- Email: [email protected]
Convex analysis of invariant sets for a class of nonlinear systems
Systems & Control Letters 54 (2005) 729 – 737 www.elsevier.com/locate/sysconle
Convex analysis of invariant sets for a class of nonlinear systems夡 Tingshu Hu1 , Zongli Lin∗ Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P. O. Box 400743, Charlottesville, VA 22904-4743, USA Received 9 April 2003; received in revised form 15 April 2004; accepted 14 November 2004 Available online 28 December 2004
Abstract We study the invariance of the convex hull of an invariant set for a class of nonlinear systems satisfying a generalized sector condition. The generalized sector is bounded by two odd symmetric functions which are convex/concave in the righthalf plane. In a recent paper, we showed that, for this class of systems, the convex hull of a group of invariant ellipsoids is invariant. This paper shows that the convex hull of a general invariant set need not be invariant, and that the convex hull of a contractively invariant set is, however, invariant. © 2004 Elsevier B.V. All rights reserved. Keywords: Convexity; Invariant set; Contractive invariance; Lyapunov stability
1. Introduction Convexity is often a desired property for a function or a set. In stability analysis, we usually use invariantlevel sets of certain Lyapunov functions to estimate the domain of attraction. If we have several invariant sets, then it is obvious that their union is also invariant. Since the convex hull of these invariant sets is potentially much larger than the union, we would like to know if the convex hull is still invariant. Of course, a positive answer should only be expected from nonlinear systems satisfying certain conditions. 夡
Supported in part by NSF Grant CMS-0324329.
∗ Corresponding author.
E-mail address: [email protected] (Z. Lin). 1 Current address: Department of Electrical and Computer En-
gineering, University of California, Santa Barbara, CA, USA. 0167-6911/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2004.11.011
In this paper, we study the convexity of invariant sets for a nonlinear system x˙ = Ax + B (F x, t),
(1)
where (·, t) belongs to a class of nonlinear functions. A block diagram for such a system is plotted in Fig. 1. In the absolute stability theory, the class of nonlinear functions is described with a conic sector, which can be used to represent a possibly uncertain and time-varying nonlinear component. The absolute stability of the Lur’e systems in Fig. 1 is a classical problem in control theory. It has been studied extensively in the nonlinear systems and control literature (see e.g. [1,8,12,14,15,17,18] and references therein), and is still attracting tremendous attentions (see [2–4,10,11,13,16] for a sample of recent literature).
730
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737 u
ψ (u, t )
v
5
−1
F( sI − A) B
v
4
v=ψ1 (u)
3
v=ψ(u)
2
Fig. 1. A system with a nonlinear component.
1
v=ψ2(u)
u
0 o
5 –1
v
4
v=k 1u
–2
3 –3
v=ψ(u)
2
–4 1
v=k 2u
–5 –8
u
0
–6
–4
–2
0
2
4
6
8
o –1
Fig. 3. The generalized sector.
–2 –3 –4 –5 –8
–6
–4
–2
0
2
4
6
8
Fig. 2. The classical linear sector.
Traditionally, the uncertain nonlinear function is assumed to be inside a sector bounded by two straight lines. The common tools for absolute stability under such a sector condition include circle criterion and Popov criterion, which give sufficient conditions for global stability over the sector. Since global absolute stability does not generally hold, another trend in the development of absolute stability theory is the study of absolute stability within a finite region (see e.g. [4,8,9,13,17]). In the case that global absolute stability does not hold, we need to restrict our attention to a finite region in the state space, where a sector that is narrower than the global sector can be used to bound the nonlinear function (u, t). Fig. 2 plots a sector between two straight lines v =k1 u and k2 u. This sector is a global bound for one of the nonlinear functions but is only a local bound for the other one, which can only be globally bounded by v = k1 u and 0. In the finite region, a guaranteed stability region can then be obtained by using some invariant level set of a quadratic or Lur’e type Lyapunov function (see e.g. [9,13]).
In an effort to give a better description for the uncertain and/or time-varying nonlinear component, we recently (in [6]) generalized the sector such that its boundary is defined by two odd symmetric nonlinear functions which are either concave or convex over [0, ∞]. For simplicity, these functions are said to be concave or convex. Fig. 3 plots a sector bounded by a convex function and a concave function. We first studied the absolutely contractively invariant (ACI) ellipsoids and developed a necessary and sufficient condition under which an ellipsoid is ACI. We then showed that the convex hull of a group of ACI ellipsoids is also ACI. With the results of [6], we are tempted to ask the question: is the convex hull of an arbitrary ACI set also ACI? This paper will give a positive answer to this question. It then follows that for a sector bounded by two concave/convex functions, the largest ACI set is convex. In fact, an ACI set is associated with a (positively) homogeneous Lyapunov function which is strictly decreasing along the trajectories of the system. Since the largest ACI set is convex, in searching for a homogeneous Lyapunov function to estimate the domain of attraction, we only need to restrict our attention to those whose level sets are convex. In the case that the Lyapunov function is homogeneous of degree greater than one, this means that the function is convex. In an attempt to further generalize the results, we would like to know if we can replace ACI with AI
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
(absolutely invariant) and still get a positive answer. However, we will use an example to show that the convex hull of an arbitrary invariant set need not be invariant even for a system with convex nonlinearity. This paper is organized as follows. Section 2 gives a review of the definitions of the generalized sector and absolute invariance. Section 3 presents some results on convex analysis. Section 4 analyzes the invariance of the convex hull of an invariant set and Section 5 contains some concluding remarks.
731
concave for u < 0. In Fig. 3, one of the boundary functions is convex and the other one is concave. Here is a simple fact about concave and convex functions. Fact 1. Let be a concave (convex) function. If we draw a straight line that is tangential to at (u0 , (u0 )), u0 0, then the straight line is above (below) (u) for all u > 0. 2.2. The generalized sector and absolute stability
Notation. For two integers k1 , k2 , k1 < k2 , we denote I [k1 , k2 ] = k1 , k1 + 1, . . . , k2 . For S ⊂ Rn , we use co{S}, int{S} and *S to denote the convex hull, the interior and the boundary of S, respectively. For S ⊂ Rn and a real number , S = {x : x ∈ S}.
