Synergetic Control for m-Parallel Connected DC-DC Buck Converters
Synergetic Control for m-Parallel Connected DC-DC Buck Converters I. Kondratiev University of South Crolina Columbia, USA Email: [email protected]
E. Santi University of South Crolina Columbia, USA Email: [email protected]
R. Dougal University of South Crolina Columbia, USA Email: [email protected]
G. Veselov University of Radio-Enginneering Taganrog, Russia Email: [email protected]
Abstract— A promising new approach, synergetic control theory, based on ideas of self-organization, is used to design control for the general case of m parallel-connected DC-DC buck converters serving resistive load. A general nonlinear synergetic PI control algorithms are derived using averaged models of the system. Using the general control laws, two particular cases are considered. They are two and three parallel-connected DC-DC buck converter systems. It is shown that the derived controls suppress unmeasured piecewise constant disturbances, nullify errors of current sharing among parallel connected converters, and ensure exponential asymptotic stability. It is pointed out that by change of control law parameters it is possible to choose between master-slave and democratic system operation regimes. Matlab/Simulink simulations of the closed loop performance are performed and characteristics of synergetic control design are discussed. Thus, it is shown that the approach allows designers to derive analytical control laws that ensure exponential asymptotic stability for nonlinear, high dimensional, and multiconnected systems.
I.
INTRODUCTION
Growing consumer power demands and advances in DC electric distribution systems have resulted in increasing interest in control strategies for parallel-connected DC converters. Implementing these converters in power distribution systems promises to fulfill the most important goals of autonomous power system design, including greater system flexibility and reliability, simpler fabrication, testing, and assembly; and minimized size, weight, operational requirements, and overall cost [1]. However, despite the seeming simplicity of DC systems the nonlinearity and multiconnectivity of the controlled system can sometimes result in chaotic behavior [2]. High dimensionality further complicates system analysis. As a result, most classical control design techniques are not sufficiently capable for this class of autonomous DC power systems. The problems of DC system control design have been addressed by such techniques as feedback linearization, sliding mode, and synergetic controls. Application of the feedback linearization technique presented in [3] is constrained by the existence of the diffeomorphic transformation and requires additional effort in order to incorporate integral action to compensate for changes in the
system operating conditions. Sliding mode control allows the incorporation of integral action into nonlinear control design, but it is hard to implement due to the requirement for fairly high controller bandwidth. The sliding control application described in [4] addresses this issue by combining sliding mode control with Lyapunov function based control using a state-space averaged model. This combination solves the problem of controller bandwidth, but requires implementation of both control algorithms and implementation of mechanisms for switching from one control strategy to another within each period of the converter switching frequency. On the other hand, the application of synergetic control, as shown by application in [5], is also able to account for nonlinearity and can incorporate integral action into the control law but requires neither a high bandwidth controller nor switching between strategies. Thus, synergetic control theory opens new horizons for efficient design of controls for nonlinear systems. Synergetic control theory [6], based on ideas of modern mathematics and synergetics, uses the capability of open systems for self-organization [7]. Synergetic control is based on dynamic control of the interaction of energy and information within the system. By introducing virtual dissipative structures [8], it induces directed self-organized motion within the system. This motion takes control of the system degrees of freedom and leads the system to a specified state. As a result, systems such as parallelconnected power converters work coherently (in unison), supplying the required amount of energy with improved quality. Thus, coordinated behavior improves the system performance and reliability. In short, utilizing dissipative structure algorithms, synergetic control theory supplies an analytical control design procedure for nonlinear, multi-connected, high dimensional systems, which includes the essential properties of the controlled plant within the task statement. As a result, synergetic control theory allows designers to state and then efficiently solve many control problems, heretofore neither solved by other known methods nor even stated due to their extended complexity. These problems relate not only, for example, to global stability of the closed loop operation or global optimization of the system behavior but also to simplifying transitions among power sharing strategies, or
minimizing system power losses. The synergetic control design procedure [6] is based on a new method for generating control laws u(ψ)=u(x) or feedback as functions of the specified macro-variables ψ(x). These controls direct the system from an arbitrary initial condition into the vicinity of manifolds ψ(x)=0 and then ensure asymptotically stable motion along these manifolds toward the end attractors or equilibrium state. On these attractors the specified properties of the controlled system are guaranteed. In the current study, this procedure is applied to a system of m parallel-connected DC buck converters. Employing a simplified state-space averaged model of the system, the paper first presents the design of a general control algorithm. Then, for two particular examples: two and three parallel-connected converter systems, the capabilities of synergetic control for power sharing are explored and the design considerations for macro-variables are discussed. The simulations performed in VTB, a freely available modeling environment with an extended library of power electronic components [9], show the strong performance of the designed control even beyond the validity limits of the system model that was used in the control design, for example in discontinuous conduction mode. II.
SYNERGETIC CONTROL DESIGN
A. System model The prototype system of paralleled buck converters presented in Fig. 1 is the focus for this research.
The primary goal of control design for the system (1) is to maintain a specified output voltage Vc1,ref (2) while the system is affected by a disturbance. The designed control laws must ensure asymptotic stability of closed loop operation. dM dt = η ⋅ (v c1 − V c1, ref ) ν c1 1 m dν c1 dt = C ∑ i Lj − R C + δ ⋅ M ext t t j =1 di L1 d c1 E 1 = − ν c1 + L1 L1 dt ... di Lm = − 1 ν + d cm E c1 dt Lm Lm
(1)
where dci is the desired switch duty of the ith converter, M(t) is the disturbance influencing the system, and δ is a coefficient that will be determined later based on stability of the closed loop system. vc1 = Vc1,ref
(2) The essence of the extended model (1) is that due to the introduced additional state variable, the system is capable of rejecting piece-wise constant disturbances and obtaining zero steady state error. Next, coordinate transformation (3) is performed: v = ν c1 − Vc1,ref i1 = iL1 i2 = iL 2 ... im = iLm
(3)
This transformation accounts for control design requirements by moving the system origin of coordinates into the output voltage equilibrium point. Later on, it also allows us to focus our attention on current sharing. The resulting system model is presented in (4).
Fig. 1. System containing m parallel-connected buck converters
Assuming that the buck converters operate in continuous conduction mode and that switching occurs at a very high frequency, sources Ei are the same, that a piece-wise constant disturbance sufficiently approximates system load and parameter changes, ignoring parasitic effects, and using the averaged capacitor voltage vc1 and inductor currents iLi as state variables, the extended state space averaged system model is shown in (1) where function M(t) represents a disturbance affecting the system.
