynamic representations of this converter have been presented (instantaneous model, average model and the hybrid model). Two controllers have been developed: i) Input-Output feedback linearization controller based on the average model and ii) hybrid predictive controller based on the hybrid model. The obtained results show the performance and robustness of the hybrid predictive controller with respect to input-output feedback linearization control. ICPEA 5 Paper ID 8 URL:

10 The nd International Conference on Power Electronics and their Applications (ICPEA 5), Djelfa on 9-3 March 5, Algeria As a future work, the case of three cells converter should be investigated. Observability and observer design for output feedback of multicellular inverter will be analyzed and developed. References [] T. Meynard, H. Foch, Electronic device for electrical energy conversion between a voltage source and a current source by means of controllable switching cells, European Patent 9/ , July 8, 99. [] G. Gateau, Contribution à la commande des convertisseurs statiques multicellulaires: Commande non linéaire et commande floue, Ph. D. dissertation, INP, Toulouse, France, 997. [3] K. Benmansour, A. Benalia, M. Djemaï, J. de Leon, Hybrid control of a multicellular converter, Nonlinear Analysis: Hybrid Systems, Volume, Issue, March 7, Pages 6-9, [4] G. Gateau, M. Fadel, P. Maussion, R. Bensaid and T. A. Meynard, Multicell Converters: Active Control and Observation of Flying-Capacitor Voltages,, IEEE [5] L. Amet, M. Ghanes, J. -P Barbot, "Direct control based on sliding mode techniques for multicell serial chopper," American Control Conference (ACC),, pp.75,756, June 9 -July. [6] D. Patino, M. Baja, H. Cormerais, P. Riedinger. Alternative control methods for DC/DC converters: An application to a four-level three-cell DC/DC converter. International Journal of Robust and Nonlinear Control, Vol., Issue, July, pages - 33, ISSN: [7] P. Hauroigne, P. Riedinger, C. Iung, "Observer-based output-feedback of a multicellular converter: Control Lyapunov function Sliding mode approach," 5st Annual Conference on Decision and Control (CDC), pp.77-73, -3 Dec., doi:.9/cdc [8] D. Kamri, R. Bourdais, J. Buisson, C. Larbes, Practical stabilization for piecewise-affine systems: A BMI approach, Nonlinear Analysis: Hybrid Systems, Volume 6, Issue 3, August, Pages [9] M. Benmiloud, A. Benalia, Hybrid Feedback for multicellular converter based on the quadratic stabilization theory, 3rd International Conference on Systems and Control (ICSC3), Algeria. Oct 3. [] H.K. Khalil. Nonlinear Systems. Prentice Hall, 3rd edition,. [] TRABELSI Mohamed Abdallah. Modélisation et Commande des Systèmes Physiques à Topologie Variable : Application au Convertisseur Multicellulaire. Thèse de doctorat en Energie et Systèmes. Lyon : Institut National des Sciences Appliquées, 9, 6p. ICPEA 5 Paper ID 8 URL: