In this paper, we present a quantitative analysis of the robustness of a reduced-order pole-assignment state-space self-tuning controller for a multivariable adaptive control system whose order of the real process is higher than that of the model used in the controller design. The result of stability analysis shows that, under a specific bounded modelling error, the adaptively controlled closed-loop real system via the reduced-order state-space self-tuner is BIBO stable in the presence of unmodelled dynamics

Chen, Zengqiang

Yuan, Zhuzhi

English

NASA Technical Reports Server

NASA-CR-200335
CONTROL THEORY AND APPLICATIONS
D
VoL 11_ No. 3
Jun. , 1994
/D'2
Robmmess of Reduced-Order Multivariable State-Space
Self-Tuning Controller
YUAN Zhuzh/ and CHEN Zengq_mg
(Dep_ment of _ and Sy,_.m _, Nanka/Univer_ty.Tlan_n, 300071, PRC)
L. S. Skieh
(_t _ Electrical Engine=inS, Un/ve_./ty elf Hc,._tm.H_.mtcn. USA)
Absu's¢/s In this paper, we present a quantitatlvc analysis of the ro0_._me_s of a reduced-oral=
pole-s._gnment sta_-spaco s,_-mn_g co_trol]_ for a multlvariab/c adaptive conU'ol s_em whose or-
der of the real process is h/ghcr than that ¢g the model used In the oontroler design. The result of sta-
billy analysis show_ that, under a spec/flc boundcdmodelling error, the adaptively oon_'oIed clo_-
loop real system via the reduced-ord_ state-spaceself-tuner is BIBO _blc in the pre_enc_of unmed-
el]ed dyrmml_.
Key words, adapt/ve controls reduced-order controll_s re_t control, state_spao_ self-tuner
1 [n_'oduction
The problem of robusmess of an adaptive control system has recently been studied by" many
autho_ D~s3. This is because the development of adaptive controllers for adaptive control systems
is based on the assumption that the model used in the controller design is an accurate representa-
tion of the real process! however, the degree of most real processes is often h/gher than that of
the model used in practice. As a result, a stability problem may occur due to a mismatch of the
orders of the modeled processes and the real processes [_]. Hence, a study of robust stability of the
utilized algorithms for the controller design is necc,ssary.
Durin8 the last decade, vast amount of research was devoted to quantitative analysis of the
robusmess of self-adaptive algorithms such as the development of con/c sector theory and normal-
lzed system scheme [z's3. In reference [4], the normalized parameter estimation approach com-
bined with a dead-zone method in which the modelling errors are treated as a bounded dismrba_nce
and utilized as a parameter adaptation stopping criterion to guarantee global stability was devel-
oped. In contrast, in reference [5], the robust stability of a multivariable adaptive controller
based on a factorization approach was established, which is useful for the robust stability analysis
of adaptive algorithms.
In this paper, we are concerned with the robust stability of the multivariabie adaptive con-
trol system v/a the reduced-order state-space self-tun/rig controUer developed in reference _6].
Our approach to quantitative analysis of the robust stability of the adaptive control system _s3 can
t,',anusa_ ra_ved _ _, 1992, _-veed M_. n, _99_.
https://ntrs.nasa.gov/search.jsp?R=19960016273 2020-06-16T04:34:26+00:00Z
278 CONTROL THEORY AND APPLICATIONS Vol. ! 1
be described as follow=. First, we utilize the normalized parameter es_tw.at/on schemeCO to carry
out the parameter estimat/on with the presence of unmodeged dynamics. Then, we use the tu_-
lyt/cal method developed in referenco [5] to resolve the robust stability of the adapt/ve control
system via the reduced-order state-space self-tuner['3. Flnagy, we determine the bound of the
modellin 8 error with which the self-tuner can be tolerated.
2 System Description
In this paper, both of the plant and the reduced-order model are assumed controUable and
observable.
