considerably smaller number of parameters. In [Thibodeau et al. (2009)] it is further shown that the stabilizability problem using truncated ellipsoidal LFs can be ...
Stabilization of constrained linear systems via smoothed truncated ellipsoids A. Balestrino ∗ , E. Crisostomi ∗ , S. Grammatico ∗ , A. Caiti ∗ ∗ Department of Energy and Systems Engineering, University of Pisa, Largo Lucio Lazzarino 1, 56122 Pisa, Italy e-mails: {a.balestrino, a.caiti}@dsea.unipi.it. {emanuele.crisostomi, grammatico.sergio}@gmail.com
Abstract: Polyhedral Lyapunov functions are convenient to solve the constrained stabilization problem of linear systems as non-conservative estimates of the domain of attraction can be obtained. Alternatively, truncated ellipsoids can be used to find an under-estimate of the feasible region, with a considerably reduced number of parameters. This paper reformulates classic geometric intersection operators in terms of R-functions, leading to a new family of smooth Lyapunov functions. This approach can be used to smooth both polyhedral and truncated ellipsoids Lyapunov functions improving control performances, as shown in several benchmark examples. Keywords: Constrained linear systems, composed Lyapunov functions, R-functions. 1. INTRODUCTION A popular approach to solve many stability/stabilizability problems is based on the search of appropriate Lyapunov Functions (LFs)/Control LFs (CLFs). The main difficulty of this approach is that in many control applications it is not clear which class of functions is the “most convenient” to search for a suitable LF/CLF. An example is the stabilization problem of constrained linear systems, which is the topic of this paper, where the same problem can be solved with different levels of complexity and accuracy, depending on the choice of the class of candidate CLFs. The exact solution of the problem corresponds to finding the largest admissible controlled invariant region of the state space, where admissibility regards both state and control constraints (if any). An approximated solution is given by the largest controlled invariant ellipsoidal set included inside the feasible region, easily computed according to standard Linear Matrix Inequality (LMI) arguments [Boyd et al. (1994)]. Such an approximated solution is obtained if quadratic LFs are exploited. On the other hand, arbitrarily accurate solutions can be computed if Polyhedral LFs (PLFs) are used [Gutman and Cwikel (1986)], [Blanchini (1995)]. PLFs are more flexible than ellipsoidal functions, as they can be shaped to fit the exact feasible region; the polyhedral region is described by a linear set of inequalities like kF xk∞ ≤ 1 and in [Blanchini and Miani (1998)] it is also shown that the polyhedral function can be smoothed by restating the set of inequalities as kF xk2p ≤ 1, for an appropriate choice of p. The smoothed solution is more convenient because the induced LF becomes everywhere differentiable and gradient-based controllers can be used for obtaining timecontinuous control signals [Petersen and Barmish (1987a)]. Apparently, there is only one (major) drawback in the use of PLFs, i.e. the computational burden to find suitable LFs.
More recently a trade-off solution has been proposed in the literature, where functions induced by truncated ellipsoidal sets are used as the basic class for the candidate LFs [O’Dell and Misawa (2002)], [Thibodeau et al. (2009)]. As shown in [O’Dell and Misawa (2002)], the advantage of using truncated ellipsoids is that on average a good approximation of the feasible region is computed with a considerably smaller number of parameters. In [Thibodeau et al. (2009)] it is further shown that the stabilizability problem using truncated ellipsoidal LFs can be casted in a quasi-LMI fashion (i.e. a set of Bilinear Matrix Inequalities (BMIs) that can be reduced to a set of LMIs once some parameters are fixed). Note that the estimated admissible region is controlled invariant under a linear state feedback control law which is designed to satisfy the physical constraints