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Semiglobal stabilization of multi-input linear systems with saturated linear state feedback
Systems & Control Letters 23 (1994) 247-254 North-Holland
247
Semiglobal stabilization of multi-input linear systems with saturated linear state feedback Jos6 Alvarez-Ramirez,
Rodolfo
Smirez and Jesfis Alvarez
Divisibn de Ciencias Bhsicas e Ingenierla, Universidad Aut6noma Metropolitana - lztapalapa, Apdo. Postal 55-534, 09340, M~xico, D.F., Mexico
Received 31 May 1993 Revised 16 October 1993 Abstract: For multi-input linear systemswith eigenvaluesin the closed left-halfcomplex plane, we address the problem of stabilization by a linear state feedback subject to saturation. Using an eigenvalue-generalized eigenvector assignment technique, we prove that such
systems can be semigloballystabilized with a saturated linear state feedback. Based on this result, we propose an algorithm to calculate an e-parametrizedfamily of state feedback gain matrices that semigloballystabilize the system. Two examples are used to illustrate the results. Keywords: Multi-input linear systems; input saturation; semiglobal stabilization; eigenstructure assignment.
1. Introduction Recently, Sontag and Sussmann E11] proved that any linear system with eigenvalues in the closed left-half complex plane can be globally asymptotically (GA) stabilized by a bounded continuous feedback control. As a consequence of a result due to Fuller [3] (see also [14]) for n t> 3, the n th-order integrator chain cannot be GA stabilized by a saturated linear state feedback (SLSF). For certain cases a nonlinear stabilizing state feedback can be constructed using the ' n e s t e d s a t u r a t i o n s ' approach developed by Teel E15] and Yang et al. [17]. These results pose an interesting question: if a design with linear state feedback is chosen, what is the best attainable result? Motivated by Fuller's result, Smirez et al. [13] have shown that an nth-order integrator chain is semiglobally stabilizable (SGS) by an SLSF. By SGS we mean that, given any bounded set D ~ R", there exists a saturated linear control which makes D _ f2(0), O(0) being the region of attraction of the origin. In this work, we prove that linear multi-input systems with eigenvalues in the closed left-half complex plane can be semiglobally (SG) stabilized by means of an SLSF. To prove this result, we construct an e-parametrized family of SLSFs which asymptotically stabilize the system. As the parameter e ~ 0, the closed-loop system approaches the open-loop one, and its corresponding region of attraction tends to be all ~". Because in most practical problems one is interested in bounded regions, it is justified to have the SG stabilizability property instead of the GA stabilizability one. At first sight, one drawback of the semiglobal approach is that large stability regions imply slow dynamics (small gains). One way to overcome this problem is to use a gain-schedule scheme. For instance, in [12] a state-dependent gain is used to globally stabilize a linear system. The paper is organized as follows. First, we show why the eigenvalue placement idea, which sufficed to solve the single-input case [13], is not sufficient to solve the multi-input case. This will lead us to address
Correspondence to: R. Smlrez, Universidad Aut6noma Metropolitana - Iztapalapa, A.P. 55-534, C.P. 09000, Mexico, D.F., Mexico.
0167-6911/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI0167-691 I(93)E0139-8
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the multi-input case with the assignment of both eigenvalues and eigenvectors, obtaining an e-parametrized family of SLSFs which semiglobally stabilize the multi-input system. The indices of the open-loop eigenstructure are maintained in the controlled (closed-loop) system. Our proof is based on the geometric ideas developed in [13] and the eigenvalue-generalized eigenvector assignment presented in [2, 4, 5, 9]. For the sake of applicability of our results, we propose an algorithm to obtain the state feedback gain matrix.
2. Statement of the problem Consider the controllable linear system x e E ", u e ~ m, m > / 2 .
=Ax+Bu,
(2.1)
Controllability implies the existence of a stabilizing linear state feedback u = G x (u s = Gjx, 1 <~j <~ m) which locates the eigenvalues of the closed-loop matrix Ac = A + B G in the open left-half complex plane. That is, a(Ac) c C-. Suppose that the input vector u is constrained to take values in a compact set Q = [ u { , u ? ] × •. • x [u;,,u+~], where u f < 0 and u f > O, 1 <<.j <<.m, are the lower and upper limits of the j t h input. Then, the closed-loop system becomes (2.2)
~: = A x + B S ~ ( G x ) ,
where 6a = (6el . . . . .
5e~(uj) =
6era) : R m -* Q c ~m is a saturation function given by
uf u~
if u s < u f , if u 7 <~ u s <~ u +,
uf
if uj > u f
I
(2.3)
for 1 ~/3, there does not exist a gain matrix G which globally asymptotically stabilizes the saturated system (2.2). Let us introduce the following definition of semiglobal stabilization (potentially globally stabilization [1 ]) by linear saturated feedback.
Definition 2.1. The system (2.1) is semiglobally stabilizable (SGS) by SLSF if, for each bounded set D c R ", there exists a gain matrix Go such that u = G o x is a local stabilizer and D is contained in the region of attraction of the origin f2(0) of system (2.2). In this work, our main objective is to prove that the system (2.1) with eigenvalues in the closed left-half complex plane is SGS by SLSF.
Remark 2.2. Since 6¢j(Gx) = G x in a neighborhood of the origin, it is clear that the origin is a locally asymptotically stable equilibrium point of the closed-loop saturated system (2.2).
3. Preliminaries Here we introduce the nomenclature and some geometric features of the saturation regions in •". 3.1. Region o f nonsaturation
For each input us, 1 ~
v; (u;)=
R": 6ix > u ; ( < u ; ) } ,
u°=
u;
6ix
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J. Alvarez-Ramirez et al. / Semiolobal stabilization of multi-input linear systems
Therefore, the saturation function 5e: ~ " ~ Q c R ~ induces a partition of R" into 3" regions given by the intersection of the regions U °, U f and U j-, 1 ~
J At U o, the system (2.2) behaves as the nonsaturated linear stable system 2 = A ~x, and therefore the origin is the unique equilibrium point of (2.2) at U °. The boundaries between the sets U f , U j- and U °, 1 ~
r? (r;)
= { x e R " : C;x = u ? ( u ; ) } ,
where F + and F 7 are referred to as the j t h upper and lower saturation boundaries. Let us consider the following Euclidean distances for each 1 ~
p+(u °) =
dist(rj+, 0),
p-(U °) =
dist(rj-, 0).
