DISCOVERING MULTIPLE LYAPUNOV FUNCTIONS FOR SWITCHED HYBRID SYSTEMS ∗ ZHIKUN SHE† AND BAI XUE‡ Abstract. We in this paper analyze local asymptotic stability of switched hybrid systems, whose subsystems have polynomial vector fields, by discovering multiple Lyapunov functions in quadratic forms. We start with an algebraizable sufficient condition for the existence of quadratic multiple Lyapunov functions. Then, since different discrete modes are considered, we apply real root classification together with a projection operator to under-approximate this sufficient condition step by step, arriving at a set of semi-algebraic sets which only involve the coefficients of the pre-assumed multiple Lyapunov function. Note that for each step, we additionally use the information on the different discrete modes to optimize our intermediate computation results. Afterwards, we compute a sample point in the corresponding semi-algebraic set for the coefficients, resulting in a multiple Lyapunov function. Finally, we test our approach on some examples using a prototypical implementation and compare it with the generic quantifier elimination based method and the sum of squares based method. These computation and comparison results show the applicability and promise of our approach. Furthermore, our approach is extendable for piecewise affine systems and non-polynomial switched systems by discovering multiple Lyapunov functions beyond quadratic forms. Key words. Multiple Lyapunov functions, real root classification, projection operator, semi-algebraic sets
1. Introduction. Asymptotic stability of switched hybrid systems has recently received growing attentions in the analysis and design of control systems [5, 22, 31, 38, 16, 27, 13], e.g., automotive powertrain systems [3] and rigid body systems [37]. During the past decade, many stability criteria have been proposed for switched and hybrid systems [57, 25]. Especially, it is well-known that a sufficient condition for verifying asymptotic stability of switched hybrid system is the existence of common Lypunov functions or multiple Lypunov functions [27, 5]. Therefore, the computation of common or multiple Lyapunov functions has also become a very interesting topic in computer science, systems control and systems science. When the differential equations in subsystems are polynomials, due to decidability of the theory of real-closed fields [61], one can verify the existence of common Lyapunov functions or multiple Lyapunov functions by quantifier elimination (QE) based method [31, 56, 14]. However, all the existing decision procedures (e.g., implemented in the software packages QEPCAD B [15, 7] or REDLOG [17]), while being able to solve impressively difficult examples, are not enough to be able to solve this problem in practice. Another choice is to linearize the subsystems and then use linear matrix inequalities (LMI) based method [4] for computing common or multiple Lyapunov functions [5, 34, 22]. Further, without linearization, one can use the sum of squares programming (SOS) based method [38, 32] to compute common or multiple Lyapunov functions by relaxation to a semidefinite problem [12], which also amounts to solving a linear matrix inequality feasibility problem, i.e., a convex optimization. However, Hilbert has already pointed out that apart from three special cases, not all non-negative polynomials can be formulated as sums of squares [41, 11]. Recently, an automatable decompositional method has been proposed for the compu∗ This work was supported by NSFC-11422111, NSFC-11371047, NSFC-11290141, NSFC-61003021 and SKLSDE-2013ZX-10. This is an extended and revised version of our earlier conference paper [50]. Different from [50], we have a new separate subsection describing about the real root classification and the adaptive CAD based semi-algebraic system solver, a new separate subsection describing the comparison results, and a new separate section discussing the extendable applicability of our current algebraic approaches. † SKLSDE, LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, China (
[email protected]). ‡ School of Mathematics and Systems Science, Beihang University, Beijing, China (
[email protected]).
1
tation of local Lyapunov functions [30, 29] by using graph-based reasoning to decompose hybrid automata into subgraphs and then solving semi-definite optimization problems. In this paper, following the algebraic idea in [52, 46], we propose a mechanisable technique for local asymptotic stability analysis of switched hybrid systems, whose subsystems also have polynomial vector fields. Note that the motivations for considering polynomial vector fields lie in the following two sides [1]: 1. Polynomial vector fields enjoy much added interest stemming from the fact that various other types of vector fields are often approximated by their Taylor expansions of some order around equilibrium points. 2. Aside from practice, polynomial vector fields have also always been at the heart of diverse branches of mathematics: some of the earliest examples of chaos (e.g. the Lorenz attractor) arise from simple low degree polynomial vector fields [26]; the still-open Hilbert’s 16th problem concerns polynomial vector fields on the plane [21]. We start with the classical definition on local asymptotic stability of switched hybrid systems, which can be assured by the existence of multiple Lyapunov functions. Then, we construct an algebraizable sufficient condition on the existence of multiple Lyapunov functions in quadratic forms, which is theoretically solvable for polynomial systems due to the theory of real-closed fields [61]. Note that since different discrete modes are considered, this sufficient condition is more complex than the one in [52, 46]. Especially, we additionally need to compute two non-collinear vectors in the corresponding state space of each mode. For efficient computation, we first transform the algebraizable constraints to equivalent ones and then under-approximate the equivalent ones step by step in a conservative way, arriving at semi-algebraic sets only involving the coefficients of the pre-assumed multiple Lyapunov function. Based on these resulting semi-algebraic sets, we can compute a sample point for the coefficients to obtain a multiple Lyapunov function. Again, since different discrete modes are considered, in addition to real root classification, our current algorithms for under-approximation are distinctly different from the ones in [52, 46]. These distinct differences are as follows: first, we apply a projection operator to assure that the eliminated state variables do not appear in the intermediate under-approximations so that all state variables can be thoroughly eliminated in the later steps; second, we use the information on each mode to optimize our intermediate computation results at each step, obtaining simpler ones. We prototypically implement our algorithms based on DISCOVERER, which is a semialgebraic set solver [63], and then test the implementation on five examples. These computation results demonstrate the applicability and promise of our approach. Note that our approach can also be potentially useful for verifying other properties of hybrid systems (e.g., verifying safety by generating invariants [19, 42]). Furthermore, our approach can be extendable for piecewise affine systems and non-polynomial switched systems. The contributions of our current method involve the following points: 1. Our method constructs multiple Lyapunov functions in quadratic forms (and even beyond quadratic forms) by sufficiently considering the local property of the vector fields. This is different from the method based on linear matrix inequalities [5, 34, 22]. That method relies on linearizing the systems and cannot work on systems whose subsystems may be degenerate (i.e., there is at least one subsystem whose linearization has one (or more) pair of conjugated purely imaginary eigenvalues or admits 0 as an eigenvalue). However, our method can handle this case. 2. Our method can find rigor polynomial multiple Lyapunov functions and thus avoid the numerical errors caused by floating computations. In this sense, compared to the sum-of-squares decompositions [38, 32] based method, our computation results are more reliable. Moreover, our method is more efficient to some extent than the 2
