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Generalized Hamiltonian realization of time-invariant nonlinear systems
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Automatica 39 (2003) 1437 – 1443 www.elsevier.com/locate/automatica
Brief Paper
Generalized Hamiltonian realization of time-invariant nonlinear systems Yuzhen Wanga; c;∗ , Chunwen Lia , Daizhan Chengb b Institute
a Department of Automation, Tsinghua University, Beijing 100084, China of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China c School of Control Science and Engineering, Shandong University, Jinan 250061, China
Received 14 December 2001; received in revised form 30 January 2003; accepted 20 March 2003
Abstract A key step in applying the Hamiltonian function method is to express the system under consideration into a generalized Hamiltonian system with dissipation, which yields the so-called generalized Hamiltonian realization (GHR). In this paper, we investigate the problem of GHR. Several new methods and the corresponding su7cient conditions are presented. A major result is that if the Jacobian matrix of a time-invariant nonlinear system is nonsingular, the system has a GHR whose structure matrix and Hamiltonian function are given in simple forms. Then the orthogonal decomposition method and a su7cient condition for the feedback dissipative realization are proposed. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Generalized Hamiltonian realization; Feedback dissipative realization; Jacobian matrix; Orthogonal decomposition method
1. Introduction In recent years, port-controlled Hamiltonian (PCH) systems, proposed by Maschke and van der Schaft (1992) and van der Schaft and Maschke (1995), have been investigated in detail by van der Schaft (1999), Maschke, Ortega, and van der Schaft (2000), Escobar, van der Schaft, and Ortega (1999), Ortega, Lor>?a, Nicklasson, and Sira-Ram>?rez (1998), Ortega, van der Schaft, Maschke, and Escobar (2002), Fujimoto and Sugie (2001), and Cheng, Spurgeon, and Xiang (2000). Indeed, the Hamiltonian function in the PCH system is considered as the total energy and can play the role of Lyapunov function for the system. Because of this, the Hamiltonian function method (the energy-based
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Hassan Khalil. Supported by Project 973 of China (G1998020307, G1998020308) and China Postdoctoral Science Foundation. ∗ Corresponding author. Department of Automation, Tsinghua University, Beijing 100084, China. E-mail addresses: [email protected] (Yuzhen Wang), [email protected] (Chunwen Li), [email protected] (Daizhan Cheng).
0005-1098/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0005-1098(03)00132-8
Lyapunov function method) has been developed (Maschke et al., 2000) and applied to many practical control problems (Cheng, Xi, Hong, & Qin, 1999; Shen, Ortega, Lu, Mei, & Tamura, (2000); Wang, Chen, & Hong, 2001; Xi, Cheng, Lu, & Mei, 2002). It has been shown in Cheng et al. (1999), Wang et al. (2001) and Xi et al. (2002) that the Hamiltonian function method has some advantages. The key point in applying the Hamiltonian function method is to express the system concerned into a Hamiltonian system with dissipation, which is called the dissipative Hamiltonian realization. In general, to complete the dissipative Hamiltonian realization, we Hrst express the system into a generalized Hamiltonian system, i.e., obtain the generalized Hamiltonian realization (GHR), and then eliminate the non-dissipative part of the GHR by a state feedback to get a Hamiltonian system with dissipation. We recall some related concepts Hrst: Denition 1 (Cheng et al., 2000). (1) A dynamic system x˙ = f(x);
x ∈ Rn
(1)
is said to have a generalized Hamiltonian realization (GHR) if there exists a suitable coordinate chart and a Hamiltonian function H such that (1) can be expressed as x˙ = T (x)∇H;
(2)
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Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
where T (x) is an n × n matrix called the structure matrix and ∇H = @H=@x. If the structure matrix can be expressed as T (x) = J (x) − R(x), with skew-symmetric J (x) and symmetric positive semi-deHnite R(x), then system (2) is called a dissipative Hamiltonian realization. Furthermore, if R(x) ¿ 0, (2) is called a strict dissipative Hamiltonian realization. (2) A controlled dynamic system x˙ = f(x) + g(x)u
(3)
is said to have a state feedback Hamiltonian realization if there exists a suitable state feedback u = (x) + v such that the closed-loop system can be expressed as x˙ = T (x)∇H + g(x)v:
(4)
If T (x) can be expressed as T (x) = J (x) − R(x), J (x) is skew-symmetric and R(x) ¿ 0(¿ 0), then (4) is called a feedback (strict) dissipative Hamiltonian realization. The GHR problem has been studied in some recent works (Hebertt Sira-Ram>?rez, 1998; Cheng et al., 2000). There are, however, no eOective methods to handle it yet. In this paper, we investigate the GHR problem of nonlinear systems. We propose several new methods to handle the problem and give some new su7cient conditions for the realization.
2. Generalized Hamiltonian realization In this section, we investigate the GHR problem of system (1) and propose several new methods and su7cient conditions for the GHR. Let Jf denote the Jacobian matrix @f=@x. In system (1), set T @f @f1 @fn Ai = ; = ;:::; @xi @xi @xi ai =
@ ; @xi
i = 1; 2; : : : ; n:
Construct two equations as follows: A2 −A1 A3 −A1 .. . .. . X1 (x) An −A1 X2 (x) A3 −A2 =0 . . . .. .. . . Xn (x) An −A2 . . .. .. An −An−1
(5)
and
a2
a3 .. . an
−a1 −a1 ..
. −a1
a3
−a2
.. .
..
.
an
−a2 .. .
.. . an
X1 (x) X2 (x) ⊗ In . . . Xn (x)
−an−1
=0;
(7)
where Xi (x) (i=1; 2; : : : ; n) are n-dimensional column vector Helds, ⊗ is the Kronecker product and In is the n×n identity matrix. Here, for an arbitrary scalar function h(x) we deHne @h(x) ; i ∈ {1; 2; : : : ; n}: (8) ai · h(x) = @xi Theorem 1. If Eqs. (6) and (7) have a solution (X1T (x); : : : ; XnT (x))T such that matrix (X1 (x) X2 (x) · · · Xn (x))n×n is nonsingular, then there exists a Hamiltonian function H (x) such that system (1) has a GHR as follows x˙ = T (x)∇H;
(9)
where T (x) = (X1 (x) X2 (x) · · · Xn (x))
−T
.
Proof. Using Poincare lemma, a straightforward computation shows the conclusion. Remark 1. The Hamiltonian function H (x) in Theorem 1 can be given the form x1 h1 (x1 ; x2 ; : : : ; x n ) d x1 H (x) = x1(0)
+
x2(0)
+
x2
xn
x(0) n
h2 (x1(0) ; x2 ; : : : ; x n ) d x2 + · · ·
(10)
hn (x1(0) ; : : : ; x(0) n−1 ; x n ) d x n ;
T where hi =XiT f(x); i=1; 2; : : : ; n, and x(0) := (x1(0) ; : : : ; x(0) n ) n ∈ R is an arbitrary initial point.
