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Approach to robust multivariable combustion control design
Computers in Industry 18 (1992) 25-39 Elsevier
25
Applications
Approach to robust multivariable combustion control design Ju§ Kocijan and Rihard Karba University of Ljubljana, Faculty of Electrical and Computer Engineering, Tr[a~ka 25, Ljubljana, Sloven& Received May 15, 1991 Accepted July 22, 1991
Two robust controller structures are presented. The first is a multivariable proportional-integral output feedback pole assignment controller and the second is a robust servomechanism scheme. These two structures affect the process in quite different ways and therefore can be combined to give more robust control than each controller separately. The mentioned structure was used for combustion control, which represents a multivariable and nonlinear problem with time delay. The simulation results of a closed-loop control system are presented and discussed. The whole procedure of designing robust multivariable control was supported with the corresponding compute~-aidcd control system design tools.
Keywords: Multivariable control, Robust control, Computeraided control system design, Combustion process.
1. Introduction
With the development of technology, system control is becoming more and more important in industry. In industry applications, multivariable and nonlinear systems are frequently met. Also uncertainties of different origin appear in nearly all control problems due to the character and properties of modelling and control design procedures. Therefore and because of lack of methods for nonlinear systems control, the robustness of the controller becomes necessary for the successful implementation in industry. Control design is based on the mathematical model of the system to be controlled [1,2]. This model is rarely completely adequate because of
different reasons. Model uncertainties occur due to incomplete knowledge of the system. The structure of the model equations can be obtained from basic laws of physics and engineering, but the numerical values of the parameters are known only within some tolerances. Even if the original system is exactly known, the model is often linearized around the corresponding operating point and reduced in order to simplify calculations or to avoid difficulties that arise from nonlinearities of the complete model. Because of the deviations between the model and original system two control design strategies are adequate, namely adaptive and robust control. Adaptive control algorithms are those which are able to respond actively to the variations of parameters by changing properties of the controller itself. After the parameter vector changes its value, the control system correspondingly adjusts its behaviour and again performs in the required way. These algorithms are efficient, but the controllers are very complex and therefore rarely used in practical a0plications. An alternative way for the control of systems with uncertainties and parameter changes are robust control algorithms [3]. A robust multivariable controller is linear and time-invariant and satisfies the design requirements for the plants with given uncertainties and parameter changes. Because of their relative simplicity such con-
0166-3615/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
26
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troller structures are widely used in industrial applications. A lot of robust multivariable controller algorithms were developed which are more or less frequently used in practice. The aim of the present work is to present a combination of two robust controllers: a multivariable proportionalintegral output feedback pole assignment controller and a robust servomechanism scheme. These two structures affect the process in quite different ways and therefore can be combined to give more robust control than each controller separately. The control scheme was used for a real problem of combustion control, which represents a multivariable and nonlinear system with time delay. Results of control with the mentioned combined structure are presented. The whole procedure of designing the robust multivariable control system was supported with the corresponding computer-aided control system design tools: a language for continuous, discrete and hybrid dynamic systems simulation, StMCOS [4], software for the design of multivariable pole as-
Ju~ Koeijan received the BS and MS degrees in electricalengineeringfrom the Faculty of Electrical and Computer Engineering, University of l~ubljana, Slovenia in 1988 and 1990 respectively. Since 1988 he has been with the University of ~ubUana as Young Researcher. In 1990 he became a TeachingAssistant at the Faculty of Electrical and Computer Engineering. At present he is pursuing his postgraduate studies. His research involves control of multivariable systems and simulationof dynamicalsystems,Currentlyhe works on robust multivariablecontrol systems, Mr. Kocijan received a Universityaward for the best student researchwork in 1989. He also attc,~d~d~he International Space UniversitySummer Session 1991 in Toulouse. France. Rihard Karba received his BS, MS and PhD degrees in electrical engineering from the Facultyof Electrical and Computer Engineering, University of Ljubljana, Slovenia. In 1977 he joined the same faculty and in 1987 became Professor of Automatic Control, where he is currentlyresponsible for lectures on: Elements in automatic control; Modeling of processes and Multivariable systems(on graduate and postgraduate level). His major research emphasis is on the fields of dynamicalsystemsmodeling and simulation; multivariable control design, and modeling and simulation in pharmacokinetics,
Computers in Industry signment controllers [5] and a program package for analysis and control design, ANA [6]. All these were developed at the Faculty of Electrical and Computer Engineering in cooperation with the Jozef Stefan Institute in Ljubljana.
2. Multivariable proportional-integral feedback pole assignment controller
output
Although sophisticated controller structures and design algorithms exist, the conventional PID controller is still frequently used in process control. It is simple and proved to be good for tracking as well as for disturbance rejection. In single-loop control, the PID controller is significantly robust against uncertainties of the plant. PID controllers have therefore been extended to multi-input multi-output, so called multivariable plants. Parameters can be obtained with different techniques. In our case a proportional-integral output feedback pole assignment controller [7] is discussed. Let us give a brief description of the design method. Consider the following linear time-invariant system:
Yt( t ) = Ax( t ) + Bu( t ) + Ew( t ),
(1)
y(t) = Cx(t) + Fw(t),
(2)
where x is a state vector of dimension (n x 1), u is a control vector of dimension ( m x 1), y is an output vector of dimension (l x 1), w is a vector of disturbances of dimension (l x 1) and A, B, C, E and F arc constant matrices of dimensions (n x n), (n x m), (l X n), (n x l) and (I x l) respectively. For this model the controller should be such that the resulting closed-loop system has a certain number of its poles on previously prescribed positions. The procedure for output feedback controller design consists of two steps. The approach in the first step is based on the equivalence of the closed-loop characteristic polynomial of a multivariable system and an equivalent system with one input and multiple outputs. This results in a controller matrix which is dyadic, i.e. has range equal to one. A closed-loop structure with a dyadic output feedback pole assignment controller can be seen in Fig. 1.
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J. Koc~ian,R. Karba / Robust multivariabte control
¥
t
I
c
II
27
>
1
Fig. 1. Closed-loop structure with dyadic output feedback pole assignment controller. If K is the product of a column and a row vector:
K=qk,
(3)
where q is a (m × 1) vector and k is a (1 x l ) vector, then the multivariable system (A, B, C) is reduced to the equivalent system with one input (A, Bq, C). One of the vectors k or q must be defined by the designer while the other is obtained from the system of linear algebraic equations. This partially free choice of parameters gives more flexibility in designing closed-loop performances. The closed-loop characteristic polynomial is given as:
H,( s) = H,,( s) + kWo( s)q,
(4)
where H0(s) is the system characteristic polynomial and Wo is the system numerators matrix. For the desired closed-loop poles we can state H~ = 0 and the closed-loop equations could be written in the form:
m r =/,
(5)
where
To maintain the achieved poles on the specified locations independently of the second step, vector k 2 in the second step must be correspondingly calculated from the following set of algebraic equations
Lk2 = e,
(6)
where L contains the rows wi(A/) of the matrix W0(Ak), k = 1. . . . . m - 1. Here one of the elements of k 2 must be specified by the designer, while the eigenvalues of the matrix A - BuC must be distinct. In the second step we specify the type of controller and obtain additional full-rank controller matrices. A multivariable output feedback PI controller designed by the pole assignment method can shift 21 + m - 1 poles to the specified locations. Shifting a larger number of poles requires a calculation of the nearest possible locations which can be attained. Because of the I-part in the direct path, the order of closed-loop system is enlarged to (n + l), while the P-part of the controller is in the output feedback loop. In this case the control law has the form:
wo(a,)q u = -Ky + o f ( v - y )
M=
Wo(ar)q and -Ho(AI) f= -Ho(Ar) where r = max(m, l) and q is specified by the designer.
dr,
(7)
where v is the (m x 1) vector of reference inputs. The closed-loop structure with the multivariable PI output feedback pole assignment controller is shown in Fig. 2. The closed-loop characteristic polynomial in the second step is:
1 H2(s ) = H l ( S ) +k2Wl(s)q2 + 7k2Wl(s)r2,
(8)
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I v I+
I I
I
L___ Fig. 2. Closed-loop structure with multivariable PI output feedback pole assignment controller.
where k2, q2, r2 are dyadic controller vectors, H 1 is ihe closed-loop characteristic polynomial and W~ is the numerators matrix of the closed-loop system from the first step. For desired closed-loop poles, again H z ---0 and the closed-loop equations can be written in the matrix form:
Mg = f , where
(9)
wL(*~)~w~(~) M=
...