generic QE based methods [31, 56, 14]. This is because our variable elimination method relies on real root classification method and this real root classification method is more efficient than the generic QE based methods [10]. 3. Our real root classification based method can also be an alternative approach to the problem of using various versions of positivstellens¨atze [20, 58, 33, 43, 44, 12]. The structure of the paper is as follows: We start with the classical definition on local asymptotic stability of switched hybrid systems in Section 2; We present an algebraizable sufficient condition on multiple Lyapunov functions for asymptotic stability analysis in Section 3 and briefly explain how to compute a multiple Lyapunov function in Section 4; We review the real root classification in Subsection 5.1 and explain how to use our real root classification and projection operator based approach for computing required under-approximative constraints in Subsection 5.2; We analyze the correctness of our approach in Section 6; We test our approach on five examples using a prototypical implementation in Subsection 7.1 and compare it in Subsection 7.2 with the LMI based method, the QE based method and the SOS based method, respectively; We make a simple extension of our approach to compute multiple Lyapunov functions beyond quadratic forms for piecewise affine systems and non-polynomial switched systems in Section 8; We conclude the paper in Section 9. 2. Problem Definitions. A switched hybrid system SHS is a system of form x˙ = fi (x), where i ∈ M, M = {1, · · · , N} is the set of finite discrete modes, x ∈ Xi , Xi ⊂ Rn is the continuous state space of mode i, and fi (x) is the vector field describing the dynamics of mode i. In addition, for two different modes i and j, a switch from mode i to mode j need to occur if the evolution in mode i hits the switching surface defined by a switching constraint over x. Moreover, we assume that there are only finite switches in finite time such that the system does not exhibit the Zeno behavior [8, 55] and a switch must occur before the evolution in mode i leaves its corresponding continuous state space Xi such that the system does not exhibit the blocking behavior [28]. A trajectory of a SHS in the sense of Carath´eodory [27] is a finite or an infinite sequence of flows r0 (t), r1 (t), · · · , r p (t), · · · , associated with a time sequence T 0 , · · · , T p , · · · , such that 1. every flow r p (t) is associated with a mode i and evolves according to r˙ p (t) = fi (r p (t)), 2. for each p > 0, r p−1 (T p ) satisfies the switching constraints and r p (0) = r p−1 (T p ). For simplicity, we denote a trajectory r0 (t), · · · , r p (t), · · · by x(t) satisfying ∑p T i ) = r p (T p ) = r p+1 (0), and 1. x(0) = r0 (0), for all p ≥ 0, x( i=0 ∑ p−1 ∑ p−1 ∑ p T i ). T i , i=0 T i ), x(t) = r p (t − i=0 2. for all t ∈ ( i=0 Note that an execution of a hybrid automaton is similarly defined in [28] by using a hybrid time trajectory (i.e., a finite or infinite sequence of time intervals). Definition 2.1 (Locally Asymptotically Stable). A switched hybrid system is called stable if for any ϵ > 0, there exists a δ(ϵ) > 0 such that ∥x(0)∥ < δ(ϵ) ⇒ ∥x(t)∥ < ϵ, ∀t > 0, and locally attractive if there exists a δ such that ∥x(0)∥ < δ ⇒ lim x(t) = 0, t→+∞
where 0 is its equilibrium point. A switched hybrid system is called locally asymptotically stable if it is both stable and locally attractive. Even if we have Lyapunov functions for each subsystem defined by x˙ = fi (x) individually, we still need to impose restrictions on switching to guarantee stability, which can be easily seen from the following example. 3
Example 1. Consider fi (x), where ( ) ( ) −x1 + 10x2 −x1 + 100x2 f1 (x) = and f2 (x) = . −100x1 − x2 −10x1 − x2 Clearly, x˙ = fi (x) is globally stable for i = 1, 2. But the switched system using f1 (x) in the second and fourth quadrants and f2 (x) in the first and third quadrants is unstable, which can be easily seen from Fig. 1. 1.2
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Fig. 1 A Trajectory of Example 1. Moreover, even if we have at least one unstable subsystem, we can still guarantee stability after imposing certain restrictions on switching, which can be seen below. Example 2. Consider fi (x), where ( ) ( ) −x1 − 100x2 x1 + 10x2 f1 (x) = and f2 (x) = . 10x1 − x2 −100x1 + x2 Clearly, x˙ = f1 (x) is globally stable and x = f2 (x) is unstable. However, if we define X1 = R2 and X2 = {x : −10x1 − x2 ≥ 0, 2x1 − x2 ≥ 0} and a switch from mode i to mode j need to occur if a trajectory in mode i hits the switching surface gTi, j x = 0, where i, j ∈ {1, 2}, i , j, g1,2 = (−10, −1) and g2,1 = (2, −1), then the switched system is globally asymptotically stable, which can be easily seen from Fig. 2. 1
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Fig. 2 A Trajectory of Example 2. In general, stability verification is undecidable [2]. In this paper, we consider a class of switched hybrid systems of form x˙ = fi (x), i ∈ M = {1, · · · , N}, x ∈ Xi ⊂ Rn ,
(2.1)
where fi is a polynomial vector field satisfying fi (0) = 0 1 and Xi is a union of polyhedral sets defined by form {x : Ei, j x ≥ 0, Ei, j is an n × n matrix} such that Xi =
li ∪ ∪ {x : Ei, j x ≥ 0, Ei, j is an n × n matrix} and Xi = Rn . j=1
i∈M
1 The assumption that f (0) = 0 for all i has already been used in the literatures [5, 32]. However, in Section 8, i this assumption is not required to hold.
4
