Kobe University Repository : Kernel

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Computat ion of Lyapunov Funct ions for Hybrid Automata viaLMIs(Hybrid Systems)(Concurrent Systems and Hybrid Systems)

著者Author(s) Masubuchi, Izumi / Yabuki, Seiji / Tsuji, Tokihisa

掲載誌・巻号・ページCitat ion

IEICE transact ions on fundamentals of electronics, communicat ionsand computer sciences,E87-A(11):2937-2943

刊行日Issue date 2004-11-01

資源タイプResource Type Journal Art icle / 学術雑誌論文

版区分Resource Version publisher

権利Rights

DOI

JaLCDOI

URL http://www.lib.kobe-u.ac.jp/handle_kernel/90001337

PDF issue: 2020-06-26

lEICE TRANS. FUNDAMENTALS. VOL.E87-A. NO. II NOVEMBER 2004 2937

I PAPER Special Section on Concurrent Systems and Hybrid Systems

Computation of Lyapunov Functions for Hybrid Automata via LMls

Izumi MASUBUCHIt a), Seiji YABUKIt , and Tokihisa TSUJIt, Nonmembers

SUMMARY This paper provides a computational method to construct a Lyapunov function to prove a stability of hybrid automata that can have nonlinear vector fields. Algebraic inequalities and equations are formu­lated. which are solved via LMI optimization. Numerical examples are presented to illustrate the proposed method. key words: hybrid automata, stability analysis, sum of squares. LMls

1. Introduction

Control systems, in general, work under physical laws that the plant dynamics obeys and logic rules that are imple­mented in the controller. Recent development of computer science enables use of high-performance control, which raises control problems that high-level (logical) and low­level (e.g. , simple error feedback) control schemes can not be designed separately for their close interaction. This is a reason for which hybrid dynamical systems (HDS) have been receiving great deal of attention recently in the field of control engineering.

Stability analysis, which is one of the most important subjects in control theory, is also needed in analysis and synthesis of hybrid dynamical systems. There are consid­erable number of results that generalize notion of Lyapunov stability to HDS, where various 'Lyapunov-like' functions are proposed to prove a certain stability of trajectories of HDS (e.g., [4], [14], [15], [18]). On the other hand, com­putational methods to obtain Lyapunov functions for HDS have been presented by using convex optimization via linear matrix inequalities (LMIs) [6], [16], [17] . However, those existing results handle only piecewise affine systems while there are methodologies for stability analysis of nonlinear systems (e.g., [5], [10]-[12] for very recent results).

In this paper, we propose a method to compute Lya­punov functions for stability analysis of hybrid automata whose vector field can be nonlinear with respect to the con­tinuous variable of possibly different dimensions. We con­sider a generalized Lyapunov stability of the 'origin' and the problem of stability analysis is formulated roughly as follows: find a function which is positive, whose derivative along continuous evolution is negative and which does not increase while discrete evolution. The computational prob­lem to find such a function is reduced to an LMI optimiza-

Manuscript received April 6, 2004. Manuscript revised June 18, 2004. Final manuscript received July 22,2004.

tThe authors are with the Graduate School of Engineering, Hiroshima University, Higashi-hiroshima-shi, 739-8527 Japan.

a) E-mail : msb@mec.hiroshima-u.ac.jp

tion problem (semidefinite programming), which is solved with efficient algorithms. Numerical examples are provided to illustrate the proposed method.

2. Problem Formulation

2.1 Hybrid Automata and Their Executions

For a finite collection V of variables v, let V (in boldface) denote the set of valuations of the variables. We refer to variables whose set of valuation is finite or countable as dis­crete and to variables whose set of valuations is a subset of a Euclidean space or a manifold as continuous. We call a discrete variable mode also. In the following, we give a def­inition of hybrid automata [1], [2]. The formulation here is based on [7] , [8], [19] and extended so that the valuation of the continuous variables can be vector spaces with different dimensions.

Definition 1 (Hybrid automaton): A hybrid automaton HA is a collection HA = {Q, X, Init,f, D, E, G, R}, where

• Q is a finite collection of discrete variables with Q be­ing a finite set;

• X is a finite collection of continuous variables with X = Rnq for q E Q, where nq is the dimension of X.

