Robustness and Optimality: A Dual Performance Index

Cop\l"iglH © IFAC 9th Triennial World Congress lludapc,,1. lIun g.-tn. 19X ..

OPTI MAL CONTROL SYSTEMS-I1

ROBUSTNESS AND OPTIMALITY: A DUAL

PERFORMANCE INDEX M. J. Grimble and M. A. Johnson Intiwlrial Control L'llil. Department 0/ Electronic ami Electrical El1gillef1111g, CI/it'fnitl' of Strathclydf. GfUlXf Str"t, Glasgow. UK

Abstract. A composite performance index is suggested for the synthesis of controllers meeting both system performance requirements and achieving good sensitivity and robustness properties. The index to be minimised comprises the usual LQG error and contro l we ighting terms, with additional sensitivity/complementary sensitivity matrix norms. A solution procedure is presented involving time domain optimality conditions and the use of po lynomial matrix methods. Keywords. Control system analysis; stochas tic systems.

control system synthesis; sensitivity analysis n

then A€:prxm(s) implies A = i~O Ais

INTRODUCTION

i

where Ai€:R

rxm

Let Rrxm(s) denote the set of real rational matrices, then G€:Rrxm(s) is proper if G(oo) is finite and strictly proper if G(oo) = 0, otherwise it is improper.

The weighting matrices of the linear-quadraticgaussian cos t fun c tional are usually chosen to achieve time domain performance specifications.

These may be r ela t e d to the transient or steadystate behaviour of th e system . Techniques such as those du~ to Solheim (1972), Harvey and Stein (1978) and Grimble ( 1981) t o achieve such a weighting mat rix selection are well known . In some

Any rational matrix G(s) can be written in the form of the matrix fractions G(s) = A-l(s)B(s) = Bl(S)A~I(s). Polynomial matrices A and B can be

situations, especia lly where energy conservation

chosen left coprime and such that A is row reduced. Similarly, Bl and Al can be chosen right coprime with Al column reduced.

is an aim, the quadratic cost index naturally arises and the weighting matrices are obtained directly from the problem description.

Adjoint operators are denoted by superscript * In particular define the frequency domain adjoint: w* = WT(-s).

The achievement of good sensitivity and robustness properties i s not an issue usually considered in

LQG control l er synthesis. Howeve r, there has been some interest in this t opi c : Doy le and Stein (1981) examined the stability r ob ustness properties of the LQG controller, Safonov e t al (1981) used

SYSTEM DESCRIPTION AND LQG/ SENSITIVITY COST FUNCTIONS

dynamic weight matrices t o accommodate r obu stness requirements and Francis ( 1983) presented the

A multivariable, linear, time-invariant, system

description is assumed. The optimal closed l oop situation is shown in Fig. I, from which the pertinent system equations are as follows:

solution of the problem to minimise the no rm of the sensitivity/comple me ntar y sensitivity functions. In the sequel , the sta ndard LQG cost functional and the sensitivity cost functionals o f Francis (1983) are combined t o form a dual perf o rmance index. Thus a balance can be achieved between the

Output equation

v + d

y

( 1)

with output y, system response v, and disturbance d.

desire to attain certain time-domain performance

specifications and th e need for good robustness characteristics. Clea rl y by choosing appr o priate limiting values of the weighting matrices either the LQG or optima l robustness problem can be recovered separate l y.

System equation v =

Wu

with W€:Rrxm(s) and input u. Disturbance subsystem

The development of the theory commences with the system description and details of the cost functionals. The time -d omai n conditions o f optimality are then derived. Transformation to the s - domain follows and a controller extracted using generalised spectral factors (Shaked (1976); Grimble (1979». To e nabl e the controller expressions to be utilised, a matrix polynomial system description (K~ce ra ( 1979» is employed to obtain diophantine equa ti ons for the controller

d = Wo~

(3)

with WoERrxqO(s) and white noise input, ~ . Observation equation z = y +

V

with measurement white noise, v.