M ′ = η ⋅ v m v ′ = 1 i − v + Vc1,ref + δ ⋅ M ∑j Ct j =1 Rext Ct 1 ′ i1 = − L ⋅ (v + Vc1,ref ) + u1 1 i ′ = − 1 ⋅ (v + V c1,ref ) + u 2 2 L2 ... 1 im′ = − ⋅ (v + Vc1,ref ) + um Lm
(4)
where ui=E/Li*dci Thus, the extension of the system model and coordinate transformation allow us to incorporate requirements of the control design and the model of the referenced disturbance into the system model. B. Design a general control algorithm A basic tenet of synergetic control design is the use of
macro-variables to define laws of interaction among the system components. These macro-variables define the properties of motion of the extended system (4) to the equilibrium state. For simplicity, in our case these macrovariables (5) are chosen to be linear combinations of the system variables (4). m
ψ i = ai1 ⋅ M + ai 2 ⋅ v + ∑ ai , j +2i j i = 1, m j =1
or
(5)
Ψ = AX − B
where
Ψ = (ψ 1 ,ψ 2 ,...ψ m )
T
a13 a A = 23 ... a m3
a14 a24 ... am 4
... a1n ... a2 n ... ... ... amn
T X = (i1 , i 2 ,...i m ) , bi = −a i1 M − a i 2 v,
i = 1, m
The number of macro-variables in the set (5) equals the number of control channels. The coefficients aij are chosen such that conditions (6) are satisfied. (6) det A ≠ 0. According to [6], under synergetic control (7) macrovariables (5) become manifolds (8). u( x ) = u(ψ ) (7) (8) Ψ = AX − B = 0 The evolution of the macro-variables towards the manifolds (8) can be defined in various ways. The possibility shown in (9) is the equation used for control design in this paper. Tiψ i′ = −ψ i i = 1, m Ti > 0 (9) The control defined by (9) moves the representing point of the system towards and along the manifolds (8) to the specified equilibrium point (2). The expression for controls is found by substituting the system model (4) into (9) and solving the resulting system jointly, yielding the controls. (10) U = A−1 ⋅ D where di = −
ψi Ti
m a ai 2 ⋅ (v + Vc1, ref ) + + ∑ ij + 2 + j =1 L R j ext ⋅ Ct
m i − η ⋅ ai1 ⋅ v − ai 2 ⋅ ∑ j + δ ⋅ M j = 1 Ct
(11)
i = 1, m
Substituting the expression for di from (11) into (10) gives: m
ui = ∑ A−1 (i, j ) ⋅ d j ,
i = 1, m
j =1 (12) where A-1(i,j) is the i, jth element of inverted matrix A The control laws (12) ensure asymptotically stable convergence towards the specified manifolds (8). The speed of convergence is regulated by value of Ti. Moreover, the synergetic controls (12) move the system-representing point onto the intersection of the manifolds (8). As a result, after the transients are finished equalities (8) are satisfied.
Equations (8) show that the state variables are no longer independent. As a result, due to contraction of state space, the system (4) is now of reduced order, which equals: (13) d =n−m where n is the order of the given system (4) and m stands for the number of control channels. In our case, the dimension of the reduced order system equals two. Thus, the result of the synergetic control design procedure is an analytical control law that ensurs asymptotically stable motion towards the intersection of the introduced manifolds (8). According to [6], the stability of the system at the manifold intersection and the equilibrium point can be checked by any appropriate method [10]. C. Stability conditions Once the representing point arrives on the manifold intersection, the equalities (8) are satisfied and the system has n-m independent variables defining its behavior. To define the dynamics of the closed system on the manifold intersection, the equations (8) are represented as an algebraic system: (14) A⋅ X = B As long as the conditions (6) are satisfied, the solution of the system (14) is as follows: (15) X = A −1 ⋅ B Thus, the closed loop system dynamics on the intersection of the manifolds based on the solution of (15) is as follows: M ψ′ = η ⋅ vψ m m vψ + V c1, ref −1 vψ′ = − R C − ∑∑ A ( j, i ) ⋅ (a i1 M ψ + a i 2 vψ ) + δ ⋅ M ψ j =1 i =1 ext t
(16)
To define the stability conditions, the system (16) is rearranged into one equation: vψ′′ + F1 ⋅ vψ′ + F2 ⋅ vψ = 0 (17) where F1 =
m m m m 1 + ∑∑ A −1 ( j, i ) ⋅ a i 2 , F2 = η ⋅ ∑∑ A −1 ( j, i ) ⋅ a i1 − δ R ext C t j =1 i =1 j i = 1 = 1
From (17) it follows that the stability conditions for the closed loop system on the intersection of the manifolds are: F1 > 0, F2 > 0, Ti > 0, i = 1, m (18) It can be seen from (17) and (18) that positive value of parameter δ makes stability conditions more strict. Hence, δ=1 is used in stability check of the closed loop system. In order to highlight the internal properties of the system motion along the manifolds, equation (17) is written in the following form: vψ′ = − F1 ⋅ vψ − F2 ⋅ ∫ vψ dt = 0
(19)
Equation (19) shows that on the intersection of the manifolds (8) a PI control law for the coordinate v is introduced. Thus, according to (18) and (19), the synthesized control laws (12), besides ensuring asymptotic stability, also suppress the piece-wise constant disturbance M(t) influencing the converters.
III.
RESULTS
The derived general algorithm has been applied to twoconverter and three-converter systems. The resulting control laws and stability conditions are presented in the appendix. Several simulations have been performed on these systems. The nominal values of system parameters are presented in Table 1.
transformation (3). To achieve the specified (22) current sharing in the system, the manifolds have this structure: 1 1 − a M − a12 v ⋅ X = 11 0 1 − 1
(23)
Corresponding to the particular choice of manifolds (23), the system diagram is shown in Fig. 3 a.
TABLE 1 NOMINAL VALUE OF SYSTEM PARAMETERS
Parameter Nominal Value rL1=rL2=rLn 0.1 Ohm L1=L2=Ln 150 µH C1=C2=Cn 2200 µF VC,ref=U0 5.0 V Switching frequency 50 kHz T1=T2=Tn 0.001 η= ηn 20000 a11=a11n 1000 a12=a12n 300 δ 1 The input voltage varies between 12-25V. The output voltage is regulated at 5.0V. The objectives for simulation are to explore current sharing and the effect of macrovariable coefficients on the dynamic response of the closed loop system.