It is clear that if u f = - u f , p + ( U °) = p - ( U ° ) . The internal radius of the set U °, say p ( U ° ) , is given by p ( U °) = m i n {p + (U°a ), p - ( U ° ) . . . . .
(3.1)
p +(U°), p- (U°) } .
3.2. Eigenstructure o f the m a t r i x A
F r o m [-6], recall the following definitions and results. Assume that the matrix A has ~ ~< n blocks of order X , in its Jordan canonical form (JCF). Let I D = {dk, 1 ~< k ~< o,U}. Suppose that r and s (r + s = J g ) of such blocks are related to real and complex eigenvalues, respectively. Then, the matrix A has J g eigenvalues: J,k = ~ k Jr- (-- 1)k(-oki6C for 1 ~< k ~< s and ,~keR for s + 1 ~< k ~< o,U (i = ~ - 1 ) , where s is even and for each odd k < s, ~k = ~k+l, Ogk = ~Ok+X. Let A = {2k: 1 ~< k ~< ~ } be the set of such eigenvalues (not necessarily distinct). Associated with A, there are o,U chains of generalized eigenvectors which satisfy dk, k = i . . . . .
(3.2)
At)k,i ~- 2kVk, i "k- Vk, i - X
for 1 ~< k ~< ~ and 1 ~< i ~< dk, where Vk, O = 0. The eigenstructure indices of the matrix A are defined as follows: EI(A) = { ~ , r, s; dl . . . . , dk}. For each odd k < s, we have that 2k = 2k+ * ~, where * denotes complex conjugation. Then Vk, i = Vk+ ~,~ = Pk, i + qk, ii, for 1 <~ i <~ dk, where Pk, i, qk, i~ R". The set of real vectors {PR, 1. . . . .
Pk,nk; qk, 1. . . . .
qk, d~: for all odd k < s; v s + L 1 , . . . , vs+l,ds+,; ' ' ' ;
V~,I . . . . .
VX,dk}
is a basis of R".
4. Main results The problem of the existence of a feedback which stabilizes a system with eigenvalues in the closed left-half complex plane can be reduced to the problem of finding a state feedback which relocates only the eigenvalues on the imaginary axis (see [-11, 13]), so that, without loss of generality, we may assume that all the eigenvalues of the matrix A are located on the imaginary axis. Let A ° be the set of eigenvalues of A, given by A°={2°}k~=x={2 °=(-1)ka~kiforl~
°=0fors+l~
Given ~ > 0, let E = [0, ~], and consider the following symmetric set of complex numbers A¢(e) = {~.k(e))k= 1 , e s ~ E, with the following properties: (a) A¢(0) = A°; (b) Re(2~,(e)) < 0 for all e ¢ E , and 1 ~< k ~< Jg; (c) 2~,(e) as a function of e is C ~ for all 1 ~< k ~< J~g.
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J. Alvarez-Ramirez et al. / Semiglobal stabilization of multi-input linear systems
Remark 4.1. A simple way to define a set satisfying properties (a)-(c), is the following: let {],~,} be a set of negative numbers and define 2~,(e) = e2~ + 09ki for all 1 ~< k ~< s and 2~(e) = E2~ for s + 1 <~ k ~< Jg.
Proposition 4.2. Let (A, B) be a controllable pair. Assume that the set A¢(e) satisfies properties (a)-(c). Then there exists a positive number ~ > O, and a state-feedback matrix G(e) such that the closed-loop matrix A¢(E) = A + BG(E) has eigenvalues A¢(E) with block indices ID = {dk, 1 <~ k <<,~ }, and [IG(E) II ~< ~ IIe II, for all ere. Proof. Let k = {1,2 . . . . . ~ } and dk = {1,2 . . . . . dk} for all k~k. For each k e k , choose a set of vectors {Uki(e), i e dk} satisfying Uki(E) = eOkl, where Okl is any constant vector such that [I Oki II ~ 0. Calculate the vectors v~(e) defined by the chains (2~,(e)I -- A)v~i(E)- V ~ , i - I ( 8 ) = BUki(8 ),
(4.1)
for all i t d k and k e k , where V~o(e)= 0. As in [2,5], the existence of a real matrix G(e) such that G(e)v~i(e) = Uu(e) for all i t d k and k t k can be proved. F r o m (4.1), it follows that G(e) also satisfies [Z~,(e)I
-
A
-
BG(E)]v~,(e)
- v~,,,_ ~ (e) = O.
(4.2)
This implies that the vectors v~a(e), i e dk and k t k, are the generalized eigenvectors of the closed-loop matrix Ac(e) associated to the set of eigenvalues A~(e). It can be seen [-2, 9] that G(E) is given by
G(E) = ~g(e)~3¢(e)- 1, where ~/(e) = e ~ and Me(e) are matrices whose column vectors are a linear combination of the Ok~'S and of the v~a(E)'s, respectively. The column vectors of @~(e) are a basis of R" (in fact, the column vectors of ~ ( e ) span the invariant subspaces associated with A~(E)). For e = 0, (4.1) and (4,2) define the generalized eigenvectors associated with the open-loop eigenvalues A °, so that @¢(0)- t exists. Since @¢(e)- ~ exists for all e t E and changes continuously, and E is compact, fl = m a x , ~ 11~¢(~)-1 II is well defined. Finally, we have that I[G(~)II ~< ~ [I~7 I[ft. [] Corollary 4.3. Let x = T(E)z be a linear e-dependent transformation of coordinates which changes continuously with e. Then, for e r e , IIa'(E)ll ~< ~'llEII, where G'(E) = G(~)T(E) and p' = ~ s u p o < , ~ II T(e)LI. []
Corollary 4.4. The internal radius of the nonsaturated region U°(e), p(U°(e)), diverges to oo when e converges to zero.
Proof. Let G(e)= (GI(E) . . . . .
Gin(E)). F r o m Proposition 4.2,
p+ (U°(E)) = u~/IL aj(e)LI >1 u~l~ IkE II, p - (U°(E)) = u j-/II Gj(E)[I >~ u;/e [IEII for all i ~
p(U°(e)) >t min {u;, lug-I}/e lit II. i
Finally, it follows that p(U°(e)) ~ ~ as IEI --, 0. By assigning eigenvalues and generalized eigenvectors that maintain the eigenstructure indices (i.e. EI(A¢(e)) = El(A)), the state-feedback matrix G(e) satisfies the property G(0) = 0 and this implies At(0) = A. That is, as e ~ 0, the closed-loop system tends to the open-loop one (with input u = 0). This is a crucial property for the validity of the main result. With the following example, we show why placement of eigenvalues, which sufficed to solve the single-input case [1 3], it is not sufficient to solve the multi-input case.
