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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 1, JANUARY 2015

Haar-Based Stability Analysis of LPV Systems

Leonardo O. de Araújo, Paulo C. Pellanda, Memb

er, IEEE, Juraci F. Galdino, and Alberto M. Sim?es

Abstract—A new gridding-based algorithm for stability analysis of Linear Parameter-Varying (LPV) systems is presented. The algorithm inherits the main features of classical gridding techniques: it can handle a vast class of parametric dependencies as well as non-convex parametric domains. Novelty of the proposed approach lies in the use of Haar wavelet transform theory to guarantee constraint satisfaction over the entire parametric domain, even for an arbitrarily sparse grid. It represents a major improvement over traditional gridding approaches, which fail to provide such a certi?cate without requiring a posteriori veri?cation tests. The resulting algorithms rely on semide?nite programing and are related to suf?cient stability conditions whose degree of conservatism decreases as the grid density and the Haar truncation level increase. Two numerical examples corroborate the validity of the proposed algorithms. Index Terms—Haar wavelet, linear parameter-varying (LPV) system, parameter-dependent Lyapunov function, stability analysis. I. I NTRODUCTION Stability analysis of Linear Parameter-Varying (LPV) systems remains a challenge, despite the remarkable recent progress in dynamical system theory. Most of the analysis methods for uncertain or timevarying linear systems based on Lyapunov stability theory turn out to be inadequate in the particular case of LPV systems. First, parameters are generally assumed to take value in some convex polytope, usually a hyperrectangle [1], [2] or a simplex [3]–[5]. Unfortunately, such an assumption is invalid for the large class of LPV systems in which the set of allowable parameter trajectories de?nes more irregular domains. To circumvent the eventual non-convexity of the parametric domain, those methods have to recur to some sort of convex covering technique, see e.g. [6]. However, such a scheme is likely to introduce conservatism since non-realistic trajectories are taken into account. Second, existing approaches can generally handle only a limited class of parametric dependencies of system matrices, basically linear [3]–[5], [7], af?ne [1], [2], [8] or rational [9]–[11]. Consequently, those methods are unable to directly address more general dependencies encountered in some applications of great practical interest, like, for example, quasi-LPV systems arising in aerospace applications where some of the endogenous parameters enter system matrices via trigonometric functions [12]. To apply the above methods to such problems, one has to either resort to some kind of linearization scheme or to embed the system into a polytopic one, which is recognizably a conservative procedure. A well-known strategy to overcome the above drawbacks is the gridding approach [13], [14]. One of its most appealing features is the much more general class of parameter-dependent Lyapunov Functions (LF) or systems that can be handled, including non-convex parametric domains. On the other hand, a major ?aw of traditional griddingbased techniques is that computed solutions are guaranteed to satisfy constraints only at those points in the grid, not necessarily over the entire parametric domain. Consequently, an over-optimistic estimate of the stability domain may follow. In practice, one must select the grid as dense as possible and then hope that constraints will be satis?ed over the entire domain. Obviously, the denser the grid, the higher the associated computational burden. In this work, new gridding-based algorithms for stability analysis of LPV systems are presented. The proposed algorithms build on the wavelet theory [15], [16] to overcome the above limitations of the classical gridding schemes. Novelty of the proposed approach lies in the use of wavelet theory to guarantee constraint satisfaction over the entire analysis domain, even when a sparse parameter grid is considered. It represents a major improvement over traditional gridding approaches, which fail to provide such a certi?cate without requiring additional veri?cation tests. To that end, the parameter-dependent state matrix is initially replaced in matrix inequalities by an arbitrarily precise approximation obtained by a truncated Haar series expansion. Then a quadratic-in-the-state Haar-based parameter-dependent LF is sought, while somehow taking into account the neglected terms of the Haar expansion of the state matrix. The resulting computational test involves solving a semide?nite program (SDP) and corresponds to a suf?cient stability condition whose degree of conservatism decreases as the grid density and/or the resolution level of the Haar decomposition increases. The algorithm also inherits the main bene?ts of gridding techniques: it can directly address a large class of systems as well as non-convex parametric domains. Notation: For a given real matrix M ∈ Rn×m , M T denotes its transpose, M its induced 2-norm, and M pq its (p, q )th entry. Notation S (M ) = M + M T is used in large matrix inequalities. Sn stands for the set of real symmetric matrices of size n. For two matrices M, N ∈ Sn , notation M N means that M ? N is positive de?nite and M N that M ? N is positive semi-de?nite. Moreover, notation ±M N means that both inequalities M N and ?M N are satis?ed. L2 (R) denotes the space of square-integrable real-valued functions. For a matrix-valued mapping M : R → Rn×m , M (·) ∈ L2 (R) means that M pq (·) ∈ L2 (R), for all entry (p, q ). For a compact domain C ? Rn , IC (·) stands for the indicator function of C IC ( θ ) =

