Rational L1-suboptimal feedback system design1

RATIONAL L1-SUBOPTIMAL FEEDBACK SYSTEM DESIGN...

14th World Congress of IFAC

F-2d-08-2

Copyright © 1999 JFAC 14th Triennial \Vorld Congress~ Beijing, P.R. China

RATIONAL ll-SUBOPTIMAL FEEDBACK SYSTEM DESIGN

1

Mark E. Halpern

Dept of Electrical and Electronic Engineering

the Univerrsity of Melbourne Grattan Si J Parkville" Vie 3052) ..4ustralia. m,. halpern@ee. mu. oz. au) i. evans (rYee. m·u. oz. au

Abstract: A difficulty v:.rith the 11 Inethodology is that optiInal solutions can be of arbitrarily high order. In this paper, the use of a rational signal model is proposed for obtaining rationalsuboptinlal solutions of low order to scalar II problerns. This model allows an arbitrary nunlber of distinct positive closed-loop poles to be incorporated. Solving an interpolation constrained II optimization problem then involves t\VO finite linear programs and gives a solution which Inake~ use of only the best out of all the prespecified poles. Using the signal model in a scalar t\V"'o-block problem gives rational suboptimal solutions in a simple procedure invol"ing four linea.r programs. Copyright © 19991FAC Ke}T\Vords: Discrete-time systems, optimal control, pole assignment, linea.r programming.

1 INTRODUCTION The problem of optinlal rejection of persistent bounded disturbances in feedback systerns \-vas posed and solved by Barabanov and

Granichin (1984) and independently formulated by \Tidyasagar (1986). A subsequent solution proposed by Dahlch and Pearson (1987) used linear programrning and duality theory to solve an II luinimiz.ation problem subject to interpolation constraints. Dahleh and Peal·son (1988) found a sirnilar rnatherrlatical problern arises in the stability robustness context. It is well kno"\vn that this one-block problem has a solution "vhich has flIlite irnpulse response (FIR). It can be obtained by solving a linear program (LP) corresponding to the primal problem or another corresponding with the dual probleul. A difficulty lh.-ith the 111ethodology i~ that the order of the optimal solution ca.n be arbitrarily high, in contrast to the case Vl-rith 12 and Hoc optimisation problerns where the 80IThis "\vork Vo.ras supported by the Australian Research Council.

lutjon order corresponds with the number of in-

terpolation constraints. Another issue is the possible nonexistence of sol utiOllS to 11 opthnization problems containing interpolation on the ::)tability

boundary; see (Barabanov and Granichiu, 1984; Vidyasagar, 1991). Controller implementation in the pre::;ence of such difficulties requires some relaxing of the requirement for II optimality. One simply obtained suboptimal solution is the best FIR solution of given order, found by solving a truncated version of the primal or dual LP. In an effort to obtain an improvement in terms of a smaller It norm than is obtained using a reduced order FIR solution, a single real positive pole \vas incorporated by Halpern, et al. (1996) into the solution of the scalar problem, giving the solution a rational rath€r than polYllQluial transfer function. Both the primal and dual optimization problems

remained finite linear programs.

The approach

could be applied directly to the problenl of l1 optimal design with one prespecified pole. Also, it Vla.s shown that, for a simple example, where the op-

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Copyright 1999 IFAC

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RATIONAL L1-SUBOPTIMAL FEEDBACK SYSTEM DESIGN...

tinlU111 cost could be calculated analyticall~y~ simple calc\lhlS CDuld be applied to the dual to find the exact pole location to mininlise the 11 norm. Other than in such simple cases, finding the best pole location required repeated solution of a LP~ using a different pole value each tirne. The case \vith two poles, real and complex is developed by HaIpern (1998). There again, results are largely restricted to pole placement. A different approach for the design of red uced order controllers in an II context is by Blanchini and Sznaier (1997).

