* Correspondence to: Jang-Lee Hong, Department of Electronic Engineering, Van-Nung Institute of Technology,
Chungli, Taiwan.
R E-mail: janglee@cc.vit.edu.tw
Contract/grant sponsor: National Science Council, R.O.C.; contract/grant number: NSC 89-2218-E238-001
H
Rcontrol for nonlinear a$ne systems:
a chain-scattering matrix description approach
Jang-Lee Hong * R and Ching-Cheng Teng
Department of Electronic Engineering, Van-Nung Institute of Technology, Chungli, Taiwan Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu, Taiwan
SUMMARY
This paper combines an alternative chain-scattering matrix description with (J, J )-lossless and a class of conjugate (!J, !J )-lossless systems to design a family of nonlinear H output feedback controllers. The present systems introduce a new chain-scattering setting, which not only o!ers a clearer expression for the solving process of the nonlinear H control problem but also removes the"ctitious signals introduced by the traditional chain-scattering approach. The intricate nonlinear a$ne control problem thus can be transformed into a simple lossless network and is easy to deal with in a network-theory context. The relationship among these (J, J ) systems, ¸-gain, and Hamilton}Jacobi equations is also given. Block diagrams are used to illustrate the central theme. Copyright 2001 John Wiley & Sons, Ltd.
KEY WORDS: nonlinear systems; ¸-gain; hamilton}Jacobi equations; state-space method
1. INTRODUCTION
Since Zames [1] proposed the concept of sensitivity minimization in the H domain, many researchers have made valuable contributions to the study of the H domain. Parameterization of all linear H-(sub) optimal output feedback controllers were given by Glover et al. [2]. Green et al. [3] and Kimura [4] then o!ered an alternative method by using J-spectral or (J, J )-lossless factorization. Also, Kimura [5] and Ball et al. [6] developed a"ctitious signals method to solve the linear 4-block control problem. Hong and Teng [7] then developed a new method which both matched the famous results of Glover et al. [2] and removed the "ctitious signals.
As in the extension of linear H control theory to nonlinear settings, the local disturbance attenuation with internal stability was "rst studied by Ball and Helton [8], Bas
'
ar and Bernhard [9], and Van der Schaft [10, 11]. Van der Schaft used the notion of dissipativity in a nonlinear
system to show that the Hamilton}Jacobi equation is the nonlinear version of the Riccati equation considered in linear systems which yields the solution of a nonlinear H-state feedback control problem. As for measurement feedback, Ball et al. [12] established the necessary conditions for the existence of a solution. Moreover, Isidori [4, 13] summarized the notion of the dissipative system and the theory of di!erential games to de"ne su$cient conditions based on two Hamilton}Jacobi equations.
An alternative approach using (J, J )-inner-outer factorization or the chain-scattering approach, Helton and James [14, 26], Baramov and Kimura [15], and Ball et al. [16, 17] solved the so-called 2-block case. Following this approach, Pavel and Fairman [18] introduced a non-linear version of the "ctitious signals method to solve the general 4-block case, which reduced the 4-block case to a simple 2-block case. However, one must then be careful to ignore the "ctitious signals when seeking the solution for the original problem.
The present paper aims to reformulate the earlier results by combining the traditional (J, J )-lossless system with a class of nonlinear conjugate (!J,!J )-lossless system to solve the 4-block nonlinear H-output feedback control problem. This new chain-scattering matrix description extends the concept of Hong and Teng's [7] to the nonlinear setting and discrads the "ctitious signals proposed recently by Pavel and Fairman [18]. Therefore, the controller thus obtained is quite straight-forward and provides deeper insight into the synthesis of the controllers.
In Section 2, we brie#y state the standard nonlinear a$ne H control problem. Section 3 proposes the relationships among the Hamiltonian system, (J, J )-lossless, conjugate (J, J )-lossless, and conjugate (!J,!J )-lossless matrices. The main results and the relation between the nonlinear H control problem and the chain-scattering matrix description are presented in Section 4.
2. NOTATIONS AND PRELIMINARY INFORMATION R denotes a real number
1L denotes n-dimensional Euclidean space
RL the set of proper real rational function matrices with no poles on the jw axis. dom(Ric) denotes the Hamiltonian matrix with no eigenvalues on the jw-axis. G&
(s) denotes G2(!s) and G*(s) denotes G2(sN).
The chain-scattering matrix description is abbreviated as CSMD. Consider a smooth nonlinear a$ne system PK given by
PK :"
xR "A(x)#B(x)u y"C(x)#D(x)uwith y31N>N, u31K>K, and x"(x, x,2,xL) are local co-ordinates for a smooth
state-space manifold M de"ned in a neighbourhood) of the origin. Also, we assume x"0 is an equilibrium point and C(0)"0.