and to maximize the size of the estimated feasible region. In this case gradient-based controllers can not be used because the LF is constructed by intersecting ellipsoidal with polyhedral sets and it is not everywhere differentiable. This paper provides two main contributions: first, the results of [Balestrino et al. (2010)], where a new class of smoothed PLFs was introduced to stabilize Linear Differential Inclusion (LDI) systems, are generalized to include the case of constrained linear systems. The proposed smoothing technique is different from the conventional 2pnorm [Blanchini and Miani (1998)] as the sublevel sets do not have the same shape everywhere inside the feasible region, but they become smoother near the origin. This property is expected to be convenient for control purposes as it is verified in several simulations. The second contribution is that the proposed smoothing technique can be extended to the case of CLFs induced by truncated ellipsoidal regions, and therefore gradient-based controllers can be used also in the case of [O’Dell and Misawa (2002)] and [Thibodeau et al. (2009)]. This is an important novelty
Table 1. Correspondence between logic functions and R-functions not ¬ α
and ∧ α
or ∨
1
R-composition
0.5
p −r
r12 + r22 − 2αr1 r2 √ 2p − 2 − 2α r1 + r2 + r12 + r22 − 2αr1 r2 √ 2 + 2 − 2α r1 + r2 −
x2
Boolean
α = 0 1.5
0 −0.5 −1
as the 2p-norm can not be used to smooth the intersection of the ellipsoidal and polyhedral regions, and gradientbased controllers provide better control performances than the simple static state feedback ones, while preserving the same estimate of the admissible region. The proposed smoothing technique is easily derived from reinterpreting the intersection of ellipsoidal and polyhedral regions in the context of R-functions, which are realvalued functions that generalize the classic pointwise min and max operators. R-functions are described in the next section, while in Section 3 the proposed control law is introduced and motivated. Section 4 shows simulation results, where the derived control law is compared with the PCLFs of [Blanchini and Miani (1998)] and the linear feedback control laws of [Thibodeau et al. (2009)]. In the last section the main results are summarized and future lines of research are outlined. 2. R-FUNCTIONS FOR CONTROL APPLICATIONS R-functions were introduced in [Rvachev (1982)] and more recently have been discussed in [Shapiro (2007)]. Definition 1. (from [Shapiro (2007)]). A function r : Fn ⊆ Rn → R is an R-function if there exists a Boolean function R : Bn → B, where B = {0, 1}, such that the following equality is satisfied: h (r (x1 , x2 , . . . , xn )) = R (h (x1 ) , h (x2 ) , . . . , h (xn )) , where h : R → B is the standard Heaviside function. The Boolean function R is also called the companion function of the R-function r. Informally, a real function r is an R-function if it can change one property (sign) only when some of its arguments change the same property (sign). The parallelism between logic functions and Rfunctions becomes more evident when classic Boolean operators are recovered as described in Table 1. For instance, according to Table 1, the interpretation of the and composition is that the composed function is positive when evaluated in x if and only if both r1 (x) and r2 (x) are positive. The result is obtained by exploiting the triangle inequality and the law of cosines, and it holds for all values of α ∈ [0, 1] ⊂ R [Balestrino et al. (2009a)]. The terms at the denominator in Table 1 are normalizing factors. 1
Remark 1. It can be easily seen that when α = 1, r1 ∧ 1
r2 = min {r1 , r2 } and similarly r1 ∨ r2 = max {r1 , r2 }. There is also a nice geometric interpretation of R-functions as illustrated in the next example. Example. Consider the truncated ellipsoid obtained by intersecting the polyhedron kF xk∞ ≤ 1 and the ellipsoid x⊤ P x ≤ 1, where
−1.5 −2
−1.5
−1
−0.5
0 x
0.5
1
1.5
2
1
Fig. 1. The R-function r∧ is calculated with α = 0 for all of the R-intersections. i i h h F =
2/3 0
0 2/3
, P =
1/3 0
0 1
.