(6)
The following is a main result in this section. Proposition 1. If the Jacobian matrix Jf is invertible, then n 1 2 @H ; H= fi (11) x˙ = Jf−T @x 2 i=1
is a GHR of system (1).
Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
Proof. Let Xi (x) = @f=@xi = (@f1 =@xi ; : : : ; @fn =@xi )T ; i = 1; 2; : : : ; n. Because T T @f @f @f @f Ai Xj = = = A j Xi ; @xi @xj @xj @xi i; j = 1; 2; : : : ; n, (X1T (x); : : : ; XnT (x))T is a solution to (6). On the other hand, @Xj @Xi @2 f @2 f = = = ; @xj @xi @xj @xj @xi xi Therefore, which implies (aj In )Xi = (ai In )Xj . (X1T (x); : : : ; XnT (x))T is also a solution to (7). Noting that (X1 (x) X2 (x) · · · Xn (x)) = Jf is nonsingular, in view of Theorem 1 we have x˙ = Jf−T ∇H , where H (x) can be detern mined by (10). Here hi (x) = XiT f(x) = j=1 (@fj =@xi )fj . From (10) we get n xi (0) H (x) = hi (x1(0) ; : : : ; xi−1 ; xi ; xi+1 ; : : : ; x n ) d xi xi(0)
i=1
n
i=1 j=1
n
j=1 i=1
− fj2 (x1(0) ; : : : ; xi(0) ; xi+1 ; : : : ; x n ) n
1 2 = fj (x1 ; x2 ; : : : ; x n ) 2 j=1
x˙ = Jf−T ∇H;
n
n
i=1
i=1
1 2 1 2 (0) fi (x) − fi (x ): 2 2
Since adding a constant to H (x) does not change the form of Hamiltonian systems, we can simply choose H (x) = n 1 2 i=1 fi (x). 2 Remark 2. We can prove that (1) Jf−T is consistent with the changing law of structure matrices under coordinate transformations. (2) If Jf is locally invertible at the equilibrium, (11) locally holds. Lemma 1. Assume J (x) is an n×n skew-symmetric matrix and R(x) is an n×n positive (semi-)de>nite matrix. If J (x)− R(x) is nonsingular, then there exist a skew-symmetric matrix J1 (x) and a positive (semi-)de>nite matrix R1 (x) such that (J (x) − R(x))−1 = J1 (x) − R1 (x):
(13)
n
H (x) =
1 2 fi (x): 2
(14)
i=1
On the other hand, JfT = 12 [JfT − Jf ] − {− 12 [JfT + Jf ]} and {− 12 [JfT + Jf ]} is positive deHnite. From Lemma 1, there is a skew-symmetric matrix J (x) and a positive deHnite matrix R(x) such that Jf−T = J (x) − R(x), from which and (14) the theorem holds.
− fj2 (x1(0) ; x2(0) ; : : : ; x(0) n ) =
Theorem 2. If JfT + Jf is negative de>nite, then system (1) has a strict dissipative Hamiltonian realization as follows:
Proof. Since Jf = 12 [Jf − JfT ] + 12 [Jf + JfT ], the Hrst part of the right-hand side is skew-symmetric and the second part is negative deHnite, whereby Jf is invertible (Remark 3). It follows from Proposition 1 that system (1) has a realization as
1 2 (0) (0) fj (x1 ; : : : ; xi−1 ; xi ; : : : ; x n ) 2 n
Remark 3. In Lemma 1, we can prove that if R(x) is positive (or negative) deHnite, then J (x) − R(x) is invertible.
where J (x) is some n × n skew-symmetric matrix, R(x) is some n × n positive de>nite matrix and H (x) = n 1 2 i=1 fi (x). 2
− fj2 (x1(0) ; : : : ; xi(0) ; xi+1 ; : : : ; x n ) =
Proof. For arbitrary 0 = x ∈ Rn , let y := (J (x) − R(x))−1 x = 0. Because J (x) is skew-symmetric and R(x) is positive (semi-)deHnite, yT (J (x) − R(x))y = −yT R(x)y ¡ 0(6 0). Therefore, [(J (x) − R(x))−1 x]T (J (x) − R(x))(J (x) − R(x))−1 x ¡ 0(6 0), which is equivalent to xT (J (x) − R(x))−T x ¡ 0(6 0). From this, we get xT [(J (x) − R(x))−1 + (J (x) − R(x))−T ]x ¡ 0(6 0), which implies that (J (x) − R(x))−1 + (J (x) − R(x))−T is negative (semi-)deHnite. On the other hand, (J (x) − R(x))−1 = 1 −1 − (J (x) − R(x))−T ] + 12 [(J (x) − 2 [(J (x) − R(x)) −1 R(x)) + (J (x) − R(x))−T ], and the Hrst part of the right-hand side is skew-symmetric and the second part is negative (semi-)deHnite. Thus, Lemma 1 holds.
x˙ = (J (x) − R(x))∇H;
1 2 (0) (0) fj (x1 ; : : : ; xi−1 = ; xi ; : : : ; x n ) 2 n
1439
(12)
In the following, we consider the case when Jf is singular. First, we give a lemma. Lemma 2 (Langson and Alleyne, 1997). If a scalar function h(x) with h(0) = 0 (x ∈ Rn ) has continuous nth-order partial derivatives, then h(x) can be expressed as h(x) = a1 (x)x1 + · · · + an (x)x n ;
(15)
where ai (x); i = 1; 2; : : : ; n, are scalar functions. Theorem 3. Consider system (1) with f(0) = 0. If the Jacobian matrix Jf is singular and has a nonsingular main diagonal block, then there exists an n × n matrix M (x) and a vector >eld g(x) with nonsingular Jacobian matrix such that f(x) = M (x)g(x). Furthermore, system (1)
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Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
has a Hamiltonian realization as follows x˙ =
where
M (x)Jg−T ∇H:
(16)
1
T1 (x) = A (x)B (x)
Now, we give a proof for Theorem 3, which itself provides a useful algorithm to Hnd M (x) and g(x). Proof. Without loss of generality, we assume that Rank Jf = k ¡ n and @(f1 ; : : : ; fk )=@(x1 ; : : : ; xk ) is a nonsingular main diagonal block. Let f1 = (f1 ; : : : ; fk )T , f2 =(fk+1 ; : : : ; fn )T , x1 =(x1 ; : : : ; xk )T ; x2 =(xk+1 ; : : : ; x n )T . Since f(0) = 0, f2 (0) = 0. From Lemma 2, there exists an (n − k) × n matrix A(x) = (A1 (x); A2 (x)) such that f2 (x) = A(x)x = A1 (x)x1 + A2 (x)x2 :
(17)