J. Alvarez-Ramirez et al. / Semiolobal stabilization of multi-input linear systems
251
Example 4.5. Consider the harmonic oscillator with two control inputs: )¢1 -~" --0~X2 @ Ul,
X2 = ~X~ + U2,
lutl, [u21 ~ 1.
The open-loop eigenvalues are A ° = { ~ i , - ~ i } . The state feedback u ~ ( x ) = ( 1 +or)x2, u 2 ( x ) = _ ( ~ + ~2 + eZ)x a _ 2ex2 places the closed-loop eigenvalues at A¢(e) = { - e + ~i, - e -- ~i}. At e = 0 the closed-loop system becomes x l = x2, x2 = - ~2Xx, and p(U °) = min{1/(1 + ~), 1/(~ + ~2)}. Although A¢(0) = A °, the closed-loop system (2.3) has bounded internal radius p(U°(e)) for all e e E. Theorem 4.6. Consider a controllable multi-input system (2.1) with eigenvalues in the closed left-half complex plane. The system (2.1) is SGS by SLSF. Proof. Without loss of generality, let us assume that the matrix A has its eigenvalues on the imaginary axis. Consider a state-feedback function u(x)= G(e)x constructed as in Proposition 4.2 and such that the closed-loop matrix Ac(e) has eigenvalues A°(e) as given in Remark 4.1. Let f2~(0) be the region of attraction of the origin of the system (2.2) for each e~E, and ~ ~ ~" be any bounded set. T o prove that (2.1) is SGS, it suffices to prove that there exists an e e E such that ~ ~ f2~(0). Consider a Jordan-like coordinate transformation x = T(e)z, e ~ E, such that, II T(~) - x II = 1 and, the closed-loop matrix de(e) = T(e)- x(A + BG(e)) T(e) is written in the form [10] -dl
dc(
A,
) =
rI)1
..
r o ]
where
Aj =
=
.
C~ Ia~ Cj I&
0 .
0
Cj
]
1 ~
~,
1 ~ j ~ s, ~j ~ fir2aj × I~ 2a~,
Cj =
where 6~ > O. Consider the following L y a p u n o v function (independent of e):
V(z) = IIz II2/2.
(4.3)
Then, the level sets LS(y) = {z E •": V(z) ~< V} are balls of radius (2V)1/2. The time derivative of (4.3) along the vector field Acz, is given by - l?(z) = ztP(e)z, where P(e) = -(see(e) + dc(e)t). With the following choice of 38: 6, = e min {Re(2~/dj)}, J
J. Alvarez-Ramirez et al. / Semiglobal stabilization of multi-input linear systems
252
the principal minors of P (e) are positive. Hence Silvester's conditions [-6, 7] are satisfied and the matrix P (e) is positive definite for all e ~ E, so that, along the trajectories generated by the vector field ~¢(e)z, I?(z) < 0 for all z ~ 0 . Now, we will show that there exist e ~ E and ~ > 0 such that ~ ' = L S ( 7 ) ~ U °', where ~ ' = T(e)- ~(~) and U °'(e) = T(e)- ~(U 0 (e)). Let p~ (~) = max { IIz II : z ~ ~ } be the radius of the set ~. F r o m Corollaries 4.3 and 4.4, p(U o,(e)) diverges to ~ when e converges to zero. Since p ~ ( ~ ' ) ~< p~(~), and LS(y) are balls centered at the origin, there exist positive numbers e and 7 such that ~ ' ~ LS(y) ~ U°'(e). On the other hand, in U°'(e) the system behaves as the nonsaturated system ~ = ~ ( e ) z , from where LS(y) is an invariant set under the flow exp(t~cc(e)). Therefore @' ~ f2~(0)', where f2~(0)' is the region of attraction of (2.2) in z-coordinates. Since T(e) is a global (linear) change of coordinates, ~ = f2,(0). [] While revising this work, we learned about a preliminary version of [8] in which Theorem 4.6 is proved along different lines.
5. An algorithm to calculate G(e) The evaluation of G(e) by simultaneous placement of eigenvalues and generalized eigenvectors is not an easy numerical problem. Let us explore an alternative procedure which is based on the idea of recovering the open-loop system (2.1) with input u = 0 when e, ~ 0 (Proposition 4.2). The following algorithmic procedure focuses on the evaluation of a gain matrix G(e) that satisfies the following property (Theorem 4.6): IIG(~)II ~< ~tl~ll
for all e E E .
(5.1)
(i) Using any performance design procedure, fix the eigenvalues ;S~,+ COki with 2~ < 0 and ~Ok = 0 for s + 1 ~< k ~ o~4£,where the ~k'S are the imaginary parts of the open-loop eigenvalues. Take the set At(e) of closed-loop eigenvalues as in Remark 4.1: 2~(e) = e2~, + Ogki for 1 ~< k ~< s and 2~,(~) = E2~, for s + 1 ~< k ~< ~'. (ii) Calculate the characteristic polynomial P,(2;e) associated to the set At(e). Once we have P,(2; e) = ~ = o a)(e) 2 j with a,¢(e) -- 1, one can see that the coefficients {a)(e)}~=o are polynomial functions of e which satisfy a~(~) = aj0 + ~j(~),
(5.2)
where {ajo }j=o , are the coefficients of the characteristic polynomial associated to the set A ° of open-loop eigenvalues and ~j(~) are polynomial functions of e, for all 0 ~
(5.3)
where (~(e) is a continuous function of e. Let P~(2; G, e) = ~y= ~ s~((~, ~)2 j be the characteristic polynomial associated to the closed-loop matrix A¢(e) = A + eBd(e). Since e multiplies at least one row of the matrix A¢(e),sj(d, e) is given by
s j ( G , e ) = a j o +8gj(G,e),
1 <~j <<.n.
(5.4)
(iv) Making the coefficients (5.2) of P,(2) equal to the coefficients (5.4) of P,(2; tJ, ~), one obtains a set of nonlinear algebraic (in fact, polynomial) equations with unknowns {t~(e)jl}, 1 ~
(5.5)
Since (~(e) ~ ~ " x ~" ~ ~ ' " , then (5.4) is a set of n equations with nm unknowns. For m = 1, (5.5) is a square system which, from the controllability of (A, B), has a unique solution. For m >/2 one has, in general, n(m - 1) degrees of freedom, which must be specified (using any suitable criteria) to obtain a square system. We propose the following procedure to fix n(m - 1) variables in equation (5.5): (a) Try a partition of (~(e) = (t~ (~), t~(e)), t~ (e) ~ I/~", (~(e) ~ ~ " - ~)", and fix (7~(~) as a constant value (~*. (b) The square system s(G~; G~', 0) - ~,(0) = 0 has at least one solution (~* (in fact, (~' = 0 is a solution).
