Copyright © IFAC Control Science and T echnology (8th Triennial World Congress) Kyoto. Japa n. 1981
ON THE DESIGN OF LINEAR TIME-INVARIANT ROBUST CONTROLLER Y. Kida and E. Masada Department of Electrical Engineering, The University of Tokyo, Tokyo , japan
Abstract. A method to get a robust controller through the design of a re gula tor and an observer is offered. The system model, with superabundant expression of disturbance to get robustness, is transformed into a particular model in which disturbance can be directly canceled by control input. Then a controller and an observer with some optimality for the model is constructed and its robustness is shown. Relation between the original disturbance and the state vector of the observer is considered and a method to coordinate controller parameter reflecting designers intention is suggested. Keywords. Robust control, linear systems, regulator theory, observers, optimal control. INTRODUCTION
Additional dimensions for the robustness still remains in the observer, but the y are distinguished from the rest and their convergence can be set separately.
Suppression of insistent disturbance is an important problem in regulator design. For disturbances des criable with a linear timeinvariant di f f e rential equation, their effects on system outputs can be asymptotically canceled by a controller with dynamic compensator. For single-input single-output system, a method to construct such a controller as optimal feedback system is also known. l
GENERALIZED MODEL AND ITS TRANSFORMATION Generalized model. Consider a linear time-invariant s ystem described by (1). Each component of the disturbance vector v satisfies differential equation (2). Every characteristic root of (2) has positive real part. We will call this model as the original model. dX xERn uERm dt = Ax + Bu +Ev yeRr v~Rh (l) [ y = Cx + Du +Fv A~ F; Const. real matrices
A single-output close loop system with such controller has a characteristic which gives asymptotic regulation of system output as long as the closed loop s ystem remains stable and disturbance satisfies the given differential equation, even if the actual system differs from the model used in the controller design. Davison et al. called it robustness and presented necessary and sufficient conditions for existence of linear time-invariant robust controller and its minimal order structure. ?' This controller for multivariable systems usually has additional dimensions which is necessary only for the robustness.
dP
d
- ( ) + ... + al at( ) + a ( O dt P
0
(2)
Then consider a model (3) with disturbances e and f, each component of which satisfies (2). If we let e=Ev, f=Fv, e and f satisfies (2) for any E and F. Using a disturbance generator capable to generate any (e, f) as lon g as they satisfy (2), generalized model (3) is described by (4). ( Ai is a characteristic root of (2) and it has multiplicity of qi' (i=l, .. ,p) Ik represents k xk unit matrix.}
In this paper, it is shown that a robust controller can be constructed as a feedback controller with an observer for the equivalent disturbance generator. To get the robust construction, supersufficient expression of disturbance is introduced, and transformed into two parts through a state transformation. One enters the s ystem through the same input matri x as control input. The other has no effect on the s ystem. It is shown that such transformation can be always achieved if there exist linear time-invariant robust controller. Then optimal feedback controller with state observer f or th e transformed s yst e m i s construc t ed and it s robus tness is shown .