2. A generalized sector and absolute invariance 2.1. Concave functions and convex functions We first give a formal definition of some functions that we will use to define the boundary of the generalized sector. Given a scalar function : R → R. Assume that (1) is continuous, piecewise differentiable, (0)= 0 and ddu > 0. u=0
(2) is odd symmetric, i.e., (−u) = −(u). A function satisfying the above assumption is said to be concave if it is concave for u > 0. That is, for any u1 , u2 > 0,
(u1 + (1 − )u2 ) (u1 ) + (1 − )(u2 ) ∀ ∈ [0, 1]. A function satisfying the above assumption is said to be convex if it is convex for u > 0. That is, for any u1 , u2 > 0,
(u1 + (1 − )u2 ) (u1 ) + (1 − )(u2 ) ∀ ∈ [0, 1]. These definitions are made for simplicity. It should be understood that by odd symmetry, a concave function is convex for u < 0 and a convex function is
Consider the system x˙ = Ax + B (F x, t),
(2)
where A ∈ Rn×n , B ∈ Rn×1 and F ∈ R1×n are given matrices. A set S ⊂ Rn is said to be invariant for (2) if every trajectory starting from it stays within it for t > 0. The domain of attraction of the origin for system (2) is an invariant set and a traditional way to estimate it is to use invariant sets that contain the origin in its interior. In stability analysis, we often use contractively invariant set to estimate the domain of attraction. Definition 1. Consider system (2). Let S ⊂ Rn be a compact set such that kS ⊂ int{S} for all k ∈ [0, 1). We say that S is contractively invariant if for every k ∈ (0, 1] and for every x ∈ *{kS}, Ax + B (F x, t) points strictly inward of kS, i.e., for every t > 0, there exists 0 > 0 such that x + (Ax + B (F x, t)) ∈ int{kS} ∀ ∈ (0, 0 ). In the above definition of contractive invariance, it is required that kS ⊂ int{S} for all k ∈ [0, 1). This guarantees that S is simply connected and contains the origin in its interior. Also, S equals the closure of int{S}. Actually, a contractively invariant set is associated with a positively homogeneous Lyapunov function which is strictly decreasing along the trajectories of system (2). Define V (x) := min{ 0 : x ∈ S},
(3)
then V (x) is positive-definite and positively homogeneous of degree one. Since kS ⊂ int{S} for all k ∈ [0, 1), for every x0 ∈ *S, we have kx 0 ∈ int{S} for all
732
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
k ∈ [0, 1). It is straightforward to verify that for any x, V (x) equals the unique 0 such that x ∈ *(S) and
*(S) = {x ∈ Rn : V (x) = }. We claim that V (x) is continuous. Because of homogeneity, we only need to verify the continuity for x ∈ *S, i.e., V (x) = 1. Given any > 0, we have (1 − )S ⊂ int{S},
S ⊂ int{(1 + )S}.
It follows that min{x1 − x2 : x1 ∈ *S, x2 ∈ (1 − )S} =: 1 > 0, min{x1 − x2 : x1 ∈ *S, x2 ∈ Rn \int{(1 + )S}} =: 2 > 0. Hence for every x0 ∈ *S, there is an open neighborhood of x0 , {x ∈ Rn : x − x0 < min{1 , 2 }}, from which V (x) ∈ (1 − , 1 + ). This verifies the continuity. It should be noted that V (x) can also be defined to be positively homogeneous of any degree q > 0 as follows: V (x) := min{q : x ∈ S, 0}. If q 1 and S is convex, then V (x) is convex. From Definition 1, it is clear that if S is contractively invariant, then kS is also contractively invariant for every k ∈ (0, 1). With a contractively invariant set S, a trajectory x(t) starting from any x0 ∈ *S will go to the interior of S and we have x(t) ∈ *{k(t)S} with k(t) strictly decreasing and converge to 0. Thus x(t) → 0. If we define a function V as in (3), then V (x(t)) is strictly decreasing and converges to 0. In Definition 1, the nonlinear function (u, t) in system (2) is assumed to be known. In practice, there always exists some degree of uncertainty about a nonlinear component. In view of this, we would like to study the invariance of a set for a class of nonlinear functions. On the other hand, some nonlinear functions (u, t) could be complicated and we would like to bound it with simpler functions 1 (u) and 2 (u). These problems arise from the same situations which motivated the problem formulation of absolute stability, where the nonlinear function (u, t) is bounded by two linear functions 1 (u) = u and 2 (u) = u. If the system is not globally absolutely stable over a linear sector [, ], we have to consider the stability on
a finite region of the state space, over which a pair of nonlinear functions 1 and 2 may better describe the property of the nonlinear component. For this reason, we introduced the generalized sector in [6]. Definition 2. Given functions 1 : R → R and 2 : R → R, each of which is concave or convex. A function (u, t), piecewise continuous in t and locally Lipschitz in u, is said to satisfy a generalized sector condition if
(u, t) ∈ co{1 (u), 2 (u)} ∀u, t ∈ R. We use co{1 , 2 } to denote the generalized sector, i.e., the set of functions that satisfy the above generalized sector condition. A set S ⊂ Rn is said to be absolutely invariant (AI) over the sector co{1 , 2 } if it is invariant for (2) under all the possible (u, t) satisfying the generalized sector condition. A set S ⊂ Rn is said to be absolutely contractively invariant (ACI) over the sector co{1 , 2 } if it is contractively invariant for (2) under all the possible (u, t) satisfying the generalized sector condition. We see that if S is ACI, then any trajectory starting from it will converge to the origin under all (u, t) satisfying the generalized sector condition. Hence S is an absolute stability region. Let us next state a simple fact. Fact 2. Given a compact convex set S and a class of functions i (u), i ∈ I [1, N ]. Suppose that for each i ∈ I [1, N ], S is (contractively) invariant for x˙ = Ax + B i (F x). Let (u, t) be a function such that (u, t) ∈ co{i (u), i ∈ I [1, N ]} for all u ∈ R and t ∈ R, then S is (contractively) invariant for x˙ = Ax + B (F x, t). This fact follows directly from the definition of invariant set and the convexity of S. By Fact 2, we see that the absolute (contractive) invariance of a convex set is equivalent to its (contractive) invariance under both 1 and 2 . Although we may use two arbitrary nonlinear functions 1 and 2 to define a generalized sector,
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
concave and convex functions appear to be simpler and easier to handle, and may lead to better properties. For example, it was shown in [6] that the invariance of an ellipsoid under a concave/convex nonlinearity is equivalent to some linear matrix inequalities. Moreover, the convex hull of a group of contractively invariant ellipsoids is also contractively invariant. With two general nonlinear functions 1 and 2 , it is hard to expect other properties beyond Fact 2. On the other hand, many commonly encountered nonlinearities are either concave or convex, for example, the tangent function, the saturation function and the dead zone function. In view of this, we will focus on the invariance of a set under a concave or convex function.
3. Some facts about convex sets For easy reference, we collect in this section some results from convex analysis (e.g. see [5]). Let S be a compact convex set. We say that x0 ∈ S is an extreme point of S if it cannot be represented as the convex combination of other points in S, i.e., x0 =
N i=1
i xi ,
N i=1
i = 1,
If *V (x) has only one element at x, then it is called the derivative of V at x. Fact 3. Consider x0 = 0. Let V (x0 ) = k0 . Then c ∈ *V (x0 ) if and only if cT x = k0 is a supporting hyperplane of k0 S at x0 . This fact can be seen as follows. For any k > 0, V (kx 0 ) = kk 0 . Suppose c ∈ *V (x0 ). Then cT (kx 0 − x0 )=(k −1)cT x0 (k −1)k0 . It follows that cT x0 =k0 . If x ∈ k0 S, then V (x) − V (x0 ) 0. Hence cT (x − x0 ) 0 for all x ∈ k0 S. Therefore, cT x = k0 is a supporting hyperplane of k0 S at x0 . On the other hand, suppose that cT x = k0 is a supporting hyperplane of k0 S at x0 . Any x ∈ Rn can be written as x = kx 1 for some x1 ∈ *{k0 S} and k > 0. Hence V (x) = kk 0 and cT x = kcT x1 kk 0 = V (x). Noting that cT x0 = k0 = V (x0 ), we have ∀x ∈ Rn .
cT y < 0
∀c ∈ *V (x0 ).
