Structured Stability Margin and the Finite Argument Principle

Copyright © IFAC Design Melhods of Control Systems. Zurich, Switzerland, 1991

STRUCTURED STABILITY MARGIN AND THE FINITE ARGUMENT PRINCIPLE A. Rantzer* and B. Bernhardsson** *Depanment o/Optimization and Systems Theory. Royal Institute o/ Technology. 5-110 44 Stockholm. Sweden **Depanment 0/ Automatic Control. Lun" Institute o/Technology. P.O. Box 118. 5-221 00 Lund. Sweden

Abstract: If the structure of the uncertainty in a lin ea r model is known. it is natural to use this information in robustness analysis. In pa rticular , \I·hen th c model depends on a number of uncertain parameters onc sometimes defines a "structured stability margin" measuring the smallest parameter deviation giving instability. There are different definitions of the structured stabili ty margin. They differ in the way the structure of the uncertainty is prescribed. In this article wc suggest a new definition that use the probabilistic distribution of the parameters. We will defin e and calculate a 'structured stabi li ty margin' which is tailor made tu make use of covariance information on parametric uncertainty. Such information is typically obtained from a parametric identification. In the calculation of stability margin it is natural to evalu ate the characteristic polynomial along the boundary of the stability region. The 'fi nite argument principle' is a tool, which can be used to reduce the number of such evaluations. The frequenci es will also automatically concentrate to critical regions . \Ve show explicitly, how the finite argument principle can be used to compute the structured stabi lity margin . An example from robustn ess analysis of a mechanical system is presented.

1.

Introduction loop system is stable. For the phase margin the system is instead perturbed to e-sTG(s) . There are also different robustn ess measures using the H = -norm. These all have in common that the uncertainty, .0.(iw), is assumed to be unstrllctured. In the complex plane the uncertainty is then restricted to have the form of circles 1.0. (iw) 1 ~ p( iw). Very often some structure of the unmodeled dynamics is known. It is then conservative to use an unstructured description of the uncertainty, since that includes cases that will not occur in reality. One way to overcome the conservatism is to use the j.t-synthesis , see (Doyle 1982, 1984). In the j.t-synthesis different structures can be imposed on the uncertainty. The theory is rather general and allows for the follo wing structure on the uncertain d ynamics:

Different measures of the robustness of a linear control system design has been suggested in the literature . Most of these measures concern stability robustness although some measures of performance robustness exists. Stability robustness measures how well the controller can maintain closed loop stability when the open loop system changes around a nominal case. This change can be due to either erroneous modeling or to a real change in the process dynamics , e.g. changed operating conditions. Different robustness measures differ in the way the process uncertainty is introduced. For the classical gain margin the open loop system G(s) is perturbed to kG(s). The gain margin is then given by the interval [kmin' kmazl around the nominal value k = 1 for which the closed

139

complex

real

diagonal

diagonal

EXAMPLE I-Uncertain Open System Assume that the linear system

fullblocks

(1) Here cSi' model parametric uncertainty, cSi' is used for uncertainty in the frequency domain and .6.i represent unmodeled dynamics.

is controlled with the linear controller

R(s)u = - S(s)y

DEFINITION I-(Structured Singular Value)

Jl-1(G) = min{o-(.6.) I det(I - G.6.) = O}

Assume that the parameters ak and bk are unknown. After identification their nominal values are a~ and b~ . The closed loop characteristic polynomial can then be written in the form (2) :

"" where .6. should vary over all matrices of the form (1). 0 A severe drawback with Jl is that in it's general form it is very hard to compute. The choice of appropriate stability robustness measure is affected by things as:

•

The more structure or probabilistic information of the uncertainty that is (correctly) used the more useful the measure will be.

•

The measure should be computable with a reasonable amount of work.

•

The measure should answer questions as: what uncertainty is most critical, what frequency is most critical?

AR + BS = AoR + BoS + 2)ak - a~)sk R+ L(b k - b~)sk S

We now exploit the uncertainty information provided by identification to form a realistic robustness measure: DEFINITIO N 2- Structured Stability Margin Consider the family of polynomials of the form (2). Let P be a positive definite matrix. The robustness measure Pstab is the largest scalar such that all polynomials of the form (2) with 2 e Tp-1e < Pstab

are stable. Relation to other robustness measures

or alternatively one single full real m x m block and the restriction that G must have rank one. During the final preparation of this paper we noticed that our stability measure can be seen to be equivalent to a special case of the real stability radius as defined in (Hinrichsen and Pritchard 1988). Our algorithm however has the advantage that we have a simple method for optimization over wand that the discontinuous case need no separate treatment.

The most basic performance criteria of a control system is stability. We will study stability of a family of characteristic polynomials of the form

(2)

k=l

Here po(s) is the nominal polynomial of degree n, Pk (s) are polynomials of degree less or equal

e

(e

1

ern)

o

The stability measure we have defined is not a standard Jl-measure. It uses an uncertainty structure which in Jl can be described by one nonsquare block

2. The Structured Stability Margm

to nand parameters.

