Simultaneous Modeling and Data-Distribution-Dependent Robust Control System Design

K :::;

L,

where

Consider the condition

3. GROUPING OF DATA as an example of F(Po , C , {W,}, 8, £) ,< I, where S is a sensitivity function and {Ws, Wes} are

In a general case, Problem 2.1 is difficult to solve strictly. One of the modified methods is to use 144

iteration between the determination of Po and the design of C. In order to analyze the relationship between design parameters in DRCP, an algorithm is induced, which is composed of as few calculation processes as possible by employing simplifications of the representation of plants or controllers. Let plants P be stable, SISO and discrete time systems or continuous time systems, and their transfer functions are represented by polynomials of a stable operator ( as

.......... ....................................

Fig. 2. Grouping of {Pj } (2)

grouping also. The constraints (7) are reduced to LMIs for the parameters {pf} by employing the canonical controllability form {A j, B j, C j, Dj} of

The following are examples of (. • Discrete time system z-l, (1-z- 1)/T, (z-I_ A)/(1-AZ- 1), where z is a time advance operator, T is sampling time and IAI < 1.

(8) as AJ

.- [ JJ

• Continuous time system

l/(s

+ a),

J j .-

> o.

Suppose that several data sets of the coefficients Pi (i = 0,1, . . . , t) corresponding to P are given by identification processes or a priori information, e.g., as

(3)

0

:lf

'

E R(W-l)XW,

(9)

where K j is a constant row vector comprised of the coefficients for Zi or Si in the denominator of (Wand the numerator of W esK , and the elements of C j and Dj are the coefficients for zi or Si in the numerator of WC~K(Pe - P j ) and they are affine for pf . Let

as {pP), p(2), ... ,p(h)}. Employing the bounded real lemma (see, e.g., (Boyd, et al., 1994; Stoorvogel, 1992)), the condition

Hence the processes of obtaining the data are not restricted for the purpose of generalization of the current discussions. Next define truncated functions corresponding to the data sets as

+ p~ ( + ... + p} (t , p~ + p~( + ... + p;(t,

PI (() := p~ P 2(() :=

[:

E R WXW

B j := [0 ... 0 1 E R wX1 , Cj(Pe,Pj ) E R 1XW , Dj(Pe,Pj ) E R, w : = max { n, t} + r, j = 1, 2, . . . r: order of W esK

I PI1, . .. , PtI} , {2 {Po, Po, PI2, ... , Pt2} , N PIN , ... , PtN} . ... , {Po,

J

1

s/(s + a), (s - a)/(s + a), where s

is a time derivative operator and Re(a)

KTf

iiWc~K(Pe - p(j))iioo < 8K, Vp(j) E



(10)

>0

(11)

is equal to

(4)

Mat(
In preparation for solving DRCP, the data sets {Pj } in (4) are divided into several groups such as

M(j):=diag(M(j),X(j)),

j=1,2, ... ,h, (12)

and M(j) corresponding to p(j) is given as -

where 81< (11: = 1, 2, . .. , L) are the radii of Cl< (Fig. 2).

M(j):=

-T

1

M(j)3l M(j)32 M(j)33 j=1,2, . .. ,h,

Let Pc be for a central model of Cl< as

Pe=pf+pf(+· .. p~(n.

[

-T

M(j)ll M(j)2l M(j)31 M(j)21 M(j)22 MG)32

(6)

(13)

M(j)ll := X(j) - AG)X(j)A(j) ,

The additive uncertainty form will be considered for U(Pe , P j , {WiK }) in the following sections; therefore the condition

M(j)21 := -BG)X(j)A(j), M(j)22 := 8K - BG)X(j)B(j) , M(j)3l :=

C(j)(Pe,p(j)),

M(j)32 := D(j)(Pe , PUl ),

(where the weighting functions Wc s" need not coincide with those of DRCP) is used for the data

(14)

145

for discrete time systems. For continuous time systems, the similar matrices are defined also and the following arguments are applicable. Note that the inequality (11) is affine for {pr}, X(j) and 0,. .

