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Probabilistic robust optimization of two LQ control problems
PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
14th World Congress of IFAC
G-2e-16-5
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTROL PROBLEMS
Hong Yue*, Weisun Jiang**
*Assistant Professor, Institute ofAutomation, Chinese Academ~v ofSciences, Be(jing 100080, P. R. China, Email: [email protected], Fax: (86-10) 62563156 ** Professor, Institute ofAutomatic Control, East China University ofScience & Technology, Shanghai 200237. P. R. China
Abstract: The probabilistic robust optImIzation (PRO) of LQ control problems are studied in the framework of soft-bound descriptions. The improved robustness and the unity design in the parameter space are achieved by optimizing a probability-weighted sum of quadratic performances of the system with respect to possible parametric uncertainties~ Tw~o PRO problems are fonnulated considering real control requirements. The approach with the construction of a probability-extended system is proposed to determine the optimal control profile and the gradient-based solution is presented to find the optimal feedback control. Numerical examples illustrate the advantages of the presented methods for systems with large parametric uncertainties. Copyright~, 1999 IFAC
Keywords: soft-bounded, probability) robustness, linear quadratic optimization
control have been obtained in recent years. Janiszowski (1987) investigated the stability of discrete-time control systems with statistical uncertainties in model parameters. Stengel, et al~ (1995) proposed a probabilistic evaluation of robustness based on expected parameter variations. They suggested that the degree of robustness required for satisfactory operation is related to the plant variations most likely to occur. Thus robustness matrices should be probabilistic, deriving from descriptions of parameter variations that are bounded by manufacturing tolerances or physical constraints. Djavdan, et al. (1989) presented a robust controller design method that achieves perfonnance robustness by maximizing the probability of the simultaneous realization of the design specifications. Schaper~ et at. (1992) introduced the probabilistic measure of a controller~s ability to reject disturbances for process control applications. The study of robust estimation and feedforward control by Sternad and Ahlen (1993) ,vas based on probabilistic descriptions of model errors. The robust design is obtained by minimizing
I. INTRODUCTION In most treatment of robustness, model errors are described by hard bounds. The controller is then designed on the worst-case situations to guarantee robust stability and certain level of perfonnances. But conditions for worst cases are always extreme and have a lov.-' probability of occurring, thereby the controller is overly conservative for more typical operating conditions that have a much higher probability of occurring. Soft-bound description avoids such limitations, in which model uncertainties are characterized by probability distributions. Not only the range of uncertainties, but also the likelihood is taken into ace·ount. Common model errors will have larger impacts on the controller than the unlikely worst cases~ The conservativeness is thus reduced and a more natural balance between control perfonnance and robustness is achieved (Schaper, et al., 1992; Sternad and Ahlen~ 1993).
Some encouraging results of soft-bound robust
3599
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
the squared estimation error, averaged both with respect to model errors and noise.
an approximate stochastic distribution with pdf P(S i)' Two probability robust LQ optimization problems are formulated.
There are a variety of ways to set up the necessary models for soft-bound robust control. Tt is possible that soft bounds are more readily obtainable in a noisy environment than hard bounds. Gaussian distribution is commonly used to describe soft-bound errors and can be justified in many practical situations. Several other probability distributions, such as Unifonn distribution, Triangle distribution and Fractile distribution, are also considered for judging system uncertainties (Frey and Rubin~ 1992)~
Problem I Determine the optimal control u(t)=Ut(t)=u2(t)=·~,==um(t)
(4)
so as to minimize the per.formance index m
J pRO = l:P(8 j )J} i=1
=
I: p(a )I~r (xT Qxi + i
uT Ru)dt
(5)
;=}
\1--'here
Q E R nxn
is a positive-semidefinite real
symmetric matrix and RE R rxr is a positive-definite real symmetric matrix.
The purpose of this paper is to study the probabilistic robust LQ optimization problems for parametric uncertain systems. Robust LQ design has long been of interests in state-space robust control. Most efforts are conducted on hard-bound minimax design. The w'ork presented here is based on soft-bound concept and yields increased robustness in cases of large uncertainties.
Problem 2 Determine the .feedback control malrix K E R rxn of the optimal control vector u J Ct) ~ - KX j (t) (6) so as to minimize the performance index m
J PRO
=;
~
p(a i )Jj
1:;;;;1
2. PROBABILISTIC ROBUST LQ OPTIMIZATION PROBLEMS
==
m t T P (fJ i )Iof(x j Qxl 2:
7'
+U j Ru,)dt
;:;:::::1
m t TT}' = :LPC8')}Of(X1 QX1 +x/ K RKxi)dt
The idea of probabilistic. robust optimization was proposed for chemical process control (Terwiesch, et al., 1994; Terwiesch and Agarwal, 1995). Given a probabi lity distribution for the uncertain parameters from either identification or prior knowledge, the optimization is carried out with the expectation of the
perfonnance index over the entire parameter range. The objective is obtained by discretizing the continuous parameter space e, so that it is presented by a finite number of grid points e i ' The expected perfonnance is then approximated through a sum of probability-weighted performances. J
pRO =
1ee J (u,S)P(9)de
;:;:; fJ(u?s;)pce
i)
(1)
J::::l
where m is the number of selected points and p(e j) is the discrete probability distribution function (pdf)~This fonnulation provides an improved solution where robustness to selected uncertainties is incorporated by design rather than by chance. In this paper, PRO design is introduced into LQ control for systems with parametric uncertainties. Consider the parametric uncertain system X = Ax + Bu, x(O) == Xo
It'here Q E R nxn is a positive-semidefinite real symmetric matrix and R E R rxr is a positive-definite real symmetric matrix. Problem 1 is proposed for robust optimization control of some chemical processes. The optimization of transient operating profiles for batch processes has been the subject of a large number of academic research contributions. As realistic applications are subject to model errors, optimal control for batch reactors assuming proper knowledge of uncertain models becomes the predominant issue in many batch unit optimization problems. In problem 1, a solution is considered from the soft-bound point of view. Problem 2 is formulated for closed-loop feedback control of systems with soft-bound models. It may fit for more general control requirements rather than the optimal profile in problem 1~
3. METI-IODS OF OPTIMIZATION SOLUTION
(2)
.A and B are coefficient matrices containing uncertainties. Suppose that this system can be represented by a set of possible systems X; (t) = Aixi (t) + B;uj Ct) , xi(O)~XO'
(7)
i=]
i=1?2,···,m (3) r \vhere Xi (t) E Rn and u~ (t) E R . Let ei denotes the set of parameters ( A j , Hi ) and assumed that it has
3~ 1
Probability-extended- System Solution
There are many algorithms available to solve nonlinear optimal control problems similar to problem]. In the work of TernFiesch and Agarwal (1995) the optimization problem is solved by a control vector iteration (CVl) algorithm based on Pontryagin's maximum principle. Ruppen et al.