,,,,,(~l)~w,,(A~)
1
1
'
. .
where j = 2m + ! - 1, and wi(i ffi 1,..., m) is the ith element of w(s)---k2Wt(s). Vector g is defined as:
gTffi[q21r21 ... q2mr2,,] and vector f is defined as:
fT=
[-/'/2(At)
... -H2(A2m+t-,)].
The final full-rank controller is obtained by the equation:
go(s) •g+
1
-fl $ 1
--
1
qk + q2k2 + sP. + sr2 2,
(10)
where Q~ is a constant matrix which assures the cyclicity of the augmented system.
3. The robust
servomechanism
problem
This problem and its solution were solved and discussed in detail by Davison et al. [8,9]. It is
presented in [2]. Let us give the following notes about this problem and its solution. Consider a linear, time-invariant system described by eqns. (1) and (2). The disturbance vector w(t) may or may not be measurable. To simplify the problem we assume that w(t) and the vector of reference inputs are step functions. In practice a step change in reference inputs occurs when the set-point is changed to a new value. We discuss the servomechanism problem, which tends to find such a controller that the resulting controlled system is stable and the steady-state error is zero for all initial values and all disturbances and reference inputs which are step functions. The solution for this problem is a robust controller which contains a certain duplicated model of its "environment", i.e. of the disturbances and the reference inputs affecting the system. A robust controller which is a solution for the servomechanism problem stated above can always be found if the following conditions arc satisfied: - (A, B) is a stabilizable pair; - (C, A) is a detectable pair; - the number of inputs is greater than or equal to number of outputs i.e. m >i I; and - it is proven that rank[A
B] 0 •n+l.
The first two conditions mean that all the unstable modes of the controlled system should be controllable and observable. If the conditions are satisfied, the system can be stabilized by dynamic output feedback. A robust controller can be designed using the available measurements y(t), such that the resulting closed-loop system is
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J. Kocijan, R. Karba / Robust multivariable control
u
servo
._%
J' +y+,+m I -I I
29
Y
complementary controUer
Fig. 3. Closed-loop structure.
stable and asymptotic regulation [2] takes place for all disturbances w(t) and reference inputs Yr(t). Such a controller consists of two parts: a servo compensator which is completely determined by the disturbances and reference inputs and a complementary controller which stabilizes the overall system. A block diagram of a closedloop system with the mentioned controller is shown in Fig. 3. The robust controller satisfies the following equations:
u(t)
= K t [ ( t ) +K2r/(t ),
(11)
where g t and K 2 are cortstant matrices, a~(t) is a vector of complementary controller states and ~:(t) is the output of the servo compensator defined for the step reference changes and disturbances as
~(t) =e(t)
(12)
and
e(t)
- y ( t ) -Yr(t).
(13)
The robust controller will achieve asymptotic regulation in the presence of any finite changes (not just arbitrarily small perturbations) in the system parameters A, B and C, in the feedback parameters K~ and K 2, in the complementary controller parameters, in the order of the assumed mathematical model describing the system and in the order of the complementary controller,
provided that the overall closed-loop system remains stable. A robust controller exists for systems (A, B, C) provided that m >1l and the outputs to be controlled can be physically measured. The complementary controller is defined by
it(t) =A~I(t) + Bu( t).
(14)
It is the model of the controlled system representing its characteristics. When A is a stable matrix, it is easy to see that the gain matrices K~ and K 2 are the only parameters which need to be found. The structure of the complementary controller enables any output feedback controllers of the given system that already exist to be retained in the final control configuration. Both the complementary controller and the servo compensator are generalizations of ~imilar concepts in classical univariable control theory. Any of the conventional techniques, such as pole assignment or linear optimal control [2,1] may be used to find appropriate values for K t and K 2.
4. The joined structure for robust multivariable closed-loop control
As has already been mentioned in the previous section, any output feedback controllers that already exist in the given system can be incorpo-
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b --
system
-.i .-~
II <°,.,<.,..°,:,, I. c°"t'°il'r / Fig. 4. Joined closed-loop robust structure consisting of a multivariable proportional-integral output feedback pole assignment controller and a robust servomechanism scheme.
rated in the complementary controller. This means that dynamic output feedback, which stabilized the system and also enables satisfactory responses to be obtained, must be primarily applied. Afterwards the servo compensator and the complementary controller can be designed for the already stabilized system. In our case we decided to apply first the multivariable proportional-integral output feedback pole assignment controller to stabilize the system and to give it satistactory robustness properties. The obtained closed-loop structure is then completed in the mentioned way. The combination of both control strategies gives the joined closedloop structure shown in Fig. 4. We consider a system described by eqns. (1) and (2). The proportional-integral controller is given by the equations: (15) (16)
u=Qq-Ky,
4 = Uo - Cx, and the robust controller by the equations uo=K,g+[K2
K3]
,
uc = Qp - KC~,
(20)
ti=uo-C~.
(21)
The vector [~,] is the output of the complementary controller: -c
o
The order of complementary controller is increased to (n + l) because of the additional states of proportional-integral controller. The joined closed-loop structure is described in state-space presentation as: .~ I q I
[A-BKC [ -C c
0
•
0
.P 1 ×
BQ
o
o
o
0
KI
K2
K3
0 0 0
0 0
0 A - BKC
0 BQ.
KI
K 2- C
K3
:]
(17)
where K3 is the gain matrix for the augmented states due to the I-part of the multivariable PI controller, ~:(t) is determined by eqn. (12) and and p are given by the following relations
JlpJ
+
(24)
[q
il--A~+Buc,
(18)
y=[C 0 0 00]/~
Yc---C . ,
(19)
[~
+[FO
]I']
Yr "
(25)
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J. Kocijan, R. Karba / Robust multivariable control
By carrying out the following coordinate transformation on the state vector:
=
P-qVl-X
/
0
0
0
0
I
0
0
;
Matrix (29) is the state matrix of the closedloop system with the state space-controller
(33)
~0 = [K 2 K3 KI] ,
(26)
-01 _01 O0 ol Ol
which is applied on the system
we obtain
[i]
=
[A-B?C BQ o
y = [ C 0 O] .P
q
[ A - BKC BQ 0
0
-- C + K 2
K3
~
o0
o0 0
0
0
--[
K1
F
× ~X
o l
K2
I
__0C oJ
I Yr] '
(27)
+IF o1[,"].