Here, to analyze the structure of the relationship between design parameters in DRCP, it is preferred to maintain linearity of parameters as much as possible; therefore a nominal model Po and the free parameter Q of a controller are also represented as

Define EP( o. Let GNj := G N - bY} , pi E EP(G N ) and define G N - 1 as

Po

(15)

4.2

Mat(EP(G Nj ))

> O.

Constraints with state space representation

diag(Mat(EP(Gd), Mat(EP(G z )), . . . , Mat(EP(GL))) > 0, (23)

(16)

For all I\, (= 1, 2, .. . , N), G,. and the minimum 0", are defined sequentially and they satisfy

Next some {G,.} are selected from {G 1 , G 2 ,

where Mat(EP(G K ) ) is similarly given by (11) (14). Note that (23) is also affine for {pil, X(j) and OK.

'"V

To derive conditions corresponding to (21), a canonical form of state space representation is employed, as in Section 3. Let the canonical controllability form {A"" B"" GK , D K } of the mixed sensitivity function of (21) be

.. • ,

G N } appropriately and renumbered as {G 1 , G z ,

. . . , Gd (e.g., their 0", have large gaps between then reset the corresponding W s'" and W es ", (I\, = 1,2, . .. , L) in DRCP.

o",+d;

A",

4. SIMULTANEOUS DETERMINATION OF NOMINAL MODEL AND CLOSED LOOP SYSTEM DESIGN

:=

1",

Let U, Fin DRCP be the additive error function and the mixed sensitivity function respectively, then , the conditions in DRCP are represented as

IIWc~,.(Po - Pj)lloo < 0"" 'VPj EG K,I\,=1,2, . .. ,L

t

V2£::;~CS 1

(18)

< 1,

:=

[0 1 °1] 0

r

E R(v-1)xv,

E R VX1 ,

G",(PO,Q,EK,O",):=

[~G"'l(Po,Q)l

E

D",(Po,Q,E""O",):=

[~D"'l(Po,Q)l 20", D",z(Q)

E R ZX1

V20K GKZ(Q)

R 2xv ,

,

v := n + m + r, (24) where K", is a constant row vector comprised of the coefficients for zi or si in the denominator of (V, Ws", and W eS "'. Note that the elements of G K1 and D",l are the coefficients for zi or s i in the numerator of Ws",(I - PoQ) and they are biaffine for pi and qi· The elements of GKZ and D",2 are the coefficients for zi or Si in the numerator of WesKQ and they are affine for qi.

1 :S,:S L (19)

(20) Then the norm condition (21) is equivalent to the inequality

Substituting Sand G into the inequalities (19), the following well-known form is obtained:

where E", = El , 1 ::; I\, ::; reduced to the following.

,

V

S .1 G Q .- 1 + PoG' := 1 - PoQ

es",Q

V xv

B", := [0 ... 0 1

where E", = El , 1 ::; I\, ::; R ::; L, and G", is a member of {G 1 , G z , ... , Gd. The sensitivity function S and stabilizing controller G in (19) for stable Po are given as a parametrized form with Q E Hoo as follows (Doyle, et al., 1992) .

V2o",W 1 1[ ~ws"'(I-PoQ)ll

[lJ KJ] E R 0

Mixed sensitivity minimization conditions

11 [

(22)

As in Section 3, the condition (18) is reduced to

subject to

4.1

= pg + pr ( + ... + p~ (n ,

Q = qO+q1(+···+qm(m.

diag(E1, E z , . .. ,EL)

> 0,

(25)

E,. := diag(E"" Y",),

<11
-

-

E"'ll EJ21 EJ31]

,

E,.:=

00

(21) R ::; L. Hence DRCP is

[

~"'Zl ~"'zz ~J32

,

E K31 E",32 E",33

E",ll := Y", - A~Y,.A""

E",Zl := -BJY,.A"" EK22 := I - B~YKBK '

Problem 4.1 Find a nominal model Po and a free parameter Q subject to the inequalities (18) and (21) simultaneously such that El is minimized.

E",31 := CK(PO, Q, 0",), E",32 := D",(Po, Q, OK)' E",33 := diag(E!, 1), 146

C,,(Po , Q, 8,,) := diag(f",l)

X

C,,(Po , Q, fIt, 8,,), Step 1 (determine a nominal model Po) ineq. (23)

D,,(Po,Q,8,,) := diag(f", 1) x D"(Po,Q,f,,,8,,). for discrete time systems. For continuous time systems, the similar matrices are defined also and the following arguments are applicable to them.