3600
Copyright 1999 IFAC
ISBN: 0 08 043248 4
PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
14th World Congress ofIFAC
(1995) reports a numericaJ solution that circumvents the expensive integration through use of collection and infeasible _path successive linear programming (SLP) optimization. These methods are suitable for optimization problems with the performance to be a sum of weighted perfonnances. However~ two shortcomings exist in some cases for these methods~ One is that they are computationally demanding. The s(}lution procedure is gradient-based and the optimization calculations are performed iteratively. The other is that the piecewise constant function fonn is used to approximate the true continuous function, \vhich may decrease the accuracy of the result. To avoid these limitations, a computationally more efficient solution strategy, based on the construction of a probability-extended system~ is proposed for problem 1.
It is obvious that Q and R accord with the conditions for LQ weighting matrices. In this case, the performance index in problem 1 is equivalent to the LQ performance of the extended system, Le. J pRO
c=
[Xl' Xl'·· . '
-
Xm]T , T
A]
0
A=
(9)
o B=[B l ,B2 ,···,Bm ]T The extended system is represented as
(10)
X :::: AX + Bu ,
(11)
X(O)
=X 0
This ~"Iarge" extended system (Il) is composed of m "small" systems in (3). For system (11) with performance ]=f~f(XT(jX+uTRu)dt
(12)
Q is a positive-semidefinite real symmetric matrix and R is a positive-definite real symmetric where
Jnatrix ~ the optimal control is u(t) = -KX(t)
(13)
K == "R- 1 S T p
(14)
v.,hen t f
= oc.
X(t) is decided by system (11) and
P is the positive symmetric solution to the Riccati equation PA +ATp+Q_PBR-1fjT p =0
(15)
By way of proper construction of the weighting matrices, the optimal control in problem 1 will be detennined through the extended system.
T
+u j Ru,)dt
I
m
T
r
m
:;::SOf(L:X,. p(er)Qxi +U ~p(ei)Ru)dt i=]
=
i=l
J~f (XT(jX + uTJiu)dt
= ] (18) Thus, by constructing the extended system (11) and defining the weighting matrices Q and R in a probability-weighting form, problem 1 is transformed into a standard LQ optimization design problem. The probability robust LQ optimization problem is solved in an analytical way.
3.2 Gradient-based Solution
(8)
X o =[xo,xo,···,xo]
T
1:;::::1
Let
x
t
m
= L,P(9 j )Jof(X j Qxj
The objective of problem 2 is to find a linear state feedback control that minimizes a sum of LQ perfonnances. This is similar to the simultaneous stabilization control proposed by Howitt and Luus (1993), who presented an efficient procedure for the solution. The emphases of the two problems are different. The system in simultaneous stabilization is mode)ed by a set of linear rnodels that might have no clear interrelations. Thus there is a lack of rules in deciding proper weighting factors for control performances. Problem 2 is studied in soft-bound framework and the weighting factors are decided by pdf of the uncertain parameters which is more reasonable. In this paper, Howitt and Luus s method is extended to provide the solution to problem 2. Calculation of the objective function and its derivatives is required for this gradient-based method. Expansion is made to the reported work so as to consider the difference of \veighting matrices. The herein solution is feasible for more general cases. A brief discussion of the procedure is given in the following. 1
The variable in the optImIzation problem is the feedback control matrix K. Write K in terms of its rows
(19)
Defining
o Pce 2 )Q
Q==
o
where k j (16)
Rn, J4 = 1, · .. ,r and then define the
vector V to be v = [k l k 2 ••• k r l~rxl (20) Thus the unknown to be chosen by optimization is v . The gradient of the objective function is given by
P(8 m )Q
R = ~p(e;)R
E
(17)
I=:I
3601
Copyright 1999 IFAC
ISBN: 0 08 043248 4
PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
8J
pRO =---a;;-
g PRO
14th World Congress ofIFAC
G~·lQX; +Gi1K T RKx f TT
(21)
== 2j;f
The gradient of the i th perfonnance index with respect to k j is
oJ, t () T T T )dt - = raf_(x- Qx- +x- K RKx· ak,
8k,
JI
.I
ri'
,
(22)
Let
;~ J:~r
dJ i 'd.t = XiT (Q + K T RK)x
(23)
dG-dt
represent the sensitivity matrix of the state X j with respect to K j • Taking the two tenns to be differentiated within the integral in (22) separately~ the first term is
a
(}xi"
T
T
--(Xi Qx;)=--QX i +(I n 0X i 8k j ok j
. . TK
+ (I ® xTK T ) n
l
J
ok.
8k
I
where
(25)
BK
T
=[!
.
x~J) t
[~
x~j) J
~
.
!1xr
(26)
Ao
T
RKx j
)
= L P(8 i )gj
!lxr
= 2G.,1'jj K T RKx i
(27)
+ 2LijRKx j
(28)
.