Here, for two vectors x, y ∈ Rn , x ≥ y means x1 ≥ y1 , . . . , xn ≥ yn . Moreover, we assume that li, j T gi, j,r x = 0, where a switch from mode i to mode j occurs on the hyperplanes defined by ∨r=1 ∪li, j n×1 T i , j, gi, j,r ∈ R and gi, j,r , 0. Letting S i, j = r=1 {x : gi, j,r x = 0} and SSi, j ⊂ S i, j be the switching set w.r.t switches from mode i to mode j, we additionally assume that 1. SSi, j ∩ SSi,k = 0 for all j , k such that the switching system is deterministic [28]; 2. sliding modes [62] on the switching hyperplanes will not take place. For simplicity, we denote the above described system of form (2.1) as PS. Then, given a system PS, to prove its local asymptotic stability, it is sufficient to verify the existence of multiple Lyapunov functions, which will be discussed in Section 3. 3. An Existence Condition for Multiple Lyapunov Functions. In this section, we will construct an algebraizable sufficient condition for local asymptotic stability analysis of switched hybrid systems. To start with, we will introduce a classical theorem on multiple Lyapunov functions for switched hybrid systems as follows. Theorem 1. [27] For a given system PS, if there exist a neighborhood U of the origin and continuously differentiable functions Vi (x) : Xi → R, i ∈ M, such that 1. for each i ∈ M, Vi (0) = 0 and dtd Vi (0) = 0; 2. for each i ∈ M and each x ∈ Xi ∩ U, if x , 0, then Vi (x) > 0; 3. for each i ∈ M and each x ∈ Xi ∩ U, if x , 0, then dtd Vi (x) = (∇Vi (x))T · fi (x) < 0; 4. for each i, j ∈ M, where i , j, if x ∈ {x ∈ U : x ∈ SSi, j }, then Vi (x) ≥ V j (x), then PS is locally asymptotically stable. Here, the family {Vi (x) : i ∈ M} is called as a multiple Lyapunov function (or, MLF for short). According to Theorem 1, for a given switched hybrid system PS, its local asymptotic stability can be guaranteed by the existence of a multiple Lyapunov function {Vi (x) : i ∈ M}. Thus, our intuitive purpose is to find a multiple Lyapunov function. However, finding such a multiple Lyapunov function is in general difficult by hands. So it is reasonable to consider automatically discovering multiple Lyapunov functions in quadratic forms. For this, we will firstly transform the conditions in Theorem 1 to be algebraizable ones such that we can automatically compute multiple Lyapunov functions in quadratic forms. ∪i For each i ∈ M, let Xi be of form lj=1 {x : Ei, j x ≥ 0, Ei, j is an n × n matrix} and Vi (x) be ∑ i ′ of quadratic from, and represent V˙ i = dtd Vi (x) = (∇Vi (x))T · fi (x) as m k=2 vik (x), where mi is ′ the degree of V˙ i and vik (x) is the sum of all monomials of degree k in V˙ i . Since a polynomial p(x) is called a homogeneous multivariate polynomial of degree k if all its monomials have the same degree k, the v′ik (x) here is a homogeneous polynomial of degree k. For constructing an algebraizable sufficient condition for local asymptotic stability analysis, we additionally need the following theorem. Theorem 2. Let p = p(x) be a polynomial of degree m ≥ 1. If p(x) is positive definite in W = {x : Ex ≥ 0, E ∈ Rn×n }, then for any integer k ≥ m + 1 and any polynomial q(x) of form ∑k i=m+1 qi (x), where qi (x) is a homogeneous multivariate polynomial of degree i, there is a neighborhood U of x = 0 such that the sum p(x) + q(x) is positive definite in {x : x ∈ U ∩ W}. Proof. First, we transform the Cartesian coordinates (x1 , · · · , xn ) to polar coordinates (r, θ1 , · · · , θn−1 ). Letting x1 = rcosθ1 x2 = rsinθ1 cosθ2 , · ·· xn = rsinθ1 sinθ2 · · · sinθn−2 sin θn−1 where 0 ≤ r < +∞, 0 ≤ θ1 , · · · , θn−2 ≤ π, 0 ≤ θn−1 ≤ 2π, we can represent p(x) as p(r, θ) = ∑ rϕ1 (θ) + r2 ϕ2 (θ) + r3 ϕ3 (θ) + · · · + rm ϕm (θ) and q(x) as q(r, θ) = ki=m+1 ri ϕi (θ). 5
Since W is defined to be {x : Ex ≥ 0, E ∈ Rn×n }, we can find θ1,1 , θ1,2 , · · · , θn−1,1 and θn−1,2 such that W is equivalent to {(r, θ) : 0 ≤ r < +∞, θ1,1 ≤ θ1 ≤ θ1,2 , · · · , θn−1,1 ≤ θn−1 ≤ θn−1,2 }, where θ = (θ1 , · · · , θn−1 ). Letting Ω0 = {θ : θ1,1 ≤ θ1 ≤ θ1,2 , · · · , θn−1,1 ≤ θn−1 ≤ θn−1,2 } and f (r, θ) = p(r, θ) + q(r, θ), for proving our theorem, it is sufficient to only prove that there is an l > 0 such that f (r, θ) is positive in {(r, θ) : 0 < r < l, θ ∈ Ω0 }. Since p(r, θ) is positive definite, then there is no θ ∈ Ω1 such that ϕ1 (θ) = 0, · · · , and ϕm (θ) = 0. Let Ωi = {θ ∈ Ω0 : ϕ1 (θ) = 0, · · · , ϕi (θ) = 0}, where 1 ≤ i ≤ m. Clearly, Ωm = ∅ and for each i ∈ {1, . . . , m − 1}, Ωi−1 is compact. Moreover, Ωm−1 ⊆ Ωm−2 ⊆ · · · ⊆ Ω1 ⊆ Ω0 and there is an index I such that 1 ≤ I ≤ m, ΩI = ∅ and ΩI−1 , ∅. If I = 1, then for all θ ∈ Ω0 , ϕ1 (θ) , 0. Due to the assumption that p(x) is positive definite, ϕ1 (θ) > 0 for all θ ∈ Ω0 . Since Ω0 is compact, there exists a ε > 0 such that ϕ1 (θ) ≥ ε, implying that for all θ ∈ Ω0 , f (r, θ) ≥ r(ε + rϕ2 (θ) + · · · + rk−1 ϕk (θ)). Thus, due to the continuity of rϕ2 (θ) + · · · + rk−1 ϕk (θ), there is an l > 0 such that f (r, θ) is positive in {(r, θ) : 0 < r < l, θ ∈ Ω0 }. So, we only need to consider the case that I > 1 as follows. 1. Firstly, we get a subset ω′ from the set ω = {0, 1, · · · , I} using the following strategy: (a) let s := I − 1 and ω′ := ω; (b) if s ≥ 1 and there exists a k such that Ω s = Ωk , where 0 ≤ k < s, then ω′ := ω′ \ {s}, s := s − 1 and return to (b). Denote ω′ as {I1 , · · · , I j , I j+1 }, where I1 < · · · < I j+1 and j > 1. Clearly, I1 = 0, I j+1 = I and for each k = 2, · · · , j + 1 and for each s satisfying Ik−1 < s < Ik , ϕ s (θ) ≡ 0 in ΩIk−1 . 2. For all θ ∈ ΩI j , p(r, θ) = r I j (ϕI j (θ) + rϕI j +1 (θ) + · · · + rm−I j ϕm (θ)). Due to the assumption that p(x) is positive definite, for all θ ∈ ΩI j , ϕI j (θ) > 0. Since ΩI j is compact, there exists a εI j > 0 such that ϕI j (θ) ≥ εI j . Thus, due to the continuity of εI ϕI j (θ), there is an open set HI j such that ΩI j ( HI j and for all θ ∈ HI j , ϕI j (θ) ≥ 2j . So, for all θ ∈ HI j ∩ ΩI j−1 , f (r, θ) = r I j (ϕI j (θ) + r I j+1 −I j ϕI j+1 (θ) + · · · + rk−I j ϕk (θ)) ≥ εI r I j ( 2j + r I j+1 −I j ϕI (θ) + · · · + rk−I j ϕk (θ)). Due to the continuity of r I j+1 −I j ϕI (θ) + · · · + k−I j r ϕk (θ), there exists rI j+1 > 0 such that f (r, θ) is positive in {(r, θ) : 0 < r < rI j+1 , θ ∈ HI j ∩ ΩI j−1 }. 3. Let Ω′I j−1 = ΩI j−1 \ HI j . Then, ϕI j−1 (θ) , 0 in Ω′I j−1 . Due to the assumption that p(x) is positive definite, ϕI j−1 (θ) > 0 in Ω′I j−1 . Since Ω′I j−1 is also compact, there exists a εI j−1 > 0 such that ϕI j−1 (θ) ≥ εI j−1 . Thus, there is an open set HI j−1 such that εI Ω′I j−1 ( HI j−1 and for all θ ∈ HI j−1 , ϕI j−1 (θ) ≥ 2j−1 . Similarly, if I j−1 > I1 , then for all θ ∈ HI j−1 ∩ ΩI j−2 , f (r, θ) = r I j−1 (ϕI j−1 (θ) + εI r I j −I j−1 ϕI j (θ) + · · · + rk−I j−1 ϕk (θ)) ≥ r I j−1 ( 2j−1 + r I j −I j−1 ϕI j (θ) + · · · + rk−I j−1 ϕk (θ)). Thus, there exists rI j > 0 such that f (r, θ) is positive in {(r, θ) : 0 < r < rI j , θ ∈ HI j−1 ∩ ΩI j−2 }. 4. Due to deduction, we can prove that there exist an open set HIi and rIi+1 > 0 such that f (r, θ) is positive in {(r, θ) : 0 < r < rIi+1 , θ ∈ HIi ∩ ΩIi−1 }, where i = 2, · · · , j. 5. According to the assumption that p(x) is positive definite, ϕI2 (θ) > 0 in ΩI1 \ j−1 j−1 ∪k=1 (ΩIk ∩ HIk+1 ). Since ΩI1 \ ∪k=1 (ΩIk ∩ HIk+1 ) is also compact, there exists a εI2 > 0 j−1 such that ϕI2 (θ) ≥ εI2 for all θ ∈ ΩI1 \ ∪k=1 (ΩIk ∩ HIk+1 ). Thus, there exists rI2 > 0 j−1 such that f (r, θ) is positive in {(r, θ) : 0 < r < rI2 , θ ∈ ΩI1 \ ∪k=1 (ΩIk ∩ HIk+1 )}. 6. Let l = min{rI2 , · · · , rI j+1 }. Since f (r, θ) is positive in {(r, θ) : 0 < r < rIi+1 , θ ∈ HIi ∩ ΩIi−1 }, where i = 2, · · · , j, f (r, θ) is positive in {(r, θ) : 0 < r < l, θ ∈ Ω0 }. Theorem 2 states that a polynomial’s local positive definiteness can be assured by the positive definiteness of its truncated lower-order terms. Thus, based on Theorems 1 and 2, we can construct an algebraizable sufficient condition — which can then be used to compute 6
