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Output feedback variable structure control design
Pergamon
00051098(94)00151-0
Output Feedback Variable Structure Control Design* REYAD
EL-KHAZALII
and RAYMOND
DECARLOt
Design techniques are described for constructing an output feedback variable structure controller to drive a system’s output trajectory to a stable sliding manifold. Algorithms for the design of the sliding manifold are also given. Key Words-Variable structure control; variable structure systems; sliding states: Lyapunov methods; linear systems; eigenfunctions.
Abstmet-This paper describes a procedure for variable structure output feedback control (VSOFC) for a class of multivariable linear time-invariant systems. The equivalent control method of Utkin in the output feedback mode yields a reduced-order system exhibiting output feedback equivalent dynamics. Using the Kimura-Davison sufficient conditions for pole-placement using output feedback, a sufficient condition is derived to assign the eigenvalues of the ‘reduced’-order system. The observability and controllability of the reduced system is discussed. Two switching surface design algorithms are introduced based on both a static output feedback and on an eigenstructure assignment techniques. A control design is discussed as well.
VSOFC (variable structure output feedback control) have been considered by Heck and Ferri (1989) and White (1990). Verghese er al. (1988) considered robust tracking by sliding mode control for the cases where the number of inputs equals the number of outputs and where the equivalent control is not unique. This paper continues the investigation of VSOFC. The design procedure of a variable structure control (VSC) system is a two-stage process. The first phase is to choose a set of switching surfaces such that the original system restricted to the intersection of the switching surfaces has a desired behavior. The second phase is to determine a switched control law that forces the system’s trajectory to and maintains it on the sliding surface (Utkin, 1978; Utkin and Yang, 1978; DeCarlo et al., 1988). Switching surface design in VSOFC is introduced based on assigning a desired set of eigenvalues to the reduced-order system. The necessary and sufficient conditions for pole-placement using output feedback (Brash and Pearson. 1970: Davison, 1970; Davison and Chatterjee, 1971; Davison and Wang, 1975; Sirisena and Choi, 1975; Kimura, 1975, 1977; Andry et al., 1983; Hanmandlu and Shantaram, 1986) are used to design the switching surface matrix. In this paper, the VSOFC switching surface has the form a(y) = Sy(t), where y(t) E R’ is the output vector and S E R”“’ is the so-called switching surface matrix. The advantage of VSOFC is that it avoids using an observer to estimate the unavailable states while achieving the desired control objective under reasonable conditions. Difficultieis do arise, however: (i) the equivalent system trajectory in the sliding mode is projected onto a lower-dimensional subspace, part of which is fixed and equal to the null space of C;
1. INTRODUCTION
Variable structure systems (VSS) (Itkis, 1976; Young et al., 1977; Utkin, 3978, 1983, 1984; Utkin and Yang, 1978; Bondarev er al., 1985; Heck and Ferri, 1989; White, 1990; El-Khazali and DeCarlo, 1991) are systems having a (high-speed) switched control law. Typical applications of VSS linear and nonlinear systems utilize full-state feedback or observers that require extra dynamics to measure the unavailable states (Bondarev et al., 1985). This increases the complexity of the design procedure, and, in some cases, duplicates the size of the controlled system. This has led to an output feedback approach (Brash and Pearson, 1970; Davison, 1970; Davison and Chatterjee, 1971: Davison and Wang, 1975; Sirisena and Choi, 1975; Kimura, 1975, 1977; Andry et al., 1983; Hanmandlu and Shantaram, 1986). Several authors have considered VSC in an output feedback format. Linear systems using * Received 23 April 1992; received in final form 9 August 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. Utkin under the direction of Editor Huibert Kwakemaak. Corresponding author Professor Raymond A. DeCarlo. Tel. +l 317 494 3523; Fax +l 317 494 3371; E-mail [email protected]. t School of Electrical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. 805
R. El-Khazali and R. DeCarlo
806
(ii) the equivalent control is not unique when the product CB is not of full column rank; and (iii) the observability of the reduced-order system is not guaranteed. Viability of VSOFC requires observability because the reduced dynamics exhibit an equivalent output feedback structure where observability is necessary for pole placement by output feedback (Davison, 1970; Kimura, 1977). Such constraints limit the class of allowable systems. The purpose of this paper is to investigate the VSOFC design for a class of linear time-invariant systems. For simplicity, the discussion here concentrates on systems linear and with ideal switching in the control, (infinitely fast) in the control law (Utkin, 1978, 1984; Utkin and Yang, 1978). The paper is organized as follows. The problem statement and system definition are introduced in Section 2. Section 3 furnishes some preliminaries and background, including the observability of the reduced-order system. Switching surface design algorithms I and II are presented in Section 4 and 6 respectively, while Section 5 investigates system eigenstructure in design is the sliding mode. The control formalized in Section 7. 2. SYSTEM
DEFINITION AND STATEMENT
PROBLEM
Consider the usual linear time-invariant model 1(f) = Ax(t) + Bu(t),
state
having state x E R”, control u E R” and output y E R’. Assumption
1. The system is assumed to be completely controllable and completely observrank[C] =r, and the rank [B] = m, able, Kimura-Davison condition is satisfied, i.e. nsm+r-1. In VSOFC (Heck and Ferri, 1989; White 1990; Verghese et al., 1988; El-Khazali and DeCarlo, 1991), the control component u, of u has two possible structures h(Y) =
u,:(y)
when a;(y) > 0,
u;(y)
when a;(y)
3. PRELIMINARIES
AND
BACKGROUND
Using the method of equivalent control (Utkin, 1978, 1983, 1984; Utkin and Yang, 1978), if the system is initially on the switching surface at to then a(y(t)) = 0 and &(y(t)) = 0 for all t 2 rO. Therefore ti = SCAx(t) + SCBu(t) = 0.
(1)
y(t) = Q(t),
tracking or regulation. This is accomplished in two phases. First phase. Given the system (1) and a symmetric set of n -m distinct numbers A,_, = {A,, , A,_,}, find a real map S E R”“’ such that n - m eigenvalues of (1) in the sliding mode are precisely those of the set A,_, (El-Khazali and DeCarlo, 1991). The first phase presupposes a reduction of order of (1) to an (n - m)-dimensional equivalent system. This is because the ndimensional state dynamics of (1) must satisfy the m algebraic equations g(y) = 0. Such constraints reduce the equivalent system from an nth-order system to an (n - m)th-order system (Itkis, 1976; Utkin, 1978, 1984; Utkin and Yang, 1978; DeCarlo et al., 1988: El-Khazali and DeCarlo, 1991). Second phase. Determine a switched control of the form (2) such that the system’s trajectory is globally attractive (stable) to the sliding surface from any point in the output space (Utkin, 1978, 1983, 1984: Utkin and Yang, 1978).
(2)
depending on the location of the system’s output representative point y(t) with respect to the switching manifold a(y) = 0, defined as (3) L(Y)l’ = SY(l) c+(Y)=[a:(Y) ... where S is an m X r real parameter matrix to be designed (Heck and Ferri, 1989; White, 1990). It is desired to choose S to stabilize the system in the sliding mode or to achieve some form of
(4)
This results in the equivalent control law (Heck and Ferri, 1989: White, 1990; Verghese et al., 1988) u,,(t) = -(SCB)-‘SCAx(t), (5) which exists uniquely if SCB is nonsingular. Clearly, the equivalent control is not unique when CB is rank-deficient or when r
2. r > m, S is full row rank, CB has full column rank, and SCB is nonsingular.
Given Assumption 2, substituting yields the equivalent system i(t) = [(1,, - B(SCB)-‘SC)A]x(t), y(t) = Cx(t),
(5) into (1) (64 (6b)
807
Output feedback VSC which represents the system dynamics in the sliding mode. The goal is to choose the function u = Sy(t) and a switched output feedback control of the form (2) to adjust the state dynamics of (6) to achieve a desired behavior of the output trajectory. To design a switching surface of the form (3), one needs to investigate the equivalent system (6a). If I = m then, by Assumption 2, S is a nonsingular transformation of the switching surface Cx(t) = 0; hence the switching surface dynamics Sy(t) = Xx(t) = 0 is invariant with respect to S (DeCarlo et al., 1988). On the other hand, if r > m then the switching surface dynamics depends on N[S] (Heck and Ferri, 1989; White, 1990; Verghese et al., 1988; El-Khazali and DeCarlo, 1991). To see the nature of this dependence, it is convenient to put the system (1) into the regular form (Luk’yanov and Utkin, 1981)
\‘J Y=
[Cl
where B, E R”“” is nonsingular and where z, E R”-“, z: E R”, A,,, Arz, &, At2, C, and C2 are real matrices of appropriate dimensions. The switching surface in the new coordinates is V(I, 9z2) = S[C,
C,l[ 3
= PI.
(8)
By Assumption 2, CB = C2B2 is of full column rank. Since Bz is an )12X m nonsingular matrix, rank [C,] = rn, i.e. CZ has full column rank. Suppose that the system’s output representative point is initially on the switching surface; then c(z) = 0 and b(z) = 0. Solving (8) for z2 and substituting into (7) yields the reduced-order dynamics i, = (A,, - &(SC2)-'SC,)Z,
Proposition 3.1. Let K E R”“’ and C2 E R’“” be full rank of rank m, r > m, and let S E R”“’ be chosen such that SC2 is nonsingular. Then a necessary and sufficient condition for there to exist a solution S of the equation (SCJ’S = K is KC2 = I,,,.
(9)
where K + (X2)-‘S represents an output feedback gain map; Al2 and C, appear respectively as input and output matrices of the reduced order system. S must be chosen such that SC2 is nonsingular. Switching surface design reduces to finding S so that the reduced-order system of (9) has a desired spectrum. i.e. cr(A,, - A,,KC,) = A,_, (El-Khazali and DeCarlo, 1991; Hashimoto and Utkin, 1991). Equation (9) exhibits an output feedback structure in K. Sufficient conditions exist for a solution to the spectral assignment problem (Kimura, 1975, 1977). However, under what conditions does there exist an S such that the
(10)
Proof: Necessiry. Let S E R”‘“’ be such that = K. Right multiplication by C2 gives
(X2)-‘S
= KC2 = I,,,.
(SC,)-‘SC, Sufficiency.
With KC2 = I,,,, let S = K. Then
(SC2)-‘S=(KC2)-,K=I,,,K=K.
0
If a solution to (SC,))‘S = K exists, it must satisfy (lo), i.e. K is a left inverse of C2. Hence switching surface design reduces to finding a gain feedback K that assigns A,_, = (A,, . . . , A,_,,,$ to the reduced-order system (9) subject to the constraint given by (10). From the proof of Proposition 3.1, S = K is an acceptable switching surface matrix. Proposition 3.2. Let C2 E R’“” and rank [C,] = m < r, and let K E R”“’ satisfy KC2 = I,,,. Then
the set K, of all solutions of (10) is nonempty and is given by K, = {K: K = CTL + TM’}
(11)
where r E R”““-“’ is an arbitrary parameter matrix, and where the rows of M’ E R”-“‘“’ are a basis for the left annihilator of C2, i.e. MC2 = [O]. Substituting the form of K given by (11) into (9) yields ’ = 61
'(A,, -A,XC,)Z,, y = (L - C:!K)C,z,.
spectrum is assigned? This constrains the class of KS. The next two propositions characterize the class of permissible K matrices.