Thus, substituting (24) and (28) into (22) yields
oJ
f( T T r ) i ak. = 2Io" (G u Qx; + Gij K RKx j + L(iRKx;)dt (29 j
Let gi be the gradient of J vectorv ~ Le. 8J g I. = ..... ov l
1
[~\~O _2~~~o __
l
[1.00 0.12] 0.05
1.63 '
d -- N(O,O.4, (-0.5,1.0)), to
Substituting (27) into (25) yields T
=
E
....
x~j) r
0
a
jth
(35)
Example 1 Consider the system x = (AD + d . E)x + Bu, x(O) ~ [l.O~O.5]1' ~
where superscript (j) denotes the jth column in the matrix. Let
Bk. i
(33) is the
4. NUMERICAL EXAMPLES
I
=
in
the performance index and its derivative necessary for gradient-based methods are obtained. In this paper, the BFGS method, used by Howitt and Luus (1993), has been taken to find the optimal feedback matrix K.
T
x· - J ok. x~j)
--(Xi K
bti ERr,j=l,·~·,r
ok)
.J
Lij
(34)
m
aK T
l
j
,
--
(I ®x· ) - - = n
T·
i=:}
.I
oK T
7'
gPRO
By matrix calculation, the middle part of the second tenn in (25) becomes
r
(33)
:J
GirQX i + GlrK RKx j + LrrRKx t
,.
T T T 8K 'l' =2G(jK RKx,+2(In ®x] )----a;;-RKx;
j'
,G;i (0) == 0
-
:
l'
J
x
T bi/Xi
+G~KT RKx i + LilRKxi]
o(RKx.)
J
(32)
column of the control coefficient matrix E i • With the integration of equations (31 )-(34) and the probability weighting operations fOT J PRO in (7),
1
8k.
J i (0) = 0
j ,
gi(O)=O
j
= ~RK"'(-
GXQx; [
ak j
~(xTKTRKx) 1
- BiK)G~i :J
dg i -2
o(Qx
(24)
T
= (Ai
--;;( -
The second term in the integral becomes ak
-_Y
i) )---
== 2G[ Qxi
(30)
It is convenient to evaluate J; and gj when integral operations are converted to the following ordinary differential operations xj(t)==(A i -BjK)xi(t), xj(O)=xo (31)
)
Gij = (
+ LflRKx i ]
: dt [ G,~QXi +Gi~KTRKxi + LirRKx i
with respect to
B
=
[~ ~l
Q= R == 12
= 0.0,
t f ;;::: 6.0 .
In this example, a two-side truncated Gaussian distribution (Yue, 1996) is assumed for the uncertain parameter d. The truncated Gaussian distribution N(Jl,cr 2 ~ Ca, b)) is fonnulated from the standard Gaussian distribution N(IJ.~(J 2) to avoid physically unrealistic values for the unbounded paf. The truncated end ( a~ b) is placed for one side or both sides of the random variable according to the practical situations. 7 points are taken into account for this design. The pdf value for each parameter point is nonnalized such that the sum of all selected pdfs is 1.
3602
Copyright 1999 IFAC
ISBN: 0 08 043248 4
PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
14th World Congress ofIFAC
Table 1 LQ perfQnnance indices for different Parametric uncertainties
i= I -0.50
i=2
;==3*
-0.25
0.00
i == 4 0.25
i=5 0.50
i=6 0.75
i == 7 1.00
0.1436
0.1815
0.1963
0.1815
0.1436
0.0972
0.0562
0.3501 0.4055
0.3988
0.4678
0.5732
0.7556
1.1508
2.5399
0.7905
6.2362
JrPRO
0.4458
0.5007
0.7065
0.9458
1.6435
0.6284
5.2279
J; (minimax)
0.6928
0.7198
0.7542
0.5801 0.7993
0.8615
0.9544
1.1273
0.8247
5.9093
d
j
P(d i
)
J i (nominal)
Simulation results are presented in Fig. 1 and Table 1.
Consider the problem of controlling the outlet concentration C A by manipulating the coolant temperature Te . These equations can be expressed in dimensionless quantities as
* denotes the nominal system. Table I shows that the best overall performance is obtained by PRO method in the whole range of uncertainties. Compared to the nominal method, PRO design increases the robustness and performance consistency to parameter changes while retaining good performance near nominal point. Compared to the minimax design~ this method is less conservative and offers beuer overall performance. This result is due to the soft-bound description for uncertainties and the formulation of the optimization problem, because the expected perfonnance is considered and highly probable models affect the design more than do very rare worst cases.
Xl X2 = -(1+
l
F V
dt
F
(-All)
dt
V
CpP
--.
E
L'A
RT
VCpP
- = - ( TF -T)+--KDLA exp(--)---(Tc -T) u, 0.1
0,·· -0.11'
.
-----·-··---1
r
.-. :-~ .~.:.-~ .-.,....-~-
-0.2/ >"~
2
,
~l]=[[-I~DaeXP(l+:~:rY)
x2
DaCn
10
DaRn]J . [Xl] +
BDaR
X
n
]
-1-13 +BDaC n
-BD a exp(-"-) 1+x /y
2
[0]
J3 ·u
en and Rn are the cone center and cone radius of the nonlinear term, which are decided by the nominal point and its bounds. d is the uncertain factor introduced by linearization and is bounded between -1 and +1 (Schaper et at, 1992). In this example, a unifonn probability distribution is assumed for d and 5 points are taken into account for PRO design. Numerical parameters chosen for this example are D a = 0,072, B :;; 1.5, ~ = O~3 , Xv == [0.3,2.45]',
-, E -CA)-KoC A exp(--) RT
dT
+Do(l-xl)exp(]+::/y)
f3 )x 2 + BDa (l-x1) exp(-1+x )+ J3u x2 / y
0 +d· [ 0
exothennic reaction A ~ B, occurring in a constant volume, continuously stirred tank reactor: dC A
=-Xl
The non linear model is transfonned into linear model by cone-sector bounded Iinearization. The resulting model is expressed in linear state equations with uncertainty.
Example 2 Consider the most commonly used reactor model that describes an irreversible,
--=~(CAF
L:J,.
