A Systematic Approach to Control System Design Using a Reverse Frame Alignment Design Technique

Cop yright © IFAC Simulation o f Control Systems. Vienna . Austria . 19H6

A SYSTEMATIC APPROACH TO CONTROL SYSTEM DESIGN USING A REVERSE FRAME ALIGNMENT DESIGN TECHNIQUE G. K. H. Pang and A. G.

J.

MacFarlane

Ca ll1bridgf L' nivn sitv Engineering Department, Trumpingtoll S treet, Ca mbridge , CB2 1 Pl, UK

Abstract . A new technique for the design of multivariable feedback control systems in the frequency domain is proposed. The technique is based on singular value decompositions and handles stability, performance and robustness aspects of the design problem. The overall approach, which is powerful, systematic, and intuitively appealing, is well suited for use in an expert system context. Keywords. Multivariable control system design; feedback ; expert system; singular value decomposition.

G is normal iff

INTRODUCTION There are several reasons why it is useful to develop feedback control system design approaches for multivariable systems which are generalisations of the classical frequency-response approach (MacFarlane, 1979):

i

= 1, ... ,m

where {gi} is the set of characteristic values and {ail is the set of singular values. Proof : See Pang (1986a) .

(i)

Fairly simple controllers are sufficient for many purposes. (H) Such techniques are intuitively appealing. ( iii) They are directly related to well established classical methods. (iv) They are well suited for interactive computing. For these reasons, a considerable effort (Edmunds, Jeanes and Maciejowski, 1983; Hung and MacFarlane, 1982; MacFarlane and Kouvaritakis, 1979; MacFarlane, 1979; Pang, 1986a) has been expended in developing such approaches. Further advantages appear, due to the powerful indicators (Hung and MacFarlane, 1982; MacFarlane and Hung, 1985) available, when these methods are used in an expert system context (Pang and Boyle, 1986b ) . To be really useful, such techniques must be completely systematic; this requirement is mandatory in an expert system context. The approach briefly outlined here is described in detail in Pang (1986a) . It gives a systematic and intuitively appealing approach to the fr equencyresponse design of multivariable controllers which satisfactorily handles the three key aspects of the problem: stability, performance and robustness . In essence it is an extension of the characteristic value techniques described in MacFarlane and Kouvaritakis (1977) using the singular value techniques described in Hung and MacFarlane (1982 ) . A particular feature of the extension is the use of a skewness indicator, or normality indicator (Hung and MacFarlane, 1982 ) , to monitor the robustness properties of the feedback loop. DEFINITIONS AND RELATIONSHIPS

Definition 2:

Let G E £mXID Decomposition G

a

Schur

Triangular

S T S* u

S ( D + T ) S* where S is unitary, Tu is upper triangular, D is diagonal containing the eigenvalues of G and T is the strictly upper triangular part of Tu' We define (1)

MS(G)

where 11

IIF denotes the Frobenius norm of a

matrix (Hung and MacFarlane , 1982). Definition 3: Let G

E

Gain ratio

~ have characteristic values gi and

singular values a . i We define the gain ratios i

1, ... ,m

for an arbitrary ordering of gi and a i . Theorem 4

be an m x m complex matrix . Then

E

can be expressed explicitly in terms of its gain ratios Pi as

£mXID E £mXID

have

The normality indicator MS (G) for a matrix G

Theorem Let G

MS(G) - A Normality Indicator

C. K. H. Pang a nd A. C . .J. MacFarlane

228 m ! i=l

MS(G)

o.

2

].

1 - Pi

2

1/2 (2)

m ! i=l

~

with k .

].

i

1, ...

,m.

real constants.

1

2 o.

When this sub-controller is cascaded with the system, and at high frequencies,

Proof: See Pang (1986a) . Prol2osition 5: Characteristic gains and phases plus principal gains are the primary indicators of a system.

The product

G(s~)K~

has the following properties

(Pang, 1986a) : 1.

The input and output gain frames are aligned and thus the system is normal at high frequencies.

2.

Since the system is normal, the principal and characteristic gains coincide with no divergence (see Theorem 1). Therefore, the principal and characteristic loci tend to coincide in a neighbour of s = ~ .

3.

Since the system is normal as Isl~ ~ , it is robust in the high frequency region.

The Simple Design Technique (SDT) will now be outlined. A detailed description of the technique is given in Pang (1986a). The design of the controller is separated into two distinct stages : (i) design for good phase properties in the high frequency region, and (ii) design for good gain properties in the low frequency region. The primary indicators are used for the assessment of stability, performance and robustness of the closed-loop system. The design objective is achieved by a manipulation of the primary indicators of the open-loop transfer function into a suitable form.