We consider functions r1 (x) = 1 − (2/3)x1 , r2 (x) = 1 + (2/3)x1 , r3 (x) = 1 − (2/3)x2 , r4 (x) = 1 + (2/3)x2 , ⊤ rq = 1 − x⊤ P x, where x = [x1 x2 ] is the state vector. For convenience the functions have been normalized so that their maximum value is 1. Then we compute the intersection α34 α23 α12 α r∧ = r1 ∧ r2 ∧ r3 ∧ r4 ∧ rq , according to the equation of Table 1, for arbitrary values of α, αij ∈ [0, 1]. The intersection function is positive inside the geometric intersection region, it is zero on the boundary, negative outside and its maximum value is 1 at the origin. The sublevel sets of the function f∧ = 1 − r∧ (for α = αij = 0) are shown in Figure 1. The composed function is the smoothed intersection between the smoothed polyhedral function introduced in [Balestrino et al. (2010)] and a quadratic function. Alternatively we can also compute the R-intersection between the (non-smoothed) polyhedral function 1−kF xk∞ , 1
1
1
rp = r1 ∧ r2 ∧ r3 ∧ f4 , and the quadratic function rq = 1 − x⊤ P x, that is α
rp ∧ rq . In this case, Figure 2 shows the sublevel sets of the α corresponding f α = 1 − rp ∧ rq with α = 1 (truncated ∧ ellipsoid) and with α = 0. Remark 2. As previously mentioned, the intersection of a polyhedral region with ellipsoids has already been used as a candidate LF in [Thibodeau et al. (2009)]. Such a LF is identical to the R-intersection of the polyhedral function and the quadratic one with α = 1, as in the example shown on the left of Figure 2. Other control applications of the R-functions can be found in [Balestrino et al. (2009a)] and [Balestrino et al. (2009b)]. 3. CONSTRAINED STABILIZATION OF LINEAR SYSTEMS 3.1 Problem statement Let us consider the constrained stabilization problem of a linear system: x(t) ˙ = Ax(t) + Bu(t) s.t. (1) x(t) ∈ X ⊂ Rn , u(t) ∈ U ⊂ Rm ,
α = 0 1.5
1
1
0.5
0.5 x2
x
2
α = 1 1.5
0
0
−0.5
−0.5
−1
−1
−1.5 −2
−1.5
−1
−0.5
0 x
0.5
1
1.5
2
−1.5 −2
−1.5
−1
−0.5
1
0 x
0.5
1
1.5
2
1
Fig. 2. On the left the intersection between the polyhedral function and the quadratic one is performed with α = 1, while on the right with α = 0. where A ∈ Rn×n , B ∈ Rn×m . The control objective is to design a control law u(t) such that x(t) asymptotically converges to the origin, in accordance to the state and control input constraints. We assume that the set X is compact, convex and 0-symmetric, and that U = {u ∈ Rm : kuk∞ ≤ 1} . (2) The first assumption is standard in constrained stabilization problems, while the second is only a simplificative assumption that does not affect the generality of the results.
The corresponding R-function R α is computed as the ∧ intersection of the 1−level set of the linear constraints forming (3). This is explained in the two-steps procedure (5)-(6) for clarity: α
R α (x) = (1 − Fi x) ∧ (1 + Fi x) , ∧,i
R α (x) = ∧
α
r ^
i = 1, 2, . . . , r,
R α (x). ∧,i
(5) (6)
i=1
The polyhedron {x ∈ Rn : R α (x) = 0} is exactly the same ∧ described by kF xk∞ = 1. The particular choice of the 1−level set does not affect the generality of the approach, because the state can be appropriately rescaled.
Polyhedral Lyapunov Functions Typically, the drawbacks of using Quadratic LFs (QLFs) are that the estimate of a positive controlled invariant set is restricted to ellipsoidal regions and that the control law is a linear state feedback.
As previously remarked, R α (x) > 0 ∀x s.t. kF xk∞ < 1 ∧ and max{R α (x)} = R α (0) = 1. Therefore the associated
A PLF V∞ : Rn → R is usually described by equation V∞ (x) = kF xk∞ = max{|Fi x|}, (3)
candidate (positive definite) LF is V α (x) = 1 − R α (x).
i
r×n
x
∧
∧
∧
∧
th
where F ∈ R is a full column rank matrix, Fi is the i row of F , and sublevel sets have the shape of a polyhedron with 2r facets. PLFs outperform QLFs in terms of nonconservative estimates of a positive controlled invariant set, but the lack of differentiability causes some difficulties in the control synthesis. Smooth PLFs circumvent both difficulties, as they provide non-conservative estimates and enable the use of gradientbased controllers. In this framework, R-functions are used as an alternative way of expressing the PLFs, so that the parameter α can be used to tune the smoothness of the sublevel sets. The proposed smoothing technique is different from conventional ones, e.g. the 2p-norm of [Blanchini and Miani (1998)] v u r uX 2p V2p (x) = kF xk2p = t (Fi x)2p , (4) i=1
because the sublevel sets become smoother close to the origin, which is usually convenient to improve control performances. The correctness of the proposed smoothing technique is proved in Section 3.2, while a comparison with the 2p-norm is provided in Section 4 (Examples 1 and 2).