dx = Ax +Bu + e, dt
y
1189
[C
Nl[~)
y = Cx + Du + f
(3)
(4)
+ Du
1190
Y. Kid a and E. Masada
here
M=[M} ... Mpl
N=[N
... N l l p Ni=[YlO .. 01
ddt
Mi =[Y 2 0 .. 01 Y =[I 01 nx (n+r) matrix n 2 Y =[O Irl ; rx(n+r) matrix 3 J=block diag. {J ... Jp } l Ji=EJB( Ai , qi' n+r) EJB( A,q,k) Alk denotes
here
[~ , ] +
BP r 0] 0 o r'
y
[C DP 01
~
,n
r
, A, -1
~'=
[~ ,]
mu (10)
+
Du
, ' n' P = Q, r '= , ' A",-l
-1
Ik Ik
q blocks
Ark
(4) is a generalized model of (1) for arbitral (E, F, v) as long as v satisfies (2). Conditions for the transformation. If following two conditions (5) and (6) are satisfied, we can choose mxr complex matrices Qi (i=l .. so that they satisfy condition (7). If all Ai have positive real part, (5) and (6) are always satisfied if there exist linear time-invariant robust controller for (1). (See reference 2)
p)
rank[:- Ail
[~
[~ , ]
In system (10), np state variables ~ ' are sepalated from the others and have no effect on y'. Deleting them, we get observing model (11). Because equivalent disturbance y is directly additional to the control input u, we can easily cancel it if ~ is observable from y. (Note t,;lx but y is the same as it of original model.) Observability of ~ is easily shown from (5) "' (7) as below. (12) d dt t,; y
dt~
Ct,; + D(u + y )
y
BP
A- A.I ~
rank
d
At,; + B(u + y )
A- A.I ~
rank 0
r - A.I
0
~
DP
C
= r~ = P~
:] = n+r
for all Ai
(5)
for all Ai
(6)
BQ A- A.I ~
C
DQ
n + rp
n
(11)
(12)
CONTROLLER rank [A:Ailn] = n
[A~\ In B~iJ
is nonsingular. (7)-1 (Qi is mxr complex matrix)
Q is complex conjugate to Q i j if Ai is complex conjugate to A •• J
Q is a real matrix if Ai is a real number. i (7)-2 Transformation and observing model. With Q chosen according to (7), state vector i transformation (8) converts system (4) into (9). (Contents of the matrix T and proof of the transformation are held in Appendix.) (:)
ddt
~ [~,]= [~, ]
rp n EC np n 'E C
t,; ERn T[:J
[~
[~, ] [~,
BQ A 0] 0 o A'
y = [C DQ 0)
+
(8)
mu
(9) Du
]+
here Q=[QIL l ..• QpLpl L.=[I 0 .. 0) (qi blocks) A'=block diag· {J
For (11) has uncontrollable unstable part ( ~ ), we can't get an ordinal optimal state feedback controller directly. It is because that it is impossible to make both u and y converge to O. But our object is to make y converge to 0, and it is inavoidable that u doesn't converge in order to cancel the effect of non-declining disturbance. So it is reasonable to separate control vector u into two parts as described in (13) and choose performance function taking account of only ul (and, of cource, t,; ). u = ul + u2
J- }
l 0. lS
po.
"
J
pS
}
From condition (7)-2, we can easily transform the complex system (9) into a real system (10) .
u l = Kl t,; ,
u2 =
K2 ~
(13)
Letting K2 = -P, we get (14) from (11). Then we can determine ul or Kl as an optimal state feedback for (14). Note that (14) is the same as original model (1) with v = O. Because (t,;,~) is not given directly , we must use estimated value (€,2). Control rule becomes (15). d
~
At,; + BU
u = Kl ~ + K2 e
r
~
A =block diag.{J
Control rule.
y = Ct,; + DUI
l [Kl K2
(14)
optimal feedback gain for (14) (15) -P
Observer. Constructing full dimensional observer for the observing model (11), we get (16). Observer gain G = [Glt G2 t l t can be determined as a optimal filter (Kalman filter) gain for the stochastic system (17), if we set the white noise ( ~ , ~ ) appropriately. Setting noise for the generalized model (4), we can
On the Design of Linear Time-Invariant Robust Controller get
(~,~)
(~ ~P) m+
ddt m y ddt
through transformation by T and
[~)
(~)u + [~~) (y - y) (16)
[C DP] m
+ Du
[~ ~p) (n
+
(~) u
+
[~)
+
Du
+
~
y = [C DP]
T.
1191
mentioned before. Here we consider A~D differ from Aa~Da. In this case, closed loop system becomes (21).
~ = Aa x + Ba u + Ea v dt
(20)
y = Cax + Da u + Fa v ~
(17)
It is important to notice that G must be chosen so that all elements of 2 converge to in order to make observer (16) work effectively for any (E,F). It may happen that this condition is not satisfied if the noise (~,~) is not chosen appropriately. We will discuss about it in later section.
A c
=[
C c
=
e
System noise ~ represents random change of the state vector and it is the request for observer's convergence. On the other hand, noise ~ can be considered penalty for G. So, this method to design an observer as a filter is useful in actual application.
y
Ac (:) + Bc v
ddt [:)
~~a [C
= Cc (:)
+ D v (21) c
BK a A + K con. GDa ] D = F a c
D ] a
a
[:;J
B c
Let G(s) be the transfer function matrix from disturbance v to output y. Then,
= Cc(sI-A c )
G(s)
-1
Bc + Dc
and following relations are held.