• Init c :=: is a set of initial values, where S = {(q, x) q E Q,x E Rnq};

• !: (q, x) E S ~ !q(x) E TX is a vector field; • D : q E Q ~ Dq E 2

R nq is a map assigning a subset

'mode-invariant set' of X for each q E Q; • E c QxQ is a set of edges, where e = (r, q) E E means

that the graph of the automaton has an edge form q to r.

• G : e = (r,q) E E ~ Ge E 2R"q is a map assigning a

subset 'guard' of Rnq for each e = (r, q);

• R : (e,x) E Ex G e ~ ReCx) E R"' with e = (r,q) is a map called reset that assigns a subset of Rn, for each e = (r, q) E E and x E Ge •

Example 1: Consider a hybrid automaton shown in Fig. 1. It has the following components: Q = {P, N}, X = R for all q E Q, Init = Q x R, !N(X) = -2x + x2, !p(x) = -2x - x2, DN = {x : x ::; I}, Dp = Ix : x ~ -l}, E = lel,e21 = {(N,P),(P,N)}, Gel = Ix: x = I}, G e2 = Ix: x = -1) and R(el, x) = R(e2, x) = x. We omit identity reset maps in the graph.

2938

x=l

x =-1

Fig. 1 Hybrid automaton example I.

~tl

u u ~~

~2 J plr p1 p r P

Fig. 2 Two-mass-spring system that can collide with wall.

Example 2: Consider a two-mass-spring system in Fig. 2 that can collide with a wall. Two objects with mass ml , m2,

displacement PI, P2, respectively, are connected to each other with a nonlinear spring that causes force kl (p I - P2) + k2(PI - P2)3. The system is controlled with input force Ui = -KpiPi - KdiPi if the right side mass is apart from the wall (mode A), while UI = -KpIPI -KdIPI, U2 = 0 ifit is on the wall (mode C). When the right object touches the wall, it stops at the point of the wall P2 = Pw and the velocity is set zero; P2 = O. The object leaves the wall when force between the object and the wall gets zero, which occurs at PI = Pw' This system is represented as a hybrid automaton as follows.

Q = {A,C}, x-{ R4, q=A - R2, q=C

PI P2 P2 r if q = A r ifq = C {

X = [ PI

X = [ PI PI

Init = {(A, x) : x E R4'P2 < Pw}

X2 _!!J...o - !L03 _ Kpl XI _ Kdl X2

Inl P ml P ml ml

X4 !!J...o + !:J.. 03 _ Kp2 X _ Kd2 X ~ P ~ p ~ 3 ~ 2

!c(X) = [ X2

_!!J...o - !L 03 _ Kpl XI - Kdl x 2 m ! P m l P m l m l

o = { XI - X3 if q = A P XI - Pw if q = C

DA = {x : X3 ~ Pw}, Dc = {x : XI ~ Pw}

E = lei, e21 = ICC, A), (A, C)}

Ge, = {x: X3 = Pw), Ge2 = {x : XI = Pwl

Re, ([ PI PI P2 P2 r) = [PI PI r ReA PI PI r) = [PI PI Pw 0 r We assume the following.

Assumption 1: 1. For every fixed q E Q, the differential equation.i = !q(x) has a solution x(t) defined on [to, 00)

IEICE TRANS. FUNDAMENTALS. VOL.E87-A, NO.II NOVEMBER 2004

- .- - - --

Fig. 3 A mode-invariant set and guards.

for any initial condition x(to) = Xo . 2. The mode-invariant set Dq is given by

Dq = (x E R"q : dqi(x) ~ 0,

i=I , ... ,Noq }, (I)

where dqi : Rnq ~ R, i = I , .. . , N Oq are smooth func­tions on Rnq.

3. The guard Ge, e = (r, q) and the mode-invariant set satisfy

(2)

i.e., the boundary of Dq coincides with the union of the guards from the mode q to certain modes.

4. The guard is represented as follows.

NGe

Ge = U G~) , (3) j = 1

G~) = (x = g~d(q) : q E G~j\ (4)

G~) = (q E R"q- I : g~)(q) ~ 0, . ' (j) 1=1, ... ,NGe }, (5)

where g~~ : Rnq-I ~ Rnq and g~) : R"q -I ~ R, i =

1, . .. , N~; are smooth functions on Rnq-I . 5. lnit c {(q,x): x E Dq,q E Q}.

The relation between a mode-invariant set and guards is depicted in Fig. 3, where there are three edges from q, say ei = (ri, q), i = 1,2,3, and NGel = 1, NGe2 = 2, NGe, = 1.