multiple Lyapunov functions in quadratic forms by a real root classification and projection operator based algebraic approach in Section 5 — as follows. Theorem 3. For a given switched hybrid system PS, if there exist functions Vi (x) in quadratic forms, i ∈ M, such that 1. for each i ∈ M and each j ∈ {1, . . . , li }, (1a) ∀x[x ∈ Xi ∧ x , 0 ⇒ Vi (x) , 0], and [ ] (1b) ∃y, z Ei, j y ≥ 0 ∧ Ei, j z ≥ 0 ∧ (y, z) , ∥y∥∥z∥ ∧ Vi (y) > 0 ∧ Vi (z) > 0 , 2 where (·, ·) is the inner product of vectors, 2. for each i ∈ M, there exists a positive integer mi,c ≤ mi 3 such that for each j ∈ {1, . . . , li }, ∑mi,c ′ (2a) ∀x[x ∈ Xi ∧ x , 0 ⇒ k=2 vi,k (x) , 0], and ∑mi,c ′ ∑mi,c ′ [ ] (2b) ∃y, z Ei, j y ≥ 0 ∧ Ei, j z ≥ 0 ∧ (y, z) , ∥y∥∥z∥ ∧ k=2 vik (y) < 0 ∧ k=2 vik (z) < 0 , where mi is the degree of V˙ i = (∇Vi (x))T · fi (x) and v′ik (x) is the sum of all monomials of degree k in V˙ i . 3. for each i, j ∈ M and each r ∈ {1, . . . , li, j }, ∀x[gTi, j,r x = 0 ⇒ Vi (x) − V j (x) ≥ 0], then the family {Vi (x) : i ∈ M} is a multiple Lyapunov function, implying that PS is locally asymptotically stable. Proof. For each i ∈ M, since Vi (x) is a quadratic form, it is clear that Vi (0) = 0 and d V (0) = 0. i dt Due to the condition (1a), for each i ∈ M, 0 is the unique solution of Vi (x) = 0 in Xi . We want to prove that for each i ∈ M and each x ∈ Xi , if x , 0, then Vi (x) > 0. 4 Suppose that for each i ∈ M, there is a point w ∈ Xi such that w , 0 and Vi (w) < 0. Thus, there is a j ∈ {1, . . . , li } such that w ∈ {x : Ei, j x ≥ 0}. For this j, according to the condition (1b), there are two points y and z such that (y, z) , ∥y∥∥z∥, Ei, j y ≥ 0, Ei, j z ≥ 0, Vi (y) > 0 and Vi (z) > 0. Letting h(t) = w + t(y − w), where h(0) = w and h(1) = y, we have Vi (h(0)) < 0 and Vi (h(1)) > 0. So there exists a t′ ∈ (0, 1) such that Vi (h(t′ )) = 0, implying that w+t′ (y−w) = 0 or w + t′ (y − w) < Xi . Since the set {x : Ei, j x ≥ 0} is convex and connected, we have: w + t′ (y − w) ∈ {x : Ei, j x ≥ 0} ⊂ Xi . Thus, w + t′ (y − w) = 0. Similarly, letting g(t) = w + t(z − w), there is a t′′ ∈ (0, 1) such that Vi (g(t′′ )) = 0, implying that w + t′′ (z − w) = 0 or w + t′′ (z − w) < Xi . Since the set {x : Ei, j x ≥ 0} is convex and connected, we have: w + t′′ (z − w) ∈ Xi . Thus, w + t′′ (z − w) = 0. Hence, (y, z) = ∥y∥∥z∥, contradicting with the assumption that (y, z) , ∥y∥∥z∥. So, for all x ∈ {x ∈ Xi |x , 0}, Vi (x) > 0. ∑ c ′ Similarly, according to the conditions (2a) and (2b), we can get mj=2 vi j (x) < 0 for all x ∈ {x : x , 0 ∧ Xi }. From Theorem 2, there is a neighborhood Ui of the origin such that for all x ∈ {x ∈ Xi ∩ U|x , 0}, V˙ i (x) < 0. Letting U = ∩i Ui , since SSi, j ⊂ S i, j , from Theorem 1, the family {Vi (x) : i ∈ M} is a multiple Lyapunov function of PS, implying that PS is locally asymptotically stable. 4. Computing Multiple Lyapunov Functions. In Section 3, for a given switched hybrid system PS, we have derived an algebraizable sufficient condition for local asymptotic Ei, j = (a)1×1 and a , 0, the condition (1b) should be replaced by ∃x[x , 0∧ax ≥ 0∧Vi (x) > 0] ∑mi,c ′ and the condition (2b) should be replaced by ∃x[x , 0 ∧ ax ≥ 0 ∧ k=2 vik (x) < 0]. 3 Due to Theorem 2, we here try to use the negative definiteness of the lower-order terms of V ˙ i to assure the local negative definiteness of V˙ i . 4 Note that for the case that E = (a) i, j 1×1 and a , 0, according to the replaced condition, we can also prove that for each i ∈ M and each x ∈ Xi , if x , 0, then Vi (x) > 0. That is, if there is a point y , 0 such that y , x, ay ≥ 0 and V(y) < 0, due to the zero point theorem, then there is a point z , 0 such that az ≥ 0 and V(z) = 0, contradicting the condition (1a). 2 Note that when
7
stability analysis, which is formulated as Theorem 3. However, there are still no efficient methods in the literature for directly finding such a multiple Lyapunov function. In this section, we will under-approximate the constraints in Theorem 3 respectively in the sense that every solution of the under-approximation is also a solution of the original condition, formulate these under-approximations as a semi-algebraic set, and then compute a sample point in the semi-algebraic set to form a multiple Lyapunov function. ∑mi,c ′ For each i ∈ M, we represent Vi (x) = xT Pi x as Li (x, pi ) and k=2 vik (x) as L˙ i,mi,c (x, pi ), where Pi is a n × n symmetric matrix and pi is a vector formed by all the entries of Pi . Let us start with solving the condition (3). Since a switch from mode i to mode j occurs li, j T on the hyperplanes defined by ∨r=1 gi, j,r x = 0, where i , j and gi, j,r , 0, the corresponding l
i, j switching condition can be over-approximated by S i, j = ∪r=1 {x : gTi, j,r x = 0}. Thus, for all x ∈ T {x : gi, j,r x = 0}, by replacing a certain xh with a linear combination of the components of z, we can formulate Vi (x, pi )−V j (x, p j ) as zT Qi jr z, where 1 ≤ h ≤ n, z = (x1 , · · · , xh−1 , xh+1 , · · · , xn ) and Qi jr is a symmetric matrix depending on pi and p j . Let
H(λ) = λn + cn−1 (pi , p j )λn−1 + · · · + c0 (pi , p j ) be the characteristic polynomial of Qi jr . To make the condition (3) hold, that is, zT Qi jr z ≥ 0 for all z ∈ Rn−1 , due to the Descartes rule [47], it is equivalent to finding pi and p j such that Λ3,i, j (pi , p j ) holds, where l
i, j Λ3,i, j (pi , p j ) = ∧r=1 ∨n−1 k=0
[[
] [ ]] n−1 n−l ∧k−1 l=0 cl (pi , p j ) = 0 ∧ ∧l=k (−1) cl (pi , p j ) > 0 .
Notice that the condition (1) and the condition (2) have similar formulae and Vi (x) in ∑mi,c ′ the condition (1) and k=2 vi,k (x) in the condition (2) have degree of at least 2. We can first consider under-approximating the conditions (2a) and (2b) and then apply the same approach for the conditions (1a) and (1b). For under-approximating the condition (2a), for each i, we equivalently transform ∀x[[x ∈ Xi ∧ x , 0] ⇒ L˙ i,mi,c (x, pi ) , 0] i to ∧lj=1 ϕi, j (pi ), where ϕi, j (pi ) = [∀x[[Ei, j x ≥ 0 ∧ x , 0] ⇒ L˙ i,mi,c (x, pi ) , 0]]. Further, for each j ∈ {1, . . . , li }, ϕi, j (pi ) is equivalently transformed to ∧nk=1 ϕi, j,k (pi ), where ϕi, j,k (pi ) = [∀xk [[Ei, j (0, · · · , 0, xk , · · · , xn ) ≥ 0 ∧ xk , 0] ⇒ L˙ i,mi,c (0, · · · , 0, xk , · · · , xn , pi ) , 0]]. Assume that, for each ϕi, j,k (pi ), we have a real root classification and projection operator based method — which will be discussed later in Subsection 5.2 — to get an under-approximative constraint m m i Λ1,i,i,cj,k (pi ) that only involves pi . Thus, we can obtain a big constraint ∧lj=1 ∧nk=1 Λ1,i,i,cj,k (pi ), m denoted as Λ1,ii,c (pi ). For solving the condition (2b), we first compute two points y and z 5 in the region defined by the set {x : Ei, j x ≥ 0} as follows: if the rank of Ei, j is less than n, let y be a nonzero solution of Ei, j x = 0 and z = −y; otherwise, let y be the solution of Ei, j x = (1, 0, . . . , 0)T and z be the solution of Ei, j x = (0, 1, 0, . . . , 0)T . Then, we simply replace the condition m (2b) by the constraint [L˙ i,mi,c (y, pi ) < 0 ∧ L˙ i,mi,c (z, pi ) < 0], denoted as Λ2,i,i,cj . Note that such a replacement will not lose information, which will be analyzed in Section 6. We denote m m i ∧lj=1 Λ2,i,i,cj as Λ2,ii,c (pi ). Similarly, for each i ∈ M, we can under-approximate the condition (1a) and (1b) in Theorem 3 to get the constraints ΛV1,ii (pi ) and ΛV2,ii (pi ), respectively.