Consider the following m-input-output block otr_erver-type discrete-time stochastic I/near
plant=
z.'(t) ---- A.'z, (t -- 1) -t- B'u(t -- 1) t- K'e'(t -- 1), (la)
y(t) = C'Tz: (k) -t- e" (t) (lb)
where
.4.'= : i "'. , a" = , =:(t)= ,
0. 0.... tz_(t)J
C: r = I'/. 0. 0.... 0.].x.,,
u(t) E R = and y(Z)E 11" are input and output vectors, respectively) block elements A=, B,_E
R "x. (i= 1,2, ... ,r) are constant matrices, z2 (t) E R'(i-- 1,2, .-. ,z) e" (k) E R = is the inno-
vation process which is a white noise proce_ with zero mean and covariance R: E R =x" and
sup II (O II 6. (2)
_O<:m
with 6o_0, K: ER "x= is the Kalman fain matrix.
A" (z -I) == I= -I" A.lz-= 4" A,,=z-' + ... -I- A.z-', (3a)
a'(z -!) = a.=z -I + B.=z -= + --- + B.,z-'. (3b)
D" (z -t) _- I= -{- D.lz -I -_- D.=z -= -{- ... -t- D..z', . (3c)
and D., = A,_-F K.,, i = 1,2,--- ,z. (3<!)
It is observed from (3d) that the Kalman gain K.. can be directly computed from the estimated
An alternate representation of the orisinal system in (1) _ be describedparameters D._ and A..
as follows =
_(t) = 0"r_ • (t) + e" (t), (4a)
where
0 °r ---- _A_l,"" ,A..,B.I,"" ,B.,,D.I, "'" ,D..], (4b)
_'(t) = [- _(t-- 1),..._ - y.(t- 1),..., - _.(t-- r),u,(t-- 1),-..,
u.(t -- 1) ,... ,u.(t -- r),e((t -- 1),...,e_ (t -- 1),...,_(t -- r)] r. (4¢)
The 0" in (4) is the parameter matrix of the original system. For a reduce-order controller de-
sign, a reduced-order observable model is required as
=.(t) = A.=,(t -- 1) + B.=(t -- 1) + K,8(t -- 1), (5a)
No. S Robustness of Reduced-Order Multivarfablo State-Space Self-Tunfn8 Controller 279
!
where
if(/:) -,, &'_.z.(/_) -J- e(E), (Sb)
- , B,= , K,-- , =,@)= ,
,4. " i "'.
Oo O.... _.(t)J
_. = [Z. O. O. -.. 0.3._.,
where m_r, and e(k) is the innovation process of the model.
The equivalent observable ARMAX model of (5) is
A(z-_);(t) == aCz-_)uCt) + DCz-_)e(k),
The alternate form of the model in (6) can also be rewritten as followst
y(t) = 0T_(k) + e(t),
where
(6)
(Ta)
0r = [A.t, "'",A=, Bot,"" ,B=,D°I,"" ,D,], (7b)
• (t) = r- vs(t- 1),.;., - v.(t- 1),..., - v.(t- s),ut(t- 1),...,
u.(_ - 1),...,=,,(t - _),el.(t - i),...,_(/= - 1) , ... ,e- (t - ,0] r. (7c)
where 0 is the parameter matrix of the reduced-order model. The e(/c) in (6) can be decomposed
into two terms,
e(k) = _(t) + e" (k), (Sa)
where e" (k) is the innovation process of the ori_nal system and
_(t) == 0"r_ . (k) -- 8r_(k), (Sb)
In reference C6"], it was assumed that _(k) in (8) is a zero-mean stochastic sequence and statisti-
cally independent of e' (/c). In th_ paper, the assumption in reference [6] is relaxed so that _(/_)
is not a zero-mean stochastic sequence and can be represented as
D(z-t)_(t) ---- _r_P(k), (ga)
where
30 r = _[A.+I,'" ,A.,B.+I,'"B,,,D.+I, "" ,D.,].x_,C,-,_, r _> _, (gb)
[0, r ---- _,
d_(k) ---- E- yl(t -- • -- 1),..., -- 1,(k-- ,= -- 1),..., -- 1.(k -- ,'),=_(t-- _ -- 1),...,
u.(k -- ,= -- 1) ,... ,=.(t -- r),el" (k -- ,, -- 1),...,e_ (k -- _ -- l),.-.,e_ (t -- ,.)l v.