In the following, the smoothing procedure of [Balestrino et al. (2010)] is summarized. It is assumed that a suitable polyhedral function for the stabilization problem is available, for instance it can be found using methods outlined in [Brayton and Tong (1980)], [Blanchini (1995)] or [Polanski (2000)].
(7)
Remark 3. Both function V2p (4) for p → +∞ and V α (7) ∧ for α = 1 coincide with the polyhedral function V∞ (3). Truncated ellipsoids The use of truncated ellipsoids provides an approximation of the maximal feasible set which is less conservative than the ellipsoidal approach, but computationally more convenient than polyhedral approximations, which might involve a very large number of vertices and facets. The use of truncated ellipsoids has been proved effective among others by [O’Dell and Misawa (2002)] and [Thibodeau et al. (2009)], where linear feedback control laws are used. In [Thibodeau et al. (2009)] the truncated ellipsoid function is described as Vte (x) = max x⊤ P x, x⊤ Ci⊤ Ci x , (8) i
where Ci ∈ R1×n , for i = 1, ..., r. The use of the max operator makes Vte non differentiable.
The same solution can be reformulated in terms of Rfunctions (i.e. via intersecting ellipsoids and polyhedral regions) and the parameter α can be used as a smoothing factor. Smoothing allows for the recovering of gradientbased controllers and average control performances are improved as it is shown in Section 4 (Examples 3 and 4). To perform the R-intersection of a polyhedral function and an ellipsoidal one, we can compute the R-function R α associated to the polyhedral function kF xk∞ (3) by ∧ following the procedure (5)-(6) of the previous subsection.
Here we denote the composed R-function with Rp , where the subscript p indicates the R-intersection associated to the polyhedral function. Then we define the R-function associated to the quadratic function x⊤ P x Rq (x) = 1 − x⊤ P x. (9) Finally the R-intersection is computed in accordance to the composition rule of Table 1 as
Rα (x) =
Rp (x) + Rq (x) −
∧
p
Rp (x)2 + Rq (x)2 − 2αRp (x)Rq (x) √ 2 − 2 − 2α (10)
and the candidate CLF is V α (x) = 1 − R α (x). ∧
∧
(11)
Remark 4. If both Rp (for all hyperplanes intersections) and the final R-intersection (10) are computed with α = 1, then the standard truncated ellipsoid (8) is obtained. In other words, the truncated ellipsoid is a special case of the smoothing performed with the R-functions framework. 3.2 Lyapunov-based control synthesis A function V is a suitable CLF if the condition x ∈ Rn : ∇V (x)B = 0⊤ ∧ ∇V (x)Ax ≥ 0 = ∅, (12) is satisfied, as stated in [Petersen and Barmish (1987b)]. In the above equations, ∇V (x) is the gradient of function V (x) and 0⊤ is a row vector of zeros of appropriate dimensions. In practice, condition (12) implies that whenever the control action is ineffective, function V should decrease just the same, in fact the time derivative of the CLF V is V˙ (x(t), u(t)) = ∇V (x(t))Ax(t) + ∇V (x(t))Bu(t). (13) If condition (12) is satisfied, then there exists a control law u(t) such that V˙ (x(t), u(t)) is always negative. The control law u(t) that minimizes V˙ (x(t), u(t)), over the set U , is the (sliding) control u (x(t)) = −sign B ⊤ ∇V (x(t))⊤ (14) so that the time derivative of V becomes m X | (∇V (x)B)i |, (15) V˙ (x(t)) = ∇V (x(t))Ax(t) − i=1
where (∇V (x)B)i is the ith component of the row vector ∇V (x)B.