Controller input equation

parameters.

e

Notation

o

= r - z + n

with refe r ence r, and noise input, n,

Let prxm(s) denote the set of polynomial matrices, 323

(5)

M. J . Grimble and M. A. Johnson

324 Reference subsystem = W

r

l

(6)

1;

with wlERrxql (s), and white noise input, 1;. The zero mean stationary white noise signals r., v, and 1; are assumed to be mutually uncorrelated, with constant covariance matrices Q, R, Q1;1; respectively. (W,W , W strictly proper). o l The LQG Cost Functional The basic formulation for the use of an LQG cost functional occurs in the time domain. The objective being t o minimise the time averaged performance index: J

l

2~ (J~(u)

= lim T->oo

with J~ = E{ (e ,Ql e >H r

(7)

)

(8)

+ (u, R l u )H } m

The usual choice for power spectrum ~n' is the identity matrix corresponding t o white noise inputs. However, the actual noise in the system at the reference node has power spectrum: ~

hence it is more appropriate to define ~n = ~cc Further, from·equations (1) t o (6) obtain (with n = 0):

e

( 17)

o

and v = T(W 1; - V - Wor.) l Hence from (16) and (17) identify th e power spectrum for eo as:

~

where tracking erro r is given by

(9)

e = r - y

the weighting matrices satisfy Q

.-k_

l

= Q1Q1ER

rXr

..-k-

= R1R1ER

(s) and RI

mXm

S~

e e o 0

cc

( 18)

S*

and ·from (16) a\1d (18).identify the power spectrum for v as

~ (s)

(16 )

cc

vv

T~

cc

( 19)

T*

Collating (15), (18) and (19) yields

and eEL~I-T,TJ; UEL;I-T, TJ; E{·} denotes

1

expecta ti on. Frequency dependent weighting matrices may be chosen t o achieve frequency response shaping as demonstrated by Gupta (1980).

z--'"

j

(tracdQ2~

1IJ. -_ j

eo eo

+ Q3~v) ds

whi c h m~y be given th e time domain form:

Sensitivity Reduction

(20)

Sensttivity considerations are usually formulated in the s-domain, thus from Fig. 1 obtain: y = (I+WC )-l d + (I+WC )-I WC (r+n-v) o

0

0

A Dual Performance Index (11)

o

(12)

0

and it is useful t o introduce the transfer between input u and references, viz:

( 13)

G c

Lemma The relationships between W, Co' S, and T include: T

=

WG

( 14a)

c

G = C S c

S C

o

0

=

I - T -1

=

G S

I - WG

=

c

c G (I - WG ) c

c

*",,*tra ce{(S Q2 Q2 S

- J'"

- J'"

(8) m (21)

+ (v,Q3 v \ r

r

*

The basic solution procedure has three steps: (i) The time domain necessary and sufficient condition of optimality is obtained via the usual gradient approach, Grimble (1979). (ii) The optimality condition is transformed to the s -d omair t o ex tract th e op timal cont r olle r equation for Co' (iii) A matrix fraction plant description is utilised to obtain a solution procedure for the controller parameters. These results are e ncapsulat ed in the following the o rems:

where ~n denotes th e power spectrum of the signal n (Fig . 1). This may be r ea rranged to give:

tra ce{Q2S~nS

E{(e 0,Q2 e o\

+ (u,R u\ l

SOLUTION PROCEDURE

j'"

joo

r

L~I-T, TJ or L;I-T, TJ respectively.

Hardy space norm. Generalising tnis to the multivariable case yields a sensitivity cost functional

I

/T

E{(e,Ql e \

( 14d)

IIQ2slI~ + IIQ3TII~ and 11 . 11; defined the

211j

wh e r e Jl T

2 . 1{ 1 hm 2T J T + J T} T->oo

and the Hilbert space inner products are defined 01

functional:

J

J

( 14c)

was chosen to minimise the composite cost

1

Time domain performance characteristics and frequency domain sensitiv~ty considerations may thus be investigated by means of the time averaged dual criterion:

( 14b) .