In this paper the approach used by Halpern, et al. (1996) is extended to allow the incorporation of more than one pole iIlto the solution and sinlultaneously to aid in the selection of poles. A signal nl0del is proposed, comprising an unknown FIR part together v;,rith an arbitrary number of real prespecified positive distinct poles under a restriction on the signs of their residues, whieh are other"\-vise unknown~ Applying the 11 optinlization approach to this signal model yields) for a probleIIl with q interpolation constraints, a solution involving at Ulost q poles, giving an element of optimal pole select-ion to the approach. A consequence is that one can obtain good results V\,rithout an lnfonned selection of the poles; a large number of candidate poles can be incorporated into the problem and solving the two linear programs automatically chooses the best nuu1.ber of poles to include and rejects the others by setting their residues to zero.

In the 11 context, the theory for multibio ck problems is less complete, and obtaining the optinlal solution can be rather difficult; they are usually obtained from a sequence of suboptimal solutions. There are several methods for obtaining suboptimal solutions. They generally involve ohtainillg upper and lov.rer bounds on the optimal cost as the order of an approximating solution is increased in some manner. \Vhen the upper and lo'\ver bounds are close enough, the solution corresponding with the llpper bound can [>e used, but tlluis may involve a high order FIR a.pproximation and these require high order controllers, even though the true optiInum controllers could be of lov.r order. The problenl was first posed by by Dahleh and Pearson (1988); Sta.ffans (1991) introdUCred the terminology, finitely many variables (F~{V) to describe the truncated prinlal miniInization and the term finitely many equations (FI\1E) to describe a truncated dual problem; sce also Staffans (1992) and used these pairs of solutiQIls to find true oprimal solutions. A detailed exposition of the issues involved is by Diaz-Bobillo and Dahleh (1993) where the delay augrnentation (DA) algorithm ~~as introduced for the general multiblock problem. A related approach is by Casavola (1996). An a1-

14th World Congress ofIFAC

ternative approach is by Barabanov and Sokolov

(1994)Finding the optinlal solution to avoid controller order inflation requires determining ,vhich components of the solution are finitely supported. There are at least two difficulties with previous approaches. Firstly, there are only limited results indicating which types of problem do have FIR. COH1ponents in their optin1a1 sol utions. Secondly, identification of the FIR parts of a solution relies on observed behaviour of iterated solutions. Nreanwhile, Staffans (1991) has shown with a simple example that the optirnaJ solution can have very high order \vith miniscule performance improvement over a much lower ol~der solution. This fact provides strong motivation for considering suboptimal low order solutions.

The luet.hod proposed in this paper is to use the signal formulation used in the earlier part of the paper; applied to the multiblock problem~ it is in effect a modificatioll of the F1\1V approach and is also along the lines of a suggestion by Diaz-Bobillo and Dahleh (1993) to incorporate the kIlO"'~ poles into the solutions_ Since it is not known a-priori which, if any~ components of the optimal solution will be rational, but is expected they could have poles from the weighting functions, the solution is allowed to contain these poles, along with others if desired. Solving the primal linear program then selects the best of the poles fi'onj those speciiied. The approach gives automat3c selection of ,vhich~ if any, parts of the solution are FIR and readily allows all components of the solution to be rational.

2 PRELI.. \II!'\ARlES The z~transform, h , of a sequence, h == {hi}~O~ is defined by it === L~o hiz i . With this definition, a stahIe transfer function has all of its poles at values of z : jzl > 1 . The symbol z alsQ denotes the unit delay. The 11 nornl of a sequence h is denoted !~hlll and is given by !lhll 1 == 2:~o Ihil· The scalar l1 optimisation problem can be (see Dahleh and Pearson 1987): •

v;;;;;=

::Xl

~

mIll I

r:p

V\7 ritten

L-t lq)~j

E l

(1)

1 i=O

subject to

4>{aj)

where aI, ... ~ a q are given distinct complex nUInbcrs, all satisfying Iai ~ < 1 and the bj '8 are given coruplex nUlnbers, not all zero. The bj 75 corre-

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Copyright 1999 IFAC

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RATIONAL L1-SUBOPTIMAL FEEDBACK SYSTEM DESIGN...