(9c)
Then, from (9a), it is reasonable to ma.ke the following assumption:
Assumption I Assume that there exists a #:>0, such dmt for k_0, the unmodelled error
satisfies the constraint:
and further, we have
(lO=)
II_(t) II < 1, II¢" @) |, (_0_)
where _° (/c) is related to the phmt Input and output se.quences. _1. (lOb) shows that the rood-
280 CONTROL THEORY AND APPLICATIONS Vol. 11
i J i i , i
elling error t(_) is relatJvely bounded! therefore,# can be considered a mea._re of the relative
magnitude of the modelling error. Note our primary Interest l= to find a relat/vebound of' the
modellin$ error with which the adapt/vo controller can be tolerated.
3 Normalized Parameler Estimation
In order to develop an aclapt/ve control law, we shall first introduce the parameter _t/mation
M$orithm for the model in (E).
Defining the parameter estimation error, _(1:)= 61(k)- O, the estimated output j (/0 = 61(k
-- 1)_(k), and the innovation proc_, e(IO----r(;O--_(k).
The reduced-order model equation to be estimated is written as follows:
_O) = _'(_ -- 1)_(_) + e(_), (lla)
61(_ -- I) = C21.t,--.,.;L.,_._,'" ,_,,bot,-." ,/_,]r, (lib)
$(]_) = ['-- yi(k-- 1),"', -- It(I;-- 1),"', -- y,t(]_-- tt),ul(];-- l),...,
_(_ -- 1),...,u_(_- n),etO -- 1),"-,_.(1_ -- 1),"" ,*.(;_- _)]r. (Ilc)
As far as the parameter identification is concerned, in the case of bounded disturbance,, the
dead zone technique is utilized to prevent parameter drift. On the other hand, in the case of un-
modelled dynamics, the identification error may grow without bound: hence, the dead zone tech-
nique of the bounded distrurbenc_ can not be applied directly. This leads to the u._ of the param-
eter normallzat/on technique which allows unmodelled dynamics to be treated as bounded distur-
bances. In this paper, a normalized Imrameter est_.ation scheme r4] is used to estimate the param-
eter A,,, B.,, and D,, in (6).
The normalized variablea el' the proo_ are defined
• (k) = max( max ]O," (_) ] ,r.), (12b)
where r. is a pre-seleeted positive eotmtant, and O: is the i-th element of 0". Using these nor-
realized variable_, the model in (Ta) can be rewritten as follows:
l_(_) = ar_(_) + _'0), (laa)
where
_(_) . (13b)
It can be shown in the following lemma that the sequence {_'(_)} in (l_b) is bounded.
Lemma _ The norm&lized sequence of the perturbed si_j_als {a'(_)) in (13b) is
bounded.
Proof
II,_a_O011 II o"(t> II
• O) + .O) :_ _s(r - t),n]_ + _"II,a*(_) II r.'
we are able to e_tinmte an upper bound of d'(IO.
Define
(13o)
• °
No. $ Robustneam of Reduced-Order Muitlvarisblo S_to-Spaco Self-'l'unlng Controller 281
a,- C_,,,'",a,.]',
where ,_>0 is an estinutte of an upper bound of JJ_(k)[, i=, l,...,m. Also, define _=,
4 _ , then _ is an estimate of an upper bound of ! d'(t) U.