Hence, the set S is contractively invariant for
c T x0 = k 0 .
The intersection of a supporting hyperplane with the set S is called an exposed face of S. A point x0 is an extreme point of S if and only if it is an extreme point of any exposed face containing it. This implies that, if x0 ∈ S is not an extreme point, then it is not an extreme point of any exposed face. If S is a compact convex set containing the origin in its interior, a Minkowski function can be defined as V (x) := min{ 0 : x ∈ S}.
*V (x) = {c ∈ Rn : cT (x˜ − x) V (x) ˜ − V (x) n ∀x˜ ∈ R }.
Consider x0 ∈ *{k0 S}. Given a vector y ∈ Rn , it points strictly inward of k0 S from x0 if and only if
A hyperplane cT x = k0 is a supporting hyperplane of S at x0 ∈ *S if ∀x ∈ S,
The subderivative of V at x, denoted as *V (x), is
cT (x − x0 ) V (x) − V (x0 )
i 0, xi ∈ S
⇒ x1 = x2 = · · · = xN = x0 .
c T x k0
733
x˙ = Ax + B (F x) if and only if cT (Ax 0 + B (F x 0 )) < 0 ∀c ∈ *V (x0 ),
x0 ∈ S\{0},
and S is invariant if and only if cT (Ax 0 + B (F x 0 )) 0
(4)
This function is convex, positive definite and positively homogeneous of degree one.
∀c ∈ *V (x0 ),
x0 ∈ *S.
By Fact 3, we can replace c ∈ *V (x0 ) with c ∈ Rn such that cT x = k0 is a supporting hyperplane of k0 S at x0 .
734
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
4. Invariance of the convex hull of an invariant set
30
20
Consider the system 10
x˙ = Ax + B (F x).
We assume that is concave. If is convex, we can replace (u) with 0 u − 1 (u), where 0 is a constant and 1 (u) is concave, and obtain x˙ = (A + 0 BF )x − B 1 (F x).
(6)
It is easy to see that if we have a group of contractively invariant sets, then their union is also contractively invariant. Hence an invariant set need not be convex. What we are interested in is the convex hull of an invariant set. In [6], we showed that the convex hull of a group of contractively invariant ellipsoids is contractively invariant. In this paper, we would like to extend this result of [6] to a more general invariant set. It turns out that we have quite different conclusions for the convex hull of an invariant set and the convex hull of a contractively invariant set. We will discuss these two situations separately. 4.1. The general invariance We know that the domain of attraction is an invariant set. It is desirable that the domain of attraction is a convex set. We may have a reason to expect this for a class of systems where the nonlinearity is convex/concave, e.g., system (5). As we have shown in [7], for the special case where is the standard saturation function, if A ∈ R2×2 and its eigenvalues have positive real part, then the domain of attraction is convex and its boundary is the unique limit cycle. However, this result cannot even be extended to all the secondorder systems, especially when A has two eigenvalues of different signs. For example, we have a system x˙ = Ax + B sat(F x), where sat(u) = sign(u) min{1, |u|} and 0 0 1 , B= , F = [ −2 A= 5 1 0
−1 ] .
The domain of attraction is not bounded, as shown in Fig. 4, where its boundary is plotted with solid lines. The boundary of the domain of attraction is generated
x2
(5) 0
(5,0)
(-5,0)
–10
–20
–30 –40
–30
–20
–10
0 x1
10
20
30
40
Fig. 4. A nonconvex domain of attraction.
by simulation. It is composed of four trajectories, two of them go from infinity toward (5, 0) and the other two go from infinity toward (−5, 0). It is obvious that this domain of attraction is not convex. Then what about its convex hull? Is it invariant? The convex hull of the domain of attraction is a strip, whose boundary is plotted as dash–dotted lines in Fig. 4. We chose an arbitrary initial state (marked with “∗”) inside the strip but outside the domain of attraction, the trajectory goes out of this strip and diverges. This gives us a counter example for the invariance of the convex hull of an invariant set. However, for the convex hull of a contractively invariant set, we have a quite different conclusion, as will be shown next. 4.2. The contractive invariance Without loss of generality, assume that d/du|u=0 = 1. Let S ⊂ Rn be a compact set such that kS ∈ int{S} for all k ∈ [0, 1). Suppose that S is contractively invariant for (5). Then by Definition 1, kS is contractively invariant for all k ∈ (0, 1]. Inside kS, as k approaches 0, the system approximates the linear system x˙ = Ax + BF x.
(7)
By taking the limit, it is easy to see that kS is invariant for the linear system (7), and hence S is also invariant for the linear system.
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
If (u) = u for an interval [0, u0 ], u0 > 0, then for sufficiently small k, the system is exactly linear inside kS. In this case, the contractive invariance of a set S implies its contractive invariance for the linear system (7). The following is the main result of the paper. Theorem 1.
Proof. With co{S}, we can define V (x) = min{ 0 : x ∈ co{S}}. (a) We will show that for all x on the boundary of co{S}, x˙ points inward of co{S}, i.e., cT (Ax + B (F x)) 0
∀c ∈ *V (x), x ∈ *co{S}.
The invariance of co{kS} for k ∈ (0, 1] follows similarly from the contractive invariance of kS. Here, we only need to consider x ∈ *co{S}\*S. Since for those x ∈ *S, x˙ points inward of S implies that it points inward of co{S}. Now, consider x0 ∈ *co{S}\*S. Let cT x = 1 be a supporting hyperplane of co{S} at x0 . We need to prove that cT (Ax 0 + B (F x 0 )) 0.
(8)
Since x0 ∈ / *S, it is not an extreme point of co{S}. Hence, this supporting hyperplane must also touch some points on *S. In other words, cT x = 1 is also a supporting hyperplane at some points in *S. Moreover, x0 can be expressed as a convex combination of x1 , x2 , . . . , xN ∈ *S and cx j = 1 for all j ∈ I [1, N]. This means that there exist j > 0, j ∈ I [1, N], such that x0 =
j =1
j xj ,
By taking k → 0, and noting that d/duu=0 = 1, we obtain cT (Ax j + BF x j ) 0
∀j ∈ I [1, N ].
N j =1
j = 1.
Since kS is contractively invariant for all k ∈ (0, 1], we have cT (Akx j + B (kF x j )) < 0, k ∈ (0, 1].
j ∈ I [1, N ], (9)
(10)
Hence, for all x ∈ co{x1 , x2 , . . . , xN }, cT (Ax + BF x) 0.
(a) If S is contractively invariant for (5), then co{kS} is invariant for all k ∈ (0, 1]. (b) If S is contractively invariant for both (5) and (7), then its convex hull is contractively invariant for (5).