Basic properties for this system which will be used in the present paper are stated in the following de"nitions (see Pavel et al. [18]).
Dexnition 1
The system PK is said to be stabilizable if there exists a continuous function F(x) with F(0)"0 such that A(x)#B(x)F(x) is asymptotically stable.
Dexnition 2
The system PK is said to be zero-state detectable if for all x31L, u"0, and y"0, ∀t*0, implies xP0 as tPR.
Dexnition 3
The Zero dynamics of system PK are de"ned as the set of state trajectories +(x(t), generated by the set of input U and initial conditions X such that the output is identically null, i.e., xR "A(x)#B(x)u, with 0"C(x)#D(x)u, ∀t*0, u3U, x(0)3X.
Dexnition 4
The system PK is said to have ¸-gain less than or equal to c if its zero state response satis"es
2#y(t)# dt)c
2#u(t)# dt with ¹'0
The following de"nition of the right-coprime factorization can be found in Pavel et al. [18] or Scherpen et al. [19]. Which is well known in linear case. As this is one of the key ideas of this paper, for completeness, we rewrite it below.
Dexnition 5
A right-coprime factorization of system PK , with (A(x), B(x)) being stabilizable, is given by two systems
N :"
xR "A(x)#B(x)F(x)#B(x);?(x)fy"C(x)#D(x)F(x)#D(x);?(x)f M :"
xR "A(x)#B(x)F(x)#B(x);?(x)f y"F(x)#;?(x)f
with ;?(x) is invertible and M\ is the inverse system of M such that:
(i) for every initial condition of PK there exist initial conditions for N and for M\ such that the input}output behaviour of PK equals the input}output behaviour of N3M\, where N3M\ denotes the system obtained by the series interconnection of M\ followed by N,
(ii) A(x)#B(x)F(x) is asymptotically stable;
(iii) N and M are right coprime, i.e. the zero dynamics of the system [,+] is asymptotically stable. The standard nonlinear azne HR
control problem
Consider the following smooth (C) nonlinear a$ne H framework
P :" i g j g k xR "A(x)#B(x)w#B(x)u z"C(x)#D(x)u y"C(x)#D(x)w (1)
where z(t)31N, y(t)31N, w(t)31K, and u(t)31Kare the error, observation, disturbance, and control input, respectively. The states x"(x, x,2, xL) are local co-ordinates for a state-space manifold M de"ned in a neighbourhood) of the origin in 1L. Assume x"0, an equilibrium point, also A (0)"0, C(0)"0, and C(0)"0. Furthermore, as in the general 4-block nonlinear H-control problem, the inequalities m'p and p'm must hold.
Figure 1.
The suboptimal nonlinear a$ne H control problem is then modelled so as to choose a controller K which connects the observation vector y to u so that K locally, asymptotically stabilizes the closed-loop system in a neighbourhood ) of the origin with internal stability. Furthermore, the closed-loop system with a local ¸-gain is less than or equal to a prescibed numberc.
Figure 1 shows a general set-up for the nonlinear a$ne H control system.
For simplicity and yet without any loss of generality of the derivations in subsequent sections, letc"1 and take the following assumptions for the 4-block nonlinear a$ne H control problem. Assumptions:
A1. (A(x), B(x)) is locally stabilizable and (C(x), A(x)) is locally detectable in a neighbourhood) of the origin. A2. D2(x)D(x)"IK
and D(x)D2(x)"IN.
D2(x)C(x)"0 and B(x)D2(x)"0
A3. Any bounded trajectory x(t) of system xR (t)"A(x(t))#B(x(t))u(t), satisfying C(x(t))#D(x(t))u(t)"0, for all t*0, in such that limR x(t)"0.
A4. rank *A *x(0)!jwI B(0) *C *x (0) D(0) " n#p, ∀u3R
Assumption A1 is necessary for the existence of stabilizing controllers. Assumptions A3 and A4 imply that the pair +A(x), C(x), is locally zero state detectable and +A(x), B(x), is locally stabilizable at the origin. These assumptions are the nonlinear version of standard assumptions usually considered in linear case (see References [2, 3, 7, 20]).
3. (J, J )-LOSSLESS SYSTEMS AND HAMILTONIAN SYSTEMS 3.1. The (J, J )-lossless system (#*J#"J )
Before discussing the (J, J )-lossless property in nonlinear system, let's consider this property in the following linear chain-scattering setting.
y y y & & #* & &uu u #*:"
xR "Ax#Bu y"Cx#Du (2)with y31N, y31O, u31K, and u31L. It is well known (see references [5, 7, 21, 22]) that, matrix#*(s)3RLN>O;
K>L is said to be a (J, J )-lossless matrix if p*m, q*n and #*(s)J J#*(s)"J for each s3jw
#*(s)*J#*(s))J for each Re[s]*0
where J"diag+IN,!IO, and J "diag+IK,!IL,. Its relevant state-space properties are stated below.
