- Email: [email protected]
Design of observer-based controllers for a class of discrete systems
Antomatica, Vol. 18, No. 3. pp. 323-328, 1982 Printed in Great Britain.
0005- 1098/821030323-06503.00/0 Persamon Press Lid. © 1982 International Federation of Automatic Control
Brief Paper Design of Observer-based Controllers For a Class of Discrete Systems* M A G D I S. M A H M O U D ?
Key Words--Discrete systems; time scale modeling; system order reduction; controllers; observers.
Abstract--Design of observers and observer-hased controllers for linear, discrete control systems with fast and slow modes are considered in this paper. The time separation is expressed in terms of an inequality relating norms of system submatrices. It is shown that reconstruction of fast and slow states can be accomplished by a full-order observer, the gains of which are computed using a two-stnge procedure. Then, provided that the fast subsystem is asymptotically stable, it is shown that a low-order observerbased controller can be designed with independent gain matrices to stabilize the origiual discrete system. The theoretical analysis is illustrated by a ninth-order boiler system model.
scale theory to a class of discrete systems having a timeseparaU.'on property has been developed (Mnhmoud, Chen and Singh, 1980). A separate eiganvalue assignment for this class of systems has been proposed (Mahmoud, Chen and Singh, 1981) using independent feedback gains. In this paper, the problem of designing observers and observer-based controllers for discrete systems with inaccessible states and having the time-separation property are considered. The structural properties of these systems are investigated in Mnhmond (1981). These results are used to build up full-order observers and to estabfish the conditions under which reduced-order observers could be designed. Then, a two-stage procedure is developed to compute the gains of the observer-hased controllers. The contributions of this paper are
1. Introduction
ONE OF the fundamental objectives in control systems design is the achievement of suitable eigenvalue locations in order to ensure Satisfactory dynamic performance. In modern control theory, linear state feedback (Woiovich, 1974) provides an appropriate compensation technique to meet this objective under the assumption that all state variables can be used in forming feedback signals. Unfortunately, this assumption is not always valid in practice. A well-known approach to overcome this difficulty is to generate the feedback control law via an estimate of the state vector 0Voiovich, 1974). The estimation is performed using an asymptotic state estimator, often called Luenberger observer, which employs only the available directly measureable input and output signals. Hence, the problem of designing controllers for systems with incomplete state measurements is equivalent to constructing observer-based controllers. Design of such controllers relies on the complete coiltrollability and observability of the system under consideration. For continuous dynamical systems, the design procedure is implemented utilizing essentially one of the poie-assignment algorithms (Munro, 1979). However, it has been pointed out recently OVillems, 1980; Mahmond and Singh, 1981a) that the problem of des'LLmingobservers for discrete control systems is not directly analogous to the continuous case. The main reason is due to the delay between measuring and processing the information. When dealing with large-scale dynamic systems, it is considered desirable (Mahmoud and Singh, 1981b) to simplify the control design efforts by exploiting the inherent structural properties. Such simplification could be obtained throul0t the use, for example, of hierarchical control (Sinuh and Tith, 1978), aggregation (Aoki, 1978) or singular perturbation Ogokotvic, O'Malley and Sannuti 1976). However, most of the existing results appear to be restricted to continuous-time systems. Recently, an extension of multi-time
(at it complements the results obtained in Mahmoud (1981) on structural properties of discrete systems with fast-slow separation; (b) it establishes the conditions under which full- and reduced,order observers can be designed to reconstruct the fast and slow states; and (c) it provides a two-staga procedure to compute the gain matrices of an observer-based stabilizing controller for discrete systems with inaccessible states.
*Received 30 March 1981; revised 28 July 1981. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by editor A. Sage. ?Control Systems Centre, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester M60 IQD, U.K.
where the (n2 × nO matrix L and (n~ × n2) matrix M satisfy
A ninth-order boiler system mode] (Wilson, 1974) is used to illustrate the theoretical analysis. 2. Time-Separation in Discrete Systems
Consider an asymptotically stable, linear, discrete control system described by the state and output equations x(0)= x0
(la)
z(k + 1) = A~t(k) + A ~ ( k ) + B2u(k), z(0)= ze
(lb)
x(k+l)--Atx(k)+A~gk)+Bmu(k),
y(k) = C,x(k) + C2z(k)
(2)
where x(k) E R", s(k) E R "2are the state vectors, and u(k) E R" and y(k)E R p are the control input and output vectors, respectively. System (1) said to possess the time-separation property if its eiganspectrum consists of nm large eigenvalues and n2 small eiganvalues within and including the unit circle (Mahmoud, Chen and Singh, 1980). We will assume from now onwards that system (1) is completely controllable and completely observable in the manner of Mahmoud (1981). Using the transformation
zt(k)] = \ L
\z(k)/
A , L - L A I + L A 2 L - A3 = 0
(4at
M ( A , + LA2) - (A1 - A 2 L ) M + A2 = 0
(4b)
and Ij is the (nj × n~) identity, then it is shown (Mahmoud, 323
324
Brief Paper
Chen and Singh, 1980) that system (1) has the time-separation property if the matrix inequality ~,I(A,- A2L)-'II <
1
il(A, + LA2)I[
(5)
is satisfied. In this case, the vector x~(k) ~ R" represents the slow modes whereas the vector zf ~ R "~ denotes the fast modes. For discrete two-time-scale systems, the fast modes are important only during a short initial period [0, T~]. After that period they are nngligible and the hehaviour of the system can be described by its slow modes. It is shown (Mahmond, 1981) that the slow and fast subsystems are given by x,(k + 1) ffi Aox,(k) + Bou,(k)
(6a)
y,(k) = Cox,(k) + Dou,(k)
(6b)
z/(k + 1) ffi A,s/(k)+ B2uf(k)
(7a)
y~(k)-- C2zf(k)
signals. We assume that there is a one-step delay between measuring and processing the information records. Thus, a full-order observer for system (1) and (2) is given by
(7b)
i(k + 1) = A l i ( k ) ÷ A2i(k) + Bin(k) + K l ~ ( k ) - Czi(k)- C~i(k)]
(10a)
t(k + 1) = A~i(k) + A4t(k) + B:u(k) + g2[y(k)- C~i(k) - C2t(k)] (lOb) where i ( k ) E R" is the estimate of x(k), t(k)E R '~ is the estimate of s(k) and KI, K2 are the desisn parameters that may be suitably selected to ensure any desired degree of convergence of the observation scheme. The purpose is to use the time-separation property in order to establish the conditions under which full-order observers (10) can be designed so as to reconstruct the state vectors of system (1), In terms of the observation error vectors i(k) ffi x(k) - / ( k ) and l(k)ffi z(k)-i(k), it follows from (I), (2) and (I0) that
where
i(k 4- I) ffi (At - K , C I ~ ( k ) + (A: - K , C2)i(k)
(I In) (l lb)
Ao = Ai + A2(12 - A4)-I A3
(8a)
i ( k + I)(A3 - K2CI)i(k) + (A4 - KzCz)rz(k).
Be = BI + A ~ ( l s - A4)-~ B2
(8b)
Co = C1 + C2(I2 - A4)-I A3
(8c)
System (11) will function as an observer for system (1) and (2) if the (n, x p) matrix K, and the (n2 x p) matrix Kz can be chosen such that system (10) is asymptotically stable. The following lemma establishes the main result.