Manuscript received May 28, 2013; revised December 30, 2013; accepted April 24, 2014. Date of publication May 7, 2014; date of current version December 22, 2014. This work was supported in part by the Brazilian Agencies CNPq under Grants 309846/2011-0 and 308046/2009-9 and FAPERJ under Grant E-26/102.269/2009. Recommended by Associate Editor F. Blanchini. The authors are with the Electrical and Defense Engineering Graduate Programs, Military Institute of Engineering, Rio de Janeiro, RJ, Brazil (e-mail: leonardo.araujo@gmail.com; pcpellanda@ieee.org; galdino@ime.eb.br; simoes@ime.eb.br). Digital Object Identi?er 10.1109/TAC.2014.2322436

Δ Δ

1, if θ ∈ C , 0, if θ ∈ C .

For a mapping f : C → R, f (·) ∈ LC 2 (R) denotes that f (·)IC (·) ∈ L2 (R). The inner product of two functions f, g ∈ L2 (R) is denoted

∞

f (θ ), g (θ ) =

?∞

Δ

f (θ)g (θ)dx.

0018-9286 ? 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 1, JANUARY 2015

193

For two mappings F : R → Rn×m and g : R → R, F (θ), g (θ) stands for a matrix whose (p, q )th entry is given by F pq (θ), g (θ) . ⊕ stands for the direct sum of disjunct subspaces. In denotes the identity matrix of dimension n × n. Dn denotes the n-dimensional unit hypercube with vertexes whose coordinates are either 0 or 1. N≥0 (R≥0 ) and N+ (R+ ) denote, respectively, nonnegative and strictly positive natural (real) numbers. II. P RELIMINARIES A. Problem Formulation Consider an LPV state-space representation x ˙ (t) = A (θ(t)) x(t) + B (θ(t)) u(t) y (t) = C (θ(t)) x(t) + D (θ(t)) u(t) (1)

ψ (θ ) =

Δ

1, if 0 ≤ θ < 0.5, ?1, if 0.5 ≤ θ < 1, 0, otherwise.

(5)

Next, consider the following sets of basis functions: φj ( θ ) = φ( θ ? j ) ,

Δ

i

Δ

j∈N , (6)

Δ

ψij (θ) = 2 2 ψ (2i θ ? j ), i ∈ N, 0 ≤ j < 2i .

Δ

Also, consider functional subspaces V0 = span(φj : j ∈ N), Wi = span(ψij : 0 ≤ j < 2i ) and let Vi+1 = Vi ⊕ Wi . An important property concerning the above subspaces is that V0 ? · · · Vi ? Vi+j ? · · · ? L2 , with Vi → L2 as i → ∞. With the above de?nitions in mind, the Haar transform fΣ∞ of a given square-integrable function f : D → R ∈ L2 (R) is such that fΣ∞ (θ) = f (θ), ?θ, and can be obtained as

∞ 2i ? 1 Δ

where x ∈ Rn , u ∈ Rm , y ∈ Rp and θ ∈ Rr denote state, input, output and time-varying parameter vectors respectively. A large class of nonlinear dynamical systems, including many systems of interest in aerospace and mechatronics ?elds, are known to admit such a statespace representation. Only a very few assumptions on LPV model (1) are made here. Firstly, parameter trajectories θ(t) are assumed to be contained in ˙(t) is a given compact domain Θ, whereas the rate of variation θ supposed to be valued in a given hyperrectangle Θd . It is also assumed that mapping θ → A(θ) ∈ LΘ 2 (R). If, on the one hand, such a general parameter dependence renders representation (1) ?exible enough to encompass a wide range of practical applications, on the other hand obtaining stability analysis tools for such a large class of systems consists in a particularly challenging task. A well-known suf?cient condition to the stability of the system (1) is the existence of a quadratic parameter-dependent LF [17] V (x, θ) = xT P (θ)x (2)

fΣ∞ (θ) = v0 φ0 (θ) +

i=0 j =0

Δ

wij ψij (θ)