Based on the identification alsorlthm in [4], we can obtain the convergence propertie_ of
the posterior estimation error and of the parameters. A posterior estimation error is defined as
j(t) A r(t) -- _(t)_(t). 04)
[,emma 2 According to the esimation algorithm in [4], we have:
i) II_(t) II is uniformly bounded, which implies that there exists a constant M_0, and
D(t)Eo= {0, II0 II <M), where D is a closed subset of R_";
e) it=[ II_(t) II'- II@0--1)II"]----00
_) l_[ IIa(t) II- II_(t-h) II]=0, h isa ltmit_positiveinteger,
iv) There exists a positive integer K, such that
II_(t) II < 2_ax II_" (t) II ,r.), _ t > K,.
This proof can be found in reference ['4].
4 Multiv_iable State-Space Self-Tuning Controller
Once the system parameters 63(k) axe obtained, the adaptive control law can be determined
follows r°]:
The estimated ._.(/0 with _(/0 can be written as
_.(t) = ._O)_.(t -- 1) + _.(t).(t -- 1) +/_.(t)_(t -- 1), (15a)
_(k) = C_'._.(k) -t- t(t), (15b)
where .Tl.(k), J_o(k) and /_.(k) axe the kth step estimation of ,4., Bo and Ko, respectively.
A._umptlon 2 Let O(6"F)==[,;l:-t_..,4:-'_.,'",,;l_.,_.]. Assume that there exists a posi-
tive real constant y._0 such that IdetG(_)[_y.. Then: the state-feedback control law is given
by
u(t) = H,r(t) -- g_.T.t.(t), (16a)
where r(t)ER" is a reference input vector with an input gain matrix H, ER "x', ,_.ER "×"
¥, = a,_ -- A_, i == l,..-m, (16b)
where .4. comes from the controllable model of (1.5), and
_o(_)= _,o._'-' = ]_(_.-P,), (1_)
det(Z/. -- _- _)j_'.T=)] = _J_det(_,, -- e(). (16d)
5 Extended Dynamic System Description
In this section, we reformulate the adaptive control system developed in previous sections in-
to a composite dynamic system which is suitable for rob/_st stability analysis using the theory de-
veloped in the next section.
First, let
"O(t) = [_'rCt),_,r(t)], (17)
282 CONTROL THEORY AND APPLICATIONS rot. 11
where _'r(t) and _(t) are defined in (4c) and (15), respectively. Next,
_'(t),
where
_p'(t + I) = s_p" (t) + _,(i) + _P,O) + _..(t),
r 0 0 0 ...
I.(,-I) 0 0 ..-
S= , 0 l.c,-n) 0 ...
• . ...
0 0 0 ...
=
0
0 ,
0
___, ,.___,t__ ,___t_._
• T(_) E'-= Vr(k) 0_T,.. n_r .,r .. n_ n'r . _rl! * tvltvlt* lVmt_lte* tl,,mjt
_$(t) = E07,...,07,ur(t),07,...,0'.v007,".,0"l_,
'T *T T .. T
_.r. (k) = Eo1",... ,oz, oT,... ,o. ,, (_),o2 ,. ,o2],
where 0_=C0,'",0]sx,. Then, from (15b) and (16a), Eq. (lSa) becomes
where
we rewrite
(1Be)
(18b)
(18c)
(1Be)
(P" (t -I- 1) = Sd_* (t) + Ot(t)_.(t) + Dl(t)i(t) -I- Ds(t)e* (t) + O4(t)r(t), (19a)
"-4" "--:.I
° ilDi = -- p_/'
0 , Dz ffi * Ds
o o I
o oJ
'o"
_o
_o
|
iO
_y
tim
L.O.
"0"
0
,q.
D_ = . (19b)
0
o
0o
Sub=tituting (IF_) into (1Sa) 8ires
i.(t + D = (,_(_) -- _.(t),_,T,)_(_) + _'.(t)i(t) + _.(t)H_'(t)
= _'_(t)_.(t) + _.(t)_(t) + h.(t)H_(_,
_'_(t) = ._.(t) - _.(t)#T't..