Do = C2( I2 - A,)-J B2
(Sd)
u(k) = u:(k) + uf(k).
(Be)
The matrix ( I s - A J is assumed invertible, which is always valid for asymptotically stable discrete systems. Fast-slow separation allows invcstiption of the structural properties of system (1) and separate feedback control designs (Mahmond, Chen and Singh, 1981; Mahmoud, 1981).
L e m m a i. Suppose that (12 - A4)-~ exists. If (Ao, Co) and (A4, C2) are observable pairs, then system (11) is asymptotically stable. The gain matrix Kt is given by K , = Ko{I + C d l z - A 4 ) - I K ~ - A2(ls - A J - t K z
(12)
where Ke and Ks are any matrices for which ( h e - K e G ) and (A4 - K s C 9 , respectively, have spectral norms less than one.
Remarks. (1) In general, permutation and scaling of states in system
(1) are necessary to isolate fast and slow states. This can be achieved by examining the row- and cohimn-dominant elements in the system matrix, assigning the dominant elements to the d i a ~ n a l in a decreasing order and then searchina for suitable dingonal transformation matrices to reduce the effects of the o f f d h ~ n a l blocks in the system matrix. Et~cient computational algorithms to perform such operatlons are presently available (Mahmoud and Singh, 1981b). (2) It can be shown that if the matrices L and M satisfying (4a) and (4b) exist, then they are approximated to a first order by L = -(12 - A4)-~A~ + O(e)
M ffi[A~ + A~(I~ - A~)-~A~]-iAz + 0(~)
(9a) (gb)
where ~ is the separation ratio • = maxJA(A,)~lmin A (.40)1.
Proof. T h e time-separation property entails that system (11) has slow and fast subsystems given by
3. Construction of observers Design of linear state feedback for discrete two-time-scale
systems is accomplished (Mahmoud, Chen and Singh 1980, 1981) under the assumption that all state variables can be used in forming feedback signals. In this section the problem of constructing observers in order to estimate the slow and fast states is considered. These observers employ only the available directly measureable input and output
(13a)
if(k + 1) = H f x f ( k )
(13b)
where H, = (AI - K I C , ) + (A2 - KIC2XI2 - A4 + K2C2)-I(A3 - K2CI)
(14a) /-//= ( A , - K2C9.
(14b)
The gain matrix K2 can be chosen so that ( A , - K 2 ¢ 9 has spectral norm less than o n e , since (,44, C7) is an observable pair by hypothesis. This means that the fast subsystem (13b), (14b) is asymptotically stable. Consider the slow subsystem. Using the matrix identities
(9c)
(3) The fast ¢igenvalues and slow eigenvahies are perturbation from the ei~nvalues of .44 and he. respectively. (4) The interpretation of (Be) is that the control input u(k) has a slow component u,(k) which derives the subsystem (6a) and a fast component uf(k) which derives the subsystem (6b).
i , ( k + 1) ffi H,x,(k)
( F + G H ) -~ = F - ' ( I + G H F - I ) -I = F - ' [ I - G ( I + H F - ' G ) - ' H F -I]
(15)
with F = (12 - .44), O =/(2, H = C2 and manipulating (14a) along with (8a) and (8¢), it follows that H~ = A o - KoCo
(16a)
where Ko = Ki + ( A 2 - K , Cz)(I2- A J - I K2[ I + C2(I2- A4)-~ K2] -l.
(16b) The (n~ x p) matrix Ko can be chosen so that (Ao- KoC0)
Brief Paper has spectral norm less than one, since (A0, Co) is an observable pair by hypothesis. Hence, the slow subsystem (14a), (14b) is asymptotically stable. From (15) and (16b), one can immediately obtain (12). Since the asymptotic stability of the fast and slow subsystems guarantees the asymptotic stability of system (11) under the time separation property (Mahmoud, 1981), thus the lemma is proved. The result derived by Lemma I is useful as it complements the results obtained in Mahmoud (1981) concerning the'structural properties of discrete control systems with slow and fast modes. It is interesting to note that the construction of full-urder observer (10) is a two-stage procedure (a) compute the gain K2 to assign n= arbitrary eigenvalues of (A, - K2C2); (b) compute the gain K0 to assign nl arbitrary eigenvalues of (Ao- KoCo). Then compute the gain K1 using (12).
CoroUary. I f A 4 i s a stable matrix, then state observer (10) reduces to
i(k + I) = Aji(k) + A=t(k) + B1u(k)
+/Co[y(k)-C,t(k)-C~(k)]
07)
Proof. Since A4 is a stable matrix (has spectral norm less than one), it is obvious that the fast subsystem is asymptotically stable. This means that /(2 = 0 is an admissible choice for /(2 and from (12) KI = Ko. Thus (17) follows immediately. It is evident that the two-stage procedure for the design of full-order observers in the dual of the procedure developed in Mahmoud, Chen and Singh (1981) for the design of stabilizing state-feedback controllers for discrete two-time scale systems. 4. Design of obser, er-based controllers It is shown (Mahmoud, Chen and Singh, 1980, 1981) that linear state feedback can be designed for discrete system (1) using separate gain matrices. In Mahmond (1981) a lowerorder stabilizing controller is derived assuming that the fast subsystem is asymptotically stable. This is due to the fact that the controllability of the slow subsystem is invariant to a class of fast controls. These results are applicable when all state variables can be directly measured. In this section we consider the problem of designing observer-based controllers for discrete system (1) when its eigenvalues are clustered into nl large eiganvalues and a2 small eigenvalues. For simplicity, it is assumed that the fast subsystem is asymptotically stable. Thus, a reduced-order observer of the slow subsystem (6) and (8) is given by i(k+l) =Aei(k)+Bou(k)+Ko[y(k)-Coi(k) -Don(k)].
(18) An observer-based controller is described by
n(k)=Goi(k)
(19)
325
and slow subsystem given by
x,(k + I) = Fox,(k)
(21b)
where
F0= (KoA~
BiGo
Ao+ BoCo- Koco-KoDoGo) + (KAob2)(/2-A4)-I[A3B2Go]. (2lc)
The unknown gain matrices K0 and Go are determined by the following lemma.
Lemma 2. If (a) ,44 is a stable matrix, (b) (Ao,B0) is a controllable pair and (c) (Ao, Co) is an observable pair; then the control law (19) is a stabilizing controller. The gains Go and K0 are any matrices for which (,4o+ BoGo) and ( A o - KoCo), respectively, are stable matrices. Proof. The assumption that A4 is a stable matrix implies that the fast subsystem (21a) can be neglected since we are able to control the slow subsystem (21b), (21c) for any fast disturbances (Mahmoud, 1981). Consider the slow subsystem matrix Fo. In view of (Sa) through (8d), F0 reduces to
Fo = (KoA~o
BoGo Ao + BoGo - KoCo/
(22)
and under the equivalence transformation, it becomes
(~
_,)0
I
( o7o o
-
A o - KoCo] (23)
so that the eigenvalues of Fo are the eigenvalues of (,4o+ BoGo), together with the eiganvalues of (Ao-KoCo). Since (Ao,/3o) and (Ao, Co) are controllable and observable pairs by hypothesis, respectively, it follows that matrices Go and K0 exist such that F0 is a stable matrix. Thus, the observerbased controller (19) is a stabilizing feedback controller for system (20), since A4 is a stable matrix by hypothesis, and the proof is completed. Lemma 2 suggests a two-stage procedure to design the observer-based controller (19). In the first stnge, the observer gain matrix Ko is computed to place nl arbitrary eigenvalues and in the second stage the controller gain Go is computed to place nl arbitrary closed-loop eiganvalues. The usefulness of this iemma lies in identifying conditions under which the common practice of neglecting feedback controllers for systems with inaccessible states. Experience indicates that a good design results ff the observer eigenvalues are selected to be smaller in their magnitudes than the desired closed-loop eigenvalues. Next, we illustrate the application of this two-stage feedback control design to a discrete-time system.
