Synergetic Synthesis Of Dc-Dc Boost Converter Controllers: Theory And Experimental Analysis

Transcription

1 Synergetic Synthesis Of Dc-Dc Boost Converter Controllers: Theory And Experimental Analysis A. Kolesnikov ( + ), G. Veselov ( + ), A. Kolesnikov ( + ), A. Monti ( ++ ), F. Ponci ( ++ ), E. Santi ( ++ ), and R. Dougal ( ++ ) ( + )Department of Automatic Control System Taganrog State University of Radio-Engineering (TSURE) 44 Nekrasovsky St., Taganrog, 34798, Russia ( ++ )Department of Electrical Engineering University of South Carolina Swearingen Center, Columbia, SC 908 U.S.A. Abstract- This paper describes a new approach to the synthesis of controllers for power converters based on the theory of synergetic control. The controller synthesis procedure is completely analytical, and is based on fully nonlinear models of the converter. Synergetic controllers provide asymptotic stability with respect to the required operating modes, invariance to load variations, and robustness to variation of converter parameters. With respect to their dynamic characteristics, synergetic controllers are superior to the existing types of PI controllers. We present here the theory of the approach, a synthesis example for a boost converter, simulation results, and experimental results. I. INTRODUCTION Design of controllers for power converter systems presents interesting challenges. In the context of system theory, power converters are non-linear time-varying systems; they represent the worst condition for control design. Much effort has been spent to define small-signal linear approximations of power cells so that classical control theory could be applied to the design. See for example [,]. Those approaches guarantee the possibility to use a simple linear controller, e.g. Proportional-Integral controller, to stabilize the system. The most critical disadvantage is that the so-determined control is suited only for operation near a specific operating point. Further analyses are then necessary to determine the response characteristics under large signal variations [3,4]. Other design approaches try to overcome the problem by using the intrinsic non-linearity and time variation for the control purpose. Significant examples of this approach include the sliding mode control, used mostly for continuous-time systems [5] and the deadbeat control, used for digital systems [6]. Those two theories have been applied to power electronics mostly because of their intrinsic capability to manage variably-structured systems. In this paper we focus on a different approach, synergetic control [7], that tries to overcome the previously described problems by using the internal dynamic characteristics of the system. The synergetic approach is not limited by any nonlinearity; instead, it capitalizes on such non-linearities. As will be discussed in the paper, this approach makes full use of the intrinsic proprieties of the system. While this is a strong point, it is also a weak point -- definition of the system model plays a more strategic role than in any other control approach. This introduces a great possibility for sensitivity to system parameters. However, as we will demonstrate with experimental results, this problem can be solved. One obvious solution is the adoption of sophisticated observers for parameter determination. This solution is reasonable only if the cost of the control is not a significant concern (e.g. high-power or high voltage applications). For situations where the control costs are of concern, we will show that suitable selection of the control macro-variables can largely resolve any sensitivity to uncertainty in system parameters. In this paper we will describe the theory of synergetic control, demonstrate its application in the case of a boost converter, describe both simulation and experimental results, and finally introduce some interesting practical considerations. II. THEORETICAL BACKGROUND Synthesis of a synergetic controller begins by defining a macro-variable, which is a function of the system state variables: ψ ( t) = ψ ( x, t) () The control objective is to force the system to operate on the manifold ψ = 0. The designer can select the characteristics of this macro-variable according to the control specifications (e.g. limitation in the control output, and so on). In the trivial case ψ is a simple linear combination of the state variables. This process is then repeated, defining as many macro-variables as there are control channels Next, the dynamic evolution of the macro-variables is fixed according to the equation: () t + ψ = 0 ; T > 0 Tψ () where T is a design parameter describing the speed of convergence to the manifold specified by the macro-