(all
-
A,J?,)z,,
(12)
where a ,, k (A,, - A,2C;LC,) and C, GMT,. The constraints imposed on K limit the available feedback. The question now is what freedom remains to assign the proper eigenvalues to the reduced-order system. This requires that the submatrices of the reducedorder system, A,2 and Cl, have certain structural properties. This will be investigated in Section 4. 3.1. Observability of the reduced-order system Equation (12) defines the reduced-order system in terms of a new output matrix Cl and a new system matrix a r,. The controllability and the observability of the triple (Cl, ii ,, , A,2) is necessary to assign a desired set of eignevalues to (12). The controllability of the pair (A, B)
808
R. El-Khazali and R. DeCarlo
guarantees
the controllability of the pair (Young et al., 1977). Since a,, can be seen as a state feedback modification of A,, through the input matrix A12, by Lemma 2.1 (Wonham. 1979) the pair (All, A12) is also controllable. Hence sufficient conditions for pole-placement by static output feedback are observability of (Cr, AlI) and the satisfaction of the Kimura-Davison condition (Kimura, 1977). Observability of the reduced system is not guaranteed. Even if the pair (C, A) is observable, the pair (C,, A,,) may not be. On the other hand, it is possible to have (C, A) not be observable with ( C1, A, 1) observable (El-Khazali and DeCarlo, 1991). In view of (12), one requires the observability of the pair (Cl, a,,) and controllability of @rl, A& which is already guaranteed. How does the observability of (C,, AlI) affect the observability of (Cr, a,,)? If (C1,All) is observable then ( C1, a 1,) is also observable, because CTLC, can be viewed as a simple output feedback map that does not affect observability. Further, if Im [C,] n Im [C,] = 0 then the pair (Cl, a ,,) is observable. This is of course a sufficient condition. It is possible to construct an example such that if Im [C,] n Im [C,] Z 0 then the pair (C, , a 11) is not necessarily observable (El-Khazali and DeCarlo, 1991; El-Khazali, 1992). If the pair (C,, a 11) is not observable but is detectable then it is possible to show that the reduced-order dynamics are stabilizable (Wonham, 1979). The implication is that one must check (C, , a, 1) for observability. (A,,, A,,)
Assumption 3. The pair (C,, a 11) is observable. 4. SWITCHING
SURFACE
DESIGN
ALGORITHM
I
From Assumption 1, it is possible to assign the spectrum of the triple (C, A, B) by output feedback (Kimura, 1975, 1977) provided that a slight modification of the poles to be assigned can be tolerated. To apply this result to (12), recall that a 1, E R(“-m)x(n-“‘), Al2 E R(n-m)xm, (+,E R(r-m)x(n-m) and n %m + r - 1 (i.e. the Kimura-Davison condition is satisfied), with m?2andr>m. Assumption 4. A,_, = {A,, . . , A,_,} metric set of n - m complex numbers.
is a sym-
Theorem 4.1. Given Assumptions 1-4, if rank [A,,] = m (full column rank) or rank [AI21 = n -m (full row rank) then there
exists a real matrix I E R”““-“‘I at least - A,,Te’,) = A,_,, 6, arbitrarily small neighborhood.
such that within an
Proof
First the pair (a 11,AIZ) is controllable (Young et al., 1977). Hence the triple (Ci,aii, A,*) is complete. The result follows from Kimura (1975) provided that n - m I rank [AI21 + rank [C,] - 1.
(13)
Since M’C, = [O], rank [C,] = m and rank [C, C,] = r > m, we have rank [C,] = rank [MC,] = r - m. Since the original system satisfies the Kimura-Davison condition, n m%r-l=m+(r-m)-1. If rank[A*J=m then (13) is satisfied; if rank [A,,] = n -m then n-mI(n-m)+(r-mm)-1. Since r-mZ1, (13) is again satisfied. !Zl Once I is found using any existing output feedback algorithm (Andry et al., 1983), we simply set S = K = CgL+ rM’ to obtain the switching surface map. Other choices of S are possible. The following proposition characterizes all possible Ss associated with a fixed gain K. Proposition 4.1. Let K E K, and rank [S] = m. A switching surface matrix S satisfies (SC,)-‘S = K if and only if
SIZI - C2K] = [0] Proof: This follows by Propositions
(14)
3.1 and 3.2. 0
If n > m + r - 1 and rank (AJ
EIGENSTRUCTURE MODE
IN THE
SLIDING
The relationship between the eigenvectors of the original system and those of the reducedorder system will allow us to construct a second switching surface design algorithm. By Theorem 4.1, there exists a constant matrix K E HIX, such that A(A,, - A12KC1) = An_,,, = A,_,}. let {e,, . . . , en-,,,} E C”-” be the FL,...,
809
Output feedback VSC associated AlzKCl)e,,
right eigenvectors, which implies
i.e. A,e, = (A,, -
Theorem
5.1. Under
Proposition
.\(A,, - A,:KC‘,)
[(Ail - All) / -&I[-&]
1I
= LO]. (15)
Define u, = [e: 1 -(KC] e;)‘]’ for i = 1. . , n - m. The set {u,, , u,__~} denotes the n -m with associated eigenvectors of (6) {h, , . . , A~_,,,} in the sliding mode, as verified in the following proposition. Proposirion 5.1. Let (C, A, B) be in the regular form as per (7). Let K E K, be such that h,e, = (All - A,2KC,)e,, and define u, = [e: ) -(KC,e,)‘]’ for i = 1, . . , n - m.
Then {u,, . , u,_,} are n - m eigenvectors (6) in the sliding mode.
of
Proof Let S = K. In the sliding mode SCz = KICI C,]z = [KC, i,,Jz =0 by Proposition 4.1.
the
assumptions
of
K E R”‘“’ be such that = A,_,. Suppose that 3;; =
5.2. let
where span 121,. , v,,_,}, u,[e: j -(KC,e,)‘l. Then N[C] c ‘r; if and only if [KC, - Z,,,]VP = [O].
N[C, j C,] = Im It’\” j Vs’]‘. and VP E R”‘xtn-r’. with definedby(15)fori=l,....n-m.
where
R,,lF,,,1XOi-r)
(17) vy E
the
e,
Proof: Sufficiency. Suppose that (17) is true. Let K = K, + K,, where K, E K,. i.e., K,Cz = l,,.
Then sufficiency follows from Proposition 5.2 by letting K, = [O]. Necessity. Let N[C] c 7% with 7; as stated above. Suppose that N[C, / C,] = span {r,,
, r,,-, 1.
where r, = [r:, 1rf?,l’. Let w = [w: ) IV>]’ satisfy
After representing (6) in the I coordinates, i.e. in the regular form, and forming the standard eigenvalues-eigenvector equation, the result 0 follows by direct calculation. Define V,, = [ui II,_.,]. Since the n - m column vectors of the matrix V, are eigenvectors of the system in the sliding mode, and since the system is projected onto N[SC] cN[C] (ElKhazali. 19921, N[SC] = ‘& where ‘& = Im [Vd] and A[(A + BF)INlscl] = A,_, for some F (Wonham. 1979). Consequently, any desired eigenstructure in the sliding mode must contain N[C]. If -& is selected properly so that N[C] c 2’, then rank [CV,] = r - m. This means that there always exists a full-row rank matrix s E R’““‘. r > m, such that SCV, = [O].
(16) If one can construct an algorithm that produces a proper set of eigenvectors. denoted by the columns of V+ such that N[C] c ‘& then there always exists a proper switching surface defined by an S satisfying (16). Equation (16) is the key to introducing a switching surface design algorithm based on constructing 2; without the need to first determine an output feedback matrix K. In order to do this. it is necessary to characterize the general class of output feedback matrices K that guarantee N[C] c 2/-,. Proposition 5.2. Given Assumptions 1-4 with m > 2 and A, @h(A, ,), consider the reducedorder system given by (12). Let K E Ki, (11). If rank [A,*] = min (m, n - m) then there exists an (n - m)-dimensional subspace y&, where 2/-,= (~1,. . . . u,~-,}, and U, = [e: 1 -(KC,ei)t]t, such that N[C] c ‘J”,.
for some constants, zero. Then
y,, i = 1,.
, n - m, not all
c, w1 + czwz = [O].
(18)
Obviously, w2 E Im (L’S). Since {e,, . . . en_,,,} are linearly independent, {u,, . , u,,+} are independent. Hence there exist constants, not all zero. such that PI>. . . 7Pn-t?,,
“I =‘y [ W,
I
p,u;
=
,=,
It follows that w:!= -KC,w,.
(19)
(18) by K and using (19) yields [KC2 - I,,,]w~ = [0] for all w? E Im [Vz]. 0
Premultiplying
Observe that the second class of output feedback matrices, (17), is larger than the first class of (31). Both classes guarantee the satisfaction of N[C] c ??Y. Proposition 4.1 and 5.1 guarantee that an admissible full-rank switching surface matrix exists if and only if K E K,. One of the objectives, detailed in the next section, is to use the result of Theorem 5.1 to algorithmically generate a set of admissible eigenvectors, denoted by the columns of V,, that contain N[C] in their span. If this is done, then rank [C&J = (n - m) - (n - r) = r - m. However, S must be a m X r matrix, full-rank in which case
810
R. El-Khazali and R. DeCarlo
dim (N[S]) = r - m. Hence (16), SCV, = [O], is satisfied if and only if N[S] is precisely Im [C&J. Therefore if one can generate a set of admissible eigenvectors whose span contains N[C] then any full-rank S whose kernel is precisely Im [CVd] is a legitimate switching surface matrix. 6. SWITCHING
SURFACE
DESIGN ALGORITHM
problem now is to determine a procedure to construct Vd1 N[C]. Once Y$ is found, a proper switching surface matrix may be selected by solving (16) for a full-rank S. The following discussion furnishes the necessary foundations for the development of the design algorithm. Consider the system (1). If the pair (A, B) is controllable and in the regular form then, for a desired set of eigenvalues A,, = {A,, . , A,}, there exists a constant state feedback matrix F E R”“” such that A(A + BF) = A, (Wonham, 1979). Let {ul, . . . , u,} be the associated right then (A + BF)u, = Aiug. Some eigenvectors; algebraic manipulation yields for i = 1, . . , n ( -I?][;]
= [O]. I
Now for each
Ai E C, let the columns
of
Th, s [N’*, ( M:J’ form a basis for the null space of [(Ail -A) 1 -B], where for i = 1, . . . , r~
[Ail - A ( -BIT,, - [Ail - A ( -B][ $1
= [O].
A,
(21) Clearly, Ui E N,,, (Andry et al., 1983). Utilizing the regular form of (A, B), (21) has the structure
(Ail,-m- AlI)
[
Proof: From (22)
-AI~][$] =[O]. A,
[(AL-m-All) 1 But
(23)
- Ail) 1 -&21[ 3) 2 = rank [-A ,2 B2] = m since B2 is an m X m nonsingular map and A,2 is an (n -m) X m matrix of rank m. Hence N,,, rl Im (B) = 0 for i = 1, . . . , n - m. 0 The following result shows that each pair of Nh, and Nh, is disjoint for distinct eigenvalues. Lemma 6.3. let (A,,, A12) be controllable, N*, be defined by (22) and (23), Ai @A(A,,) and n - m 1 m. If rank [Al21 = m then (i) N,,nN,,=O for Ai#Ai; i,Z=l,... , n-m; (24) (ii) rank [N:] = m for i = 1,. . . , n -m; (iii) rank [N:] = n - m. Proof The proof of (ii) and (iii) is straightforward. To prove (i), suppose that for two distinct eigenvalues Ai # Aj there exists a nonzero vector w E N,, n N,,,. Then, from (23),
](AiZ,-, - A,J 1 -A121~ = PI, [(A$n-m -All)
I -A121~ = [Ol.
Subtracting yields
-A,2
--A21
6.2. Let (A, B) be in the regular form and complete, rank [B] = m, n -m zm, and Ai g A(A,,). If rank (A,,) = m then Im (N*,) fl Im (B) = 0 for i = 1, . . . , n - m.
II
The
[Ail -A
Lemma
=
[Ol,
[(A; - Aj)Zn_, 1O]W= [O]. Thus
(22) where Nn, = [(N:,)’ ] (NfJ]‘; N:, E R(“-m)xm and N;, E R”““. The following results will be useful in the development of the design algorithm. Lemma 6.1. Let (A, B) be complete, m. Suppose that A, z A(A), i = 1,.
rank [B] = , n. Then the columns of NA,are linearly independent. Proof. As the columns of NA, are given by the 0 eigenvectors u,, its columns are independent. Note that, owing to the reduction of order in the sliding mode, the condition A, e A(A) can be relaxed to require only Ai @ A(A,,) for i = 1, . . . , n - m. This restriction is necessary for the following results.
w E Ker [(A, - Ai)Z,_,,, IO]. Hence w = [O*I ~$1~and, from (23), [(Mw,,
- AlI) 1 -A,21[ 3
= -A12~2 = [Ol.
This implies the contradiction linearly dependent columns. The following important the construction of V&
that
Al2 has q
result is the key for
Theorem 6.1. Let (All,A12) be complete, n - m rm, A,,-,,, be a symmetric set of eigenvalues with Ai e A(A,,), i = 1, . . . , n - m, m 2 2, Nh, be defined by (22) and (23), and
N,, k [Nh, (. . . I NA._,J If rank [AI21 = m then rank [NJ = n.
(25)
811
Output feedback VSC Proof.
Step 1. The
first 2(r - m) vectors chosen such that
See the Appendix.
It is possible to show that the result holds even ifn-m
cur-,+, = cu, cu,-,+* = cu*
(i) N,,, n Nlh,= 0,
N1* E W’x’n-m).
’
i Zj;
Step 2. The last n + m - 2r vectors,
(27)
are chosen in an arbitrary fashion as a linear combination of the first r - m vectors. One possible choice is
C%(r-m+,) = cu, + cu3, cl&,
Proof It follows from (23) and (26) A12No = [O]. Moreover, u, E Im (N,J for 1, . . , n - m. Applying Theorem 6.1
that i =
and Lemmas 6.2 and 6.3 to this case yields the above corollary.