5 When
Ei, j = (a)1×1 , we can choose ±1 as two points if a = 0 and sgn(a) as a point if a , 0. 8
Thus, we get the following conjunction ] [ V m m N ∧i=1 Λ1,ii ∧ ΛV2,ii ∧ Λ1,ii,c ∧ Λ2,ii,c ∧ [∧Nj=1 Λ3,i, j ] , which can be formulated as a set of semi-algebraic sets. Note that for a given semi-algebraic set, we can compute a sample point in this semi-algebraic set by SASsolver, which will be described in Subsection 5.1. Therefore, we have Algorithm 1 for computing multiple Lyapunov functions in quadratic forms as follows. Algorithm 1 Computing MLFs in quadratic form Input: A given switched hybrid system PS. Output: Asymptotically stable or Unknown. 1: for each mode i ∈ M, choose a quadratic form Vi (x, pi ), where pi is a parameter. 2: for m1,c = 2 : 1 : m1 do 3: ··· 4: for mN,c = 2 : 1 : mN do [ V ] m m N 5: compute the conjunction ∧i=1 Λ i ∧ ΛVi ∧ Λ i,c ∧ Λ i,c ∧ [∧Nj=1 Λ3,i, j ] . [ V 1,i V 2,i m 1,i m 2,i ] N 6: apply SASsolver to ∧i=1 Λ1,ii ∧ Λ2,ii ∧ Λ1,ii,c ∧ Λ2,ii,c ∧ [∧Nj=1 Λ3,i, j ] . 7: if SASsolver returns a sample point over p1 , . . . pN then 8: put it into Vi and return {Vi (x) : i ∈ M} as a MLF, associated with “Locally Asymptotically Stable”, and halt. 9: end if 10: end for 11: ··· 12: end for 13: return Unknown. Remark 1. In Algorithm 1, m1,c , · · · , mN,c are required to increase with 1 each step instead of 2 used in [52] since V˙ i (x) is required to be negative definite in Xi instead of Rn , where i = 1, · · · , N. 5. Our Real Root Classification based Algebraic Approach. In this section, we will explain how to use the real root classification with a projection operator to get our underm approximative constraints Λ1,i,i,cj,k required in Algorithm 1 and how to solve a constant semialgebraic system with an adaptive CAD. 5.1. Real Root Classification and our Adaptive CAD. In this subsection, we will describe the real root classification for solving parametric semi-algebraic systems and the adaptive cad method for solving constant semi-algebraic systems. A semi-algebraic system (or short, sas) is a system of form {p1 (x, u) = 0, · · · , p s (x, u) = 0, g1 (x, u) ≤ 0, · · · , gr (x, u) ≤ 0, gr+1 (x, u) < 0, · · · , gt (x, u) < 0, h1 (x, u) , 0, · · · , hm (x, u) , 0}, where the variable x = (x1 , · · · , xn ) ranges in Rn , the parameter u = (u1 , · · · , ud ) ranges in Rd , and pi , g j , hk are polynomials in Q[x, u]. We can write the system as [P, G1 , G2 , H],
(5.1)
where P, G1 ,G2 and H stand for [p1 , · · · , p s ], [g1 , · · · , gr ], [gr+1 , · · · , gt ] and [h1 , · · · , hm ], respectively. An sas is called a constant semi-algebraic system if it contains no parameters, i.e., d = 0; otherwise, a parametric semi-algebraic system. 9
For a given parametric sas S , one can decompose the parametric space Rd into finitely many semi-algebraic sets such that on each of those semi-algebraic sets S has either positive dimensional real solutions in Rn or a constant number of real solutions [66, 67]. Therefore, one can determine the sufficient and necessary condition on the parameters such that S has a prescribed number of distinct real solutions and the condition on the parameters such that the dimension of real solutions of S in Rn is positive. A collection of all possible cases of the real solutions of S , together with corresponding sufficient and necessary conditions on the parameter for all cases, is called a complete real root classification (CRRC) of S . To compute a CRRC for a given parametric sas S , we first compute a parametric polynomial BP(u), called border polynomial,6 such that in each connected component of BP(u) , 0 in the parametric space, S has a constant number of real solutions. We can further decompose the parameter on the “border” BP(u) = 0 by adding the equality into the original system S and calling a similar procedure. By so doing, we can obtain a new border polynomial BP1 (u) for the new system such that on each connected component of the constructible set {BP(u) = 0, BP(u)1 , 0} in the parametric space, S has a constant number of real solutions. Repeating this procedure, we can obtain a decomposition of the parametric space. Then, to determine the number of distinct real solutions of S in each cell of the decomposition, it suffices to check the situation at one point of this cell. So we may take a sample point from each cell and isolate the real solutions of S at the sample point. Finally, the defining formulae of the cells together with the numbers of real solutions of S at the sample points form a complete real root classification of S . The interested reader may consult [66, 67] for details. Note that DISCOVERERis a software developed by Xia [63] using Maple, which can solve the CRRC problem for parametric sass. DISCOVERER has been integrated into the Maple package RegularChains [9] since Maple 13. To use the features of solving CRRC problem in Maple, one may first type in the following commands orderly: > with(RegularChains): > with(ParametricSystemTools): > with(SemiAlgebraicSetTools): > infolevel[RegularChains]:=1: Then, for a parametric sas S of the form (5.1) and a given non-negative integer N , to determine the necessary and sufficient conditions on u such that the number of distinct real solutions of S equals N , one can orderly type in > R := PolynomialRing([x, u]); > rrc := RealRootClassification(P, G1, G2, H, d, N, R); where R is a structure encoding the polynomial ring involving x and u, and d is the number of variables that we want to treat as parameters. Typically, d is the size of the u vector. The output of RealRootClassification is a quantifier-free formula Φ in parameters and a border polynomial BP(u) such that, provided BP(u) , 0, S has exactly N distinct real solutions if and only if Φ holds. Further, letting P′ = [p1 , ..., p s , BP], one can call > rrc1 := RealRootClassification(P’, G1, G2, H, d, N, R); to find conditions on u when the parameters are on the “border” BP(u) = 0. For better understanding the theory and implementation of real root classification, we provide an example with an intuitive illustration of the border polynomial as follows. 6 BP(u) is called a border polynomial if for two arbitrary but fixed points a and b in the same connected component of BP(u) , 0 in the parametric space Rd , the corresponding constant semi-algebraic systems, obtained by replacing the parameter u with the fixed points a and b respectively , have the same constant number of real solutions.