4.

Since the characteristic gains behave asymptotically as

Sub-Controllers Based on Singular Value Decoml2osition (Hung and MacFarlane, 1982)

Low frequency sub-controller Let G(s) be real at s - 0 and take the form

Interl2retation of the I2rimary indicators (i) The characteristic gains and phases give an indication of the closed-loop stability of the system. (ii) The principal gains give an indication of the performance of the system. (iii) The divergences between characteristic and principal gains give an indication of the robustness of the system. SIMPLE DESIGN TECHNIQUE

the phases of the characteristic gain loci will approach ~ r n/ 2 . i 5.

The principal gains can be balanced up by a suitable choice of k~ if the orders of 1

infinite zeros, ri' are the same.

In this section, we give the two main types of sub-controller used in the Simple Design Technique. Let G(s) E. [DlXDI have an singular value decomposition(SVD) at one specific frequency Sw

G(O)

= Y0!0 0 U *

where Y and U are orthogonal matrices. o o Let K=uzKy* o

0

0

0

(6)

(7)

(3)

and we define a low frequency sub-controller for G(s) as

where Y and U are unitary w w and!

(8)

i = 1, ... ,m .

w

High frequency sub-controller Asymptotically, as Isl ~ ~, G(s) takes the form

where zK = diag{ k . o

].

o

}

1, ... ,m

with k . o and a real constants . 1

G( Sco ) = Y00 !G U*00 00 where

Y~

and

U~

(4) At high frequencies, G(s)KL(s) tends to G(s) and

are real orthogonal matrices

therefore KL(s) will not affect the system in the high frequency region.

and

As Isl~ 0, ~

o . are real and r. 1

zeros of

1

are the orders of infinite

G(s~) .

which approximates

We define a high frequency sub-controller for G(s) as

(5)

and has the following properties :

229

A Reverse Frame Alignme nt Design Technique 1.

Since

Y0I0 0 U Y I o

0

* u iK

Y *a/s

000

diag{ a k.o/s } y 1

0

*

the input and output gain frames are aligned and the system is normal at very low frequencies.

2.

Since the system is normal. the principal and characteristic gain loci tend to coincide in a neighbourhood of s = O.

3.

Since the system is normal as Isl~ O. it is robust in the low frequency region.

4.

The principal gains can be balanced up by a suitable choice of the gain parameters k . o 1

5.

The phases of the characteristic gain loci of G(O)·a·Ko/s will approach -w/2 because of the effect of integrator lis.

6.

The effect of the integral action can be controlled by a suitable choice of the parameter a .

7.

Steady-state error and low frequency interaction are eliminated because of the presence of the integrator.

Matrix P+I Controller Let the sub-controllers for the high frequency region design be K~Ki(s) and the sub-controllers for the low frequency region design be

Method developed by MacFarlane and Kouvari takis (1977). The new technique is used to deal with systems which require more phase compensation or gain adjustment in an intermediate frequency region than can be provided by the Simple Design Technique. This is done by the use of a Reverse Frame Approximation (RFA) sub-controller. A detailed description of the construction of an RFA sub-controller and the design technique is given in Pang (1986a). Reverse Frame Approximation Sub-controller Let G(s) e [mxm have an SVD at one specific intermediate frequency Sw G(s ) w

= Yw I w U*w

(10)

where Y and U are unitary w w and I w

diag{ a~1 }

i = 1 •...• m.

The RFA sub-controller specific form

matrix

= M xk(s)

K(s)

K(s)

has

M-I

the

( 11)

where M is a real matrix with

U*w M '" J

(12)

is a diagonal matrix with unit modulus elements. M is an approximate real inverse of the complex matrix U* and is orthogonal if the

J

w

approximation is good. xk(s) is a rational matrix of the following form :

diagonal

1, .. . ,rn.

xk(s) Ki(s) and Kj(s) are basic types of sub-controller which may be needed for further manipulation of the primary indicators. The final controller K(s) is the product

The effects of the high and low frequency sub-controllers are combined and each comes into proper operation in the appropriate frequency region. The complete controller becomes a matrix-proportional- plus-integral controller. The presence of the intergral term l i s ensures that

The diagonal elements are simple classical compensators like phase lead (a i < bi.c i = di = 0). phase lag (ai >b • c = d i = 0) and phase lag-lead i i compensators. The appropriate choice of parameters a • b • c and d will manipulate the i i i i principal gains and phases of the loci. Asymptotically. as I s I K(s~)

~ ~

•

M xk(s~) M-I M diag{l •...• ll M-I I .

at low frequencies and

Therefore. when this sub-controller is cascaded with the system. at high frequencies. it will not affect the properties of the system . The Final Controller

at high frequencies.