6.1. Algorithm development The previous discussion suggests an alternative approach to the switching surface matrix design introduced in Section 4. To avoid determining K. one may select an appropriate eigenspace, cl/^d=~ N[C]. The construction of the desired subspace ‘7; = span {u,, . , II,_,}, is done so that u, E Im (N*,) and the set {u,. . . , u,-,,,} is linearly independent. Since Theorem 5.1 and Proposition 5.1 verify the existence of 2’, 3 N[C] and the independence of {ul, , u,_,,,}, the designer must guarantee. by construction, that “r/-, is an (n -m)dimensional subspace. Recall that dim (N[C]) = n - r. Let V, = [ul, . . . , u,_,]. There exist nonzero vectors gj E R” such that u, = N,,g,. In order to have N[C] I.=‘J$ and guarantee the existence of a proper S, (16), one must require rank [CV,,] = r - m. This is simply satisfied if , Cu,,_,) C Im (Cu,, .
. . * C~n-ml,
CuZ(r-m,+1 = cu, + cuz,
(ii) N,,, rl Im (B) = 0 for i = 1, . , n -m; (iii) rank [NA,, . . , NAn__]= n.
Im(Cu,-,+,,
(29)
= cur_,. CUZ(r-m)
IC%(r-m)+l,
where NoAE IWnx(*“--“) and Then for distinct eigenvalues
in (28) are
, Cu,_,),
=
(30)
cu, + . + cl&_,.
Since u, = N,,,g,, if r - m = 1 then (28) and (29) yield CN*,g, = CN,,g,, for i = 2, , n - m. In matrix form,
The matrix on the left is r(n - r) X m(n - m), has full row rank and has r - m = 1 more columns than rows. Thus it has a nontrivial solution in which CN,,g, # [0] for i = 1, ,n m, because u, er N[C]. Since (31) has a nontrivial solution (all other nontrivial solutions are scalar multiples of it), by Theorem 5.1 and Proposition 5.1 the vectors {u,, , u, _} are linearly independent. On the other hand, when r -m 2 2. the matrix-vector equation that results from (29) and (30) takes the following general form:
(28) provided that span {u,, . . . , u,_,} O N[C] = 0. One must also ensure that rank [Vd] = n - m, i.e. any solution of (28) must guarantee the independence of {v,, . . . , u,_,}. Since u, E Im (NA,), (20) and (21), by Proposition 5.1 a linearly independent set of eigenvectors of the system in the sliding mode does exist. How does one constructively achieve the independence of the u,? To satisfy (28), the relationships between the various vectors may be chosen as follows.
g1 ET2
&m+l
=
gr-m+z &T-m
_
(32)
812
R. El-Khazali and R. DeCarlo
The number of nontrivial solutions to (32) is greater than or equal to r - m. Since ‘& exists by Theorem 5.1. there exists an appropriate solution. i.e. an acceptable set of g,, that satisfies (32) and ensures the independence of the vectors {u,, . . , u,, -,,,). In particular. let p0 be any nontrivial solution and let P be a matrix whose columns are a basis for the null space of the matrix on the left-hand side of (32). Generally, PO + PcY
for a randomly chosen LYwill yield an acceptable solution. The following summarizes the discussion of this section and introduces steps of the design algorithm. The switching surface design algorithm.
the triple (C,, a ,,, A,2) is and A,, have full ranks, nsmfr-I. and whether r >m. If so, record the dimensions of A, I, and N[C]. For a given symmetric set of n -m distinct eigenvalues. A,, ,), = {Al,. . , A,,_,}, determine N,, from (22) or (23). Depending on 11- r and r - m, consider the following cases: (i) if r - m = 1 then solve (31) for the gi; (ii) if r - m ~2 then solve (32) for the g,. Construct ‘Ii, = {u,, . . , II_,,,}, where u, = JV&,. Finally. solve (16) for a full-row-rank S.
1. Check whether complete, CB
2.
3.
4. 5.
An example of the algorithm can be found in El-Khazali (1992). 7. CONTROL
DESIGh’
The second phase of VSOFC design is to determine a switched control that drives the system’s output trajectory to the switching manifold. In the subsequent discussion, we assume that the linear system (1) satisfies the conditions n 5 m t r - 1, r > rn, rank (AJ = and min (rrt, n - m ). the triples (C, A, B), (c,, A,, , A ,:) are complete, and rank (CB) = tn. With these structural properties, the system (1) is stabilizable using static output feedback (Kimura. lY77). In addition, there exists a switching surface matrix S such that iZ(A,, A,,(SC:) ‘SC,) = :I,,_,,: and. with the constraints u = 5~. = IO]. we have n(A - B(SCB)
‘(SCA))
= {A_,
0,.
. , 0},
where :2,,_,,, is a stable symmetric set of n -m distinct real eigenvalues. by Theorem 4.1. A traditional approach in VSS control design is to use the equivalent control ucq as part of the control vector (Utkin. 1978), i.e. the control
takes the form ti = l&Zq +f(y),
(33)
where f(y) is a suitable switched function that guarantees convergence to the switching surface. ueq = -(SCB)-‘SCAX Implementing alone would cause the system’s output trajectory to slide onto a hyperplane parallel to the switching surface. However, the requirement of output feedback precludes the use of ueqr as now discussed. If there exists a real matrix G E R’“’ such that CA = GC then one can write u eq = -(.SCB)-‘SGCx = -(SCB)-‘SGy, and the equivalent control in this case is an explicit output feedback control. Sufficient conditions for the existence of G are given by iak and DeCarlo (1987). Such conditions require in essence that the row vectors of the output matrix C be the left eigenvectors of the system matrix A. A necessary and sufficient condition for the existence of G is given in Proposition 7.1. The sufficiency part is equivalent to the one given by iak and DeCarlo (1987). Proposition 7.1. Let the earlier assumptions hold. Define v0 = N[C]. There exists G E R’“’ such that CA = CC if and only if AY’,’
ZrO.
(34)
Proof Sufficiency. Let (34) be true. Let V, = [u, . u,_,] be a matrix whose columns are
bases for Cy-,= N[C]. where by hypothesis rank [C] = r. Let {c,, . , c,} denote the columns of C’. Then {c,, . , c,, uI, , u,_,} is a basis for R”. Consider that CA[c,
..
c,
u1 = [CAc,
u,_,] .
CAc,
0
. .
0]
= [CAC’ / 01. Thus (34) is solvable for G = CAC’(CC’)-‘. Necessity. Let G E R”’ be such that GC = CA. Then GCV, = CA,V = [O]. This implies that 0 Im (A V,) c 2<,or A V, c “u;,. Satisfaction of the condition (34) implies that 2’, is A-invariant; i.e., for G to exist such that GC = CA, the pair (C, A) must be unobservable. Since (C, A) is observable by hypothesis, no G exists such that GC = CA; i.e. the equivalent control cannot depend explicitly on the output variables. In constructing the output feedback control, let us decompose ueq into two parts: one in terms of some of the measurable output variables and
813
Output feedback VSC the other in terms of the unmeasurable state variables. This can be accomplished by introducing another coordinate transformation that will eventually simplify the control design. 7.1. System in normal form Consider the system (7) already in the regular form and suppose that a proper switching surface S is selected such that A(A,, matrix with AIZ(SCZ)-,SC,) = A,_, = {A,, . . , A,_,} Re [Ai] < 0. The control design is simplified if one views the system (7) in a new coordinate frame called the normal form. This is achieved by the linear state transformation (Young et al., 1977)
7.2. Output feedback control design The control design in VSOFC requires the convergence of the system’s representative point to the switching surface: thus a switched control must be used to drive the system’s trajectory to the switching manifold. The switched controller is also used to achieve robustness. model reduction and possible disturbance rejection (Utkin, 1978; Heck and Ferri, 1989: Corless and Leitmann, 1981; Madani-Esfahani et al.. 1990). Before discussing the design procedure. let us introduce some necessary definitions. For i, j = 1, 2 let plj = IIA,,,II denote the spectral norm of A ,,“, i.e. IPLII = max {IIA,,,xll : /Ix// = 1). Also define A,,, = min{lh,l, . , /A,_,I}, where A, E 4,-m.
Let us propose the following output feedback control candidate: Since SC2 is nonsingular, T is nonsingular and T-’ is straightforward to compute. The system in the new coordinates is
A 22n(+ +
u, = -(SC2&_’
P12P21~mx
u + ff
sgn u
hn,n~m,n
(Ilull >a (ll~ll = Oh
AI,”
Am
A 2,n
.,,,1[:1
(38)
+ [S&&
Y =
[Ur -- c2(sG--‘w’
(+=
to1z$]
S],
I c*w2r’I[
(36)
,
where (Y> 0, sgn u = [sgn u1 . . sgn u,,,],, sgn u is the usual Signum function. and u_ and u,,, are the maximum and minimum singular values of a nonsingular matrix U E C(nPm)u(nP”‘, that diagonalizes A, ,“, i.e. U-‘A,,U=D=diag(A
where the subscript ‘n’ denotes normal form and A ,,“-A,,
-A&G-‘SC,,
A ,2n--A,z(SCJ1, A 2,n * SC,A,, -
-t SC>Az,
(Sc,A,l + SCzAz)(ScJ’SC,,
A zzn+ (SC,A,l
+ SCzA&SCz)-,,
while the equivalent coordinates is given by u eq = -(SG&-‘&,nq s &&)
Theorem matrix.
in
S
be
the
new
- (SWJ1&n(+
+ %2((+).
The equivalent system o=O)yieldsrj=A,,,qanda=O. Remark. Let AIZ(SCZ)-iSC,)
control
dynamics
(37) ((9) with
such that A(A,, as per Section 7.1 and 4.1. It follows that A,,” is a stability = A,_,
,,...
.A,_,,)
(39)
The first and third terms of (38) were introduced by Madani-Esfahani et al. (1990), who made use of u_, (37), directly. It is not possible to use ueq of (37) because the term A,,,q contains state information that is not directly accessible at the output. The term is inserted into the (~12~21UmaxlhrnlnUrnin)(+ control structure of (38) to overcome this problem. Note also that for u = 0, II, = &u, which represents a relay control. An alternate to the proposed output feedback control structure of (38) was introduced by Heck and Ferri (1989). The control is of the form u = -(SCB)-‘[SCANy + sat (a)], where sat (.) is the usual saturation function. However, this controller requires the existence of a matrix N E R”“’ such that (SC)‘(SC)A(I - NC) is negative-semidefinite. The justification of the control (38) proceeds more smoothly via a state transformation
[:I=g-l-+y~l
C40)
which diagonalizes A,,” for U chosen as per (39).
814
R. El-Khazali and R. DeCarlo
The system and switching surface coordinates are
in the new
where cy> 0 and F0 = A&M, M2 + S), with M2 a maximal-rank (orthogonal) matrix such that M,C, = 0 and with M, chosen as the leastsquares solution to A 21”
-
A2znMM2C,
=
0.
With MI so chosen, define c = llA21n - A22nM,M2C, IIF.
(43)
It follows that (41)
F,Y = &@I
7.1. The
asymptotically control (38).
system (41) is globally stable with the output feedback
Proof
Consider the following candidate Lyapunov function (Rosenbrock, 1963; Weissenberger, 1973):
v, =-=,a,*
P21
nun
+
Using the system (41) with the control follows that + ~(TmaxP21 SL
u-~A
ll%Ill
ku”
F,Y = &J%MzCI
(P12/“min)
PX~,,,~~
and ll9nII llDll llqnllII~II, it followsthat
since a’sgn cr/l)(~ll > 1.
l,a,l
(44
v, = & llqnll+ 11~11. Differentiating
with respect to time yields
(38), it + jf$ ~
(AzlnQn
+ A22ng
-_,y-K,a-asgna)
12n
=
W--‘A,,,
4J’qn
K
qf,Z.-‘A,2,a
I
~‘&n&n
5
WA:,,
-K,
<_-CUP
A drawback of using u, is that it does not utilize the full output vector y, possibly leading the due to A control that utilizes the complete output vector is u* =
I
u
llul,-a+
5 -K&n,,
Cl
ll(l”ll
- C’, M;M: Ai2”)
llflll
u’ sgn u v,l--a
uq, + A22nu.
Let a Lyapunov function be given by
+ qhDq, 5 -Amin llqnl12,
‘~9
Theorem 7.2. The system described by (35) is globally asymptotically stable with the control given by (42).
(’ I/q,//+qL [ Since
uq,
which reduces to
l/(+11.
The objective is to show that VI is negative on the system’s trajectory. For /(q I(f 0 and I((~11# 0 we have
~tnaxP2lqn ‘D qn 1’1=--_AIIll” llqnll
+ A22nW1 M2 + W2WJ
Proof: lhll
+ S)[& - CZ(SCZ)-~SIG
= &n(MMz
The following theorem verifies the viability of the control structure proposed in (38). Theorem
M2 + S)Y
lhnll + E!e (+m,nIlull
+ (+max IIA:,, - GM;M:A:,,lI lhnll -K,
llul+a~
= (cmax~ - K,A,in)
llqnll+ (F-
KI) II~II
u’ sgn u -aI/a.