Fig. 2. Structure of the system feedback on the manifolds
a)
B. Current Sharing on the Manifolds 1) Two-converter system Directed by synergetic control laws (12), the representing point of the two-converter system inevitably reaches the intersection of the manifolds (8) after (3-5)•T. On the manifolds the motion of the system is defined by the following equations: a13 a14 − a M − a12 v ⋅ X = 11 (20) a 23 a 24 − a 21 M − a 22 v The system currents calculated from (20) using (15) are functions of output voltage error. Moreover, coefficients of matrices A and B in (15) define the properties of current sharing among converters on the manifolds. In systems with two parallel-connected converters the only basic current sharing regimes possible are master-slave and democratic. When the system is on the manifolds either of these can be achieved by a particular choice of the coefficients in (20). Equations (20) define the structure of the feedback influencing the system on the manifolds, which is presented in Fig. 2. Master-slave current sharing occurs when the master converter maintains output voltage and all other converters supply current according to the assigned rating [11], usually, defined for the slave converter by the master. The conditions for master-slave current sharing are summarized in the following mathematical expressions: v = Vc1,ref (21) i1 = i2 (22) Requirement (21) has been accounted for by coordinate
x0
b) x0
Equilibrium x0
Manifold 2
Manifold 1
x0
Fig. 3. System under master-slave current sharing: a) feedback structure b) state portrait (x0 are various initial conditions, arrows on trajectories show direction of the system motion)
Fig. 3 b shows the motion of the system representing point, which starts from various initial conditions x0. Influenced by the synergetic control it moves along the integral curves (24).
m
ψ = ∑ a 2, j + 2i j + C1
j =1 (24) where C1 is a constant corresponding to the initial conditions. The curves described by (24) are solutions of the differential equation (25) and lie parallel to manifold_2 in Fig. 3: T ⋅ψ ′ = −ψ (25) Following the solution (24) the representing point reaches manifold_1 in Fig. 3 b and moves along to the equilibrium state, where the static current sharing is defined by (22). As a result, on manifolds (23) the system exhibits equal dynamic and static current sharing. On the other hand, democratic current sharing occurs when converters provide current according to their current rating (22) and each converter accounts for the output voltage error [12]. The specifications are the same as for master-slave sharing (21) and (22). The structure of the manifolds is shown in (26).
− a M − a12 v 1 1 ⋅ X = 11 − a 22 v 1 1 −
(26)
a)
same pattern as for master-slave current sharing. However, integral curve (27) differs from (24) in that it depends on output voltage error. This dependence allows change of the system current sharing. m
ψ = a22 ⋅ v + ∑ ai , j + 2i j + C2
j =1 (27) where C2 is a constant depending on the initial conditions. During the transition to the equilibrium, coefficient a22 influences the rate of change of converter currents. Since the output voltage error equals zero at equilibrium, it does not affect the static current sharing in the system. Thus, the major differences between master-slave and democratic current sharing are that master-slave sharing provides constant dynamic and static current sharing, while democratic current sharing can have varying dynamic and static current sharing. In summary, for the system containing two converters, democratic and master-slave are the only possible current sharing methods. The structure of manifolds (20) defines the regime of current sharing in the system. Moreover, the transition among regimes can be made by a simple change of coefficients in the manifold equations (20). 2) Three-converter system The behavior of the closed loop system with three converters is similar to that of the two-converter system explained above, except that any combination of democratic and master-slave current sharing can be applied. A few possible combinations are presented in Table 2
TABLE 2 A FEW POSSIBLE SYNERGETIC CONTROL STRATEGIES FOR THE THREE-CONVERTER SYSTEM
Control Strategy Master-slave
Democratic
x0
b) x0
Equilibrium x0
Manifold 2
Manifold structure 1 1 1 − a11 M − a12 v 0 1 − 1 0 ⋅ I = 1 0 − 1 0
1 − a 11 M − a 12 v 1 1 − a 22 v 1 − 1 0 ⋅ I = 1 0 − 1 − a v 13
Two masters with one 1 1 1 − a 11 M − a 12 v − a 22 v slave 1 − 1 0 ⋅ I = 1
x0 Manifold 1
Fig. 4. System under democratic current sharing: a) feedback structure b) state portrait (x0 are various initial conditions and arrows on trajectories show direction of the system motion)
The system diagram reflecting the structure of feedback, when the system is on the manifold intersection (26), is shown in Fig. 4 a. The difference between feedback structures (23) and (26) corresponds to the use of output voltage error in the second converter. The motion of the system representing point follows the
0
− 1
0
The control strategies presented in Table 2 provide various static and dynamic current sharing capabilities. As shown in the previous section, the master-slave strategy provides equal static and dynamic current sharing and the democratic strategy allows the two to be different. The twomaster with one slave strategy represents a mixture of the two basic methods. Table 2 does not present all of the possibilities; however it shows that by changing coefficients in the manifold equations it is possible to choose different current sharing methods, even dynamically. Thus, a greater number of converters in the system yields more flexibility in current sharing. The structure of manifolds (8) defines the regime of current sharing in the
system and, if converters are fed from different sources, dynamic current sharing provides a way to regulate the dynamic contributions of the sources according to their properties. Moreover, the transition among regimes can be made by a simple change of coefficients in the manifold equations (8). In this way, certain coefficients of the manifolds (8) define static or dynamic current sharing properties of the system. C. Dynamic response and manifold coefficients In addition to the relations among the structure of manifolds and the properties of current sharing explained in the previous section, a key concept of synergetic control design is that manifolds define the closed loop system behavior. Specifically, the manifold coefficients Ti, a11, and a12 also affect the system transients. The correspondence between these coefficients and the transient response of the closed loop system is shown by a simulation of the twoconverter system, when the load resistance is changed from 2 to 1 Ω. While a particular coefficient is varied the remaining parameters are held equal to their nominal values which are presented in Table 1 and are referred later on in figures by subscript n. Fig. 5 shows how varying parameters T1 and T2 by 70% from their nominal values Tn affect the transient response. Fig. 6 and Fig. 7 illustrate the system transients for a11 and a12 respectively, to highlight the influence on the system transients the parameter a11 is changed by 60% and a12 is varied by 50%.
a)
a)
b)
Fig. 7. Impact of a12 on transient response of the two-converter system a)output voltage error b) inductor currents (1:a12=1.5a12n, 2:a12=a12n, and 3:a12=0.5a12n)
Control law parameters T, a11 and a12 affect the transient response of the closed loop system and allow designer not only to achieve specified performance but also to limit rate of increase of the input current withdrawn from the bus or energy source by proper choice of parameters T. D. Simulation of the closed loop system The closed loop system was modeled in VTB [9] using the averaged model of the buck converter accounting for Discontinuous Conduction Mode (DCM) operation. The goal of the simulation experiment was to check the performance of the system under varying load conditions and to inspect the capability of the synergetic control to withstand discontinuities in operation of the buck converters. The results of the simulation are presented in Fig. 8, which shows dynamic and steady state performances of two-converter system when parameters of the system equal to their nominal values and load resistance changes from 10 kΩ (DCM operation) to 1 Ω (full load).
b)
Fig. 5. Impact of T on transient response of the two-converter system a)output voltage error b) inductor currents (1:T1=T2=0.3Tn, 2:T1=T2=Tn, and 3:T1=T2=1.7Tn)
a)
to current overshoots. Thus, due to the complex interconnection among the coefficients and the system settling time and transient response, a compromise needs to be found.