Controller and closed loop system. From (15) and (16), controller is described by (18). (18) with (11) yields closed loop system (19). (19) is described by (E,~) and observation error (eE,e C). In (19), unstable uncontrollable variables ( C) are separated. Then, E, e , e go to zero as time goes to E KCand G are chosen so that all infinity, 1f the eigenvalues of [A + BKl] and Aobs . have negative real parts. d =A con.).J + Gy ~ A con.
).J =m [K
K
l
(Ac-AiI) is nonsingular for Ai with positive or zero real part as long as closed loop system remains stable. Then,
= na + n + rp •
rank(Ac-AiI)
And combining it with (22), we get (23). u = K).J
(18)
[A+BKl-GlC-GlDKl -G C-G DK 2 2 l
rank G(A ) i
~)
=
rank [Ac
~Ai I c
:c) - (n c
a
+ n + rp) (23)
K ] 2 A+BK 0 0 0
y
. [.C,I:A,.,r).l)J ~] [lA'::"J :']
l
0 BKl BK2 r 0 0 0 Aobs • 0
E -
c
e
e
E
(19)
C
(C+DKl)E + Oc + DKle
E
+ DK 2e C
(n
a
+ n + rp)
However, we can show (24) is held for any (Aa~Fa). (Detail was omitted.) Ac -Ai I rank [ C
B ) Dc
c
~
na + r
c
A = [A-GlC -BK 2+GI DK 2 ] obs. -G C r +G DK 2 2 2 Generalized model (4) contains original model (1) for any (E,F), and observing mole (11) is equivalent to (4) as far as y is concerned. So, connecting controller (18) to original model (1), we can make y converge to zero for any (E,F).
And, from the contents of Acon .' we also rank(A
con.
- AiI)j
<
hav~
n + rank(r- AiI)j (25)
= n + r(p-j) ZEROS OF TRANSFER FUNXTION MATRIX Suppose original model (1) is not accurate and actual system is described by (20). If difference between (1) and (20) lies only in (E,F), controller (18) remains effective as
From (23), (24), and (25), we get (26). rank
[dj~~lG(S») ds
_ s-A
=
i
0
j-l ) ~(s) [ S=A , ds] i (26)
o
Y. Kida and E. Masada
11 92
for
EXAMPLE IN A MAGLEV DESIGN
j = 1, •.. qi
(26) shows that any unstable or insistent mode o f disturbance (Ai) is a zero (with multiplicity of qi) of each element of G(s). It means that as ymptotic regulation occur in closed loop system (26) as far as (21) remains s table and v satisfies (2). In other words, (21) has the robustness. In addition, this characteristic depends on (25), and it's not necessary that observer has full dimension.
(27) is a linearized 3 order model of an electro-magnetic levitation system of a vehicle. (Fig. 1) Only vertical movement is considered and variables are deviations from normal operating points. Gap length y must be regulated to keep levitation.