(20a)
(20b)
and combining the resulting equations (19a) and (20a) yields the composite dynamic equation of
the closed-loop system:
Z(t + 1) ---- B_(t)Z(t) -I- Bz(t)_(/O 4- B_(k)e'(t) "4-g,(t)r(t) (21a)
where
Fs _,(t)l _,(t)l F_,(t)l
Lo L oJ'
In the followin$ section, we carry out the robuststability analysis using (21a):
S The Robustness of Self-Tuning Ad_ptive Controller
In this section, we state two lemmas as follows.
Lemnm _ _der the time-varyin8 difference equation
=(_ + I) = A(O=(O +.f(¢,z), x(O = z_ _- R',
r l
(21b)
" (22a)
NO. $ Robustness of Rcduce_l-Order MulHvsrhkb/e Sta(e-Space $elt-Tunln$ Controller 283
° ,
where f(l,s) is bounded as
| .t'(g,s) | _ (4 "1- tit(O) U x(O U "l- r:(O, (22b)
,_0, 0<d:(t) E _, and0<rz(O C-r".
Supple that the zero-input system tn (22) of theform
z(t -t- 1) =, A(Oz(O (23)
is exponentially st;,ble, I.e., there exist some constants at_l and l_az_0 such that
O(t,tl) I] _ al_'-'P (24)
for any gt_0 and t_t, where O_t, lt) is the state transition matrix of the system in 423).
Then, If the following ineqtutl/ty is sa_Lisfieds
0 _ _ < (1 --az_ ), (25)
ul
we can conclude that
i) =(OEt", and in addition, _ z(O _ _v(t) as _--,-oo, where v(t) is the output of the
system with the transfer function (7 (z -t) =az/(z--(10al-i-a2)), driven by rz(O.
if) Whenever rz(t)E_', and pE[l,oo), then z(I)EP which implies I! z(_) II-*0 as
--4POO,
Proof see reference [5].
Lemma 4[r3 Consider the system
z(t + 1) =. A(Oz(t) (26)
which has the followin8 properties s
a) _ A(t) _ is uniformly bounded!
b) There exist 0<_e4<_l, such that
maxl_,_(.A(O)l_l--eo<l forall C_te,
c) sup J]A(t+l)--A(t) ]] is sufficiently small.
where te is a positive constant, and _,j(A(I)) denotes the jth ei$envalue of the matrix A(I).
Then, the system is exponentJaJly stable.
In order to show the system in 421a) to be exponentially stable, we proceed throush the fol-
lowins raelr..
Step 1 Show that gs(E) satisfies part a) of Lemma 4.
From Lenuna 2-1), we know thai: II2.0) g and II_.(t) II ue bo,:_ded.From Assump-
tion 2, we knowthatIIr._ II and _ T, II are_ bounds.Thus, g_,(g) II ts bounded.
Show that 8s(t) satisfi_ part b) of Lemma 4.Step 2
Since
det(_- gl(t)) =det(_L/- S)det(_ -- El(t))
m
=det(_L/-- 8)Hdet(_,,- P._ 427)
i=.!
the roots of det(_L/--8) are zero. If we choose P,, i==l,...,m such that det(_L/--P,) has no
roo_ lye8. outside the ch'cular disk, i.e., O<_l;.l<l-,.. and 0<_<1, then gt(t) satisfies
284 CONTROLTHEORYANDAPPLICATIONS' Vol.11
partb) of Lemma 4.
Step 3 Show that gt(k) satisfies prat ¢) of Lemma 4.
The above fact can obviously be verified from Lemma 2-11/) and Assumption 2.
Based on the resuits shown in the above three steps, we co¢lude that Bt(E) is exponentially
stable. This implies thatan associationof the al and at with the expontmtlally stable gi(k) is wi-
den°.