I

X(j),

pi,

I

6,.

1

Step 2 (design Q such that is minimized) ineq. (25)

Note 4.1 The left-hand side of (25) is affine for {pi, 8", Y", f~} or {qi, Y f~} and biaffine for {pi, 8,IJ and qi in both discrete time systems and continuous time systems. K ,

I

Y K , f~,

I

qi

t iteration

4.3 Algorithm

Step 3 (move Po such that is minimized) ineq. (23) & (25)

The inequalities (23) and (25) are combined to determine a nominal model and to design a controller simultaneously. However some elements of these inequalities are, unfortunately, biaffine for the parameters, and iterative calculations must be employed to determine a local optimal solution.

El

1.1 biaffine El

I

1

I)

Fig. 3. Schematic diagram of Algorithm 4.1 Let PI '" P4 be truncated functions of discrete time systems corresponding to data sets as

Algorithm 4.1

PI = 10.0 + 6.0z- 1 + 1.0z- 2 , P2 = 11.0 + 5.2z- 1 + 0.3z- 2 , P3 = 10.5 + 6.3z - 1 + 1.2z- 2 , P4 = 7.5 + 4.3z- 1 + 0.7z- 2 •

Step 1: Solve (23) for pi such that max{8,,} is minimized. Set fa = O. Step 2: Fix pI and OK' and solve (25) for Y K, Q and fK such that f1 is minimized. Step 3: Fix Q, and solve (23) and (25) simultaneously for pi, X(j), YK , 8K and f~ such that f1 is minimized. If If 1 - fol < 0 for a given tolerance 0, exit. Else set fa = f1 and return to Step 2.

The central nominal model Po (1), the radius of the largest uncertainty 8(1) and the robust performance f1(1) in the mixed sensitivity function under W s .. = W es .. =l are

PO(I) = 9.4401

Fig. 3 shows a flow of Algorithm 4.1.

+ 5.0434z- 1 + 0.7664z- 2 ,

8(1) = 2.75, f1(1) = 0.06366.

Note 4.2 Step 1 is a process which gives an initial central nominal model Po such that all G" are contained in a ball. Step 2 solves DRCP for the fixed nominal model Po. In Step 3, the free parameter Q given in Step 2 is fixed, and the nominal model Po is moved so as to minimize f1 such that all G K are contained in the corresponding balls with the center of Po. Then returning to Step 2, Q is redesigned for the renewed Po. Refer to (Nesterov, et al., 1994) for solving LMI.

Next, divide PI'" P4 into G 1 = {P1 ,P2 ,P3 } and G 2 = {P1 ,P2 ,P3 ,P4 } using the grouping algorithm given in Section 3, and set W St = 1, WeSt =1 , W S2 = 1/2 and W es2=1, respectively. These settings mean that for all data sets in G 2 , the weight on the performance is relaxed as W S2 = 1/2. On the other hand, the corresponding weight for the data sets in a small area G 1 is large and high performance is expected for them. After 300 iterations of Step 2 and Step 3, the parameters converge to the following and the value of f1(300) after the iterations is about 18 % of the value of f1(1)'

The aim of the algorithm is to include as few processes as possible for solving DRCP under a few conditions in order to clarify the relation between parameters in DRCP. Algorithm 4.1 is a successful example. Note that f1 to be optimized is a free parameter in Step 2 and Step 3 with minimum violation of linearities of all free parameters. Therefore it follows that the sequence of f1 does not increase and the convergence of the algorithm is guaranteed.

Po (300) = 10.0085

+ 4.6250z- 1 + 1.0660z- 2

82 (300) = 7.4584, 81(300) = 2.3003 f1(300) = 0.01161 The gains of Pj , PO (l), Po (300) are shown in Fig. 4. Note that the gain of the central nominal model Po(l) passes through the center of the plots of Pj • On the other hand, that of Po (300) deviates from the center and in particular, in the high-frequency band, it is outside the range of Pj (j = 1 '" 4).

5. SIMULATION In this section, it will be demonstrated that the difference between robust control system design in the conventional fashion and the modified one discussed in this work by showing simple numerical simulations.