(28)
LP - q J
,=[c
o o o
o]
31
.
0][x] 0
q
+
[i]
uo,
(34)
o
(35)
Through the mentioned coordinate transformation the structure of the combined system can be clearly seen also from the mathematical model. Matrices K, Q, K 1, £2 and K 3 should be selected to ensure closed-loop stability. Matrices K and Q are obtained with the output feedback pole assignment niethod. Matrices KI, K 2 and K 3 can be designed by any classical control design method, i.e. they can be calculated also by the aid of pole assignment method (proportional state feedback pole assignment). These joined structure was developed separately to give a more robust controller. 5. Example
The eigenvalues of the state matrix in eqns. (27) and (28) are because of transformation (26) equal to those of matrices
A - BKC BQ O ] -Cc+K2
K3
01
(29)
0
and
Matrix (30) is the state ma,*rix of the closedloop system obtained by applying the proportional-integral output feedback p,31e assignment controller to the system (A, B, C). The closedloop system is described by the equations:
[q]=[A-BKC -C
BOQ][~]
+[O].o,
(31)
In the previous chapters we nave discussed two methods for multivariable robust control design and coupled structure to obtain even better robustness performances. Controller robustness is an essential property in practical control engineering where nonlinear, time-variant and delayed systems are met. The joined robust controller structure was applied to combustion process which represents multivariable and nonlinear problems with time delay. In the first step the multivariable proportional-integral output feedback pole assignment controller was designed which yields also a partial linearization of the nonlinear system. In the second step the structure was completed with the servomechanism scheme. The system under investigation is a process of combustion in a steam boiler PK 401 at Cinkarna Celje Company, Celje, Slovenia. It is controlled by the multiloop microcomputer controller MMC
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32
90, which has been developed at the Jozef Stefan Institute. This device enables the testing of numerous controll algorithms and the verification of results obtained with simulation. The discussed multivariable robust controll represents one of possible control algorithms which could be applied to the combustion process.
02
0
5.1. Model of the combustion process According to [10] the combustion process model with two inputs (flow of air and fuel) and two outputs (O 2 and CO) can be described with the following equations: dxo 2
--:
dt
Fig. 5. Analog scheme of the combustion process model.
1
vZ{-xo2[+z + +:(vd - Vo)] + 2 1 ~ - 100Vo@~},
-
dxco dt -
-
-
1
+
(36)
vo)]
+ (1 - a)1.866Cq~g},
(37)
A more detailed look on eqn. (38) shows that it represents a bilinear (nonlinear) model. The analog scheme of this model is shown in Fig. 5.
5.2. Air trap-door model
where Xo2 is the percentage of 0 2 (vol%), Xco is the percentage of CO (vol%), Vk is the volume of the combustion chamber (ma), ~ is the normalized flow of fuel (kg/s), q~ is the normalized flow of air (Nma/kg), Vo is the theoretically needed air volume for the combustion of one unit of fuel (Nma/kg), Va is the theoretically obtained gas volume from one unit of fuel (Nm3/kg), (1 - a) is the relative portion of carbon converted into CO (%/100), and C is the relative portion of carbon in fuel (%/100). Equations (36) and (37) can be expressed in the form:
The air flow is controlled through an air trapdoor. As it is a part of the closed-loop, it has to be modelled and added to the combustion model. The air trap-door is a nonlinear dynamical system. Its gain is described by the relations:
it(t) = AlU( t )x( t ) + A2v( t )x( t )
where K z is the air trap-door gain, ~ is the angle of the air trap-door, and qb~m,~x is the maximum flow of air. Air trap-door dynamics are described by the first-order transfer function:
+B;u(t) +B~v(t), y ( t ) • x ( t - Ta), where
(39)
Kzffi ~
exp
2-exp
--
/
0 ° ~<~b < 45 °,
~
45 ° < ~b < 90 o,
0.5
xT = [Xo2 Xco], u =,t,z,
(38)
K~ -- - - ~
G(s) ~ K ~ - -
s +0.5 "
45'))
(40)
(41)
(42)
v
At=-l/Vk,
A2=(Vo-Vo)/Vk
B ; T = [ B , 0] = [21/V k 0]
B~T _- [B2 B4]
= [ - IOOVo/Vk (1 -a)1.866C/Vk], and Td is the delay, transport time.
5.3. Linearization of the model The most common step in control system design is linearization of the nonlinear model around the corresponding operating point. Among the many methods for the linearization of nonlinear systems we chose the approach with
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J. Kocijan, R. Karba / Robust muitivariable control
H~'~. ~
~
~
~
~I
~l~-li-~l I T ~
~
~
33
I~
0.52 0.52
0.51
f.' C~5!
/
7'
O,5O 0.50 0.50 0."19
0,49 O.48 I
0'180.00
O.80
I
I
'
I.~0 ~'-"
I
[
2.40
~:.20 _
I
I
I
'
I
4.O0 4.BO 5.~0 ~.40 I SltlCOS F'~" ~ IJS LJL~LJ~N~
I
7,20 ......
I
)
8,O0 I ×10 2
Fig. 6. Closed-loop r e s p o n s e of the nonlinear system without time delay a n d reference i n p u t s - - O 2.
identification. The "Black box" identification method with least-squares algorithm from the program package for system analysis and control design ANA was used for identification of our nonlinear model. A pseudo random binary signal (PNBS) around the chosen operating point, which was angle of air trap-door at 46 ° and flow of fuel 1 kg/s, was used as input signal. The simulated responses of the nonlinear model, obtained by the aid of the simulation language s~ucos [4], represented the other data needed for identification. As a result the following system matrices A, B, C were obtained: A =
B __
C_-
- 1.415240 -0.663208 1.3703x10 -2
1.506356 0.594137 -1.5252X10 -2
- 2.009392 X 10 -3 5.169228 X 10 -2 -4.250321 × 10 -4 0
oo] 0
"
0.586975 ] - 0.438879 /' -5.1391X10-zJ
-4.160395 ] -3.231999 3.869638 × 10 -2
These matrices determine controllable and observable linear time-invariant process.
5. 4. Control design Control design is an interactive and iterative procedure which through comparison between alternative control algorithms can lead to an acceptable solution of the control problem. In the first step the proportional-integral output feedback pole assignment controller was designed with software for the design of multivariable pole assignment controllers [5]. After an iterative and interactive procedure the desired closed-loop poles were chosen to be: -0.1 +j0, - 0.3 + jo,
-0.2+j0, - 0.5 + j0,
-0.7 +j0, and the controller matrices were: K = [ -3.862247 -408.9212 ] - 0.2130442 - 3.622375 J -0.2972958 -31.07514 ] Q = - 0.04225255 0.8685015 J
34
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x,o-{)~CZ]~ - I
~
~
~l
~
~
~
~
I~-FF~'--I F~-~---I
0.45 0.45 0.44
f.j-"
0.44 0.44
/
0.43
/
/
/
/
,/
/'
0.43 0.43 0.43 0.42
0.420.60
I
Ol ~0
I
I160 +'
I
2140
!
~:m20
I
~.O0
I
4180
I
~1 ~0
I
~140
i si.co8 FE ~ IJS LJUBLJ~N~
(
I
I
~120
)
8& O0
~ )
×|0
2
I
:'
Fig. 7. Closed-loop response of the nonlinear system without time delay and reference inputsmCO.