(7)

where coef?cients wij and v0 of the above Haar series expansion are given by the inner products wij = f, ψij and v0 = f, φ0 . The Haar transform can be seen as a function which is piecewise constant in in?nitely many intervals. As the resolution index i increases in (7), each new projection of the original function f (θ) onto the subspace Wi , added to its projection onto the subspace Vi , provides further information for its Haar representation. If the outer sum in the second term on the righthand side of (7) is truncated, then one has only an approximation to f (·) instead of an exact representation. In the sequel, notation fΣ∞ (·) is used to denote the precise representation of a function f (·) given by its Haar transform (7), whereas fΣI (·) stands for the approximation to f obtained by truncating the expansion (7) as follows:

I 2i ? 1

with mapping P : Θ → Sn piecewise differentiable. Time dependence has been omitted in (2) for notational simplicity. It can be easily shown that V (·, ·) in (2) represents an LF for (1) if and only if P (θ) is a feasible solution to the following set of Parameterized Linear Matrix Inequalities (PLMI): P (θ ) 0, ?θ ∈ Θ, ?θ ∈ Θ, ˙ ∈ Θd . ?θ (3) (4)

fΣI (θ) = v0 φ0 (θ) +

i=0 j =0

Δ

wij ψij (θ).

(8)

?P (θ) ˙ θ + S (P (θ)A(θ)) ? 0, ?θ

Obviously, fΣI (·) → fΣ∞ (·) = f (·) as I → ∞. The Haar transform presents an energy conservation property similar to that in Parseval theorem. The energy in Haar coef?cients tends to decrease as the resolution i increases [20]. At resolution level I , there 2I ? 1 ∈ R+ such that exist scalars {τj }j =0 |wIj | ≤ 2?I (s+ 2 ) τj

1

The feasibility problem de?ned by PLMI (3), (4) is recognizably hard to solve by virtue of the in?nitely many constraints involved. Here the originally in?nite-dimensional feasibility program (3), (4) is solved by dint of a gridding scheme in which the PLMI problem is reduced to a ?nite-dimensional program. As stressed before, most of the techniques proposed in the literature to solve this problem require stronger assumptions on the parametric dependence of state and Lyapunov matrices, or also on the shape of domain Θ, see, e.g., [18], [19]. B. The Haar Transform The Haar transform [15], [16] is a discrete wavelet transform providing an in?nite-dimensional representation of square-integrable functions. To begin with, consider the so-called father and mother Haar functions, φ : R → R and ψ : R → R respectively, given by φ( θ ) =

Δ

(9)

where s ∈ R≥0 represents the H?lder exponent of f (·). Now, let Γi,j ? R denote the sub-domain where ψij (·) is different from zero, as illustrated in Fig. 1 for the particular case of ψ21 (·). For higher decomposition levels (i > I ), it can be shown that there exist scalars κij such that |wij | ≤ 2?i(s+ 2 ) κij

1

(10)

where κij ≤ τm , for m ∈ {0, 1, . . . , 2I ? 1} representing the index for which the inclusion Γi,j ? ΓI,m holds. Hence, if f represented a signal, then its energy would be mostly concentrated in low-resolution coef?cients. On the basis of the scaling property of the Haar father function, it can be shown that for any decomposition level i > I

2i ? 1 2I ? 1

1, 0,

if 0 ≤ θ < 1, otherwise,

φj (2i θ) =

j =0 j =0

φj (2I θ) = φ0 (θ),

?θ ∈ D .

(11)

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 1, JANUARY 2015

Following the notation in Section II-B, let AΣ∞ (θ) represent the Haar transform of state matrix A(θ) A Σ∞ ( θ ) = A ΣI ( θ ) + A E ( θ )

Δ

(16)

where AΣI (θ) corresponds to a truncated series expansion similar to (8), and AE (θ) represents the residual matrix given by

∞ 2i ? 1

A E (θ ) =

i=I +1 j =0

Aij ψij (θ).

(17)

Fig. 1. Haar basis functions.