14th World Congress of IFAC

sponding with real Zj '8 are also real. Any complex aj ~s occur in conjugate pairs~ as do the corresponding bj ~s. It is well kno\vn that the solution to (1 ~2) has an FIR z-trallsform, namely ~rith the

form

1Vo

rj,(z) ~

L ~izi i=O

where No is sorne integer greater than or equal to q -1. The value of ]'10 can, in some eases be much larger than q - 1 and it is this fact that motivates the work here; solutions vlith orders in the range q -l~ ... 1 No -1 are sought, with II norms as small as possible. One simple approach is to insist the solution has the forill ~(z) :::::: 2::0 d;,'izi for some chosen N E {q - 1, ... ,Jo7\To - I} and then to solve a linear progranl for the values of CPo, ... ,r.P1V. In thi~ paper an approach, using rational rather than polynornial solutions J is presented. 2 .. 1 One real pole One real pole was incorporated by Halpern, et al. (1996) into z) by insisting ;p (z) took th c form

J(

....

'1fl,-l

L

==

4'(z)

.

c:Pi Zl

i=O

4Jrn zm

+ -1- + VIZ

(3)

Here rn, is chosen by the designer in the range m E {q - 1 ~ ... , ]\70 - 1} to give a desil'ed aIllount of overparametrisation. A positive stable pole has VI E (-1,0]. The l1 norm of rP of (3) is given by

(4) FOT

given

Vi

,.rn,

the problern of mininlising ~l cPl\ 1

is V

=

nlin ,

q;o, .

4

•

;

~

Yrn

~ll.-f"r ~ I~i( i=O

11Jmi + VI

4- 1

setting m-I

J;(z) =

n

(p;Zi

-+

Z"m

i=O

L

1

i=l

Ti

+ ViZ

,

(7)

"vhel'e~ for convenience, the v~s arc ordered according to -1 < 'VI < < V n ::; 0 and collected into a vector .Q ~ {VI, , V n }.

Unfortunately? if n > 1 1 there is no single expression corresponding V\.rith (4) for 1I q)1~ 1. For the tVlO cases where all Ti 2: 0 or all Ti ~ 0 and only those, m-I

114>lh

=

L

fr.f

n

l1>il +

i=O

L

1 +t V

i=l

.

(8)

1-

Accordingly, in thi8 paper, the signs of the residues t\VO cases where all r i 2:: 0 or all ri ::; O. Halpern (1998) has shown that there is a considerable increase in problem complexity and a large nurnber of different LP's nlay be needed if different signed residues are allowed, even with only t\VO poles, Le. n == 2.

are restricted to the

3.2 One-block problelll Here the signal model (7) is incorporated into the solution of a scalar il optimisation problem subject to q interpolation constraints. The problem ofminirnising ~1m-l, rl ~ ... ~ T n · The problem ,~.rith ri :::;

(5)

L

all

Ti

2:: 0 is named LP 1a, and that vvith all

0, LPlh. Problenl LPla is shown below.

LPla:

subject to interpolation constraints,

~1

i,

~ cPtaj -ri=O

r.Pmaj" 1+

viaj

(9) == b j ~

j = 1, 2, ... ,q. (6)

This can be v,rritten as a finite LP and solved numerically. 11oreover it has the same form as (1,2) and is a weighted II norm minimisation. This fact allo\vs its dual problerll to be simply obtained as sho1vn by Halpern et al. (1996). For simple problems~ using duality enables closed form expressions for the optilual pole location to be obtained.

3 RESULTS

subject to

where)

==

Similarly

1 2? .. ,q. 7

Vlb

is defined as the optimal cost for

LPl b, obtained by changing the constraints on all ri to ri ::; o. The final result is obtained after solving both LPla and LPl b: VI

3.1 Signal Model Here n real, distinct, stable positive poles fixed at z == -llvI, -11v2 ... ~ -l/v n are incorporated, by

= min

[Vla, Vlb}

and the optinlal variables are those corresponding with the smaller of

VIa, Vlb.