On the other hand, f1 = f1 (x1 ; x2 ); f1 (0; 0) = 0, and @f1 =@x1 is nonsingular. It follows from the implicit function theorem that there exists a function such that x1 = (f1 ; x2 ); (0; 0) = 0. From Lemma 2 again, there is a k × n matrix B(x) = (B1 (x); B2 (x)) such that x1 can be expressed as 1 f 1 = B1 (x)f1 + B2 (x)x2 : x = B(x) (18) x2 Substituting (18) into (17), we get f2 = A1 (x)B1 (x)f1 + (A1 (x)B2 (x) + A2 (x))x2 . Therefore, 1 1 Ik 0 f f = : × A1 (x)B1 (x) A1 (x)B2 (x) + A2 (x) f2 x2 Let M (x) =
Ik
0
A1 (x)B1 (x)
A1 (x)B2 (x) + A2 (x)
is nonsingular. Thus, the Hrst part of the theorem holds. Furthermore, from Proposition 1 we have
2
−T
2
−(A (x)B (x) + A (x))
@f1 @x2
T
@f1 @x1
−T ;
T2 (x) = A1 (x)B2 (x) + A2 (x) and H (x) =
k n 1 2 1 2 fi + xj : 2 2 i=1
(20)
j=k+1
3. Feedback dissipative Hamiltonian realization In this section, we investigate the feedback dissipative Hamiltonian realization. First, we propose a new approach to the GHR called the orthogonal decomposition method. Then we use the method to study the feedback dissipative realization. Let us introduce the following concept. A function V (x) is called a regular positive deHnite function if V (x) ¿ 0 (x = 0), V (0)=0, x=0 =0 and @V=@x|x=0 = 0. For example, @V=@x| n 2 x H (x) = 12 i=1 i is a regular positive deHnite function on Rn . Consider system x˙ = f(x);
f(0) = 0;
x ∈ Rn :
(21)
Proposition 2. If there exists a function H (x) (∇H = 0) such that Lf H = 0, ∀x ∈ Rn , then system (21) can be expressed as x˙ = J (x)∇H , where 1 J (x) = [f∇H T − ∇HfT ] (22) ∇H 2
Proof. J (x)∇H =
1 [f∇H T − ∇HfT ]∇H ∇H 2
=
1 1 f∇H T ∇H − ∇HfT ∇H ∇H 2 ∇H 2
=
1 1 f ∇H 2 − ∇HLf H 2 ∇H ∇H 2
= f; which means that the proposition holds.
x˙ = M (x)
T1 (x)
1
@f1 @x1
is an n × n skew-symmetric matrix.
and g(x) = ((f1 )T ; (x2 )T )T , then f(x) = M (x)g(x) and T @f1 0 T @x1 @g = T @x 1 @f In−k @x2
−T @g @x −T @f1 ×∇H = @x1
1
0 T2 (x)
∇H;
(19)
Remark 4. If H (x) in Proposition 2 is replaced by a regular positive deHnite function, then (22) should be modiHed as 1 [f∇H T − ∇HfT ] x = 0; 2 ∇H J (x) = (23) 0 x = 0:
Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
Proposition 2 indicates that if the directional derivative of H (x) along vector Held f(x) is zero everywhere, i.e., f(x) ⊥ ∇H holds for all x, then the system can be expressed as a traditional Hamiltonian system. Motivated by this, we propose a new approach to the GHR of system (21): the orthogonal decomposition method. Choose a regular positive deHnite function H (x) and write out the equi-value surfaces of H (x). At arbitrary point x = 0, we decompose f(x) along the gradient direction ∇H and the tangential direction of the equi-value surface as follows: f(x) = fgd (x) + ftd (x);
(24)
fgd (x) =
f; ∇H ∇H; ∇H 2
ftd (x) = f(x) − fgd (x)
(25)
and ·; · denotes the inner product. Because Lftd H = ftd ; ∇H = f − fgd ; ∇H = f; ∇H − fgd ; ∇H = f; ∇H − (f; ∇H = ∇H 2 )∇H; ∇H = 0, ftd ⊥ ∇H . From Proposition 2 we know
x˙ = T (x)∇H + g(x)u; where
T (x) =
(30)
J (x) + S(x);
x = 0;
0;
x=0
and J (x), S(x) are given by (26) and (27). Decompose S(x) as follows:
1 T [ftd ∇H T − ∇Hftd ]: ∇H 2
(26)
f; ∇H I; ∇H 2
(27)
where I is the n × n identity matrix, then when x = 0, x˙ = fgd (x) + ftd (x) = (J (x) + S(x))∇H . Thus, system (21) has a GHR as follows: J (x) + S(x); x = 0; x˙ = T (x)∇H; T (x) = (28) 0; x = 0: We call (28) the orthogonal decomposition Hamiltonian realization. Because there are always regular positive deHnite functions, we have the following result. Theorem 4. An arbitrary system x=f(x), ˙ f(0)=0, always has the orthogonal decomposition Hamiltonian realization (28). From Theorem 4, we readily get the following corollary. Corollary 1. If system x˙ = f(x) with f(0) = 0 is asymptotically stable, then it has a strict dissipative Hamiltonian realization with its Hamiltonian function positive de>nite. In the following, we investigate the feedback dissipative Hamiltonian realization. Consider the single-input system x˙ = f(x) + g(x)u;
f(0) = 0;
x ∈ Rn :
(29)
(32)
and substitute it into (30), then we get x˙ = (J (x) − R1 (x) + R2 (x))∇H 1 g[Lg Hv − ∇H T R2 (x)∇H ] Lg H
= (J (x) − R1 (x))∇H + R2 (x)∇H −
Let
(31)
where R1 (x) ¿ 0 and R2 (x) is symmetric. When x = 0, set 1 [Lg Hv − (∇H )T R2 (x)∇H ] u= Lg H
+
ftd (x) = J (x)∇H;
S(x) =
Assume there exists a regular positive deHnite function H (x) such that Lg H = 0, for all x = 0. With this H (x), from the orthogonal decomposition realization, system (29) can be expressed as
S(x) = −R1 (x) + R2 (x);
where
J (x) =
1441
1 g∇H T R2 (x)∇H + gv Lg H
= (J (x) − R1 (x))∇H +
1 [R2 (x)∇HLg H Lg H
− g∇H T R2 (x)∇H ] + gv = (J (x) − R1 (x))∇H +
1 [R2 (x)∇HgT Lg H
− g∇H T R2 (x)]∇H + gv = (J (x) + J˜ (x) − R1 (x))∇H + gv; where J˜ (x) = (1=Lg H )[R2 (x)∇HgT − g(∇H )T R2 (x)] is skew-symmetric. When x = 0, we set u = v. Thus, system (30) can be expressed as x˙ = M (x)∇H + gv; where M (x) =
(33)
J (x) + J˜ (x) − R1 (x);
x = 0;
0;
x = 0:
Since J (x) + J˜ (x) is skew-symmetric and R1 (x) ¿ 0, (33) is a strict dissipative Hamiltonian realization. From the above, we get the following result. Proposition 3. If there exists a regular positive de>nite function H (x) such that Lg H (x) = 0 (x = 0), then system (29) has a feedback strict dissipative Hamiltonian realization with H (x) as its Hamiltonian function.
