A Geometric Approach to Large-Signal Stability of ... - Semantic Scholar
A Geometric Approach to Large-Signal Stability of Switching Converters under Sliding Mode Control and Synergetic Control E. Santi, D. Li, A. Monti, A. M. Stanković # #
Department of Electrical Engineering University of South Carolina Columbia, SC 29208 [email protected]
Department of Electrical Engineering Northeastern University Boston, MA
Abstract - The objective is to study the large-signal stability of switching converters operating in sliding mode or under synergetic control. Under both control approaches, the system state reaches a desired control manifold and then stays on that manifold at all times. The question of stability on the control manifold is examined, i.e., whether the system will converge to the desired steady-state point. The proposed geometric method allows the determination of the direction of evolution at any point on a control manifold, therefore providing large-signal stability information. Stability conditions with a clear geometric interpretation can be established for any point on a control manifold. In particular, if the method is applied to the steady-state operating point, it yields stability conditions that exactly match conditions obtained from a small-signal linearized analysis. This validates and demonstrates the power of the proposed approach. The method is fairly general and it can be applied to any second-order converter. The paper discusses the cases of Buck, Boost and Buck-Boost converters under resistive load and under constant power load, a case of particular interest in multi-converter systems.
I.
II.
PROBLEM STATEMENT
Let us consider a second-order converter such as a Buck, Boost, or Buck-Boost converter. The state of the converter is given by the vector
i x= (1) v where i is the inductor current and v is the capacitor voltage, which is also the output voltage. The converter averaged model has the form
x& = A(d ) x + B(d )Vg where d is the duty cycle and
(2)
Vg is the input voltage.
The system is controlled either in sliding mode [1] or using synergetic control [2]. In both cases the control forces the system to operate on a desired control manifold. For simplicity a linear control manifold is selected, even if the described stability method is general and can be applied to nonlinear manifolds as well. A linear control manifold containing a desired steady-state point
INTRODUCTION
The objective is to study the large-signal stability of switching converters operating in sliding mode or under synergetic control. Under both control approaches, the system state reaches a desired control manifold and then stays on that manifold at all times. The question of stability on the control manifold is examined, i.e., whether the system will converge to the desired steady-state point. The proposed geometric method allows the determination of the direction of evolution at any point on a control manifold, therefore providing large-signal stability information. Stability conditions with a clear geometric interpretation can be established for any point on a control manifold. In particular, if the method is applied to the steady-state operating point, it yields stability conditions that exactly match conditions obtained from a small-signal linearized analysis. This validates and demonstrates the power of the proposed approach. The method is fairly general and it can be applied to any second-order converter. The paper discusses the cases of Buck, Boost and Buck-Boost converters under resistive load and under constant power load, a case of particular interest in multi-converter systems.
I ref X ref = Vref
(3)
is given by
ψ = K T ( x − X ref ) = 0
(4)
with
K T = [k 1]
(5) The design parameter to be chosen is the value of the constant k , which determines the slope of the manifold in the state plane. The state plane with a control manifold ψ = 0 is shown in Fig. 1. Once a manifold has been selected, three questions arise: 1. Reachability, i.e., whether the system starting from a generic point will eventually reach the manifold. 2. Existence of a control that keeps the system on the manifold. In the case of sliding mode control this is the question of existence of a sliding mode on the manifold.
1
3.
Stability of the reduced-order system evolution on the manifold, i.e., whether the system moving on the manifold will move towards the desired steadystate point X ref or not.
Notice that, if such a function can be found,
depend on duty cycle. This gives information on the system evolution as shown in Fig. 2. In the region g ( x ) > 0
i
function F (x ) must increase and in the region g ( x) < 0
X ref
I ref
slope −
function F (x) must decrease. Given a control manifold, we can verify whether movement on the manifold will converge to the desired steady-state point X ref or not.
1 k
The case of a Boost converter is now examined to illustrate the method. As shown in Fig. 2, two different types of load will be considered: resistive load and constant power load.
i
Fig. 1 State plane with control manifold ψ
= 0.
0 < d eq ( x) < 1
C
R
i
v -
(a) L
+ i=Pout/v
Vg
C
(b)
v -
Fig. 2 Boost converter with resistive load (a) and constant power load (b).