2 variable. Finally, the control law (evolution in time of the control output) is synthesized according to equation () and the dynamic model of the system. Briefly, any manifold introduces a new constraint on the domain of the state space, and thereby reduces the order of the system and forces it in the direction of global stability. The procedure summarized above can be easily implemented as a computer program for automatic synthesis of the control law or it can be performed by hand for simple systems, such as for the boost converter, that have a small number of state variables. By suitable selection of macro-variables the designer can obtain interesting characteristics for the final system such as: Global stability Parameter insensitivity Noise suppression These results are obtained while working on the full nonlinear system and the designer does not need to introduce simplifications in the modeling process to obtain a linear description as is required for classical control theory. III. SYNTHESIS OF A SYNERGETIC CONTROLLER FOR A BOOST CONVERTER We now synthesize a controller for a DC-DC boost converter (see Fig. ). The classical time-averaged model of the converter is: x xc () t = ( u) + Vg ; L L (3) x x xc () t = ( u), C RC 0 u (4) where x is the inductor current, x the capacitor voltage and u the switch duty cycle. Our objective is to obtain the control law u ( x, x ) as a function of state co-ordinates x, x, which provides the required values of converter output voltage x = xs and, therefore, current x = x for S various operating modes, while satisfied limitation (4). Fig. : Boost Converter scheme According to this method, we introduce the following macro variable ψ = x β x β 0 (5) ; > Substitution of ψ (3) into the functional equation: T ψ () t + ψ = 0; T > 0 (6) yields: x () t β x () t + ψ = 0 (7) T Now substituting the derivatives x () t and x () t from (3) and (4), the control law is obtained: LC β x U = u = + Vg + ψ (8) Cx + βlx RC L T The expression for u is the control action for the converter controller. Substituting macro variable ψ and T = λ RC into (8), we obtain the control law as: ( λ ) LC x β u = + x + Vg (9) Cx + β Lx λ RC λ RC L When λ =, i.e. T = RC, we get: LC x u = + Vg (0) Cx + β Lx RC L Control laws (8), (9), or (0), according to (6), inevitably move the representing point (RP) of object () firstly to invariant manifold ψ = 0 (3), and then along this manifold to the converter s steady state: x = x ; s x = xs. Let us study the behavior of the closed loop system: Cx β x () t = V V Lx Cx + g + ψ g RC L T + ; β + L () Lx β x x () t = + Vg + ψ β Lx + Cx RC L T RC on the manifold ψ = 0 (3). For this purpose, we substitute relation x = β x into (). This results in: xψ β Vg ψ () t = + ; R( β L + C) β L + C () β Vg ψ () t = x +. ψ R( β L + C) β L + C Each separate equation of () describes the behavior of the corresponding converter coordinate x or x on the manifold ψ = 0. Evidently, equation () is asymptotically stable with respect to the converters steady state: x s β RVg ; xs = β = RV (3) g From relation (3) we see that converter s steady state operating point depends on the power source voltage

3 Vg and on the load resistance R. After we set the required reference value of the converter s output voltage x s, (3) gives us a possibility to find β, present in macro variable ψ, i.e. xs xs = = RVg RVg β. (4) The steady state value of control u s, which provides the converter s steady state (3), will be determined by the following equation: β u. (5) S = Knowing β in (4) and u S in (5), we can find the steady state parameters of the converter. So, the synthesized control law u after a time interval approximately equal to 3 T moves the RP of the plant to the manifold ψ = 0, and then, according to equation (), provides asymptotically stable movement along ψ = 0 to the converter s steady state (3). According to (), the time to move RP along ψ = 0 is determined by the R β L + C. expression ( ) Control law u provides converter motion from an arbitrary initial state x 0, x 0 to the steady state 0 0 > x s, x s. In other words, the synthesized control law u with T > 0 and β > 0 guarantees asymptotic stability (in the whole) of the closed loop system with respect to the converter s steady state. IV. OTHER POSSIBLE CONTROL SYNTHESIS FOR THE BOOST CASE The previous case illustrated a very simple case of control synthesis that transformed the boost circuit into a first order system always working in the manifold described by the macro-variable. This case does not cover all the possible situations we could face in reality, where more complex macro-variables must be introduced. One classical problem is accounting for limitation of one of the state variables, for example, limiting the maximum input current. This problem can be simply solved by defining a new macro-variable: ψ = x Atanh( β x ) (6) where A = x max. This defines a new manifold where the current is naturally limited. In the rest of paper other assumptions will bring us to the definition of other possible macro-variables. V. SYSTEM MODELING RESULTS Extensive simulation analysis has been conducted to verify the control performance. The simulations have been performed using both Matlab and the VTB simulator [8]. Fig. and Fig. 3 show the transients created by changes in the load and in the power source amplitude, as predicted by Matlab models. Fig. 4 shows the phase portrait of the system and the stability characteristics of the control system as demonstrated by convergence to the manifold. Other simulation results, obtained in the VTB environment, are shown in Fig. 5 and Fig. 6. This second step was useful insofar as guiding construction of the real converter because of the possibility to use more detailed models of the power cell. For example, the capacitance model in VTB contains also the equivalent series resistance, giving the opportunity to explore more realistic problems. Fig. : The voltage transients Fig. 3: The load changing