For a matrix-valued function M : D → Rn×m ∈ L2 (Rn×m ), its Haar transform is de?ned as

∞ 2i ? 1

M Σ∞ ( θ ) = M 0 φ 0 ( θ ) +

i=0 j =0

Δ

Mij ψij (θ),

(12)

with M0 = M, φ0 , Mij = M, ψij and MΣ∞ (θ) = M (θ). Although the Haar transform formulation presented here involves a unidimensional domain, it can be easily extended to the multidimensional case [15]. Consider momentarily the af?ne mapping fa : θ → aθ + b, with a, b ∈ R, and θ ∈ D. The Haar transform of such an af?ne mapping presents some interesting properties that play a central role in this work. First, the coef?cients of the Haar transform of fa (θ) can be shown to be given by wij = fa (θ), ψij (θ) = ?2 a ?i(1+ 1 2) 4 . (13)

As previously mentioned, a fundamental property of the Haar transform is that the coef?cients of highest resolution gathered in AE (θ) are those of lowest energy level. Therefore, for a suf?cient high decomposition level I in (16) the residue AE (θ) carries poor or irrelevant information about the parametric dependence of the state matrix. In such a case, the ?nite dimensional series expansion AΣI (θ) becomes a very precise representation of A(θ). Notice that AΣI (θ) is piecewise constant, and hence can be interpreted as some sort of discretization of A(θ). A central idea here consists in considering in PLMI (4) the truncated Haar series expansion AΣI (θ) instead of the state matrix A(θ), while taking into account some bound on the norm of residue AE (θ). To begin with, consider such a bound provided in the following lemma. Lemma 1: Let AE (θ) represent the residual matrix (17) appearing in the Haar transform (16) of A(θ). Then there exist real-valued 2I ? 1 such that, for all θ ∈ D piecewise constant functions {αj (θ)}j =0

2I ? 1

A E (θ ) ≤

j =0

αj (θ ).

(18)

Proof: According to (6), AE (θ) can be rewritten as

∞ 2i ? 1

A E (θ ) =

i=I +1 j =0

2 2 Aij ψ (2i θ ? j ).

i

Thus, for a given resolution level i the Haar coef?cients of the af?ne mapping fa (θ) are the same for all indexes j . Another particular property for the af?ne mapping fa that follows from (13) is that w ( i +m ) j = 2 ?

Δ ?1 {θk }2 k=0

I +1 3m 2

For each entry (p, q ) of matrices Aij , consider the upper bounds given pq ?i(s+(1/2)) , where s ∈ N+ and κpq by (9), (10): |Apq ij | ≤ κij 2 ij ∈ R+ . Next, the following sets of parameter-dependent matrices are de?ned:

pq i Mij ∈ Rn×n , i ≥ I, 0 ≤ j < 2i : Mij = 2 2 Apq ij ψ (2 θ ? j ) Δ pq ?is ? ij ∈ Rn×n , i ≥ I, 0 ≤ j < 2i : M ? pq = M κij 2 ψ (2i θ ? j ) ij Δ

i

wij .

I +1

(14)

, .

? D denote a set of 2 linearly spaced Let DI = points, where each point is located at the center of intervals de?ned by subspace VI +1 , i.e. θk = k + 1 2 1 2k + 1 = I +2 . 2I +1 2 (15)

pq ? pq (θ), ?θ. Moreover (θ ) ≤ M Notice that Mij ij ∞ 2i ? 1 ∞ 2i ? 1

A E (θ ) ≤

i=I +1 j =0

Mij (θ) ≤

i=I +1 j =0

? ij (θ) . M

(19)

Finally, let fΣI represent the truncated Haar series expansion of fa . It can be veri?ed that for any resolution level I ∈ N+ it follows that fΣI (θk ) = fa (θk ), ?θk ∈ DI . In other words, at every point in DI the truncated Haar series expansion of such an af?ne mapping always provides the exact value of the function. III. A N EW H AAR -BASED S UFFICIENT C ONDITION FOR Q UADRATIC S TABILITY In this section, a new suf?cient condition for quadratic stability of LPV system (1) on the basis of a parameter-independent LF is obtained. For the sake of clarity, the scalar case θ ∈ D is discussed.

The (p, q )th entry of the matrix on the right-hand side of (19) can be rewritten as follows:

∞ 2i ? 1 ?is κpq ψ (2i θ ? j ) ij 2 i=I +1 j =0 ∞ 2(I +i+1) ?1 ?(I +i+1)s κpq φj (2I +i+1 θ) (I +i+1)j 2 i=0 j =0 ∞ 2(I +i+1) ?1

=

(20)

= 2?(I +1)s

i=0

2?is

j =0

I +i+1 κpq θ) (I +i+1)j φj (2

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 1, JANUARY 2015

195

∞

2I ? 1

≤

2

?(I +1)s i=0 2I ? 1

2

?is m=0

pq τm φm (2I θ)

(21)

2?(I +1)s = 1 ? 2? s

pq τm φm (2I θ). m=0

(22)