Condition (12) is not sufficient to guarantee that the time derivative (15) is always negative, because here the control u (14) is constrained in U . To overcome this feasibility problem, it is possible to derive a Petersen-like condition that guarantees that V is a suitable CLF by means of a constrained state feedback control (ku(t)k∞ ≤ 1): ) ( m X | (∇V (x)B)i | ≥ 0 = ∅. (16) x ∈ X : ∇V (x)Ax − i=1
A possible way of checking equation (16) is by solving an optimization problem. For instance, if it is required that the state should be constrained inside the intersection of a polyhedron represented by matrix F and an ellipsoid
represented by the positive definite matrix P , the optimization problem ) ( m X | (∇V (x)B)i | s.t. max ∇V (x)Ax − x∈X i=1 (17) X = x ∈ Rn : kF xk∞ ≤ 1, x⊤ P x ≤ 1 has to be solved. In particular, condition (16) is satisfied if and only if the solution of (17) is negative. In this work we solve problem (17) to prove the correctness of the proposed smoothed CLF. The drawback of the control law (14) is that it is highly discontinuous over time and often not implementable on real actuators. The discontinuity caused by the sign function can be avoided by approximating the control law (14) with arbitrary precision by using u (x(t)) = −sat κB ⊤ ∇V (x(t))⊤ , (18)
for κ ∈ R+ sufficiently large [Blanchini (2009)], where sat is the component-wise vector saturation function. It is particularly convenient to associate a gradient-based control to an everywhere differentiable CLF, because the corresponding control law is continuous over time [Petersen and Barmish (1987a)]. 4. SIMULATIONS 4.1 Comparison with smoothed polyhedral functions In this subsection, we consider the examples proposed in [Blanchini and Miani (1998)], where the control law is a gradient-based control associated to a PLF, smoothed with standard 2p-norms. The framework of R-functions is used to smooth the inner sublevel sets of the polyhedral function kF xk∞ . The candidate CLFs are computed by following the procedure (5)-(6) with α = 0 and they have been proved to be suitable CLFs for the corresponding constrained control problem by solving (17). Example 1 concerns the constrained stabilization of the the dynamical system 1 1 1 x˙ 1 (t) x1 (t) 1 0 u1 (t) = + . (19) x˙ 2 (t) x2 (t) 0 1 u2 (t) 4 1 1
The CLF proposed in [Blanchini and Miani (1998)] is the polyhedral function kF xk2p , where i h F =
with p = 3.
1.5 −0.5 −0.5 1.5
,
(20)
The controlled region is kF xk∞ ≤ 1, that it is larger than the one (kF xk2p ≤ 1) provided in [Blanchini and Miani (1998)], because also the corners of the polyhedral region are included. In the table, values of typical control indices are shown: ISE is the Integral of the Squared Error values and it should be small to avoid large state errors; ISTE is the Integral Square Time Error and it also should be small to avoid large state errors or slow convergence; T represents the time of convergence (2-norm of the state vector smaller
Table 2. Average control performances, normalized with respect to the results of [Blanchini and Miani (1998)]. Results have been obtained averaging over 100 simulations starting from random initial states inside the intersection region.
Table 3. Average control performances, normalized with respect to the results of [Thibodeau et al. (2009)] (Example 3) and [O’Dell and Misawa (2002)] (Example 4). Results have been obtained averaging over 100 simulations starting from random initial states inside the intersection region.
Example 1 kF xk2p R-function (α = 0)
Example 3
ISE
ISTE
T
IADU
1 0.2832
1 0.0823
1 0.2908
1 0.9408
ISE
ISTE
T
IADU
1 0.9147
1 0.5997
1 0.6557
1 0.7612
Truncated Ellipsoid R-function (α = 0)
Example 2 kF xk2p R-function (α = 0)
ISE
ISTE
T
IADU
1 0.9478
1 0.9090
1 0.9560
1 0.9928
Example 4 Truncated Ellipsoid R-function (α = 0)
ISE
ISTE
T
IADU
1 1.0059
1 0.9808
1 0.9603
1 0.8533
1 0.8 0.6
computed by following the procedure (5)-(6) with α = 0 to compute Rp , together with the procedure (9)-(11) again with α = 0. Finally V α has been proved to be ∧ a suitable CLF for the constrained control problem by solving problem (17).
0.4
x
2
0.2 0 −0.2 −0.4 −0.6 −0.8 −1
−1
−0.8
−0.6
−0.4
−0.2
0 x
0.2
0.4
0.6
0.8
1
1
Fig. 3. Controlled state trajectories converging to the origin according to the R-function CLF.
Example 3. The constrained stabilization of the dynamical system characterized by matrices # " # " A=
−1 1 0
0 −2 1
0 −1 0
, B=
0 0 1
, F = I3 ,
(23)
than 10−3 ); finally IADU is the Integral of the Absolute value of the time Derivative of the control signal u and it is desired to be small to avoid stress of the control actuator.
has been also addressed in [Thibodeau et al. (2009)]. The control synthesized in [O’Dell and Misawa (2002)] is linear, i.e. u = Kx, where
With the use of R-functions all performance indices are improved, since the non-homothetic sublevel sets provide a smoother state convergence with respect to high-order 2pnorms. Normalized simulation results, averaged over 100 simulations, are shown in Table 2.