A. Resistive Load Case Consider a Boost converter with resistive load. The converter averaged model is given by
(6)
a sliding mode locally exists. This result provides a method to check for the existence of a sliding mode. It can be shown that the existence of a sliding mode on a manifold implies that a synergetic control exists that keeps the system on the same manifold.
di = Vg − v(1 − d ) dt dv v = i (1 − d ) − C dt R L
STABILITY METHOD
(8)
Let us apply the procedure.
The stability method consists of the following steps:
Step 1. The locus of steady-state operating points is obtained by setting all derivatives in (8) equal to zero. Eliminating the duty cycle between the two equations we obtain
Identify the locus of steady-state operating points for the converter. This identifies a curve g ( x ) = 0 in the state plane parameterized by duty cycle d . Find a function F (x) such that its directional derivative along system trajectories is
dF ∂F = • x& = b ⋅ g (x) dt ∂x where b is a constant.
+
Vg
This work focuses mostly on question 3. The system is assumed to move on the manifold and the question of stability of the reduced-order system on the manifold is examined. Around the steady-state operating point local stability can be established by studying the linearized system as in [3-4]. Comparison with the large-signal method developed here will provide a verification of the proposed large-signal method. In the following we will assume that condition 2 is verified, and therefore a sliding mode exists on the manifold. It has been shown [5] that if the sliding mode equivalent control satisfies the condition
III.
L
v
Vref
2.
BOOST CONVERTER CASE
IV.
ψ =0
1.
dF does not dt
g ( x) = g (i, v) = i −
1 2 v =0 RVg
This equation represents a parabola in the state plane.
(7)
Step 2. Let us choose
2
(9)
F ( x) = F (i, v) =
1 2 1 2 Li + Cv 2 2
i
(10)
g(i,v) = 0
Let us verify that this equation satisfies (7).
dF ∂F di ∂F dv di dv = + = Li + Cv dt ∂i dt ∂v dt dt dt
(11)
Substituting (8)
[
]
dF v = i Vg − v(1 − d ) + v i (1 − d ) − dt R
C
(12)
D
which simplifies to
v2 dF = Vg i − = Vg ⋅ g (i, v) R dt
(13)
Therefore, equation (7) is satisfied with
v
b = Vg . Notice
Fig. 4 State plane showing possible movement directions for various duty cycles at two different points.
how duty cycle cancels out in equation (12).
i
i
g(i,v) = 0
i
ψ=0
g(i,v) > 0 F(i,v)
Xref D ψ=0
C g(i,v) < 0
Xref
F(i,v)
STABLE
v
1 2 1 2 Li + Cv = Const 2 2
v
(b)
Fig. 5 Two choices of control manifold ψ
F (i, v) .
=0
leading to stable and
unstable behavior, respectively.
The situation is shown in Fig. 3. The parabola g (i, v) = 0 divides the state plane in two regions. In one region g (i, v) > 0 and therefore F (i, v) increases with time. In the other region the opposite happens. Notice that the curves
F (i, v) =
UNSTABLE
(a)
Fig. 3 State plane showing steady-state locus and regions of increasing and decreasing
v
shows how stability of a given manifold can be established by examining the state plane. Given a control manifold ψ = 0 of the form (4), the intersection with the
Fig. 5
g (i, v) = 0 identifies the steady-state operating point X ref = ( I ref ,Vref ) . We assume that a steady-state locus
(14)
are ellipses in the state plane. Fig. 4 shows two generic points C and D with the corresponding ellipses (14) going through them. The vectors coming out of the two points represent trajectories for different values of the control (duty cycle). The interpretation is that at any point C such that g (i, v) > 0 the trajectories do depend on the control action (duty cycle), but all trajectories move towards increasing F (i, v) , i.e., all trajectories move towards larger ellipses.
sliding mode exists on the manifold. This can be easily checked using condition (6). The question of stability on the manifold is whether the system trajectory on the manifold will converge to the steady-state point or will move away from it. To establish stability around the steady-state point, the F (i, v) = Const curve that goes through point X ref is
g (i, v) < 0 trajectories move
g (i, v) > 0 region and movement along the manifold must be in the direction of increasing F (i, v) , and therefore towards steady-state point X ref . On the other hand, point D is in the g (i, v) < 0 region, so movement must result in a
At any point D such that towards decreasing ellipses.