1442
Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
Example 1. Find a feedback dissipative Hamiltonian realization of the following system: x 1 + x2 x1 + x12 x2 3 x˙ = −x1 + x2 + x2 − x1 u x3 x3 := f(x) + g(x)u:
(34)
First, using the orthogonal decomposition method, we express (34) as a generalized Hamiltonian system. We choose the regular positive deHnite function as follows: H (x) = 1 2 2 2 2 (x1 + x2 + x3 ). When x = 0, from (25) we get fgd =
f; ∇H ∇H = (x1 ; x2 ; x3 )T ; ∇H 2
and when x = 0, we let u = v. Substituting the control u into (35), we have x˙ = M (x)∇H + g(x)v; where M (x) =
J (x) + J˜ (x)
= x2 x3
−x1 x3 ; 0
S(x) = (f; ∇H = ∇H 2 )I = Diag{1; 1; 1}. So system (34) can be expressed as
x = 0; x = 0:
(35)
Second, we design a control law to get the desired dissipative realization. Because Lg H = x12 + x22 + x32 = 0 (x = 0), we know from Proposition 3 that system (34) has a feedback strict dissipative realization with H (x) as its Hamiltonian function. Decompose S(x) as follows: S(x)=−R1 +R2 , where R1 =Diag{1; 1; 1} ¿ 0, R2 =Diag{2; 2; 2} ¿ 0. When x = 0, according to (32) we set u= =
1 [Lg Hv − ∇H T R2 (x)∇H ] Lg H 2 1 (x1 + x22 + x32 )v x12 + x22 + x32 2 0 0 x1 −(x1 ; x2 ; x3 ) 0 2 0 x2 0 0 2 x3
= −2 + v;
x = 0;
0; x = 0; 1 [R2 (x)∇HgT − g∇H T R2 (x)] J˜ (x) = Lg H 0 x12 + x22 −2 −x2 − x2 = 2 0 2 x1 + x22 + x32 1 −x2 x3 x1 x3
Therefore,
x˙ = T (x)∇H + g(x)u; J (x) + S(x); T (x) = 0;
J (x) + J˜ (x) − R1 (x);
x2 x 3
−x1 x3 : 0
Since
ftd = f − fgd = (x12 x2 ; −x13 ; 0)T :
1 T J (x) = [ftd ∇H T − ∇Hftd ] ∇H 2 0 x12 + x22 2 x1 −x2 − x2 = 2 0 2 x1 + x22 + x32 1 −x2 x3 x1 x 3
(36)
0
2 x12 − 2 −x − x2 2 2 2 2 x1 + x2 + x3 1 −x2 x3
x12 + x22 0 x1 x3
x 2 x3
−x1 x3 0
is skew-symmetric and R1 ¿ 0, (36) is a strict dissipative Hamiltonian realization. 4. Conclusion We have investigated the Hamiltonian realization problem of time-invariant nonlinear systems, proposed several operable and comparatively systematic methods and given some su7cient conditions for the realization. The main results are as follows. (1) If the Jacobian matrix of a system is nonsingular, the system has a GHR whose structure matrix and Hamiltonian function are given in very simple forms. (2) If the Jacobian matrix of the system is singular and has a nonsingular main diagonal block, then the system also has a GHR whose structure matrix and Hamiltonian function can be easily gotten. (3) A new method of GHR called the orthogonal decomposition method has been obtained. It has been proved that an arbitrary nonlinear system has an orthogonal decomposition Hamiltonian realization, and the corresponding realization form can be given by a set of formulas. References Cheng, D., Spurgeon, S., & Xiang, J. (2000). On the development of generalized Hamiltonian realizations. Proceedings of the 39th IEEE conference on decision and control, Vol. 5, Sydney, Australia (pp. 5125 –5130). Cheng, D., Xi, Z., Hong, Y., & Qin, H. (1999). Energy-based stabilization of forced Hamiltonian systems with its application to power systems.
Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443 Proceedings of the 14th IFAC world congress, Vol. O, Beijing, People’s Republic of China (pp. 297–303). Escobar, G., van der Schaft, A. J., & Ortega, R. (1999). A Hamiltonian viewpoint in the modelling of switching power converters. Automatica, 35(3), 445–452. Fujimoto, K., & Sugie, T. (2001). Canonical transformations and stabilization of generalized Hamiltonian systems. Systems and Control Letters, 42(3), 217–227. Sira-Ram>?rez, H. (1998). A general canonical form of feedback passivity of nonlinear systems. International Journal of Control, 71(5), 891–905. Langson, W., & Alleyne, A. (1997). InHnite horizon optimal control of a class of nonlinear systems. Proceedings of IEEE American control conference, Vol. 5, Albuquerque, New Mexico (pp. 3017–3022). Maschke, B. M., Ortega, R., & van der Schaft, A. J. (2000). Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Transactions on Automatic Control, 45(8), 1498–1502. Maschke, B. M., & van der Schaft, A. J. (1992). Port-controlled Hamiltonian systems: Modelling origins and system theoretic properties. Proceedings of the IFAC symposium on NOLCOS, Bordeaux, France (pp. 282–288). Ortega, R., Lor>?a, A., Nicklasson, P. J., & Sira-Ram>?rez, H. (1998). Passivity-based control of Euler–Lagrangian systems. Communications and control engineering. Berlin: Springer. Ortega, R., van der Schaft, A. J., Maschke, B., & Escobar, G. (2002). Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica, 38(4), 585–596. Shen, T., Ortega, R., Lu, Q., Mei, S., & Tamura, K. (2000). Adaptive L2 -disturbance attenuation of Hamiltonian systems with parameter perturbations and application to power systems. Proceedings of the 39th IEEE conference on decision and control, Vol. 5, Sydney, Australia (pp. 4939 – 4944). van der Schaft, A. J. (1999). L2 -gain and passivity techniques in nonlinear control. Berlin: Springer. van der Schaft, A. J., & Maschke, B. M. (1995). The Hamiltonian formulation of energy conserving physical systems with external ports. D Archive fDur Elektronik und Ubertragungstechnik, 49, 362–371. Wang, Y., Cheng, D., & Hong, Y. (2001). Stabilization of synchronous generators with Hamiltonian function approach. International Journal of Systems Science, 32(8), 971–978. Xi, Z., Cheng, D., Lu, Q., & Mei, S. (2002). Nonlinear decentralized controller design for multimachine power systems using Hamiltonian function method. Automatica, 38(3), 527–534.