b)
Fig. 6 Impact of a11 on transient response of the two-converter system a)output voltage error b) inductor currents( 1:a11=0.4a11n, 2:a11=a11n, and 3:a11=1.6a11n)
Results indicate that these parameters have similar impacts on the system transients with some particularities. Parameters T define the speed of convergence of the macrovariables (5) towards the manifolds (8). Lowering those parameters improves the settling time of the system. While the shape of the voltage transients is preserved, passing a particular threshold for T causes current overshoots. On the other hand, varying a11 has an impact on both the shape of the transients and the settling time. In contrast to a11, the decrease of parameter a12 improves settling time and leads
a)
c)
b)
d)
Fig. 8. Transients of the two-converter system a) and c) output voltage error b) and d) converters currents a) and b) values of the inductances equal to Ln c) and d) values of inductances are L1=1.1Ln (line 1) and L2=0.9Ln (line 2)
The output steady state voltage in DCM has a periodic form which amplitude is within 3% of nominal voltage. The oscillation of the output voltage corresponds to the inadequacy of the models to predict system behavior in DCM operation. However, by incorporating integral action into synergetic control law the closed loop system can maintain output voltage within specified limits. Moreover, the magnitude of output voltage oscillations can be controlled by the value of time constant T. By decreasing this parameter the specified accuracy of output voltage and
current sharing in DCM can be achieved. From Fig. 8 it follows that two-converter system under synergetic control exhibits an ability to nullify output voltage and current sharing errors and show the strong performance even beyond the validity limits of the system model used in the control design which correspond to operation of the converters in discontinuous conduction mode. IV.
CONCLUSIONS
APPENDIX B. Vector controls for two and three-converter system Substitution of m=2 and m=3 into equations (12) gives the vector controls for two and three-converter systems, which respectively are (27) and (32). 1 1 ⋅ (a24 d1 − a23d 2 ) u2 = ⋅ (− a14 d1 + a13d 2 ) det A det A where : det A = a13 ⋅ a24 − a23 ⋅ a14
u1 =
ψ1
d2 = −
T1
a a a + 13 + 14 + 12 ⋅ (v + Vc1, ref ) − η ⋅ a11 ⋅ v − a12 ⋅ k L1 L2 Rext Ct
− a 23 ⋅ a14 ⋅ a 35 − a 25 ⋅ a 34 ⋅ a13 − a 33 ⋅ a 24 ⋅ a15 d1 = − d2 = − d3 = −
A robust control algorithm for m-parallel-connected buck converters has been presented together with important considerations for manifold design. This algorithm, derived using synergetic control theory, includes the capability to account for nonlinear properties of the controlled system, to suppress unmeasured piece-wise disturbances and ensures asymptotically stable motion toward the intersection of the manifolds. It also defines the stability conditions for motion along the intersection to the equilibrium state. The algorithm provides such benefits as nullification of output voltage and current sharing errors by incorporating integral action into the controls and design simplicity based on the independence of manifolds. It also allows system transition among current sharing methods by a simple change of control law coefficients. In addition, the closed loop system shows a robust transient response despite variations of system parameters. Using the algorithm will help to improve power distribution systems particularly with regard to flexibility and reliability.
d1 = −
1 ⋅ ((a 24 a 35 − a 34 a 25 ) ⋅ d 1 + (a 33 a 25 − a 23 a 35 ) ⋅ d 2 + (a 23 a 34 − a 33 a 24 ) ⋅ d 3 ) det A 1 ⋅ ((a 34 a15 − a14 a 35 ) ⋅ d 1 + (a13 a 35 − a 33 a15 ) ⋅ d 2 + (a 33 a14 − a13 a 34 ) ⋅ d 3 ) u2 = det A 1 ⋅ ((a14 a 25 − a 24 a15 ) ⋅ d 1 + (a 23 a15 − a13 a 25 ) ⋅ d 2 + (a13 a 24 − a 23 a14 ) ⋅ d 3 ) u3 = det A where : det A = a13 ⋅ a 24 ⋅ a 35 + a14 ⋅ a 25 ⋅ a 33 + a 23 ⋅ a 34 ⋅ a15
u1 =
(28)
ψ2
a a a + 23 + 24 + 22 ⋅ (v + Vc1, ref ) − η ⋅ a21 ⋅ v − a22 ⋅ k T2 L1 L2 Rext Ct
i i k = 1 + 2 + δ ⋅ M Ct Ct
C. Stability conditions The systems behavior on the manifolds is defined by equation (17). The coefficients F1 and F2 corresponding to two and three-converter system respectively presented in (30) and (31) are found from (17) using m=2 and m=3. If the conditions (18) are satisfied, the synthesized vector control laws will ensure asymptotic stability of the closedloop system and suppress the uncontrolled piecewiseconstant disturbance M (t ) = const .
ψ1
a a a a + 13 + 14 + 15 + 12 T1 L1 L2 L3 Rext C t
ψ2 T2
ψ3 T3
F1 =
i i i ⋅ (v + Vc1,ref ) − η ⋅ a11 ⋅ v − a12 1 + 2 + 3 + δ ⋅ M Ct Ct Ct
a a a a + 23 + 24 + 25 + 22 L2 L3 Rext C t L1
i i i ⋅ (v + Vc1,ref ) − η ⋅ a 21 ⋅ v − a 22 1 + 2 + 3 + δ ⋅ M Ct Ct Ct
a a a a + 33 + 34 + 35 + 32 L2 L3 Rext C t L1
i i i ⋅ (v + Vc1,ref ) − η ⋅ a 31 ⋅ v − a 32 1 + 2 + 3 + δ ⋅ M Ct Ct Ct
a12 ⋅ (a 24 − a14 ) + a 22 ⋅ (a13 − a 23 ) 1 + Rext Ct det A
a ⋅ (a − a14 ) + a 21 ⋅ (a13 − a 23 ) − δ ⋅η F2 = 11 24 det A F1 =
(a24 a35 − a34 a25 + a34 a15 − a14 a35 + a14 a25 − a 24 a15 ) ⋅ a12 + 1 1 ⋅ (a33a 25 − a23a35 + a13a35 − a33a15 + a23a15 − a13a25 ) ⋅ a22 + + det A Rext Ct (a23a34 − a33a24 + a33a14 − a13a34 + a13a24 − a23a14 ) ⋅ a32
(a a − a34 a25 + a34 a15 − a14 a35 + a14 a25 − a 24 a15 ) ⋅ a11 + 1 24 35 F2 = ⋅ (a33a25 − a23a35 + a13a35 − a33a15 + a23a15 − a13a25 ) ⋅ a21 + − δ ⋅ η det A (a a − a a + a a − a a + a a − a a ) ⋅ a 33 24 33 14 13 34 13 24 23 14 31 23 34
(29)
(30)
(31)
ACKNOWLEDGMENT The authors acknowledge the support of ONR grants N00014-02-1-0623, N00014-03-1-0434. REFERENCES [1]
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E. Santi University of South Crolina Columbia, USA Email: [email protected]
R. Dougal University of South Crolina Columbia, USA Email: [email protected]
G. Veselov University of Radio-Enginneering Taganrog, Russia Email: [email protected]
Abstract— A promising new approach, synergetic control theory, based on ideas of self-organization, is used to design control for the general case of m parallel-connected DC-DC buck converters serving resistive load. A general nonlinear synergetic PI control algorithms are derived using averaged models of the system. Using the general control laws, two particular cases are considered. They are two and three parallel-connected DC-DC buck converter systems. It is shown that the derived controls suppress unmeasured piecewise constant disturbances, nullify errors of current sharing among parallel connected converters, and ensure exponential asymptotic stability. It is pointed out that by change of control law parameters it is possible to choose between master-slave and democratic system operation regimes. Matlab/Simulink simulations of the closed loop performance are performed and characteristics of synergetic control design are discussed. Thus, it is shown that the approach allows designers to derive analytical control laws that ensure exponential asymptotic stability for nonlinear, high dimensional, and multiconnected systems.