F,y + K,fl+ a sgna (Ilall >O), (Ilflll = 01,
-w2~2,-1{ o
(42)
Choosing K0 = vmaxElh,in and K1 = KOp1Jurnin and observing that CT’sgn a/ IICTI(2 1 gives Cl ri, 5 -CY, verifying the theorem.
815
Output feedback VSC The control of (42) can be shown to be of lower norm than the control of (38). 7.3. Control design with boundary layer Both (38) and (39) yield control vectors that are discontinuous on the switching surface. This of course causes chattering due to the fast switching, which might excite unmodeled highfrequency modes (Utkin, 1978; DeCarlo et al., 1988). The boundary-layer approach introduced by Corless and Leitmann (1988) reduces the effect of chattering in the sliding mode by implementing a linear control within a 6neighborhood of the switching surface. Thus the control (39) can be modified to have the form F,Y +
(+maxWlZ
g
+
Q
sgn u
hminamin
(Ilall ‘6))
u3 = -(SC*z?J’ F,Y
+
~maxEPl2 hmingmin
a Cl+--a 6 (Ilall
I
5
6).
(45) The convergence of the system’s output trajectory to the switching surface with the control (45) follows by Theorem 7.2. 8. CONCLUSIONS
AND CLOSING
REMARKS
This paper has investigated static output feedback variable structure control for linear state models from the switching surface design to the actual construction of the variable structure controller. Switching surface design has been considered in depth, with two algorithms for switching surface design being given. The first algorithm converts the problem to an output feedback problem to achieve a prescribed spectrum for a restricted class of output feedback matrices giving rise to a valid switching surface matrix. The second algorithm builds a valid eigenspace around a prescribed set of eigenvalues for the system in a sliding mode from which one can specify a valid switching surface matrix. Two different variable structure controls have been given that guarantee global asymptotic stability of the system. REFERENCES Andry, Jr, A. N., E. Y. Shapiro and J. C. Chung (1983). Eigenstructure assignment for linear systems. IEEE Trans. Aerospace Electron. Svst., AES-19, 71 l-728. Bondarev, A. G., S. A. Bondarev, N. E. Kostyleva and V. I. Utkin (1985). Sliding modes in systems with asymptotic state observers. Automatica, 21, 6-11. Brash, F. M. and J. B. Pearson (1970). Pole placement using dynamic compensators. IEEE Trans. Autom. Control, AC-E,
34-43,
Corless, M. J. and G. Leitmann (1981). Continuous state feedback guaranteeing uniform ultimate boundedness for
uncertain dynamic systems. IEEE Trans. Autom. Controt. AC-M, 1139-1144. Davison, E. J. (1970). On pole assignment in linear systems with incomplete state feedback. IEEE Trans. Autom. Control.
AC-15
348-351.
Davison. E. .I. and R. Chatterjee (1970). A note on pole assignment m linear systems with incomplete state feedback. IEEE Trans. Autom. Control, AC-16,98-99. Davison. E. J. and S. H. Wang (1975). On pole assignment in linear multivariable systems using output feedback. IEEE Trans. Autom.
Control,
AC-20,
Si6-Si8.
DeCarlo. R. A.. S. H. Zak and G. P. Matthews (1988). Variable structure control of nonlinear multivariable systems: a tutorial. IEEE Proc., 76, 212-232. Drazenovic. B. (1%9). The invariance conditions in variable structure systems. Automatica, 5, 287-295. El-Ghezawi. 0. M. E., A. S. Zinober and S. A. Billings (1983). Analysis and design of variable structure systems using a geometric approach. Int. J. Control, 38, 1121-1134. El-Khazali, R. (1992). Variable structure output feedback control with applications to a chemical process and a power system. PhD Thesis, Purdue University, School of Electrical Engineering. El-Khazali, R. and R. A. DeCarlo (1991). Variable structure output feedback control: switching surface design. Proc. 29th Annual Aherton Conf on Communications, and Computing. Monticello, IL. September
Control,
1991, pp.
430-439. El-Khazali, R. and R. A. DeCarlo (1993). Output feedback variable structure control using dynamic compensation for linear systems. In Proc. 1993 American Control Conf., San Francisco, June 1993, pp. 954-958. Hanmandlu, M. and V. Shantaram (1986). Eigenvalue assignment by unitary rank output feedback. Int. 1. Control,
44, 17-42.
Hashimoto. H. and V. Utkin (1991). Output feedback sliding mode control. In Proc. Korean Automatic Control Conf, October 1991, pp. 1412-1414. Heck, B. S. and A. A. Ferri (1989). Application of output feedback to variable structure systems. J. Guidance, l2, 932-935.
Itkis, U. (1976). Control Systems of Variable Structure. Wiley, New York. Kimura, H. (1975). Pole assignment by gam output feedback. IEEE
Trans. Autom.
Control,
AC-U), 509-516.
Kimura, H. (1977). A further result on the problem of pole assignment by output feedback. IEEE Trans. Autom. Control,
AC-22,
458-463.
Luk’yanov, A. G. and V. I. Utkin (1981). Methods of reducing equations of dynamic systems to regular form. Autom.
Rem. Control,
pp 5-13.
Madani-Esfahani, S. M., M. Hached and S. H. Zak (1990). Estimation of sliding mode domains of uncertain variable structure systems with bounded controllers. JEEE Trans. Autom.
Control,
AC-35,
446-449.
Rosenbrock. H. H. (1963). A method of investigating stability. In IFAC Proc., Base& pp. 590-594. Sirisena, H. R. and S. S. Choi (1975). Optimal pole placement in linear multivariable systems using dynamic output feedback. Int. 1. Control, 21, 661-671. Utkin, V. I. (1978). Shding Modes and Their Application in Variable Structure Systems. MIR, Moscow. Utkin, V. I. (1983). Variable structure systems. Automatica, 19. 5-25.
Utkin, V. I. (1984). Variable structure systems-present and future. Autom. Rem. Control. 44. 1105-1120. Utkin, V. I. and K. D. Yang (1978). Methods for constructing discontinuity planes in multidimensional variable structure systems. Automatica, 14, 72-77. Verghese, G. C.. R. B. Fernandez and J. K. Hedrick (1988). Stable, robust tracking by sliding mode control. Syst. Control Lett., 10, 27-34. Weissenberger, S. (1973). Stability regions of large-scale systems. Automatica, 9, 563-663. White, B. A. (1990). Applications of output feedback in variable structure control, In A. S. I. Zinober (Ed.), Deterministic Control of Uncertain Systems, Chap. 8, pp. 145169. Royal Military College of Science. Shriverham, U.K.
816
R. El-Khazali and R. DeCarlo
Wonham.
W. M. (1979). Linear Multivariable Conrrol: A Approach. 3rd ed. Springer-Verlag. New York. Young, K. K. D.. P. V. Kokotovic and V. 1. Utkin (1977). A singular perturbation analysis of high-gain feedback systems. IEEE Trans. Aurom. Control, AC-22, 931-938. &k. S. H. and R. A. DeCarlo (1987). Sliding mode properties of min-max controller and observers for uncertain systems. In Proc. 2% Annual Al&won Conf: on Communications, Control, and Computing. Monticello, IL, September 1987. Vol. II, pp. 813-818.
Define n-m -AI) - 2 o,(A, - A,). ,=?
Geornefric
a,*(Az
Thus there are two cases to consider: (i) a? = 0 and (ii) a3 # 0. If as = 0 then (A.13) yields ,;
Now, for n - m > m 2 2, suppose that N* is rank-deficient. Then there exists a nonzero vector x such that
, n -m.
n
row space [(A,1 -A,,)
In particular, there n - m) such that
exist arbitrary
/ -A,J
vectors
Let i = 1; then for 2 5 j 5 n -m,
(A4
u, (i = 1,.
,
-Alz].
(A.4)
+ (A, - A,)[u; j 01.
n--m 3 a4.r’ = c a, n (A, - A,)[u: 101. ,=‘l I=,
(AS)
-m.
gl,(A,I-A,,)=(A,-A,)u:. glP,z = [O]. 2sj%n
Equation
(A.6)
2sjsn-m.
(A.7)
-mm.
(A.8)
Equation (A.8) implies that there exist n - m - 1 row vectors that annihilate A,,. However, since Al2 is full-rank and of rank m, the maximum number of left annihilators of Al2 is n ~ m - m. Since m 2 2 by hypothesis, n -m - 1 > n - m m. This implies that the set of row vectors Is:,),“=;” is linearly dependent. i.e. there exist constants aj, , a,_,,, not all zero. such that n--m g12 = c +, (A.9) ,=.1 and using (A.7) yields
Again there are two cases to consider: (i) a,, = 0 and (ii) al#O. Using the same arguments as in the previous step leads to the conclusion that “mm 3 ,; o, ,y (A, - A,@: IO]= 101. Continuing in this fashion, and assuming that an_,,, ., f 0, the previous steps described by (A.12)-(A.18) imply a,-, u;_,_,
=
“mm-1 ,F, (A,-,, -A,) u:, n,.
n-m-2
a,-+z
,E, (Lmml -A,)
or, for some scalar p, U;_m_, kpuL_“,.
4=$$$(“,-A,)N:. I
(A.19)
(A.lO)
From the definition (A.3) of x’ and (A.10). we conclude that x’ = &[(A21 - A,,)
~4 e [ ax(A? - A,)(Ax - A>)- ;T; o, ,fi (A, - A,)]x’.
I -A,,1 = (A,- A,)[4 101. (A.6)
Define gl, * u\ - uj for all 21jsn then implies
Postmultiplying (A.9) by Ail-A,,
Again, from the definition conclude that
/ -A,z]
(A.3) of x’ and (A.19), we
x’ = ul- ,,-,](A,m,m,l =~4-,n[(Lm~
-A,,) -A,,)
= P’ + p(L-“t-1
n, 101
- &,)[J&
mm/ 01,
or
q(A, - A,)(,$- Adb: j 01.
Moving the first term on the right-hand side of (A.ll) left and simplifying yields
(A.ll)
to the
(1 - p)x’=p(A,_,_,
- A,.~,,,)[&,
IO].
(A.20)
Clearly, if p = 1 then u,,_, = [0] and x = (01, i.e. rank [NJ = n. But, if p # 1 then (1 - p)x’N, = p(A,-,,_,
[(AZ -A,) - ;&(A,
I -A,21
1 -A121
+ PC& -m--1 - L,)[u:,
-g&f: I
(A.18)
where
Subtracting (A.5) from (A.4) gives
(ui - u:)[(A,l -A,,)
(A.17)
o,(A, - Ai)(A3 - A,)
Observe that (A.17) and (A.ll) propagate in a similar fashion, i.e., substituting from (A.17) into (A.1 1) yields
But _r’= u;[(h,l - A,,) 1-A,J
(A.16)
n-m 2 a,(,$ - A,)(,$ - A&: ,=4
,n -mm. (A.3)
(A.3) yields
x’=u’l[(&I-A,,)[
(A.15)
This implies that the two cases lead to the same conclusion, i.e. the vectors {uj};Za”’are linearly dependent. If a, = 0 for j 5 3 then uz = 0 and _r= 0, i.e. rank (NJ = n. Suppose that as # 0 (if a‘3= 0, one can solve for uq). Solving for u; from (A.15) yields
u: = for i= l,..
/ -Alz]
x’=u:[(h,l-A,,)
a,(A, - A,)(A, - A,)u;N: = [O].
(A.1)
n-m
,=I
,=3
n--m ,z o,(A, - Ai)(A, - A2b: = (01.
Since (A,,.A,,) is complete, and, since A, o w(A,,), it follows that ((A$ - A,,) 1-A,*] is a maximum-rank map. Equations (A.l) and (22) imply that x must satisfy X' =
u3x1NA= c
and (22). we have
However, by Lemma 6.3, the matrix NL is of full row rank. This means that
= 2.m = n.
x’N,, = [0] for i = 1,.
(A.14)
n,(A, - A,)(A, - A&: = ]O].
while if a, # 0 then, from (A.1). (A.ll) APPENDIX A-PROOF OF THEOREM 6.1 First consider the case where n -m = m. or n = 2m. By Lemma 6.2, rank [N*, @NJ
(A.13)
- A,_,)u;_N:
= [O]. (A.21)
- Ai)],’ n-n3 = ,z @,(A,- A,)(A, - A#
101. (A.13
However, N: is a full-row-rank matrix (Lemma Thus u,_, = [0], which means that x = [O]. Therefore [N*] = n.