J. Alvarez-Ramirez et al. / Semi#lobal stabilization of multi-input linear systems
253
If the d e r i v a t i v e Ds(e,)/DG1[(~ = GLe =0) is nonsingular, the implicit function theorem implies the existence of 0 < e* ~< ~ such that the solution G1 (0) can be extended to all e e [0, e*]. It is clear that if such solution G(e) exists, G(e) satisfies property (5.1). (c) If the selected partition of G(e) does not satisfy the derivative requirement, try another partition. Example 5.1. Let us construct a family of state feedbacks which SG stabilize the harmonic oscillator of Example 4.5. Take A~(e)= { - e - ~i, - e + ~i}. Then the polynomial associated to A~(e) is P,(2; ~)= A2 + 2e2 + e 2 + ~2; therefore, yo(e) = 1, 71(e) = - 2 and y2(e) = e. If uj(x) = e(81xXl + 0i2x2), the characteristic polynomial of the matrix A~(e) = A + eBG is given by /~,(2; G, e) = 25 + s~(6, e)2 + So(6, e), where s I ( G , e) = --e_.(011 "-~ 022) and so(G, e) = e 2 0 1 1 0 2 2 --F eo~021 + e2812021 - eo~012 -~ o~2. From (5.4) and (5.5), the gain matrix G must satisfy the following set of algebraic equations: 011 + 022 = - 2 ,
(5.6)
~811022 + ~851 + e#ls#s1 - ~812 = e. For e = 0, this system becomes 011 + g22 = - 2 ,
021 -- g12 = 0 .
Let us try the partition (71 = (011,021), G2 = (812,822). ThenDs(e)/DG11(G*., = o~ = I, which is nonsingular. Therefore, the solution (~l(0) can be extended for e~E*. If G* = (1, 1), the solution of system (5.6) is 011(e) ~--- --3e,
012(e) = e,
021(e) = e(4e + 00/(0~ - e),
022(e) = e.
The corresponding matrix G(e) satisfies property (5.1) for 0 ~< e < e* < ~.
[0101 [, 01
Example 5.2. Consider the control system ~ = A x + Bu, B = [B1,B2], x E ~3, u E •2, where A=
-1
0 1 0
0 0
,
B=
1 1
1
.
(5.7)
-1
The matrix A has eigenvalues A ° = {0, +i, --i}. Thus, the system (5.7) is SGS via SLSF. The characteristic polynomial P°(2) associated to the eigenvalues A ° is given by P°(2) = 2 a + 2. Let us choose the dosed-loop eigenvalues as At(e) = { - e , - e + i, - e - i}, e > 0. Then the characteristic polynomial associated to A c(e) is given by Pn(2)=A 3 + 3 e 2 2 + ( 1 + 3 e E ) 2 + e 3 + e . Therefore ~,o(e)--1 + e 2, 7 i ( e ) = 3 e , ~'2(e)=3 and ~3(e) = 0. After calculating the coefficients {g~(6, e)}~.-ot associated to the polynomial /-~n(2; t~, e), the following set of three equations for the elements of G is obtained (system (5.4)): g l l "[- g12 "[- 022 "[- 013 - - 023 = - 3 , E(012 "~ 0 2 2 ) ( 0 1 3 -- 0 2 3 ) "~ e 0 1 1 ( 0 1 2 "~- 022) "~ 8 0 1 1 ( 0 1 3 -- 023) - - (I -[- E011 -- g 0 2 1 ) 0 1 3 - - e(012 -- 0 2 2 ) ( 0 1 3 -- 023) -- (1 "4- '~012)(011 + 021) -- 012 = 3e, E 2 0 1 1 ( 0 1 2 + 0 2 2 ) ( 8 1 3 - - 023) "4- /~013(Egl1 + E021 -- 1)(812 - - g22)
(5.8)
"q- (l "['- 8 0 1 2 ) ( 0 1 3 -- 0 2 3 ) ( 1 + /~011 -- e 0 2 1 ) - - g 0 1 3 ( 0 1 2 "q- 0 2 2 ) ( 1 + e011 - - g021) -- /32011(012 - - 0 2 2 ) ( 0 1 3 - - 023)
-- (1 + e012)(,~011 "~ e821 - -
1)(813 --
g23) = 1 + e 2.
Let us try the following partition: G1 = (011,812,053) and (~5 = (813,021,822)- Fix G2 to some constant value G*. At e = 0, the system (5.8) is reduced to the set of linear equations 011 +812 - 023 = - 3 811 ~ -- 013 - - 2 0 1 2 , 2 0 2 3 = 2 8 1 3 -- 1.
0"2-0"3, (5.9)
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J. Alvarez-Ramirez et al. / Semiglobal stabilization of multi-input linear systems
This set of equations has a unique solution (~1. On the other hand, one can see from (5.8) that Ds(e)/D(J1116,,= = 0)is equal to the matrix constructed with the left-hand side of (5.9) ( in fact, all the terms of (5.8) having e vanish). Therefore, Ds(e)/D(~lll61 =8",~=o) is nonsingular and, for all e e E the solution (~1 (e = 0) can be extended.