I.
INTRODUCTION
Growing consumer power demands and advances in DC electric distribution systems have resulted in increasing interest in control strategies for parallel-connected DC converters. Implementing these converters in power distribution systems promises to fulfill the most important goals of autonomous power system design, including greater system flexibility and reliability, simpler fabrication, testing, and assembly; and minimized size, weight, operational requirements, and overall cost [1]. However, despite the seeming simplicity of DC systems the nonlinearity and multiconnectivity of the controlled system can sometimes result in chaotic behavior [2]. High dimensionality further complicates system analysis. As a result, most classical control design techniques are not sufficiently capable for this class of autonomous DC power systems. The problems of DC system control design have been addressed by such techniques as feedback linearization, sliding mode, and synergetic controls. Application of the feedback linearization technique presented in [3] is constrained by the existence of the diffeomorphic transformation and requires additional effort in order to incorporate integral action to compensate for changes in the
system operating conditions. Sliding mode control allows the incorporation of integral action into nonlinear control design, but it is hard to implement due to the requirement for fairly high controller bandwidth. The sliding control application described in [4] addresses this issue by combining sliding mode control with Lyapunov function based control using a state-space averaged model. This combination solves the problem of controller bandwidth, but requires implementation of both control algorithms and implementation of mechanisms for switching from one control strategy to another within each period of the converter switching frequency. On the other hand, the application of synergetic control, as shown by application in [5], is also able to account for nonlinearity and can incorporate integral action into the control law but requires neither a high bandwidth controller nor switching between strategies. Thus, synergetic control theory opens new horizons for efficient design of controls for nonlinear systems. Synergetic control theory [6], based on ideas of modern mathematics and synergetics, uses the capability of open systems for self-organization [7]. Synergetic control is based on dynamic control of the interaction of energy and information within the system. By introducing virtual dissipative structures [8], it induces directed self-organized motion within the system. This motion takes control of the system degrees of freedom and leads the system to a specified state. As a result, systems such as parallelconnected power converters work coherently (in unison), supplying the required amount of energy with improved quality. Thus, coordinated behavior improves the system performance and reliability. In short, utilizing dissipative structure algorithms, synergetic control theory supplies an analytical control design procedure for nonlinear, multi-connected, high dimensional systems, which includes the essential properties of the controlled plant within the task statement. As a result, synergetic control theory allows designers to state and then efficiently solve many control problems, heretofore neither solved by other known methods nor even stated due to their extended complexity. These problems relate not only, for example, to global stability of the closed loop operation or global optimization of the system behavior but also to simplifying transitions among power sharing strategies, or
minimizing system power losses. The synergetic control design procedure [6] is based on a new method for generating control laws u(ψ)=u(x) or feedback as functions of the specified macro-variables ψ(x). These controls direct the system from an arbitrary initial condition into the vicinity of manifolds ψ(x)=0 and then ensure asymptotically stable motion along these manifolds toward the end attractors or equilibrium state. On these attractors the specified properties of the controlled system are guaranteed. In the current study, this procedure is applied to a system of m parallel-connected DC buck converters. Employing a simplified state-space averaged model of the system, the paper first presents the design of a general control algorithm. Then, for two particular examples: two and three parallel-connected converter systems, the capabilities of synergetic control for power sharing are explored and the design considerations for macro-variables are discussed. The simulations performed in VTB, a freely available modeling environment with an extended library of power electronic components [9], show the strong performance of the designed control even beyond the validity limits of the system model that was used in the control design, for example in discontinuous conduction mode. II.
SYNERGETIC CONTROL DESIGN
A. System model The prototype system of paralleled buck converters presented in Fig. 1 is the focus for this research.
The primary goal of control design for the system (1) is to maintain a specified output voltage Vc1,ref (2) while the system is affected by a disturbance. The designed control laws must ensure asymptotic stability of closed loop operation. dM dt = η ⋅ (v c1 − V c1, ref ) ν c1 1 m dν c1 dt = C ∑ i Lj − R C + δ ⋅ M ext t t j =1 di L1 d c1 E 1 = − ν c1 + L1 L1 dt ... di Lm = − 1 ν + d cm E c1 dt Lm Lm
(1)
where dci is the desired switch duty of the ith converter, M(t) is the disturbance influencing the system, and δ is a coefficient that will be determined later based on stability of the closed loop system. vc1 = Vc1,ref
(2) The essence of the extended model (1) is that due to the introduced additional state variable, the system is capable of rejecting piece-wise constant disturbances and obtaining zero steady state error. Next, coordinate transformation (3) is performed: v = ν c1 − Vc1,ref i1 = iL1 i2 = iL 2 ... im = iLm
(3)
This transformation accounts for control design requirements by moving the system origin of coordinates into the output voltage equilibrium point. Later on, it also allows us to focus our attention on current sharing. The resulting system model is presented in (4).
Fig. 1. System containing m parallel-connected buck converters
Assuming that the buck converters operate in continuous conduction mode and that switching occurs at a very high frequency, sources Ei are the same, that a piece-wise constant disturbance sufficiently approximates system load and parameter changes, ignoring parasitic effects, and using the averaged capacitor voltage vc1 and inductor currents iLi as state variables, the extended state space averaged system model is shown in (1) where function M(t) represents a disturbance affecting the system.