0.50 '~''--~m. "45.6 "$$,(5 "I~O "200 "250
~~
"~0
.,oo \\ -4~ -r-~)o o,~ (
I
\,~.~ , 0.80
, 1.60
2. 0
~
3. 0
I
4,00
I
4,80
I
5.60
I
~,40
I Slti~.', FE ~* IJS LJO~JANA Fig. 8. Control signals for the nonlinear system without time delay.
I
7.20
8.00
I xlO 2
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J. Kocijan, R. Karba / Robust multivariable control
x10 0.52 0.52 0.51 0.51 0.50 0.50 0.50 0.49
O.~J9 0.48
00"
: 0.80
0.4 .00
. ,+ 1.60
J 2,40
I 3.20
I
'
J
I
)
I
5.~ ~;.40 7.20 ...... [ 5IMCOSF~ ~ I J ~ LJVBLJRNfl . . . . . 4.00
4.B0
I'
8.00 "7 xlO 2
Fig. 9. Closed-loop response of the nonlinear system with time delay and reference inputs--O 2.
0.45
0,4~ O.44
~.~---'-~ .m
v,,r'~ 0,44
¢
~48 0.43 0.43
i
0,42 0.42 0.41 ....0.00 I
. 0.80
. i.~0. --
.2.40 . -
. . 4.00. 3..'..,o 4.~0 5,~0 ..... L~,icos rE ~ 1~s L-J~,~J~
~.~0
7.'-,0
8.00 i~:io 2
Fig. 10. Closed-loop response of the nonlinear system with time delay and reference inputs--CO.
3~
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[Load
xlO-i O.5O
-4.55
-&60
-I 4.? -19,7 -24.?
t
-25.8
-3#.8, -3&9
't N
-44.9' I
o
0.80
I
1.60
'
I
2.40
....
3."0
I
I
#
i
I
I
4.00
4.80
~.6o
4.40
7.20
o.oo
,l S1~05 FE ~t IJ$ L.JUI3LJAW~
I
.J
xiO
"i
2
Fig. 11. Control signals for the nonlinear system with time delay.
The closed-loop responses of the nonlinear system without time delay and reference inputs are shown in Fig. 6 for 02 and in Fig. 7 for CO, while control signals are shown in Fig. 8. The closed-loop responses of the nonlinear system with time delay (0.8 s) and reference inputs are shown in Fig. 9 for 02 and in Fig. 10 for CO, while control signals are shown in Fig. 11 This controller shows satisfactory dynamics and, applied on a nonlinear model, causes partial linearization of the nonlinear system. However, digital simulation with SIMCOS [4] shows that robustness against parameter perturbation does not mean always that the system has the same robustness against time delay. In our case the relation between these two properties was inverse: more satisfactory control of the nonlinear model caused less satisfactory response of the delayed system. In the next step, the second part of the proposed joined robust structure was developed to increase robustness. The servo compensator includes a model of step changes of the reference and disturbance vectors and is described by eqn. (12).
The complementary controller should include the model of the process, but for the sake of simplicity of the control design procedure the linearized model of the closed-loop process with the proportlonal-integral output feedback pole assignment controller was used. Matrices, g l, K 2 and g 3 were obtained by the proportional state feedback pole assignment as described in the previous section, using the mentioned software. The overall closed-loop poles were chosen to be: - 0 . 0 5 + jO, -O.08+jO, - 0 . 1 + j 0 , -0.15 +jO, - 0 . 2 +jO, - 0 . 2 5 + j0, and controller matrices were: = [ 0.0065 0.1372] KI [ - 0.0001 0.0055/' = [ 12.4201 -9.6756 296.0749] K2 0.0038 I - 0.0003 1.5746 ' /(3=[-0.7002 -7.8728] 0.0015 0.0055 " - 0 . 0 3 + jO,
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0,52 0.52 0.51
/
0.51 0-50 0.50 0.50 0.49
/
I
0.49 0.48
0.480.
I
I
0.80
1.60 '
l
I
2.40
3.2~
i
I
I
I
I
4.00 4.80 5.60 &40 [ SlflCO$ D" 0 IJS LJLIL:~.&I:~¥1
I
I
7.~
~.
8.00 I xlO 2
Fig. 12. Closed-loop response of the nonlinear system with time delay and reference inputs--O 2.
xl0 "i 0.46 0.45
/,2 /
0.45 0.44 0.44
i
0.43 0,43 0.42 0.42
"\\. ---
_T.z.-,--
// /
0.41 0.410-00
I
I
0.80
,,
I
"
1.6.0
I
I
2.40
3.20
|
I
I
4.00 4.80 5.60 I Sl~0S rr ~ iJs L&UBUA~
i
!
6.40
7,20
Fig. 13. Closed-loop response of the nonlinear system with time delay and reference inputs--CO.
I
&oo
I
xl0
2
37
38
Applications
Computersin Industry
-.~.,.__.,_,_____
0.50
"r~.55
\
-It.6
/
-17.7
I
-2:::,?
1
-29. ?
/
-~. 8
11
-41,~
i
)
-47,9
I
-53. 9
ou.8.bo
~ -
L
-
0.80
"
J,"
I.~0
2,'I0
L
3.20
I
4.00
4.80
5. 0
6.40
[ SIIICOSFE ~ IJ~ LJUE:LJ~
?.20
8.00 I xl0 2
Fig. 14. Control signals for the nonlinear system with time delay.
The closed-loop responses of the nonlinear system with time delay (0.8 s) and reference inputs are shown in Fig. 12 for 0 2 and in Fig. 13 for CO, while control signals are shown in Fig. 14. From these figures it can be seen that overall robustness structure has made the response faster but more smooth. Robustness against parameter deviations has been enlarged with the augmented controller structure while robustness against time delay remained on the same level. Larger time delay causes high amplitude oscillations. As the combustion process is controlled by a microcomputer controller, the question about loss of information because of discretization should be stated. Extra simulation runs with a discretized controller are planned for future work.
6. Conclusion Control system design appears to become more and more important in industry for obtaining a higher level of technology, Multivariable and non-
linear systems are usually met in a real-life environment. But there is still lack of efficient methods for nonlinear control system design. Robust controllers are a possible solution for this problem. Robust multivariable control system design represents a widely used set of different approaches and methods. Two of them have been presented and an attempt of joining together these two structures in order to increase robustness has been shown in this work. The proportional-integral-differential family of controllers are very well known representatives of robust controllers. The multivariable proportional-integral output feedback pole assignment controller is just one from this set of controllers. The other controller is a robust controller for solving the multivariable servomechanism problem. This controller achieves asymptotic regulation even in the presence of perturbations in the system and controller parameters, which are not necessarily small. The procedure of robust controller design enables any output feedback con-
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trollers that already exist in the given system to be retained in the final control configuration. Both controller structures have been joined together and used in combustion process controll. The combustion process represents a multivariable and nonlinear problem with time delay and controller robustness properties could be seen in this process control. Results of control system design were obtained by the aid of computer simulation of a nonlinear process model and control system. From these results it can be seen that robustness against parameter perturbations does not always mean the same as robustness against time delay. In our case this was an inverse relation. All disturbances have to be known and included in the structure of the robust controller. Since all disturbances are not always known, this fact represents a disadvantage in control system design. Also questions about loss of information because of discretization when a microcomputer controller is applied in practice should be asked. However, robustness against parameter changes has been shown to be enlarged with the augmented controller structure and that was the intention of the present work.