The (2nt+n2)th-order system (20) exhibits the timeseparation property with fast subsystem described by
5. Example The state variables of a boiler system comprising a superheater and riser in series with each other (Wilson, 1974) are: density and temperature of output steam flow, temperature of the superheater, riser outlet mixture quality, water flow in riser, pressure, riser tube-wall temperature, and temperature and level of water in boiler. The variables which can be manipulated are input fuel and input water flows whereas the directly measurable output variables are temperature of output steam flow, riser outlet mixture quality, pressure and water level. By simulating the ninth-order linear continuous model (Wilson, 1974) and its discretized version, using first-order approximation, it is found that a sampling period of 0.5 s yields a discrete model whose response matches very closely that of the continuous model. Using the permutation matrix
zi(k x 1) = A,~/(k)
p = {eg, es, e2, es, eT, e6, e,, e3, ej}
where the (ntx p) matrix K0 and the (m x n~) matrix Go are unknown gains to be determined. Consider the composite system (1), (2), (18) and (19)
-(k +
t(k+ 1)/= roc,
s, ao
Ao+ SoCo- roCo- roUoao/
+ (KAo~2)z(k) - . / x ( k ) ~] + A , z ( k) . z(k + 1) = [A3 - B2(-iol~ft(k)
(20a) (20b)
(21a)
Brief Paper
326
and the scaling matrix C,= S = diag {0.015, 0.015, 0.05, 0. I, 2.0.5 x 10-4, 0.15, 5, 0.2 x 104} where e~ is the elementary column vector whose ith entry I. the transformed discrete system has the eisenvalues {1.0, 0.1452-+ 0.0726i, 0.2298, 0.89, 0.996, 0.9741 ± 0.0905j, 0.8461} and it is estimated to have six slow and three fast variables. In terms of model (1) and (2), the subsystem matrices are I AI=
1.0 0 0 0 0 0
-0.1489 x 10-~ 0.105 x 10-3 0.1051 x 10-s 0.9866 - 0.355 x 10.3 - 0.2745 x 10-3 -0.1389 x 104 0.9686 0.3156 x 10-3 0.8048 x 10-~ 0.2856 x 10-2 0.9057 -0.2065 x 10-2 0.3328 X 10 -2 0.7091 x 10-3 0.7152 x 10-2 0.2589 x 10-' 0.1980 x I0 -I
r_0.2667 x 10-s _0.5914 x 10-~ | -0.1585 x 10-7 0.4712 × 10 -2 A-/ 0.8717+10-4 0.9676x10 -s 2-[ 0.1169 x 10-~ 0.3265 x 10-5 | -0.1071 x 10-4 -0.9028 x 10-5 L °1445 x !°-5 0.1345 x 10-4
I i
a 3=
I A4 =
I Bt =
B2 =
I
0.2375 0.67 x 10-m -0.4447 x 10-4 0.1998 0.2825 x !0 -3 0.1777x10-4 -0.3191 x 10-3 0.2177 x 10-3 -0.6494 x 10-4 -~0.1159 x 10-3 -0.7698 x 10-3
2.308
-0.7292
Bo=
0.1651
Co= I i.0
× 10 -2
00 0 0.3012 × 103
J
O0
-0.2894 0.9544 -0.3907 -0.7275 0.8829 -0.8358
x x x x
10-I 0.3127 x 10-3 | lO-5 -0.1949 × I0 -j 10-I 0.2572 x 10-* 10-4 0.1951 0.1479 x 10-I .x 10-3 0.8705
The slow subsystem is described by the variables: water level, temperature of the superheater, temperature of water in the boiler, temperature of riser, riser outlet mixture quality and pressure, in order of dominance. On the other hand, the fast subsystem is represented by the variables: water flow in riser, temperature and density of output steam flow. "3 -0.1006 x 103 l -0.3105 x 102 0.3291 x 103
J
I|Adl : 1.0099 /|A~I : 0.3664 x 10J
JlA~ = 0.1307 × 10-3
IAdl = I.o3o2 Levi = 0.4529 x lIP
[[Ae-'[[ = 1.2475
IIA~
=
0.2668
IIMdt = 0.1538 x i0 -3
The solution of (4a) after seven iterations has the norm Jl/-~ : 0.4963 x 103, while the solution of (4b) after one iteration has the norm IIMtH: 0.1539 x 10-3. Thus, equations (9a) and (9b) are satisfied with • : 0.2069, which implies that the discrete model possesses the time-separation property. In terms of (7), the fast subsystem is described by the triple (A4,/h, C2) and with reference to (6), the slow subsystem is characterized by
0.5951× -0.2997 x 0.9721 0.2983 x 0.2994 x 0.2994 x 0.2697 x
10 -3
10-~ 10-2 l0 -2 l0 -2 10-I
0.8872 -0.2274 0.1350 0.9058 0.8083 0.8083 0.2045
x 10-4 x l04 x 10-3 × 10-3 x 10-3 x 10-I
0.331 x 10-5 -] 0.1168 x 10-I | 0.9704 x 10-4/ 0.1109 [ - 0.2026 x I 0-5 [ 0.1232 x 10-2J 0 0 0 -0.3318
00
Evaluation of the spectral norms of the different submatices gives
-0.2622 x I0-' "7 0.8275 x l0 -~
-0.5085 x 10-*
x 10-s x 10-3 x 10-s x 10-3 × 10-3
Iioo l °° t 1 0 0 I 0 0
0 1 0 '
-0.231 x l i p 0.2.564 x 10-~ -0.1692
l
1.8098
I
0.2953 -0.6407 0.4760 -0.1528 -0.2016 -0.1355
-0.1336 x I0' -0.6724 0.4815 x 101
0.4490xlO-Sql 0.1159 x 1 0 - 1 | 0.3889 x 10-4 0.1109 0.2689 x 10-4| 0.1239 x 10-2.]