drawn in Fig. 5. As shown in Fig. 5a, let us consider two points C and D in a neighborhood of point X ref . Point C is in the
F (i, v) and therefore towards smaller
3
decrease of point
the
F (i, v) and therefore it is towards steady-state
manifold
is
stable
as
long
as
the
curves
F (i, v) = Const are steeper than the manifold curve, as
X ref . We can conclude that in the case of Fig. 5a the
shown in Fig. 7. Notice also that the manifold does not need to be linear in order to apply the method.
manifold is stable. Using a similar argument we can show that in the case of Fig. 5b the manifold is unstable, because the system will move away from steady-state point X ref .
i
From the argument above we can derive stability limits at point X ref . In order to be in the stable case of Fig. 5a and not in the unstable case of Fig. 5b, the manifold must belong to the stable sector shown in Fig. 6. The condition is that the slope of the manifold, which is equal to − 1 / k , must be larger than the slope of the curve F (i, v) = const going through point curve
ψ=0
X ref , and it must be smaller than the slope of
g (i, v) = 0 at point X ref . These are the slopes of
tangent lines S F and
S g , respectively, as shown in Fig. 6.
STABLE
Therefore, the stability conditions are
Slope( S F ) < −
1 < Slope( S g ) k
Fig. 7 Large-signal stable manifold.
(15)
B. Energy Interpretation For the Boost converter case there is an interesting interpretation of the method in terms of stored energy and power balance. Notice that multiplying equation (9) by Vg we obtain that
It is straightforward to show that the conditions are
−
C Vref 1 2V < − < ref L I ref k RVg
(16)
Notice that these are necessary and sufficient conditions for stability at point X ref , therefore they are tight. It will be
Vg ⋅ g (i, v) is equal to the difference between input and output power (17).
shown that a small-signal stability analysis gives exactly the same stability conditions as (16), validating the proposed method.
i F(i,v) =const
v2 Vg ⋅ g (i, v) = Vg ⋅ i − = Pin − Pout = 0 R i Pin=Pout
g(i,v) = 0 Xref Stable sector
(17)
Pin>Pout
Stable sector Sg
v
Stored energy
SF v
Pin Pout the stored energy F (i, v) must increase, and in
proposed method can be applied at any point of the manifold. Given any point on the manifold, the geometric argument establishes the direction of evolution with time, providing large-signal stability information. For example, in the case of a manifold with negative slope (the more interesting case, because it gives faster response), motion on
the region Pin
< Pout it must decrease. This interpretation
applies only to the Boost case, and not to the Buck and
4
xˆ& = Axˆ + Bd dˆ + Bvˆg
Buck-Boost cases discussed later in the paper, therefore the proposed stability analysis method is more general.
Differentiating the manifold equation (4) and substituting the linearized system equations (22) one obtains for vˆg = 0
C. Constant Power Load Case The averaged equations for the converter of Fig. 2b are
(
dˆ = − K T Bd
di = Vg − v(1 − d ) dt dv P C = i (1 − d ) − out dt v L
[
This identifies a horizontal line in the state plane. Step 2. Choosing the same F (i, v ) function as before (10), it can be easily shown that (7) is satisfied (20)
shows the two regions identified by the steady-state locus g (i, v ) = 0 . Fig. 9b shows a stable manifold. The
(21)
λ=−
As expected, this condition is more restrictive than in the resistive case (16).
i
g(i,v)=0 F(i,v) (a)
ψ=0 v
STABLE
]
K T A xˆ
(24)
(25)
(26)
kCVref − LI ref
OTHER CONVERTERS
The proposed geometric stability method has been discussed in detail for the case of a Boost converter. The cases of resistive load and constant power load have been examined. The method applies to Buck and Buck-Boost converters as well and the derivations reported above for the Boost converter can be repeated for these other converters as well. Table 1 lists system equations, functions F (i, v) and
Xref
g(i,v) < 0
−1
Vg
V.