References [1] A. Baccioti, Potential global stabilizability, IEEE Trans. Automat. Control AC-31 (1986) pp. 974-976. [2] C.T. Chen, Introduction to Linear System Theory (Holt, Reinhart and Winston, New York, 1970). [3] A.T. Fuller, On the large stability of relay and saturated control systems with linear controllers, lnternat. J. Control 10 (1969) 457-480. [4] T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, NJ, 1980). [5] G. Klen and B.C. Moore, Eigenvalue-generalized eigenvector assignment with state feedback. IEEE Trans. Automat. Control (1977) 140-141. [6] P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic Press, New York, 1985). 17] S. Lefschetz, Stability of Nonlinear Control Systems (Academic Press, New York, 1965). [8] Z. Lin and A. Saberi, Semi-global exponential stabilization of linear systems subject to input saturation via linear feedbacks, Systems Control Lett., to appear. [9] B.C. Moore, On the flexibility offered by state feedback in multivariable systems beyond closed loop eigenvalue assignment, IEEE Trans. Automat. Control (1976) 689-692. [10] J. Palis Jr. and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction (Springer, New York, 1982). [11] E.D. Sontag and H.J. Sussmann, Nonlinear output feedback design for linear systems with saturating controls, Proc. 29th IEEE Conf. Dec. Control (1990) 3414-3416. [12] R. Su~irez,J. Solis and J. Alvarez, Stabilization of linear controllable systems by means of bounded continuous nonlinear feedback control, Systems Control Lett., to appear. [13] R. Su~irez, J. Alvarez and J. Alvarez, Stability regions of closed loop linear systems with saturated linear feedback Internal Report CBI. UAM-Iztapalapa (1991). [14] H.S. Sussmann and Y. Yang, On the stabilizability of multiple integrators by means of bounded feedback controls, Proc. 30th IEEE Conf. Dec. Control (1991) 70-73. [15] A.R. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls, Systems Control Lett. 18(3) (1992) 165-171. [16] Y.Ch. Yang Lu, Singularity Theory (Springer, New York, 1976). [17] Y. Yang, HJ. Sussmann and E.D. Sontag, Global stabilization of linear systems with bounded feedback, Proc. IFAC Nonlinear Control Systems Design Symposium (1992) 15-20.
247
Semiglobal stabilization of multi-input linear systems with saturated linear state feedback Jos6 Alvarez-Ramirez,
Rodolfo
Smirez and Jesfis Alvarez
Divisibn de Ciencias Bhsicas e Ingenierla, Universidad Aut6noma Metropolitana - lztapalapa, Apdo. Postal 55-534, 09340, M~xico, D.F., Mexico
Received 31 May 1993 Revised 16 October 1993 Abstract: For multi-input linear systemswith eigenvaluesin the closed left-halfcomplex plane, we address the problem of stabilization by a linear state feedback subject to saturation. Using an eigenvalue-generalized eigenvector assignment technique, we prove that such
systems can be semigloballystabilized with a saturated linear state feedback. Based on this result, we propose an algorithm to calculate an e-parametrizedfamily of state feedback gain matrices that semigloballystabilize the system. Two examples are used to illustrate the results. Keywords: Multi-input linear systems; input saturation; semiglobal stabilization; eigenstructure assignment.
1. Introduction Recently, Sontag and Sussmann E11] proved that any linear system with eigenvalues in the closed left-half complex plane can be globally asymptotically (GA) stabilized by a bounded continuous feedback control. As a consequence of a result due to Fuller [3] (see also [14]) for n t> 3, the n th-order integrator chain cannot be GA stabilized by a saturated linear state feedback (SLSF). For certain cases a nonlinear stabilizing state feedback can be constructed using the ' n e s t e d s a t u r a t i o n s ' approach developed by Teel E15] and Yang et al. [17]. These results pose an interesting question: if a design with linear state feedback is chosen, what is the best attainable result? Motivated by Fuller's result, Smirez et al. [13] have shown that an nth-order integrator chain is semiglobally stabilizable (SGS) by an SLSF. By SGS we mean that, given any bounded set D ~ R", there exists a saturated linear control which makes D _ f2(0), O(0) being the region of attraction of the origin. In this work, we prove that linear multi-input systems with eigenvalues in the closed left-half complex plane can be semiglobally (SG) stabilized by means of an SLSF. To prove this result, we construct an e-parametrized family of SLSFs which asymptotically stabilize the system. As the parameter e ~ 0, the closed-loop system approaches the open-loop one, and its corresponding region of attraction tends to be all ~". Because in most practical problems one is interested in bounded regions, it is justified to have the SG stabilizability property instead of the GA stabilizability one. At first sight, one drawback of the semiglobal approach is that large stability regions imply slow dynamics (small gains). One way to overcome this problem is to use a gain-schedule scheme. For instance, in [12] a state-dependent gain is used to globally stabilize a linear system. The paper is organized as follows. First, we show why the eigenvalue placement idea, which sufficed to solve the single-input case [13], is not sufficient to solve the multi-input case. This will lead us to address
Correspondence to: R. Smlrez, Universidad Aut6noma Metropolitana - Iztapalapa, A.P. 55-534, C.P. 09000, Mexico, D.F., Mexico.
0167-6911/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI0167-691 I(93)E0139-8
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the multi-input case with the assignment of both eigenvalues and eigenvectors, obtaining an e-parametrized family of SLSFs which semiglobally stabilize the multi-input system. The indices of the open-loop eigenstructure are maintained in the controlled (closed-loop) system. Our proof is based on the geometric ideas developed in [13] and the eigenvalue-generalized eigenvector assignment presented in [2, 4, 5, 9]. For the sake of applicability of our results, we propose an algorithm to obtain the state feedback gain matrix.
2. Statement of the problem Consider the controllable linear system x e E ", u e ~ m, m > / 2 .
=Ax+Bu,
(2.1)
Controllability implies the existence of a stabilizing linear state feedback u = G x (u s = Gjx, 1 <~j <~ m) which locates the eigenvalues of the closed-loop matrix Ac = A + B G in the open left-half complex plane. That is, a(Ac) c C-. Suppose that the input vector u is constrained to take values in a compact set Q = [ u { , u ? ] × •. • x [u;,,u+~], where u f < 0 and u f > O, 1 <<.j <<.m, are the lower and upper limits of the j t h input. Then, the closed-loop system becomes (2.2)
~: = A x + B S ~ ( G x ) ,
where 6a = (6el . . . . .
5e~(uj) =
6era) : R m -* Q c ~m is a saturation function given by
uf u~
if u s < u f , if u 7 <~ u s <~ u +,
uf
if uj > u f
I
(2.3)
for 1 ~
Definition 2.1. The system (2.1) is semiglobally stabilizable (SGS) by SLSF if, for each bounded set D c R ", there exists a gain matrix Go such that u = G o x is a local stabilizer and D is contained in the region of attraction of the origin f2(0) of system (2.2). In this work, our main objective is to prove that the system (2.1) with eigenvalues in the closed left-half complex plane is SGS by SLSF.
Remark 2.2. Since 6¢j(Gx) = G x in a neighborhood of the origin, it is clear that the origin is a locally asymptotically stable equilibrium point of the closed-loop saturated system (2.2).