1.0 -0.1334 x 10-4 0 0.9959 0 0.3450 x 10-2 0 0.7419 x 10-2 0 -0.6295 x 10-2 0 -0.6295 x 10-2 0 0.1967 x 10-3
I
J
-6.0165 0.3120 x 102 0.2490 x lip -0.8749 -0.5153 x lip 6.2408
-0.4393
A0=
_0.3823 x 10-5 q 0.5030 x 10-4 -0.1144x10 -5 0.1673 x 10.4 0.1334 x 10-4 0.1143×10 -3
C2=
I000 0000 0000 0000
0 0 0 = 0.2520
0 !.0 1.0 0.1771 x 10-~
-0.2813 0.1616 -0.6551 -0.1132 0.8862 0.8862 -0.130~
x 10-~ x 10-~ x l0 -I x I0 -s x 10-2
-0.7785 0.6003 0.1263 0.2015 0.2149 0.2149 0.9145
x 10-3 -] x l0 -5 × 10-j x l0 -I x 10-I
Brief Paper It is easy to check that both slow and fast subsystems are completely controllable and observable. A full-order observer can be designed to reconstruct the slow and fast states. Assigning three eigenvalues at {0.15, 0.13, 0.11} yields I! K2=
0 0 0
0 0 0
-61.6373"7 -0.1962 | -0.9500 J
and positioning six eigenvalues at {0.99, 0.97, 0.95, 0.93, 0.91, 0.98} gives [--0.3992 |-0.1301 | 0.4738 KI - - / 0.6404 0.9804 [_-0.3736
/
x 10-I × 10-2 x 10-1 x 10-I
0.8283 -0.2002 -0.1646 0.1308 0.1859 -0.4547
x 10-2 x l0 -3 x 10-1 x 10-' × 10-:
-0.7771 0.3398 -0.3697 -0.2290 -0.2819 0.5312
x 10-2 x l0 -3 x 10-4 × 10-t x 10-I
327
values. Experience indicates that a good design results if the observer eigenvalues are selected to he smaller in their magnitudes than the desired closed-loop eigenvalues. A detailed analysis of a ninth-order boiler system model is given to illustrate the theoretical developments. The results obtained in this paper complement those of (Mahmoud, 1981) concerning the structural properties of discrete control systems with fast and slow modes. One of the main contributions of this paper is that it emphasizes the usefulness of the fast-slow separation in simplifying the design of stabilizing feedback controllers for discrete systems with inaccessible states. 0.1325 x 10-4 "] -0.2193 x 1 0 - 3 |
|
0.1758 x 1 0 - 3 | 0.3910 x 1 0 - 4 | -0.1472 × 10-3 J
From (16b), the gain matrix KI is computed as [--0.3992 × 10-I
|-0.1301
x 10 -2
T,- | "~ = |
0.4738 x 10-I 0.404 | 0.8904 x 10-t L-0.3736
0.8283 -0.2002 0.3095 0.1308 0.1859 -0.4547
X 10 -2 × 10 -3
x 10-I x 10-t × 10-I
-0.7771 0.3398 -0.1646 -0.2290 -0.2819 0.5312
× 10 -2
x l0 -3 x i0 -I x 10-1 × 10-I
In determining Ke and K2, a pole-assignment algorithm of the spectral factorization type has been used (Munro, 1979). This completes the construction procedure of the fullorder state observer. Since A4 is a stable matrix, (Ae, Be) is a controllable pair and (Ao, Co) is an observable pair, then Lemma 2 can he used to design a lower-order observer-based controller. Placing the observer eigenvalues at{0.83, 0.82, 0.81, 0.80, 0.79, 0.78} gives [- - 0.5879 | - 0 . 9 0 6 3 × 10-j . ] - 0 . 1 2 4 9 x 103 Ko=| 0.3669 x 102 [-0.1932 × 10t [_-0.1232 × 102
0.2398 03282 x 10-I 0.6905 × 102 -0.1807+102 0.9968 0.7272 x 10I
-0.2538 -0.6228 -0.8445 0.1703 -0.1554 -0.1058
x x x x
10-~ 10-2 10~ 10I
x 10j
-0.1996 -0.2585 0.7004 0.2321 -0.8178 -0.5173
x 10-3 "7 X 10 -3 [
x 10-2 [ x 10-3 l x 10-3 | × 10-4.J
Several extensions of the present work that deserve further research are possible. First is the problem of designing output feedback controller using separate gain matrices. Second is the optimal discrete regulator problem.lnvestigations into these interesting problems are presently in progress.
0.2100 x 10-4 "7 -0.6453 x 10-3 [ 0.6829 x 10-2 | -0.1813 x 1 0 - 2 / 0.1285 x 10-3[ 0.7215 x IO-3 ..J
and selected the desired closed-loop eigenvalues to he {0.99, 0.97, 0.95, 0.93, 0.91, 0.89} results in Go
- 0.2035 x 103 L -0.3760 x 122
- 0.7295 x ! 02 -0.8199 x 101
0.6145 x 102 0.5847 × 101
The matrices Ko and Go are then the required gains to implement the dynamic state feedback controller (19) to the discrete boiler system model.
6. Conclusions A class of linear, discrete control systems having a timeseparation property has been considered. It is shown that when an inequality relating norms of system submatrices is satisfied, the hehaviour of the original system is approximated by two lower-order sybsystems: a slow subsystem with large eigenvalues and a fast subsystem with small eigenvalues. Control problems of such systems with inaccessible states are investigated. Conditions under which full- and reduced-order observers can he designed to reconstruct the slow and fast states are derived. The design procedure is accomplished by computing two independent gain matrices. Under the assumption that the fast subsystem is asymptotically stable, a lower-order observer-based controller (dynamic state feedback controfier) is designed by separate placement of the observer and closed-loop eigen-
0.6797 x 101 0.8749
0.7904 x 101 0.5298 x 101
0.3152 x 102 ] 0.6536x l01 J
References Aoki, M. (1978). Some approximation methods for estimation and control of large scale systems, IEEE Trons AuL Control AC-23, 173. Kokotovic, P. V., R. E. O'Maliey, Jr and P. Sannuti (1976). Singular perturbations and order reduction in control theory--An overview, Automat|ca, 12, 123. Mahmoud, M. S. (1981). Structural properties of discrete systems with slow and fast modes, Control Systems Centre Report No. 505, UMIST, Manchester (to appear
in Large Scale Systems). Mahmoud, M. S., Y. Chen and M. G. Singh (1980). Discrete two-time-scale systems, Control Systems Centre Report No. 497, UMIST, Manchester (to appear in Z
Computers & Elect. Engng). Mahmoud, M. S., Y. Chen and M. G. Singh (1981). On the eigenvalue assignment in discrete systems with fast and slow modes, Control Systems Centre Report No. 499, UMIST, Manchester (submitted for publication). Mahmoud, M. S. and M. G. Singh (1981a). Decentralized
328
Brief Paper
state reconstruction of interconnected discrete systems. Large Scale Systems 2, 151. Mahmoud, M. S. and M. G. Singh (1981b). Large Scale Systems Modelling. Pergamon Press Oxford. Munro, N. (1979). Pole assignment. Pro¢. lEE. 126349. Porter, B, B, (1977). Singular perturbation methods in the design of full-order observers for multivariable linear systems. Int. J. Control, 26, 589.
Singh, M. O. and A. Titli (1978). Systems: DecompositionControl and Optimization. Pergamon Press, Oxford. WiUems, J. L. (1980). Design of state observers for linear discrete-time systems. Int..l. Systems Sci., 11, 139. Wilson. D. A. (1974). Model reduction for multivariable systems. Int. J. Control, 20, 57. Wolovich, W. A. (1974). Linear Multioariable Systems. Springer-Verlag New York.