Francis (1983) formulated the opti mal sensitivity problem for th e scalar case wher e the controller

211j

(21 )

T

( 10)

hence define the sensitivity matrix:

compleme ntary sensitivity matrix T = (I + WC ) -I WC = I_SERrxr(s)

with J2

( 15'

Theorem 1 Time domain condition of optimality The necessary and sufficient cond iti on of op timality for the cost functional (22) with definitions (8) and (21) subject t o th e c l osed loop configuration of Fig. 1 is the g radient

325

Robustness and Optimality: A Dual Performance Index

(35)

condition: «W*QW+R )G'" 1 c cc

W*(Q1'" o + Q2'" cc )(t)

0

Further, write

A-'1A

\it >. 0 where

'" cc '" '"

A A- 1ER mxm (s)

1C

QC

(37)

ro

and define Q1 + Q2 + Q3

Q

and

r

(23)

0

Proof:

(24)

(25)

+ '" vv

0

(26)

rr + "'dd

Wiener-Hopf equation: s-domain solutions

The optimal transfer Gc arising from the timedom~in optimality conditions (23) to (26) satisfies: (27)

G c

and the optimal controller Comay be determined from + G

C

c

o

(I -

WG )-1 = c

(I -

G W) -1 G c

c

(28)

where generalised spectral factors Y, Y satisfy 1 * "'cc and Y-:~ Y = W*--QW + RI YY (29) 1 1 and {.} denotes the causal contribution of the two side Laplace transform. Proof:

Then using equation (29), the control spectral factor may be defined via

* = AroB1 * * [Q1n * Q1n + Q2n * Q2n + Q* Q lB1Aro D1D1 3n 3n

o

(see Appendix 1)

Theorem 2

(38)

o

(see Append ix 1)

+

A* R* R A

(39)

oc n n oc

and the spectral factors Y, Y may be identified 1 as: (40) 1 and Y A- D. The spectral factor D1 can be taken to be column reduced assuming an appropriate choice of plant coprime decomposition. The simplification of equation (27) is as follows: YY

* = '" = '" cc 0

+

R

thus'" (y*)-1 = Y o

(42)

Substituting (42) in (27) yields: Gc

y-1{(y*)-1W*[(Q +Q2)Y-Q R(y*)-1}+y-

1 (43)

To facilitate the calculation of a controller, a matrix fraction representation for the system equations is utilised. Recall (1) to (3) as y

= Wu

(30)

+ Wo~

and reference subsystem (4) as r

= W1l;

(31 )

then in matrix fraction form: Ay = Bu + and Ar = El;

The polynomial system solution procedure has previously been detailed by Ku~era (1980) . The following analysis uses the same basic technique of solution but incorporates certain modifications

appropriate to this particular problem. An innovations plant description can be obtained from (32) and (33) as: o

= DE - Bu

where D is the stable spectral factor satisfying: DD *

+ CQC * + ARA *

(34)

The route to a control spectral factor requires equation (29) to be transformed into a polynomial equation . This is achieved as follows: The cost functional weighting matrices are given the matrix fraction representation.

Q 1

Q~Q1

Q 1

.-...rk_

Q 2

Q Q2 2

Q2

.-...rk ......

Q 3

QQ 3 3 _k_

Q 3

R1

R1R1

R1

Gc

-1 * -1 * -1 Y1 {(Y 1) W(Q1+ Q2)Y}+Y

(44)

Substituting from equations (35) to (41), in reduced form (44) yields:

C~

where AEprxr(s), BEprxm(s), CEprxqO(s) and EEp rxq 1 The matrix fraction A- 1B, A- 1C and A-lE are not necessarily left coprime but the greatest common left divisor of A and B can be assumed stable .

Ae

If one of the following conditions holds: (i) measurements noise null; viz R = 0, (ii) error weighting Q1 and noise weighting R are constant matrices, then the last term in the causal bracket of (43) can be omitted, giving,

Q1nA:1E Rr xr(s) rxr Q A- 1ER (s) 2n c r xr 1 Q A- ER (s) 3n c mxm 1 R A- ER (s) n r

where A Eprxr(s), A Epmxm(s) and A ,A are stable. e r e r rXm mXm (s) Right coprime matrices A EP (s) and B EP 1 1c are introduced so that

G c

-1 * -1 * *-1-1 Y1 {(D 1 ) AroB1Q12D2A2}Y

(45)

with

and A2D;1 i~ a right coprime matrix fraction representation.