10
Example 3. Consider the following simple system: p1 (x1 , x2 , a) = x1 + a, p2 (x1 , x2 , a) = x22 + ax1 + 1, where x1 and x2 are variables and a is a parameter. For this system, the border polynomial is BP(a) = (a − 1)(a + 1). Letting N = 2 and following the above commands, we can obtain that the system has two distinct real roots if and only if a > 1 or a < −1. According to Algorithm 1, we first use real root classification based method to obtain [ V ] m m N the conjunction ∧i=1 Λ1,ii ∧ ΛV2,ii ∧ Λ1,ii,c ∧ Λ2,ii,c ∧ [∧Nj=1 Λ3,i, j ] ; then, we have to determine whether the conjunction holds and compute a sample point if it holds. Note that a sample point here is a point which meets the conjunction. For determining whether the conjunction holds, we can equivalently reformulate it as a disjunction of conjunctions, denoted as ∨ s ∧t h s,t (p)∆0, where h s,t (p)s are polynomials and ∆ ∈ {=, ≤, 0∧16c2 −12b2 +9a2 > 0∧[3b2 −8a2 ≥ 0∨2b2 −3a2 < 0∨ 4a2 c2 −b22 > 0]]∧ [12c2 −65b2 +152a2 > 0∧40c2 −79b2 +96a2 > 0∧[169b2 −376a2 < 0 ∨ c22 + 2b2 c2 − 34a2 c2 + 10b22 + 2a2 b2 + a22 < 0 ∨ 85b2 − 292a2 > 0 ∨ c2 + b2 − 17a2 > 9 4 16 4 9 a1 − 34 b1 + c1 − 16 a2 + 34 b2 − c2 ≤ 0 ∧ 16 0]] ∧ [ 16 9 a2 − 3 b2 + c2 − 9 a1 + 3 b1 − c1 ≤ 0] for Example 4; 18
2. the program returns [5b1 − a1 < 0 ∧ 100c21 − 2004a1 c1 + 1001b21 + 10000a21 < 0 ∧ 100c2 + 40b2 − 9a2 < 0 ∧ 4c2 + 2b2 + a2 > 0 ∧ 28c2 + 4b2 − 3a2 > 0 ∧ a1 − 10b1 + 100c1 − a2 + 10b2 − 100c2 ≥ 0 ∧ a2 + 2b2 + 4c2 − a1 − 2b1 − 4c1 ≥ 0] for Example 5; 3. the program returns “aborted” for Example 6 in several seconds; 4. the program cannot halt for Example 7 and 8 within five hours. Thirdly, according to the theory in [38], we use a bilinear programming solver PENBMI [23] to compute {Vi (x) : i ∈ M} with the precision 10−6 such that for all i ∈ M, j ∈ {1, · · · , li }, h ∈ M and r ∈ {1, · · · , li,h }, s1,i, j (x)Vi (x) −
n ∑ k=1
s1,i, j,k ei, j,k x − εi, j
n ∑ k=1
′
xkd ∈
∑ , n
−[(1 − Vi (x))s2,i, j (x) + (∇Vi (x)) · fi (x)s3,i, j (x)+ n n ∑ ∑ ∑ ′′ s4,i, j,k (x)ei, j,k x + ε′i, j xkd ] ∈ , T
k=1
k=1
s′i,h,r (x)Vi (x)
(7.2)
n
∑ + pi,h,r (x)gi,h,r x − Vh (x) ∈ , n
where εi, j > 0, ε′i, j > 0, d′ is the degree of s1,i, j (x)Vi (x), d′′ is the degree of Vi (x)s2,i, j (x), ei, j,k ∑ is the k-th row of Ei, j , pi,h,r is a polynomial, n is the set of polynomials that can be expressed ∑ as a sum of squares of polynomials, and s1,i, j , s1,i, j,k , s2,i, j , s3,i, j , s4,i, j,k and s′i,h,r ∈ n . By applying PENBMI for the above formulation (7.2) with the degrees listed in the appendix, we obtained the following computation results: 1. the program returns V1 (x1 , x2 ) = 1.6176 × 103 x12 + 0.3872 × 103 x1 x2 + 2.0425 × 103 x22 and V2 (x1 , x2 ) = 1.8017 × 103 x12 + 1.0178 × 103 x1 x2 + 1.0089 × 103 x22 for Example 4; 2. the program returns V1 (x1 , x2 ) = 0.1435 × 103 x12 − 0.0174 × 103 x1 x2 + 1.3721 × 103 x22 and V2 (x1 , x2 ) = 8.3604 × 103 x12 + 5.0908 × 103 x1 x2 + 3.6582 × 103 x22 for Example 5; 3. the program returns “PENBMI failed” for Example 6; 4. for Example 7, the program returns V1 (x1 , x2 ) = 212.4043x12 + 419.6675x22 and V2 (x1 , x2 ) = 238.5895x12 − 278.8058x1 x2 + 762.2790x22 . Also, we get s2,2,1 = 0.0851 × 10−9 x12 − 0.2232 × 10−9 x1 x2 − 0.1689 × 10−9 x22 and s2,2,2 = 0.0851 × ∑ 10−9 x12 − 0.2232 × 10−9 x1 x2 − 0.1689 × 10−9 x22 . However, s2,2,1 , s2,2,2 < n and thus the computation is numerically unreliable; 5. the program returns “PENBMI failed” for Example 8. Summing up the above computation results, we know that our approach is more applicable to some extent than the LMI based method, more efficient to some extent than the generic QE based method, and more reliable and applicable to some extent than the SOS based method. The reasons may lie in the following facts: 1. LMI is unpractical for degenerate systems; 2. the generic QE techniques [17, 7] can be theoretically used to produce an equivalent and quantifier-free formula and our method is applied to a special structure for producing a sufficient condition. Further, QE uses CAD while DISCOVERER uses an adaptive CAD where some variables can be eliminated by Ritt-Wu’s method. 3. LMI and SOS both use floating computation such that certain non-negative polynomials cannot be practically formulated as sums of squares. Moreover, the results obtained by floating computation may be numerically unreliable, e.g., Example 7. 8. Discussions. According to the classical theorem on multiple Lyapunov functions [27], ∑mi,c ∑mi,c when we replace k=2 by k=1 in Theorem 3 and let m1,c , . . . , mN,c start from 1 in Algorithm 1 instead of 2, our proposed algebraic approach is also applicable for systems that do not sat19
isfy fi (0) = 0, e.g., switched affine systems [27]. This can also be obtained from the proof of ∑mi,c ′ Theorem 3 since there is no constant term in V˙ i and k=1 vik (0) = 0 if Vi (x) is of quadratic form. Such an applicability can be seen from the following example. Example 9. Consider a simple switched affine system: x˙ = fi (x), i ∈ M = {1, 2, 3, 4}, ) ( ) ( ) ( ) x1 + x2 − 1 x1 + x2 + 1 x1 + x2 + 1 x1 + x2 − 1 where f1 (x) = , f (x) = ; f (x) = ; f (x) = ; x1 − x2 − 1 2 x1 − x2 + 1 4 x1 − x2 − 34 3 x1 − x2 + 43 X1 = {x( : E1,1) x ≥ 0}, X(2 = {x): E2,1 x ≥ (0}, X3 =) {x : E3,1 x( ≥ 0}, )X4 = {x : E4,1 x ≥ 0}, 1 0 1 0 −1 0 −1 0 E1,1 = , E2,1 = ; E3,1 = , E4,1 = ; gT1,2 = (0, 1), gT2,3 = 0 1 0 −1 0 −1 0 1 (1, 0), gT3,4 = (0, 1), gT4,1 = (1, 0). Clearly, this systems is globally stable, which can be seen from Fig. 3. (
0.15
0.1
0.05
x2
0
−0.05
−0.1
−0.15
−0.2 −0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
x1
Fig. 3. A trajectory of Example 9 starting from (0.1,0.1). Suppose that Vi (x) = xT Pi x, where Pi (i = 1, 2, 3, 4) is symmetric. Using our just menT tioned approach, we √can √ Lyapunov function, get {Vi (x) = x P√i x, i = 1, 2,3, 4} as√a multiple 3 23 13 3 7 14 7 13 23 7 − − 8 4 4 16 , P2 = √8 , P3 = √16 , P4 = 32√ where P1 = 3 √8 7 . 13 3 23 14 9 9 − 1 1 − 8 8 4 4 16 8 Thus, for a given PS, the assumption that fi (0) = 0 for all i may not be required in this ∑i section. Moreover, let Vi (x) = dk=2 vik (x) be a polynomial of degree di with di ≥ 2, where vik (x) is the sum of all terms of degree k in Vi (x), and represent V˙ i = dtd Vi (x) = (∇Vi (x))T ·fi (x) ∑ i ′ ′ ˙ as m k=1 vik (x), where mi is the degree of Vi (x) and vik (x) is the sum of all terms of degree k in ˙ Vi . Then, we can construct an algebraizable sufficient condition for the existence of multiple Lyapunov functions beyond quadratic forms as follows. Theorem 5. For a given PS that may not satisfy fi (0) = 0 for all i, if there exist functions ∑i Vi (x) = dj=2 vi j (x) of degree di with di ≥ 2, i ∈ M, such that 1. for each i ∈ M, there exists a positive integer di,c ≤ di such that for each j ∈ {1, . . . , li }, ∑di,c (1a) ∀x[x ∈ Xi ∧ x , 0 ⇒ k=2 vik (x) , 0], and ∑di,c ∑di,c [ ] (1b) ∃y, z Ei, j y ≥ 0 ∧ Ei, j z ≥ 0 ∧ (y, z) , ∥y∥∥z∥ ∧ k=2 vik (y) > 0 ∧ k=2 vik (z) > 0 , 8 where (·, ·) is the inner product of vectors, 2. for each i ∈ M, there exists a positive integer mi,c ≤ mi such that for each j ∈ {1, . . . , li }, ∑di,c