0005- 1098/821030323-06503.00/0 Persamon Press Lid. © 1982 International Federation of Automatic Control
Brief Paper Design of Observer-based Controllers For a Class of Discrete Systems* M A G D I S. M A H M O U D ?
Key Words--Discrete systems; time scale modeling; system order reduction; controllers; observers.
Abstract--Design of observers and observer-hased controllers for linear, discrete control systems with fast and slow modes are considered in this paper. The time separation is expressed in terms of an inequality relating norms of system submatrices. It is shown that reconstruction of fast and slow states can be accomplished by a full-order observer, the gains of which are computed using a two-stnge procedure. Then, provided that the fast subsystem is asymptotically stable, it is shown that a low-order observerbased controller can be designed with independent gain matrices to stabilize the origiual discrete system. The theoretical analysis is illustrated by a ninth-order boiler system model.
scale theory to a class of discrete systems having a timeseparaU.'on property has been developed (Mnhmoud, Chen and Singh, 1980). A separate eiganvalue assignment for this class of systems has been proposed (Mahmoud, Chen and Singh, 1981) using independent feedback gains. In this paper, the problem of designing observers and observer-based controllers for discrete systems with inaccessible states and having the time-separation property are considered. The structural properties of these systems are investigated in Mnhmond (1981). These results are used to build up full-order observers and to estabfish the conditions under which reduced-order observers could be designed. Then, a two-stage procedure is developed to compute the gains of the observer-hased controllers. The contributions of this paper are
1. Introduction
ONE OF the fundamental objectives in control systems design is the achievement of suitable eigenvalue locations in order to ensure Satisfactory dynamic performance. In modern control theory, linear state feedback (Woiovich, 1974) provides an appropriate compensation technique to meet this objective under the assumption that all state variables can be used in forming feedback signals. Unfortunately, this assumption is not always valid in practice. A well-known approach to overcome this difficulty is to generate the feedback control law via an estimate of the state vector 0Voiovich, 1974). The estimation is performed using an asymptotic state estimator, often called Luenberger observer, which employs only the available directly measureable input and output signals. Hence, the problem of designing controllers for systems with incomplete state measurements is equivalent to constructing observer-based controllers. Design of such controllers relies on the complete coiltrollability and observability of the system under consideration. For continuous dynamical systems, the design procedure is implemented utilizing essentially one of the poie-assignment algorithms (Munro, 1979). However, it has been pointed out recently OVillems, 1980; Mahmond and Singh, 1981a) that the problem of des'LLmingobservers for discrete control systems is not directly analogous to the continuous case. The main reason is due to the delay between measuring and processing the information. When dealing with large-scale dynamic systems, it is considered desirable (Mahmoud and Singh, 1981b) to simplify the control design efforts by exploiting the inherent structural properties. Such simplification could be obtained throul0t the use, for example, of hierarchical control (Sinuh and Tith, 1978), aggregation (Aoki, 1978) or singular perturbation Ogokotvic, O'Malley and Sannuti 1976). However, most of the existing results appear to be restricted to continuous-time systems. Recently, an extension of multi-time
(at it complements the results obtained in Mahmoud (1981) on structural properties of discrete systems with fast-slow separation; (b) it establishes the conditions under which full- and reduced,order observers can be designed to reconstruct the fast and slow states; and (c) it provides a two-staga procedure to compute the gain matrices of an observer-based stabilizing controller for discrete systems with inaccessible states.
*Received 30 March 1981; revised 28 July 1981. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by editor A. Sage. ?Control Systems Centre, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester M60 IQD, U.K.
where the (n2 × nO matrix L and (n~ × n2) matrix M satisfy
A ninth-order boiler system mode] (Wilson, 1974) is used to illustrate the theoretical analysis. 2. Time-Separation in Discrete Systems
Consider an asymptotically stable, linear, discrete control system described by the state and output equations x(0)= x0
(la)
z(k + 1) = A~t(k) + A ~ ( k ) + B2u(k), z(0)= ze
(lb)
x(k+l)--Atx(k)+A~gk)+Bmu(k),
y(k) = C,x(k) + C2z(k)
(2)
where x(k) E R", s(k) E R "2are the state vectors, and u(k) E R" and y(k)E R p are the control input and output vectors, respectively. System (1) said to possess the time-separation property if its eiganspectrum consists of nm large eigenvalues and n2 small eiganvalues within and including the unit circle (Mahmoud, Chen and Singh, 1980). We will assume from now onwards that system (1) is completely controllable and completely observable in the manner of Mahmoud (1981). Using the transformation
zt(k)] = \ L
\z(k)/
A , L - L A I + L A 2 L - A3 = 0
(4at
M ( A , + LA2) - (A1 - A 2 L ) M + A2 = 0
(4b)
and Ij is the (nj × n~) identity, then it is shown (Mahmoud, 323
324
Brief Paper
Chen and Singh, 1980) that system (1) has the time-separation property if the matrix inequality ~,I(A,- A2L)-'II <
1
il(A, + LA2)I[
(5)
is satisfied. In this case, the vector x~(k) ~ R" represents the slow modes whereas the vector zf ~ R "~ denotes the fast modes. For discrete two-time-scale systems, the fast modes are important only during a short initial period [0, T~]. After that period they are nngligible and the hehaviour of the system can be described by its slow modes. It is shown (Mahmond, 1981) that the slow and fast subsystems are given by x,(k + 1) ffi Aox,(k) + Bou,(k)
(6a)
y,(k) = Cox,(k) + Dou,(k)
(6b)
z/(k + 1) ffi A,s/(k)+ B2uf(k)
(7a)
y~(k)-- C2zf(k)
signals. We assume that there is a one-step delay between measuring and processing the information records. Thus, a full-order observer for system (1) and (2) is given by
(7b)
i(k + 1) = A l i ( k ) ÷ A2i(k) + Bin(k) + K l ~ ( k ) - Czi(k)- C~i(k)]
(10a)
t(k + 1) = A~i(k) + A4t(k) + B:u(k) + g2[y(k)- C~i(k) - C2t(k)] (lOb) where i ( k ) E R" is the estimate of x(k), t(k)E R '~ is the estimate of s(k) and KI, K2 are the desisn parameters that may be suitably selected to ensure any desired degree of convergence of the observation scheme. The purpose is to use the time-separation property in order to establish the conditions under which full-order observers (10) can be designed so as to reconstruct the state vectors of system (1), In terms of the observation error vectors i(k) ffi x(k) - / ( k ) and l(k)ffi z(k)-i(k), it follows from (I), (2) and (I0) that
where
i(k 4- I) ffi (At - K , C I ~ ( k ) + (A: - K , C2)i(k)
(I In) (l lb)
Ao = Ai + A2(12 - A4)-I A3
(8a)
i ( k + I)(A3 - K2CI)i(k) + (A4 - KzCz)rz(k).
Be = BI + A ~ ( l s - A4)-~ B2
(8b)
Co = C1 + C2(I2 - A4)-I A3
(8c)
System (11) will function as an observer for system (1) and (2) if the (n, x p) matrix K, and the (n2 x p) matrix Kz can be chosen such that system (10) is asymptotically stable. The following lemma establishes the main result.