Expression (20) comes from the equivalence |ψ (θ ? j )| = φj (θ), whereas (21) follows from property (11) and the de?nitions in (9), (10). Expression (22) is easily obtained after identifying the ?rst summation in (21) as the sum of a geometric series. Consider now the following set of matrices: M j ( θ ) ∈ R n× n , 0 ≤ j < 2 I : M j ( θ ) ?(I +1)s pq τj Δ 2 = φj (2I θ) ? 1?2 s

pq

(1). Notice that the conservatism of the condition of Theorem 1 tends to decrease with increasing decomposition level I of the state matrix, since AΣI (θ) → AΣ∞ (θ) and AE (θ) → 0 as I → ∞. The number of decision variables on program (24), (25) depends only on the number n of states and is given by 1 + (n(n + 1)/2). On the other hand, an increasing decomposition level I implies an increasing grid density and an exponentially increasing number of n × n matrix inequalities constraints: 2I +2 . IV. H AAR -BASED PARAMETER -D EPENDENT LF FOR Q UADRATIC S TABILITY A NALYSIS Conservatism in Theorem 1 can be reduced by adopting a parameter-dependent LF. To that end, consider initially the following piecewise constant function constructed by Haar basis functions:

G 2g ? 1

. (23)

Inequality (18) ?nally follows from (19), (22) and the de?nition Δ αj (θ ) = M j (θ ) . I +1 Δ ?1 ?D For a prescribed resolution level I , let DI = {θk }2 k=0 I +1 linearly spaced denote as in Section II-B the grid consisting of 2 points in (15). The next theorem states the main result of this section. Thmeorem 1: Consider the LPV system (1) and the upper bound (18) on the residue of the state matrix. Then the LPV system is quadratically asymptotically stable if there exist γ ∈ R+ and P ∈ Sn 0 such that the following set of LMI holds: P ? γIn 0,

2I ? 1

Q ( θ ) = Q 0 φ0 ( θ ) +

g =0 h=0

Δ

Qgh ψgh (θ)

(28)

with Q0 , {Qgh } ∈ Sn . The candidate Lyapunov matrix P (θ) is then selected as

G 2g ? 1

P (θ ) =

Δ

?0 (θ) + Q(θ)?θ = Q0 φ

g =0 h=0

?gh (θ) + Q ? (29) Qgh ψ

(24) αj (θk )In ? 0, ?θk ∈ DI .

j =0

? ∈ Sn and where Q

Δ ?0 (θ) = φ Δ ?gh (θ) = φ0 (θ)?θ, ψ

S (P AΣI (θk ))+2γ Proof: Since P ∈ Sn Therefore, for all θ ∈ D S (P AE (θ))

(25) P ≤ γ.

ψgh (θ)?θ, g ∈ N, 0 ≤ h < 2g . (30)

0, constraint (24) implies

Alternatively to (29), P (θ) can also be precisely represented by the in?nite series expansion corresponding to its Haar transform P Σ∞ ( θ ) = P ΣI ( θ ) + P E ( θ )

Δ

S (P AE (θ)) In

2 ?1

I

2 P

I

A E (θ ) I n

2 ?1

(31)

2 P

j =0

αj (θ )I n

2γ

j =0

αj (θ )I n . (26)

where PΣI (θ) represents, as before, the truncated series expansion

I 2i ? 1

P ΣI ( θ ) = P 0 φ 0 +

i=0 j =0

Pij ψij (θ)

Next, since AΣI (·) is by construction piecewise constant, then for all θ ∈ D there exist at least one discrete value θk ∈ DI such that

2I ? 1

and PE (θ) corresponds to the residue

∞ 2i ? 1

S (P AΣI (θ)) + 2γ

j =0

αj (θ )I n

2I ? 1

PE (θ ) =

i=I +1 j =0

Pij ψij (θ)

(32)

= S (P AΣI (θk )) + 2γ

j =0

αj (θk )In ? 0. (27)

Hence, for all θ ∈ D, it follows that: S (P A(θ)) = S (P AΣ∞ (θ)) = S (P AΣI (θ) + P AE (θ))

2I ? 1

with P0 = P (θ), φ0 (θ) and Pij = P (θ), ψij (θ) . Notice that P (θ) in (29) is piecewise af?ne. Hence, it follows by construction that, for any resolution level I ≥ G, it presents the following key property: PΣI (θk ) = P (θk ), ?θk ∈ DI . (33)

S (P AΣI (θ)) + 2γ

j =0

αj (θ )I n ? 0 .