·J pRO
,;t'
>
:;::" /
en == 12.51 ~ Rn = I 1. 0 I. Take y ::;;:; x l ~ Q = I , R = I, Tf == 10.0. Simulation results are shown in
r-
Fig. 2. The designed feedback control matrices jn terms of 3 different methods are: K nominal::;:: [~O.16,O.43] ,
~::l
~O.7
.0.8)
I
-0.9 l
L
o
0.5
_
1IlOmlna4..2:eRQ•. :brurumax.f ~ 3.5 4 4.5 5 5.5 6
. . . ..J_ ._....
1
1,5
2
:2.6
K minimax = [-1.59,3.86] K PRO ~ [0.19,1. 71]. The uncertainty range is large in this case. It is not acceptab le to use nominal design because it leads to unstable system responses in some cases (Fig.2 (a)). Both PRO design and minimax design are able to guarantee robust stability and provide consistent perfonnance in the whole uncertainty range (Fig.2 (b) and (c). However, rninimax design is more conservative than PRO design in most cases considered. In Fig. 2 (d), the LQ perfonnances from PRO design are better than the results from minimax design in the range from -1.0 to O.85~ with only small loses in the worst cases nearby (0.85 to 1.0). 'l
t
(a) U1 vs time U2
0.1
0 1
-0.1:
,:1'
:::/ /
:::1 / / I -O.S
i
_0.7
-o.ef
I
-0.9'
o
\no~~~~ ?:.~~9.~p~~~~i~~.J
L.
0.5
1
1.5
2
2.5
(b) U2
3
3.5
4
4.5
5
5.5
6
vs time
Fig.I. Optimal control protile in example 1
3603
Copyright 1999 IFAC
ISBN: 0 08 043248 4
PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
2.5-
5i
2\
.t
14th World Congress ofIFAC
The proposed methods are suitable for systenls with large parametric uncertainties. Although it is computationally intensive compared to hard-bound robust optimization design, requirements are well within the capabilities of existing computers, It should be mentioned that PRO design might be effective even when the soft-bound information is not exact, because the selected uncertainties are incorporated and the overall objective is considered at the time of the problem fonnulation.
1.5i';~a1~~~
"~/l~'~ 2.
0.5.
.
o _~ ... J:..~_~_ .._...
-"O
I -0.5'-"
__ .L._---
o
! __ .., ----. -,--.-
--, ----.
1
2
4
:3
.~ - - - - - " - _ .. __ --"5 6 7
-'
I
--_._.1-.
8
9
10 t
(a) nominal design
_.- .. - - - "-- ..., :d...-1. 2:d=-O.5, 3:d""O. 4.d=,O.5, 5:d~1
,--.
REFERENCES '.' 5
Djavdan, P., H. J. A. F.. Tulleken) M. H. Voetter, H.. B. Verbruggen and G. J. Olsder (1989). Probabilistic robust controller design. In: Proceedings o.f the 28'h Conference on Decision and Control, Tampa, Florid~ pp. 2164-2172. Frey, H. C. and E. S. Rubin (1992). Evaluate uncertainties in advanced process technologies. Chem. Eng. Prog.~ 88(5), pp. 63-70. Howitt, G. D. and R. LUllS (1993). Control of a co Ilection of linear systems by linear state feedback control. Int. J. Control, 58 (I), pp. 79-96. Janiszowski, K. B (1987). Sufficient condition for stable controJ of discrete-time systems with statistical process modeL Int. J. Control, 45(2») pp. 693-700. Ruppen, D., C, Benthack and D. Bonvin (1995). Optimization of batch reactor operation under parametric uncertainty--computationaI aspects. J Proc. Contr., 5(4), pp. 235-240. Schaper, C. D., D.E. Seborg and D. A. Mellichamp (1992). ProbabiJistic approach to robust controL Ind. Eng. Chem. Res., 31(7), pp. 1694 - \704. Stenget R. F.~ L. R. Ray and C. I. Marrison (1995). Probabi listic evaJ uation 0 f contro1 system robustness. 1nl. J. Systems Sci.~ 26(7), pp. 1363]382. Sternad.. M. and A. Ahlen (1993). Robust filtering and feedforward control based on probabilistic descriptions of model errors. Automatica:> 29(3), pp. 661-679. Terwiesch P. and M. AgarwaJ (J995). Robust input polices for batch reactors under parametric uncertainty. Chem. Eng. Comm., 131, pp. 33-52. Terwiesch P., M. Agarwal and D. w. T~ Rippin (1994). Batch optimization with imperfect modelling: a survey. J. Proc. Cont!:, 4(4), pp. 238258. Vue H. and W. S. Jiang (1996). A new probabilistic robust optimization method. In: Proceedings of/he IEEE International Conference on Systems, Man and Cyberne!ics~ Beijing, Chin~ VoJ.3, pp. 22052209.
I
___ (
_------.: ..
3
I
4
0
' - - - __ ...l.. ~----L--l-.-----1
6
7
8
9
10
(h) PRO design y 2 y
'.5.
f-
5
It -{
1'.··'~·
'~r' '~\4
05~·
•.·"
ol"~~'-'--'::~., -----'
2
1:r-
3
_
4
... ..L...._.__ J.
5
. .....L.
I 1
• _ _ ._.....L.•. _.,.!
a
6
9
10
Cc) minimax des;gn
.J
1:liom;naJ, 2:PRO, 3:T1'!.inJmax
80
70 .
:r 30
1(1c
o
_ _-
~2
'--~_"'----J-_.L--.........L-.~-'----..L.-~.l..----I..-_ ..----1
-1
-0.8
-O.S
-0.4
-0.2
0
0.2
0.4
0.6
(d) J vs uncertainty
0.8
d
Fig.2 Simulation results in example, 2
5. CONCLUSION
In this paper~ two probability robust LQ optimization problems are fonnulated and solutions are provided. 1t combines the following important features. Firstly:> probabilistic perfonnance tradeoffs are made between high likelihood conditions and low likeljhood conditions. in the uncertainty range. The design emphasis is thus more reasonable. Secondly, the design in the whole parameter space is taken in its entirety so that the best overall perfonnance is achieved, The results obtained from numerical examples demonstrate the vje~rpoints.