Do = C2( I2 - A,)-J B2
(Sd)
u(k) = u:(k) + uf(k).
(Be)
The matrix ( I s - A J is assumed invertible, which is always valid for asymptotically stable discrete systems. Fast-slow separation allows invcstiption of the structural properties of system (1) and separate feedback control designs (Mahmond, Chen and Singh, 1981; Mahmoud, 1981).
L e m m a i. Suppose that (12 - A4)-~ exists. If (Ao, Co) and (A4, C2) are observable pairs, then system (11) is asymptotically stable. The gain matrix Kt is given by K , = Ko{I + C d l z - A 4 ) - I K ~ - A2(ls - A J - t K z
(12)
where Ke and Ks are any matrices for which ( h e - K e G ) and (A4 - K s C 9 , respectively, have spectral norms less than one.
Remarks. (1) In general, permutation and scaling of states in system
(1) are necessary to isolate fast and slow states. This can be achieved by examining the row- and cohimn-dominant elements in the system matrix, assigning the dominant elements to the d i a ~ n a l in a decreasing order and then searchina for suitable dingonal transformation matrices to reduce the effects of the o f f d h ~ n a l blocks in the system matrix. Et~cient computational algorithms to perform such operatlons are presently available (Mahmoud and Singh, 1981b). (2) It can be shown that if the matrices L and M satisfying (4a) and (4b) exist, then they are approximated to a first order by L = -(12 - A4)-~A~ + O(e)
M ffi[A~ + A~(I~ - A~)-~A~]-iAz + 0(~)
(9a) (gb)
where ~ is the separation ratio • = maxJA(A,)~lmin A (.40)1.
Proof. T h e time-separation property entails that system (11) has slow and fast subsystems given by
3. Construction of observers Design of linear state feedback for discrete two-time-scale
systems is accomplished (Mahmoud, Chen and Singh 1980, 1981) under the assumption that all state variables can be used in forming feedback signals. In this section the problem of constructing observers in order to estimate the slow and fast states is considered. These observers employ only the available directly measureable input and output
(13a)
if(k + 1) = H f x f ( k )
(13b)
where H, = (AI - K I C , ) + (A2 - KIC2XI2 - A4 + K2C2)-I(A3 - K2CI)
(14a) /-//= ( A , - K2C9.
(14b)
The gain matrix K2 can be chosen so that ( A , - K 2 ¢ 9 has spectral norm less than o n e , since (,44, C7) is an observable pair by hypothesis. This means that the fast subsystem (13b), (14b) is asymptotically stable. Consider the slow subsystem. Using the matrix identities
(9c)
(3) The fast ¢igenvalues and slow eigenvahies are perturbation from the ei~nvalues of .44 and he. respectively. (4) The interpretation of (Be) is that the control input u(k) has a slow component u,(k) which derives the subsystem (6a) and a fast component uf(k) which derives the subsystem (6b).
i , ( k + 1) ffi H,x,(k)
( F + G H ) -~ = F - ' ( I + G H F - I ) -I = F - ' [ I - G ( I + H F - ' G ) - ' H F -I]
(15)
with F = (12 - .44), O =/(2, H = C2 and manipulating (14a) along with (8a) and (8¢), it follows that H~ = A o - KoCo
(16a)
where Ko = Ki + ( A 2 - K , Cz)(I2- A J - I K2[ I + C2(I2- A4)-~ K2] -l.
(16b) The (n~ x p) matrix Ko can be chosen so that (Ao- KoC0)
Brief Paper has spectral norm less than one, since (A0, Co) is an observable pair by hypothesis. Hence, the slow subsystem (14a), (14b) is asymptotically stable. From (15) and (16b), one can immediately obtain (12). Since the asymptotic stability of the fast and slow subsystems guarantees the asymptotic stability of system (11) under the time separation property (Mahmoud, 1981), thus the lemma is proved. The result derived by Lemma I is useful as it complements the results obtained in Mahmoud (1981) concerning the'structural properties of discrete control systems with slow and fast modes. It is interesting to note that the construction of full-urder observer (10) is a two-stage procedure (a) compute the gain K2 to assign n= arbitrary eigenvalues of (A, - K2C2); (b) compute the gain K0 to assign nl arbitrary eigenvalues of (Ao- KoCo). Then compute the gain K1 using (12).
CoroUary. I f A 4 i s a stable matrix, then state observer (10) reduces to
i(k + I) = Aji(k) + A=t(k) + B1u(k)
+/Co[y(k)-C,t(k)-C~(k)]
07)
Proof. Since A4 is a stable matrix (has spectral norm less than one), it is obvious that the fast subsystem is asymptotically stable. This means that /(2 = 0 is an admissible choice for /(2 and from (12) KI = Ko. Thus (17) follows immediately. It is evident that the two-stage procedure for the design of full-order observers in the dual of the procedure developed in Mahmoud, Chen and Singh (1981) for the design of stabilizing state-feedback controllers for discrete two-time scale systems. 4. Design of obser, er-based controllers It is shown (Mahmoud, Chen and Singh, 1980, 1981) that linear state feedback can be designed for discrete system (1) using separate gain matrices. In Mahmond (1981) a lowerorder stabilizing controller is derived assuming that the fast subsystem is asymptotically stable. This is due to the fact that the controllability of the slow subsystem is invariant to a class of fast controls. These results are applicable when all state variables can be directly measured. In this section we consider the problem of designing observer-based controllers for discrete system (1) when its eigenvalues are clustered into nl large eiganvalues and a2 small eigenvalues. For simplicity, it is assumed that the fast subsystem is asymptotically stable. Thus, a reduced-order observer of the slow subsystem (6) and (8) is given by i(k+l) =Aei(k)+Bou(k)+Ko[y(k)-Coi(k) -Don(k)].
(18) An observer-based controller is described by
n(k)=Goi(k)
(19)
325
and slow subsystem given by
x,(k + I) = Fox,(k)
(21b)
where
F0= (KoA~
BiGo
Ao+ BoCo- Koco-KoDoGo) + (KAob2)(/2-A4)-I[A3B2Go]. (2lc)
The unknown gain matrices K0 and Go are determined by the following lemma.
Lemma 2. If (a) ,44 is a stable matrix, (b) (Ao,B0) is a controllable pair and (c) (Ao, Co) is an observable pair; then the control law (19) is a stabilizing controller. The gains Go and K0 are any matrices for which (,4o+ BoGo) and ( A o - KoCo), respectively, are stable matrices. Proof. The assumption that A4 is a stable matrix implies that the fast subsystem (21a) can be neglected since we are able to control the slow subsystem (21b), (21c) for any fast disturbances (Mahmoud, 1981). Consider the slow subsystem matrix Fo. In view of (Sa) through (8d), F0 reduces to
Fo = (KoA~o
BoGo Ao + BoGo - KoCo/
(22)
and under the equivalence transformation, it becomes
(~
_,)0
I
( o7o o
-
A o - KoCo] (23)
so that the eigenvalues of Fo are the eigenvalues of (,4o+ BoGo), together with the eiganvalues of (Ao-KoCo). Since (Ao,/3o) and (Ao, Co) are controllable and observable pairs by hypothesis, respectively, it follows that matrices Go and K0 exist such that F0 is a stable matrix. Thus, the observerbased controller (19) is a stabilizing feedback controller for system (20), since A4 is a stable matrix by hypothesis, and the proof is completed. Lemma 2 suggests a two-stage procedure to design the observer-based controller (19). In the first stnge, the observer gain matrix Ko is computed to place nl arbitrary eigenvalues and in the second stage the controller gain Go is computed to place nl arbitrary closed-loop eiganvalues. The usefulness of this iemma lies in identifying conditions under which the common practice of neglecting feedback controllers for systems with inaccessible states. Experience indicates that a good design results ff the observer eigenvalues are selected to be smaller in their magnitudes than the desired closed-loop eigenvalues. Next, we illustrate the application of this two-stage feedback control design to a discrete-time system.