6.3). rank 0
00051098(94)00151-0
Output Feedback Variable Structure Control Design* REYAD
EL-KHAZALII
and RAYMOND
DECARLOt
Design techniques are described for constructing an output feedback variable structure controller to drive a system’s output trajectory to a stable sliding manifold. Algorithms for the design of the sliding manifold are also given. Key Words-Variable structure control; variable structure systems; sliding states: Lyapunov methods; linear systems; eigenfunctions.
Abstmet-This paper describes a procedure for variable structure output feedback control (VSOFC) for a class of multivariable linear time-invariant systems. The equivalent control method of Utkin in the output feedback mode yields a reduced-order system exhibiting output feedback equivalent dynamics. Using the Kimura-Davison sufficient conditions for pole-placement using output feedback, a sufficient condition is derived to assign the eigenvalues of the ‘reduced’-order system. The observability and controllability of the reduced system is discussed. Two switching surface design algorithms are introduced based on both a static output feedback and on an eigenstructure assignment techniques. A control design is discussed as well.
VSOFC (variable structure output feedback control) have been considered by Heck and Ferri (1989) and White (1990). Verghese er al. (1988) considered robust tracking by sliding mode control for the cases where the number of inputs equals the number of outputs and where the equivalent control is not unique. This paper continues the investigation of VSOFC. The design procedure of a variable structure control (VSC) system is a two-stage process. The first phase is to choose a set of switching surfaces such that the original system restricted to the intersection of the switching surfaces has a desired behavior. The second phase is to determine a switched control law that forces the system’s trajectory to and maintains it on the sliding surface (Utkin, 1978; Utkin and Yang, 1978; DeCarlo et al., 1988). Switching surface design in VSOFC is introduced based on assigning a desired set of eigenvalues to the reduced-order system. The necessary and sufficient conditions for pole-placement using output feedback (Brash and Pearson. 1970: Davison, 1970; Davison and Chatterjee, 1971; Davison and Wang, 1975; Sirisena and Choi, 1975; Kimura, 1975, 1977; Andry et al., 1983; Hanmandlu and Shantaram, 1986) are used to design the switching surface matrix. In this paper, the VSOFC switching surface has the form a(y) = Sy(t), where y(t) E R’ is the output vector and S E R”“’ is the so-called switching surface matrix. The advantage of VSOFC is that it avoids using an observer to estimate the unavailable states while achieving the desired control objective under reasonable conditions. Difficultieis do arise, however: (i) the equivalent system trajectory in the sliding mode is projected onto a lower-dimensional subspace, part of which is fixed and equal to the null space of C;
1. INTRODUCTION
Variable structure systems (VSS) (Itkis, 1976; Young et al., 1977; Utkin, 3978, 1983, 1984; Utkin and Yang, 1978; Bondarev er al., 1985; Heck and Ferri, 1989; White, 1990; El-Khazali and DeCarlo, 1991) are systems having a (high-speed) switched control law. Typical applications of VSS linear and nonlinear systems utilize full-state feedback or observers that require extra dynamics to measure the unavailable states (Bondarev et al., 1985). This increases the complexity of the design procedure, and, in some cases, duplicates the size of the controlled system. This has led to an output feedback approach (Brash and Pearson, 1970; Davison, 1970; Davison and Chatterjee, 1971: Davison and Wang, 1975; Sirisena and Choi, 1975; Kimura, 1975, 1977; Andry et al., 1983; Hanmandlu and Shantaram, 1986). Several authors have considered VSC in an output feedback format. Linear systems using * Received 23 April 1992; received in final form 9 August 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. Utkin under the direction of Editor Huibert Kwakemaak. Corresponding author Professor Raymond A. DeCarlo. Tel. +l 317 494 3523; Fax +l 317 494 3371; E-mail [email protected]. t School of Electrical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. 805
R. El-Khazali and R. DeCarlo
806
(ii) the equivalent control is not unique when the product CB is not of full column rank; and (iii) the observability of the reduced-order system is not guaranteed. Viability of VSOFC requires observability because the reduced dynamics exhibit an equivalent output feedback structure where observability is necessary for pole placement by output feedback (Davison, 1970; Kimura, 1977). Such constraints limit the class of allowable systems. The purpose of this paper is to investigate the VSOFC design for a class of linear time-invariant systems. For simplicity, the discussion here concentrates on systems linear and with ideal switching in the control, (infinitely fast) in the control law (Utkin, 1978, 1984; Utkin and Yang, 1978). The paper is organized as follows. The problem statement and system definition are introduced in Section 2. Section 3 furnishes some preliminaries and background, including the observability of the reduced-order system. Switching surface design algorithms I and II are presented in Section 4 and 6 respectively, while Section 5 investigates system eigenstructure in design is the sliding mode. The control formalized in Section 7. 2. SYSTEM
DEFINITION AND STATEMENT
PROBLEM
Consider the usual linear time-invariant model 1(f) = Ax(t) + Bu(t),
state
having state x E R”, control u E R” and output y E R’. Assumption
1. The system is assumed to be completely controllable and completely observrank[C] =r, and the rank [B] = m, able, Kimura-Davison condition is satisfied, i.e. nsm+r-1. In VSOFC (Heck and Ferri, 1989; White 1990; Verghese et al., 1988; El-Khazali and DeCarlo, 1991), the control component u, of u has two possible structures h(Y) =
u,:(y)
when a;(y) > 0,
u;(y)
when a;(y)
3. PRELIMINARIES
AND
BACKGROUND
Using the method of equivalent control (Utkin, 1978, 1983, 1984; Utkin and Yang, 1978), if the system is initially on the switching surface at to then a(y(t)) = 0 and &(y(t)) = 0 for all t 2 rO. Therefore ti = SCAx(t) + SCBu(t) = 0.
(1)
y(t) = Q(t),
tracking or regulation. This is accomplished in two phases. First phase. Given the system (1) and a symmetric set of n -m distinct numbers A,_, = {A,, , A,_,}, find a real map S E R”“’ such that n - m eigenvalues of (1) in the sliding mode are precisely those of the set A,_, (El-Khazali and DeCarlo, 1991). The first phase presupposes a reduction of order of (1) to an (n - m)-dimensional equivalent system. This is because the ndimensional state dynamics of (1) must satisfy the m algebraic equations g(y) = 0. Such constraints reduce the equivalent system from an nth-order system to an (n - m)th-order system (Itkis, 1976; Utkin, 1978, 1984; Utkin and Yang, 1978; DeCarlo et al., 1988: El-Khazali and DeCarlo, 1991). Second phase. Determine a switched control of the form (2) such that the system’s trajectory is globally attractive (stable) to the sliding surface from any point in the output space (Utkin, 1978, 1983, 1984: Utkin and Yang, 1978).
(2)
depending on the location of the system’s output representative point y(t) with respect to the switching manifold a(y) = 0, defined as (3) L(Y)l’ = SY(l) c+(Y)=[a:(Y) ... where S is an m X r real parameter matrix to be designed (Heck and Ferri, 1989; White, 1990). It is desired to choose S to stabilize the system in the sliding mode or to achieve some form of
(4)
This results in the equivalent control law (Heck and Ferri, 1989: White, 1990; Verghese et al., 1988) u,,(t) = -(SCB)-‘SCAx(t), (5) which exists uniquely if SCB is nonsingular. Clearly, the equivalent control is not unique when CB is rank-deficient or when r
2. r > m, S is full row rank, CB has full column rank, and SCB is nonsingular.
Given Assumption 2, substituting yields the equivalent system i(t) = [(1,, - B(SCB)-‘SC)A]x(t), y(t) = Cx(t),
(5) into (1) (64 (6b)
807
Output feedback VSC which represents the system dynamics in the sliding mode. The goal is to choose the function u = Sy(t) and a switched output feedback control of the form (2) to adjust the state dynamics of (6) to achieve a desired behavior of the output trajectory. To design a switching surface of the form (3), one needs to investigate the equivalent system (6a). If I = m then, by Assumption 2, S is a nonsingular transformation of the switching surface Cx(t) = 0; hence the switching surface dynamics Sy(t) = Xx(t) = 0 is invariant with respect to S (DeCarlo et al., 1988). On the other hand, if r > m then the switching surface dynamics depends on N[S] (Heck and Ferri, 1989; White, 1990; Verghese et al., 1988; El-Khazali and DeCarlo, 1991). To see the nature of this dependence, it is convenient to put the system (1) into the regular form (Luk’yanov and Utkin, 1981)
\‘J Y=
[Cl
where B, E R”“” is nonsingular and where z, E R”-“, z: E R”, A,,, Arz, &, At2, C, and C2 are real matrices of appropriate dimensions. The switching surface in the new coordinates is V(I, 9z2) = S[C,
C,l[ 3
= PI.
(8)
By Assumption 2, CB = C2B2 is of full column rank. Since Bz is an )12X m nonsingular matrix, rank [C,] = rn, i.e. CZ has full column rank. Suppose that the system’s output representative point is initially on the switching surface; then c(z) = 0 and b(z) = 0. Solving (8) for z2 and substituting into (7) yields the reduced-order dynamics i, = (A,, - &(SC2)-'SC,)Z,
Proposition 3.1. Let K E R”“’ and C2 E R’“” be full rank of rank m, r > m, and let S E R”“’ be chosen such that SC2 is nonsingular. Then a necessary and sufficient condition for there to exist a solution S of the equation (SCJ’S = K is KC2 = I,,,.
(9)
where K + (X2)-‘S represents an output feedback gain map; Al2 and C, appear respectively as input and output matrices of the reduced order system. S must be chosen such that SC2 is nonsingular. Switching surface design reduces to finding S so that the reduced-order system of (9) has a desired spectrum. i.e. cr(A,, - A,,KC,) = A,_, (El-Khazali and DeCarlo, 1991; Hashimoto and Utkin, 1991). Equation (9) exhibits an output feedback structure in K. Sufficient conditions exist for a solution to the spectral assignment problem (Kimura, 1975, 1977). However, under what conditions does there exist an S such that the
(10)
Proof: Necessiry. Let S E R”‘“’ be such that = K. Right multiplication by C2 gives
(X2)-‘S
= KC2 = I,,,.
(SC,)-‘SC, Sufficiency.
With KC2 = I,,,, let S = K. Then
(SC2)-‘S=(KC2)-,K=I,,,K=K.
0
If a solution to (SC,))‘S = K exists, it must satisfy (lo), i.e. K is a left inverse of C2. Hence switching surface design reduces to finding a gain feedback K that assigns A,_, = (A,, . . . , A,_,,,$ to the reduced-order system (9) subject to the constraint given by (10). From the proof of Proposition 3.1, S = K is an acceptable switching surface matrix. Proposition 3.2. Let C2 E R’“” and rank [C,] = m < r, and let K E R”“’ satisfy KC2 = I,,,. Then
the set K, of all solutions of (10) is nonempty and is given by K, = {K: K = CTL + TM’}
(11)
where r E R”““-“’ is an arbitrary parameter matrix, and where the rows of M’ E R”-“‘“’ are a basis for the left annihilator of C2, i.e. MC2 = [O]. Substituting the form of K given by (11) into (9) yields ’ = 61
'(A,, -A,XC,)Z,, y = (L - C:!K)C,z,.
spectrum is assigned? This constrains the class of KS. The next two propositions characterize the class of permissible K matrices.