J. Kocijan, R Karba / Robust multivariable control
39
References [1] B. Friedland, Control System Design, Mc Graw-Hill, New York, 1986. [2] R.V. Patel and N. Munro, Multicariable Systems Theory and Design, Pergamon Press, Oxford, 1982. [3] J. Lunze, Robust Multivariable Feedback Control, Prentice-Hall, Englewood Cliffs, NJ, 1988. [4] B. Zupan~i~:, D. Matko, R. Karba and M. Sega, "SIMCOS --Digital simulation language with hybrid capabilities", Proc. 4th Syrup. Simulationstechnik, Ziirich, Switzerland, 1987, pp. 205-212. [5] J. Kocijan, "Computer aided multivariable controller design", Work prepared for University Award for Best Student Research Work, Faculty of Electrical and Computer Engineering, Ljubljana, 1989 (in Slovene). [6] M Sega, S. Strm~nik, R. Karba and D. Matko, "Interactive program package ~'qA for system analysis and control design", Proc. 3rd 1FAC/1FIP Int. Symposium CADCE'85, Copenhagen, Denmark, 1985, pp. 145-150. [7] H. Seraji, Design of proportional-plus-integral controllers for multivariable systems, lnt. J. Control, Vol. 29, No. 1, 1979, pp. 49-63. [8] E.J. Davison and A. Goldenberg, "Robust control of a general servomechanism problem: The servo compensator", Automatica, Vol. 11, 1975, pp. 461-471. [9] E.J. Davison, "The robust control of a servomechanism problem for linear time-invariant multivariable systems", IEEE Trans. Autom. Control, Vol. AC-21, 1976, pp. 2534. [10] J. (~retnik, S. Strm~:nik and B. Zupan~i~, "A model for combustion of fuel in the boiler", Proc. 3rd Symp. Simuiationstechnik, Bad Miinster, Amstein, 1985, pp. 469-473.
25
Applications
Approach to robust multivariable combustion control design Ju§ Kocijan and Rihard Karba University of Ljubljana, Faculty of Electrical and Computer Engineering, Tr[a~ka 25, Ljubljana, Sloven& Received May 15, 1991 Accepted July 22, 1991
Two robust controller structures are presented. The first is a multivariable proportional-integral output feedback pole assignment controller and the second is a robust servomechanism scheme. These two structures affect the process in quite different ways and therefore can be combined to give more robust control than each controller separately. The mentioned structure was used for combustion control, which represents a multivariable and nonlinear problem with time delay. The simulation results of a closed-loop control system are presented and discussed. The whole procedure of designing robust multivariable control was supported with the corresponding compute~-aidcd control system design tools.
Keywords: Multivariable control, Robust control, Computeraided control system design, Combustion process.
1. Introduction
With the development of technology, system control is becoming more and more important in industry. In industry applications, multivariable and nonlinear systems are frequently met. Also uncertainties of different origin appear in nearly all control problems due to the character and properties of modelling and control design procedures. Therefore and because of lack of methods for nonlinear systems control, the robustness of the controller becomes necessary for the successful implementation in industry. Control design is based on the mathematical model of the system to be controlled [1,2]. This model is rarely completely adequate because of
different reasons. Model uncertainties occur due to incomplete knowledge of the system. The structure of the model equations can be obtained from basic laws of physics and engineering, but the numerical values of the parameters are known only within some tolerances. Even if the original system is exactly known, the model is often linearized around the corresponding operating point and reduced in order to simplify calculations or to avoid difficulties that arise from nonlinearities of the complete model. Because of the deviations between the model and original system two control design strategies are adequate, namely adaptive and robust control. Adaptive control algorithms are those which are able to respond actively to the variations of parameters by changing properties of the controller itself. After the parameter vector changes its value, the control system correspondingly adjusts its behaviour and again performs in the required way. These algorithms are efficient, but the controllers are very complex and therefore rarely used in practical a0plications. An alternative way for the control of systems with uncertainties and parameter changes are robust control algorithms [3]. A robust multivariable controller is linear and time-invariant and satisfies the design requirements for the plants with given uncertainties and parameter changes. Because of their relative simplicity such con-
0166-3615/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
26
Applications
troller structures are widely used in industrial applications. A lot of robust multivariable controller algorithms were developed which are more or less frequently used in practice. The aim of the present work is to present a combination of two robust controllers: a multivariable proportionalintegral output feedback pole assignment controller and a robust servomechanism scheme. These two structures affect the process in quite different ways and therefore can be combined to give more robust control than each controller separately. The control scheme was used for a real problem of combustion control, which represents a multivariable and nonlinear system with time delay. Results of control with the mentioned combined structure are presented. The whole procedure of designing the robust multivariable control system was supported with the corresponding computer-aided control system design tools: a language for continuous, discrete and hybrid dynamic systems simulation, StMCOS [4], software for the design of multivariable pole as-
Ju~ Koeijan received the BS and MS degrees in electricalengineeringfrom the Faculty of Electrical and Computer Engineering, University of l~ubljana, Slovenia in 1988 and 1990 respectively. Since 1988 he has been with the University of ~ubUana as Young Researcher. In 1990 he became a TeachingAssistant at the Faculty of Electrical and Computer Engineering. At present he is pursuing his postgraduate studies. His research involves control of multivariable systems and simulationof dynamicalsystems,Currentlyhe works on robust multivariablecontrol systems, Mr. Kocijan received a Universityaward for the best student researchwork in 1989. He also attc,~d~d~he International Space UniversitySummer Session 1991 in Toulouse. France. Rihard Karba received his BS, MS and PhD degrees in electrical engineering from the Facultyof Electrical and Computer Engineering, University of Ljubljana, Slovenia. In 1977 he joined the same faculty and in 1987 became Professor of Automatic Control, where he is currentlyresponsible for lectures on: Elements in automatic control; Modeling of processes and Multivariable systems(on graduate and postgraduate level). His major research emphasis is on the fields of dynamicalsystemsmodeling and simulation; multivariable control design, and modeling and simulation in pharmacokinetics,
Computers in Industry signment controllers [5] and a program package for analysis and control design, ANA [6]. All these were developed at the Faculty of Electrical and Computer Engineering in cooperation with the Jozef Stefan Institute in Ljubljana.
2. Multivariable proportional-integral feedback pole assignment controller
output
Although sophisticated controller structures and design algorithms exist, the conventional PID controller is still frequently used in process control. It is simple and proved to be good for tracking as well as for disturbance rejection. In single-loop control, the PID controller is significantly robust against uncertainties of the plant. PID controllers have therefore been extended to multi-input multi-output, so called multivariable plants. Parameters can be obtained with different techniques. In our case a proportional-integral output feedback pole assignment controller [7] is discussed. Let us give a brief description of the design method. Consider the following linear time-invariant system:
Yt( t ) = Ax( t ) + Bu( t ) + Ew( t ),
(1)
y(t) = Cx(t) + Fw(t),
(2)
where x is a state vector of dimension (n x 1), u is a control vector of dimension ( m x 1), y is an output vector of dimension (l x 1), w is a vector of disturbances of dimension (l x 1) and A, B, C, E and F arc constant matrices of dimensions (n x n), (n x m), (l X n), (n x l) and (I x l) respectively. For this model the controller should be such that the resulting closed-loop system has a certain number of its poles on previously prescribed positions. The procedure for output feedback controller design consists of two steps. The approach in the first step is based on the equivalence of the closed-loop characteristic polynomial of a multivariable system and an equivalent system with one input and multiple outputs. This results in a controller matrix which is dyadic, i.e. has range equal to one. A closed-loop structure with a dyadic output feedback pole assignment controller can be seen in Fig. 1.