3604
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
G-2e-16-5
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTROL PROBLEMS
Hong Yue*, Weisun Jiang**
*Assistant Professor, Institute ofAutomation, Chinese Academ~v ofSciences, Be(jing 100080, P. R. China, Email: [email protected], Fax: (86-10) 62563156 ** Professor, Institute ofAutomatic Control, East China University ofScience & Technology, Shanghai 200237. P. R. China
Abstract: The probabilistic robust optImIzation (PRO) of LQ control problems are studied in the framework of soft-bound descriptions. The improved robustness and the unity design in the parameter space are achieved by optimizing a probability-weighted sum of quadratic performances of the system with respect to possible parametric uncertainties~ Tw~o PRO problems are fonnulated considering real control requirements. The approach with the construction of a probability-extended system is proposed to determine the optimal control profile and the gradient-based solution is presented to find the optimal feedback control. Numerical examples illustrate the advantages of the presented methods for systems with large parametric uncertainties. Copyright~, 1999 IFAC
Keywords: soft-bounded, probability) robustness, linear quadratic optimization
control have been obtained in recent years. Janiszowski (1987) investigated the stability of discrete-time control systems with statistical uncertainties in model parameters. Stengel, et al~ (1995) proposed a probabilistic evaluation of robustness based on expected parameter variations. They suggested that the degree of robustness required for satisfactory operation is related to the plant variations most likely to occur. Thus robustness matrices should be probabilistic, deriving from descriptions of parameter variations that are bounded by manufacturing tolerances or physical constraints. Djavdan, et al. (1989) presented a robust controller design method that achieves perfonnance robustness by maximizing the probability of the simultaneous realization of the design specifications. Schaper~ et at. (1992) introduced the probabilistic measure of a controller~s ability to reject disturbances for process control applications. The study of robust estimation and feedforward control by Sternad and Ahlen (1993) ,vas based on probabilistic descriptions of model errors. The robust design is obtained by minimizing
I. INTRODUCTION In most treatment of robustness, model errors are described by hard bounds. The controller is then designed on the worst-case situations to guarantee robust stability and certain level of perfonnances. But conditions for worst cases are always extreme and have a lov.-' probability of occurring, thereby the controller is overly conservative for more typical operating conditions that have a much higher probability of occurring. Soft-bound description avoids such limitations, in which model uncertainties are characterized by probability distributions. Not only the range of uncertainties, but also the likelihood is taken into ace·ount. Common model errors will have larger impacts on the controller than the unlikely worst cases~ The conservativeness is thus reduced and a more natural balance between control perfonnance and robustness is achieved (Schaper, et al., 1992; Sternad and Ahlen~ 1993).
Some encouraging results of soft-bound robust
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PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
the squared estimation error, averaged both with respect to model errors and noise.
an approximate stochastic distribution with pdf P(S i)' Two probability robust LQ optimization problems are formulated.
There are a variety of ways to set up the necessary models for soft-bound robust control. Tt is possible that soft bounds are more readily obtainable in a noisy environment than hard bounds. Gaussian distribution is commonly used to describe soft-bound errors and can be justified in many practical situations. Several other probability distributions, such as Unifonn distribution, Triangle distribution and Fractile distribution, are also considered for judging system uncertainties (Frey and Rubin~ 1992)~
Problem I Determine the optimal control u(t)=Ut(t)=u2(t)=·~,==um(t)
(4)
so as to minimize the per.formance index m
J pRO = l:P(8 j )J} i=1
=
I: p(a )I~r (xT Qxi + i
uT Ru)dt
(5)
;=}
\1--'here
Q E R nxn
is a positive-semidefinite real
symmetric matrix and RE R rxr is a positive-definite real symmetric matrix.
The purpose of this paper is to study the probabilistic robust LQ optimization problems for parametric uncertain systems. Robust LQ design has long been of interests in state-space robust control. Most efforts are conducted on hard-bound minimax design. The w'ork presented here is based on soft-bound concept and yields increased robustness in cases of large uncertainties.
Problem 2 Determine the .feedback control malrix K E R rxn of the optimal control vector u J Ct) ~ - KX j (t) (6) so as to minimize the performance index m
J PRO
=;
~
p(a i )Jj
1:;;;;1
2. PROBABILISTIC ROBUST LQ OPTIMIZATION PROBLEMS
==
m t T P (fJ i )Iof(x j Qxl 2:
7'
+U j Ru,)dt
;:;:::::1
m t TT}' = :LPC8')}Of(X1 QX1 +x/ K RKxi)dt
The idea of probabilistic. robust optimization was proposed for chemical process control (Terwiesch, et al., 1994; Terwiesch and Agarwal, 1995). Given a probabi lity distribution for the uncertain parameters from either identification or prior knowledge, the optimization is carried out with the expectation of the
perfonnance index over the entire parameter range. The objective is obtained by discretizing the continuous parameter space e, so that it is presented by a finite number of grid points e i ' The expected perfonnance is then approximated through a sum of probability-weighted performances. J
pRO =
1ee J (u,S)P(9)de
;:;:; fJ(u?s;)pce
i)
(1)
J::::l
where m is the number of selected points and p(e j) is the discrete probability distribution function (pdf)~This fonnulation provides an improved solution where robustness to selected uncertainties is incorporated by design rather than by chance. In this paper, PRO design is introduced into LQ control for systems with parametric uncertainties. Consider the parametric uncertain system X = Ax + Bu, x(O) == Xo
It'here Q E R nxn is a positive-semidefinite real symmetric matrix and R E R rxr is a positive-definite real symmetric matrix. Problem 1 is proposed for robust optimization control of some chemical processes. The optimization of transient operating profiles for batch processes has been the subject of a large number of academic research contributions. As realistic applications are subject to model errors, optimal control for batch reactors assuming proper knowledge of uncertain models becomes the predominant issue in many batch unit optimization problems. In problem 1, a solution is considered from the soft-bound point of view. Problem 2 is formulated for closed-loop feedback control of systems with soft-bound models. It may fit for more general control requirements rather than the optimal profile in problem 1~
3. METI-IODS OF OPTIMIZATION SOLUTION
(2)
.A and B are coefficient matrices containing uncertainties. Suppose that this system can be represented by a set of possible systems X; (t) = Aixi (t) + B;uj Ct) , xi(O)~XO'
(7)
i=]
i=1?2,···,m (3) r \vhere Xi (t) E Rn and u~ (t) E R . Let ei denotes the set of parameters ( A j , Hi ) and assumed that it has
3~ 1
Probability-extended- System Solution
There are many algorithms available to solve nonlinear optimal control problems similar to problem]. In the work of TernFiesch and Agarwal (1995) the optimization problem is solved by a control vector iteration (CVl) algorithm based on Pontryagin's maximum principle. Ruppen et al.