K = [ 0.360 −0.053 −0.671 ] .
In [Thibodeau et al. (2009)] the same control of [O’Dell and Misawa (2002)] is used, as solving the optimization procedure to find both optimal P and K matrices would require the solution of a BMI problem.
Example 2 [Blanchini and Miani (1998)] concerns the constrained stabilization of the dynamical system characterized by matrices 0 0 −1 1 0 0
Example 4 [O’Dell and Misawa (2002)]. The dynamical system is the double integrator 0 0 1 u(t), (25) x(t) + x(t) ˙ = 1 0 0
A=
1 0 0
−1 0 0
1 0 1
1 0 , B= 0 1 0 0
0 . 0 1
(21)
The state constraint is kxk∞ ≤ 1 and a suitable polyhedral CLF kF xk∞ with 1.000 0 ⊤ 0 0 0 0.610 0 0.328 0 F =
0 0 0
1.250 0 0 0 0 0.762 0 0.225 . 0 1.000 0 1.250 0.610 0.762 0.983 1.126 0 0 1.000 1.250 0 0.762 0.656 1.126 (22)
Also in this examples, the use of R-functions yields better control performances. The comparison is shown in Table 2. Some controlled state trajectories, starting from random initial points are shown in Figure 3. 4.2 Comparison with truncated ellipsoids Examples 3 and 4 are taken from [O’Dell and Misawa (2002)]. For each example, the candidate CLF V α is ∧
(24)
with state constraints |x1 | ≤ 25, |x2 | ≤ 5 and control constraint |u| ≤ 1. A truncated ellipsoid and the static controller are synthesized in [O’Dell and Misawa (2002)]. Such static state feedback control is compared with the gradient-based control associated to the truncated ellipsoid smoothed with R-functions. The gradient-based control is smoother and it yields much faster convergence. Moreover, the use of a static state feedback control yields an undesirable oscillating behavior of the state trajectory, as it is shown in Figure 4. Finally, Table 4 shows that the computational time required by the nonlinear control law (18) associated to the smooth R-function CLF is actually comparable with the case in which (18) is associated to usual CLFs. The linear state feedback control is not considered in the comparison as it is obviously faster.
inite functions. Future work will also focus on casting the proposed Petersen-like condition into an easily-tractable LMI problem.
8 6 4
x
2
2
REFERENCES
0 −2 −4 −6 −8
−25
−20
−15
−10
−5
0 x1
5
10
15
20
25
−25
−20
−15
−10
−5
0 x
5
10
15
20
25
8 6 4
x
2
2 0 −2 −4 −6 −8
1
Fig. 4. Controlled state trajectories converging to the origin according to the truncated ellipsoid CLF together with a static state feedback controller (top) and according to the R-function CLF together with the use of a gradient-based controller (bottom). Table 4. Average required computational time of equation (18), normalized with respect to the use of a PCLF. Results have been obtained averaging over 10000 state points. Example Example Example Example
1 2 3 4
V∞
V2p
Vte
Vα
1 1 1 1
1.39 1.59 1.48 1.39
1.51 1.85 1.63 1.54
0.72 1.52 0.95 0.74
∧
5. CONCLUSION In this paper a solution for the constrained stabilization problem of linear systems has been proposed. The control law minimizes the time derivative of a smoothed control Lyapunov function within the set of bounded controls. The novelty of the approach follows from the reinterpretation of polyhedral functions and truncated ellipsoids in terms of R-functions, providing a general framework to smooth both polyhedral and truncated ellipsoidal functions in a new non-homothetic way, which it has been shown to be convenient in terms of control performances. A Petersen-like condition has been proposed to check if a smoothed Lyapunov function candidate is a suitable control Lyapunov function by means of a gradient-based bounded control. The main contribution is the use of a gradient-based control together with the introduced smoothed truncated ellipsoidal Lyapunov function for constrained stabilization of linear systems, since the use of everywhere differentiable functions allow the control to be continuous over time. Although only quadratic and polyhedral functions have been investigated in this paper, the proposed framework allows for a flexible composition of arbitrary positive def-
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