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Yuzhen Wang graduated from Tai’an Teachers College in 1986, received his M.S. degree from Shandong University of Science & Technology in 1995 and his Ph.D. degree from the Institute of Systems Science, Chinese Academy of Sciences in 2001. He is now the Postdoctoral Fellow in Tsinghua University, Beijing, China. His research interests include nonlinear control systems, Hamiltonian systems and robust control. Dr. Wang received the Prize of Guan Zhaozhi in 2002, and the Prize of Huawei from the Chinese Academy of Sciences in 2001. Chunwen Li received his B.S. degree and Ph.D. degree from Department of Automation, Tsinghua University in 1982 and 1989, respectively. Since 1994, he has been a Professor with Department of Automation, Tsinghua University. His research interests include nonlinear control systems, inverse systems, CAD and simulation of nonlinear systems, and robust control. Prof. Li received the National Youth Prize in 1991 and the Prize of Chinese Outstanding Ph.D. Degree Receiver in 1992. Daizhan Cheng graduated from Tsinghua University in 1970, and received the M.S. degree from Graduate School, the Chinese Academy of Sciences and the Ph.D. degree from Washington University, St. Louis, MO, in 1981 and 1985, respectively. Since 1990, he has been a Professor with the Institute of Systems Science, Chinese Academy of Sciences. He was an Associate Editor of Mathematical Systems, Estimator and Control (91-93), and Automatica (98-02). He is an Associate Editor of Asia J. Control. Deputy Chief Editor of Control and decision, and a member of the editorial board of Systems Science and Complex, Systems Science and Mathematics, Control Theory and Applications. He is the Chairman of the Control Theoretical Committee, Chinese Automation Association. His research interests include nonlinear system and control, Hamiltonian systems and numerical method in system and control.
Automatica 39 (2003) 1437 – 1443 www.elsevier.com/locate/automatica
Brief Paper
Generalized Hamiltonian realization of time-invariant nonlinear systems Yuzhen Wanga; c;∗ , Chunwen Lia , Daizhan Chengb b Institute
a Department of Automation, Tsinghua University, Beijing 100084, China of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China c School of Control Science and Engineering, Shandong University, Jinan 250061, China
Received 14 December 2001; received in revised form 30 January 2003; accepted 20 March 2003
Abstract A key step in applying the Hamiltonian function method is to express the system under consideration into a generalized Hamiltonian system with dissipation, which yields the so-called generalized Hamiltonian realization (GHR). In this paper, we investigate the problem of GHR. Several new methods and the corresponding su7cient conditions are presented. A major result is that if the Jacobian matrix of a time-invariant nonlinear system is nonsingular, the system has a GHR whose structure matrix and Hamiltonian function are given in simple forms. Then the orthogonal decomposition method and a su7cient condition for the feedback dissipative realization are proposed. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Generalized Hamiltonian realization; Feedback dissipative realization; Jacobian matrix; Orthogonal decomposition method
1. Introduction In recent years, port-controlled Hamiltonian (PCH) systems, proposed by Maschke and van der Schaft (1992) and van der Schaft and Maschke (1995), have been investigated in detail by van der Schaft (1999), Maschke, Ortega, and van der Schaft (2000), Escobar, van der Schaft, and Ortega (1999), Ortega, Lor>?a, Nicklasson, and Sira-Ram>?rez (1998), Ortega, van der Schaft, Maschke, and Escobar (2002), Fujimoto and Sugie (2001), and Cheng, Spurgeon, and Xiang (2000). Indeed, the Hamiltonian function in the PCH system is considered as the total energy and can play the role of Lyapunov function for the system. Because of this, the Hamiltonian function method (the energy-based
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Hassan Khalil. Supported by Project 973 of China (G1998020307, G1998020308) and China Postdoctoral Science Foundation. ∗ Corresponding author. Department of Automation, Tsinghua University, Beijing 100084, China. E-mail addresses: [email protected] (Yuzhen Wang), [email protected] (Chunwen Li), [email protected] (Daizhan Cheng).
0005-1098/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0005-1098(03)00132-8
Lyapunov function method) has been developed (Maschke et al., 2000) and applied to many practical control problems (Cheng, Xi, Hong, & Qin, 1999; Shen, Ortega, Lu, Mei, & Tamura, (2000); Wang, Chen, & Hong, 2001; Xi, Cheng, Lu, & Mei, 2002). It has been shown in Cheng et al. (1999), Wang et al. (2001) and Xi et al. (2002) that the Hamiltonian function method has some advantages. The key point in applying the Hamiltonian function method is to express the system concerned into a Hamiltonian system with dissipation, which is called the dissipative Hamiltonian realization. In general, to complete the dissipative Hamiltonian realization, we Hrst express the system into a generalized Hamiltonian system, i.e., obtain the generalized Hamiltonian realization (GHR), and then eliminate the non-dissipative part of the GHR by a state feedback to get a Hamiltonian system with dissipation. We recall some related concepts Hrst: Denition 1 (Cheng et al., 2000). (1) A dynamic system x˙ = f(x);
x ∈ Rn
(1)
is said to have a generalized Hamiltonian realization (GHR) if there exists a suitable coordinate chart and a Hamiltonian function H such that (1) can be expressed as x˙ = T (x)∇H;
(2)
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Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
where T (x) is an n × n matrix called the structure matrix and ∇H = @H=@x. If the structure matrix can be expressed as T (x) = J (x) − R(x), with skew-symmetric J (x) and symmetric positive semi-deHnite R(x), then system (2) is called a dissipative Hamiltonian realization. Furthermore, if R(x) ¿ 0, (2) is called a strict dissipative Hamiltonian realization. (2) A controlled dynamic system x˙ = f(x) + g(x)u
(3)
is said to have a state feedback Hamiltonian realization if there exists a suitable state feedback u = (x) + v such that the closed-loop system can be expressed as x˙ = T (x)∇H + g(x)v:
(4)
If T (x) can be expressed as T (x) = J (x) − R(x), J (x) is skew-symmetric and R(x) ¿ 0(¿ 0), then (4) is called a feedback (strict) dissipative Hamiltonian realization. The GHR problem has been studied in some recent works (Hebertt Sira-Ram>?rez, 1998; Cheng et al., 2000). There are, however, no eOective methods to handle it yet. In this paper, we investigate the GHR problem of nonlinear systems. We propose several new methods to handle the problem and give some new su7cient conditions for the realization.
2. Generalized Hamiltonian realization In this section, we investigate the GHR problem of system (1) and propose several new methods and su7cient conditions for the GHR. Let Jf denote the Jacobian matrix @f=@x. In system (1), set T @f @f1 @fn Ai = ; = ;:::; @xi @xi @xi ai =
@ ; @xi
i = 1; 2; : : : ; n:
Construct two equations as follows: A2 −A1 A3 −A1 .. . .. . X1 (x) An −A1 X2 (x) A3 −A2 =0 . . . .. .. . . Xn (x) An −A2 . . .. .. An −An−1
(5)
and
a2
a3 .. . an
−a1 −a1 ..
. −a1
a3
−a2
.. .
..
.
an
−a2 .. .