The diophantine matrix equ~tion solution is now derived for the particular case where the complementary sensitivity is not costed in the cost functional, namely, Q3 = O. (This is the case examined by Francis (1983». Introduce the polynomial matrices H, G and F as the solutions (degriF < degriDj, i = 1,2, .. . ,m) with respect to F of the bilateral diophantine equations: * *(47) D* G + FA2 AroB 1Q12 D2 1

D~H

FB3

A* R R D oc n n 3

(48)

where B3 and D3 are defined through the right coprime representation

D- 1BA

r

=

B3D;1ERrxm(s)

(49)

A suitable solution is obtained by first calculat ing any solution of these equations and then using a division algorithm to calculate a suitable F which satisfies the row degree requirement. The general technique has been described by Ku~era (1979) . -1

Postmultiplying equation (47) by D2 B 1A and ro -1 equation (48) by D3 Aoc yields:

M. J. Grimble and M. A. Johnson

326

(50) A

DC

*n

(51)

R R A

n oc

Note that -I

D-IBA

A2D2 BIAro

A I c ro

-I

B3 D3 Aoc (52)

Hence adding equations (50) and (51) and using (52) obtains

*

-I

From equation (39), where Q has been assumed null: 3 D - GD-IB A + HD-IA (53) I 2 I ro 3 oc The solution for the optimal controller is derived as follows: Substitute from (47) into (45) then G

c

YI

* -I * -1-1 {( DI ) [D I G + FA2l A2 } + Y

G = y-IGA-Iy- I 2

From (46) obtain G = A D-IGD-IA- I I I

c

2

which simplifies using (53) to M-IN C 0

ab

0

D-lAA

D-IBA

c'

D-lAA

r

and A-lA r

1c

are obtained using:

c

D-IBA r

Ic

Then the dual criterion (22) can be minimised subject to appropriate weighting matrix assumptions. Control spectral factor DI obtains from (39) and filter spectral factor D (used above) obtains from (34). In the case where complementary sensitivity is not included the optimal controller is obtained as follows.

* + FA2 DIG

(55)

0

HD-IA- I and N 3 r 0

Right coprime polynomials A and BI satisfy lc A-IA-IB = B A-I . c I le Right coprime matrix fraction repres e ntations o f

* - FB3 DIH

where M

Controller solution for the dual criterion

A solution for (H,G,F) such that degriF < degriD* is determined from the bilateral di ophantine equation:

c

and from (28) the optimal controller is given C = A (D - GD-IB A }-I GD-I A-I o I ro 2 c

0

Theorem J

r

Recall that F. is calculated using a division * algorithm (Kucera (1979» so that degriF
surranarised as:

A-lA (54)

c

With the details of the last step of the solution procedure established, the results may be

Assuming a matrix fraction representation equation (32) and (33), the following relations are required

-I

DI (GD 2 BIAro + HD 3 Aoc)

-I

and (48). The positivity constraints on RI and R guarantee the existence of spectral factors D and DI (Youla (1961}).

GD-IA- I 2 c

(56)

A left coprime matrix fraction representation for Mo and No may be defin~d -1 , -1 [Mo,Nol = [N(A D )- , G(A D } 1 = D [Ho,Gol r 3 f c 2

Left coprime matrix fraction repr esen tati on is determined using:

and the optimal controller follows as:

o

(57)

so that the op timal controller is given as C

H-IG

o

0

(58) 0

(The return difference ma y be used to verify that this is th e stabilising solution, viz: 1+ C W

o

- I

-I

Ho (HoAlc + GoAcBI }A lc H-ID D A A-I = A D H-ID A-IA- I o f oc r r 3 I oc r

wher: D , DIor D , DI are stable polynomial f 3 matri ces.) The condi ti ons for the solvability of the optimal problem are: (a) The greatest common left divis or Uo of A and B is a stable polynomial matrix . (b) The plant is free o f unstable hidden modes. (c) The matrices Mo' No yield an asymptotically stable cau sal transfer. Whilst conditions (a) and (b) are required to ensure stability of the control problem, condition (a) also ensures the existence of a unique minimal degree solution t o the diophantine equations (47)

It is also possible to develop similar theory for the case of more general weighting functional, where 03 1 0, and both sensitivity and comp l eme ntary sensitivity are costed. The analysis for this general case is more complicated, but no t unduly restrictive.