Analogously to Lemma 1, the next lemma provides a bound on the residual matrix PE (θ). First, let

Δ ?0 (θ), ψIj (θ) , λj ( θ ) = φ Δ ?gh (θ), ψIj (θ) χghj (θ) = ψ

Thus, PLMI (3), (4) are satis?ed for all θ ∈ D, which implies the quadratic stability of LPV system (1). In summary, feasibility of the ?nite-dimensional program on Theorem 1 implies feasibility of the in?nite-dimensional problem given by PLMI (3), (4), which in turn allows to infer stability of the LPV system

(34)

represent the Haar coef?cients at level I of expansion set (30). Lemma 2: Consider P (θ) in (29) and its Haar transform (31). For G ?1 any decomposition level I ≥ G ∈ N+ , there exist scalars {πh }2 h=0 ∈

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 1, JANUARY 2015

R+ such that the following statements are equivalent: 1) There exist matrices Q0 , {Qgh } ∈ Sn such that

2 ?1

I

which leads to the conclusion that

∞ 2i ? 1 ∞ 2i ? 1

G 2 ?1

g

± Qgh χghj φj (2 θk )

I i=I +1 j =0

Pij |ψij (θ)|

i=I +1 j =0

Pij ψij (θ) = PE (θ). (44)

±2

I 2

Q 0 λj +

j =0 g =0 h=0

G

2 ?1

πh φh (2G θk )In ,

h=0

?θk ∈ DI ;

(35)

2) The following PLMI holds:

2G ? 1

PE (θ )

h=0

πh φh (2G θ)In ,

?θ ∈ D .

(36)

Finally, (36) follows from (44), considering (38), (39) and (43). The bound on PE (θ) obtained on Lemma 2 will be used in the next theorem, in which the main result of this section is presented. Theorem 2: Consider the LPV system (1) and upper bound (18) on ˙ (t)| ≤ ρ ∈ the residue of the state matrix. Suppose that θ(t) ∈ D, |θ R, ?t. Also, consider the candidate matrix P (θ) de?ned in (29) and its partial derivative Q(θ) in (28). Then the LPV system is quadrati? ∈ Sn , cally asymptotically stable if there exist matrices Q0 , {Qgh }, Q

?1 2 ?1 and scalars {πh }2 h=0 , {γj }j =0 ∈ R+ , such that LMI (35) plus the following set of LMI hold for all θk ∈ DI and given resolution levels G ≤ I ∈ N+ : 2G ? 1

G I

Proof: Only implication (1) =? (2) is proved. Consider P (θ) in (29). The coef?cients at level I of PΣ∞ (θ) are given by

G 2g ? 1

P (θk ) Qgh χghj . (37) P (θk ) +

h=0 h=0 2G ? 1

πh φh (2G θk )In ,

2I ? 1 G

(45)

PIj (θ) =

P (θ), ψIj (θ)

= Q 0 λj +

g =0 h=0

πh φh (2 θk )In

j =0

γj φj (2I θk )In ,

(46) (47)

In (37), the equality then be rewritten as

2I ? 1

? ψIj (θ) Q,

= 0 was used. Inequality (35) can

2G ? 1

± ρQ(θk ) + S (P (θk )AΣI (θk )) + ξ (θk )In ? 0 where

±2 2

j =0

I

PIj φj (2I θk )

h=0

πh φh (2G θk )In .

(38)

2G ? 1 2I ? 1

Next, since {φj } are piecewise constant, then for all θ ∈ D there exist at least one discrete value θk ∈ DI such that

2I ? 1 2I ? 1

ξ (θk ) = 2

h=0

Δ

AΣI (θk ) πh φh (2G θk )+2

j =0

γj αj (θk )φj (2I θk ).

2

I 2

PIj φj (2 θk ) = 2

j =0

I

I 2

PIj φj (2 θ).

j =0

I

(39)

Proof: First, notice that since P (·) is piecewise af?ne, then PΣI (θk ) = P (θk ), ?θk ∈ DI . Consequently, (45) can be rewritten in the equivalent form

2G ? 1

The right-hand side of (39) can be rewritten as follows: 2 2 2

I 2

2I ? 1

∞

PIj φj (2I θ) =

j =0 ∞ 2I ? 1

I

i=0

1 2i

2 2

I 2

2I ? 1

PΣI (θk )

h=0

πh φh (2G θk )In

0.