4 Fig. 5: VTB schematic Fig. 4: System phase portrait VI. LABORATORY EXPERIENCE Following theoretical analysis, a laboratory prototype was designed and built. Since synergetic control is well suited for digital implementation a DSP-based platform was selected for migration from the VTB environment to the real world. The small-scale power converter system has the following nominal characteristics: - Rated Input Voltage: V - Rated Output Voltage: 40 V - Maximum Load: 00 W - Input Inductance: L = 46 mh - Output Filter Capacitance:.360 mf - Main Switch: IRF540N The main targets of the experimental analysis were: Verification of the control theory Analysis of problems related to the model parameter sensitivity By defining the controller in Simulink, we were able to easily export the control algorithms to both a dspace platform for control of the real hardware, and to the VTB environment for system simulation. The ease of inserting the Simulink controller into both hardware model allowed unique opportunities to rapidly experiment with a wide variety of macro-variable definitions in order to identify and resolve significant early problems. One interesting observation common to both experimental environments (simulation and hardware) was the possibility to introduce any kind of transient in the output voltage reference without requiring any soft-start option. The system easily remained stable under large non-linear transients. Fig. 6: VTB results On the other hand, adoption of the simplest macro-variable definitions revealed significant problems with respect to parameter sensitivity. This sensitivity mostly affected the steady-state value of the output voltage -- which resulted to be different from the reference value. For this reason, after the first set of experiments, a new macro-variable was defined: ψ = (x - x ) + ref k (x - xref) (7) This new macro-variable significantly reduced the problem of parameter sensitivity and allowed for the steady state to be set more accurately. Using this approach two main parameters had to be tuned for control performance: The value T involved in the main synergetic equation () The value of K involved in the macro-variable definition.