The (2nt+n2)th-order system (20) exhibits the timeseparation property with fast subsystem described by
5. Example The state variables of a boiler system comprising a superheater and riser in series with each other (Wilson, 1974) are: density and temperature of output steam flow, temperature of the superheater, riser outlet mixture quality, water flow in riser, pressure, riser tube-wall temperature, and temperature and level of water in boiler. The variables which can be manipulated are input fuel and input water flows whereas the directly measurable output variables are temperature of output steam flow, riser outlet mixture quality, pressure and water level. By simulating the ninth-order linear continuous model (Wilson, 1974) and its discretized version, using first-order approximation, it is found that a sampling period of 0.5 s yields a discrete model whose response matches very closely that of the continuous model. Using the permutation matrix
zi(k x 1) = A,~/(k)
p = {eg, es, e2, es, eT, e6, e,, e3, ej}
where the (ntx p) matrix K0 and the (m x n~) matrix Go are unknown gains to be determined. Consider the composite system (1), (2), (18) and (19)
-(k +
t(k+ 1)/= roc,
s, ao
Ao+ SoCo- roCo- roUoao/
+ (KAo~2)z(k) - . / x ( k ) ~] + A , z ( k) . z(k + 1) = [A3 - B2(-iol~ft(k)
(20a) (20b)
(21a)
Brief Paper
326
and the scaling matrix C,= S = diag {0.015, 0.015, 0.05, 0. I, 2.0.5 x 10-4, 0.15, 5, 0.2 x 104} where e~ is the elementary column vector whose ith entry I. the transformed discrete system has the eisenvalues {1.0, 0.1452-+ 0.0726i, 0.2298, 0.89, 0.996, 0.9741 ± 0.0905j, 0.8461} and it is estimated to have six slow and three fast variables. In terms of model (1) and (2), the subsystem matrices are I AI=
1.0 0 0 0 0 0
-0.1489 x 10-~ 0.105 x 10-3 0.1051 x 10-s 0.9866 - 0.355 x 10.3 - 0.2745 x 10-3 -0.1389 x 104 0.9686 0.3156 x 10-3 0.8048 x 10-~ 0.2856 x 10-2 0.9057 -0.2065 x 10-2 0.3328 X 10 -2 0.7091 x 10-3 0.7152 x 10-2 0.2589 x 10-' 0.1980 x I0 -I
r_0.2667 x 10-s _0.5914 x 10-~ | -0.1585 x 10-7 0.4712 × 10 -2 A-/ 0.8717+10-4 0.9676x10 -s 2-[ 0.1169 x 10-~ 0.3265 x 10-5 | -0.1071 x 10-4 -0.9028 x 10-5 L °1445 x !°-5 0.1345 x 10-4
I i
a 3=
I A4 =
I Bt =
B2 =
I
0.2375 0.67 x 10-m -0.4447 x 10-4 0.1998 0.2825 x !0 -3 0.1777x10-4 -0.3191 x 10-3 0.2177 x 10-3 -0.6494 x 10-4 -~0.1159 x 10-3 -0.7698 x 10-3
2.308
-0.7292
Bo=
0.1651
Co= I i.0
× 10 -2
00 0 0.3012 × 103
J
O0
-0.2894 0.9544 -0.3907 -0.7275 0.8829 -0.8358
x x x x
10-I 0.3127 x 10-3 | lO-5 -0.1949 × I0 -j 10-I 0.2572 x 10-* 10-4 0.1951 0.1479 x 10-I .x 10-3 0.8705
The slow subsystem is described by the variables: water level, temperature of the superheater, temperature of water in the boiler, temperature of riser, riser outlet mixture quality and pressure, in order of dominance. On the other hand, the fast subsystem is represented by the variables: water flow in riser, temperature and density of output steam flow. "3 -0.1006 x 103 l -0.3105 x 102 0.3291 x 103
J
I|Adl : 1.0099 /|A~I : 0.3664 x 10J
JlA~ = 0.1307 × 10-3
IAdl = I.o3o2 Levi = 0.4529 x lIP
[[Ae-'[[ = 1.2475
IIA~
=
0.2668
IIMdt = 0.1538 x i0 -3
The solution of (4a) after seven iterations has the norm Jl/-~ : 0.4963 x 103, while the solution of (4b) after one iteration has the norm IIMtH: 0.1539 x 10-3. Thus, equations (9a) and (9b) are satisfied with • : 0.2069, which implies that the discrete model possesses the time-separation property. In terms of (7), the fast subsystem is described by the triple (A4,/h, C2) and with reference to (6), the slow subsystem is characterized by
0.5951× -0.2997 x 0.9721 0.2983 x 0.2994 x 0.2994 x 0.2697 x
10 -3
10-~ 10-2 l0 -2 l0 -2 10-I
0.8872 -0.2274 0.1350 0.9058 0.8083 0.8083 0.2045
x 10-4 x l04 x 10-3 × 10-3 x 10-3 x 10-I
0.331 x 10-5 -] 0.1168 x 10-I | 0.9704 x 10-4/ 0.1109 [ - 0.2026 x I 0-5 [ 0.1232 x 10-2J 0 0 0 -0.3318
00
Evaluation of the spectral norms of the different submatices gives
-0.2622 x I0-' "7 0.8275 x l0 -~
-0.5085 x 10-*
x 10-s x 10-3 x 10-s x 10-3 × 10-3
Iioo l °° t 1 0 0 I 0 0
0 1 0 '
-0.231 x l i p 0.2.564 x 10-~ -0.1692
l
1.8098
I
0.2953 -0.6407 0.4760 -0.1528 -0.2016 -0.1355
-0.1336 x I0' -0.6724 0.4815 x 101
0.4490xlO-Sql 0.1159 x 1 0 - 1 | 0.3889 x 10-4 0.1109 0.2689 x 10-4| 0.1239 x 10-2.]