Definition 2 (Hybrid time trajectory [7), [8), [19]): A hy­brid time trajectory is a sequence of intervals I = (h }~=o with N < 00 or N = 00 such that

• Ik = [tb t~l for all 0 ~ k < N, • if N < 00 then IN = [tN, t~] or IN = [tN, t~), • tk ~ t~ and t~ = tk+ I for all 0 ~ k < N.

For a hybrid time trajectory I, let (I) denote the se­quence of integers {O, 1,2, .. . , Nl and let Ihl = t~ - tk.

Definition 3 (Execution [7] , [8], [19]): If a triplet X =

MASUBUCHI et a!.: COMPUTATION OF LYAPUNOV FUNCTIONS FOR HYBRID AUTOMATA VIA LMIS

{I, q, xl, where I = {hl~=o is a hybrid time trajectory, q : k E (I) - qk E Q and x = (XkO : k E (I) l with each Xk(t) being a C i map from h to Rnq, , satisfies the fol­lowing conditions,X is an execution of the hybrid automaton HA, or HA accepts X:

1. (Initial condition) (qO, xo(to» E Init. 2. (Continuous evolution) For each k E (I) such that Ih I >

0, it holds that

{ Xk(t) = fq, (Xk(t», t E h Xk(t) E D q" t E [tk> t~)

3. (Discrete evolution) For each k E (I), it holds that

{

ek := (qk+i,qd E E Xk(t~) E Ge,

Xk+ \ (t~) = Re, (Xk(t~»

2.2 A Stability of Executions

(6)

(7)

Stability of trajectories of continuous systems t is defined for a positively invariant set, which is generalized to HA as follows:

Definition 4 (Positively invariant set): The set PI c Init is called a positively invariant set if every execution X = {I, q, xl with (qO , xo(to» E PI satisfies (% Xk(t» E PI, VXEh,VkE(I).

In this paper we consider a stability of executions that can be seen as a generalization of the stability of an equilib­rium of nonlinear systems.

Definition 5 (Equilibrium set): Let Qo be a subset of Q. A set Equi = ((q,xq) : q E Qo ,Xq E Dql is an equilibrium set if

• fq(xq) = 0, q E Qo. • If e = (r, q) E E and q E Qo, then r E Qo and Re(xq) =

Xr ·

It is easy to see that Equi is a positively invariant set. Note that the above definition implies that nq's, q E Equi are identical to each other. Without loss of generality, we can assume that Xq = 0 for all q E Qo; then Equi = {(q,O) : q E Qol and we refer to Equi as origin.

Definition 6 (Stability of the origin): The origin is said to be stable if for any t: > 0 there exists 6 > 0 such that the following condition holds for every execution X = {I, q, xl:

Ilxo(to)1I < 6 ::::} IIXk(t)1I < S, Vk E (I), Vt E h (8)

Note that the above definition, which is frequently adopted in stability analysis of hybrid dynamical systems, implicitly assumes that the norm II . II provides a meaningful distance from origin to each point of Dq for every q E Qo. It is seen that Dq does not contain the stable origin if q (/. Qo.

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2.3 Lyapunov Stability Criterion

Generalization of Lyapunov functions has been extensively studied in various ways, e.g., in [4], [14], [15], [18]. The sta­bility condition below in Proposition 1 is based on Petterson & Lennartson [14], [15] , which allows a hybrid automaton to have modes that do not have an equilibrium point.

Definition 7: Denote by R+ the set of nonnegative real numbers. A continuous function a : R+ - R+ is said to be a class-X function if a(O) = 0 and a(r\) < a(r2) for any 0 ::; rl < r2.

Proposition 1: The origin of HA is stable if there exists a function Vq(x) : (q, x) E =: - R satisfying the following conditions.

Vq(X) is a Ci map when q is fixed .

a(l lxID::; Vq(x) ::;,B(llxID, Vq E Q, Vx E Dq (9) for some class-X functions a(·),,B(·).

aVq(x) -ax-- fq(x) ::; 0, Vq E Q, Vx E D q. (10)

Vr(Re(x» ::; Vq(x),

Ve = (r, q) E E, Vx E Ge . (11)

This proposition is proved similarly to those found in [4] . We call a function Vq(x) Lyapunov function if it satisfies the conditions in Proposition 1.