¸emma 1
Let#*(s)3RLN>O;
K>Lwith (A, B) controllable, (C, A) detectable. Then#* is (J, J )-lossless if:
(i) A2X#XA#C2JC"0; (ii) XB#C2JD"0;
(iii) D is (J, J )-unitary (i.e. D2JD"J and DJ D2"J); (iv) X*0.
Obviously, from the above lemma, one has
#**J#*"(D#C(sI!A)\B)*J(D#C(sI!A)\B)
"D2JD#B2(s*I!A2)\C2JD#D2JC(sI!A)\B
#B2(s*I!A2)\C2JC(sI!A)\B
"J !B2(s*I!A2)\(s*X#sX)(sI!A)\B (3) That is#*(s)I J#*(s)"J , ∀s3jw and #*(s)*J#*(s))J , ∀Re [s]*0.
Since y"#*u and #**J#*)J , one has
u*R#**J#*u)u*J uNy*Jy)u*J uN#y#!#y#)#u#!#u# N#y###u#)#u###y# (4) where #u#:"
\#u(t)# dt " \ u(t)*u(t) dt This implies that the output energy is less then or equal to the input energy.Furthermore, from Lemma 3 in Hong et al. [7], one has the following lemma for a conjugate (J, J )-lossless system. For simplifying the mathematical narration, we use&if ' instead of &if and only if ' in these lemmas for linear case.
Lemma 2
Let#A*(s)3RLK>L;
N>Owith (C, A) observable, (A, B) stabilizable. Then#A* is conjugate (J, J )-lossless if
(i) A>#>A2#BJ B2"0; (ii) DJ B2#C>"0; (iii) D is (J, J )-unitary; (iv) >*0.
Remark 1
As a similar computation as in Equation (3), one obtains that #A*J #JA*"J for each s3jw #A*J #*A*)J for eachRe[s]*0
Besides, if X in Lemma 1 is invertible, then (i), (ii) in Lemma 1 and (i), (ii) in Lemma 2 are related by reciprocity: >"X\. Furthermore, from the (J, J )-unitary of D, this indicates that #* in Lemma 1 also is a conjugate, (J, J )-lossless system (i.e. #*(s)J #*(s)J"J,∀s3jw and #*(s)J #*(s)*)J, ∀ Re[s]*0).It immediately shows that, if > in Lemma 2 is invertible, then
#A* is conjugate (J, J )-lossless implies that#A* also in (J, J )-lossless. By a similar computation as in Equation (4), one obtains that#A* also has the property of output energy is less then or equal to the input energy.
Now, consider the following de"nition of (J, J )-losslessness for the nonlinear system. This de"nition is a modi"ed version of the well-known results of the dissipative system while applied to the chain-scattering setting (see Willems [23] and Pavel et al. [18]).
Dexnition 6
A nonlinear C chain-scattering system # given by
y y y & & # & &uu u # :"
xR "a(x)#b(x)u y"c(x)#d(x) u (5)with y31N, y31O, u31K, and u31L, is called a (J, J )-lossless system, if # has an equilibrium point at x"0 with c(0)"0 and if there exists a storage function <(x)*0, such that:
<(x(¹))!<(x(0))"1 2
2
(u2(t)J u(t)!y2(t)Jy(t)) dt*0 (6)
with x(0)"0, <(0)"0, and ¹*0, where J"diag+IN,!IO, and J "diag+IK,!IL,
Obviously, from Equation (6), one can see that this (J, J )-lossless system has a same property as it is in linear case, (i.e., the output energy is less than or equal to the input energy). Furthermore, if <(x) is di!erentiable then Equation (6) becomes
<V(x)[a(x)#b(x)u]"u2J u![c(x)#d(x)u]2J[c(x)#d(x)u]
Direct computation yields the following lemma for the (J, J )-lossless system (see Ball et al. [17] or Pavel et al. [18]).
Lemma 3
System# is (J, J )-lossless with respect to a smooth function <(x), if: (i) <V(x)a(x)#c2(x)Jc(x)"0;
(ii) <V(x)b(x)#c2(x)Jd(x)"0; (iii) d2(x)Jd(x)"J ;
Remark 2
If one further de"nes the input vector u in# as u"[XYUY] and the output y as y"[XU], then from Equation (6), it is obvious that:
2 [(#z (t)#!#w (t)#)!(#z(t)#!#w(t)#)] dt*0 That is 2 (#z (t)#!#w (t)#) dt)0 N2 (#z(t)#!#w(t)#) dt)0Furthermore, as proposed by Crouch et al. [24], the Hamiltonian extension of system# is xR "a(x)#b(x)u pR "!