(all
-
A,J?,)z,,
(12)
where a ,, k (A,, - A,2C;LC,) and C, GMT,. The constraints imposed on K limit the available feedback. The question now is what freedom remains to assign the proper eigenvalues to the reduced-order system. This requires that the submatrices of the reducedorder system, A,2 and Cl, have certain structural properties. This will be investigated in Section 4. 3.1. Observability of the reduced-order system Equation (12) defines the reduced-order system in terms of a new output matrix Cl and a new system matrix a r,. The controllability and the observability of the triple (Cl, ii ,, , A,2) is necessary to assign a desired set of eignevalues to (12). The controllability of the pair (A, B)
808
R. El-Khazali and R. DeCarlo
guarantees
the controllability of the pair (Young et al., 1977). Since a,, can be seen as a state feedback modification of A,, through the input matrix A12, by Lemma 2.1 (Wonham. 1979) the pair (All, A12) is also controllable. Hence sufficient conditions for pole-placement by static output feedback are observability of (Cr, AlI) and the satisfaction of the Kimura-Davison condition (Kimura, 1977). Observability of the reduced system is not guaranteed. Even if the pair (C, A) is observable, the pair (C,, A,,) may not be. On the other hand, it is possible to have (C, A) not be observable with ( C1, A, 1) observable (El-Khazali and DeCarlo, 1991). In view of (12), one requires the observability of the pair (Cl, a,,) and controllability of @rl, A& which is already guaranteed. How does the observability of (C,, AlI) affect the observability of (Cr, a,,)? If (C1,All) is observable then ( C1, a 1,) is also observable, because CTLC, can be viewed as a simple output feedback map that does not affect observability. Further, if Im [C,] n Im [C,] = 0 then the pair (Cl, a ,,) is observable. This is of course a sufficient condition. It is possible to construct an example such that if Im [C,] n Im [C,] Z 0 then the pair (C, , a 11) is not necessarily observable (El-Khazali and DeCarlo, 1991; El-Khazali, 1992). If the pair (C,, a 11) is not observable but is detectable then it is possible to show that the reduced-order dynamics are stabilizable (Wonham, 1979). The implication is that one must check (C, , a, 1) for observability. (A,,, A,,)
Assumption 3. The pair (C,, a 11) is observable. 4. SWITCHING
SURFACE
DESIGN
ALGORITHM
I
From Assumption 1, it is possible to assign the spectrum of the triple (C, A, B) by output feedback (Kimura, 1975, 1977) provided that a slight modification of the poles to be assigned can be tolerated. To apply this result to (12), recall that a 1, E R(“-m)x(n-“‘), Al2 E R(n-m)xm, (+,E R(r-m)x(n-m) and n %m + r - 1 (i.e. the Kimura-Davison condition is satisfied), with m?2andr>m. Assumption 4. A,_, = {A,, . . , A,_,} metric set of n - m complex numbers.
is a sym-
Theorem 4.1. Given Assumptions 1-4, if rank [A,,] = m (full column rank) or rank [AI21 = n -m (full row rank) then there
exists a real matrix I E R”““-“‘I at least - A,,Te’,) = A,_,, 6, arbitrarily small neighborhood.
such that within an
Proof
First the pair (a 11,AIZ) is controllable (Young et al., 1977). Hence the triple (Ci,aii, A,*) is complete. The result follows from Kimura (1975) provided that n - m I rank [AI21 + rank [C,] - 1.
(13)
Since M’C, = [O], rank [C,] = m and rank [C, C,] = r > m, we have rank [C,] = rank [MC,] = r - m. Since the original system satisfies the Kimura-Davison condition, n m%r-l=m+(r-m)-1. If rank[A*J=m then (13) is satisfied; if rank [A,,] = n -m then n-mI(n-m)+(r-mm)-1. Since r-mZ1, (13) is again satisfied. !Zl Once I is found using any existing output feedback algorithm (Andry et al., 1983), we simply set S = K = CgL+ rM’ to obtain the switching surface map. Other choices of S are possible. The following proposition characterizes all possible Ss associated with a fixed gain K. Proposition 4.1. Let K E K, and rank [S] = m. A switching surface matrix S satisfies (SC,)-‘S = K if and only if
SIZI - C2K] = [0] Proof: This follows by Propositions
(14)
3.1 and 3.2. 0
If n > m + r - 1 and rank (AJ
EIGENSTRUCTURE MODE
IN THE
SLIDING
The relationship between the eigenvectors of the original system and those of the reducedorder system will allow us to construct a second switching surface design algorithm. By Theorem 4.1, there exists a constant matrix K E HIX, such that A(A,, - A12KC1) = An_,,, = A,_,}. let {e,, . . . , en-,,,} E C”-” be the FL,...,
809
Output feedback VSC associated AlzKCl)e,,
right eigenvectors, which implies
i.e. A,e, = (A,, -
Theorem
5.1. Under
Proposition
.\(A,, - A,:KC‘,)
[(Ail - All) / -&I[-&]
1I
= LO]. (15)
Define u, = [e: 1 -(KC] e;)‘]’ for i = 1. . , n - m. The set {u,, , u,__~} denotes the n -m with associated eigenvectors of (6) {h, , . . , A~_,,,} in the sliding mode, as verified in the following proposition. Proposirion 5.1. Let (C, A, B) be in the regular form as per (7). Let K E K, be such that h,e, = (All - A,2KC,)e,, and define u, = [e: ) -(KC,e,)‘]’ for i = 1, . . , n - m.
Then {u,, . , u,_,} are n - m eigenvectors (6) in the sliding mode.
of
Proof Let S = K. In the sliding mode SCz = KICI C,]z = [KC, i,,Jz =0 by Proposition 4.1.
the
assumptions
of
K E R”‘“’ be such that = A,_,. Suppose that 3;; =
5.2. let
where span 121,. , v,,_,}, u,[e: j -(KC,e,)‘l. Then N[C] c ‘r; if and only if [KC, - Z,,,]VP = [O].
N[C, j C,] = Im It’\” j Vs’]‘. and VP E R”‘xtn-r’. with definedby(15)fori=l,....n-m.
where
R,,lF,,,1XOi-r)
(17) vy E
the
e,
Proof: Sufficiency. Suppose that (17) is true. Let K = K, + K,, where K, E K,. i.e., K,Cz = l,,.
Then sufficiency follows from Proposition 5.2 by letting K, = [O]. Necessity. Let N[C] c 7% with 7; as stated above. Suppose that N[C, / C,] = span {r,,
, r,,-, 1.
where r, = [r:, 1rf?,l’. Let w = [w: ) IV>]’ satisfy
After representing (6) in the I coordinates, i.e. in the regular form, and forming the standard eigenvalues-eigenvector equation, the result 0 follows by direct calculation. Define V,, = [ui II,_.,]. Since the n - m column vectors of the matrix V, are eigenvectors of the system in the sliding mode, and since the system is projected onto N[SC] cN[C] (ElKhazali. 19921, N[SC] = ‘& where ‘& = Im [Vd] and A[(A + BF)INlscl] = A,_, for some F (Wonham. 1979). Consequently, any desired eigenstructure in the sliding mode must contain N[C]. If -& is selected properly so that N[C] c 2’, then rank [CV,] = r - m. This means that there always exists a full-row rank matrix s E R’““‘. r > m, such that SCV, = [O].
(16) If one can construct an algorithm that produces a proper set of eigenvectors. denoted by the columns of V+ such that N[C] c ‘& then there always exists a proper switching surface defined by an S satisfying (16). Equation (16) is the key to introducing a switching surface design algorithm based on constructing 2; without the need to first determine an output feedback matrix K. In order to do this. it is necessary to characterize the general class of output feedback matrices K that guarantee N[C] c 2/-,. Proposition 5.2. Given Assumptions 1-4 with m > 2 and A, @h(A, ,), consider the reducedorder system given by (12). Let K E Ki, (11). If rank [A,*] = min (m, n - m) then there exists an (n - m)-dimensional subspace y&, where 2/-,= (~1,. . . . u,~-,}, and U, = [e: 1 -(KC,ei)t]t, such that N[C] c ‘J”,.
for some constants, zero. Then
y,, i = 1,.
, n - m, not all
c, w1 + czwz = [O].
(18)
Obviously, w2 E Im (L’S). Since {e,, . . . en_,,,} are linearly independent, {u,, . , u,,+} are independent. Hence there exist constants, not all zero. such that PI>. . . 7Pn-t?,,
“I =‘y [ W,
I
p,u;
=
,=,
It follows that w:!= -KC,w,.
(19)
(18) by K and using (19) yields [KC2 - I,,,]w~ = [0] for all w? E Im [Vz]. 0
Premultiplying
Observe that the second class of output feedback matrices, (17), is larger than the first class of (31). Both classes guarantee the satisfaction of N[C] c ??Y. Proposition 4.1 and 5.1 guarantee that an admissible full-rank switching surface matrix exists if and only if K E K,. One of the objectives, detailed in the next section, is to use the result of Theorem 5.1 to algorithmically generate a set of admissible eigenvectors, denoted by the columns of V,, that contain N[C] in their span. If this is done, then rank [C&J = (n - m) - (n - r) = r - m. However, S must be a m X r matrix, full-rank in which case
810
R. El-Khazali and R. DeCarlo
dim (N[S]) = r - m. Hence (16), SCV, = [O], is satisfied if and only if N[S] is precisely Im [C&J. Therefore if one can generate a set of admissible eigenvectors whose span contains N[C] then any full-rank S whose kernel is precisely Im [CVd] is a legitimate switching surface matrix. 6. SWITCHING
SURFACE
DESIGN ALGORITHM
problem now is to determine a procedure to construct Vd1 N[C]. Once Y$ is found, a proper switching surface matrix may be selected by solving (16) for a full-rank S. The following discussion furnishes the necessary foundations for the development of the design algorithm. Consider the system (1). If the pair (A, B) is controllable and in the regular form then, for a desired set of eigenvalues A,, = {A,, . , A,}, there exists a constant state feedback matrix F E R”“” such that A(A + BF) = A, (Wonham, 1979). Let {ul, . . . , u,} be the associated right then (A + BF)u, = Aiug. Some eigenvectors; algebraic manipulation yields for i = 1, . . , n ( -I?][;]
= [O]. I
Now for each
Ai E C, let the columns
of
Th, s [N’*, ( M:J’ form a basis for the null space of [(Ail -A) 1 -B], where for i = 1, . . . , r~
[Ail - A ( -BIT,, - [Ail - A ( -B][ $1
= [O].
A,
(21) Clearly, Ui E N,,, (Andry et al., 1983). Utilizing the regular form of (A, B), (21) has the structure
(Ail,-m- AlI)
[
Proof: From (22)
-AI~][$] =[O]. A,
[(AL-m-All) 1 But
(23)
- Ail) 1 -&21[ 3) 2 = rank [-A ,2 B2] = m since B2 is an m X m nonsingular map and A,2 is an (n -m) X m matrix of rank m. Hence N,,, rl Im (B) = 0 for i = 1, . . . , n - m. 0 The following result shows that each pair of Nh, and Nh, is disjoint for distinct eigenvalues. Lemma 6.3. let (A,,, A12) be controllable, N*, be defined by (22) and (23), Ai @A(A,,) and n - m 1 m. If rank [Al21 = m then (i) N,,nN,,=O for Ai#Ai; i,Z=l,... , n-m; (24) (ii) rank [N:] = m for i = 1,. . . , n -m; (iii) rank [N:] = n - m. Proof The proof of (ii) and (iii) is straightforward. To prove (i), suppose that for two distinct eigenvalues Ai # Aj there exists a nonzero vector w E N,, n N,,,. Then, from (23),
](AiZ,-, - A,J 1 -A121~ = PI, [(A$n-m -All)
I -A121~ = [Ol.
Subtracting yields
-A,2
--A21
6.2. Let (A, B) be in the regular form and complete, rank [B] = m, n -m zm, and Ai g A(A,,). If rank (A,,) = m then Im (N*,) fl Im (B) = 0 for i = 1, . . . , n - m.
II
The
[Ail -A
Lemma
=
[Ol,
[(A; - Aj)Zn_, 1O]W= [O]. Thus
(22) where Nn, = [(N:,)’ ] (NfJ]‘; N:, E R(“-m)xm and N;, E R”““. The following results will be useful in the development of the design algorithm. Lemma 6.1. Let (A, B) be complete, m. Suppose that A, z A(A), i = 1,.
rank [B] = , n. Then the columns of NA,are linearly independent. Proof. As the columns of NA, are given by the 0 eigenvectors u,, its columns are independent. Note that, owing to the reduction of order in the sliding mode, the condition A, e A(A) can be relaxed to require only Ai @ A(A,,) for i = 1, . . . , n - m. This restriction is necessary for the following results.
w E Ker [(A, - Ai)Z,_,,, IO]. Hence w = [O*I ~$1~and, from (23), [(Mw,,
- AlI) 1 -A,21[ 3
= -A12~2 = [Ol.
This implies the contradiction linearly dependent columns. The following important the construction of V&
that
Al2 has q
result is the key for
Theorem 6.1. Let (All,A12) be complete, n - m rm, A,,-,,, be a symmetric set of eigenvalues with Ai e A(A,,), i = 1, . . . , n - m, m 2 2, Nh, be defined by (22) and (23), and
N,, k [Nh, (. . . I NA._,J If rank [AI21 = m then rank [NJ = n.
(25)
811
Output feedback VSC Proof.
Step 1. The
first 2(r - m) vectors chosen such that
See the Appendix.
It is possible to show that the result holds even ifn-m
cur-,+, = cu, cu,-,+* = cu*
(i) N,,, n Nlh,= 0,
N1* E W’x’n-m).
’
i Zj;
Step 2. The last n + m - 2r vectors,
(27)
are chosen in an arbitrary fashion as a linear combination of the first r - m vectors. One possible choice is
C%(r-m+,) = cu, + cu3, cl&,
Proof It follows from (23) and (26) A12No = [O]. Moreover, u, E Im (N,J for 1, . . , n - m. Applying Theorem 6.1
that i =
and Lemmas 6.2 and 6.3 to this case yields the above corollary.