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J. Koc~ian,R. Karba / Robust multivariabte control
¥
t
I
c
II
27
>
1
Fig. 1. Closed-loop structure with dyadic output feedback pole assignment controller. If K is the product of a column and a row vector:
K=qk,
(3)
where q is a (m × 1) vector and k is a (1 x l ) vector, then the multivariable system (A, B, C) is reduced to the equivalent system with one input (A, Bq, C). One of the vectors k or q must be defined by the designer while the other is obtained from the system of linear algebraic equations. This partially free choice of parameters gives more flexibility in designing closed-loop performances. The closed-loop characteristic polynomial is given as:
H,( s) = H,,( s) + kWo( s)q,
(4)
where H0(s) is the system characteristic polynomial and Wo is the system numerators matrix. For the desired closed-loop poles we can state H~ = 0 and the closed-loop equations could be written in the form:
m r =/,
(5)
where
To maintain the achieved poles on the specified locations independently of the second step, vector k 2 in the second step must be correspondingly calculated from the following set of algebraic equations
Lk2 = e,
(6)
where L contains the rows wi(A/) of the matrix W0(Ak), k = 1. . . . . m - 1. Here one of the elements of k 2 must be specified by the designer, while the eigenvalues of the matrix A - BuC must be distinct. In the second step we specify the type of controller and obtain additional full-rank controller matrices. A multivariable output feedback PI controller designed by the pole assignment method can shift 21 + m - 1 poles to the specified locations. Shifting a larger number of poles requires a calculation of the nearest possible locations which can be attained. Because of the I-part in the direct path, the order of closed-loop system is enlarged to (n + l), while the P-part of the controller is in the output feedback loop. In this case the control law has the form:
wo(a,)q u = -Ky + o f ( v - y )
M=
Wo(ar)q and -Ho(AI) f= -Ho(Ar) where r = max(m, l) and q is specified by the designer.
dr,
(7)
where v is the (m x 1) vector of reference inputs. The closed-loop structure with the multivariable PI output feedback pole assignment controller is shown in Fig. 2. The closed-loop characteristic polynomial in the second step is:
1 H2(s ) = H l ( S ) +k2Wl(s)q2 + 7k2Wl(s)r2,
(8)
28
Applications
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I v I+
I I
I
L___ Fig. 2. Closed-loop structure with multivariable PI output feedback pole assignment controller.
where k2, q2, r2 are dyadic controller vectors, H 1 is ihe closed-loop characteristic polynomial and W~ is the numerators matrix of the closed-loop system from the first step. For desired closed-loop poles, again H z ---0 and the closed-loop equations can be written in the matrix form:
Mg = f , where
(9)
wL(*~)~w~(~) M=
...
,,,,,(~l)~w,,(A~)
1
1
'
. .
where j = 2m + ! - 1, and wi(i ffi 1,..., m) is the ith element of w(s)---k2Wt(s). Vector g is defined as:
gTffi[q21r21 ... q2mr2,,] and vector f is defined as:
fT=
[-/'/2(At)
... -H2(A2m+t-,)].
The final full-rank controller is obtained by the equation:
go(s) •g+
1
-fl $ 1
--
1
qk + q2k2 + sP. + sr2 2,
(10)
where Q~ is a constant matrix which assures the cyclicity of the augmented system.
3. The robust
servomechanism
problem
This problem and its solution were solved and discussed in detail by Davison et al. [8,9]. It is
presented in [2]. Let us give the following notes about this problem and its solution. Consider a linear, time-invariant system described by eqns. (1) and (2). The disturbance vector w(t) may or may not be measurable. To simplify the problem we assume that w(t) and the vector of reference inputs are step functions. In practice a step change in reference inputs occurs when the set-point is changed to a new value. We discuss the servomechanism problem, which tends to find such a controller that the resulting controlled system is stable and the steady-state error is zero for all initial values and all disturbances and reference inputs which are step functions. The solution for this problem is a robust controller which contains a certain duplicated model of its "environment", i.e. of the disturbances and the reference inputs affecting the system. A robust controller which is a solution for the servomechanism problem stated above can always be found if the following conditions arc satisfied: - (A, B) is a stabilizable pair; - (C, A) is a detectable pair; - the number of inputs is greater than or equal to number of outputs i.e. m >i I; and - it is proven that rank[A
B] 0 •n+l.
The first two conditions mean that all the unstable modes of the controlled system should be controllable and observable. If the conditions are satisfied, the system can be stabilized by dynamic output feedback. A robust controller can be designed using the available measurements y(t), such that the resulting closed-loop system is
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J. Kocijan, R. Karba / Robust multivariable control
u
servo
._%
J' +y+,+m I -I I
29
Y
complementary controUer
Fig. 3. Closed-loop structure.
stable and asymptotic regulation [2] takes place for all disturbances w(t) and reference inputs Yr(t). Such a controller consists of two parts: a servo compensator which is completely determined by the disturbances and reference inputs and a complementary controller which stabilizes the overall system. A block diagram of a closedloop system with the mentioned controller is shown in Fig. 3. The robust controller satisfies the following equations:
u(t)
= K t [ ( t ) +K2r/(t ),
(11)
where g t and K 2 are cortstant matrices, a~(t) is a vector of complementary controller states and ~:(t) is the output of the servo compensator defined for the step reference changes and disturbances as
~(t) =e(t)
(12)
and
e(t)
- y ( t ) -Yr(t).
(13)
The robust controller will achieve asymptotic regulation in the presence of any finite changes (not just arbitrarily small perturbations) in the system parameters A, B and C, in the feedback parameters K~ and K 2, in the complementary controller parameters, in the order of the assumed mathematical model describing the system and in the order of the complementary controller,
provided that the overall closed-loop system remains stable. A robust controller exists for systems (A, B, C) provided that m >1l and the outputs to be controlled can be physically measured. The complementary controller is defined by
it(t) =A~I(t) + Bu( t).
(14)
It is the model of the controlled system representing its characteristics. When A is a stable matrix, it is easy to see that the gain matrices K~ and K 2 are the only parameters which need to be found. The structure of the complementary controller enables any output feedback controllers of the given system that already exist to be retained in the final control configuration. Both the complementary controller and the servo compensator are generalizations of ~imilar concepts in classical univariable control theory. Any of the conventional techniques, such as pole assignment or linear optimal control [2,1] may be used to find appropriate values for K t and K 2.
4. The joined structure for robust multivariable closed-loop control
As has already been mentioned in the previous section, any output feedback controllers that already exist in the given system can be incorpo-
30
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b --
system
-.i .-~
II <°,.,<.,..°,:,, I. c°"t'°il'r / Fig. 4. Joined closed-loop robust structure consisting of a multivariable proportional-integral output feedback pole assignment controller and a robust servomechanism scheme.
rated in the complementary controller. This means that dynamic output feedback, which stabilized the system and also enables satisfactory responses to be obtained, must be primarily applied. Afterwards the servo compensator and the complementary controller can be designed for the already stabilized system. In our case we decided to apply first the multivariable proportional-integral output feedback pole assignment controller to stabilize the system and to give it satistactory robustness properties. The obtained closed-loop structure is then completed in the mentioned way. The combination of both control strategies gives the joined closedloop structure shown in Fig. 4. We consider a system described by eqns. (1) and (2). The proportional-integral controller is given by the equations: (15) (16)
u=Qq-Ky,
4 = Uo - Cx, and the robust controller by the equations uo=K,g+[K2
K3]
,
uc = Qp - KC~,
(20)
ti=uo-C~.