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

RATIONAL L1-SUBOPTIMAL FEEDBACK SYSTEM DESIGN...

14th World Congress of IFAC

A sufficient condition to guarantee the existence of a solution to both LPla and LPlb is that the number of unsigned variables is greater than or equal to the nUlnber of equality constraints, that

and

is

Tbjs problcnl can be written, see (Dahleh and

m

~q.

(11)

Note that in some cases, sol utions will exist to LPla or LP} b \vith m < q~ l'he existence of 80lution~ to both LPla and LPlb is not needed; indeed a low order solution viola.ting (11) may be desirable; ho\vever the sign constraints on the ri ~s make it difficult to guarantee a-priori the existence of any solution unless (11) is satisfied.

PC

'T(z) == 1 + PC

Pear~on,

1988; Staffans, 1991,1992) rau'

v2

or

=

min

~I, ~2 E L1

Ifep ~ I

3 . 3 Mixed sensitivity miIlilnization The problem of minimizing the II norm of mixed sensitivity was first posed by Dahleh and Pcarson (1988). For this problem, a rational plant P(z) \vith rt p distinct unstable poles al, ... ,anp and n z distinct nonminimum phase zeros anp+l,"" anp+n~ is given. Also given is a pair of stable rational weighting functions ~Vl (z), Ml2 (z), which, to simplify the presentation~ is taken to be all-paler that is

W 1 (z) = 1-V2 (z)

=

D1~Z)'

z = 9?l,:~nE it Il:~ 111

subject to q

:::=

n p + n z interpolation constraints,

(14)

Of,

and to a set of ne convolution constraints obtained by equating like powers of 2 in

Note that t as pointed out by Staffans (1991) the constraints (15) which are equivalent to S+T ::::: 1, when taken together \vith the interpolation constraints (14) on S, imply T satisfies corresponding required interpolation constraints. To incorporate the signal paper~ set

n~odel

proposed in this

and

1

1 D (z) 2

min 11;F.. ...... 'I1 '±"l ':1:"2 ' 1 Z b I~l"IZIIlg C()

i1

where :l"n.~ p) n, V1 ,Vn are prespecified and the variables are (<1.>1)0, ... , (])11~-]' (rl)l,' .. ' (rl).n~ (q,2)O, ... ,(cP2)p-l and (r2)1, ... ,(r2)n, all real scalars.

\vhere DJ (z) is a polynomial of order ndl and D 2 (z) is a polynomial of order nd2. The problem is to compute a suboptiulal solution to the t,"'o-hlock ro\v problem

~ta

jJ

:2

1

V

It is easy to show that, like for the standard II norm minimisation (where there are no restrictions on the signs of the variables), at most q out of the variables t/Jo, ... , rpm-I, rI, ... 1 r n can be nonzero in the solution of LPla or LPlb. This means that if enough poles are incorporated so that n > q, then some of the prespecified poles will not appear in the solution since their residues v.rill be zero. In other "vords, solving the LPl's automatically selects the number of poles and the best ones froIn all the Vi to comprise the solution.

1

(12)

•••

r-.rhe interpolation constraints (14) are like those (10) for the one-block problenl. To obtain the convolution constraints~ (16, 17) are substituted into (15) to obtain, after multiplying both sides by all denominator terlns~ the following equation \vhich is polynomial in z:

alternatively,a suboptimal solution to the two-

block column problem

v 20! = where

~l

stabili:~: C(z) /I:~ 111

= W j Sand 5(z)

cP2

=

(13)

== W 2 "T where

1 +lpC

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RATIONAL L1-SUBOPTIMAL FEEDBACK SYSTEM DESIGN...

14th World Congress of IFAC

subject to

J>CO.95)

==:

0.8,

~(O.8) ~ O.7 J

nCl + n

==

ViZ)~

(18)

The optinlal solution

i=l

Equating like powers of z gives ne convolution constraints 1vherc ne ~

1

n -

+ max [ndl + m) nd2 + pJ .