l evel --------------~--~
base rail
rail surface leve l gap l ength xl; magnet position
magnet COMPROMISE BETWEEN ROBUSTNESS AND CLOSED LOOP DYNAMICS Disturbance generator with dimension hp is needed to emulate all v which satisfies (2). (h is dimension of disturbance vector v.) But actuall y , v is often caused by a system of lower order. Let the order of the system actually be g, then p ~ g ~ hp. In observing model (11), equivalent disturban ce is described by a generator of rp dimension. In case that g ~ rp, (11) has (rp-g) dimensional superfluous dynamics . Because of this, convergence of these additional dynamics isn't guaranteed by the noise derived f rom a noise set for the generator of v (and for (1». In fact, we must connect these unstable dynamics to the object system in order to make them stable. And when connected, they have bad effect on closed loop system dynamics. So, in case that original model (1) is accurate, convergence of this additional part of the observer is not only unnecessary but also harmful. But it is necessary to observe all rp dynamics of equivalent disturbance in order to make the closed loop system robust. (In fact, this part in our observer has the same construction as the minimal order robust controller has.) Then, we must make compromise between the dynamics of closed loop system itself (in case that the model is accurate) and the robust suppression of disturbance (in case that the model is not accurate). In this problem, we can use additional noise. At first, we set noise for the original model itsel f . Let it be ~ O. Then we set noise for the generator of v, and translate it for the generalized model (4), then into the observing model (11). Let it be Wl. At last, we set additional noise ~2 for the additional dynamics in (11). Using ~ = ~ O + ~ l + ~ 2' we can derive robust controller. W2 will be chosen so that it moves independently to (WO + ~ l) and covers co-space of (WO + ~l)· ~O represents random change of x, and ~l represents that of disturbance. They coordinate observer gain G on the assumption that model (1) is accurate. On the other hand, W2 represents request for robust suppression in case that model isn't accurate. In this way, convergence of the parallel structure in the observer can be separately controlled, and G can be coordinated reflecting our intention.
-
bogy
u
Fig .l
x
u; applied voltage
A MAGLEV system
Ax+ Bu +Ev
[-t
A
C= [ 1
1
0 -a 0
l
(0
-u
l
Cx + Fv
Y
B
0
=
I
o
(27) E
bJ
( 0
= la~!ala2
]
F = -1
D= 0
Here we consider a disturbance v=sin(wt). Then, p=p=2, ql=q2=1, Al=j w, A2=-j w. Generalized model becomes (28). It has 4 independent sine-wave generators. Initial state equivalent to v=sin(wt) is (29). They yield observing model (30) and its initial state (31) through the transformation by T and T.
(~)
=
r~ ~) (~)
+
(~)u
y = [C N]
1 0 0 0 1 0 0 0] M = 0 1 000 100 [ 001 000 1 0
(~)
[jWI 4
J
o
(28) 0
]
-j wI
4
N = [0 0 0 1 0 0 0 1) [0,0,0,0,-a ,-a +a a ,1, l O l 2 0,a ,a -a a ,-1]/2 (29) l O l 2 AE; + B(u + y ) CE;
E; y
r =
(_~2 ~)
r: y
rr: Pr:
(30)
P = [1 1)
t
[ E; t 1;0 ) = [O,-w,O, w3 /b,a w3 /b) 2 0
(31)
System nise ~ l is chosen to be a parallel vector with (31). Then, with a delta-function replacing white noise PO' ~ l represents the disturbance v=sin(wt). On the other hand WO is chosen to be parallel with a initial state which causes the same y as v=l (step) does. Additional noise ~ 2 isn't needed, for this system has only one output. ljil
[E;O
t
t 1;0 )
vsin Pl
On the Design of Linear Time-Invariant Robust Controller Responses for v=sin(wt) and v=l are simulated for various G derived from weightening parameters shown in table 1. Results are shown in Fig. 2 and 3. Dashed lines are response with direct state feedback without observer.
TABLE 1
Weightening Parame t ers
No.
1
2
3
4
5
50
100
200
400
800
v v
step sin
= 100 ,
J J 93
In this method, superfluous expression of disturbance is introduced to get robustness. System was transformed so that equivalent disturbance enters the system through the same input matrix as the control input. Equivalent disturbance is observable from system output, so it can be easily cancelled using observer mechanism. This transformation can be done if there exists linear time invariant robust controller. Many works have done on regulator design problem with an observer, and their results can be applied to robust controller design. In this paper, full dimensional observer in Kalman filter style is adopted. And controller parameter is chosen to be optimal according to a performance index which is directly related with the system model through the state transformation.
Ave.( q,2 ) = 1
Because the equivalent disturbance is based on the superabundant expression, it also has additional dynamics in most cases. In such cases, controller parameters should be yielded by trade-off between dynamics of closed loop itself and robust suppression of disturbance. A method to coordinate controller parameter reflecting designer's intention by additional noise is suggested.
o
0.2 Fig.2
0.4
O.6sec.