6.1. Algorithm development The previous discussion suggests an alternative approach to the switching surface matrix design introduced in Section 4. To avoid determining K. one may select an appropriate eigenspace, cl/^d=~ N[C]. The construction of the desired subspace ‘7; = span {u,, . , II,_,}, is done so that u, E Im (N*,) and the set {u,. . . , u,-,,,} is linearly independent. Since Theorem 5.1 and Proposition 5.1 verify the existence of 2’, 3 N[C] and the independence of {ul, , u,_,,,}, the designer must guarantee. by construction, that “r/-, is an (n -m)dimensional subspace. Recall that dim (N[C]) = n - r. Let V, = [ul, . . . , u,_,]. There exist nonzero vectors gj E R” such that u, = N,,g,. In order to have N[C] I.=‘J$ and guarantee the existence of a proper S, (16), one must require rank [CV,,] = r - m. This is simply satisfied if , Cu,,_,) C Im (Cu,, .
. . * C~n-ml,
CuZ(r-m,+1 = cu, + cuz,
(ii) N,,, rl Im (B) = 0 for i = 1, . , n -m; (iii) rank [NA,, . . , NAn__]= n.
Im(Cu,-,+,,
(29)
= cur_,. CUZ(r-m)
IC%(r-m)+l,
where NoAE IWnx(*“--“) and Then for distinct eigenvalues
in (28) are
, Cu,_,),
=
(30)
cu, + . + cl&_,.
Since u, = N,,,g,, if r - m = 1 then (28) and (29) yield CN*,g, = CN,,g,, for i = 2, , n - m. In matrix form,
The matrix on the left is r(n - r) X m(n - m), has full row rank and has r - m = 1 more columns than rows. Thus it has a nontrivial solution in which CN,,g, # [0] for i = 1, ,n m, because u, er N[C]. Since (31) has a nontrivial solution (all other nontrivial solutions are scalar multiples of it), by Theorem 5.1 and Proposition 5.1 the vectors {u,, , u, _} are linearly independent. On the other hand, when r -m 2 2. the matrix-vector equation that results from (29) and (30) takes the following general form:
(28) provided that span {u,, . . . , u,_,} O N[C] = 0. One must also ensure that rank [Vd] = n - m, i.e. any solution of (28) must guarantee the independence of {v,, . . . , u,_,}. Since u, E Im (NA,), (20) and (21), by Proposition 5.1 a linearly independent set of eigenvectors of the system in the sliding mode does exist. How does one constructively achieve the independence of the u,? To satisfy (28), the relationships between the various vectors may be chosen as follows.
g1 ET2
&m+l
=
gr-m+z &T-m
_
(32)
812
R. El-Khazali and R. DeCarlo
The number of nontrivial solutions to (32) is greater than or equal to r - m. Since ‘& exists by Theorem 5.1. there exists an appropriate solution. i.e. an acceptable set of g,, that satisfies (32) and ensures the independence of the vectors {u,, . . , u,, -,,,). In particular. let p0 be any nontrivial solution and let P be a matrix whose columns are a basis for the null space of the matrix on the left-hand side of (32). Generally, PO + PcY
for a randomly chosen LYwill yield an acceptable solution. The following summarizes the discussion of this section and introduces steps of the design algorithm. The switching surface design algorithm.
the triple (C,, a ,,, A,2) is and A,, have full ranks, nsmfr-I. and whether r >m. If so, record the dimensions of A, I, and N[C]. For a given symmetric set of n -m distinct eigenvalues. A,, ,), = {Al,. . , A,,_,}, determine N,, from (22) or (23). Depending on 11- r and r - m, consider the following cases: (i) if r - m = 1 then solve (31) for the gi; (ii) if r - m ~2 then solve (32) for the g,. Construct ‘Ii, = {u,, . . , II_,,,}, where u, = JV&,. Finally. solve (16) for a full-row-rank S.
1. Check whether complete, CB
2.
3.
4. 5.
An example of the algorithm can be found in El-Khazali (1992). 7. CONTROL
DESIGh’
The second phase of VSOFC design is to determine a switched control that drives the system’s output trajectory to the switching manifold. In the subsequent discussion, we assume that the linear system (1) satisfies the conditions n 5 m t r - 1, r > rn, rank (AJ = and min (rrt, n - m ). the triples (C, A, B), (c,, A,, , A ,:) are complete, and rank (CB) = tn. With these structural properties, the system (1) is stabilizable using static output feedback (Kimura. lY77). In addition, there exists a switching surface matrix S such that iZ(A,, A,,(SC:) ‘SC,) = :I,,_,,: and. with the constraints u = 5~. = IO]. we have n(A - B(SCB)
‘(SCA))
= {A_,
0,.
. , 0},
where :2,,_,,, is a stable symmetric set of n -m distinct real eigenvalues. by Theorem 4.1. A traditional approach in VSS control design is to use the equivalent control ucq as part of the control vector (Utkin. 1978), i.e. the control
takes the form ti = l&Zq +f(y),
(33)
where f(y) is a suitable switched function that guarantees convergence to the switching surface. ueq = -(SCB)-‘SCAX Implementing alone would cause the system’s output trajectory to slide onto a hyperplane parallel to the switching surface. However, the requirement of output feedback precludes the use of ueqr as now discussed. If there exists a real matrix G E R’“’ such that CA = GC then one can write u eq = -(.SCB)-‘SGCx = -(SCB)-‘SGy, and the equivalent control in this case is an explicit output feedback control. Sufficient conditions for the existence of G are given by iak and DeCarlo (1987). Such conditions require in essence that the row vectors of the output matrix C be the left eigenvectors of the system matrix A. A necessary and sufficient condition for the existence of G is given in Proposition 7.1. The sufficiency part is equivalent to the one given by iak and DeCarlo (1987). Proposition 7.1. Let the earlier assumptions hold. Define v0 = N[C]. There exists G E R’“’ such that CA = CC if and only if AY’,’
ZrO.
(34)
Proof Sufficiency. Let (34) be true. Let V, = [u, . u,_,] be a matrix whose columns are
bases for Cy-,= N[C]. where by hypothesis rank [C] = r. Let {c,, . , c,} denote the columns of C’. Then {c,, . , c,, uI, , u,_,} is a basis for R”. Consider that CA[c,
..
c,
u1 = [CAc,
u,_,] .
CAc,
0
. .
0]
= [CAC’ / 01. Thus (34) is solvable for G = CAC’(CC’)-‘. Necessity. Let G E R”’ be such that GC = CA. Then GCV, = CA,V = [O]. This implies that 0 Im (A V,) c 2<,or A V, c “u;,. Satisfaction of the condition (34) implies that 2’, is A-invariant; i.e., for G to exist such that GC = CA, the pair (C, A) must be unobservable. Since (C, A) is observable by hypothesis, no G exists such that GC = CA; i.e. the equivalent control cannot depend explicitly on the output variables. In constructing the output feedback control, let us decompose ueq into two parts: one in terms of some of the measurable output variables and
813
Output feedback VSC the other in terms of the unmeasurable state variables. This can be accomplished by introducing another coordinate transformation that will eventually simplify the control design. 7.1. System in normal form Consider the system (7) already in the regular form and suppose that a proper switching surface S is selected such that A(A,, matrix with AIZ(SCZ)-,SC,) = A,_, = {A,, . . , A,_,} Re [Ai] < 0. The control design is simplified if one views the system (7) in a new coordinate frame called the normal form. This is achieved by the linear state transformation (Young et al., 1977)
7.2. Output feedback control design The control design in VSOFC requires the convergence of the system’s representative point to the switching surface: thus a switched control must be used to drive the system’s trajectory to the switching manifold. The switched controller is also used to achieve robustness. model reduction and possible disturbance rejection (Utkin, 1978; Heck and Ferri, 1989: Corless and Leitmann, 1981; Madani-Esfahani et al.. 1990). Before discussing the design procedure. let us introduce some necessary definitions. For i, j = 1, 2 let plj = IIA,,,II denote the spectral norm of A ,,“, i.e. IPLII = max {IIA,,,xll : /Ix// = 1). Also define A,,, = min{lh,l, . , /A,_,I}, where A, E 4,-m.
Let us propose the following output feedback control candidate: Since SC2 is nonsingular, T is nonsingular and T-’ is straightforward to compute. The system in the new coordinates is
A 22n(+ +
u, = -(SC2&_’
P12P21~mx
u + ff
sgn u
hn,n~m,n
(Ilull >a (ll~ll = Oh
AI,”
Am
A 2,n
.,,,1[:1
(38)
+ [S&&
Y =
[Ur -- c2(sG--‘w’
(+=
to1z$]
S],
I c*w2r’I[
(36)
,
where (Y> 0, sgn u = [sgn u1 . . sgn u,,,],, sgn u is the usual Signum function. and u_ and u,,, are the maximum and minimum singular values of a nonsingular matrix U E C(nPm)u(nP”‘, that diagonalizes A, ,“, i.e. U-‘A,,U=D=diag(A
where the subscript ‘n’ denotes normal form and A ,,“-A,,
-A&G-‘SC,,
A ,2n--A,z(SCJ1, A 2,n * SC,A,, -
-t SC>Az,
(Sc,A,l + SCzAz)(ScJ’SC,,
A zzn+ (SC,A,l
+ SCzA&SCz)-,,
while the equivalent coordinates is given by u eq = -(SG&-‘&,nq s &&)
Theorem matrix.
in
S
be
the
new
- (SWJ1&n(+
+ %2((+).
The equivalent system o=O)yieldsrj=A,,,qanda=O. Remark. Let AIZ(SCZ)-iSC,)
control
dynamics
(37) ((9) with
such that A(A,, as per Section 7.1 and 4.1. It follows that A,,” is a stability = A,_,
,,...
.A,_,,)
(39)
The first and third terms of (38) were introduced by Madani-Esfahani et al. (1990), who made use of u_, (37), directly. It is not possible to use ueq of (37) because the term A,,,q contains state information that is not directly accessible at the output. The term is inserted into the (~12~21UmaxlhrnlnUrnin)(+ control structure of (38) to overcome this problem. Note also that for u = 0, II, = &u, which represents a relay control. An alternate to the proposed output feedback control structure of (38) was introduced by Heck and Ferri (1989). The control is of the form u = -(SCB)-‘[SCANy + sat (a)], where sat (.) is the usual saturation function. However, this controller requires the existence of a matrix N E R”“’ such that (SC)‘(SC)A(I - NC) is negative-semidefinite. The justification of the control (38) proceeds more smoothly via a state transformation
[:I=g-l-+y~l
C40)
which diagonalizes A,,” for U chosen as per (39).
814
R. El-Khazali and R. DeCarlo
The system and switching surface coordinates are
in the new
where cy> 0 and F0 = A&M, M2 + S), with M2 a maximal-rank (orthogonal) matrix such that M,C, = 0 and with M, chosen as the leastsquares solution to A 21”
-
A2znMM2C,
=
0.
With MI so chosen, define c = llA21n - A22nM,M2C, IIF.
(43)
It follows that (41)
F,Y = &@I
7.1. The
asymptotically control (38).
system (41) is globally stable with the output feedback
Proof
Consider the following candidate Lyapunov function (Rosenbrock, 1963; Weissenberger, 1973):
v, =-=,a,*
P21
nun
+
Using the system (41) with the control follows that + ~(TmaxP21 SL
u-~A
ll%Ill
ku”
F,Y = &J%MzCI
(P12/“min)
PX~,,,~~
and ll9nII llDll llqnllII~II, it followsthat
since a’sgn cr/l)(~ll > 1.
l,a,l
(44
v, = & llqnll+ 11~11. Differentiating
with respect to time yields
(38), it + jf$ ~
(AzlnQn
+ A22ng
-_,y-K,a-asgna)
12n
=
W--‘A,,,
4J’qn
K
qf,Z.-‘A,2,a
I
~‘&n&n
5
WA:,,
-K,
<_-CUP
A drawback of using u, is that it does not utilize the full output vector y, possibly leading the due to A control that utilizes the complete output vector is u* =
I
u
llul,-a+
5 -K&n,,
Cl
ll(l”ll
- C’, M;M: Ai2”)
llflll
u’ sgn u v,l--a
uq, + A22nu.
Let a Lyapunov function be given by
+ qhDq, 5 -Amin llqnl12,
‘~9
Theorem 7.2. The system described by (35) is globally asymptotically stable with the control given by (42).
(’ I/q,//+qL [ Since
uq,
which reduces to
l/(+11.
The objective is to show that VI is negative on the system’s trajectory. For /(q I(f 0 and I((~11# 0 we have
~tnaxP2lqn ‘D qn 1’1=--_AIIll” llqnll
+ A22nW1 M2 + W2WJ
Proof: lhll
+ S)[& - CZ(SCZ)-~SIG
= &n(MMz
The following theorem verifies the viability of the control structure proposed in (38). Theorem
M2 + S)Y
lhnll + E!e (+m,nIlull
+ (+max IIA:,, - GM;M:A:,,lI lhnll -K,
llul+a~
= (cmax~ - K,A,in)
llqnll+ (F-
KI) II~II
u’ sgn u -aI/a.