PIj φj (2I θ)

j =0

Since PΣI (·) is piecewise constant, then for all θ ∈ D there exist one discrete value θk ∈ DI such that PΣI (θ) = PΣI (θk ), and hence

2G ? 1

=

i=0 j =0 ∞ 2I ? 1

2 2 PIj φj (2I θ) 2i+1

I +i+1 2

P ΣI ( θ )

h=0

πh φh (2G θ)In

0,

?θ ∈ D .

(48)

=

i=0 j =0

2

P(I +i+1)j φj (2I θ)

(40)

Now, let PE (θ) represent the unknown residue (32) associated to P (θ). Thus, from (48) it follows that:

2G ? 1

∞ 2I +i+1 ?1

=

i=0 ∞ 2 j =0

I +i+1

2

?1

I +i+1 2

P(I +i+1)j φj (2

I +i+1

θ)

(41)

P ΣI ( θ ) + P E ( θ ) = P ( θ )

h=0

πh φh (2G θ)In + PE (θ).

(49)

=

i=0 ∞ j =0 2i ? 1

P(I +i+1)j ψ(I +i+1)j (θ)

(42)

=

i=I +1 j =0

Pij |ψij (θ)|.

(43)

Since (35) is veri?ed by hypothesis, then (36) holds according to G ?1 Lemma 2. Consequently, since {πh }2 h=0 ∈ R+ , it follows from (49) and (36) that P (θ) 0, ?θ ∈ D, which is exactly PLMI (3). Next, following the same arguments leading to (48), it follows from (46) that

2G ? 1 2I ? 1

Step (40) follows from property (14) of af?ne mapping P (·) in (29), whereas (41) follows from property (11). Step (42) follows from the equivalence φj (2I +m θ) = 2?((I +m)/2) |ψ(I +m)j (θ)|. Now, notice that Pij ψij (θ) = Pij |ψij (θ)| , ?Pij |ψij (θ)| , 0,

1 if j 2i1 +1 ≤ θ < (j + 1) 2i+1 , if (j + 1) 2i1 ≤ θ < ( j + 2) 2i1 +1 +1 , otherwise

PΣI (θ)+

h=0

πh φh (2 θ)In

j =0

G

γj φj (2I θ)In , ?θ ∈ D. (50)

Thus, from (36) one can conclude that, ?θ ∈ D

2G ? 1 2I ? 1

P (θ )

P ΣI ( θ ) +

h=0

πh φh (2G θ)In

j =0

γj φj (2I θ)In . (51)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 1, JANUARY 2015

197

Finally, once again by the arguments leading to (48) and (50), it follows that inequality (47) also holds for all θ ∈ D. Consequently, from (36) and (51) it follows that: 0 ?P (θ) + S (PΣI (θ)AΣI (θ)) + ξ (θ)In ?θ ?P (θ) + S (PΣI (θ)AΣI (θ)) ±ρ ?θ ±ρ + 2 PE (θ ) ±ρ ?P (θ ) ?θ ?P (θ) + S (P (θ)A(θ)) . ?θ A ΣI ( θ ) I n + 2 P ( θ ) A E (θ ) I n

Fig. 2. (a) Estimates of ζ ? in Example 1 for different resolution levels I given by Theorem 1 and the necessary condition (54); the value given by the necessary and suf?cient condition in [11] is also shown; (b) Estimates of ρ? in Example 2 for different resolution levels {I, G} given by Theorem 2.

+ S ((PΣI (θ) + PE (θ)) AΣI (θ) + P (θ)AE (θ)) , = ±ρ (52)

˙(t)| ≤ ρ ∈ R, ?t, then it follows from Now, since, by assumption, |θ (52) that (4) is also satis?ed. The quadratic stability of LPV system (1) ?nally comes from the fact that both PLMI (3), (4) are satis?ed. The quantity ξ (θk ) appearing in LMI (47) provides a measure of the conservatism resulting from the Haar truncation process. Notice that ξ (θ), and hence the conservatism, tends to decrease with increasing resolution levels I and G. Indeed, increasing resolution I corresponds to taking more information about the system into account, whereas increasing resolution G corresponds to providing more degree of freedom to the Lyapunov candidate with respect to its parametric dependence. On the other hand, increasing level I leads to an increase in the number of LMI constraints, whereas increasing G implies more decision variables. The total number of decision variables is 2I + 2G (1 + n(n + 1)) + n(n + 1)/2, whereas the number of n × n matrix constraints is given by 6(2I +1 ). For resolution levels I and G tending to in?nity, ξ (θ) vanishes asymptotically, and hence the only source of conservatism remaining in Theorem 2 comes from the use of a quadratic-in-the-state LF. If, on the one hand, it can be argued that the quadratic state dependence is too restrictive, on the other hand, the parametric dependence of the considered LF is much more general. It encompasses as particular instances the great majority of dependencies in the literature, e.g., the popular homogeneous polynomial case. In the limit, the stability condition on Theorem 2 involves a quadratic-in-the-state LΘ 2 -in-theparameter LF. Although the discussion has been limited to the scalar case θ ∈ D, the above results can also be extended to the multi-dimensional case [15]. In such a case, a different truncation level can be considered for each one of the parameters, hence allowing for a better balancing between the required computational effort and the conservatism of the stability analysis test. V. N UMERICAL E XPERIMENTS A. Example 1 Consider the following model introduced in [11]: A (θ(t)) =