.. . an
X1 (x) X2 (x) ⊗ In . . . Xn (x)
−an−1
=0;
(7)
where Xi (x) (i=1; 2; : : : ; n) are n-dimensional column vector Helds, ⊗ is the Kronecker product and In is the n×n identity matrix. Here, for an arbitrary scalar function h(x) we deHne @h(x) ; i ∈ {1; 2; : : : ; n}: (8) ai · h(x) = @xi Theorem 1. If Eqs. (6) and (7) have a solution (X1T (x); : : : ; XnT (x))T such that matrix (X1 (x) X2 (x) · · · Xn (x))n×n is nonsingular, then there exists a Hamiltonian function H (x) such that system (1) has a GHR as follows x˙ = T (x)∇H;
(9)
where T (x) = (X1 (x) X2 (x) · · · Xn (x))
−T
.
Proof. Using Poincare lemma, a straightforward computation shows the conclusion. Remark 1. The Hamiltonian function H (x) in Theorem 1 can be given the form x1 h1 (x1 ; x2 ; : : : ; x n ) d x1 H (x) = x1(0)
+
x2(0)
+
x2
xn
x(0) n
h2 (x1(0) ; x2 ; : : : ; x n ) d x2 + · · ·
(10)
hn (x1(0) ; : : : ; x(0) n−1 ; x n ) d x n ;
T where hi =XiT f(x); i=1; 2; : : : ; n, and x(0) := (x1(0) ; : : : ; x(0) n ) n ∈ R is an arbitrary initial point.
(6)
The following is a main result in this section. Proposition 1. If the Jacobian matrix Jf is invertible, then n 1 2 @H ; H= fi (11) x˙ = Jf−T @x 2 i=1
is a GHR of system (1).
Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
Proof. Let Xi (x) = @f=@xi = (@f1 =@xi ; : : : ; @fn =@xi )T ; i = 1; 2; : : : ; n. Because T T @f @f @f @f Ai Xj = = = A j Xi ; @xi @xj @xj @xi i; j = 1; 2; : : : ; n, (X1T (x); : : : ; XnT (x))T is a solution to (6). On the other hand, @Xj @Xi @2 f @2 f = = = ; @xj @xi @xj @xj @xi xi Therefore, which implies (aj In )Xi = (ai In )Xj . (X1T (x); : : : ; XnT (x))T is also a solution to (7). Noting that (X1 (x) X2 (x) · · · Xn (x)) = Jf is nonsingular, in view of Theorem 1 we have x˙ = Jf−T ∇H , where H (x) can be detern mined by (10). Here hi (x) = XiT f(x) = j=1 (@fj =@xi )fj . From (10) we get n xi (0) H (x) = hi (x1(0) ; : : : ; xi−1 ; xi ; xi+1 ; : : : ; x n ) d xi xi(0)
i=1
n
i=1 j=1
n
j=1 i=1
− fj2 (x1(0) ; : : : ; xi(0) ; xi+1 ; : : : ; x n ) n
1 2 = fj (x1 ; x2 ; : : : ; x n ) 2 j=1
x˙ = Jf−T ∇H;
n
n
i=1
i=1
1 2 1 2 (0) fi (x) − fi (x ): 2 2
Since adding a constant to H (x) does not change the form of Hamiltonian systems, we can simply choose H (x) = n 1 2 i=1 fi (x). 2 Remark 2. We can prove that (1) Jf−T is consistent with the changing law of structure matrices under coordinate transformations. (2) If Jf is locally invertible at the equilibrium, (11) locally holds. Lemma 1. Assume J (x) is an n×n skew-symmetric matrix and R(x) is an n×n positive (semi-)de>nite matrix. If J (x)− R(x) is nonsingular, then there exist a skew-symmetric matrix J1 (x) and a positive (semi-)de>nite matrix R1 (x) such that (J (x) − R(x))−1 = J1 (x) − R1 (x):
(13)
n
H (x) =
1 2 fi (x): 2
(14)
i=1
On the other hand, JfT = 12 [JfT − Jf ] − {− 12 [JfT + Jf ]} and {− 12 [JfT + Jf ]} is positive deHnite. From Lemma 1, there is a skew-symmetric matrix J (x) and a positive deHnite matrix R(x) such that Jf−T = J (x) − R(x), from which and (14) the theorem holds.
− fj2 (x1(0) ; x2(0) ; : : : ; x(0) n ) =
Theorem 2. If JfT + Jf is negative de>nite, then system (1) has a strict dissipative Hamiltonian realization as follows:
Proof. Since Jf = 12 [Jf − JfT ] + 12 [Jf + JfT ], the Hrst part of the right-hand side is skew-symmetric and the second part is negative deHnite, whereby Jf is invertible (Remark 3). It follows from Proposition 1 that system (1) has a realization as
1 2 (0) (0) fj (x1 ; : : : ; xi−1 ; xi ; : : : ; x n ) 2 n
Remark 3. In Lemma 1, we can prove that if R(x) is positive (or negative) deHnite, then J (x) − R(x) is invertible.
where J (x) is some n × n skew-symmetric matrix, R(x) is some n × n positive de>nite matrix and H (x) = n 1 2 i=1 fi (x). 2
− fj2 (x1(0) ; : : : ; xi(0) ; xi+1 ; : : : ; x n ) =
Proof. For arbitrary 0 = x ∈ Rn , let y := (J (x) − R(x))−1 x = 0. Because J (x) is skew-symmetric and R(x) is positive (semi-)deHnite, yT (J (x) − R(x))y = −yT R(x)y ¡ 0(6 0). Therefore, [(J (x) − R(x))−1 x]T (J (x) − R(x))(J (x) − R(x))−1 x ¡ 0(6 0), which is equivalent to xT (J (x) − R(x))−T x ¡ 0(6 0). From this, we get xT [(J (x) − R(x))−1 + (J (x) − R(x))−T ]x ¡ 0(6 0), which implies that (J (x) − R(x))−1 + (J (x) − R(x))−T is negative (semi-)deHnite. On the other hand, (J (x) − R(x))−1 = 1 −1 − (J (x) − R(x))−T ] + 12 [(J (x) − 2 [(J (x) − R(x)) −1 R(x)) + (J (x) − R(x))−T ], and the Hrst part of the right-hand side is skew-symmetric and the second part is negative (semi-)deHnite. Thus, Lemma 1 holds.