F,y + K,fl+ a sgna (Ilall >O), (Ilflll = 01,
-w2~2,-1{ o
(42)
Choosing K0 = vmaxElh,in and K1 = KOp1Jurnin and observing that CT’sgn a/ IICTI(2 1 gives Cl ri, 5 -CY, verifying the theorem.
815
Output feedback VSC The control of (42) can be shown to be of lower norm than the control of (38). 7.3. Control design with boundary layer Both (38) and (39) yield control vectors that are discontinuous on the switching surface. This of course causes chattering due to the fast switching, which might excite unmodeled highfrequency modes (Utkin, 1978; DeCarlo et al., 1988). The boundary-layer approach introduced by Corless and Leitmann (1988) reduces the effect of chattering in the sliding mode by implementing a linear control within a 6neighborhood of the switching surface. Thus the control (39) can be modified to have the form F,Y +
(+maxWlZ
g
+
Q
sgn u
hminamin
(Ilall ‘6))
u3 = -(SC*z?J’ F,Y
+
~maxEPl2 hmingmin
a Cl+--a 6 (Ilall
I
5
6).
(45) The convergence of the system’s output trajectory to the switching surface with the control (45) follows by Theorem 7.2. 8. CONCLUSIONS
AND CLOSING
REMARKS
This paper has investigated static output feedback variable structure control for linear state models from the switching surface design to the actual construction of the variable structure controller. Switching surface design has been considered in depth, with two algorithms for switching surface design being given. The first algorithm converts the problem to an output feedback problem to achieve a prescribed spectrum for a restricted class of output feedback matrices giving rise to a valid switching surface matrix. The second algorithm builds a valid eigenspace around a prescribed set of eigenvalues for the system in a sliding mode from which one can specify a valid switching surface matrix. Two different variable structure controls have been given that guarantee global asymptotic stability of the system. REFERENCES Andry, Jr, A. N., E. Y. Shapiro and J. C. Chung (1983). Eigenstructure assignment for linear systems. IEEE Trans. Aerospace Electron. Svst., AES-19, 71 l-728. Bondarev, A. G., S. A. Bondarev, N. E. Kostyleva and V. I. Utkin (1985). Sliding modes in systems with asymptotic state observers. Automatica, 21, 6-11. Brash, F. M. and J. B. Pearson (1970). Pole placement using dynamic compensators. IEEE Trans. Autom. Control, AC-E,
34-43,
Corless, M. J. and G. Leitmann (1981). Continuous state feedback guaranteeing uniform ultimate boundedness for
uncertain dynamic systems. IEEE Trans. Autom. Controt. AC-M, 1139-1144. Davison, E. J. (1970). On pole assignment in linear systems with incomplete state feedback. IEEE Trans. Autom. Control.
AC-15
348-351.
Davison. E. .I. and R. Chatterjee (1970). A note on pole assignment m linear systems with incomplete state feedback. IEEE Trans. Autom. Control, AC-16,98-99. Davison. E. J. and S. H. Wang (1975). On pole assignment in linear multivariable systems using output feedback. IEEE Trans. Autom.
Control,
AC-20,
Si6-Si8.
DeCarlo. R. A.. S. H. Zak and G. P. Matthews (1988). Variable structure control of nonlinear multivariable systems: a tutorial. IEEE Proc., 76, 212-232. Drazenovic. B. (1%9). The invariance conditions in variable structure systems. Automatica, 5, 287-295. El-Ghezawi. 0. M. E., A. S. Zinober and S. A. Billings (1983). Analysis and design of variable structure systems using a geometric approach. Int. J. Control, 38, 1121-1134. El-Khazali, R. (1992). Variable structure output feedback control with applications to a chemical process and a power system. PhD Thesis, Purdue University, School of Electrical Engineering. El-Khazali, R. and R. A. DeCarlo (1991). Variable structure output feedback control: switching surface design. Proc. 29th Annual Aherton Conf on Communications, and Computing. Monticello, IL. September
Control,
1991, pp.
430-439. El-Khazali, R. and R. A. DeCarlo (1993). Output feedback variable structure control using dynamic compensation for linear systems. In Proc. 1993 American Control Conf., San Francisco, June 1993, pp. 954-958. Hanmandlu, M. and V. Shantaram (1986). Eigenvalue assignment by unitary rank output feedback. Int. 1. Control,
44, 17-42.
Hashimoto. H. and V. Utkin (1991). Output feedback sliding mode control. In Proc. Korean Automatic Control Conf, October 1991, pp. 1412-1414. Heck, B. S. and A. A. Ferri (1989). Application of output feedback to variable structure systems. J. Guidance, l2, 932-935.
Itkis, U. (1976). Control Systems of Variable Structure. Wiley, New York. Kimura, H. (1975). Pole assignment by gam output feedback. IEEE
Trans. Autom.
Control,
AC-U), 509-516.
Kimura, H. (1977). A further result on the problem of pole assignment by output feedback. IEEE Trans. Autom. Control,
AC-22,
458-463.
Luk’yanov, A. G. and V. I. Utkin (1981). Methods of reducing equations of dynamic systems to regular form. Autom.
Rem. Control,
pp 5-13.
Madani-Esfahani, S. M., M. Hached and S. H. Zak (1990). Estimation of sliding mode domains of uncertain variable structure systems with bounded controllers. JEEE Trans. Autom.
Control,
AC-35,
446-449.
Rosenbrock. H. H. (1963). A method of investigating stability. In IFAC Proc., Base& pp. 590-594. Sirisena, H. R. and S. S. Choi (1975). Optimal pole placement in linear multivariable systems using dynamic output feedback. Int. 1. Control, 21, 661-671. Utkin, V. I. (1978). Shding Modes and Their Application in Variable Structure Systems. MIR, Moscow. Utkin, V. I. (1983). Variable structure systems. Automatica, 19. 5-25.
Utkin, V. I. (1984). Variable structure systems-present and future. Autom. Rem. Control. 44. 1105-1120. Utkin, V. I. and K. D. Yang (1978). Methods for constructing discontinuity planes in multidimensional variable structure systems. Automatica, 14, 72-77. Verghese, G. C.. R. B. Fernandez and J. K. Hedrick (1988). Stable, robust tracking by sliding mode control. Syst. Control Lett., 10, 27-34. Weissenberger, S. (1973). Stability regions of large-scale systems. Automatica, 9, 563-663. White, B. A. (1990). Applications of output feedback in variable structure control, In A. S. I. Zinober (Ed.), Deterministic Control of Uncertain Systems, Chap. 8, pp. 145169. Royal Military College of Science. Shriverham, U.K.
816
R. El-Khazali and R. DeCarlo
Wonham.
W. M. (1979). Linear Multivariable Conrrol: A Approach. 3rd ed. Springer-Verlag. New York. Young, K. K. D.. P. V. Kokotovic and V. 1. Utkin (1977). A singular perturbation analysis of high-gain feedback systems. IEEE Trans. Aurom. Control, AC-22, 931-938. &k. S. H. and R. A. DeCarlo (1987). Sliding mode properties of min-max controller and observers for uncertain systems. In Proc. 2% Annual Al&won Conf: on Communications, Control, and Computing. Monticello, IL, September 1987. Vol. II, pp. 813-818.
Define n-m -AI) - 2 o,(A, - A,). ,=?
Geornefric
a,*(Az
Thus there are two cases to consider: (i) a? = 0 and (ii) a3 # 0. If as = 0 then (A.13) yields ,;
Now, for n - m > m 2 2, suppose that N* is rank-deficient. Then there exists a nonzero vector x such that
, n -m.
n
row space [(A,1 -A,,)
In particular, there n - m) such that
exist arbitrary
/ -A,J
vectors
Let i = 1; then for 2 5 j 5 n -m,
(A4
u, (i = 1,.
,
-Alz].
(A.4)
+ (A, - A,)[u; j 01.
n--m 3 a4.r’ = c a, n (A, - A,)[u: 101. ,=‘l I=,
(AS)
-m.
gl,(A,I-A,,)=(A,-A,)u:. glP,z = [O]. 2sj%n
Equation
(A.6)
2sjsn-m.
(A.7)
-mm.
(A.8)
Equation (A.8) implies that there exist n - m - 1 row vectors that annihilate A,,. However, since Al2 is full-rank and of rank m, the maximum number of left annihilators of Al2 is n ~ m - m. Since m 2 2 by hypothesis, n -m - 1 > n - m m. This implies that the set of row vectors Is:,),“=;” is linearly dependent. i.e. there exist constants aj, , a,_,,, not all zero. such that n--m g12 = c +, (A.9) ,=.1 and using (A.7) yields
Again there are two cases to consider: (i) a,, = 0 and (ii) al#O. Using the same arguments as in the previous step leads to the conclusion that “mm 3 ,; o, ,y (A, - A,@: IO]= 101. Continuing in this fashion, and assuming that an_,,, ., f 0, the previous steps described by (A.12)-(A.18) imply a,-, u;_,_,
=
“mm-1 ,F, (A,-,, -A,) u:, n,.
n-m-2
a,-+z
,E, (Lmml -A,)
or, for some scalar p, U;_m_, kpuL_“,.
4=$$$(“,-A,)N:. I
(A.19)
(A.lO)
From the definition (A.3) of x’ and (A.10). we conclude that x’ = &[(A21 - A,,)
~4 e [ ax(A? - A,)(Ax - A>)- ;T; o, ,fi (A, - A,)]x’.
I -A,,1 = (A,- A,)[4 101. (A.6)
Define gl, * u\ - uj for all 21jsn then implies
Postmultiplying (A.9) by Ail-A,,
Again, from the definition conclude that
/ -A,z]
(A.3) of x’ and (A.19), we
x’ = ul- ,,-,](A,m,m,l =~4-,n[(Lm~
-A,,) -A,,)
= P’ + p(L-“t-1
n, 101
- &,)[J&
mm/ 01,
or
q(A, - A,)(,$- Adb: j 01.
Moving the first term on the right-hand side of (A.ll) left and simplifying yields
(A.ll)
to the
(1 - p)x’=p(A,_,_,
- A,.~,,,)[&,
IO].
(A.20)
Clearly, if p = 1 then u,,_, = [0] and x = (01, i.e. rank [NJ = n. But, if p # 1 then (1 - p)x’N, = p(A,-,,_,
[(AZ -A,) - ;&(A,
I -A,21
1 -A121
+ PC& -m--1 - L,)[u:,
-g&f: I
(A.18)
where
Subtracting (A.5) from (A.4) gives
(ui - u:)[(A,l -A,,)
(A.17)
o,(A, - Ai)(A3 - A,)
Observe that (A.17) and (A.ll) propagate in a similar fashion, i.e., substituting from (A.17) into (A.1 1) yields
But _r’= u;[(h,l - A,,) 1-A,J
(A.16)
n-m 2 a,(,$ - A,)(,$ - A&: ,=4
,n -mm. (A.3)
(A.3) yields
x’=u’l[(&I-A,,)[
(A.15)
This implies that the two cases lead to the same conclusion, i.e. the vectors {uj};Za”’are linearly dependent. If a, = 0 for j 5 3 then uz = 0 and _r= 0, i.e. rank (NJ = n. Suppose that as # 0 (if a‘3= 0, one can solve for uq). Solving for u; from (A.15) yields
u: = for i= l,..
/ -Alz]
x’=u:[(h,l-A,,)
a,(A, - A,)(A, - A,)u;N: = [O].
(A.1)
n-m
,=I
,=3
n--m ,z o,(A, - Ai)(A, - A2b: = (01.
Since (A,,.A,,) is complete, and, since A, o w(A,,), it follows that ((A$ - A,,) 1-A,*] is a maximum-rank map. Equations (A.l) and (22) imply that x must satisfy X' =
u3x1NA= c
and (22). we have
However, by Lemma 6.3, the matrix NL is of full row rank. This means that
= 2.m = n.
x’N,, = [0] for i = 1,.
(A.14)
n,(A, - A,)(A, - A&: = ]O].
while if a, # 0 then, from (A.1). (A.ll) APPENDIX A-PROOF OF THEOREM 6.1 First consider the case where n -m = m. or n = 2m. By Lemma 6.2, rank [N*, @NJ
(A.13)
- A,_,)u;_N:
= [O]. (A.21)
- Ai)],’ n-n3 = ,z @,(A,- A,)(A, - A#
101. (A.13
However, N: is a full-row-rank matrix (Lemma Thus u,_, = [0], which means that x = [O]. Therefore [N*] = n.
6.3). rank 0