?1+θ (t)?θ 2 (t) 1+θ (t)

Fig. 2(a) depicts estimates of ζ ? provided by Theorem 1 for different resolution levels I . As expected, the estimates approach ζ ? from below as I increases, which represents a key property of the proposed approach. For I ≥ 8, the conservatism in Theorem 1 becomes negligible, and hence the estimated ζ ? virtually coincide with the true bound ζ ? = 4.256 for quadratic stability. For each truncation level I , the estimated ζ ? is found by a bisection algorithm on the normalization factor ζ de?ning D. Also in Fig. 2(a) are depicted estimates of ζ ? obtained by checking inequalities (3), (4) solely at those points in grid DI . A necessary condition to the quadratic stability of (1) is that there exists P ∈ Sn 0 such that the following set of LMI is satis?ed: P A(θk ) + AT (θk )P ? 0, ?θk ∈ DI . (54)

For a given resolution level I , the estimate of ζ ? obtained via (54) consists in an upper bound on the one provided by Theorem 1, so the former serves to assess the conservatism of the latter. B. Example 2 Consider now the following LPV model taken from [8]: A (θ(t)) = 8 ? 108θ(t) 120 ? 120θ(t) ? 9 + 9 θ (t) ?18 + 17θ(t)

0

1 ?1

(53)

where the parameter θ(t) ∈ [0, ζ ], ?t. The objective in this example is to determine the maximum ζ , denoted ζ ? , such that the origin is asymptotically stable, even for unbounded parameter variation rate. In [11], a necessary and suf?cient condition for the stability of a class of LPV systems including (53) has been obtained on the basis of a homogeneous polynomial LF. When applied to (53) considering a quadratic LF, the resulting LMI condition determines ζ ? = 4.256.

˙(t)| ≤ ρ. The problem here is to determine where θ(t) ∈ [0, 1] and |θ the maximum ρ, denoted ρ? , such that the origin is asymptotic stable. In [8], the value ρ? = 66.81 has been determined. Fig. 2(b) depicts estimates of ρ? obtained by the LMI condition on Theorem 2 for different pairs of resolution levels {I, G}. As expected, increasing resolution level I leads to a less conservative estimate of ρ? , as it implies increasing the grid density. Additionally, increasing resolution level G also leads to better results by providing more degree of freedom to Lyapunov matrix P (θ). For I = 12 and G = 10, the value ρ? = 252.49 is determined. It consists in a signi?cant improvement over the result obtained in [8] on the basis of quadratic-in-the-parameter quadratic-in-the-state LF. Remark: By comparing (13) with (10), it follows that in the af?ne case considered in Example 2, there is no conservatism in upper bound 2I ? 1 (10) for s = 1 and κij = a/4, ?i, j , and hence scalars {αj (θ)}j =0 in Lemma 1 can be obtained straightforwardly. In Example 1, since the denominator is always positive and the Lyapunov matrix is parameterindependent, the analysis problem can be simpli?ed to involve polynomial dependence only, so computing the scalars αj (θ) is once again straightforward. A systematic procedure for computing these scalars with as little conservatism as possible for a general parameter dependence remains an area for research.

198

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 1, JANUARY 2015

VI. C ONCLUSION New algorithms for stability analysis of LPV systems have been presented. The proposed approach relies on Haar transform theory to solve the original in?nite-dimensional feasibility problem by means of ?nite dimensional semide?nite programing. One appealing feature of the proposed approach is that it can handle a very large class of parametric dependencies as well as non-convex parametric domains. Moreover, realistic parameter variation rates can be easily taken into account. It is worth pointing out that all these features are rarely jointly present in existing methods. Extensions of the proposed method for performance analysis and for LF with more general state dependencies are currently under investigation.

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