CONCLUSION The usual LQG problem may have its criterion extended to include sensitivity and complemen t ary weighting matrix terms. Thus the frame work for assessing both time " domain performance and robustness issue s in one problem formulation has been created . The solution r ou t es available include the usual state-space problem description or the polynomial system methods described here. The remaining issues to be addressed includ e the construction of th e solution software, and th e problem of choosing dynamic weighting matrices to achieve desirabl e features in system performance . Both of these areas are currently under investigation.

327

Robustness and Optimality: A Dual Performance Index Solheim, O.A. (1972). Design of optimal control systems with prescribed eigenvalues. Int. Journal of Control, 15, No. I, 143-166-.--Harvey, C.A and Stein, G~(1978). Quadratic weight for asymptotic regulator properties. IEEE Trans Automatic Control, AC-23, No.3, 378-387. Grimble, M.J. (1981). Design of optimal output regulators using multivariable root loci. lEE Proc. 128, Part D, No. 2,41-49. --Doyl~.C~nd G. Stein (1981). Multivariable feedback design: Concepts for a classicalmodern synthesis. IEEE Trans on Automatic Control, AC-26, No. I, 4-16. Safonov, M.G.~. Laub and G.L. Hartmann (1981) Feedback properties of multivariable systems: The ~ole and use of the return difference matrix. IEEE Trans on Automatic Control, AC-26 No. I, 47-65. -Francis, B.A. (1983). On the Wiener-Hopf approach to optimal feedba~k design. System & Control Letters, Vol.2, No. 4, 197-201 Shak~( 1976). A general transfer function approach to linear stationary filtering and steady-state optimal control problems. Int. Journal of Control, 24, No.6, 741-770. ---Grimble, M.J. (1979). 'SoT;;"tion of the stcchastic optimal control problem in the s-domain for systems with time delay. Proc lEE, 125, No.7, 697-704. Kucera, V. (1979). Discrete linear cont r ol: The polynomial equation approach. John Wiley & Sons, Chichester . Grimble , M.J. (1979). Solution of the discrete time stochastic optimal control problem in the z-d omain. Int. Journal of Systems Science 10, No.12, 1329-1390. Gupt~ N.K. (1980). Frequency shaped cost functionals: Extension of linear quadratic gaussian design methods. Guidance & Control, 3, No.6, 529-535. Ku~e~a, G. (1980) . Stochastic multivariable control: A polynomial equatio n approach. IEEE Trans on Automatic Control, AC-25, No.5, 913 - 919. -Youla, D.C. (1961). On the factorisation of rational matrices. IRE Trans on Information Theory, 2, 172-189.

(A.6)

e = (I - WG ) (r - d) + WG V c c then ~

G

uu

G*

~

(A.7)

c cc c

and ~

+ WG

ee

~

c vv

(WG) * c

(A.8) Thus using (18), (19), (A.7) and (A.8) yields:

* trace{(W *QIWGc~cc - 2W * QI~o)Gc} + trace{QI~o} (~o

= ~rr

+

~dd; ~cc

= ~o

+

~vv)

trace{Rl~uu} = trace{RIGC~ccG:} trace{Q2~ e

o

e

} = trace{Q2s~ 0

cc

S*} = trace{s*Q2S~

trace{(I-WG )*Q2(I-WG)~ c

c

trace{(w*Q2WG ~

c cc

cc

- 2W*Q2~

cc

}

}

cc

)G*} c

+ trace{Q2~cc} tra ce{Q3~v)

* = tra ce{T *Q3T~cc} = trace{W * Q3WGc~ccGc}

Collating terms

trace{QI~ee +Q2~e e + Q3~v) + trace{Rl~u) o 0

trace{[w*(QI+Q2+Q3)W+RI1Gc cc G: 1 - 2W* (Ql~

o

+Q2~ cc )G*C + QI~ 0 + Q2~ cc }

(A.9)

and th~ full cost functional (22), with (8) and (21) maybe written:

APPENDIX J

Proof of Theor em 1

I

*_

T

*

lim 2T {j [trace{( [W QW+RI lG ~ G)( t) T->oo -T c cc c

Recal l from equa ti ons (22) with (8) and (21) J = lim T->oo

2~ {J~(u)

+

J~(u)} + trace{(Ql~0+Q2~cc)(t)}1 dt + (u, Rlu\ }

(A.IO) where Q = Q + Q2+ Q I 3 The necessary and sufficient condition for optimality follows the usual variational argument based on perturbing the operator Gc by an arbitrary physical realisation €G c ' evaluting the condition

m

Individuall y the terms become:

3J/3€1€=o

T

J-T trace {(Q l~ ee )(t)}

dt

(A . I)

2

3 J /3 €2'

=

0, and assessing the nature of The analysis closely follows that in

Grimble (1979), to yield the required time domain optimality condition as:

T

( trace (RI~uu)(t)} dt

(A.2)

*

*-

\it

T

J

-T

trace{(Q2~e e )(t)} dt 0

G(r-d-v) c

~

0

(A.

11)

(A.3) Proof of Theorem 2

0

where ~ denot es the usual correlation function. From Fig. I and equation (9) obtain: u

o

«W QW+RI)Gc~cc-W [QI~0+Q2~ccl)(t)

-T

(A.5)

In the s-domain gradient equation (A. 11) becomes T G(s) = (W ( -s )QW(s)+RI)Gc(s)~cc(s) T

W ( -s )[QI~o(s)+Q2~cc(s)1

(A.12)

M. J. Grimble and M. A. Johnson

328

where G(s}€Rmxr(s} is the transform of a causal time function.

at say frequency wo' To minimise the sensitivity to low frequency disturbances and plant variations, the sensitivity function must be limited in the frequency range of importance. To achieve zero sensitivity to modelling er r ors at zero frequency, integral control action is of course required which can be achieved by including an integrator in the error costing matrix.

Generalised spectral factors (Shaked (1979}) are defined : YY *

~

cc

and Y* Y

1 1

Then (A.12) becomes

A possible design approach is to choose the LQG weighting terms by the usual performance requirements relating to transient r es ponse on energy m1n1misation. Although recently some work with examples by Gupta (1980) demonstrated low frequency shaped weighting matrices can be incorporated in the usual LQG formulation. Thus the dual weighting terms can then be incr eased in significance until the transient perfo rmance starts to deteriorate significantly. As in all design problems an iterative approach is necessary which includes the va ri ation· of weighting matrix frequency responses to achieve certain singular

to give

thus

o

*

*

Y G Y-{(Y } -1 W 1 c 1

(Ql~ +Q2~ o

cc

}(y*}-I}

+

to yield result for Gc and hence Co (by Lemma I).

value requirements.

o APPENDIX 2 Choice of Weighting Matrices The performance index to be optimised assigns, as in the usual LQG problem, frequency dependent penalties to the sizes of the singular values of the sensit~vity and complementary sensitivity weighting matrices. The noise intensity matrices also, of course, affect these singular values but in the present discussions these matrices are

assumed to be fixed by the problem description. The advantage of the dual criterion is that by separating the LQG and sensitivity costing terms a more direct relationship applies between the costing terms and the singular values of interest.

To design the optimal controller via the specification of frequency dependent weighting matrices the design requirements must be agreed. Typically the singular values of the clused loop transfer function matrix might be required to have roughly equal bandwidths. The example presented in Safonov, Laub and Hartman (1981) demonstrates how this might be achieved in the LQG problem. A similar technique can be applied in the dual criterion problem, namely: If the plant model includes uncertainties at high frequencies then high frequency r o ll off is necessary, there being at l e ast X dB attenuation

Reference subsystem

r

Disturbance subsystems

'..J (3)

n

o

Controller transfer

System

C (s) o

W(s}

(Measurement noise) v

~----,-----~~

(outputs)

Fig. 1.

Closed loop optimal feedback control system

z

observations