M ′ = η ⋅ v m v ′ = 1 i − v + Vc1,ref + δ ⋅ M ∑j Ct j =1 Rext Ct 1 ′ i1 = − L ⋅ (v + Vc1,ref ) + u1 1 i ′ = − 1 ⋅ (v + V c1,ref ) + u 2 2 L2 ... 1 im′ = − ⋅ (v + Vc1,ref ) + um Lm
(4)
where ui=E/Li*dci Thus, the extension of the system model and coordinate transformation allow us to incorporate requirements of the control design and the model of the referenced disturbance into the system model. B. Design a general control algorithm A basic tenet of synergetic control design is the use of
macro-variables to define laws of interaction among the system components. These macro-variables define the properties of motion of the extended system (4) to the equilibrium state. For simplicity, in our case these macrovariables (5) are chosen to be linear combinations of the system variables (4). m
ψ i = ai1 ⋅ M + ai 2 ⋅ v + ∑ ai , j +2i j i = 1, m j =1
or
(5)
Ψ = AX − B
where
Ψ = (ψ 1 ,ψ 2 ,...ψ m )
T
a13 a A = 23 ... a m3
a14 a24 ... am 4
... a1n ... a2 n ... ... ... amn
T X = (i1 , i 2 ,...i m ) , bi = −a i1 M − a i 2 v,
i = 1, m
The number of macro-variables in the set (5) equals the number of control channels. The coefficients aij are chosen such that conditions (6) are satisfied. (6) det A ≠ 0. According to [6], under synergetic control (7) macrovariables (5) become manifolds (8). u( x ) = u(ψ ) (7) (8) Ψ = AX − B = 0 The evolution of the macro-variables towards the manifolds (8) can be defined in various ways. The possibility shown in (9) is the equation used for control design in this paper. Tiψ i′ = −ψ i i = 1, m Ti > 0 (9) The control defined by (9) moves the representing point of the system towards and along the manifolds (8) to the specified equilibrium point (2). The expression for controls is found by substituting the system model (4) into (9) and solving the resulting system jointly, yielding the controls. (10) U = A−1 ⋅ D where di = −
ψi Ti
m a ai 2 ⋅ (v + Vc1, ref ) + + ∑ ij + 2 + j =1 L R j ext ⋅ Ct
m i − η ⋅ ai1 ⋅ v − ai 2 ⋅ ∑ j + δ ⋅ M j = 1 Ct
(11)
i = 1, m
Substituting the expression for di from (11) into (10) gives: m
ui = ∑ A−1 (i, j ) ⋅ d j ,
i = 1, m
j =1 (12) where A-1(i,j) is the i, jth element of inverted matrix A The control laws (12) ensure asymptotically stable convergence towards the specified manifolds (8). The speed of convergence is regulated by value of Ti. Moreover, the synergetic controls (12) move the system-representing point onto the intersection of the manifolds (8). As a result, after the transients are finished equalities (8) are satisfied.
Equations (8) show that the state variables are no longer independent. As a result, due to contraction of state space, the system (4) is now of reduced order, which equals: (13) d =n−m where n is the order of the given system (4) and m stands for the number of control channels. In our case, the dimension of the reduced order system equals two. Thus, the result of the synergetic control design procedure is an analytical control law that ensurs asymptotically stable motion towards the intersection of the introduced manifolds (8). According to [6], the stability of the system at the manifold intersection and the equilibrium point can be checked by any appropriate method [10]. C. Stability conditions Once the representing point arrives on the manifold intersection, the equalities (8) are satisfied and the system has n-m independent variables defining its behavior. To define the dynamics of the closed system on the manifold intersection, the equations (8) are represented as an algebraic system: (14) A⋅ X = B As long as the conditions (6) are satisfied, the solution of the system (14) is as follows: (15) X = A −1 ⋅ B Thus, the closed loop system dynamics on the intersection of the manifolds based on the solution of (15) is as follows: M ψ′ = η ⋅ vψ m m vψ + V c1, ref −1 vψ′ = − R C − ∑∑ A ( j, i ) ⋅ (a i1 M ψ + a i 2 vψ ) + δ ⋅ M ψ j =1 i =1 ext t
(16)
To define the stability conditions, the system (16) is rearranged into one equation: vψ′′ + F1 ⋅ vψ′ + F2 ⋅ vψ = 0 (17) where F1 =
m m m m 1 + ∑∑ A −1 ( j, i ) ⋅ a i 2 , F2 = η ⋅ ∑∑ A −1 ( j, i ) ⋅ a i1 − δ R ext C t j =1 i =1 j i = 1 = 1
From (17) it follows that the stability conditions for the closed loop system on the intersection of the manifolds are: F1 > 0, F2 > 0, Ti > 0, i = 1, m (18) It can be seen from (17) and (18) that positive value of parameter δ makes stability conditions more strict. Hence, δ=1 is used in stability check of the closed loop system. In order to highlight the internal properties of the system motion along the manifolds, equation (17) is written in the following form: vψ′ = − F1 ⋅ vψ − F2 ⋅ ∫ vψ dt = 0
(19)
Equation (19) shows that on the intersection of the manifolds (8) a PI control law for the coordinate v is introduced. Thus, according to (18) and (19), the synthesized control laws (12), besides ensuring asymptotic stability, also suppress the piece-wise constant disturbance M(t) influencing the converters.
III.
RESULTS
The derived general algorithm has been applied to twoconverter and three-converter systems. The resulting control laws and stability conditions are presented in the appendix. Several simulations have been performed on these systems. The nominal values of system parameters are presented in Table 1.
transformation (3). To achieve the specified (22) current sharing in the system, the manifolds have this structure: 1 1 − a M − a12 v ⋅ X = 11 0 1 − 1
(23)
Corresponding to the particular choice of manifolds (23), the system diagram is shown in Fig. 3 a.
TABLE 1 NOMINAL VALUE OF SYSTEM PARAMETERS
Parameter Nominal Value rL1=rL2=rLn 0.1 Ohm L1=L2=Ln 150 µH C1=C2=Cn 2200 µF VC,ref=U0 5.0 V Switching frequency 50 kHz T1=T2=Tn 0.001 η= ηn 20000 a11=a11n 1000 a12=a12n 300 δ 1 The input voltage varies between 12-25V. The output voltage is regulated at 5.0V. The objectives for simulation are to explore current sharing and the effect of macrovariable coefficients on the dynamic response of the closed loop system.