N
735
(11)
Assume that F x 0 0. (If F x 0 0, then by the odd symmetry of (·), we can use similar argument to prove (8).) First, we suppose that F x j 0 for all j ∈ I [1, N ]. In this case, F x 0 for all x ∈ co{x1 , x2 , . . . , xN }. If cT B 0, then by the assumption that F x 0 0 and by the concavity of the function (·), we have (F x 0 ) F x 0 , and hence, cT (Ax 0 + B (F x 0 )) cT (Ax 0 + BF x 0 ) 0.
(12)
If cT B 0, then also by the concavity of (·), cT Ax + cT B (F x) is a convex function for x ∈ co{x1 , x2 , . . . , xN }. Hence we also have (8) by (9). If F x j 0 does not hold for all j ∈ I [1, N ], then we can get an intersection of the set co{x1 , x2 , . . . , xN } with the half space F x 0. This intersection is also a polygon and can be denoted as co{y1 , y2 , . . . , yN1 }. Since F x 0 0, we have x0 ∈ co{y1 , y2 , . . . , yN1 }. Some yj ’s belong to {x1 , x2 , . . . , xN }, others are not. For those yj ∈ / {xi : i ∈ I [1, N ]}, we must have F y j = 0 and yj ∈ co{xi : i ∈ I [1, N ]}. It follows from (11) that cT (Ay j + BF y j ) 0. Since (0) = 0, for those yj ’s such that F y j = 0, we have cT (Ay j + B (F y j )) = cT (Ay j + BF y j ) 0. In summary, we have cT (Ay j + B (F y j )) 0
∀j ∈ I [1, N1 ].
(13)
Because of this, we can work on the set co{y1 , y2 , . . . , yN1 } instead of co{x1 , x2 , . . . , xN }. Since F y j 0 for all j ∈ I [1, N1 ], same arguments can be used to prove (8) by using (13) instead of (9). (b) The procedure of the proof is very similar to the proof for (a). The only difference is to replace “ ” in the inequalities with “<”. This is guaranteed by the additional condition that S is contractively invariant for the linear system. Because of this, instead of (10),
736
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737 15
10 8
10
6 4
5
x2
x2
2 0
0
–2 –5
–4 –6
–10
–8 –10 –8
–6
–4
–2
0 x1
2
4
6
8
10 8 6 4
x2
2 0 –2 –4 –6 –8
–6
–4
–2
0 x1
2
4
6
8
Fig. 6. Convex hull of the two contractively invariant sets.
we have cT (Ax j + BF x j ) < 0
–10
–5
0 x1
5
10
15
Fig. 7. Convex hull and the domain of attraction.
Fig. 5. Union of two contractively invariant sets.
–10 –8
–15 –15
∀j ∈ I [1, N].
This leads to “<” for all the remaining inequalities. We note that the statement of (a) is stronger than the simple invariance of S. From Theorem 1, we can conclude that the largest contractively invariant set for system (5) is convex. Recall that a contractively invariant set is associated with a positively homogeneous Lyapunov function. By Theorem 1, we can restrict
our attention to convex Lyapunov function candidates. Furthermore, with several contractively invariant level sets of different Lyapunov functions, we can obtain a larger contractively invariant set as their convex hull. One may be tempted to extend Theorem 1 to systems with more than one nonlinear components, i.e., to the case where (·) is a vector function and B has more than one column. However, it is difficult to see such a possibility from the proof of Theorem 1, which relies on the fact that for a fixed c, the function cT (Ax + B (F x)) is either convex or concave in x. For the case that (·) is a vector function, even if all of the components of (·) are concave, their linear combination cT B (·) could be neither concave nor convex. Example 1. Consider the same system as the one in Section 4.1. Using LMI-based technique, we detected two contractively invariant sets, each of which is the convex hull of two ellipsoids. Fig. 5 plots the boundaries of these two invariant sets along with line segments marking the direction of the vector x. ˙ The line segments clearly show that the union of these sets is invariant. Fig. 6 plots the convex hull of these two contractively invariant sets along with line segments marking the direction of the vector x. ˙ Also plotted in the figure in dashed curves are the boundaries of the two invariant sets as in Fig. 5. It is clearly seen that the convex hull is also invariant. As a comparison, we plotted the convex hull against the boundary of the domain of attraction in Fig. 7.
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
5. Conclusions This paper studies the invariance of the convex hull of an invariant set for a class of nonlinear systems satisfying a generalized sector condition. We focused on the invariance of a set for a system with concave nonlinearities. We used an example to show that the convex hull of an invariant set need not be invariant but the convex hull of a contractively invariant set is invariant. References [1] M.A. Aizerman, F.R. Gantmacher, Absolute Stability of Regulator Systems, Holden-Day, San Francisco, CA, 1964. [2] M. Arcak, A. Teel, Input-to-state stability for a class of Lurie systems, Automatica 38 (2002) 1945–1949. [3] K. Gu, Absolute stability of systems under block diagonal memoryless uncertainties, Automatica 31 (4) (1995) 581–584. [4] W.M. Haddad, V. Kapila, V.S. Chellabonia, Guaranteed domain of attraction for multivariable Lur’e systems via open Lyapunov surfaces, Proceedings of 13th IFAC World Congress, San Francisco, 1996, pp. 13–18. [5] J.-B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer, Berlin, 1993. [6] T. Hu, B. Huang, Z. Lin, Absolute stability with a generalized sector condition, AC-48 (4) (2004) 535–548. [7] T. Hu, Z. Lin, L. Qiu, Stabilization of exponentially unstable linear systems with saturating actuators, IEEE Trans. Automat. Control 46 (6) (2001) 973–979.
737
[8] H. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, 1996. [9] T. Kiyama, T. Iwasaki, On the use of multi-loop circle for saturating control synthesis, Systems Control Lett. 41 (2000) 105–114. [10] D. Liu, A. Molchanov, Criteria for robust absolute stability of time-varying nonlinear continuous-time systems, Automatica 38 (2002) 627–637. [11] A. Megretski, A. Rantzer, System analysis via integral quadratic constraints, IEEE Trans. Automat. Control 42 (6) (1997) 819–830. [12] K.S. Narendra, J. Taylor, Frequency Domain Methods for Absolute Stability, Academic Press, New York, 1973. [13] T. Pare, H. Hindi, J. How, D. Benjerdpongchai, Local control design for systems with saturating actuators using Popov criteria, Proceedings of the American Control Conferences, San Diego, 1999, pp. 3211–3215. [14] J.-J.E. Slotine, W. Li, Applied Nonlinear Control, PrenticeHall, Englewood Cliffs, NJ, 1991. [15] M.M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1993. [16] T. Wada, M. Ikeda, Y. Ohta, D.D. Siljak, Parametric absolute stability of Lur’e systems, IEEE Trans. Automat. Control AC43 (11) (1998) 1649–1653. [17] J.L. Willems, The computation of finite stability regions by means of open Lyapunov surfaces, Int. J. Control 10 (5) (1969) 537–544. [18] V.A. Yakubovich, Frequency conditions for the absolute stability of control systems with several nonlinear and linear nonstationary blocks, Automat. Remote Control 28 (1967) 857–880.