Let the sub-controllers for the high frequency

It also ensures that there will be zero steady-state error in the closed-loop steady state response. since all the gain loci tend to infinity as Isl tends to zero.

design

be

sub-controllers region be

for

the

n

_k

REVERSE FRAME ALIGNMENT DESIGN TECHNIQUE The Reverse Frame Alignment design technique in the extended form presented here is a redevelopment of the Characteristic Locus Design

region

K~Ki

(s) .

Also.

let

the

intermediate frequency 1

11 M.r-(s.)M- . j=l J J J

and the sub-controller for region design be

the

low frequency

G. K. H. Pang and A. C . .J. Ma cFarlallc

230

The final controller K(s) is the product K(s) (13) The effects of the different sub-controllers are combined and each comes into proper operation in an appropriate frequency region. Ki(s) and Kl(s) are basic types of sub-controllers which may be needed for further manipulation of the primary indicators. The complete controller is a dynamic controller with integral action. At high frequencies,

At low frequencies, K(s) The presence of the integral term will ensure zero steady-state error in the closed-loop steady state response. EXAMPLE The example is a 2- input, 12-state, 2-output gas turbine model which will be referred to as AUTO ( See Edmunds, Jeanes and Maciejowski (1983) for detai Is). The Simple Design Technique is used first to design a controller for the system. The Reverse Frame Alignment technique is then used, and the result is compared with those obtained using the Characteristic Locus Design Method (Edmunds, Jeanes and Maciejowski, 1983). It is found that the Reverse Frame Alignment technique provides a more systematic method for the design of the controller. Simple Design Technique High frequency region design The primary indicators of AUTO is shown in Fig . 1. The system has different roll-off rates at high frequencies and the gains are balanced at around the bandwidth frequency. A high frequency gain of 1.5 is chosen. The primary indicators (Fig. 2) show that the gains are balanced at around s = 10j. Low frequency region design A low frequency sub-controller is added together with a scalar gain of 4 (Fig. 3). The indicator MS is given in Fig. 4. The low frequency sub-controller is

The closed-loop step response of the compensated system is given in Fig. 5. The overshoot is 30% and interaction is around 18%. One of the loci gives a rather sluggish response. Reverse Frame Alignment Technique DESIGN ONE High frequency region design The high frequency sub-controller used in the SDT is kept. The primary indicators show that one of the loci is rather flat near the bandwidth frequency. Therefore, we need to obtain gain adjustment using a RFA sub-controller.

Intermediate frequency region design An RFA sub-controller is used at s = 3j. Figure 6 gives the accuracy of approximation over the intermediate frequency region by finding the condition number of M. A phase-lag network is used to manipulate the gain of the corresponding locus. The primary indicators show that the RFA sub-controller has achieved its objective (Fig. 7). The RFA sub-controller is Kl(s)

= Ml ' diag(

-1

1 ,(s+lO)/(s+l»'Ml Low frequency region design A low frequency sub-controller is added and a scalar gain of 4 is used to obtain a good performance for the system (Fig. 8). MS is given in Fig . 9. The low frequency sub-controller is

The closed-loop step response of the design is given in Fig. 10. Both loci show a fast response to a step. The loci with an overshoot of 30% before remains the same. DESIGN TWO High Frequency region design The high frequency sub-controller used in SDT is kept. Intermediate frequency region design The RFA sub- controller used in DESIGN ONE is ~ept. A second RFA sub-controller is used at s = 7j to introduce phase lead into one of the loci.

However, the gain of that locus has been lowered (Fig. 11) and a further gain balancing is needed. The primary indicators (Fig. 12) now show that the system has a good gain balancing and acceptable phase margins for both loci at high and intermediate frequencies. What remains is the low frequency design. Low frequency region design The low frequency sub-controller is

The primary indicators and MS are given in Fig. 13 and 14. The closed-loop step response of the final design is given in Fig. 15. Because of the extra phase introduced at the bandwidth frequency, the 30% overshoot of one of the loci has been brought down to around 13%. CONCLUSION A two-stage multivariable frequency-response design technique has been described and illustrated by an example. The way in which it has been based on singular value decompositions automatically ensures that the resulting designs are robust. Extensions to the case of systems having different numbers of inputs and outputs, and to systems requiring a larger amount of intermediate-frequency-range phase compensation than can be provided by the Reverse Frame Alignment technique are given in Pang (1986a). The overall approach, as described in Pang (1986a), is considered to give a powerful, systematic and intuitively appealing approach to the design of linear multivariable feedback controllers which satisfactorily handles stability, performance and robustness, and which is well-adapted for use in an expert system context.