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PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
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(1995) reports a numericaJ solution that circumvents the expensive integration through use of collection and infeasible _path successive linear programming (SLP) optimization. These methods are suitable for optimization problems with the performance to be a sum of weighted perfonnances. However~ two shortcomings exist in some cases for these methods~ One is that they are computationally demanding. The s(}lution procedure is gradient-based and the optimization calculations are performed iteratively. The other is that the piecewise constant function fonn is used to approximate the true continuous function, \vhich may decrease the accuracy of the result. To avoid these limitations, a computationally more efficient solution strategy, based on the construction of a probability-extended system~ is proposed for problem 1.
It is obvious that Q and R accord with the conditions for LQ weighting matrices. In this case, the performance index in problem 1 is equivalent to the LQ performance of the extended system, Le. J pRO
c=
[Xl' Xl'·· . '
-
Xm]T , T
A]
0
A=
(9)
o B=[B l ,B2 ,···,Bm ]T The extended system is represented as
(10)
X :::: AX + Bu ,
(11)
X(O)
=X 0
This ~"Iarge" extended system (Il) is composed of m "small" systems in (3). For system (11) with performance ]=f~f(XT(jX+uTRu)dt
(12)
Q is a positive-semidefinite real symmetric matrix and R is a positive-definite real symmetric where
Jnatrix ~ the optimal control is u(t) = -KX(t)
(13)
K == "R- 1 S T p
(14)
v.,hen t f
= oc.
X(t) is decided by system (11) and
P is the positive symmetric solution to the Riccati equation PA +ATp+Q_PBR-1fjT p =0
(15)
By way of proper construction of the weighting matrices, the optimal control in problem 1 will be detennined through the extended system.
T
+u j Ru,)dt
I
m
T
r
m
:;::SOf(L:X,. p(er)Qxi +U ~p(ei)Ru)dt i=]
=
i=l
J~f (XT(jX + uTJiu)dt
= ] (18) Thus, by constructing the extended system (11) and defining the weighting matrices Q and R in a probability-weighting form, problem 1 is transformed into a standard LQ optimization design problem. The probability robust LQ optimization problem is solved in an analytical way.
3.2 Gradient-based Solution
(8)
X o =[xo,xo,···,xo]
T
1:;::::1
Let
x
t
m
= L,P(9 j )Jof(X j Qxj
The objective of problem 2 is to find a linear state feedback control that minimizes a sum of LQ perfonnances. This is similar to the simultaneous stabilization control proposed by Howitt and Luus (1993), who presented an efficient procedure for the solution. The emphases of the two problems are different. The system in simultaneous stabilization is mode)ed by a set of linear rnodels that might have no clear interrelations. Thus there is a lack of rules in deciding proper weighting factors for control performances. Problem 2 is studied in soft-bound framework and the weighting factors are decided by pdf of the uncertain parameters which is more reasonable. In this paper, Howitt and Luus s method is extended to provide the solution to problem 2. Calculation of the objective function and its derivatives is required for this gradient-based method. Expansion is made to the reported work so as to consider the difference of \veighting matrices. The herein solution is feasible for more general cases. A brief discussion of the procedure is given in the following. 1
The variable in the optImIzation problem is the feedback control matrix K. Write K in terms of its rows
(19)
Defining
o Pce 2 )Q
Q==
o
where k j (16)
Rn, J4 = 1, · .. ,r and then define the
vector V to be v = [k l k 2 ••• k r l~rxl (20) Thus the unknown to be chosen by optimization is v . The gradient of the objective function is given by
P(8 m )Q
R = ~p(e;)R
E
(17)
I=:I
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PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
8J
pRO =---a;;-
g PRO
14th World Congress ofIFAC
G~·lQX; +Gi1K T RKx f TT
(21)
== 2j;f
The gradient of the i th perfonnance index with respect to k j is
oJ, t () T T T )dt - = raf_(x- Qx- +x- K RKx· ak,
8k,
JI
.I
ri'
,
(22)
Let
;~ J:~r
dJ i 'd.t = XiT (Q + K T RK)x
(23)
dG-dt
represent the sensitivity matrix of the state X j with respect to K j • Taking the two tenns to be differentiated within the integral in (22) separately~ the first term is
a
(}xi"
T
T
--(Xi Qx;)=--QX i +(I n 0X i 8k j ok j
. . TK
+ (I ® xTK T ) n
l
J
ok.
8k
I
where
(25)
BK
T
=[!
.
x~J) t
[~
x~j) J
~
.
!1xr
(26)
Ao
T
RKx j
)
= L P(8 i )gj
!lxr
= 2G.,1'jj K T RKx i
(27)
+ 2LijRKx j
(28)
.