Convex analysis of invariant sets for a class of nonlinear systems夡 Tingshu Hu1 , Zongli Lin∗ Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P. O. Box 400743, Charlottesville, VA 22904-4743, USA Received 9 April 2003; received in revised form 15 April 2004; accepted 14 November 2004 Available online 28 December 2004
Abstract We study the invariance of the convex hull of an invariant set for a class of nonlinear systems satisfying a generalized sector condition. The generalized sector is bounded by two odd symmetric functions which are convex/concave in the righthalf plane. In a recent paper, we showed that, for this class of systems, the convex hull of a group of invariant ellipsoids is invariant. This paper shows that the convex hull of a general invariant set need not be invariant, and that the convex hull of a contractively invariant set is, however, invariant. © 2004 Elsevier B.V. All rights reserved. Keywords: Convexity; Invariant set; Contractive invariance; Lyapunov stability
1. Introduction Convexity is often a desired property for a function or a set. In stability analysis, we usually use invariantlevel sets of certain Lyapunov functions to estimate the domain of attraction. If we have several invariant sets, then it is obvious that their union is also invariant. Since the convex hull of these invariant sets is potentially much larger than the union, we would like to know if the convex hull is still invariant. Of course, a positive answer should only be expected from nonlinear systems satisfying certain conditions. 夡
Supported in part by NSF Grant CMS-0324329.
∗ Corresponding author.
E-mail address: [email protected] (Z. Lin). 1 Current address: Department of Electrical and Computer En-
gineering, University of California, Santa Barbara, CA, USA. 0167-6911/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2004.11.011
In this paper, we study the convexity of invariant sets for a nonlinear system x˙ = Ax + B (F x, t),
(1)
where (·, t) belongs to a class of nonlinear functions. A block diagram for such a system is plotted in Fig. 1. In the absolute stability theory, the class of nonlinear functions is described with a conic sector, which can be used to represent a possibly uncertain and time-varying nonlinear component. The absolute stability of the Lur’e systems in Fig. 1 is a classical problem in control theory. It has been studied extensively in the nonlinear systems and control literature (see e.g. [1,8,12,14,15,17,18] and references therein), and is still attracting tremendous attentions (see [2–4,10,11,13,16] for a sample of recent literature).
730
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737 u
ψ (u, t )
v
5
−1
F( sI − A) B
v
4
v=ψ1 (u)
3
v=ψ(u)
2
Fig. 1. A system with a nonlinear component.
1
v=ψ2(u)
u
0 o
5 –1
v
4
v=k 1u
–2
3 –3
v=ψ(u)
2
–4 1
v=k 2u
–5 –8
u
0
–6
–4
–2
0
2
4
6
8
o –1
Fig. 3. The generalized sector.
–2 –3 –4 –5 –8
–6
–4
–2
0
2
4
6
8
Fig. 2. The classical linear sector.
Traditionally, the uncertain nonlinear function is assumed to be inside a sector bounded by two straight lines. The common tools for absolute stability under such a sector condition include circle criterion and Popov criterion, which give sufficient conditions for global stability over the sector. Since global absolute stability does not generally hold, another trend in the development of absolute stability theory is the study of absolute stability within a finite region (see e.g. [4,8,9,13,17]). In the case that global absolute stability does not hold, we need to restrict our attention to a finite region in the state space, where a sector that is narrower than the global sector can be used to bound the nonlinear function (u, t). Fig. 2 plots a sector between two straight lines v =k1 u and k2 u. This sector is a global bound for one of the nonlinear functions but is only a local bound for the other one, which can only be globally bounded by v = k1 u and 0. In the finite region, a guaranteed stability region can then be obtained by using some invariant level set of a quadratic or Lur’e type Lyapunov function (see e.g. [9,13]).
In an effort to give a better description for the uncertain and/or time-varying nonlinear component, we recently (in [6]) generalized the sector such that its boundary is defined by two odd symmetric nonlinear functions which are either concave or convex over [0, ∞]. For simplicity, these functions are said to be concave or convex. Fig. 3 plots a sector bounded by a convex function and a concave function. We first studied the absolutely contractively invariant (ACI) ellipsoids and developed a necessary and sufficient condition under which an ellipsoid is ACI. We then showed that the convex hull of a group of ACI ellipsoids is also ACI. With the results of [6], we are tempted to ask the question: is the convex hull of an arbitrary ACI set also ACI? This paper will give a positive answer to this question. It then follows that for a sector bounded by two concave/convex functions, the largest ACI set is convex. In fact, an ACI set is associated with a (positively) homogeneous Lyapunov function which is strictly decreasing along the trajectories of the system. Since the largest ACI set is convex, in searching for a homogeneous Lyapunov function to estimate the domain of attraction, we only need to restrict our attention to those whose level sets are convex. In the case that the Lyapunov function is homogeneous of degree greater than one, this means that the function is convex. In an attempt to further generalize the results, we would like to know if we can replace ACI with AI
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
(absolutely invariant) and still get a positive answer. However, we will use an example to show that the convex hull of an arbitrary invariant set need not be invariant even for a system with convex nonlinearity. This paper is organized as follows. Section 2 gives a review of the definitions of the generalized sector and absolute invariance. Section 3 presents some results on convex analysis. Section 4 analyzes the invariance of the convex hull of an invariant set and Section 5 contains some concluding remarks.
731
concave for u < 0. In Fig. 3, one of the boundary functions is convex and the other one is concave. Here is a simple fact about concave and convex functions. Fact 1. Let be a concave (convex) function. If we draw a straight line that is tangential to at (u0 , (u0 )), u0 0, then the straight line is above (below) (u) for all u > 0. 2.2. The generalized sector and absolute stability
Notation. For two integers k1 , k2 , k1 < k2 , we denote I [k1 , k2 ] = k1 , k1 + 1, . . . , k2 . For S ⊂ Rn , we use co{S}, int{S} and *S to denote the convex hull, the interior and the boundary of S, respectively. For S ⊂ Rn and a real number , S = {x : x ∈ S}.