(19)

For the Ininimization of either (12) or (13)~ the optinlal solution with structure constrained to the form (16~ 17) is then obtained by solving four linear programs, and choosing the best solutioIl. These linear prograrns are narned LP2a-d and are obtained by choosing all possible sign restrictions on the residues 1 namely

~(z)

(r2)i

~ O}~

LP2d: {(rl)i ~ 0, (r2)i

::s O}.

(22)

=

2.862 - 4.000z - 2.026z 3

(23)

glvlng I: lfJII 1 ::= 8.887. Applying the approach in this paper, m is set to m ::= 3 and !!.. to 1: ~ {-.9,-~8,-~7,-.6,-~5,-.4,-.3~-.2}. Solving LPla and LPlb gives

LP2a: {('l)i 2:: O~ (T'2)i ~ a},

~ 01

+ O.8785z 16

giving !I d>i! 1 ==::: 2.700. _t\ny other solution satisfying the three given interpolation constraints lnust have a larger l1 norm. v\Ti th IV === 3, the optimal truncated FIR solution is

';"

LP2c: {(rl)i

i~

ri(z) ~ 1.0636 - O.7584z 3

Q:J(Z) , gIvIng i~q)lll

LP2b: {(rl)i 2: 0, ('2)i SO},

Q;(O.6);::::: 0.9.

and

~ 1.288 -

:::

1.211z

2

.1023z3 ~ 1 - .9z·

+

3.52 lvith an 11 norm considerably

less than that of (23). 4.2 2-block problem This subsection contains several examples which use the follov.ring plant and \veighting functions:

As an example, problem LP2a for the row problem; cost function (12) is outlined belo\v:

P(z) = (z

_Z.(;)0 ·~).5Z)

LP2a:

L

{('1)o,··-,(~~;!'Jp-l'

h 2:: D, .

< •

, (

l(tPl)il+

i=O

)11.~O, 1""2 )y:;! 2: 0 }

(rll12: 0 ,·· .. (7"1 ( 7'2

UT ( ) _ 0.02 VV () 1 z - 1 _ .2z' 2 z

m-I

ITlin

, .. row _ V2a -

subject to q interpolation constraints (14) and ne convolution constraints from (18).

Like for the one-block problcID in order to gual·antee the existence of solutions to all four problems, sufficient unsigned variables are needed, namely (assuming al11~i = 0), 1

+p

2: q

~

p

where p is a positive scalar. Clearly, q = 3 and == n'd2 == 1. For all the follo\"ving examples Q is set to Q :=: { - .6, -.2} so that n ::::;; 2. Satisfying (21) requires m ~ 4 and p ~ 4.

The first example considered here was posed by Dahleh and Pearson (1988) and Staffans (1991) of minimizing [JcPl~2111 with P ~ 1. Setting m ::::= p == 4, and solving LP2a--d gives the exact optimal solution~ found by StafI"ans (1991):

~ C?') - ~ _ 113z 1 -

+ rnax [ndl + rn, nd2 + p]

~

q

+ ndl,

O.004p 1 _ .6z

ndl

(20)

rn

~

m

~

q

+ nd2·

-

50

~ (z) _ 287z _ 1531z

(21)

4 EXA1JIPLE

2

-

5625

450

2

18750

~

2

+

23z 45 ~ 3

8029z 281250(1 - 3z/5)

The second example is that by Staffans (1992) and Diaz-Bobillo and Dahleh (1993) of minimizing

4.1 i-block problem The following simple problem is considered here: this tiIne 'vith p = 6. Setting rn :=:: p ~ 3, (violating (21)~ but nonetheless allo\ving a solution to

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ISBN: 0 08 043248 4

RATIONAL L1-SUBOPTIMAL FEEDBACK SYSTEM DESIGN...

14th World Congress of IFAC

be found) and solving the LP2's gives v:= 1.04135 with A

~l(Z)

:=:

0.02 - .258547z + .598851z 2

...