Fig. 5 shows the convergence of the performance f1 and Fig. 6 shows the changes of the radii 81 and 82 versus increasing number of iterations of Step 2

147

~',---------------~----~

I

i

:!f ........ / •

th

.,

J!!,f

O"t 3r· ... ... .

."'1

5~ - - - ---- :

2~

,,'

0,

'0

I O~

Frequency (radl.)

i

150

200

250

300

Fig. 6. The values of 61 and 62 versus number of iterations of Step 2 and Step 3

-------------~--__,

of a nominal model and L is the number of groups

O(f'~ 0.05-

100

Number of Iteration

Fig. 4. Gain plots of P o (I), P o (300) and Pj : P o (I), - - : P o (300), ... : Pj 0 .01 ',

50

GK •

".

5 0 .04 " ~

REFERENCES

wO.03:

Boyd, S. P., L. El Ghaoui, E. Feron and V. Balakrishnan (1994). Linear matrix inequalities in system and control theory, SIAM, Philadelphia. Chen, J., C. N. Nett and M. K. H. Fan (1992). Worst-case system identification in Hoo: Validation of a priori information, essentially optimal algorithms, and error bounds. In: Proc. American Control Conj., Chicago, pp. 251257. Chen, J. and C. N. Nett (1993). The Caratheodory-Fejer problem and Hoo identification: A time domain approach. In: Proc. IEEE Conj. on Decision and Control, San Antonio, pp. 68-73. Doyle, J. C., B. A. Francis and A. R. Tannenbaum (1992). Feedback Control Theory. Macmillan, New York. Gahinet, P . (1994). A new parametrization of Hoo suboptimal controllers. Int. J. Control, 59, pp. 1031-1051. Helmicki, A. J., C. A. Jacobson and C. N. Nett (1992). Worst-case / deterministic identification in Hoc: The continuous-time case. IEEE Trans. Autom. Control, AC-37, pp. 604610. Lee, W. S., B. D. O. Anderson, R. L. Kosut and I. M. Y. Mareels (1992). On adaptive robust control and control-relevant system identification. In: Proc. American Control Conj., Chicago, pp. 2834-2841. Nesterov, Y. and A. Nemirovskii (1994). Interiorpoint polynomial algorithms in convex programming. SIAM, Philadelphia. Schrama, R. J. P. and P. M. J. Van den Hof (1992). An iterative scheme for identification and control design based on coprime factorizations. In: Proc. American Control Conf., Chicago, pp. 2842-2846. Stoorvogel, A. (1992). The Hoc Control Problem. Prentice Hall, London. Zang, Z., R. R. Bitmead, and M. Gevers (1992). Disturbance rejection: on-line refinement of controllers by closed loop modelling. In: Proc. American Control Conf., Chicago, pp. 28292833.

0.02O.01 t

Fig. 5. The value of €1 versus number of iterations of Step 2 and Step 3 and Step 3. The matlab M-files for this simulation are available from our anonymous ftp server mec1. icsd7. tj. chiba-u. ac.jp/pub/matlab/indl. 6. CONCLUSIONS

In this work, the importance of the introduced data-distribution-dependent robust control problem (DRCP) in practical fields was emphasized and an example of simultaneous determination of a nominal model and designing a controller for solving DRCP was demonstrated. By simplifying the representation of plants and controllers, a duality of nominal models and controllers was shown in the closed loop system design. This simplification may result in insufficient ability of the model to represent real cases. If rational models are employed, there are many violations of the linearity in the matrix inequalities and some rank conditions are needed (Gahinet, 1994). The extension of the parametrizations or functions in the algorithm to general cases without loss of the obtained simple structure of parameters will be difficult, and a strict solution of DRCP is still open to investigation. One problem of the proposed algorithm is the large number of free parameters. For example, if t ~ n, the number of free parameters in the inequality (23) is calculated using

(t+r) x (t+r+1) ( 2 xj.L+ n+1)+L,

(26)

where t + 1 is the size of a data sequence, r is the order of the weighting functions, j.L is the sum of the number of data sets in EP(Gd, EP(G 2 ), ... , EP(Gd, n + 1 is the number of parameters 148