Example 3: Consider HA in Example 1. A Lyapunov function is given by Vq(x) = x2/2, q E Q, for which

V { -x2(2 - x)::; - x2, q = N q(x) = - x2(2 + x) ::; -x2, q = P

and Vr(R(r,q) (x» == Vq(x) . This proves the stability of the origin of HA in Example 1.

3. Sum of Squares

We will utilize 'sum of squares' as a tool to obtain Lya­punov functions. First, we define sum of squares (SOS in short) and provide formulae to derive linear matrix inequal­ity (LMI) conditions for positivity of functions, based on Nesterov [9]. Next, we refer to a result from real algebraic geometry, called Positivstellensatz [3] , [13], which gives a necessary and sufficient condition for feasibility of polyno­mial equality and inequality conditions.

3.1 Representations of Sum of Squares

Consider a set of functions of x E Rn, denoted as S = {b(l)(x), ... , b(M)(x)l. Let r be the set of linear combina­tions of the functions in S:

tWe mean by 'continuous system' a nonlinear system x = f(x) .

2940

= {s(X) = t c(i) b(i)(x) : c (i) E R}. (12)

Define XeS) the set of sum of squares of finite elements in 'Y(S):

XeS) = {W(X) = t S(i)(X)2 : / i) E 'Y(S)} , (13)

where the number of summands Kw depends on each w. For the set S 2 = {b(i)b(j) : i, j = I, . . . , M}, derive independent functions t<i)(x) so that

S2 c span {l(l )(x) , . . . , I(N) (x)}. (14)

Denote b(x) [ b(' )(x) b(M)(x) r, I(x) = [ l(l )(x) I(N)(x) r and define a linear map A(v) : v E

RN ~ W E R M x M satisfying the following condition:

A(l(x» =: b(x)b(x)T , Vx ERn .

Define inner products of RM and R NXN by

(V"V2) = VTV2, V',V2 E RM,

(W" W2) = Tr(W, W2), WI, W2 E RNXN,

(15)

(16)

(17)

respectively. Denote the adjoint map of A(·) by 1\*0.

Lemma 1 (Nesterov [9]): For any p(x) E XeS), there ex­ists W E RMXM, W = WT ~ 0 such that

p(x ) = b(x)TWb(x) = (N(W), I(x» . (18)

Conversely, any p(x) represented as (18) belongs to XeS).

Remark 1: The above lemma gives a representation of a sum of squares in the quadratic form p(x) = b(x)TWb(x) as well as by the inner product (A *(W), I(x», which will be uti­lized later to obtain LMI conditions. The latter form raises an optimization problem in a convex cone {p = A *(W) : W ~ O}; properties of this cone is investigated in Nesterov [9].

3.2 Positivstellensatz

In the next section, the problem to seek a Lyapunov func­tion Vq(x) in Theorem 1 is reduced to a problem to find SOS functions satisfying several equalities and inequalities re­quired to hold in a certain subset of Rnq. If the functions de­scribing HA such as fq(x), dqi(x) and the bases for SOS are polynomials, one can obtain a necessary and sufficient con­dition for positivity of functions via Positivstellensatz [3], [ 13].

Definition 8: Denote by ~ the set of sum of squares of polynomials cr(x) = s, (x? + ... + Sk(X? with some fi­nite integer k and polynomials Si(X) on Rn. Let fix) , j = I , . . . , m be polynomials on Rn . Then cone generated by h(x), j = 1, . .. , m is the set of polynomials of the form cro + Ll=, cribi, where I is a nonnegative integer, cri E ~ and bi is an element of the multiplicative monoid generated by h(x), j = 1, . .. , m.

IEICE TRANS. FUNDAMENTALS, VOL.E87- A, NO." NOVEMBER 2004

Let P(f" .. . , fm) denote the cone generated by f" . . . , fm .

Lemma 2: 1. P(f" ... ,fm) = (cro+ Li cr! f;+ Lt.) crf/;!) 2 A A ,

+ ... + Li*-) cr'j)- f, .. . k .. h ... fm + Li cr'j- f, ...

]; ... fm + ~ f, . .. f" : 01) ... E ~}, where]; means that f; is omitted in the product.

2. P(f" . . . ,fm, g) = P(f,, ··· ,fm) + gP(f" ... , fm). Proof Simple manipulations prove these equalities. Note that cr 1;2k+l, cr E ~ is rewritten as cr'l;l , cr' = cr 1;2k E ~. Thus we need finite elements of the multiplicative monoid to represent a cone.