x˙ = (J (x) − R(x))∇H;
1 2 (0) (0) fj (x1 ; : : : ; xi−1 = ; xi ; : : : ; x n ) 2 n
1439
(12)
In the following, we consider the case when Jf is singular. First, we give a lemma. Lemma 2 (Langson and Alleyne, 1997). If a scalar function h(x) with h(0) = 0 (x ∈ Rn ) has continuous nth-order partial derivatives, then h(x) can be expressed as h(x) = a1 (x)x1 + · · · + an (x)x n ;
(15)
where ai (x); i = 1; 2; : : : ; n, are scalar functions. Theorem 3. Consider system (1) with f(0) = 0. If the Jacobian matrix Jf is singular and has a nonsingular main diagonal block, then there exists an n × n matrix M (x) and a vector >eld g(x) with nonsingular Jacobian matrix such that f(x) = M (x)g(x). Furthermore, system (1)
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Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
has a Hamiltonian realization as follows x˙ =
where
M (x)Jg−T ∇H:
(16)
1
T1 (x) = A (x)B (x)
Now, we give a proof for Theorem 3, which itself provides a useful algorithm to Hnd M (x) and g(x). Proof. Without loss of generality, we assume that Rank Jf = k ¡ n and @(f1 ; : : : ; fk )=@(x1 ; : : : ; xk ) is a nonsingular main diagonal block. Let f1 = (f1 ; : : : ; fk )T , f2 =(fk+1 ; : : : ; fn )T , x1 =(x1 ; : : : ; xk )T ; x2 =(xk+1 ; : : : ; x n )T . Since f(0) = 0, f2 (0) = 0. From Lemma 2, there exists an (n − k) × n matrix A(x) = (A1 (x); A2 (x)) such that f2 (x) = A(x)x = A1 (x)x1 + A2 (x)x2 :
(17)
On the other hand, f1 = f1 (x1 ; x2 ); f1 (0; 0) = 0, and @f1 =@x1 is nonsingular. It follows from the implicit function theorem that there exists a function such that x1 = (f1 ; x2 ); (0; 0) = 0. From Lemma 2 again, there is a k × n matrix B(x) = (B1 (x); B2 (x)) such that x1 can be expressed as 1 f 1 = B1 (x)f1 + B2 (x)x2 : x = B(x) (18) x2 Substituting (18) into (17), we get f2 = A1 (x)B1 (x)f1 + (A1 (x)B2 (x) + A2 (x))x2 . Therefore, 1 1 Ik 0 f f = : × A1 (x)B1 (x) A1 (x)B2 (x) + A2 (x) f2 x2 Let M (x) =
Ik
0
A1 (x)B1 (x)
A1 (x)B2 (x) + A2 (x)
is nonsingular. Thus, the Hrst part of the theorem holds. Furthermore, from Proposition 1 we have
2
−T
2
−(A (x)B (x) + A (x))
@f1 @x2
T
@f1 @x1
−T ;
T2 (x) = A1 (x)B2 (x) + A2 (x) and H (x) =
k n 1 2 1 2 fi + xj : 2 2 i=1
(20)
j=k+1
3. Feedback dissipative Hamiltonian realization In this section, we investigate the feedback dissipative Hamiltonian realization. First, we propose a new approach to the GHR called the orthogonal decomposition method. Then we use the method to study the feedback dissipative realization. Let us introduce the following concept. A function V (x) is called a regular positive deHnite function if V (x) ¿ 0 (x = 0), V (0)=0, x=0 =0 and @V=@x|x=0 = 0. For example, @V=@x| n 2 x H (x) = 12 i=1 i is a regular positive deHnite function on Rn . Consider system x˙ = f(x);
f(0) = 0;
x ∈ Rn :
(21)
Proposition 2. If there exists a function H (x) (∇H = 0) such that Lf H = 0, ∀x ∈ Rn , then system (21) can be expressed as x˙ = J (x)∇H , where 1 J (x) = [f∇H T − ∇HfT ] (22) ∇H 2
Proof. J (x)∇H =
1 [f∇H T − ∇HfT ]∇H ∇H 2
=
1 1 f∇H T ∇H − ∇HfT ∇H ∇H 2 ∇H 2
=
1 1 f ∇H 2 − ∇HLf H 2 ∇H ∇H 2
= f; which means that the proposition holds.
x˙ = M (x)
T1 (x)
1
@f1 @x1
is an n × n skew-symmetric matrix.
and g(x) = ((f1 )T ; (x2 )T )T , then f(x) = M (x)g(x) and T @f1 0 T @x1 @g = T @x 1 @f In−k @x2
−T @g @x −T @f1 ×∇H = @x1
1
0 T2 (x)
∇H;
(19)
Remark 4. If H (x) in Proposition 2 is replaced by a regular positive deHnite function, then (22) should be modiHed as 1 [f∇H T − ∇HfT ] x = 0; 2 ∇H J (x) = (23) 0 x = 0:
Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
Proposition 2 indicates that if the directional derivative of H (x) along vector Held f(x) is zero everywhere, i.e., f(x) ⊥ ∇H holds for all x, then the system can be expressed as a traditional Hamiltonian system. Motivated by this, we propose a new approach to the GHR of system (21): the orthogonal decomposition method. Choose a regular positive deHnite function H (x) and write out the equi-value surfaces of H (x). At arbitrary point x = 0, we decompose f(x) along the gradient direction ∇H and the tangential direction of the equi-value surface as follows: f(x) = fgd (x) + ftd (x);
(24)
fgd (x) =
f; ∇H ∇H; ∇H 2
ftd (x) = f(x) − fgd (x)
(25)
and ·; · denotes the inner product. Because Lftd H = ftd ; ∇H = f − fgd ; ∇H = f; ∇H − fgd ; ∇H = f; ∇H − (f; ∇H = ∇H 2 )∇H; ∇H = 0, ftd ⊥ ∇H . From Proposition 2 we know
x˙ = T (x)∇H + g(x)u; where
T (x) =
(30)
J (x) + S(x);
x = 0;
0;
x=0
and J (x), S(x) are given by (26) and (27). Decompose S(x) as follows:
1 T [ftd ∇H T − ∇Hftd ]: ∇H 2
(26)
f; ∇H I; ∇H 2
(27)
where I is the n × n identity matrix, then when x = 0, x˙ = fgd (x) + ftd (x) = (J (x) + S(x))∇H . Thus, system (21) has a GHR as follows: J (x) + S(x); x = 0; x˙ = T (x)∇H; T (x) = (28) 0; x = 0: We call (28) the orthogonal decomposition Hamiltonian realization. Because there are always regular positive deHnite functions, we have the following result. Theorem 4. An arbitrary system x=f(x), ˙ f(0)=0, always has the orthogonal decomposition Hamiltonian realization (28). From Theorem 4, we readily get the following corollary. Corollary 1. If system x˙ = f(x) with f(0) = 0 is asymptotically stable, then it has a strict dissipative Hamiltonian realization with its Hamiltonian function positive de>nite. In the following, we investigate the feedback dissipative Hamiltonian realization. Consider the single-input system x˙ = f(x) + g(x)u;
f(0) = 0;
x ∈ Rn :
(29)
(32)
and substitute it into (30), then we get x˙ = (J (x) − R1 (x) + R2 (x))∇H 1 g[Lg Hv − ∇H T R2 (x)∇H ] Lg H
= (J (x) − R1 (x))∇H + R2 (x)∇H −
Let
(31)
where R1 (x) ¿ 0 and R2 (x) is symmetric. When x = 0, set 1 [Lg Hv − (∇H )T R2 (x)∇H ] u= Lg H
+
ftd (x) = J (x)∇H;
S(x) =
Assume there exists a regular positive deHnite function H (x) such that Lg H = 0, for all x = 0. With this H (x), from the orthogonal decomposition realization, system (29) can be expressed as
S(x) = −R1 (x) + R2 (x);
where
J (x) =
1441
1 g∇H T R2 (x)∇H + gv Lg H
= (J (x) − R1 (x))∇H +
1 [R2 (x)∇HLg H Lg H
− g∇H T R2 (x)∇H ] + gv = (J (x) − R1 (x))∇H +
1 [R2 (x)∇HgT Lg H
− g∇H T R2 (x)]∇H + gv = (J (x) + J˜ (x) − R1 (x))∇H + gv; where J˜ (x) = (1=Lg H )[R2 (x)∇HgT − g(∇H )T R2 (x)] is skew-symmetric. When x = 0, we set u = v. Thus, system (30) can be expressed as x˙ = M (x)∇H + gv; where M (x) =
(33)
J (x) + J˜ (x) − R1 (x);
x = 0;
0;
x = 0:
Since J (x) + J˜ (x) is skew-symmetric and R1 (x) ¿ 0, (33) is a strict dissipative Hamiltonian realization. From the above, we get the following result. Proposition 3. If there exists a regular positive de>nite function H (x) such that Lg H (x) = 0 (x = 0), then system (29) has a feedback strict dissipative Hamiltonian realization with H (x) as its Hamiltonian function.