*a(x) *x # *b(x) *x u 2 p!*c(x) *x # *d(x) *x u 2 u? (7) y"c(x)#d(x)u y?"b2(x)p#d2(x)u?Imposing u?"Jy in Equation (7) leads to the following Hamiltonian system for #*J# (with input u and output y?).
#*J# :
xR "a(x)#b(x)u pR "!*a(x) *x # *b(x) *x u 2 p!*c(x) *x # *d(x) *x u 2 J(c(x)#d(x)u) y?"b2(x)p#d2(x)Jc(x)#d2(x)Jd(x)u (8)This Hamiltonian system can also be denoted by
#*J#:
xR " *HK *p (x, p, u) 2 pR "!*HK *x(x, p, u)2 y?" *HK*u (x, p, u)2with Hamiltonian function HK (x, p, u)"p2(a(x)#b(x)u)#(c(x)#d(x)u)2J(c(x)#d(x)u). Recalling from Proposition 7.1.3 in Van der Schaft [25] that, +(x, p) : p"<2V, being an invariant mainfold for#*J# (with u"0) if and only if the smooth function <(x) is such that the Hamilton}Jacobi equation HK (x, <2V(x), 0)"0, which is equal to condition (i) in Lemma 3. It immediately follows that, if d2(x)Jd(x)"J and the smooth function <(x) satis"es conditions (ii) and (iv) in Lemma 3 then the system# is (J, J )-lossless. Also, from the local properties in Ball and
Van der Schaft [17, 11] such an invariant manifold exists if the Jacobian matrix of the Hamiltonian #ow associated with#*J# (with u"0) at equilibrium belongs to dom(Ric).
For discussing the ((J, J )-lossless)-(minimal-phase) factorization for a nonlinear a$ne system G so that G can be factorized as G"#% (with % minimal phase and # being (J, J )-lossless), let's consider the following nonlinear a$ne system G:
G :
xNQ "a(xN)#b(xN)u y"c(xN )#d(xN)uwhere xN "0 is an equilibrium point and c(0)"0. As shown in Ball and Van der Schaft [17, 11] that, for such a nonlinear a$ne system, while the Hamiltonian system of (G*JG)\ is given by (G*JG)\ :
xNQ " *HK ; *p (xN , p, y?)2 pR "!*HK ; *xN (xN , p, y?) 2 u" *HK ; *y? (xN , p, y?) 2then G has a ((J, J )-lossless)-(minimal-phase) factorization, suppose there exists an invariant manifold +(x, p) : p"<2V, for (G*JG)\ (with y?"0) so that the Hamilton}Jacobi equation HK ;(xN , <2VN(xN ), 0)"0 with the stability side condition *HK;/*x (x, <2xN(xN ), 0) is Lyapunov stable.
Furthermore, such an invariant manifold does exist if the Jacobian matrix of the Hamiltonian #ow associated with (G*JG)\ (with y?"0) at equilibrium belongs to dom(Ric).
The idea behind this local result is easy to understand when one considers the following characteristics for linear systems. One can further compare these characteristics with the related nonlinear Hamiltonian system.
If the linear chain-scattering system G* is denoted by G*"C(sI!A)\B#D"
ACB D
"xNQ "AxN#Bu y"CxN #Du
with R"D2JD being invertible, then the linear Hamiltonian system G**JG* given by
G**JG*" A 0 B !C2JC !A2 !C2JD D2JC B2 D2JD is contrasted with G*JG:
xNQ " *HK *p (xN , p, u) 2 pR "!*HK *xN (xN , p, u) 2 y?" *HK*u (xN , p, u)2Furthermore, the (G**JG*)\ denoted by (G**JG*)\"
A%C%;; B%; D%; is contrasted with (G*JG)\ : xNQ " *HK ; *p (xN , p, y?) 2 pR "!*HK ; *xN (xN , p, y?) 2 y?" *HK ; *y? (x, p, y?)2 where HK ;(xN , p, y?) is the Hamiltonian function for (G*JG)\.It immediately follows that A%;"
A!BR\D2JC !BR\B2
!C2(J!JDR\D2J)C !(A!BR\D2JC)2
is contrasted with the Hamiltonian #ow induced by HK ;(xN , p, y?) (with y?"0) or the Hamiltonian #ow associated with (G*JG)\ (with y?"0). Now, introducing a similarity transformation matrix ¹"['6 '] into A%;, one has
¹\A%;¹"
I !X O IA%; I 0 X I" P R Q !P2 where P"A!BR\(D2JC#B2X) R"!BR\B2 Q"X(A!BR\D2JC)#(A!BR\D2JC)2X!XBR\B2X#C2(J!JDR\D2J)C Obviously, the stability side condition*HK;/*xN(xN, <2VN(xN ), 0) is contrasted with P"A#BF (withthe state-feedback gain F"!R\(D2JC#B2X)).