Thus, substituting (24) and (28) into (22) yields
oJ
f( T T r ) i ak. = 2Io" (G u Qx; + Gij K RKx j + L(iRKx;)dt (29 j
Let gi be the gradient of J vectorv ~ Le. 8J g I. = ..... ov l
1
[~\~O _2~~~o __
l
[1.00 0.12] 0.05
1.63 '
d -- N(O,O.4, (-0.5,1.0)), to
Substituting (27) into (25) yields T
=
E
....
x~j) r
0
a
jth
(35)
Example 1 Consider the system x = (AD + d . E)x + Bu, x(O) ~ [l.O~O.5]1' ~
where superscript (j) denotes the jth column in the matrix. Let
Bk. i
(33) is the
4. NUMERICAL EXAMPLES
I
=
in
the performance index and its derivative necessary for gradient-based methods are obtained. In this paper, the BFGS method, used by Howitt and Luus (1993), has been taken to find the optimal feedback matrix K.
T
x· - J ok. x~j)
--(Xi K
bti ERr,j=l,·~·,r
ok)
.J
Lij
(34)
m
aK T
l
j
,
--
(I ®x· ) - - = n
T·
i=:}
.I
oK T
7'
gPRO
By matrix calculation, the middle part of the second tenn in (25) becomes
r
(33)
:J
GirQX i + GlrK RKx j + LrrRKx t
,.
T T T 8K 'l' =2G(jK RKx,+2(In ®x] )----a;;-RKx;
j'
,G;i (0) == 0
-
:
l'
J
x
T bi/Xi
+G~KT RKx i + LilRKxi]
o(RKx.)
J
(32)
column of the control coefficient matrix E i • With the integration of equations (31 )-(34) and the probability weighting operations fOT J PRO in (7),
1
8k.
J i (0) = 0
j ,
gi(O)=O
j
= ~RK"'(-
GXQx; [
ak j
~(xTKTRKx) 1
- BiK)G~i :J
dg i -2
o(Qx
(24)
T
= (Ai
--;;( -
The second term in the integral becomes ak
-_Y
i) )---
== 2G[ Qxi
(30)
It is convenient to evaluate J; and gj when integral operations are converted to the following ordinary differential operations xj(t)==(A i -BjK)xi(t), xj(O)=xo (31)
)
Gij = (
+ LflRKx i ]
: dt [ G,~QXi +Gi~KTRKxi + LirRKx i
with respect to
B
=
[~ ~l
Q= R == 12
= 0.0,
t f ;;::: 6.0 .
In this example, a two-side truncated Gaussian distribution (Yue, 1996) is assumed for the uncertain parameter d. The truncated Gaussian distribution N(Jl,cr 2 ~ Ca, b)) is fonnulated from the standard Gaussian distribution N(IJ.~(J 2) to avoid physically unrealistic values for the unbounded paf. The truncated end ( a~ b) is placed for one side or both sides of the random variable according to the practical situations. 7 points are taken into account for this design. The pdf value for each parameter point is nonnalized such that the sum of all selected pdfs is 1.
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PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
14th World Congress ofIFAC
Table 1 LQ perfQnnance indices for different Parametric uncertainties
i= I -0.50
i=2
;==3*
-0.25
0.00
i == 4 0.25
i=5 0.50
i=6 0.75
i == 7 1.00
0.1436
0.1815
0.1963
0.1815
0.1436
0.0972
0.0562
0.3501 0.4055
0.3988
0.4678
0.5732
0.7556
1.1508
2.5399
0.7905
6.2362
JrPRO
0.4458
0.5007
0.7065
0.9458
1.6435
0.6284
5.2279
J; (minimax)
0.6928
0.7198
0.7542
0.5801 0.7993
0.8615
0.9544
1.1273
0.8247
5.9093
d
j
P(d i
)
J i (nominal)
Simulation results are presented in Fig. 1 and Table 1.
Consider the problem of controlling the outlet concentration C A by manipulating the coolant temperature Te . These equations can be expressed in dimensionless quantities as
* denotes the nominal system. Table I shows that the best overall performance is obtained by PRO method in the whole range of uncertainties. Compared to the nominal method, PRO design increases the robustness and performance consistency to parameter changes while retaining good performance near nominal point. Compared to the minimax design~ this method is less conservative and offers beuer overall performance. This result is due to the soft-bound description for uncertainties and the formulation of the optimization problem, because the expected perfonnance is considered and highly probable models affect the design more than do very rare worst cases.
Xl X2 = -(1+
l
F V
dt
F
(-All)
dt
V
CpP
--.
E
L'A
RT
VCpP
- = - ( TF -T)+--KDLA exp(--)---(Tc -T) u, 0.1
0,·· -0.11'
.
-----·-··---1
r
.-. :-~ .~.:.-~ .-.,....-~-
-0.2/ >"~
2
,
~l]=[[-I~DaeXP(l+:~:rY)
x2
DaCn
10
DaRn]J . [Xl] +
BDaR
X
n
]
-1-13 +BDaC n
-BD a exp(-"-) 1+x /y
2
[0]
J3 ·u
en and Rn are the cone center and cone radius of the nonlinear term, which are decided by the nominal point and its bounds. d is the uncertain factor introduced by linearization and is bounded between -1 and +1 (Schaper et at, 1992). In this example, a unifonn probability distribution is assumed for d and 5 points are taken into account for PRO design. Numerical parameters chosen for this example are D a = 0,072, B :;; 1.5, ~ = O~3 , Xv == [0.3,2.45]',
-, E -CA)-KoC A exp(--) RT
dT
+Do(l-xl)exp(]+::/y)
f3 )x 2 + BDa (l-x1) exp(-1+x )+ J3u x2 / y
0 +d· [ 0
exothennic reaction A ~ B, occurring in a constant volume, continuously stirred tank reactor: dC A
=-Xl
The non linear model is transfonned into linear model by cone-sector bounded Iinearization. The resulting model is expressed in linear state equations with uncertainty.
Example 2 Consider the most commonly used reactor model that describes an irreversible,
--=~(CAF
L:J,.
·J pRO
,;t'
>
:;::" /
en == 12.51 ~ Rn = I 1. 0 I. Take y ::;;:; x l ~ Q = I , R = I, Tf == 10.0. Simulation results are shown in
r-
Fig. 2. The designed feedback control matrices jn terms of 3 different methods are: K nominal::;:: [~O.16,O.43] ,
~::l
~O.7
.0.8)
I
-0.9 l
L
o
0.5
_
1IlOmlna4..2:eRQ•. :brurumax.f ~ 3.5 4 4.5 5 5.5 6
. . . ..J_ ._....