2. A generalized sector and absolute invariance 2.1. Concave functions and convex functions We first give a formal definition of some functions that we will use to define the boundary of the generalized sector. Given a scalar function : R → R. Assume that (1) is continuous, piecewise differentiable, (0)= 0 and ddu > 0. u=0
(2) is odd symmetric, i.e., (−u) = −(u). A function satisfying the above assumption is said to be concave if it is concave for u > 0. That is, for any u1 , u2 > 0,
(u1 + (1 − )u2 ) (u1 ) + (1 − )(u2 ) ∀ ∈ [0, 1]. A function satisfying the above assumption is said to be convex if it is convex for u > 0. That is, for any u1 , u2 > 0,
(u1 + (1 − )u2 ) (u1 ) + (1 − )(u2 ) ∀ ∈ [0, 1]. These definitions are made for simplicity. It should be understood that by odd symmetry, a concave function is convex for u < 0 and a convex function is
Consider the system x˙ = Ax + B (F x, t),
(2)
where A ∈ Rn×n , B ∈ Rn×1 and F ∈ R1×n are given matrices. A set S ⊂ Rn is said to be invariant for (2) if every trajectory starting from it stays within it for t > 0. The domain of attraction of the origin for system (2) is an invariant set and a traditional way to estimate it is to use invariant sets that contain the origin in its interior. In stability analysis, we often use contractively invariant set to estimate the domain of attraction. Definition 1. Consider system (2). Let S ⊂ Rn be a compact set such that kS ⊂ int{S} for all k ∈ [0, 1). We say that S is contractively invariant if for every k ∈ (0, 1] and for every x ∈ *{kS}, Ax + B (F x, t) points strictly inward of kS, i.e., for every t > 0, there exists 0 > 0 such that x + (Ax + B (F x, t)) ∈ int{kS} ∀ ∈ (0, 0 ). In the above definition of contractive invariance, it is required that kS ⊂ int{S} for all k ∈ [0, 1). This guarantees that S is simply connected and contains the origin in its interior. Also, S equals the closure of int{S}. Actually, a contractively invariant set is associated with a positively homogeneous Lyapunov function which is strictly decreasing along the trajectories of system (2). Define V (x) := min{ 0 : x ∈ S},
(3)
then V (x) is positive-definite and positively homogeneous of degree one. Since kS ⊂ int{S} for all k ∈ [0, 1), for every x0 ∈ *S, we have kx 0 ∈ int{S} for all
732
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
k ∈ [0, 1). It is straightforward to verify that for any x, V (x) equals the unique 0 such that x ∈ *(S) and
*(S) = {x ∈ Rn : V (x) = }. We claim that V (x) is continuous. Because of homogeneity, we only need to verify the continuity for x ∈ *S, i.e., V (x) = 1. Given any > 0, we have (1 − )S ⊂ int{S},
S ⊂ int{(1 + )S}.
It follows that min{x1 − x2 : x1 ∈ *S, x2 ∈ (1 − )S} =: 1 > 0, min{x1 − x2 : x1 ∈ *S, x2 ∈ Rn \int{(1 + )S}} =: 2 > 0. Hence for every x0 ∈ *S, there is an open neighborhood of x0 , {x ∈ Rn : x − x0 < min{1 , 2 }}, from which V (x) ∈ (1 − , 1 + ). This verifies the continuity. It should be noted that V (x) can also be defined to be positively homogeneous of any degree q > 0 as follows: V (x) := min{q : x ∈ S, 0}. If q 1 and S is convex, then V (x) is convex. From Definition 1, it is clear that if S is contractively invariant, then kS is also contractively invariant for every k ∈ (0, 1). With a contractively invariant set S, a trajectory x(t) starting from any x0 ∈ *S will go to the interior of S and we have x(t) ∈ *{k(t)S} with k(t) strictly decreasing and converge to 0. Thus x(t) → 0. If we define a function V as in (3), then V (x(t)) is strictly decreasing and converges to 0. In Definition 1, the nonlinear function (u, t) in system (2) is assumed to be known. In practice, there always exists some degree of uncertainty about a nonlinear component. In view of this, we would like to study the invariance of a set for a class of nonlinear functions. On the other hand, some nonlinear functions (u, t) could be complicated and we would like to bound it with simpler functions 1 (u) and 2 (u). These problems arise from the same situations which motivated the problem formulation of absolute stability, where the nonlinear function (u, t) is bounded by two linear functions 1 (u) = u and 2 (u) = u. If the system is not globally absolutely stable over a linear sector [, ], we have to consider the stability on
a finite region of the state space, over which a pair of nonlinear functions 1 and 2 may better describe the property of the nonlinear component. For this reason, we introduced the generalized sector in [6]. Definition 2. Given functions 1 : R → R and 2 : R → R, each of which is concave or convex. A function (u, t), piecewise continuous in t and locally Lipschitz in u, is said to satisfy a generalized sector condition if
(u, t) ∈ co{1 (u), 2 (u)} ∀u, t ∈ R. We use co{1 , 2 } to denote the generalized sector, i.e., the set of functions that satisfy the above generalized sector condition. A set S ⊂ Rn is said to be absolutely invariant (AI) over the sector co{1 , 2 } if it is invariant for (2) under all the possible (u, t) satisfying the generalized sector condition. A set S ⊂ Rn is said to be absolutely contractively invariant (ACI) over the sector co{1 , 2 } if it is contractively invariant for (2) under all the possible (u, t) satisfying the generalized sector condition. We see that if S is ACI, then any trajectory starting from it will converge to the origin under all (u, t) satisfying the generalized sector condition. Hence S is an absolute stability region. Let us next state a simple fact. Fact 2. Given a compact convex set S and a class of functions i (u), i ∈ I [1, N ]. Suppose that for each i ∈ I [1, N ], S is (contractively) invariant for x˙ = Ax + B i (F x). Let (u, t) be a function such that (u, t) ∈ co{i (u), i ∈ I [1, N ]} for all u ∈ R and t ∈ R, then S is (contractively) invariant for x˙ = Ax + B (F x, t). This fact follows directly from the definition of invariant set and the convexity of S. By Fact 2, we see that the absolute (contractive) invariance of a convex set is equivalent to its (contractive) invariance under both 1 and 2 . Although we may use two arbitrary nonlinear functions 1 and 2 to define a generalized sector,
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
concave and convex functions appear to be simpler and easier to handle, and may lead to better properties. For example, it was shown in [6] that the invariance of an ellipsoid under a concave/convex nonlinearity is equivalent to some linear matrix inequalities. Moreover, the convex hull of a group of contractively invariant ellipsoids is also contractively invariant. With two general nonlinear functions 1 and 2 , it is hard to expect other properties beyond Fact 2. On the other hand, many commonly encountered nonlinearities are either concave or convex, for example, the tangent function, the saturation function and the dead zone function. In view of this, we will focus on the invariance of a set under a concave or convex function.
3. Some facts about convex sets For easy reference, we collect in this section some results from convex analysis (e.g. see [5]). Let S be a compact convex set. We say that x0 ∈ S is an extreme point of S if it cannot be represented as the convex combination of other points in S, i.e., x0 =
N i=1
i xi ,
N i=1
i = 1,
If *V (x) has only one element at x, then it is called the derivative of V at x. Fact 3. Consider x0 = 0. Let V (x0 ) = k0 . Then c ∈ *V (x0 ) if and only if cT x = k0 is a supporting hyperplane of k0 S at x0 . This fact can be seen as follows. For any k > 0, V (kx 0 ) = kk 0 . Suppose c ∈ *V (x0 ). Then cT (kx 0 − x0 )=(k −1)cT x0 (k −1)k0 . It follows that cT x0 =k0 . If x ∈ k0 S, then V (x) − V (x0 ) 0. Hence cT (x − x0 ) 0 for all x ∈ k0 S. Therefore, cT x = k0 is a supporting hyperplane of k0 S at x0 . On the other hand, suppose that cT x = k0 is a supporting hyperplane of k0 S at x0 . Any x ∈ Rn can be written as x = kx 1 for some x1 ∈ *{k0 S} and k > 0. Hence V (x) = kk 0 and cT x = kcT x1 kk 0 = V (x). Noting that cT x0 = k0 = V (x0 ), we have ∀x ∈ Rn .
cT y < 0
∀c ∈ *V (x0 ).