(21)
The vector [~,] is the output of the complementary controller: -c
o
The order of complementary controller is increased to (n + l) because of the additional states of proportional-integral controller. The joined closed-loop structure is described in state-space presentation as: .~ I q I
[A-BKC [ -C c
0
•
0
.P 1 ×
BQ
o
o
o
0
KI
K2
K3
0 0 0
0 0
0 A - BKC
0 BQ.
KI
K 2- C
K3
:]
(17)
where K3 is the gain matrix for the augmented states due to the I-part of the multivariable PI controller, ~:(t) is determined by eqn. (12) and and p are given by the following relations
JlpJ
+
(24)
[q
il--A~+Buc,
(18)
y=[C 0 0 00]/~
Yc---C . ,
(19)
[~
+[FO
]I']
Yr "
(25)
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J. Kocijan, R. Karba / Robust multivariable control
By carrying out the following coordinate transformation on the state vector:
=
P-qVl-X
/
0
0
0
0
I
0
0
;
Matrix (29) is the state matrix of the closedloop system with the state space-controller
(33)
~0 = [K 2 K3 KI] ,
(26)
-01 _01 O0 ol Ol
which is applied on the system
we obtain
[i]
=
[A-B?C BQ o
y = [ C 0 O] .P
q
[ A - BKC BQ 0
0
-- C + K 2
K3
~
o0
o0 0
0
0
--[
K1
F
× ~X
o l
K2
I
__0C oJ
I Yr] '
(27)
+IF o1[,"].
(28)
LP - q J
,=[c
o o o
o]
31
.
0][x] 0
q
+
[i]
uo,
(34)
o
(35)
Through the mentioned coordinate transformation the structure of the combined system can be clearly seen also from the mathematical model. Matrices K, Q, K 1, £2 and K 3 should be selected to ensure closed-loop stability. Matrices K and Q are obtained with the output feedback pole assignment niethod. Matrices KI, K 2 and K 3 can be designed by any classical control design method, i.e. they can be calculated also by the aid of pole assignment method (proportional state feedback pole assignment). These joined structure was developed separately to give a more robust controller. 5. Example
The eigenvalues of the state matrix in eqns. (27) and (28) are because of transformation (26) equal to those of matrices
A - BKC BQ O ] -Cc+K2
K3
01
(29)
0
and
Matrix (30) is the state ma,*rix of the closedloop system obtained by applying the proportional-integral output feedback p,31e assignment controller to the system (A, B, C). The closedloop system is described by the equations:
[q]=[A-BKC -C
BOQ][~]
+[O].o,
(31)
In the previous chapters we nave discussed two methods for multivariable robust control design and coupled structure to obtain even better robustness performances. Controller robustness is an essential property in practical control engineering where nonlinear, time-variant and delayed systems are met. The joined robust controller structure was applied to combustion process which represents multivariable and nonlinear problems with time delay. In the first step the multivariable proportional-integral output feedback pole assignment controller was designed which yields also a partial linearization of the nonlinear system. In the second step the structure was completed with the servomechanism scheme. The system under investigation is a process of combustion in a steam boiler PK 401 at Cinkarna Celje Company, Celje, Slovenia. It is controlled by the multiloop microcomputer controller MMC
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Applications
32
90, which has been developed at the Jozef Stefan Institute. This device enables the testing of numerous controll algorithms and the verification of results obtained with simulation. The discussed multivariable robust controll represents one of possible control algorithms which could be applied to the combustion process.
02
0
5.1. Model of the combustion process According to [10] the combustion process model with two inputs (flow of air and fuel) and two outputs (O 2 and CO) can be described with the following equations: dxo 2
--:
dt
Fig. 5. Analog scheme of the combustion process model.
1
vZ{-xo2[+z + +:(vd - Vo)] + 2 1 ~ - 100Vo@~},
-
dxco dt -
-
-
1
+
(36)
vo)]
+ (1 - a)1.866Cq~g},
(37)
A more detailed look on eqn. (38) shows that it represents a bilinear (nonlinear) model. The analog scheme of this model is shown in Fig. 5.
5.2. Air trap-door model
where Xo2 is the percentage of 0 2 (vol%), Xco is the percentage of CO (vol%), Vk is the volume of the combustion chamber (ma), ~ is the normalized flow of fuel (kg/s), q~ is the normalized flow of air (Nma/kg), Vo is the theoretically needed air volume for the combustion of one unit of fuel (Nma/kg), Va is the theoretically obtained gas volume from one unit of fuel (Nm3/kg), (1 - a) is the relative portion of carbon converted into CO (%/100), and C is the relative portion of carbon in fuel (%/100). Equations (36) and (37) can be expressed in the form:
The air flow is controlled through an air trapdoor. As it is a part of the closed-loop, it has to be modelled and added to the combustion model. The air trap-door is a nonlinear dynamical system. Its gain is described by the relations:
it(t) = AlU( t )x( t ) + A2v( t )x( t )
where K z is the air trap-door gain, ~ is the angle of the air trap-door, and qb~m,~x is the maximum flow of air. Air trap-door dynamics are described by the first-order transfer function:
+B;u(t) +B~v(t), y ( t ) • x ( t - Ta), where
(39)
Kzffi ~
exp
2-exp
--
/
0 ° ~<~b < 45 °,
~
45 ° < ~b < 90 o,
0.5
xT = [Xo2 Xco], u =,t,z,
(38)
K~ -- - - ~
G(s) ~ K ~ - -
s +0.5 "
45'))
(40)
(41)
(42)
v
At=-l/Vk,
A2=(Vo-Vo)/Vk
B ; T = [ B , 0] = [21/V k 0]
B~T _- [B2 B4]
= [ - IOOVo/Vk (1 -a)1.866C/Vk], and Td is the delay, transport time.
5.3. Linearization of the model The most common step in control system design is linearization of the nonlinear model around the corresponding operating point. Among the many methods for the linearization of nonlinear systems we chose the approach with
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J. Kocijan, R. Karba / Robust muitivariable control
H~'~. ~
~
~
~
~I
~l~-li-~l I T ~
~
~
33
I~
0.52 0.52
0.51
f.' C~5!
/
7'
O,5O 0.50 0.50 0."19
0,49 O.48 I
0'180.00
O.80
I
I
'
I.~0 ~'-"
I
[
2.40
~:.20 _
I
I
I
'
I
4.O0 4.BO 5.~0 ~.40 I SltlCOS F'~" ~ IJS LJL~LJ~N~
I
7,20 ......
I
)
8,O0 I ×10 2
Fig. 6. Closed-loop r e s p o n s e of the nonlinear system without time delay a n d reference i n p u t s - - O 2.
identification. The "Black box" identification method with least-squares algorithm from the program package for system analysis and control design ANA was used for identification of our nonlinear model. A pseudo random binary signal (PNBS) around the chosen operating point, which was angle of air trap-door at 46 ° and flow of fuel 1 kg/s, was used as input signal. The simulated responses of the nonlinear model, obtained by the aid of the simulation language s~ucos [4], represented the other data needed for identification. As a result the following system matrices A, B, C were obtained: A =
B __
C_-
- 1.415240 -0.663208 1.3703x10 -2
1.506356 0.594137 -1.5252X10 -2
- 2.009392 X 10 -3 5.169228 X 10 -2 -4.250321 × 10 -4 0
oo] 0
"
0.586975 ] - 0.438879 /' -5.1391X10-zJ
-4.160395 ] -3.231999 3.869638 × 10 -2
These matrices determine controllable and observable linear time-invariant process.