3. Preliminaries Here we introduce the nomenclature and some geometric features of the saturation regions in •". 3.1. Region o f nonsaturation
For each input us, 1 ~
v; (u;)=
R": 6ix > u ; ( < u ; ) } ,
u°=
u;
6ix
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J. Alvarez-Ramirez et al. / Semiolobal stabilization of multi-input linear systems
Therefore, the saturation function 5e: ~ " ~ Q c R ~ induces a partition of R" into 3" regions given by the intersection of the regions U °, U f and U j-, 1 ~
J At U o, the system (2.2) behaves as the nonsaturated linear stable system 2 = A ~x, and therefore the origin is the unique equilibrium point of (2.2) at U °. The boundaries between the sets U f , U j- and U °, 1 ~
r? (r;)
= { x e R " : C;x = u ? ( u ; ) } ,
where F + and F 7 are referred to as the j t h upper and lower saturation boundaries. Let us consider the following Euclidean distances for each 1 ~
p+(u °) =
dist(rj+, 0),
p-(U °) =
dist(rj-, 0).
It is clear that if u f = - u f , p + ( U °) = p - ( U ° ) . The internal radius of the set U °, say p ( U ° ) , is given by p ( U °) = m i n {p + (U°a ), p - ( U ° ) . . . . .
(3.1)
p +(U°), p- (U°) } .
3.2. Eigenstructure o f the m a t r i x A
F r o m [-6], recall the following definitions and results. Assume that the matrix A has ~ ~< n blocks of order X , in its Jordan canonical form (JCF). Let I D = {dk, 1 ~< k ~< o,U}. Suppose that r and s (r + s = J g ) of such blocks are related to real and complex eigenvalues, respectively. Then, the matrix A has J g eigenvalues: J,k = ~ k Jr- (-- 1)k(-oki6C for 1 ~< k ~< s and ,~keR for s + 1 ~< k ~< o,U (i = ~ - 1 ) , where s is even and for each odd k < s, ~k = ~k+l, Ogk = ~Ok+X. Let A = {2k: 1 ~< k ~< ~ } be the set of such eigenvalues (not necessarily distinct). Associated with A, there are o,U chains of generalized eigenvectors which satisfy dk, k = i . . . . .
(3.2)
At)k,i ~- 2kVk, i "k- Vk, i - X
for 1 ~< k ~< ~ and 1 ~< i ~< dk, where Vk, O = 0. The eigenstructure indices of the matrix A are defined as follows: EI(A) = { ~ , r, s; dl . . . . , dk}. For each odd k < s, we have that 2k = 2k+ * ~, where * denotes complex conjugation. Then Vk, i = Vk+ ~,~ = Pk, i + qk, ii, for 1 <~ i <~ dk, where Pk, i, qk, i~ R". The set of real vectors {PR, 1. . . . .
Pk,nk; qk, 1. . . . .
qk, d~: for all odd k < s; v s + L 1 , . . . , vs+l,ds+,; ' ' ' ;
V~,I . . . . .
VX,dk}
is a basis of R".
4. Main results The problem of the existence of a feedback which stabilizes a system with eigenvalues in the closed left-half complex plane can be reduced to the problem of finding a state feedback which relocates only the eigenvalues on the imaginary axis (see [-11, 13]), so that, without loss of generality, we may assume that all the eigenvalues of the matrix A are located on the imaginary axis. Let A ° be the set of eigenvalues of A, given by A°={2°}k~=x={2 °=(-1)ka~kiforl~
°=0fors+l~
Given ~ > 0, let E = [0, ~], and consider the following symmetric set of complex numbers A¢(e) = {~.k(e))k= 1 , e s ~ E, with the following properties: (a) A¢(0) = A°; (b) Re(2~,(e)) < 0 for all e ¢ E , and 1 ~< k ~< Jg; (c) 2~,(e) as a function of e is C ~ for all 1 ~< k ~< J~g.
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Remark 4.1. A simple way to define a set satisfying properties (a)-(c), is the following: let {],~,} be a set of negative numbers and define 2~,(e) = e2~ + 09ki for all 1 ~< k ~< s and 2~(e) = E2~ for s + 1 <~ k ~< Jg.
Proposition 4.2. Let (A, B) be a controllable pair. Assume that the set A¢(e) satisfies properties (a)-(c). Then there exists a positive number ~ > O, and a state-feedback matrix G(e) such that the closed-loop matrix A¢(E) = A + BG(E) has eigenvalues A¢(E) with block indices ID = {dk, 1 <~ k <<,~ }, and [IG(E) II ~< ~ IIe II, for all ere. Proof. Let k = {1,2 . . . . . ~ } and dk = {1,2 . . . . . dk} for all k~k. For each k e k , choose a set of vectors {Uki(e), i e dk} satisfying Uki(E) = eOkl, where Okl is any constant vector such that [I Oki II ~ 0. Calculate the vectors v~(e) defined by the chains (2~,(e)I -- A)v~i(E)- V ~ , i - I ( 8 ) = BUki(8 ),
(4.1)
for all i t d k and k e k , where V~o(e)= 0. As in [2,5], the existence of a real matrix G(e) such that G(e)v~i(e) = Uu(e) for all i t d k and k t k can be proved. F r o m (4.1), it follows that G(e) also satisfies [Z~,(e)I
-
A
-
BG(E)]v~,(e)
- v~,,,_ ~ (e) = O.
(4.2)
This implies that the vectors v~a(e), i e dk and k t k, are the generalized eigenvectors of the closed-loop matrix Ac(e) associated to the set of eigenvalues A~(e). It can be seen [-2, 9] that G(E) is given by
G(E) = ~g(e)~3¢(e)- 1, where ~/(e) = e ~ and Me(e) are matrices whose column vectors are a linear combination of the Ok~'S and of the v~a(E)'s, respectively. The column vectors of @~(e) are a basis of R" (in fact, the column vectors of ~ ( e ) span the invariant subspaces associated with A~(E)). For e = 0, (4.1) and (4,2) define the generalized eigenvectors associated with the open-loop eigenvalues A °, so that @¢(0)- t exists. Since @¢(e)- ~ exists for all e t E and changes continuously, and E is compact, fl = m a x , ~ 11~¢(~)-1 II is well defined. Finally, we have that I[G(~)II ~< ~ [I~7 I[ft. [] Corollary 4.3. Let x = T(E)z be a linear e-dependent transformation of coordinates which changes continuously with e. Then, for e r e , IIa'(E)ll ~< ~'llEII, where G'(E) = G(~)T(E) and p' = ~ s u p o < , ~ II T(e)LI. []
Corollary 4.4. The internal radius of the nonsaturated region U°(e), p(U°(e)), diverges to oo when e converges to zero.