Hence, the set S is contractively invariant for
c T x0 = k 0 .
The intersection of a supporting hyperplane with the set S is called an exposed face of S. A point x0 is an extreme point of S if and only if it is an extreme point of any exposed face containing it. This implies that, if x0 ∈ S is not an extreme point, then it is not an extreme point of any exposed face. If S is a compact convex set containing the origin in its interior, a Minkowski function can be defined as V (x) := min{ 0 : x ∈ S}.
*V (x) = {c ∈ Rn : cT (x˜ − x) V (x) ˜ − V (x) n ∀x˜ ∈ R }.
Consider x0 ∈ *{k0 S}. Given a vector y ∈ Rn , it points strictly inward of k0 S from x0 if and only if
A hyperplane cT x = k0 is a supporting hyperplane of S at x0 ∈ *S if ∀x ∈ S,
The subderivative of V at x, denoted as *V (x), is
cT (x − x0 ) V (x) − V (x0 )
i 0, xi ∈ S
⇒ x1 = x2 = · · · = xN = x0 .
c T x k0
733
x˙ = Ax + B (F x) if and only if cT (Ax 0 + B (F x 0 )) < 0 ∀c ∈ *V (x0 ),
x0 ∈ S\{0},
and S is invariant if and only if cT (Ax 0 + B (F x 0 )) 0
(4)
This function is convex, positive definite and positively homogeneous of degree one.
∀c ∈ *V (x0 ),
x0 ∈ *S.
By Fact 3, we can replace c ∈ *V (x0 ) with c ∈ Rn such that cT x = k0 is a supporting hyperplane of k0 S at x0 .
734
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
4. Invariance of the convex hull of an invariant set
30
20
Consider the system 10
x˙ = Ax + B (F x).
We assume that is concave. If is convex, we can replace (u) with 0 u − 1 (u), where 0 is a constant and 1 (u) is concave, and obtain x˙ = (A + 0 BF )x − B 1 (F x).
(6)
It is easy to see that if we have a group of contractively invariant sets, then their union is also contractively invariant. Hence an invariant set need not be convex. What we are interested in is the convex hull of an invariant set. In [6], we showed that the convex hull of a group of contractively invariant ellipsoids is contractively invariant. In this paper, we would like to extend this result of [6] to a more general invariant set. It turns out that we have quite different conclusions for the convex hull of an invariant set and the convex hull of a contractively invariant set. We will discuss these two situations separately. 4.1. The general invariance We know that the domain of attraction is an invariant set. It is desirable that the domain of attraction is a convex set. We may have a reason to expect this for a class of systems where the nonlinearity is convex/concave, e.g., system (5). As we have shown in [7], for the special case where is the standard saturation function, if A ∈ R2×2 and its eigenvalues have positive real part, then the domain of attraction is convex and its boundary is the unique limit cycle. However, this result cannot even be extended to all the secondorder systems, especially when A has two eigenvalues of different signs. For example, we have a system x˙ = Ax + B sat(F x), where sat(u) = sign(u) min{1, |u|} and 0 0 1 , B= , F = [ −2 A= 5 1 0
−1 ] .
The domain of attraction is not bounded, as shown in Fig. 4, where its boundary is plotted with solid lines. The boundary of the domain of attraction is generated
x2
(5) 0
(5,0)
(-5,0)
–10
–20
–30 –40
–30
–20
–10
0 x1
10
20
30
40
Fig. 4. A nonconvex domain of attraction.
by simulation. It is composed of four trajectories, two of them go from infinity toward (5, 0) and the other two go from infinity toward (−5, 0). It is obvious that this domain of attraction is not convex. Then what about its convex hull? Is it invariant? The convex hull of the domain of attraction is a strip, whose boundary is plotted as dash–dotted lines in Fig. 4. We chose an arbitrary initial state (marked with “∗”) inside the strip but outside the domain of attraction, the trajectory goes out of this strip and diverges. This gives us a counter example for the invariance of the convex hull of an invariant set. However, for the convex hull of a contractively invariant set, we have a quite different conclusion, as will be shown next. 4.2. The contractive invariance Without loss of generality, assume that d/du|u=0 = 1. Let S ⊂ Rn be a compact set such that kS ∈ int{S} for all k ∈ [0, 1). Suppose that S is contractively invariant for (5). Then by Definition 1, kS is contractively invariant for all k ∈ (0, 1]. Inside kS, as k approaches 0, the system approximates the linear system x˙ = Ax + BF x.
(7)
By taking the limit, it is easy to see that kS is invariant for the linear system (7), and hence S is also invariant for the linear system.
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
If (u) = u for an interval [0, u0 ], u0 > 0, then for sufficiently small k, the system is exactly linear inside kS. In this case, the contractive invariance of a set S implies its contractive invariance for the linear system (7). The following is the main result of the paper. Theorem 1.
Proof. With co{S}, we can define V (x) = min{ 0 : x ∈ co{S}}. (a) We will show that for all x on the boundary of co{S}, x˙ points inward of co{S}, i.e., cT (Ax + B (F x)) 0
∀c ∈ *V (x), x ∈ *co{S}.
The invariance of co{kS} for k ∈ (0, 1] follows similarly from the contractive invariance of kS. Here, we only need to consider x ∈ *co{S}\*S. Since for those x ∈ *S, x˙ points inward of S implies that it points inward of co{S}. Now, consider x0 ∈ *co{S}\*S. Let cT x = 1 be a supporting hyperplane of co{S} at x0 . We need to prove that cT (Ax 0 + B (F x 0 )) 0.
(8)
Since x0 ∈ / *S, it is not an extreme point of co{S}. Hence, this supporting hyperplane must also touch some points on *S. In other words, cT x = 1 is also a supporting hyperplane at some points in *S. Moreover, x0 can be expressed as a convex combination of x1 , x2 , . . . , xN ∈ *S and cx j = 1 for all j ∈ I [1, N]. This means that there exist j > 0, j ∈ I [1, N], such that x0 =
j =1
j xj ,
By taking k → 0, and noting that d/duu=0 = 1, we obtain cT (Ax j + BF x j ) 0
∀j ∈ I [1, N ].
N j =1
j = 1.
Since kS is contractively invariant for all k ∈ (0, 1], we have cT (Akx j + B (kF x j )) < 0, k ∈ (0, 1].
j ∈ I [1, N ], (9)
(10)
Hence, for all x ∈ co{x1 , x2 , . . . , xN }, cT (Ax + BF x) 0.
(a) If S is contractively invariant for (5), then co{kS} is invariant for all k ∈ (0, 1]. (b) If S is contractively invariant for both (5) and (7), then its convex hull is contractively invariant for (5).