1
1,5
2
:2.6
K minimax = [-1.59,3.86] K PRO ~ [0.19,1. 71]. The uncertainty range is large in this case. It is not acceptab le to use nominal design because it leads to unstable system responses in some cases (Fig.2 (a)). Both PRO design and minimax design are able to guarantee robust stability and provide consistent perfonnance in the whole uncertainty range (Fig.2 (b) and (c). However, rninimax design is more conservative than PRO design in most cases considered. In Fig. 2 (d), the LQ perfonnances from PRO design are better than the results from minimax design in the range from -1.0 to O.85~ with only small loses in the worst cases nearby (0.85 to 1.0). 'l
t
(a) U1 vs time U2
0.1
0 1
-0.1:
,:1'
:::/ /
:::1 / / I -O.S
i
_0.7
-o.ef
I
-0.9'
o
\no~~~~ ?:.~~9.~p~~~~i~~.J
L.
0.5
1
1.5
2
2.5
(b) U2
3
3.5
4
4.5
5
5.5
6
vs time
Fig.I. Optimal control protile in example 1
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Copyright 1999 IFAC
ISBN: 0 08 043248 4
PROBABILISTIC ROBUST OPTIMIZATION OF TWO LQ CONTRO...
2.5-
5i
2\
.t
14th World Congress ofIFAC
The proposed methods are suitable for systenls with large parametric uncertainties. Although it is computationally intensive compared to hard-bound robust optimization design, requirements are well within the capabilities of existing computers, It should be mentioned that PRO design might be effective even when the soft-bound information is not exact, because the selected uncertainties are incorporated and the overall objective is considered at the time of the problem fonnulation.
1.5i';~a1~~~
"~/l~'~ 2.
0.5.
.
o _~ ... J:..~_~_ .._...
-"O
I -0.5'-"
__ .L._---
o
! __ .., ----. -,--.-
--, ----.
1
2
4
:3
.~ - - - - - " - _ .. __ --"5 6 7
-'
I
--_._.1-.
8
9
10 t
(a) nominal design
_.- .. - - - "-- ..., :d...-1. 2:d=-O.5, 3:d""O. 4.d=,O.5, 5:d~1
,--.
REFERENCES '.' 5
Djavdan, P., H. J. A. F.. Tulleken) M. H. Voetter, H.. B. Verbruggen and G. J. Olsder (1989). Probabilistic robust controller design. In: Proceedings o.f the 28'h Conference on Decision and Control, Tampa, Florid~ pp. 2164-2172. Frey, H. C. and E. S. Rubin (1992). Evaluate uncertainties in advanced process technologies. Chem. Eng. Prog.~ 88(5), pp. 63-70. Howitt, G. D. and R. LUllS (1993). Control of a co Ilection of linear systems by linear state feedback control. Int. J. Control, 58 (I), pp. 79-96. Janiszowski, K. B (1987). Sufficient condition for stable controJ of discrete-time systems with statistical process modeL Int. J. Control, 45(2») pp. 693-700. Ruppen, D., C, Benthack and D. Bonvin (1995). Optimization of batch reactor operation under parametric uncertainty--computationaI aspects. J Proc. Contr., 5(4), pp. 235-240. Schaper, C. D., D.E. Seborg and D. A. Mellichamp (1992). ProbabiJistic approach to robust controL Ind. Eng. Chem. Res., 31(7), pp. 1694 - \704. Stenget R. F.~ L. R. Ray and C. I. Marrison (1995). Probabi listic evaJ uation 0 f contro1 system robustness. 1nl. J. Systems Sci.~ 26(7), pp. 1363]382. Sternad.. M. and A. Ahlen (1993). Robust filtering and feedforward control based on probabilistic descriptions of model errors. Automatica:> 29(3), pp. 661-679. Terwiesch P. and M. AgarwaJ (J995). Robust input polices for batch reactors under parametric uncertainty. Chem. Eng. Comm., 131, pp. 33-52. Terwiesch P., M. Agarwal and D. w. T~ Rippin (1994). Batch optimization with imperfect modelling: a survey. J. Proc. Cont!:, 4(4), pp. 238258. Vue H. and W. S. Jiang (1996). A new probabilistic robust optimization method. In: Proceedings of/he IEEE International Conference on Systems, Man and Cyberne!ics~ Beijing, Chin~ VoJ.3, pp. 22052209.
I
___ (
_------.: ..
3
I
4
0
' - - - __ ...l.. ~----L--l-.-----1
6
7
8
9
10
(h) PRO design y 2 y
'.5.
f-
5
It -{
1'.··'~·
'~r' '~\4
05~·
•.·"
ol"~~'-'--'::~., -----'
2
1:r-
3
_
4
... ..L...._.__ J.
5
. .....L.
I 1
• _ _ ._.....L.•. _.,.!
a
6
9
10
Cc) minimax des;gn
.J
1:liom;naJ, 2:PRO, 3:T1'!.inJmax
80
70 .
:r 30
1(1c
o
_ _-
~2
'--~_"'----J-_.L--.........L-.~-'----..L.-~.l..----I..-_ ..----1
-1
-0.8
-O.S
-0.4
-0.2
0
0.2
0.4
0.6
(d) J vs uncertainty
0.8
d
Fig.2 Simulation results in example, 2
5. CONCLUSION
In this paper~ two probability robust LQ optimization problems are fonnulated and solutions are provided. 1t combines the following important features. Firstly:> probabilistic perfonnance tradeoffs are made between high likelihood conditions and low likeljhood conditions. in the uncertainty range. The design emphasis is thus more reasonable. Secondly, the design in the whole parameter space is taken in its entirety so that the best overall perfonnance is achieved, The results obtained from numerical examples demonstrate the vje~rpoints.
3604
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