5. 4. Control design Control design is an interactive and iterative procedure which through comparison between alternative control algorithms can lead to an acceptable solution of the control problem. In the first step the proportional-integral output feedback pole assignment controller was designed with software for the design of multivariable pole assignment controllers [5]. After an iterative and interactive procedure the desired closed-loop poles were chosen to be: -0.1 +j0, - 0.3 + jo,
-0.2+j0, - 0.5 + j0,
-0.7 +j0, and the controller matrices were: K = [ -3.862247 -408.9212 ] - 0.2130442 - 3.622375 J -0.2972958 -31.07514 ] Q = - 0.04225255 0.8685015 J
34
Applications
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x,o-{)~CZ]~ - I
~
~
~l
~
~
~
~
I~-FF~'--I F~-~---I
0.45 0.45 0.44
f.j-"
0.44 0.44
/
0.43
/
/
/
/
,/
/'
0.43 0.43 0.43 0.42
0.420.60
I
Ol ~0
I
I160 +'
I
2140
!
~:m20
I
~.O0
I
4180
I
~1 ~0
I
~140
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J. Kocijan, R. Karba / Robust multivariable control
x10 0.52 0.52 0.51 0.51 0.50 0.50 0.50 0.49
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36
Applications
Computers in Industry
[Load
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The closed-loop responses of the nonlinear system without time delay and reference inputs are shown in Fig. 6 for 02 and in Fig. 7 for CO, while control signals are shown in Fig. 8. The closed-loop responses of the nonlinear system with time delay (0.8 s) and reference inputs are shown in Fig. 9 for 02 and in Fig. 10 for CO, while control signals are shown in Fig. 11 This controller shows satisfactory dynamics and, applied on a nonlinear model, causes partial linearization of the nonlinear system. However, digital simulation with SIMCOS [4] shows that robustness against parameter perturbation does not mean always that the system has the same robustness against time delay. In our case the relation between these two properties was inverse: more satisfactory control of the nonlinear model caused less satisfactory response of the delayed system. In the next step, the second part of the proposed joined robust structure was developed to increase robustness. The servo compensator includes a model of step changes of the reference and disturbance vectors and is described by eqn. (12).
The complementary controller should include the model of the process, but for the sake of simplicity of the control design procedure the linearized model of the closed-loop process with the proportlonal-integral output feedback pole assignment controller was used. Matrices, g l, K 2 and g 3 were obtained by the proportional state feedback pole assignment as described in the previous section, using the mentioned software. The overall closed-loop poles were chosen to be: - 0 . 0 5 + jO, -O.08+jO, - 0 . 1 + j 0 , -0.15 +jO, - 0 . 2 +jO, - 0 . 2 5 + j0, and controller matrices were: = [ 0.0065 0.1372] KI [ - 0.0001 0.0055/' = [ 12.4201 -9.6756 296.0749] K2 0.0038 I - 0.0003 1.5746 ' /(3=[-0.7002 -7.8728] 0.0015 0.0055 " - 0 . 0 3 + jO,
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J. Kocijan, R. Karba / Robust multivariable control
0,52 0.52 0.51
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38
Applications
Computersin Industry
-.~.,.__.,_,_____
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The closed-loop responses of the nonlinear system with time delay (0.8 s) and reference inputs are shown in Fig. 12 for 0 2 and in Fig. 13 for CO, while control signals are shown in Fig. 14. From these figures it can be seen that overall robustness structure has made the response faster but more smooth. Robustness against parameter deviations has been enlarged with the augmented controller structure while robustness against time delay remained on the same level. Larger time delay causes high amplitude oscillations. As the combustion process is controlled by a microcomputer controller, the question about loss of information because of discretization should be stated. Extra simulation runs with a discretized controller are planned for future work.
6. Conclusion Control system design appears to become more and more important in industry for obtaining a higher level of technology, Multivariable and non-
linear systems are usually met in a real-life environment. But there is still lack of efficient methods for nonlinear control system design. Robust controllers are a possible solution for this problem. Robust multivariable control system design represents a widely used set of different approaches and methods. Two of them have been presented and an attempt of joining together these two structures in order to increase robustness has been shown in this work. The proportional-integral-differential family of controllers are very well known representatives of robust controllers. The multivariable proportional-integral output feedback pole assignment controller is just one from this set of controllers. The other controller is a robust controller for solving the multivariable servomechanism problem. This controller achieves asymptotic regulation even in the presence of perturbations in the system and controller parameters, which are not necessarily small. The procedure of robust controller design enables any output feedback con-
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trollers that already exist in the given system to be retained in the final control configuration. Both controller structures have been joined together and used in combustion process controll. The combustion process represents a multivariable and nonlinear problem with time delay and controller robustness properties could be seen in this process control. Results of control system design were obtained by the aid of computer simulation of a nonlinear process model and control system. From these results it can be seen that robustness against parameter perturbations does not always mean the same as robustness against time delay. In our case this was an inverse relation. All disturbances have to be known and included in the structure of the robust controller. Since all disturbances are not always known, this fact represents a disadvantage in control system design. Also questions about loss of information because of discretization when a microcomputer controller is applied in practice should be asked. However, robustness against parameter changes has been shown to be enlarged with the augmented controller structure and that was the intention of the present work.
J. Kocijan, R Karba / Robust multivariable control
39
References [1] B. Friedland, Control System Design, Mc Graw-Hill, New York, 1986. [2] R.V. Patel and N. Munro, Multicariable Systems Theory and Design, Pergamon Press, Oxford, 1982. [3] J. Lunze, Robust Multivariable Feedback Control, Prentice-Hall, Englewood Cliffs, NJ, 1988. [4] B. Zupan~i~:, D. Matko, R. Karba and M. Sega, "SIMCOS --Digital simulation language with hybrid capabilities", Proc. 4th Syrup. Simulationstechnik, Ziirich, Switzerland, 1987, pp. 205-212. [5] J. Kocijan, "Computer aided multivariable controller design", Work prepared for University Award for Best Student Research Work, Faculty of Electrical and Computer Engineering, Ljubljana, 1989 (in Slovene). [6] M Sega, S. Strm~nik, R. Karba and D. Matko, "Interactive program package ~'qA for system analysis and control design", Proc. 3rd 1FAC/1FIP Int. Symposium CADCE'85, Copenhagen, Denmark, 1985, pp. 145-150. [7] H. Seraji, Design of proportional-plus-integral controllers for multivariable systems, lnt. J. Control, Vol. 29, No. 1, 1979, pp. 49-63. [8] E.J. Davison and A. Goldenberg, "Robust control of a general servomechanism problem: The servo compensator", Automatica, Vol. 11, 1975, pp. 461-471. [9] E.J. Davison, "The robust control of a servomechanism problem for linear time-invariant multivariable systems", IEEE Trans. Autom. Control, Vol. AC-21, 1976, pp. 2534. [10] J. (~retnik, S. Strm~:nik and B. Zupan~i~, "A model for combustion of fuel in the boiler", Proc. 3rd Symp. Simuiationstechnik, Bad Miinster, Amstein, 1985, pp. 469-473.