N
735
(11)
Assume that F x 0 0. (If F x 0 0, then by the odd symmetry of (·), we can use similar argument to prove (8).) First, we suppose that F x j 0 for all j ∈ I [1, N ]. In this case, F x 0 for all x ∈ co{x1 , x2 , . . . , xN }. If cT B 0, then by the assumption that F x 0 0 and by the concavity of the function (·), we have (F x 0 ) F x 0 , and hence, cT (Ax 0 + B (F x 0 )) cT (Ax 0 + BF x 0 ) 0.
(12)
If cT B 0, then also by the concavity of (·), cT Ax + cT B (F x) is a convex function for x ∈ co{x1 , x2 , . . . , xN }. Hence we also have (8) by (9). If F x j 0 does not hold for all j ∈ I [1, N ], then we can get an intersection of the set co{x1 , x2 , . . . , xN } with the half space F x 0. This intersection is also a polygon and can be denoted as co{y1 , y2 , . . . , yN1 }. Since F x 0 0, we have x0 ∈ co{y1 , y2 , . . . , yN1 }. Some yj ’s belong to {x1 , x2 , . . . , xN }, others are not. For those yj ∈ / {xi : i ∈ I [1, N ]}, we must have F y j = 0 and yj ∈ co{xi : i ∈ I [1, N ]}. It follows from (11) that cT (Ay j + BF y j ) 0. Since (0) = 0, for those yj ’s such that F y j = 0, we have cT (Ay j + B (F y j )) = cT (Ay j + BF y j ) 0. In summary, we have cT (Ay j + B (F y j )) 0
∀j ∈ I [1, N1 ].
(13)
Because of this, we can work on the set co{y1 , y2 , . . . , yN1 } instead of co{x1 , x2 , . . . , xN }. Since F y j 0 for all j ∈ I [1, N1 ], same arguments can be used to prove (8) by using (13) instead of (9). (b) The procedure of the proof is very similar to the proof for (a). The only difference is to replace “ ” in the inequalities with “<”. This is guaranteed by the additional condition that S is contractively invariant for the linear system. Because of this, instead of (10),
736
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737 15
10 8
10
6 4
5
x2
x2
2 0
0
–2 –5
–4 –6
–10
–8 –10 –8
–6
–4
–2
0 x1
2
4
6
8
10 8 6 4
x2
2 0 –2 –4 –6 –8
–6
–4
–2
0 x1
2
4
6
8
Fig. 6. Convex hull of the two contractively invariant sets.
we have cT (Ax j + BF x j ) < 0
–10
–5
0 x1
5
10
15
Fig. 7. Convex hull and the domain of attraction.
Fig. 5. Union of two contractively invariant sets.
–10 –8
–15 –15
∀j ∈ I [1, N].
This leads to “<” for all the remaining inequalities. We note that the statement of (a) is stronger than the simple invariance of S. From Theorem 1, we can conclude that the largest contractively invariant set for system (5) is convex. Recall that a contractively invariant set is associated with a positively homogeneous Lyapunov function. By Theorem 1, we can restrict
our attention to convex Lyapunov function candidates. Furthermore, with several contractively invariant level sets of different Lyapunov functions, we can obtain a larger contractively invariant set as their convex hull. One may be tempted to extend Theorem 1 to systems with more than one nonlinear components, i.e., to the case where (·) is a vector function and B has more than one column. However, it is difficult to see such a possibility from the proof of Theorem 1, which relies on the fact that for a fixed c, the function cT (Ax + B (F x)) is either convex or concave in x. For the case that (·) is a vector function, even if all of the components of (·) are concave, their linear combination cT B (·) could be neither concave nor convex. Example 1. Consider the same system as the one in Section 4.1. Using LMI-based technique, we detected two contractively invariant sets, each of which is the convex hull of two ellipsoids. Fig. 5 plots the boundaries of these two invariant sets along with line segments marking the direction of the vector x. ˙ The line segments clearly show that the union of these sets is invariant. Fig. 6 plots the convex hull of these two contractively invariant sets along with line segments marking the direction of the vector x. ˙ Also plotted in the figure in dashed curves are the boundaries of the two invariant sets as in Fig. 5. It is clearly seen that the convex hull is also invariant. As a comparison, we plotted the convex hull against the boundary of the domain of attraction in Fig. 7.
T. Hu, Z. Lin / Systems & Control Letters 54 (2005) 729 – 737
5. Conclusions This paper studies the invariance of the convex hull of an invariant set for a class of nonlinear systems satisfying a generalized sector condition. We focused on the invariance of a set for a system with concave nonlinearities. We used an example to show that the convex hull of an invariant set need not be invariant but the convex hull of a contractively invariant set is invariant. References [1] M.A. Aizerman, F.R. Gantmacher, Absolute Stability of Regulator Systems, Holden-Day, San Francisco, CA, 1964. [2] M. Arcak, A. Teel, Input-to-state stability for a class of Lurie systems, Automatica 38 (2002) 1945–1949. [3] K. Gu, Absolute stability of systems under block diagonal memoryless uncertainties, Automatica 31 (4) (1995) 581–584. [4] W.M. Haddad, V. Kapila, V.S. Chellabonia, Guaranteed domain of attraction for multivariable Lur’e systems via open Lyapunov surfaces, Proceedings of 13th IFAC World Congress, San Francisco, 1996, pp. 13–18. [5] J.-B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer, Berlin, 1993. [6] T. Hu, B. Huang, Z. Lin, Absolute stability with a generalized sector condition, AC-48 (4) (2004) 535–548. [7] T. Hu, Z. Lin, L. Qiu, Stabilization of exponentially unstable linear systems with saturating actuators, IEEE Trans. Automat. Control 46 (6) (2001) 973–979.
737
[8] H. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, 1996. [9] T. Kiyama, T. Iwasaki, On the use of multi-loop circle for saturating control synthesis, Systems Control Lett. 41 (2000) 105–114. [10] D. Liu, A. Molchanov, Criteria for robust absolute stability of time-varying nonlinear continuous-time systems, Automatica 38 (2002) 627–637. [11] A. Megretski, A. Rantzer, System analysis via integral quadratic constraints, IEEE Trans. Automat. Control 42 (6) (1997) 819–830. [12] K.S. Narendra, J. Taylor, Frequency Domain Methods for Absolute Stability, Academic Press, New York, 1973. [13] T. Pare, H. Hindi, J. How, D. Benjerdpongchai, Local control design for systems with saturating actuators using Popov criteria, Proceedings of the American Control Conferences, San Diego, 1999, pp. 3211–3215. [14] J.-J.E. Slotine, W. Li, Applied Nonlinear Control, PrenticeHall, Englewood Cliffs, NJ, 1991. [15] M.M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1993. [16] T. Wada, M. Ikeda, Y. Ohta, D.D. Siljak, Parametric absolute stability of Lur’e systems, IEEE Trans. Automat. Control AC43 (11) (1998) 1649–1653. [17] J.L. Willems, The computation of finite stability regions by means of open Lyapunov surfaces, Int. J. Control 10 (5) (1969) 537–544. [18] V.A. Yakubovich, Frequency conditions for the absolute stability of control systems with several nonlinear and linear nonstationary blocks, Automat. Remote Control 28 (1967) 857–880.