F(i,v)
)
The stability conditions on k coincide with conditions (21). This constitutes a validation of the proposed stability approach.
i g(i,v) > 0
Pout Vg
(
It can be easily shown that the conditions on k under which this eigenvalue is negative are identical to (16). A similar small-signal analysis for the constant power load case gives an eigenvalue
stability condition at steady-state point X ref is
C Vref 1
Department of Electrical Engineering University of South Carolina Columbia, SC 29208 [email protected]
Department of Electrical Engineering Northeastern University Boston, MA
Abstract - The objective is to study the large-signal stability of switching converters operating in sliding mode or under synergetic control. Under both control approaches, the system state reaches a desired control manifold and then stays on that manifold at all times. The question of stability on the control manifold is examined, i.e., whether the system will converge to the desired steady-state point. The proposed geometric method allows the determination of the direction of evolution at any point on a control manifold, therefore providing large-signal stability information. Stability conditions with a clear geometric interpretation can be established for any point on a control manifold. In particular, if the method is applied to the steady-state operating point, it yields stability conditions that exactly match conditions obtained from a small-signal linearized analysis. This validates and demonstrates the power of the proposed approach. The method is fairly general and it can be applied to any second-order converter. The paper discusses the cases of Buck, Boost and Buck-Boost converters under resistive load and under constant power load, a case of particular interest in multi-converter systems.
I.
II.
PROBLEM STATEMENT
Let us consider a second-order converter such as a Buck, Boost, or Buck-Boost converter. The state of the converter is given by the vector
i x= (1) v where i is the inductor current and v is the capacitor voltage, which is also the output voltage. The converter averaged model has the form
x& = A(d ) x + B(d )Vg where d is the duty cycle and
(2)
Vg is the input voltage.
The system is controlled either in sliding mode [1] or using synergetic control [2]. In both cases the control forces the system to operate on a desired control manifold. For simplicity a linear control manifold is selected, even if the described stability method is general and can be applied to nonlinear manifolds as well. A linear control manifold containing a desired steady-state point
INTRODUCTION
The objective is to study the large-signal stability of switching converters operating in sliding mode or under synergetic control. Under both control approaches, the system state reaches a desired control manifold and then stays on that manifold at all times. The question of stability on the control manifold is examined, i.e., whether the system will converge to the desired steady-state point. The proposed geometric method allows the determination of the direction of evolution at any point on a control manifold, therefore providing large-signal stability information. Stability conditions with a clear geometric interpretation can be established for any point on a control manifold. In particular, if the method is applied to the steady-state operating point, it yields stability conditions that exactly match conditions obtained from a small-signal linearized analysis. This validates and demonstrates the power of the proposed approach. The method is fairly general and it can be applied to any second-order converter. The paper discusses the cases of Buck, Boost and Buck-Boost converters under resistive load and under constant power load, a case of particular interest in multi-converter systems.
I ref X ref = Vref
(3)
is given by
ψ = K T ( x − X ref ) = 0
(4)
with
K T = [k 1]
(5) The design parameter to be chosen is the value of the constant k , which determines the slope of the manifold in the state plane. The state plane with a control manifold ψ = 0 is shown in Fig. 1. Once a manifold has been selected, three questions arise: 1. Reachability, i.e., whether the system starting from a generic point will eventually reach the manifold. 2. Existence of a control that keeps the system on the manifold. In the case of sliding mode control this is the question of existence of a sliding mode on the manifold.
1
3.
Stability of the reduced-order system evolution on the manifold, i.e., whether the system moving on the manifold will move towards the desired steadystate point X ref or not.
Notice that, if such a function can be found,
depend on duty cycle. This gives information on the system evolution as shown in Fig. 2. In the region g ( x ) > 0
i
function F (x ) must increase and in the region g ( x) < 0
X ref
I ref
slope −
function F (x) must decrease. Given a control manifold, we can verify whether movement on the manifold will converge to the desired steady-state point X ref or not.
1 k
The case of a Boost converter is now examined to illustrate the method. As shown in Fig. 2, two different types of load will be considered: resistive load and constant power load.
i
Fig. 1 State plane with control manifold ψ
= 0.
0 < d eq ( x) < 1
C
R
i
v -
(a) L
+ i=Pout/v
Vg
C
(b)
v -
Fig. 2 Boost converter with resistive load (a) and constant power load (b).