8 Note
that when Ei, j = (a)1×1 and a , 0, the condition (1b) should be replaced by ∃x[x , 0 ∧ ax ≥ 0 ∧ ∑mi,c ′ v (x) < 0]. k=1 ik
v (x) > 0] and the condition (2b) should be replaced by ∃x[x , 0 ∧ ax ≥ 0 ∧ k=2 ik 20
∑mi,c ′ vik (x) , 0], and (2a) ∀x[x ∈ Xi ∧ x , 0 ⇒ k=1 ∑mi,c ′ ∑mi,c ′ [ ] (2b) ∃y, z Ei, j y ≥ 0 ∧ Ei, j z ≥ 0 ∧ (y, z) , ∥y∥∥z∥ ∧ k=1 vik (y) < 0 ∧ k=1 vik (z) < 0 , 3. for each i, j ∈ M, ∀x[x ∈ S i, j ⇒ Vi (x) − V j (x) ≥ 0], then the family {Vi (x) : i ∈ M} is a multiple Lyapunov function, implying that PS is locally asymptotically stable. Proof. Although fi (0) , 0 may hold for some certain i, since all terms in Vi (x) have ∑mi,c ′ degree no less than 2, there is no constant term in V˙ i and k=1 vi,k (0) = 0. Thus, following the proof of Theorem 3, we can directly prove this theorem. ∑i Remark 4. For certain i such that fi (0) = 0, we can let Vi (x) = dk=1 vik (x) be a polyno∑di,c ∑di,c mial of degree di with di ≥ 1. Accordingly, we need to replace k=2 by k=1 in the conditions ∑di,c (1a) and (1b) of Theorem 5. Since k=1 vik (0) = 0, there is no constant term in V˙ i and ∑mi,c ′ (0) = 0, by applying Theorem 2 and following the proof of Theorem 3, we can also v k=1 ik obtain that such a replacement is still correct. For the conditions (1a), (1b), (2a) and (2b) in Theorem 5, after applying our real root classification based introduced in] Subsection 5.2, we can get an under-approximative [ V method m m Vi N i constraint ∧i=1 Λ1,i ∧ Λ2,i ∧ Λ1,ii,c ∧ Λ2,ii,c . For solving the condition (3) in Theorem 5, we start with the following preparations: 1. for every gi, j,r used for defining S i, j , we first conservatively replace ∀x[gTi, j,r x = 0 ⇒ Vi (x) − V j (x) ≥ 0] by (a) ∀x[gTi, j,r x = 0 ∧ x , 0 ⇒ Vi (x) − V j (x) > 0], or (b) ∀x[gTi, j,r x = 0 ⇒ Vi (x) − V j (x) ≡ 0]. 2. for all x ∈ {x : gTi, j,r x = 0}, by replacing a certain xh with the linear combination of the components of z, we then formulate Vi (x, pi ) − V j (x, p j ) as Vi, j,r (z, pi , p j ), where z = (x1 , · · · , xh−1 , xh+1 , · · · , xn ). Then, for the condition ∀x[gTi, j,r x = 0 ∧ x , 0 ⇒ Vi (x) − V j (x) > 0], we can apply our real root classification based method introduced in Subsection 5.2 or the method in [52] to get an under-approximative constraint Λ3,i, j,r,1 for ∀x[x , 0 ⇒ Vi, j,r (z, pi , p j ) > 0]; for the condition ∀x[gTi, j,r x = 0 ⇒ Vi (x) − V j (x) = 0], we can get an equivalent constraint Λ3,i, j,r,2 for l
i, j the identity Vi, j,r (z, pi , p j ) ≡ 0. So, we get Λ3,i, j,r = Λ3,i, j,r,1 ∨ Λ3,i, j,r,2 and Λ3,i, j = ∧r=1 Λ3,i, j,r . [ V ] m m N Clearly, if a sample point {p1 , · · · , pN } makes ∧i=1 Λ1,ii ∧ ΛV2,ii ∧ Λ1,ii,c ∧ Λ2,ii,c ∧ [∧Nj=1 Λ3,i, j ] holds, then {p1 , · · · , pN } forms a multiple Lyapunov function. Furthermore, our current algebraic approach is also applicable for non-polynomial systems by considering their corresponding Taylor expansions as follows. Theorem 6. For a given non-polynomial system of form x˙ = fi (x), where i ∈ M, such that ∑i fi ∈ C ∞ and fi (0) = 0, Let Vi (x) = dk=1 vik (x) be a polynomial of degree di with di ≥ 1 and its corresponding time-derivative w.r.t mode i can be expressed by V˙ i = dtd Vi (x) = (∇Vi (x))T · ∑ ′ ′ fi (x) = ∞ k=1 vik (x), where vik (x) is the sum of all terms of degree k in the Taylor expansion of d ˙ Vi = dt Vi (x) at the origin. If these polynomials Vi and their corresponding time-derivatives V˙ i w.r.t mode i satisfy: 1. for each i ∈ M, there exists a positive integer di,c ≤ di such that for each j ∈ {1, . . . , li }, ∑di,c (1a) ∀x[x ∈ Xi ∧ x , 0 ⇒ k=1 vik (x) , 0], and ∑di,c ∑di,c [ ] (1b) ∃y, z Ei, j y ≥ 0 ∧ Ei, j z ≥ 0 ∧ (y, z) , ∥y∥∥z∥ ∧ k=1 vik (y) > 0 ∧ k=1 vik (z) > 0 , 9 where (·, ·) is the inner product of vectors, 2. for each i ∈ M, there exists a positive integer mi,c such that for each j ∈ {1, . . . , li },
∑di,c
9 Note
that when Ei, j = (a)1×1 and a , 0, the condition (1b) should be replaced by ∃x[x , 0 ∧ ax ≥ 0 ∧ ∑mi,c ′ v (x) < 0]. k=1 ik
v (x) > 0] and the condition (2b) should be replaced by ∃x[x , 0 ∧ ax ≥ 0 ∧ k=1 ik 21
∑mi,c ′ vik (x) , 0], and (2a) ∀x[x ∈ Xi ∧ x , 0 ⇒ k=1 ∑mi,c ′ ∑mi,c ′ [ ] (2b) ∃y, z Ei, j y ≥ 0 ∧ Ei, j z ≥ 0 ∧ (y, z) , ∥y∥∥z∥ ∧ k=1 vik (y) < 0 ∧ k=1 vik (z) < 0 , 3. for each i, j ∈ M, ∀x[x ∈ S i, j ⇒ Vi (x) − V j (x) ≥ 0], then the family {Vi (x) : i ∈ M} is a multiple Lyapunov function, implying that PS is locally asymptotically stable. Example 10. Consider a simple non-polynomial system: x˙ = fi (x), i ∈ M = {1, 2, 3, 4}, x ∈ Xi ⊂ R2 , ( ( ) ) −x2 − ln(1 + x1 )x12 + x1 x24 x2 where f1 = f3 = , f = f = ; X1 = {x : E1,1 x ≥ 0}, 2 4 −x2 + sin(x1 ) −x13 − x25 ( ) 1 0 X2 = {x : E2,1 x ≥ 0}; X3 = {x : E3,1 x ≥ 0}, X4 = {x : E4,1 x ≥ 0}; E1,1 = −E3,1 = , 0 1 ( ) −1 0 E2,1 = −E4,1 = ; g1,2 = g2,1 = (1, 0), g2,3 = g3,2 = (0, 1), g3,4 = g4,3 = (1, 0), 0 1 g4,1 = g1,4 = (0, 1). Clearly, the subsystem x˙ = f1 (x) is asymptotically stable and the subsystem x˙ = f2 (x) is unstable. Moreover, the linearization of the first subsystem has one eigenvalue equal to zero. Pre-suppose that Vi (x) = ai x1 + bi x2 , where i = 1, · · · , 4. According to Theorem 6 and Algorithm 1, we get {V1 , V2 , V3 , V4 } as a multiple Lyapunov function, where V1 (x) = x1 + x2 , V2 (x) = −x1 + x2 , V3 (x) = −x1 − x2 and V4 (x) = x1 − x2 . Remark 5. A direct extension of [52, 46] only fits systems whose subsystems are all asymptotically stable. But, for systems with certain unstable subsystems, if the projection region {x[i−1] : E[i−1] x[i−1] ≥ 0} of the state space {x[i] : E[i] x[i] ≥ 0} defined for a certain unstable subsystem is Ri−1 , our proposed approach may also fail. Moreover, there actually exists a polynomial multiple Lyapunov function for a system with a partition of the whole state space defined by certain polyhedral sets; however, since we here use under-approximative solving with the projection operator, our proposed approach possibly cannot find this polynomial. For overcoming this problem, an alternative method is to use the cylindrical algebraic decomposition [14] for obtaining an equivalent formula. But, every general algorithm for cylindrical algebraic decomposition has at least a double exponential complexity. 