Next, we explore the norm bounded property of the forcing terms, 8_(k), E_" (k), and
E,r(k), in (21a) as follows.
By virtue of Lemma 2-i), Assumption 2, Lcmma 2-iv), and (2), we have
_A', II_O) II + r, IIo"O) II + A', II,(_) I}
_<2r,7_[ II_" (k) II + ,.] + K_. + K, II_(k) II, (28a)
where R'I----- sup H E2(k) ]1 , K2-- sup I] g_(k) ]] . Ka----- sup ]] E_(/t) I]. From (13¢), we
know that pC3(r--n),n3_+t_./r_s an upper bound of II,r(_) II, and X is an estimate of an up-
per bound of II a'(k) II. Thus, if we can choose _l_l_[3.(r--tt)m]_'i._-d°/r. for the estimation
in reference [43, then
IIE_(_) + E_e"{_) + E,,(_) II _<2xl[g[s(, - ,),,,3_ + ,A][ II ¢" O) II + 1'o]
+ r_Zo + r, II,-(_) II (28b)
<_2r1[_,C3(,. - =)m]a + ,A3C IIz(_) II + Tel
+ r,6. + r, II,"(_) II.
Now, comp,_.ring (22b)with (28b)yields
,to ---- 2S'_[,,[a(r -- _},_]S + h],
• po
_o
600 -- 0, ,'20,) ffi 2r,E_,Es(,.- ,,),,,IS ..4-";:,3,'.+ x_z. + K, II ,'0) II.
Then, applying Lemma 3. we have the following main results for the adaptive controller.
Theorem ] With Assumptions 1 and 2, if the/_ in (10a) (a measure of the relative mag-
nitude of the modelling error) is bounded as
0 _</_ < 2Ki[30" -- .)m]_ 01 _" ], (29)
_d :he __ '.a :he estimation algorithm [q satisfies
0< 3, <_ ,Is(, - ,),,]a + £.. (3o)
then the adaptively controlled close-loop system via the reduced-order controller is BIBO stable for
any initial condition in both the plant and the adaptive controller irrespective of the presence of
unmodelled dynamics.
Remerk 1 The BIBO stable system has the following properties t
No, 3 Robus[n©n of Reduced-Order Multlvarlabl©State-Space Se!f-Tunln8 Controller 285
i) z(t)Er', thee v(t)Er', and u(t)Er',
U Z(/_) _ _v(k), as t-_oo, where v(t) is the output of the system with the transfer
driven by r_(k).
_)
(I I
@,(z -t) : :_ (old o + aD'
al _)1 (2A'ld.r. -t- K._o) (31)IIz(t) II _< v, (t) + - (a_d0+ a
is the output of the system with the transfer function G_ (=-i), driven by
Remark 2 The value 2Kir3(r--a)m]_ L at _, j in (29) is obviously a measure of
the robusmess of the adaptive controller. It implies that the actaptive controller is allowed to be
perturbed by the modelling error a(t) satisfying the ,u in (29). For the pole-a._ignment aJgori:hm
in this paper once we have selected the desired closed-loop polynomial matrix _().) in (16c), it
is possible to determine at and a= a priori. As a result, we have the knowledge of the degree of
robust stability for the controller to be designed.
7 Conclusions
This paper has demonstrated that the state-feedback pole-assigrm_ent self-tuaLng cona-oller[6._
has a certain stabili_ robustness. As a result, the rech:ced-orcler model _ be used to design a re-
duced-order self-tuner with suitable conditions ar,d the adaptively controi:ed closed-loop original
system via the designed reduced-drder self-tuner is BIBO stable in the presence of unmodelled dy-
nan'd(::;.
Acknowledgments This work w-_ supported in part by the US Army Research Office, un-
der grant DAAL-03-91-G0[06 and NASA Johnson Space Center under grant NAG 9-380.
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Robustness of reduced-order multivariable state-space self-tuning controller

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