A. Resistive Load Case Consider a Boost converter with resistive load. The converter averaged model is given by
(6)
a sliding mode locally exists. This result provides a method to check for the existence of a sliding mode. It can be shown that the existence of a sliding mode on a manifold implies that a synergetic control exists that keeps the system on the same manifold.
di = Vg − v(1 − d ) dt dv v = i (1 − d ) − C dt R L
STABILITY METHOD
(8)
Let us apply the procedure.
The stability method consists of the following steps:
Step 1. The locus of steady-state operating points is obtained by setting all derivatives in (8) equal to zero. Eliminating the duty cycle between the two equations we obtain
Identify the locus of steady-state operating points for the converter. This identifies a curve g ( x ) = 0 in the state plane parameterized by duty cycle d . Find a function F (x) such that its directional derivative along system trajectories is
dF ∂F = • x& = b ⋅ g (x) dt ∂x where b is a constant.
+
Vg
This work focuses mostly on question 3. The system is assumed to move on the manifold and the question of stability of the reduced-order system on the manifold is examined. Around the steady-state operating point local stability can be established by studying the linearized system as in [3-4]. Comparison with the large-signal method developed here will provide a verification of the proposed large-signal method. In the following we will assume that condition 2 is verified, and therefore a sliding mode exists on the manifold. It has been shown [5] that if the sliding mode equivalent control satisfies the condition
III.
L
v
Vref
2.
BOOST CONVERTER CASE
IV.
ψ =0
1.
dF does not dt
g ( x) = g (i, v) = i −
1 2 v =0 RVg
This equation represents a parabola in the state plane.
(7)
Step 2. Let us choose
2
(9)
F ( x) = F (i, v) =
1 2 1 2 Li + Cv 2 2
i
(10)
g(i,v) = 0
Let us verify that this equation satisfies (7).
dF ∂F di ∂F dv di dv = + = Li + Cv dt ∂i dt ∂v dt dt dt
(11)
Substituting (8)
[
]
dF v = i Vg − v(1 − d ) + v i (1 − d ) − dt R
C
(12)
D
which simplifies to
v2 dF = Vg i − = Vg ⋅ g (i, v) R dt
(13)
Therefore, equation (7) is satisfied with
v
b = Vg . Notice
Fig. 4 State plane showing possible movement directions for various duty cycles at two different points.
how duty cycle cancels out in equation (12).
i
i
g(i,v) = 0
i
ψ=0
g(i,v) > 0 F(i,v)
Xref D ψ=0
C g(i,v) < 0
Xref
F(i,v)
STABLE
v
1 2 1 2 Li + Cv = Const 2 2
v
(b)
Fig. 5 Two choices of control manifold ψ
F (i, v) .
=0
leading to stable and
unstable behavior, respectively.
The situation is shown in Fig. 3. The parabola g (i, v) = 0 divides the state plane in two regions. In one region g (i, v) > 0 and therefore F (i, v) increases with time. In the other region the opposite happens. Notice that the curves
F (i, v) =
UNSTABLE
(a)
Fig. 3 State plane showing steady-state locus and regions of increasing and decreasing
v
shows how stability of a given manifold can be established by examining the state plane. Given a control manifold ψ = 0 of the form (4), the intersection with the
Fig. 5
g (i, v) = 0 identifies the steady-state operating point X ref = ( I ref ,Vref ) . We assume that a steady-state locus
(14)
are ellipses in the state plane. Fig. 4 shows two generic points C and D with the corresponding ellipses (14) going through them. The vectors coming out of the two points represent trajectories for different values of the control (duty cycle). The interpretation is that at any point C such that g (i, v) > 0 the trajectories do depend on the control action (duty cycle), but all trajectories move towards increasing F (i, v) , i.e., all trajectories move towards larger ellipses.
sliding mode exists on the manifold. This can be easily checked using condition (6). The question of stability on the manifold is whether the system trajectory on the manifold will converge to the steady-state point or will move away from it. To establish stability around the steady-state point, the F (i, v) = Const curve that goes through point X ref is
g (i, v) < 0 trajectories move
g (i, v) > 0 region and movement along the manifold must be in the direction of increasing F (i, v) , and therefore towards steady-state point X ref . On the other hand, point D is in the g (i, v) < 0 region, so movement must result in a
At any point D such that towards decreasing ellipses.