1442
Yuzhen Wang et al. / Automatica 39 (2003) 1437 – 1443
Example 1. Find a feedback dissipative Hamiltonian realization of the following system: x 1 + x2 x1 + x12 x2 3 x˙ = −x1 + x2 + x2 − x1 u x3 x3 := f(x) + g(x)u:
(34)
First, using the orthogonal decomposition method, we express (34) as a generalized Hamiltonian system. We choose the regular positive deHnite function as follows: H (x) = 1 2 2 2 2 (x1 + x2 + x3 ). When x = 0, from (25) we get fgd =
f; ∇H ∇H = (x1 ; x2 ; x3 )T ; ∇H 2
and when x = 0, we let u = v. Substituting the control u into (35), we have x˙ = M (x)∇H + g(x)v; where M (x) =
J (x) + J˜ (x)
= x2 x3
−x1 x3 ; 0
S(x) = (f; ∇H = ∇H 2 )I = Diag{1; 1; 1}. So system (34) can be expressed as
x = 0; x = 0:
(35)
Second, we design a control law to get the desired dissipative realization. Because Lg H = x12 + x22 + x32 = 0 (x = 0), we know from Proposition 3 that system (34) has a feedback strict dissipative realization with H (x) as its Hamiltonian function. Decompose S(x) as follows: S(x)=−R1 +R2 , where R1 =Diag{1; 1; 1} ¿ 0, R2 =Diag{2; 2; 2} ¿ 0. When x = 0, according to (32) we set u= =
1 [Lg Hv − ∇H T R2 (x)∇H ] Lg H 2 1 (x1 + x22 + x32 )v x12 + x22 + x32 2 0 0 x1 −(x1 ; x2 ; x3 ) 0 2 0 x2 0 0 2 x3
= −2 + v;
x = 0;
0; x = 0; 1 [R2 (x)∇HgT − g∇H T R2 (x)] J˜ (x) = Lg H 0 x12 + x22 −2 −x2 − x2 = 2 0 2 x1 + x22 + x32 1 −x2 x3 x1 x3
Therefore,
x˙ = T (x)∇H + g(x)u; J (x) + S(x); T (x) = 0;
J (x) + J˜ (x) − R1 (x);
x2 x 3
−x1 x3 : 0
Since
ftd = f − fgd = (x12 x2 ; −x13 ; 0)T :
1 T J (x) = [ftd ∇H T − ∇Hftd ] ∇H 2 0 x12 + x22 2 x1 −x2 − x2 = 2 0 2 x1 + x22 + x32 1 −x2 x3 x1 x 3
(36)
0
2 x12 − 2 −x − x2 2 2 2 2 x1 + x2 + x3 1 −x2 x3
x12 + x22 0 x1 x3
x 2 x3
−x1 x3 0
is skew-symmetric and R1 ¿ 0, (36) is a strict dissipative Hamiltonian realization. 4. Conclusion We have investigated the Hamiltonian realization problem of time-invariant nonlinear systems, proposed several operable and comparatively systematic methods and given some su7cient conditions for the realization. The main results are as follows. (1) If the Jacobian matrix of a system is nonsingular, the system has a GHR whose structure matrix and Hamiltonian function are given in very simple forms. (2) If the Jacobian matrix of the system is singular and has a nonsingular main diagonal block, then the system also has a GHR whose structure matrix and Hamiltonian function can be easily gotten. (3) A new method of GHR called the orthogonal decomposition method has been obtained. It has been proved that an arbitrary nonlinear system has an orthogonal decomposition Hamiltonian realization, and the corresponding realization form can be given by a set of formulas. References Cheng, D., Spurgeon, S., & Xiang, J. (2000). On the development of generalized Hamiltonian realizations. Proceedings of the 39th IEEE conference on decision and control, Vol. 5, Sydney, Australia (pp. 5125 –5130). Cheng, D., Xi, Z., Hong, Y., & Qin, H. (1999). Energy-based stabilization of forced Hamiltonian systems with its application to power systems.
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Yuzhen Wang graduated from Tai’an Teachers College in 1986, received his M.S. degree from Shandong University of Science & Technology in 1995 and his Ph.D. degree from the Institute of Systems Science, Chinese Academy of Sciences in 2001. He is now the Postdoctoral Fellow in Tsinghua University, Beijing, China. His research interests include nonlinear control systems, Hamiltonian systems and robust control. Dr. Wang received the Prize of Guan Zhaozhi in 2002, and the Prize of Huawei from the Chinese Academy of Sciences in 2001. Chunwen Li received his B.S. degree and Ph.D. degree from Department of Automation, Tsinghua University in 1982 and 1989, respectively. Since 1994, he has been a Professor with Department of Automation, Tsinghua University. His research interests include nonlinear control systems, inverse systems, CAD and simulation of nonlinear systems, and robust control. Prof. Li received the National Youth Prize in 1991 and the Prize of Chinese Outstanding Ph.D. Degree Receiver in 1992. Daizhan Cheng graduated from Tsinghua University in 1970, and received the M.S. degree from Graduate School, the Chinese Academy of Sciences and the Ph.D. degree from Washington University, St. Louis, MO, in 1981 and 1985, respectively. Since 1990, he has been a Professor with the Institute of Systems Science, Chinese Academy of Sciences. He was an Associate Editor of Mathematical Systems, Estimator and Control (91-93), and Automatica (98-02). He is an Associate Editor of Asia J. Control. Deputy Chief Editor of Control and decision, and a member of the editorial board of Systems Science and Complex, Systems Science and Mathematics, Control Theory and Applications. He is the Chairman of the Control Theoretical Committee, Chinese Automation Association. His research interests include nonlinear system and control, Hamiltonian systems and numerical method in system and control.