As we know, in linear system, the above Hamiltonian matrix A%;is related to the algebraic
Riccati equation:
X(A!BR\D2JC)#(A!BR\D2JC)2X!XBR\B2X#C2(J!JDR\D2J)C"0 Furthermore, this algebraic Riccati equation can be solved if the eigenvalues of the related Hamiltonian matrix A%;are not on the jw-axis (i.e. the Hamiltonian matrix belongs to dom(Ric)).
3.2. Conjugate (!J,!J )-lossless system (!#)J #K*"!J) Consider the following linear chain-scattering system#)* given as
y y y & & #)* & &uu u #K*:"
xR "Ax#Bu y"Cx#Du (9)Carefully comparing the chain-scattering structure of#)* with #*'s are shown in Equation (2) reveals that the directions of the arrow signals in#)* are contrary to #*'s. One thus has the
following lemma for conjugate (!J,!J )-lossless or conjugate (J, J )-expansive system (see Hong et al. [7]).
Lemma 4
Let#)*(s)3RLK>L;
N>Owith m)p, n)q, (C, A) observable, and (A, B) stabilizable. Then #)* is conjugate (!J,!J )-lossless if
(i) !A>!>A2#BJ B2"0; (ii) DJ B2!C>"0;
(iii) D is (J, J )-unitary;
(iv) >*0, where J"diag+IK,!IL, and J "diag+IN,!IO,. The above lemma indicates that
!#)*J #K**"!(D#C(sI!A)\B)J (D#C(sI!A)\B)*
"!J!C(sI!A)\(s>#s*>)(s*I!A2)\C2
That is !#)*(s)J #)*(s)"!J, ∀s3jw and !#)*(s)J #)*(s)*)!J, ∀Re[s]*0 or #)*(s)J #)*(s)**J, ∀Re[s]*0.
Remark 3
This conjugate (!J,!J )-lossless system also have the same property as the statement in Remark 1. That is, if > is invertible, then#)* is conjugate (!J,!J )-lossless implies that #)* also is (!J,!J )-lossless (i.e., #)**J#K**J ).
Since y"#K*u and #)**J#K**J , one has
y*Jy"u*#K**J#K*u*u*J uNy*Jy*u*J u
N#y#!#y#*#u#!#u#
N#y###u#*#u###y#
From Equation (9), this implies that the output energy is less than or equal to the input energy. Now, supposing the interconnection law u in Equation (7) is as u"!J y?, and substituting #) for #, one has the following Hamiltonian system for !#)J #K* (with input u? and output y).
!#)J #K* :
xR "a(x)!b(x)J b2(x)p!b(x)J d2(x)u? pR "!*a(x) *x ! *b(x) *x J b2(x)p! *b(x) *x J d2(x)u? 2 p !*c(x) *x ! *d(x) *x J b2(x)p! *d(x) *x J d2(x)u? 2 u? y"c(x)!d(x)J b2(x)p!d(x)J d2(x)u? (10)The Hamiltonian function HM is such that !#)J #K* :
xR " *HM *p (x, p, u?) 2 pR "!*HM *x(x, p, u?) 2 y" *HM *u?(x, p, u?) 2 is thus given as HM (x, p, u?)"p2a(x)!p2b(x)J b2(x)p!p2b(x)J d2(x)u?#c2(x)u?!u2?d(x)J d2(x)u? (11) The following de"nition gives the property for a nonlinear a$ne system #) to be conjugate (!J,!J )-lossless which is well known in linear case (see e.g. Hong et al. [7]). The same as the (J, J )-lossless system, this conjugate (!J,!J )-lossless system also involves the validity of an energy storage balance equality in integral form. The nonlinear (I, I ) case was proposed by Scherpen and Van der Schaft [19], who called it &co-inner'.Dexnition 7
A nonlinear C chain-scattering system #) given by y y y & & #) & &uu u #K :"
xR "a(x)#b(x)u y"c(x)#d(x)uwith y31K, y31L, u31N, and u31O, is called a conjugate (!J,!J )-lossless system, if #) has an equilibrium point at x"0, with c(0)"0, and the input} output map of system Equation (10) from u? to y, with x(0)"0 and p(0)"0, is equal to !J (i.e., !#)J #K*"!J), and there exists a smooth storage function =(x)*0, =(0)"0, and ¹*0 such that:
=(x(¹))!=(x(0))"1 2
2
( y2(t)Jy(t)!u2(t)J u(t)) dt*0 (12) Equation (12) shows that the output energy is less than or equal to the input energy.