drawn in Fig. 5. As shown in Fig. 5a, let us consider two points C and D in a neighborhood of point X ref . Point C is in the
F (i, v) and therefore towards smaller
3
decrease of point
the
F (i, v) and therefore it is towards steady-state
manifold
is
stable
as
long
as
the
curves
F (i, v) = Const are steeper than the manifold curve, as
X ref . We can conclude that in the case of Fig. 5a the
shown in Fig. 7. Notice also that the manifold does not need to be linear in order to apply the method.
manifold is stable. Using a similar argument we can show that in the case of Fig. 5b the manifold is unstable, because the system will move away from steady-state point X ref .
i
From the argument above we can derive stability limits at point X ref . In order to be in the stable case of Fig. 5a and not in the unstable case of Fig. 5b, the manifold must belong to the stable sector shown in Fig. 6. The condition is that the slope of the manifold, which is equal to − 1 / k , must be larger than the slope of the curve F (i, v) = const going through point curve
ψ=0
X ref , and it must be smaller than the slope of
g (i, v) = 0 at point X ref . These are the slopes of
tangent lines S F and
S g , respectively, as shown in Fig. 6.
STABLE
Therefore, the stability conditions are
Slope( S F ) < −
1 < Slope( S g ) k
Fig. 7 Large-signal stable manifold.
(15)
B. Energy Interpretation For the Boost converter case there is an interesting interpretation of the method in terms of stored energy and power balance. Notice that multiplying equation (9) by Vg we obtain that
It is straightforward to show that the conditions are
−
C Vref 1 2V < − < ref L I ref k RVg
(16)
Notice that these are necessary and sufficient conditions for stability at point X ref , therefore they are tight. It will be
Vg ⋅ g (i, v) is equal to the difference between input and output power (17).
shown that a small-signal stability analysis gives exactly the same stability conditions as (16), validating the proposed method.
i F(i,v) =const
v2 Vg ⋅ g (i, v) = Vg ⋅ i − = Pin − Pout = 0 R i Pin=Pout
g(i,v) = 0 Xref Stable sector
(17)
Pin>Pout
Stable sector Sg
v
Stored energy
SF v
Pin Pout the stored energy F (i, v) must increase, and in
proposed method can be applied at any point of the manifold. Given any point on the manifold, the geometric argument establishes the direction of evolution with time, providing large-signal stability information. For example, in the case of a manifold with negative slope (the more interesting case, because it gives faster response), motion on
the region Pin
< Pout it must decrease. This interpretation
applies only to the Boost case, and not to the Buck and
4
xˆ& = Axˆ + Bd dˆ + Bvˆg
Buck-Boost cases discussed later in the paper, therefore the proposed stability analysis method is more general.
Differentiating the manifold equation (4) and substituting the linearized system equations (22) one obtains for vˆg = 0
C. Constant Power Load Case The averaged equations for the converter of Fig. 2b are
(
dˆ = − K T Bd
di = Vg − v(1 − d ) dt dv P C = i (1 − d ) − out dt v L
[
This identifies a horizontal line in the state plane. Step 2. Choosing the same F (i, v ) function as before (10), it can be easily shown that (7) is satisfied (20)
shows the two regions identified by the steady-state locus g (i, v ) = 0 . Fig. 9b shows a stable manifold. The
(21)
λ=−
As expected, this condition is more restrictive than in the resistive case (16).
i
g(i,v)=0 F(i,v) (a)
ψ=0 v
STABLE
]
K T A xˆ
(24)
(25)
(26)
kCVref − LI ref
OTHER CONVERTERS
The proposed geometric stability method has been discussed in detail for the case of a Boost converter. The cases of resistive load and constant power load have been examined. The method applies to Buck and Buck-Boost converters as well and the derivations reported above for the Boost converter can be repeated for these other converters as well. Table 1 lists system equations, functions F (i, v) and
Xref
g(i,v) < 0
−1
Vg
V.
F(i,v)
)
The stability conditions on k coincide with conditions (21). This constitutes a validation of the proposed stability approach.
i g(i,v) > 0
Pout Vg
(
It can be easily shown that the conditions on k under which this eigenvalue is negative are identical to (16). A similar small-signal analysis for the constant power load case gives an eigenvalue
stability condition at steady-state point X ref is
C Vref 1