9. Conclusions. We in this paper proposed an algebraic approach for analyzing local asymptotic stability of switched hybrid systems, whose subsystems have polynomial vector fields. That is, we used a real root classification and projection operator based algebraic approach to arrive at a semi-algebraic set which only involves the coefficients of the preassumed quadratic forms and then solved the resulting semi-algebraic set for the coefficients. We prototypically implemented our algorithm based on DISCOVERER and tested it on five examples with comparisons to the LMI based method, the generic QE based method and the SOS based method. The computation and comparison results demonstrated the applicability and promise of our algebraic approach. Moreover, the distinguishing differences between the present paper and [52, 46] can be summarized as follows: 1. A direct extension of [52, 46] only fits systems whose subsystems are all asymptotically stable. Moreover, our current approach can be used for systems where the assumption that fi (0) = 0 for all i may not hold, e.g., Example 9. 2. Our current algebraizable sufficient condition is more complex than the one in [52, 46]. Especially, we additionally need to compute two non-collinear vectors in the corresponding state space of each mode. 3. For under-approximatively solving the constraints, we additionally apply a projection operator to assure that the eliminated state variables do not appear in the in22
termediate under-approximations so that all state variables can be thoroughly eliminated in the later steps; moreover, we use the information on each mode to further optimize our intermediate computation results at each step, obtaining simpler ones. Our short-term goal is to further optimize our algorithm and make it fully automatic, associated with complexity analysis of the algorithm [43, 44, 47, 54, 48, 1]. Our long-term goal is to verify the stability of general nonlinear hybrid systems [16] in the Lyapunov sense by computing multiple Lyapunov functions [5], or in the practical sense [24] by computing multiple Lyapunov-like functions [40, 49, 51] or constructing transition systems [39, 35, 53, 45]. Moreover, it is also interesting to investigate whether our current approach works or not for systems which may exhibit the Zeno behavior [8, 55] and admit sliding modes [62] and Filippov solutions [18, 36]. Acknowledgments. The authors deeply thank the four anonymous reviewers of this current submission for their numerous detailed suggestions and comments on improving the presentation of this paper. Also, the authors would like to thank the other four anonymous reviewers for their helpful comments on the earlier HSCC submission. REFERENCES [1] A. A. Ahmadi and P. A. Parrilo. Stability of Polynomial Differential Equations: Complexity and Converse Lyapunov Questions. http://arxiv.org/abs/1308.6833. [2] V. D. Blondel, O. Bournez, P. Koiran and J. N. Tsitsiklis. The stability of saturated linear dynamical systems is undecidable. Journal of Computer and System Sciences, 62(3): 442–462, 2001. [3] A. Beydoun, L. Y. Wang, J. Sun and S. Sivashankar. Hybrid control of automotive powertrain systems: a case study. In T. A. Henzinger, S. Sastry (Eds.), Hybrid Systems: Computation and Control, Vol. 1386, pp. 33–48, 1998. [4] S. Boyd, L. El. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [5] M. S. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transaction on Automatic Control, 43(4): 475–482, 1998. [6] C. W. Brown. Improved projection for cylindrical algebraic decomposition. Journal of Symbolic Computation, 32(5): 447–465, 2001. [7] C. W. Brown. QEPCAD B: a system for computing with semi-algebraic sets via cylindrical algebraic decomposition. ACM SIGSAM Bulletin, 38(1): 23–24, 2004. [8] M. K. Camlibel, J. S. Pang and J. L. Shen. Conewise Linear Systems: Non-Zenoness and Observability. SIAM Journal on Control and Optimization, 45(5): 1769–1800, 2006. [9] C. Chen, F. Lemaire, L. Li, M. Moreno Maza, W. Pan and Y. Xie. The ConstructibleSetTools and ParametricSystemsTools modules of the RegularChains library in Maple. In Proceedings of the International Conference on Computational Science and Applications, pp. 342–352, IEEE Computer Society Press, 2008. [10] Y. Chen, B. Xia, L. Yang and N. Zhan. Generating polynomial invariants with DISCOVERER and QEPCAD. In Proceedings of Formal Methods and Hybrid Real- Time Systems, LNCS 4700, pp. 67–82, 2007. [11] G. Chesi. On the Gap Between Positive Polynomials and SOS of Polynomials. IEEE Transaction on Automatic Control, 52(6): 1066-1072, 2007. [12] G. Chesi. Domain of Attraction: Analysis and Control via SOS Programming. Springer, 2011. [13] G. Chesi, P. Colaneri, J. C. Geromel, R. H. Middleton, R. Shorten. A Nonconservative LMI Condition for Stability of Switched Systems with Guaranteed Dwell Time. IEEE Transactions on Automatic Control, 57(5), 1297-1302, 2012. [14] G. E. Collins. Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. Leture Notes in Computer Science, Vol. 33, pp. 134–183, 1975. [15] G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12: 299–328, 1991. [16] R. A. Decarlo, M. S. Branicky, S. Pettersson and B. Lennartson. Perspective and results on the stability annd stabilizability of hybrid systems. Proceedings of the IEEE, 88(7): 1069–1081, 2000. [17] A. Dolzmann and T. Sturm. REDLOG: computer algebra meets computer logic. ACM SIGSAM Bulletin, 31(2): 2–9, 1997. [18] A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht, 1988. [19] S. Gulwani and A. Tiwari. Constraint-based approach for analysis of hybrid systems. In Proceedings of the 20th International Conference on Computer Aided Verification, pp. 190–203, 2008. 23
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25
Table 1: Degrees used for the SOS based method Example 3 4 5 6 7
d s1,1,1 2 0 2 2 2
d s1,1,1,1 2 2 2 2
d s1,1,1,2 2 2 2 -
d s1,1,2 2 2 2 -
d s1,1,2,1 2 2 2 -
d s1,1,2,2 2 2 2 -
d s1,2,1 2 2 2 2 0
d s1,2,1,1 2 2 2 2 -
d s1,2,1,2 2 2 2 2 -
Example 3 4 5 6 7
d s1,2,2 2 2 2 -
d s1,2,2,1 2 2 2 -
d s1,2,2,2 2 2 2 -
d s2,1,1 2 2 2 2 2
d s3,1,1 0 0 0 0 0
d s4,1,1,1 2 2 2 2
d s4,1,1,2 2 2 2 -
d s2,1,2 2 2 2 -
d s3,1,2 0 0 0 -
Example 3 4 5 6 7
d s4,1,2,1 2 2 2 -
d s4,1,2,2 2 2 2 -
d s2,2,1 2 2 2 2 4
d s3,2,1 0 0 0 0 0
d s4,2,1,1 2 2 2 2 -
d s4,2,1,2 2 2 2 2 -
d s2,2,2 2 2 2 -
d s3,2,2 0 0 0 -
d s4,2,2,1 2 2 2 -
d s′1,2,1 0 0 0 0 0
d s′1,2,2 0 -
d s′2,1,1 0 0 0 0 0
d s′2,1,2 0 -
d p1,2,1 1 1 1 1 1
Example 3 4 5 6 7
d s4,2,2,2 2 2 2 -
26
d p1,2,2 1 -
d p2,1,1 1 1 1 1 1
d p2,1,2 1 -