Lemma 3 (Positivstellensatz [3], [13]): Consider polyno­mials hex), j = 1, ... , s, gk(X), k = I, ... , t and hiCx), I = I, ... , u, of x E Rn . Let P, M, I be the cone generated by f lex), j = 1, ... , s, the multiplicative monoid generated by gk> k = I, . .. , t and the ideal generated by hi, I = I, .. . , u, respectively. Then the following statements are equivalent.

1. The following set is empty.

{

flex) ~ 0, ... ,fs(x) ~ 0 ) X g, (x) '* 0, . . . , g/(x) '* 0

h, (x) = 0, .. . , hu(x) = 0 (19)

2. There existf E P, gEM and hE I such that f+g2+h = O.

For D = {x E Rn : d, (x) ~ 0, . . . , dm(x) ~ A}, we define P(D) = P(d" .. . , dm) and call an element of P(D) 'SOS on D.' Below is a direct application of the above two lemmata.

Lemma 4: Let Vex) and di(x) be polynomials. Then the following statements are equivalent.

1. Vex) ~ 0 for all xED. 2. bV = a for some a,b E P(D).

Proof From Lemma 3 and the second statement of Lemma 2, the condition 1 is equivalent that a + b( - V) + ( - V)2k = 0 holds for some a, b E P(D) and nonnegative integer k. It is easily seen that this is equivalent to the statement 2. of this lemma.

Note that these results hold only for polynomials and it is not always tractable to seek Lyapunov functions Vq sat­isfying such as bq Vq = aq, aq, bq E P(D) with other con­ditions required in Proposition I. Nevertheless, application of Lemma 4 even with restricted freedom on such as aq , bq

provides useful sufficient conditions for positivity of func­tions. For example, the result of stability analysis by using piecewise quadratic Lyapunov functions in [6], [16] can be explained with a special choice of SOS functions ; see Re­mark 2.

Lastly in this section we note that elements of P(d" ... , dm) are represented as an SOS with appropriate basis S defined on D. For m = 2, p E P(d" d2) is given by p = cro + cr, d, + cr2d2 + cr)d, d2, cri E L. By setting cri = e; Piei, we see p = e T Pe with

MASUBUCHI et al.: COMPUTATION OF LYAPUNOV FUNCTIONS FOR HYBRID AUTOMATA VIA LMIS

e=

eo ....;d;el Ydze2

ydl d2e3

P = diag{P;}.

We can construct bases similarly for m ~ 3.

4. Computation of Lyapunov Functions

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In thi s section, we provide LMI conditions to derive a Lya­punov function. For simplicity, we consider

Vq(X) ~ O,q E Q,x E Dq (21)

aVq(x) Wq(x) := -----a;- !q(x) ~ 0, q E Q, x E Dq (22)

Vq(x) - Vr(Re(x» ~ 0,

Ve = (r,q) E E, Vx E Ge . (23)

It is easy to modify the following result to handle positive definiteness of Vq strictly.

Let us set the classes of Vq and Wq to seek in by P(Dq) with bases S q and S q, respectively. Functions (b, I, A)

i~ S~ct:.. 3.1 are de~oted for S q and S q as (br Iq, Aq) and (bq, Iq, Aq~, respec~vely. Then Vq(x) = bq(x) Pqbq(x) and Wq(x) = bq(x)T Qqbq(x) for some positive semidefinite ma­trices Pq and Qq. Next, obtain matrices F q, Zq and a vector of functions zqCx) for each q so that

atqCx) _ ----a;- !qCx) = Fqlq(x) + ZqZq(x), (24)

Zq(X) rt. span (/(x) } and the elements of zq(x) are linearly in­dependent. Then, from (24), we have

aVq(x) ( . atq(x) ) ----a;- !q(x) = A qCPq), ----a;- !q(x)

= (A;(Pq), Fglq(x) + ZqZq(x» )

= (F;A;(Pq), lq(x» ) + (Z;A;(Pq),Zq(x»).

Since Iq(x) and Zq(x) are independent and Wq(x) =

(A;(Qq), lq(x» ) , the equality in (22) is equivalent to

F;A;(Pq) = -A;(Qq), Z;A;(Pq) = 0. (25)

Next, consider the condition (23) for edges e = (r, q) E

E. The inequality (23) is equivalent to

y~j)(~) := Vq(g~d(~» - Vr(Re(g~~ (g») ~ 0, '(j) .