A Reverse Fra me Alig nme nt Design T ec hnique

23 1

REFERENCES Edmunds, J.M., S.E. Jeanes, and J.M. Maciejowski (1983). CLADP : The Cambridge linear analysis and design programs - user reference manual. Engineering dept., University of Cambridge. Hung, Y.S., and A.G.J. MacFarlane (1982). Multicariable Feedback : Quasi-Classical Approach. Springer-Verlag, Berlin. MacFarlane, A. G.J., and B. Kouvaritakis (1977). A design technique for linear multivariable feedback systems, Int. J. Control, 25 , 81-127. MacFarlane, A.G.J. (~d.) (1979) . Frequency Response Methods in Control Systems, IEEE Press. MacFarlane, A.G.J., and Y.S. Hung (1985). Indicators for multivariab1e feedback systems. International Conference Control 85, University of Cambridge. Pang, G.K.H. (1986a). A systematic approach to multi variable control system design. Internal report, Engineering dept., University of Cambridge. Pang, G. K.H., and J-M. Boyle (1986b). An expert system for analytical and interactive design of control systems. To appear in The Second International Expert Systems Conference, London.

- Cc' r - -- -- - - - - - -- - -- -,--,-- -,

- 180 .01L--------------,---~----~----~~==~ .1

Fig. 3 .J

The primary indicators after SDT.

I I

.25

..

I

11/

I I

if>

" ."

.1

t

.0 5

E--~

o

1

\

\

i.

I

·r

,.,I

1

I !

i

f

I

'"

,

i:

10

,01

Fig . 4

ICO

The normality indicator MS after SDT .

A \

-

/.

\

t------

'c

/

.6

f!11/ I ,

.

i :~t~~ ,:. ~~~ .~ ~~ : ~:-++-.~.------f_.~._ ..

f ",. ..... H

10

I---

I

.2

,

- .2

100

o

10

15

12

o

5 ~

The primary indicators of AUTO.

--

W/f-'

Fig. 5

Fig.

I

~

.4

~::L~__________~___·~-·~-~·-·~ _- _-·~-·~-_··~~ -·~·~·~· .01

---

Closed- loop step response after SDT.

i !I

11111

i

11 ill

f

I

9

I

I

I

I1

II

: I 1r::1

I

6

1

I

I

1

I I I WII A

--r

.1

Fig. 6

11

11

frlf "'-. f--I

\..l 10

The accuracy of approximation using an RFA sub- controller.

:1 ~-~ I ~ ~iL__________._.__. _"_._-_:~_: _.'_:_'~. _._~'_-__~~~ .rJ '

.

:,:0

w

; _;~~I; .· .•••.•. ~.'-.O.= _"-. -, ~ ~

- 150

~

- 100

.

. . . •. ....... . . .• . • ;: •... .

•. •

• :.

.01

Fig. 2

The primary indicators after high frequency design.

: . -" ••.••... L L·.:: . . . . L

....• .. •. : • . . .

Fig. 7

.: .1

.:. 1

.. ' 10

. .

: . . .. ....

:

10 0

The primary indicators after an RFA sub- controller.

232

C. K. H . Pang and A. G.

J.

Mac Farla ne

;j..0 1

.1

10

lOO

j-:~E_ .01

.1

Fig. 8

10

t

100

:;~.01

The primary indicators after DESIGN ONE.

.1

Fig . 12

1

10

100

The primary indicators after gain balancing at intermediate frequencies.

'111lll~lI1111111111 i.::mll .0 1

10

Fig . 9

A

1.2

.3

I~

C

I

.4

I

:\ J

I

I

I

I

I

,i

I

I

,

I

o

10

~

':IIII Ullr.l~ .01

T

Fig. 10

Closed- loop step response after DESIGN ONE .

f

_00

..... .

,,---t_ - - - ' - ' - - - - - - ' - -_ _ _- - - - '_

.0 1

.1

_

---'---'- - - ' - - _ . . _•.

10

.0 I

"'_'---.J'

I I

l

-.so ~

I

.4

.2

r

Fig. 11

The primary indicators after a second RFA sub-controller .

I

i I

I

I

I

,~

-.2

I

I

lCO

lOO

The normality indicator MS after DESIGN TWO.

""

~ - '0

10

.1

Fig . 14

20

,'3

The primary indicators after DESIGN TWO .

I

I

\~/

-00

10 0

I ;

I

z

to

.01

The normality indicator MS after DESIGN ONE.

Fig . 13 f f

- .2

lQO

I

o

Fig . 15

10

Closed-loop step response after DESIGN TWO.