1.0 -0.1334 x 10-4 0 0.9959 0 0.3450 x 10-2 0 0.7419 x 10-2 0 -0.6295 x 10-2 0 -0.6295 x 10-2 0 0.1967 x 10-3
I
J
-6.0165 0.3120 x 102 0.2490 x lip -0.8749 -0.5153 x lip 6.2408
-0.4393
A0=
_0.3823 x 10-5 q 0.5030 x 10-4 -0.1144x10 -5 0.1673 x 10.4 0.1334 x 10-4 0.1143×10 -3
C2=
I000 0000 0000 0000
0 0 0 = 0.2520
0 !.0 1.0 0.1771 x 10-~
-0.2813 0.1616 -0.6551 -0.1132 0.8862 0.8862 -0.130~
x 10-~ x 10-~ x l0 -I x I0 -s x 10-2
-0.7785 0.6003 0.1263 0.2015 0.2149 0.2149 0.9145
x 10-3 -] x l0 -5 × 10-j x l0 -I x 10-I
Brief Paper It is easy to check that both slow and fast subsystems are completely controllable and observable. A full-order observer can be designed to reconstruct the slow and fast states. Assigning three eigenvalues at {0.15, 0.13, 0.11} yields I! K2=
0 0 0
0 0 0
-61.6373"7 -0.1962 | -0.9500 J
and positioning six eigenvalues at {0.99, 0.97, 0.95, 0.93, 0.91, 0.98} gives [--0.3992 |-0.1301 | 0.4738 KI - - / 0.6404 0.9804 [_-0.3736
/
x 10-I × 10-2 x 10-1 x 10-I
0.8283 -0.2002 -0.1646 0.1308 0.1859 -0.4547
x 10-2 x l0 -3 x 10-1 x 10-' × 10-:
-0.7771 0.3398 -0.3697 -0.2290 -0.2819 0.5312
x 10-2 x l0 -3 x 10-4 × 10-t x 10-I
327
values. Experience indicates that a good design results if the observer eigenvalues are selected to he smaller in their magnitudes than the desired closed-loop eigenvalues. A detailed analysis of a ninth-order boiler system model is given to illustrate the theoretical developments. The results obtained in this paper complement those of (Mahmoud, 1981) concerning the structural properties of discrete control systems with fast and slow modes. One of the main contributions of this paper is that it emphasizes the usefulness of the fast-slow separation in simplifying the design of stabilizing feedback controllers for discrete systems with inaccessible states. 0.1325 x 10-4 "] -0.2193 x 1 0 - 3 |
|
0.1758 x 1 0 - 3 | 0.3910 x 1 0 - 4 | -0.1472 × 10-3 J
From (16b), the gain matrix KI is computed as [--0.3992 × 10-I
|-0.1301
x 10 -2
T,- | "~ = |
0.4738 x 10-I 0.404 | 0.8904 x 10-t L-0.3736
0.8283 -0.2002 0.3095 0.1308 0.1859 -0.4547
X 10 -2 × 10 -3
x 10-I x 10-t × 10-I
-0.7771 0.3398 -0.1646 -0.2290 -0.2819 0.5312
× 10 -2
x l0 -3 x i0 -I x 10-1 × 10-I
In determining Ke and K2, a pole-assignment algorithm of the spectral factorization type has been used (Munro, 1979). This completes the construction procedure of the fullorder state observer. Since A4 is a stable matrix, (Ae, Be) is a controllable pair and (Ao, Co) is an observable pair, then Lemma 2 can he used to design a lower-order observer-based controller. Placing the observer eigenvalues at{0.83, 0.82, 0.81, 0.80, 0.79, 0.78} gives [- - 0.5879 | - 0 . 9 0 6 3 × 10-j . ] - 0 . 1 2 4 9 x 103 Ko=| 0.3669 x 102 [-0.1932 × 10t [_-0.1232 × 102
0.2398 03282 x 10-I 0.6905 × 102 -0.1807+102 0.9968 0.7272 x 10I
-0.2538 -0.6228 -0.8445 0.1703 -0.1554 -0.1058
x x x x
10-~ 10-2 10~ 10I
x 10j
-0.1996 -0.2585 0.7004 0.2321 -0.8178 -0.5173
x 10-3 "7 X 10 -3 [
x 10-2 [ x 10-3 l x 10-3 | × 10-4.J
Several extensions of the present work that deserve further research are possible. First is the problem of designing output feedback controller using separate gain matrices. Second is the optimal discrete regulator problem.lnvestigations into these interesting problems are presently in progress.
0.2100 x 10-4 "7 -0.6453 x 10-3 [ 0.6829 x 10-2 | -0.1813 x 1 0 - 2 / 0.1285 x 10-3[ 0.7215 x IO-3 ..J
and selected the desired closed-loop eigenvalues to he {0.99, 0.97, 0.95, 0.93, 0.91, 0.89} results in Go
- 0.2035 x 103 L -0.3760 x 122
- 0.7295 x ! 02 -0.8199 x 101
0.6145 x 102 0.5847 × 101
The matrices Ko and Go are then the required gains to implement the dynamic state feedback controller (19) to the discrete boiler system model.
6. Conclusions A class of linear, discrete control systems having a timeseparation property has been considered. It is shown that when an inequality relating norms of system submatrices is satisfied, the hehaviour of the original system is approximated by two lower-order sybsystems: a slow subsystem with large eigenvalues and a fast subsystem with small eigenvalues. Control problems of such systems with inaccessible states are investigated. Conditions under which full- and reduced-order observers can he designed to reconstruct the slow and fast states are derived. The design procedure is accomplished by computing two independent gain matrices. Under the assumption that the fast subsystem is asymptotically stable, a lower-order observer-based controller (dynamic state feedback controfier) is designed by separate placement of the observer and closed-loop eigen-
0.6797 x 101 0.8749
0.7904 x 101 0.5298 x 101
0.3152 x 102 ] 0.6536x l01 J
References Aoki, M. (1978). Some approximation methods for estimation and control of large scale systems, IEEE Trons AuL Control AC-23, 173. Kokotovic, P. V., R. E. O'Maliey, Jr and P. Sannuti (1976). Singular perturbations and order reduction in control theory--An overview, Automat|ca, 12, 123. Mahmoud, M. S. (1981). Structural properties of discrete systems with slow and fast modes, Control Systems Centre Report No. 505, UMIST, Manchester (to appear
in Large Scale Systems). Mahmoud, M. S., Y. Chen and M. G. Singh (1980). Discrete two-time-scale systems, Control Systems Centre Report No. 497, UMIST, Manchester (to appear in Z
Computers & Elect. Engng). Mahmoud, M. S., Y. Chen and M. G. Singh (1981). On the eigenvalue assignment in discrete systems with fast and slow modes, Control Systems Centre Report No. 499, UMIST, Manchester (submitted for publication). Mahmoud, M. S. and M. G. Singh (1981a). Decentralized
328
Brief Paper
state reconstruction of interconnected discrete systems. Large Scale Systems 2, 151. Mahmoud, M. S. and M. G. Singh (1981b). Large Scale Systems Modelling. Pergamon Press Oxford. Munro, N. (1979). Pole assignment. Pro¢. lEE. 126349. Porter, B, B, (1977). Singular perturbation methods in the design of full-order observers for multivariable linear systems. Int. J. Control, 26, 589.
Singh, M. O. and A. Titli (1978). Systems: DecompositionControl and Optimization. Pergamon Press, Oxford. WiUems, J. L. (1980). Design of state observers for linear discrete-time systems. Int..l. Systems Sci., 11, 139. Wilson. D. A. (1974). Model reduction for multivariable systems. Int. J. Control, 20, 57. Wolovich, W. A. (1974). Linear Multioariable Systems. Springer-Verlag New York.