Fig. 2. Structure of the system feedback on the manifolds
a)
B. Current Sharing on the Manifolds 1) Two-converter system Directed by synergetic control laws (12), the representing point of the two-converter system inevitably reaches the intersection of the manifolds (8) after (3-5)•T. On the manifolds the motion of the system is defined by the following equations: a13 a14 − a M − a12 v ⋅ X = 11 (20) a 23 a 24 − a 21 M − a 22 v The system currents calculated from (20) using (15) are functions of output voltage error. Moreover, coefficients of matrices A and B in (15) define the properties of current sharing among converters on the manifolds. In systems with two parallel-connected converters the only basic current sharing regimes possible are master-slave and democratic. When the system is on the manifolds either of these can be achieved by a particular choice of the coefficients in (20). Equations (20) define the structure of the feedback influencing the system on the manifolds, which is presented in Fig. 2. Master-slave current sharing occurs when the master converter maintains output voltage and all other converters supply current according to the assigned rating [11], usually, defined for the slave converter by the master. The conditions for master-slave current sharing are summarized in the following mathematical expressions: v = Vc1,ref (21) i1 = i2 (22) Requirement (21) has been accounted for by coordinate
x0
b) x0
Equilibrium x0
Manifold 2
Manifold 1
x0
Fig. 3. System under master-slave current sharing: a) feedback structure b) state portrait (x0 are various initial conditions, arrows on trajectories show direction of the system motion)
Fig. 3 b shows the motion of the system representing point, which starts from various initial conditions x0. Influenced by the synergetic control it moves along the integral curves (24).
m
ψ = ∑ a 2, j + 2i j + C1
j =1 (24) where C1 is a constant corresponding to the initial conditions. The curves described by (24) are solutions of the differential equation (25) and lie parallel to manifold_2 in Fig. 3: T ⋅ψ ′ = −ψ (25) Following the solution (24) the representing point reaches manifold_1 in Fig. 3 b and moves along to the equilibrium state, where the static current sharing is defined by (22). As a result, on manifolds (23) the system exhibits equal dynamic and static current sharing. On the other hand, democratic current sharing occurs when converters provide current according to their current rating (22) and each converter accounts for the output voltage error [12]. The specifications are the same as for master-slave sharing (21) and (22). The structure of the manifolds is shown in (26).
− a M − a12 v 1 1 ⋅ X = 11 − a 22 v 1 1 −
(26)
a)
same pattern as for master-slave current sharing. However, integral curve (27) differs from (24) in that it depends on output voltage error. This dependence allows change of the system current sharing. m
ψ = a22 ⋅ v + ∑ ai , j + 2i j + C2
j =1 (27) where C2 is a constant depending on the initial conditions. During the transition to the equilibrium, coefficient a22 influences the rate of change of converter currents. Since the output voltage error equals zero at equilibrium, it does not affect the static current sharing in the system. Thus, the major differences between master-slave and democratic current sharing are that master-slave sharing provides constant dynamic and static current sharing, while democratic current sharing can have varying dynamic and static current sharing. In summary, for the system containing two converters, democratic and master-slave are the only possible current sharing methods. The structure of manifolds (20) defines the regime of current sharing in the system. Moreover, the transition among regimes can be made by a simple change of coefficients in the manifold equations (20). 2) Three-converter system The behavior of the closed loop system with three converters is similar to that of the two-converter system explained above, except that any combination of democratic and master-slave current sharing can be applied. A few possible combinations are presented in Table 2
TABLE 2 A FEW POSSIBLE SYNERGETIC CONTROL STRATEGIES FOR THE THREE-CONVERTER SYSTEM
Control Strategy Master-slave
Democratic
x0
b) x0
Equilibrium x0
Manifold 2
Manifold structure 1 1 1 − a11 M − a12 v 0 1 − 1 0 ⋅ I = 1 0 − 1 0
1 − a 11 M − a 12 v 1 1 − a 22 v 1 − 1 0 ⋅ I = 1 0 − 1 − a v 13
Two masters with one 1 1 1 − a 11 M − a 12 v − a 22 v slave 1 − 1 0 ⋅ I = 1
x0 Manifold 1
Fig. 4. System under democratic current sharing: a) feedback structure b) state portrait (x0 are various initial conditions and arrows on trajectories show direction of the system motion)
The system diagram reflecting the structure of feedback, when the system is on the manifold intersection (26), is shown in Fig. 4 a. The difference between feedback structures (23) and (26) corresponds to the use of output voltage error in the second converter. The motion of the system representing point follows the
0
− 1
0
The control strategies presented in Table 2 provide various static and dynamic current sharing capabilities. As shown in the previous section, the master-slave strategy provides equal static and dynamic current sharing and the democratic strategy allows the two to be different. The twomaster with one slave strategy represents a mixture of the two basic methods. Table 2 does not present all of the possibilities; however it shows that by changing coefficients in the manifold equations it is possible to choose different current sharing methods, even dynamically. Thus, a greater number of converters in the system yields more flexibility in current sharing. The structure of manifolds (8) defines the regime of current sharing in the
system and, if converters are fed from different sources, dynamic current sharing provides a way to regulate the dynamic contributions of the sources according to their properties. Moreover, the transition among regimes can be made by a simple change of coefficients in the manifold equations (8). In this way, certain coefficients of the manifolds (8) define static or dynamic current sharing properties of the system. C. Dynamic response and manifold coefficients In addition to the relations among the structure of manifolds and the properties of current sharing explained in the previous section, a key concept of synergetic control design is that manifolds define the closed loop system behavior. Specifically, the manifold coefficients Ti, a11, and a12 also affect the system transients. The correspondence between these coefficients and the transient response of the closed loop system is shown by a simulation of the twoconverter system, when the load resistance is changed from 2 to 1 Ω. While a particular coefficient is varied the remaining parameters are held equal to their nominal values which are presented in Table 1 and are referred later on in figures by subscript n. Fig. 5 shows how varying parameters T1 and T2 by 70% from their nominal values Tn affect the transient response. Fig. 6 and Fig. 7 illustrate the system transients for a11 and a12 respectively, to highlight the influence on the system transients the parameter a11 is changed by 60% and a12 is varied by 50%.
a)
a)
b)
Fig. 7. Impact of a12 on transient response of the two-converter system a)output voltage error b) inductor currents (1:a12=1.5a12n, 2:a12=a12n, and 3:a12=0.5a12n)
Control law parameters T, a11 and a12 affect the transient response of the closed loop system and allow designer not only to achieve specified performance but also to limit rate of increase of the input current withdrawn from the bus or energy source by proper choice of parameters T. D. Simulation of the closed loop system The closed loop system was modeled in VTB [9] using the averaged model of the buck converter accounting for Discontinuous Conduction Mode (DCM) operation. The goal of the simulation experiment was to check the performance of the system under varying load conditions and to inspect the capability of the synergetic control to withstand discontinuities in operation of the buck converters. The results of the simulation are presented in Fig. 8, which shows dynamic and steady state performances of two-converter system when parameters of the system equal to their nominal values and load resistance changes from 10 kΩ (DCM operation) to 1 Ω (full load).
b)
Fig. 5. Impact of T on transient response of the two-converter system a)output voltage error b) inductor currents (1:T1=T2=0.3Tn, 2:T1=T2=Tn, and 3:T1=T2=1.7Tn)
a)
to current overshoots. Thus, due to the complex interconnection among the coefficients and the system settling time and transient response, a compromise needs to be found.