Ve = (q, r) E E, V~ E Ge , J = 1, . .. , Nee. (26)

The positivity of y~j)(~) on (;~j) is ensured by SOS on (;~j) with basis S~j) = (W\ i.e., y~j)(f,) = (W~({»TT~j)b~)Sf)' T~J) ~ 0. To handle this, derive 'tj (~) and A~J)O from S j), as I(x) and A(·) are derived from S in Sect. 3.1, respectively. Then obtain matrices E~j) and R~) satisfying

Iq(g;;{?(g» = E~)~j)(D,

Ir(Re (g~d(~))) = R~W>C~). (27)

(28)

2941

Note that one can always choose basis S~) to satiate these identities. Since we have (27), (28), y~j)(~) = (A~j)· (T~j», 'tj) (~») and

Vq(g~~(~» = (A;(Pq), Iq(g~~(~» ) ,

Vr(ReCg~d(~))) = (A; (Pr), Ir(Re(g~ci(~)))),

the equality in (26) is guaranteed by

A~j)·(T~j» = (E~j» T A;(Pq) - (R~»T A ;(Pr)

We summarize the result.

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Theorem 1: The inequalities (21), (22) and (23) hold for

Vq(x) = bq(x)T Pqbq(x), a~;X) = -hq(X)T Qqhq(x), x E D q,

q E Q and Vq(x) - Vr(Re(x» = (W)fT~j)W), x E G~), j = 1, ... , Nee' e E E if and only if (25) and (29) hold for

Pq ~ 0, Qq ~ 0, q E Q and T~j) ~ 0, j = 1, . . . , Nee' e E E.

Remark 2: This result is a generalization of [10], [11] to hybrid automata and a generalization of [6] to one that han­dles nonlinear vector fields. To see the latter, let !q(x) be piecewise linear functions (linear for each q) and let Vq(x) be a piecewise quadratic Lyapunov function (PQLF), namely,

T -T - - - - T Vq(x) = x PqX, Pq = FqMFq, M = M , (30)

where F q satisfies Frx = Fqx for x on the boundary of ad­jacent regions of mode q and r, which guarantees the conti­nuity of Vq(x). In [6] , the condition Vq(x) ~ ° on polyhedral set D q = {x : K;lx ~ 0, ... , K:mx ~ O} with n-vectors Kq; is guaranteed by the following LMI condition ((11) in [6] , second inequality):

Pq - K;S qK q = Qq > 0,

Kq = [ Kq I . .. K qm r (31)

for a symmetric matrix S q with nonnegative elements [S q];j . Setting h;(x) = Kqix, we see that the equality condition in (31) yields Vq = 0"0 + 'f,i j O"Tjh;hj , where 0"0 = XT Qqx +

'f,;[S q];;(K;iX)2 E :E and O"Tj = [S q] ij E :E. This implies that Vq(x) belongs to P(D q) and thus the methodology in [6] to assure positivity of Vq(x) is explained through SOS formulation.

4 .1 Numerical Examples

Example 4: Consider a bimodal hybrid automaton with the following components.

X = R 2 Q={A,B},

!A(X) = [ - X I + X2 -4xIX2 - x3 +5~

-3xI - X2 + 2x2

hex) = [ -xl-3x2 - X~ -XI X2 ] XI - X2 + 3x~ - X I X2

Dq = {x E R 2 : dql := XI ~ OJ, q E Q

E = {e1,e2} = {(B, A),(A, B)}

2942

G el = G e2 = Ix: XI = OJ

Rei (X) = Re2 (x) = X, luit = I(q,x): X ~ OJ

The expression of the guards Gei, i = 1,2 as in (3)-(5) are

given with NCe = I, N~1 = 0, g;~(~) = [0 ~ f, ~ E RI,

for e E E. Bases for Vq(x) and B;;X) are chosen as

bA(x) = hA(X) = [ XI 2 ]T X2 X2

bs(x) = [XI X2 XIX2 X~ f [

T T ]T hs(x) = bs(x) VXibs(X)

and b~ ' )(~) = [~ ~2 f, e E E. From these bases,

we set lq(x), A q(·), lq(x), Aqo, q E Q and ~I)(~), A~ ' \.), e E E. Then we derive matrices (Fq,Zq) and