From the Hamiltonian system for !#)J #K* in Equation (10), since u"!J y? and the conjugate (!J,!J )-lossless system #) has !#)J #K*"!J (i.e. y"!Ju?), the above equation is equal to =(x(¹))!=(x(0))"1 2
2 (u2?(t)Ju?(t)!y2?(t)J y?(t))dt*0 (13) Also, the following theorem gives a state-space criterion for a nonlinear a$ne system #) to be conjugate (!J,!J )-lossless.Theorem 1
System#) is conjugate (!J,!J )-lossless with respect to a smooth function =(x), if there is an invariant manifold+(x, p) : p"=2V, for (#)J #K*)\ (with y"0) such that:
(ii) !c(x)#d(x)J b2(x)=2V(x)"0; (iii) d(x)J d2(x)"J;
(iv) =(x)*0, =(0)"0.
Proof. Replacing the right-hand side of y"c(x)!d(x)J b2(x)p!d(x)J d2(x)u? in Equa-tion (10) with (ii), (iii), and p"=2V, one obtains y"!Ju? (i.e. !#)J #K*"!J).
Furthermore, from Equation (7), since y?"b2(x)p#d2(x)u?"b2(x)=2V(x)#d2(x)u?, one has y2?J y?"(u2?d(x)#=V(x)b(x))J (b2(x)=2V(x)#d2(x)u?)
"
2=V(x)b(x)J d2(x)u?#=V(x)b(x)J b2(x)=2V(x)#u2?d(x)J d2(x)u? From (i) and (iii), this implies that
=V(x)[a(x)!b(x)J b2(x)=2V(x)!b(x)J d2(x)u?]"(u2?Ju?!y2?J y?)
Integrating both side with respect to t (from 0 to ¹), together with xR "a(x)!b(x)J b2(x)p !b(x)J d2(x)u? in Equation (10), Equation (13) follows immediately.
Note that, the di!erentiability for <(x) or =(x) is an arti"cial hypothesis imposed for all solutions of the Hamilton}Jacobi equations in this paper; however, there might exist some viscosity solutions to admit nonsmooth <(x) or =(x) (see e.g. Bas
'
ar et al. [9] or Van der Schaft [25]).
4. THE CSMD APPROACH FOR DERIVING HR
CONTROLLERS
This paper proposes an alternative method for designing nonlinear H controllers. This method is based on a combination of a chain-scattering matrix description (CSMD) together with the (J, J )-lossless and conjugate (!J,!J )-lossless properties.
From Equation (1) and the properties in De"nition 5, let P"NM\ be a right-coprime factorization, in which one chooses F(x) to be a stabilizing feedback control for the pair (A(x), B(x)), and hence N and M are stable. This is analogous to linear system theory, thus giving
N :"
xR "A(x)#B(x)F(x)#B(x);?(x)z w z y"C(x)#D(x)F(x)#D(x);?(x) z w M :" xR "A(x)#B(x)F(x)#B(x);?(x)z w w u"F(x)#;?(x) z wFigure 2.
where
B(x)"[B(x) B(x)], D(x)"
D(x)0 D(x)0 , C(x)"C(x)C(x), F(x)"F(x)F(x) and;?(x)"
;?(x) ;?(x);?(x) ;?(x) One further de"nes G and G asG:"
xR "A(x)#B(x)F(x)#B(x);?(x)z w z w"C(x)0 # 0 I D(x)0 F(x)F(x)# 0 I D(x)0 ;?(x) z w (14) G:" xR "A(x)#B(x)F(x)#B(x);?(x)z w u y" 0 C(x)# 0 D(x) I 0F(x)F(x)# 0 D(x) I 0;?(x) z w (15)It is obvious that the standard nonlinear H set-up as shown in Figure 1 is thus transformed into the chain-scattering matrix description as in Figure 2.
Remark 4
If one rewrites G in Equation (14) as
G :"
xR "AK(x)#B(x);?(x)z w z w"CK (x)#DK(x);?(x) z wthen, from Assumptions A1}A4 and Lemma 3, G will be a (J, J )-lossless system if the following properties hold (an equivalent version of these properties can be found in Isidori [4]).
(i) One chooses ;?(x)"[' \'] such that ;?(x)2DK(x)2JDK(x);?(x)"J .