b)
Fig. 6 Impact of a11 on transient response of the two-converter system a)output voltage error b) inductor currents( 1:a11=0.4a11n, 2:a11=a11n, and 3:a11=1.6a11n)
Results indicate that these parameters have similar impacts on the system transients with some particularities. Parameters T define the speed of convergence of the macrovariables (5) towards the manifolds (8). Lowering those parameters improves the settling time of the system. While the shape of the voltage transients is preserved, passing a particular threshold for T causes current overshoots. On the other hand, varying a11 has an impact on both the shape of the transients and the settling time. In contrast to a11, the decrease of parameter a12 improves settling time and leads
a)
c)
b)
d)
Fig. 8. Transients of the two-converter system a) and c) output voltage error b) and d) converters currents a) and b) values of the inductances equal to Ln c) and d) values of inductances are L1=1.1Ln (line 1) and L2=0.9Ln (line 2)
The output steady state voltage in DCM has a periodic form which amplitude is within 3% of nominal voltage. The oscillation of the output voltage corresponds to the inadequacy of the models to predict system behavior in DCM operation. However, by incorporating integral action into synergetic control law the closed loop system can maintain output voltage within specified limits. Moreover, the magnitude of output voltage oscillations can be controlled by the value of time constant T. By decreasing this parameter the specified accuracy of output voltage and
current sharing in DCM can be achieved. From Fig. 8 it follows that two-converter system under synergetic control exhibits an ability to nullify output voltage and current sharing errors and show the strong performance even beyond the validity limits of the system model used in the control design which correspond to operation of the converters in discontinuous conduction mode. IV.
CONCLUSIONS
APPENDIX B. Vector controls for two and three-converter system Substitution of m=2 and m=3 into equations (12) gives the vector controls for two and three-converter systems, which respectively are (27) and (32). 1 1 ⋅ (a24 d1 − a23d 2 ) u2 = ⋅ (− a14 d1 + a13d 2 ) det A det A where : det A = a13 ⋅ a24 − a23 ⋅ a14
u1 =
ψ1
d2 = −
T1
a a a + 13 + 14 + 12 ⋅ (v + Vc1, ref ) − η ⋅ a11 ⋅ v − a12 ⋅ k L1 L2 Rext Ct
− a 23 ⋅ a14 ⋅ a 35 − a 25 ⋅ a 34 ⋅ a13 − a 33 ⋅ a 24 ⋅ a15 d1 = − d2 = − d3 = −
A robust control algorithm for m-parallel-connected buck converters has been presented together with important considerations for manifold design. This algorithm, derived using synergetic control theory, includes the capability to account for nonlinear properties of the controlled system, to suppress unmeasured piece-wise disturbances and ensures asymptotically stable motion toward the intersection of the manifolds. It also defines the stability conditions for motion along the intersection to the equilibrium state. The algorithm provides such benefits as nullification of output voltage and current sharing errors by incorporating integral action into the controls and design simplicity based on the independence of manifolds. It also allows system transition among current sharing methods by a simple change of control law coefficients. In addition, the closed loop system shows a robust transient response despite variations of system parameters. Using the algorithm will help to improve power distribution systems particularly with regard to flexibility and reliability.
d1 = −
1 ⋅ ((a 24 a 35 − a 34 a 25 ) ⋅ d 1 + (a 33 a 25 − a 23 a 35 ) ⋅ d 2 + (a 23 a 34 − a 33 a 24 ) ⋅ d 3 ) det A 1 ⋅ ((a 34 a15 − a14 a 35 ) ⋅ d 1 + (a13 a 35 − a 33 a15 ) ⋅ d 2 + (a 33 a14 − a13 a 34 ) ⋅ d 3 ) u2 = det A 1 ⋅ ((a14 a 25 − a 24 a15 ) ⋅ d 1 + (a 23 a15 − a13 a 25 ) ⋅ d 2 + (a13 a 24 − a 23 a14 ) ⋅ d 3 ) u3 = det A where : det A = a13 ⋅ a 24 ⋅ a 35 + a14 ⋅ a 25 ⋅ a 33 + a 23 ⋅ a 34 ⋅ a15
u1 =
(28)
ψ2
a a a + 23 + 24 + 22 ⋅ (v + Vc1, ref ) − η ⋅ a21 ⋅ v − a22 ⋅ k T2 L1 L2 Rext Ct
i i k = 1 + 2 + δ ⋅ M Ct Ct
C. Stability conditions The systems behavior on the manifolds is defined by equation (17). The coefficients F1 and F2 corresponding to two and three-converter system respectively presented in (30) and (31) are found from (17) using m=2 and m=3. If the conditions (18) are satisfied, the synthesized vector control laws will ensure asymptotic stability of the closedloop system and suppress the uncontrolled piecewiseconstant disturbance M (t ) = const .
ψ1
a a a a + 13 + 14 + 15 + 12 T1 L1 L2 L3 Rext C t
ψ2 T2
ψ3 T3
F1 =
i i i ⋅ (v + Vc1,ref ) − η ⋅ a11 ⋅ v − a12 1 + 2 + 3 + δ ⋅ M Ct Ct Ct
a a a a + 23 + 24 + 25 + 22 L2 L3 Rext C t L1
i i i ⋅ (v + Vc1,ref ) − η ⋅ a 21 ⋅ v − a 22 1 + 2 + 3 + δ ⋅ M Ct Ct Ct
a a a a + 33 + 34 + 35 + 32 L2 L3 Rext C t L1
i i i ⋅ (v + Vc1,ref ) − η ⋅ a 31 ⋅ v − a 32 1 + 2 + 3 + δ ⋅ M Ct Ct Ct
a12 ⋅ (a 24 − a14 ) + a 22 ⋅ (a13 − a 23 ) 1 + Rext Ct det A
a ⋅ (a − a14 ) + a 21 ⋅ (a13 − a 23 ) − δ ⋅η F2 = 11 24 det A F1 =
(a24 a35 − a34 a25 + a34 a15 − a14 a35 + a14 a25 − a 24 a15 ) ⋅ a12 + 1 1 ⋅ (a33a 25 − a23a35 + a13a35 − a33a15 + a23a15 − a13a25 ) ⋅ a22 + + det A Rext Ct (a23a34 − a33a24 + a33a14 − a13a34 + a13a24 − a23a14 ) ⋅ a32
(a a − a34 a25 + a34 a15 − a14 a35 + a14 a25 − a 24 a15 ) ⋅ a11 + 1 24 35 F2 = ⋅ (a33a25 − a23a35 + a13a35 − a33a15 + a23a15 − a13a25 ) ⋅ a21 + − δ ⋅ η det A (a a − a a + a a − a a + a a − a a ) ⋅ a 33 24 33 14 13 34 13 24 23 14 31 23 34
(29)
(30)
(31)
ACKNOWLEDGMENT The authors acknowledge the support of ONR grants N00014-02-1-0623, N00014-03-1-0434. REFERENCES [1]
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