2


Casavola, A. (1996). A PolynolniaJ Approach to the 11-1v1ixed Sensistivity Optimal Control Problem. IEEE Trans. Automat. Contr., 41~ pp. 751-756.

~131163z3 -

----

1- .2z

.0538629z 3

1-

~62

Dahleh~ I\1.A.

and J.B. PearsoIl (1987). [1_ Optimal Feedback Controllers for :rv1IlVI0 Discrete-Time Systems. IEEE Trans. ..I4utomato Contr., AC-32~ pp. 314-322.

.

The corresponding controller is of second order, nanlely:

Dahleh, I\1.i\. and J.B. PearsoIl (1988). OptiUlal

C(z) = 2.625 - 2.567z + 0.627z 2 1 - 3.127z + 1.255z 2

Rejection of Persistent Disturbances, Robust Stability, and Mixed Sensitivity l\1in-

iznization. IEEE Trunsp Auto1natp Contr., 33, pp. 722-731,

Finally, using rn :=::: p ~ 4 gives v ~ 1.00263 with a third order controller as by Diaz-Bobillo and Dahleh (1993).

Diaz-Bobillo, I.J. and l'v1. ..L\. Dahleh (1993)~ 1\1inimization of the Jvla...ximum Peak-ta-Peak Gain: The General l\tlultiblock Problem. IEEE Trans. Autornat. Contr., 38, pp.

Since the approach is based on FMV, there is no Heed to order the components of the solution. Also, there is no need to determine which rows in the solution are active.

5

1459-1482.

Halper 11, lvLE., R.J. Evans and R.D. Hill (1996). Optimal Pole Placement Design for 8180 Discrete-Time Systems. IEEE Trans. Aut01nat. ContT., 41, pp. 1322-1326.

CONCLUSIO~

A scheme for incorporating an arbitrary number of prespecified real ucandidate n closed-loop poles into the scalar II norm optimisation framework has been given. For onc-block problelns the approach can be used to obtain reduced order suboptimal ra.tional solutions, obtained by solving a pair of linear prograuls, which in effect select the best poles out of the prespecified set. For twoblock problems, the Inethod is a fairly natural one for allo1ving solutions with infinite support and allows rational suboptimal solutions to be obtained rather HiInply by t:lolving four linear progranlS.

Halpern, 1-1.E. (1998). Optirnal Pole Placelnent -in an I} Framevlork. In: Proceed1:ngs of the 37th IEEE Conference on Decision and Control, Tampa, FL, pp. 3311-3314.

j

6 REFERENCES

Barabanov 1 A.E. and O.N. Granichin (1984). Optimal Controller with Linear Plant 1.vith Bounded I\-oise. Translated from Automatika i Telemcchanica, 5, pp. 39-46.

Staffans1 O.J. (1991). i\1ixed Sensitivity" Mininlization Problern~ with Rational l1_ Optimal Solutions. J. Optimiz. Theory Appl.~ 70, pp. 173-189, Staffans: O.J. (1992). 1\111\·10 ll-0 p timization with a Scalar Control. J. Optimiz. Theory Appl.~ 74, pp. 545-564.

Vidyasagar, 1\1. (1986). Optimal Rejection of Persistent Bounded Disturbances, IEEE Trans. Automat. Contr.; AC-31, pp~ 527534. ~1. (1991). Further Results on the Optimal Rejection of Persistent Bounded Disturbances. IEEE Trans. Automat. Contr. ~ 36 pp. 642-652.

Vidyasagar,

A.E. and A ..A.. Sokolov (1994). Geornetrical Solution to ll-Optimization Problem with Combined Conditions. In: Proceedings 1 st Asian Control Conference, Tokyo, 3, pp. 327-330.

Barabanov,

1

Blanchini: F. and lYI. Sznaier (1997). A Convex Optimization Approach to F'ixed-Order Controller Design for Disturbance Rejection in SISO Systems. In: Proceedings 36th IEEE Conference. on Decision, and Control~ San Diego, CA, pp. 1540-1545.

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