APPENDIX
Re sponse for v=sin( wt)
Contents of the transformation matrix T. T = USRIl
y
0
lr---r-~~~r---r---r-~
U
[:n
V_} P W_ } P ) [0 I nqi
block diag. {VI block diag. {W
V = [Irq . 0) i
l
W.
1
1
[~n block diag. {SI 0
-1~~~-1---+---+--~--~
S
O.6sec. Fig. 3
Si
Resp ons e for v=l (st ep) R
With small weight for step (~O), rapid convergence for v=sin(wt) is gained, but the response for v=l has large over- and undershoot. Making Vstep large, we can improve the step response, but convergence speed for sine wave becomes slower. Note additional dynamics only for the robustness isn't contained in this example. Another trade-off will be taken for multi-output systems (using ~2)'
=
[~n Ri
Sp} ) Y }) l q . blocks Y } 1 2
block diag. {Y l (block diag'{Y 2
block dia:. {R ... Rp }) l block diag,{Zi
Zi } q.1 blocks
TIl ••••••••••••••• TIp )
11
block diag.{ TI
l
•.. TIp! Y X qi)
CONCLUSION A method to design a robust controller as a state feedback controller with observer is proposed.
2 i
Y. Kida and E. Masada
1194
In+r
X 2 .i I . X n+r i
Xi
ITi
X qc l .i . 2 Xi Substituting latter three into the first,
In+r Xi In+r
Y1 X = i [:n :]Zi
Calculating S-l
[:n block
= st
In
-1
dia:.{R~l
°1 ...............
[ o block 0i
DQiYIZi
(A7 )
diag'{~l
•.•
YIX
o
i
(AB)
Co.,
N. ~ . 111
COi = [-CY X 0 ... 0] 2 i
]t,. transpose
Ni~i=
R~l}]
[Y l -YIXi 0 ... 0]
Then using (A7) and (AB), we get, COi + Ni~i = [DQiYIZi 0 ... 0]
-1 -1-1 Ri = block diag. {Zi ... Zi } 71
(A6)
And from the definition,
Inverces of them are as follows. Ut ,
BQiYIZi
op]
.... (AS)
AOi = [-AY X 0 .•. 0] 2 i
~p}
Mioi= [Y
[-Y X 0 .....• 0] 2 i
7liJi~i
In+r -Xi 0 ..• 0
-Y Xi 0 ... 0] 2
= [A i Y2Xi Y2 Xi 0
0]
Then using (A6) and (AB), we get, AOi+Mi~i+71iJi ~ i
= [BQiYlZi 0 ... 0] .... (A4)
REFERENCES Transformation by T. Because transformation by U and S are only exchanges of state vector elements, we omit them. Then we show (AI) and (A2).
Rl1[~ ~)IT-lR-l M = i
=
[~ ~)
M=
[BQiYl 0 •.. 0]
[C N]IT-lR- l = [C N] N = [DQiYl 0 i
N 0]
[~
... M_] p (AI)
nx (n+r)qi matrix [NI •.. N_] p (A2) rx(n+r)qi matrix
From definition of R, IT, it can be easily shown that (AI) and (A2) are equivalent to the following three. (A3) -1 (Aoi+Mi~i+TIiJi~i)Ri -1 (Coi+Ni~i)Ri = N
Mi
(A4) (AS)
(A3) is clear from definitions of those matrices. We will show (A4) and (AS) below.
Pearson, J.B., and C.Y. Ding (1969). Compensator design for multivariable linear systems IEEE Trans. Autom. Control, 14, 130-134. Davison, E.~ (1976). The robust control of a servomechanism problem for linear time invariant multivariable systems. IEEE Trans. Autom. Control, 21, 25-33. Davison, E.J. (1975). A generali;ation of the output control of linear multivariable systems with unmeasurable arbitrary disturbances. IEEE Trans. Autom. Control, 20, 7BB-79l. Bongiorno, J.J., and ~C. Youla (196B). On observers in multivariable control system. Int. J. Control, B, 221-243. Newmann, M.N. (1970). SpeCific-optimal control of the linear regulator using a dynamical controller based on the minimal-order Luenberger observer. Int. J. Control, 9, 33-4B. Rosenbrock (1973)~ The zeros of a system. Int. J. Control, ~, 297-299.