5 The role of T is extremely interesting. As far as equation () is concerned, T defines the speed with which we reach the manifold. On the other hand, this parameter also plays an interesting role in noise reduction. In the case of the boost control, the state vector is easily accessible and so we can assume that the error introduced in evaluation of the macro-variable is quite limited. On the other hand, its derivative is obtained by means of the state equations so then the system parameters play a significant role. Let us suppose that we have a systematic constant error in the evaluation of the derivative. If we check for the steadystate condition of this equation we will have: We want now to show some comparisons between simulated and experimental results that confirm the theoretical discussion presented in the previous paragraph. These results show the transient that follows step change of the reference voltage from 0 to 40 V. () t + e) + ψ 0 T ( ψd = and then in steady state: ψ = Te This means that by decreasing T, we decrease the time with which the manifold is reached. But also we reduce the steady-state error that is introduced by wrong estimation of the system parameters. During the experiments we found that a reduction of T from ms to 0. ms yielded a significant increase in accuracy of the steady-state. K plays a significant role after reaching the manifold; it determines the way that errors in the main state variable are canceled by using the error on the current. Decreasing K increases the control performance but also calls for a higher current peak during any transient. This situation, as pointed out in the introduction, could be solved by definition of a more sophisticated macrovariable which can account for current limitation. The synergetic approach also gives an opportunity to solve the steady-state problem by introducing a new state variable that represents the integral of the referencefeedback error. This is analogous to an integral term in a standard linear controller. We decided to avoid this option in order to keep the system simpler and to better exploit the possibilities offered by parameter tuning and macro-variable definition. However, the introduction of the integral term is always possible to force the error to go zero at steady state. According to the laboratory experience, we also figured out that this option should be considered eventually as a refining option working in the direction of keeping the integral charge as small as possible. VII. COMPARISON BETWEEN SIMULATION AND EXPERIMENTAL RESULTS All the results shown in the following have been obtained using the macro-variable definition reported in (7). Fig. 7: Output Voltage (Simulation) Fig. 8: Output Voltage (experiment) Fig. 7 and Fig. 8 show the output voltage from experiment and simulation. One can clearly see that the synergetic control transformed the second-order system into a firstorder system. This can be easily justified by considering that when we are on the manifold we have a linear relation between two state variables. Introducing the constraint, the order is reduced. This is always true for synergetic applications and it constitutes a similarity with the sliding mode approach. Fig. 9 and Fig. 0 focus on the evolution in time of the macro-variable. The two transients looks very similar in the first part moving in the direction of the manifold with the same speed. In the experimental results, anyway, a

6 second transient starts when we are close to steady state: this can be considered another side effect of the imperfect system modeling. Fig. : Input Current (averaged-simulated value) Fig. 9: macro-variable as function of time (simulated data) Fig. : Input current (filtered-experimental data) Fig. 0: macro-variable as function of time real data) Finally in Fig. and in Fig. the results for the input current are presented. Also in this case the simulation results and the experimental data match perfectly. VIII. CONCLUSIONS This paper introduced a new and interesting control approach called Synergetic Control. The main feature of this approach is to manage with the same level of simplicity both linear and non-linear systems. The main aspect of control design is definition of a macrovariable that specifies a manifold for the space variables. We have discussed several different definitions of the macro-variable and described the practical consequences of the different selections. The theoretical aspects have been discussed and then confirmed through experiment and simulation. ACKNOWLEDGEMENT This work was supported by the US Office of Naval Research (ONR) under grant N REFERENCES [] S. Sanders, J. Noworolski, Xiaojun Z. Liu, and G. C. Verghese, Generalized Averaging Method for Power Conversion Circuits, in IEEE Trans. on Power Electronics, vol. 6. N., April 99, pp [] D.M. Mitchell, "DC-DC switching regulator analysis", McGraw Hill Book Company, 988 [3] R. W. Erickson, S. Cuk, and R.D. Middlebrook, Large-scale modelling and analysis of switching regulators, in IEEE PESC Rec., 98, pp [4] P. Maranesi, M. Riva, A. Monti, A. Rampoldi, "Automatic Synthesis of Large Signal Models for

7 Power Electronic Circuits", IEEE-PESC99, Charleston (USA), July 999 [5] V.I. Utkin, "Variable Structure system with Sliding modes". IEEE Trans. on Ind. Electronics, vol AC, no., pp. -, 977. [6] L.Ben-Brahim, A. Kawamura, Digital Control of Induction Motor Current with Deadbeat Response Using Predictive State Observer, IEEE Trans. On Power Electronics, vol. 7, N. 3, July 99, pp [7] A. Kolesnikov, G. Veselov, A. Kolesnikov, et al. Modern applied control theory: Synergetic Approach in Control Theory, vol.. (in Russian) Moscow Taganrog, TSURE press, 000 [8] R. Dougal, T. Lovett, A. Monti, E. Santi, A Multilanguage Environment for Interactive Simulation and Development of Controls for Power Electronics, IEEE PESC0, Vancouver (Canada).