Proof. Let G(e)= (GI(E) . . . . .
Gin(E)). F r o m Proposition 4.2,
p+ (U°(E)) = u~/IL aj(e)LI >1 u~l~ IkE II, p - (U°(E)) = u j-/II Gj(E)[I >~ u;/e [IEII for all i ~
p(U°(e)) >t min {u;, lug-I}/e lit II. i
Finally, it follows that p(U°(e)) ~ ~ as IEI --, 0. By assigning eigenvalues and generalized eigenvectors that maintain the eigenstructure indices (i.e. EI(A¢(e)) = El(A)), the state-feedback matrix G(e) satisfies the property G(0) = 0 and this implies At(0) = A. That is, as e ~ 0, the closed-loop system tends to the open-loop one (with input u = 0). This is a crucial property for the validity of the main result. With the following example, we show why placement of eigenvalues, which sufficed to solve the single-input case [1 3], it is not sufficient to solve the multi-input case.
J. Alvarez-Ramirez et al. / Semiolobal stabilization of multi-input linear systems
251
Example 4.5. Consider the harmonic oscillator with two control inputs: )¢1 -~" --0~X2 @ Ul,
X2 = ~X~ + U2,
lutl, [u21 ~ 1.
The open-loop eigenvalues are A ° = { ~ i , - ~ i } . The state feedback u ~ ( x ) = ( 1 +or)x2, u 2 ( x ) = _ ( ~ + ~2 + eZ)x a _ 2ex2 places the closed-loop eigenvalues at A¢(e) = { - e + ~i, - e -- ~i}. At e = 0 the closed-loop system becomes x l = x2, x2 = - ~2Xx, and p(U °) = min{1/(1 + ~), 1/(~ + ~2)}. Although A¢(0) = A °, the closed-loop system (2.3) has bounded internal radius p(U°(e)) for all e e E. Theorem 4.6. Consider a controllable multi-input system (2.1) with eigenvalues in the closed left-half complex plane. The system (2.1) is SGS by SLSF. Proof. Without loss of generality, let us assume that the matrix A has its eigenvalues on the imaginary axis. Consider a state-feedback function u(x)= G(e)x constructed as in Proposition 4.2 and such that the closed-loop matrix Ac(e) has eigenvalues A°(e) as given in Remark 4.1. Let f2~(0) be the region of attraction of the origin of the system (2.2) for each e~E, and ~ ~ ~" be any bounded set. T o prove that (2.1) is SGS, it suffices to prove that there exists an e e E such that ~ ~ f2~(0). Consider a Jordan-like coordinate transformation x = T(e)z, e ~ E, such that, II T(~) - x II = 1 and, the closed-loop matrix de(e) = T(e)- x(A + BG(e)) T(e) is written in the form [10] -dl
dc(
A,
) =
rI)1
..
r o ]
where
Aj =
=
.
C~ Ia~ Cj I&
0 .
0
Cj
]
1 ~
~,
1 ~ j ~ s, ~j ~ fir2aj × I~ 2a~,
Cj =
where 6~ > O. Consider the following L y a p u n o v function (independent of e):
V(z) = IIz II2/2.
(4.3)
Then, the level sets LS(y) = {z E •": V(z) ~< V} are balls of radius (2V)1/2. The time derivative of (4.3) along the vector field Acz, is given by - l?(z) = ztP(e)z, where P(e) = -(see(e) + dc(e)t). With the following choice of 38: 6, = e min {Re(2~/dj)}, J
J. Alvarez-Ramirez et al. / Semiglobal stabilization of multi-input linear systems
252
the principal minors of P (e) are positive. Hence Silvester's conditions [-6, 7] are satisfied and the matrix P (e) is positive definite for all e ~ E, so that, along the trajectories generated by the vector field ~¢(e)z, I?(z) < 0 for all z ~ 0 . Now, we will show that there exist e ~ E and ~ > 0 such that ~ ' = L S ( 7 ) ~ U °', where ~ ' = T(e)- ~(~) and U °'(e) = T(e)- ~(U 0 (e)). Let p~ (~) = max { IIz II : z ~ ~ } be the radius of the set ~. F r o m Corollaries 4.3 and 4.4, p(U o,(e)) diverges to ~ when e converges to zero. Since p ~ ( ~ ' ) ~< p~(~), and LS(y) are balls centered at the origin, there exist positive numbers e and 7 such that ~ ' ~ LS(y) ~ U°'(e). On the other hand, in U°'(e) the system behaves as the nonsaturated system ~ = ~ ( e ) z , from where LS(y) is an invariant set under the flow exp(t~cc(e)). Therefore @' ~ f2~(0)', where f2~(0)' is the region of attraction of (2.2) in z-coordinates. Since T(e) is a global (linear) change of coordinates, ~ = f2,(0). [] While revising this work, we learned about a preliminary version of [8] in which Theorem 4.6 is proved along different lines.
5. An algorithm to calculate G(e) The evaluation of G(e) by simultaneous placement of eigenvalues and generalized eigenvectors is not an easy numerical problem. Let us explore an alternative procedure which is based on the idea of recovering the open-loop system (2.1) with input u = 0 when e, ~ 0 (Proposition 4.2). The following algorithmic procedure focuses on the evaluation of a gain matrix G(e) that satisfies the following property (Theorem 4.6): IIG(~)II ~< ~tl~ll
for all e E E .
(5.1)
(i) Using any performance design procedure, fix the eigenvalues ;S~,+ COki with 2~ < 0 and ~Ok = 0 for s + 1 ~< k ~ o~4£,where the ~k'S are the imaginary parts of the open-loop eigenvalues. Take the set At(e) of closed-loop eigenvalues as in Remark 4.1: 2~(e) = e2~, + Ogki for 1 ~< k ~< s and 2~,(~) = E2~, for s + 1 ~< k ~< ~'. (ii) Calculate the characteristic polynomial P,(2;e) associated to the set At(e). Once we have P,(2; e) = ~ = o a)(e) 2 j with a,¢(e) -- 1, one can see that the coefficients {a)(e)}~=o are polynomial functions of e which satisfy a~(~) = aj0 + ~j(~),
(5.2)
where {ajo }j=o , are the coefficients of the characteristic polynomial associated to the set A ° of open-loop eigenvalues and ~j(~) are polynomial functions of e, for all 0 ~
(5.3)
where (~(e) is a continuous function of e. Let P~(2; G, e) = ~y= ~ s~((~, ~)2 j be the characteristic polynomial associated to the closed-loop matrix A¢(e) = A + eBd(e). Since e multiplies at least one row of the matrix A¢(e),sj(d, e) is given by
s j ( G , e ) = a j o +8gj(G,e),
1 <~j <<.n.