(ii) there exists a C nonnegative di!erentiable function < (x) (with < (0)"0) that is locally de"ned in a neighbourhood of the origin and <(x) and satis"es the Hamilton}Jacobi equation
<V(x)AK(x)#CK(x)JCK(x)"0 (16) such that
<V(x)B(x);?(x)#CK2(x)JDK(x);?(x)"0 (17) This also implies that the stabilizing state feedback gain F(x) can also be obtained from Equation (17). That is <V(x) B(x)#CK2(x)JDK(x)"0 N <V(x)B(x)#
C(x)0 #DK (x)F(x) 2 JDK (x)"0 NB2(x)<2V(x)#DK2(x)JC(x) 0 #DK 2(x)JDK(x)F(x)"0 NF(x)"!R\(x)B2(x)<2V(x)# 0 D2(x)C(x) NF(x) F(x)" B2(x)<2V(x) !B2(x)<2V(x)#D2(x)C(x) where R(x)"DK 2(x)JDK(x)"!IK 0 0 D2(x)D(x) and DK (x)" 0 I D0However, as stated on p.11 in Section 3.1, the existence of <2V(x) such that A(x)#B(x)F(x) is locally asymptotically stable corresponds to the Jacobian matrix of the Hamiltonian #ow associated with G*
JG (with z "0, w "0) at equilibrium belonging to dom(Ric). Direct computation yields that such a Jacobian matrix is as
H"
!CA2C BB2!BB2!A2which is equal to the Hamiltonian matrix &H' proposed by Doyle et al. [20]. 4.1. Local disturbance attenuation by measurement feedback
Before discussing the nonlinear output-feedback control problem, "rst consider the linear case. As proposed by Hong and Teng [7], the linear 4-block H controllers are obtained directly by inverting one of the (J, J )-coprime factors of G. That is, if the linear version of G in Equation (15) has an outer-(conjugate (J, J )-inner) factorization of G"%\# so that # is
conjugate (!J,!J )-lossless and both % and %\ are stable, then the linear 4-block H controllers can be described as K"F*(%I, '), i.e.
where #'#)1 and F*(),)) indicates left CSMD. De"nitions of left and right CSMD are reported in Reference [7].
For a nonlinear system, as shown in Crouch et al. [24], Scherpen et al. [19], and Van der Schaft [10, 11, 25], there locally exists an outer-(conjugate (J, J )-inner) factorization for G (Equa-tion (15)), assuming that there exists solu(Equa-tions of the relevant Hamilton}Jacobi equa(Equa-tions. However, the outer-(conjugate (J, J )-inner) factorization does not exist in nonlinear systems in general. Hence, it is natural to replace x by some estimate m provided by a proper auxiliary dynamics. One then "nds an appropriate nonlinear system% constructed by this estimate state m such that %G satis"es the conjugate (!J,!J )-lossless properties. This also implies that locally one has the outer-(conjugate (J, J )-inner) factorization for G.
Now, one rewrites G as
G:"
xR "AK(x)#B(x);?(x)z w u y"CI (x)#DI(x);?(x) z wand de"ne system% given as follows:
% :"
mQ"AK(m)#H(m)CI(m)#H(m)u y v p";X(m)CI(m)#;X(m) u y (18) wherem is an an estimate of x, H( ) )"[H() ) H() )], CI() )"CCI ())I ())"C() )#D( ) )F( ) )F() ) and c"'(p) ('(p) is a free stable system with #'(p)#*)1).
Therefore, the state-space representation of%G is given by
%IG:"
xR mQ" AK (x) AK (m)#H(m)CI(m)#H(m)CI(x)#H(B(x);?(x)m)DI(x);?(x) z w v p";X(m)[CI(x)#CI(m)]#;X(m)DI(x);?(x) z wRewrite it as %IG:"
xR C"AC(xC)#BC(xC)z w v p"CC(xC)#DC(xC) z w Remark 5From Assumptions A1}A4 and Theorem 1, if one chooses ;X(x)"[' '] such that DC(xC)JD2C(xC)";X(x)DI(x);?(x)J;2?(x)DI2(x);2X(x)"J
and if there exists a C non-negative function =(xC)"Q(x!m) locally de"ned in a neighbour-hood of (x,m)"(0, 0) with =(0)"0 and =(xC) satis"es the Hamilton}Jacobi equation
=VCAC(xC)!=VCBC(xC)JB2C(xC)=2VC"0 (19)
such that
C(xC)!DC(xC)JB2C(xC)=2VC"0 (20)
where
=VC"[=V(xC) =K(xC)]"[QV(x!m) QK(x!m)]
then %G will be a conjugate (!J,!J )-lossless system. Together with the properties in Equation (12), this also implies that
2
(#v(t)#!#p(t)#) dt)0 N
2(#z (t)#!#w (t)#) dt)0
As we know, from Ball and Van der Schaft [10, 17] if the corresponding Jacobian matrix of the Hamiltonian #ow associated with (GJG*)\ (with y?"0) at equilibrium belongs to dom(Ric), then there exists such =(xC)"Q(x!m) so that AC(xC) is locally asymptotically stable in a neighbourhood of (x,m)"(0, 0). Direct computation yields that such a Jacobian matrix indicated by A&Xis similar to the Hamiltonian matrix J given by Doyle et al. [20], where
J"
!ABB22 C2C!C2C!A )Furthermore, suppose ZK represents the solution of such A&
X, then, comparing Q(x!m) with ZK, it
follows that
ZK \"1 2
*Q *x
V and Q(x!m)"(x!m)2ZK\(x!m) is one of such solutions.For this reason, although the nonlinear function =(xC) can be expanded as =(xC)"Q(x!m)"(x!m)2ZK\(x!m)#O(x!m) locally de"ned in a neighbourhood of (x,m)"(0, 0) with O(0)"0 and =(xC) satis"es Equations (19) and (20), we take =(xC)"Q(x!m)"(x!m)2ZK\(x!m) as its quadratic approximation at the origin.