(E~j), R~) satisfying (24) and (27)-(28), respectively. We show a few of these materials: for instance, we can

[ 2 2 x2 x3 4]T;( I) set IA(X) = XI XIX2 XIX2 2 2 X2 ' l ei (~) =

[e e ~4 r. Since

O 0 0 /:2 /:3 c4]T IA(geIO(~» = ., ., s

we derive E~ : ) = [03X3 13x3 r. which satisfies (27). We

set T~j) = 0 in (29), which together with Rei (x) = X yields Vq(x) = V,(x), X E Gei . The LMI problem for this example is to find positive semidefinite matrices Pq, Qq, q E Q satis­fying equations (25) , (29) . The obtained Lyapunov function is

VA(X) = xi - 0.14xlx2 + 0 . 89x~ - 1 .3 3xlx~ +0.02x~ + 0.833xi,

Vs(x) = 0.59xi + 0.04XIX2 + 0.89x~ + 0 . 18x~ -0.28xix2 + 0.54xlx~ + 0.02x~ + 0.94xi

+0.22x~X2 + 1.22xix~ + 0.2xlx~ + 0.83xi.

In Fig. 4, an execution for initial condition qo = A, xo(O) = (3 , 3) and level sets of Vq(x) are plotted with solid and bro­ken lines, respectively.

Example 5: Consider HA in Example 2. We set param­eters Pw = I, ml = m2 = I, kl = k2 = 5 and gains K pl = Kdl = 5, Kp2 = K d2 = 2. The guards G el and G e2 are represented as in (3)-(5) with

and

IEICE TRANS. FUNDAMENTALS. VOL.E87- A, NO.II NOVEMBER 2004

Fig. 4 An execution and level sets.

Vc(x(t))

D.S 1.5 2.5 3.5

Time[s)

Fig. 5 Lyapunov function .

respectively. Bases are chosen as

bA(x) = [ Xl X2 X3 X4 (XI - X3)2 f hA(x) = [ XI X2 X3 X4 (XI - X3)2

(XI - X3)3 XI X2 XI X3

XIX4 X2X3 X3 X4 ]T

bcCx) = [XI X2 (XI - 1)2 1 f X3 x2]T

I 2 hcCx) = [ XI

b~: ) (~) = [ ~ I

b~:)(D = [ 1

(~1_1)2 f ,'; ER3

We derive a Lyapunov function as

? 2 2 VA(x) = Xj- XIX3 + 0.lx2 + 0.7x3

+0.1 x~ + 0.25(xI - X3)4,

VcCx) = xT - XI + O . lx~ + 0.25(xI - 1)4 + 0.7 .

MASUBUCHI et al.: COMPUTATION OF LYAPUNOV FUNCTIONS FOR HYBRLD AUTOMATA VIA LMIS

A trajectory of this Lyapunov function is plotted in Fig. 5, where the solid segment is in mode A and the broken seg­ment is in mode C. The value of the Lyapunov function jumps when the mode changes from A to C.

S. Conclusion

In this paper, a computational method is provided for sta­bility analysis of hybrid automata that can have nonlinear vector fields. A stability criteria is investigated and an LMI problem is formulated to derive a sum-of-squares Lyapunov function.

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Izumi Masubuchi was born in 1969 in . Kyoto, Japan. He received the B.E., M.E. and

Dr.Eng. degrees from Osaka University in 1991, 1993 and 1996, respectively. He was a research associate of Japan Advanced Institute of Science and Technology and Kobe University in 1996-1998 and 1998-200 I, respectively. He is cur­rently with Graduate School of Mechanical Sys­tem Engineering, Hiroshima University, as an associate professor. His research interests in-clude systems and control theory, robust control

design and theory of hybrid dynamical systems. He is a recipient of Takeda Paper Prize of SICE in 1995 and Pioneer Prize of SICE Control Division in 2003. He is a member of SICE, ISCIE, JSME and IEEE.

Seiji Yabuki received the B.E. and M.E. degrees from Hiroshima University in 2002 and 2004, respectively. He is currently with J.S.T. Mfg. Co., Ltd.

Tokihisa Tsuji received the B.E. degree in 2004 from Hiroshima University. He is now a student of Graduate School of Mechanical Sys­tem Engineering, Hiroshima University. His re­search interests include stability analysis of hy­brid dynamical systems.