(5.4)
(iv) Making the coefficients (5.2) of P,(2) equal to the coefficients (5.4) of P,(2; tJ, ~), one obtains a set of nonlinear algebraic (in fact, polynomial) equations with unknowns {t~(e)jl}, 1 ~
(5.5)
Since (~(e) ~ ~ " x ~" ~ ~ ' " , then (5.4) is a set of n equations with nm unknowns. For m = 1, (5.5) is a square system which, from the controllability of (A, B), has a unique solution. For m >/2 one has, in general, n(m - 1) degrees of freedom, which must be specified (using any suitable criteria) to obtain a square system. We propose the following procedure to fix n(m - 1) variables in equation (5.5): (a) Try a partition of (~(e) = (t~ (~), t~(e)), t~ (e) ~ I/~", (~(e) ~ ~ " - ~)", and fix (7~(~) as a constant value (~*. (b) The square system s(G~; G~', 0) - ~,(0) = 0 has at least one solution (~* (in fact, (~' = 0 is a solution).
J. Alvarez-Ramirez et al. / Semi#lobal stabilization of multi-input linear systems
253
If the d e r i v a t i v e Ds(e,)/DG1[(~ = GLe =0) is nonsingular, the implicit function theorem implies the existence of 0 < e* ~< ~ such that the solution G1 (0) can be extended to all e e [0, e*]. It is clear that if such solution G(e) exists, G(e) satisfies property (5.1). (c) If the selected partition of G(e) does not satisfy the derivative requirement, try another partition. Example 5.1. Let us construct a family of state feedbacks which SG stabilize the harmonic oscillator of Example 4.5. Take A~(e)= { - e - ~i, - e + ~i}. Then the polynomial associated to A~(e) is P,(2; ~)= A2 + 2e2 + e 2 + ~2; therefore, yo(e) = 1, 71(e) = - 2 and y2(e) = e. If uj(x) = e(81xXl + 0i2x2), the characteristic polynomial of the matrix A~(e) = A + eBG is given by /~,(2; G, e) = 25 + s~(6, e)2 + So(6, e), where s I ( G , e) = --e_.(011 "-~ 022) and so(G, e) = e 2 0 1 1 0 2 2 --F eo~021 + e2812021 - eo~012 -~ o~2. From (5.4) and (5.5), the gain matrix G must satisfy the following set of algebraic equations: 011 + 022 = - 2 ,
(5.6)
~811022 + ~851 + e#ls#s1 - ~812 = e. For e = 0, this system becomes 011 + g22 = - 2 ,
021 -- g12 = 0 .
Let us try the partition (71 = (011,021), G2 = (812,822). ThenDs(e)/DG11(G*., = o~ = I, which is nonsingular. Therefore, the solution (~l(0) can be extended for e~E*. If G* = (1, 1), the solution of system (5.6) is 011(e) ~--- --3e,
012(e) = e,
021(e) = e(4e + 00/(0~ - e),
022(e) = e.
The corresponding matrix G(e) satisfies property (5.1) for 0 ~< e < e* < ~.
[0101 [, 01
Example 5.2. Consider the control system ~ = A x + Bu, B = [B1,B2], x E ~3, u E •2, where A=
-1
0 1 0
0 0
,
B=
1 1
1
.
(5.7)
-1
The matrix A has eigenvalues A ° = {0, +i, --i}. Thus, the system (5.7) is SGS via SLSF. The characteristic polynomial P°(2) associated to the eigenvalues A ° is given by P°(2) = 2 a + 2. Let us choose the dosed-loop eigenvalues as At(e) = { - e , - e + i, - e - i}, e > 0. Then the characteristic polynomial associated to A c(e) is given by Pn(2)=A 3 + 3 e 2 2 + ( 1 + 3 e E ) 2 + e 3 + e . Therefore ~,o(e)--1 + e 2, 7 i ( e ) = 3 e , ~'2(e)=3 and ~3(e) = 0. After calculating the coefficients {g~(6, e)}~.-ot associated to the polynomial /-~n(2; t~, e), the following set of three equations for the elements of G is obtained (system (5.4)): g l l "[- g12 "[- 022 "[- 013 - - 023 = - 3 , E(012 "~ 0 2 2 ) ( 0 1 3 -- 0 2 3 ) "~ e 0 1 1 ( 0 1 2 "~- 022) "~ 8 0 1 1 ( 0 1 3 -- 023) - - (I -[- E011 -- g 0 2 1 ) 0 1 3 - - e(012 -- 0 2 2 ) ( 0 1 3 -- 023) -- (1 "4- '~012)(011 + 021) -- 012 = 3e, E 2 0 1 1 ( 0 1 2 + 0 2 2 ) ( 8 1 3 - - 023) "4- /~013(Egl1 + E021 -- 1)(812 - - g22)
(5.8)
"q- (l "['- 8 0 1 2 ) ( 0 1 3 -- 0 2 3 ) ( 1 + /~011 -- e 0 2 1 ) - - g 0 1 3 ( 0 1 2 "q- 0 2 2 ) ( 1 + e011 - - g021) -- /32011(012 - - 0 2 2 ) ( 0 1 3 - - 023)
-- (1 + e012)(,~011 "~ e821 - -
1)(813 --
g23) = 1 + e 2.
Let us try the following partition: G1 = (011,812,053) and (~5 = (813,021,822)- Fix G2 to some constant value G*. At e = 0, the system (5.8) is reduced to the set of linear equations 011 +812 - 023 = - 3 811 ~ -- 013 - - 2 0 1 2 , 2 0 2 3 = 2 8 1 3 -- 1.
0"2-0"3, (5.9)
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J. Alvarez-Ramirez et al. / Semiglobal stabilization of multi-input linear systems
This set of equations has a unique solution (~1. On the other hand, one can see from (5.8) that Ds(e)/D(J1116,,= = 0)is equal to the matrix constructed with the left-hand side of (5.9) ( in fact, all the terms of (5.8) having e vanish). Therefore, Ds(e)/D(~lll61 =8",~=o) is nonsingular and, for all e e E the solution (~1 (e = 0) can be extended.
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