Since Q(x!m)"(x!m)2ZK\(x!m), and then [QV(x!m) QK(x!m)]"[QV !QV], one has the following derivation for the measurement feedback gain H(m). From Equation (20), one has
!
C(xC)#DC(xC)JB2C(xC)
!QQ2V2V "0N!(CI (x)#CI(m)#DI(x);?(x)J;2?(x)(B2(x)!DI2(x)H2(m))Q2V"0 and multiplying the right-hand side by Q\2
V , one obtains !(C (x)#CI(m))Q\2 V #DI (x);?(x)J;2?(x)(B2(x)!DI2(x)H2(m))"0 N!Q\ V (CI 2(x)#CI2(m))#B(x);?(x)J;2?(x)DI2(x)!H(m)DI(x);?(x)J;2?(x)DI2(x)"0 NH(m)DI(x);?(x)J;2?(x)DI2"!Q\ V (CI 2(x)#CI2(m))#B(x);?(x)J;2?(x)DI2(x) N [H(m) H(m)]"[Q\V (CM 2(x)#CM2(m))!B(x) Q\V (CI 2(x)#CI2(m))#B(x)D2(x)]RI\ where RI "
I 0 0 ! D(x)D2(x) " I 0 0 !I "JSpeci"cally, H(m)"[Q\V (CI 2(x)#CM2(m))#B(x)D2(x)]RI\ is the measurement feedback gain of the central controller as shown by Isidori [4].
As a summary of the discussion so far, we state the following theorem as a conclusion. Theroem 2
Under Assumptions A1}A4.
(i) Suppose there exists a C nonnegative function <(x) locally de"ned in a neighbourhood of the origin (with <(0)"0), and <(x) satis"es Equations (16), (17) such that G as shown in Remark 4 is (J, J )-lossless.
(ii) Suppose there exists a C nonnegative function =(xC)"Q(x!m) that is locally de"ned in a neighbourhood of (x,m)"(0, 0) and vanishes at (x, m)"(0, 0), and =(xC) satis"es Equations (19), (20) such that%G as shown in Remark 5 is conjugate (!J,!J )-lossless. Then, the problem of local disturbance attenuation with internal stability is solved by a family of output feedback controllers% (with a free stable system '(p)) as shown in Equation (18). A brief sketch of the proof From Equation (18), since v"'(p) with #'(p)#*
)1 and%G is
conjugate (!J,!J )-lossless, it immediately follows from Remark 5 that
2
(#v(t)#!#p(t)# dt)0 N
2(#z (t)#!#w (t)#) dt)0 Furthermore, as proposed in Remark 2, having G is (J, J )-lossless implies
2
(#z (t)#!#w (t)#) dt)0 N
2(#z(t)#!#w(t)# dt)0
This means that the ¸-gain of the closed-loop system (from w to z) is less than or equal to a prescribed numberc (c"1).
Figure 3.
To prove the internal stability, it su$ces to prove the exponential stability of the closed-loop system, which is obtained from the linear approximation result (see Hong and Teng [7]). )
Theorem 2 can be illustrated in Figure 3.
5. CONCLUSION
We have extended a class of the chain-scattering approach from the linear H control problem to the case of local disturbance attenuation with internal stability, via measurement feedback, in nonlinear a$ne systems. We have also stated the su$cient conditions for the existence of output feedback controllers. Because the "ctitious signals introduced by traditional chain-scattering approach are thought super#uous in the H control problem, a nonlinear outer-(conjugate (J, J )-inner) coprime factorization is proposed. As shown in the block diagrams, the nonlinear plant is thus described as serial energy-losslessness systems, which simpli"es the solving process and provides deeper insight in the synthesis of the controllers.
ACKNOWLEDGEMENT
To authors would like to thank the anonymous reviewers for their valuable comments which have made substantial improvements on this paper. This work was supported by the National Science Council, R.O.C., under contract NSC 89-2218-E238-001.
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