= po(s) + L ekPk(S) o

In section 2 we present a robustness measure Pstab that is based on parameter covariances. This measure gives very useful information about robust stability in connection with parametric identification. In section 3 we present a fast algorithm for calculating pstab using the 'finite argument principle'. Explicit formulas in the frequency domain make the computation time rather insensitive to the nurnber of uncertain parameters. In section 4 the theory is illustrated on an example from mechanics.

p(s,e)=po(S) + LekPds)

+ T(s)u c

Ellipsoidal uncertainty

are uncertain The ellipsoidal type of uncertainty is motivated by some general results from identification theory. For prediction error methods the asymptotic (in the number of data) uncertainty is given by : 140

will by definition give a stable closed loop system. Integration over this ellipsoid gives (4). 0

I-Asymptotic Parameter Variance Consider the Prediction Error Method

THEOREM

Remark. There might be parameters outside the ellipsoid (5) that also giYe a stable system and the yalue is therefore only a lower bound. In the rest of the paper we will work in continuous time and the stability region will be the left half plane. The generalization to other stability regions, for instance the unit circle, is direct and present little difficulty.

where ZN is the data set, E is the prediction errors, and the parameters. Assume that the model structure is linear and that there is a unique value eo giving perfect model fit such that

e

eN

->

with prob. 1 as N

8-,

3.

-> 00

Then, under some technical conditions we have

.Jii(e N

-

eO)

Calculation of Stability Margin

The structured stability margin can be calculated based on the following result

E As N(O, Pe)

PropositioIl.- Imaginary axis sweep Proof;

o

See (Ljung 1987, pp. 241).

Pstab

The covariance matrix Pe will depend on the data set and different variables in the identification algorithm. Pe is influenced by e.g. signal excitation, choice of prefilters and choice of control law. Good estimates of Pe can be obtained based on a finite number of data points. As an example one can for quadratic criteria use ~

PN=(jN

[

N

N

(jN

=

~

1",

~

T

N~'Ij;(t,eN)'Ij; (t,e N )

1",2

8 = {theta:

]-1

~

N ~E (t,8 N ) t=l

where 'Ij; are the regressors. Theorem 1 is based on the hypothesis that there exists a parameter in the parameter set giving a perfect model fit. If this is not true, the model error will contain both a bias term and a variance term. A preliminary result in estimating the variance term for this case is given in (Hjalmarsson 1990). 2-Probabilistic Stability Assume that the parameters are normally distributed with

THEOREM

e E N(8 0 ' P)

(3)

and that Pstab is calculated as in Definition 2. The closed loop system will then be stable with pro bability prob stab ~ Proof:

1

rrc.

v 271"

jP"ab e- 2/2dx z

= sup{p:

(4)

-00

All parameters in the ellipsoid

141

0 (j. p(jw,p8) for

e

T

p-

1

e:s

wE

R}

I}

Proof; T\'o Hurwitz polynomial has a zero on the imaginary axis , so 0 (j. p(jw, Pstab8) for w E R. On the other hand, by continuity any famil y that contains both stable and unstable poly nomials , must also contain a polynomial with a zero on the imaginary axis. This proves the statement. 0 A difficulty in using the proposition is that an infinite number of frequencies must be considered. Instead we will use the finite argument principle below, to reduce the calculation to a small, finite number of frequencies. In the following \\'c use the principle branch of arguments, i.e. - 7r < arg :S 7r THEORE !\I 3- Finite Argument Principle Suppose p is a polynomial of degree n with complex coefficients. If there are frequencies - 00 = Wl < W2 < ... < WN = +00 such that the equality N

L arg 1= 2

p(jwt) p(jWl-d

= 2'7rn

is well defined and true, then p is Hurwitz. Conversely, if p is a Hurwitz polynomial of degree n, then there are such that the equality holds.

{wdi:t1

Proof: Suppose p( s) = Po (s Since arg(ul" ·u n ) ::; largull with strict inequality if arg Uk the equality implies that

,8d ... (s

- ,8n).

+ ... + <

l argunl 0 for some k,

r-;

27rn =

2: arg 1=2

=

N

2: arg 1=2

p(

P

(jwL))

JWI-l

ITn (jw'l - ,8k) (j:..vl_l - ;'h )

Figure 2. p(s) is the size of the smallest ellipsoid touching the origin

1=1'

-.8d I <~ L ~ L arg ---'(j..JI -','-----~ I'

THEORDI 4-Ellipsoidal 8 Su ppose

(J .....'I-l - (3k)

-

1= 2 1= 1 i

=

~ ~ arg (jU:I-l -

" IS ("

J..JI -

rl)1 = {3d

t-/k

27rn

p(s, e) = Po(s) + eT PK(S) 8 = {e: eT p- 1 e ::; I}

Strict inequality is impossible, so all factors in the products lllu st ha\'e positive argument and P is lluf\I'itz. To prove the second part note that if P is II urwi tz, \I'e can choose {wz} as the zeros of Re p(i:..v, 0) and lm p(iw, 0).

If PK(S) has full rank, then

l;2

Pars) =

o

Re (Po ( s )1a.) (Re (p K

( S)

1a.) T

P Re (p K (s )1a.))

1/2

Let Ci(s) = PO( S)[PK(s)T PpK(S)]-I. Then

p(s) = sup Pa(s) = P",(s)(s) ",,to

The stability margin at a fixed frequency Suppose p (s, 0) is a polynomial of degree n in s depending linearly on e. Let 8 c R rrt be convex and bounded. Define for any complex number s

Proof: First note that Re (ul a.) = Re (un) 1 [0'[2 Let P = LT L and use CauchySc:hwarz inequality:

= ii,Ci T 110'1 2 .

(OT PK(s)Ci T )2 = (eT L- 1LpK(s)a T )2

Pars) = sup{p : 0 rf. Re [P(s, p8)la]} p(s) = sup{p: 0 rf. p(s,p8)}

::; eT (LT L )-le . CiPK( s f

See figures 1 and 2.

::; LT LpK(S )aT

with equality when the vector eT L -1 is parallel to CiPK(s)T LT. The definition of p",(s) now gives

.. '1 p", (s)

.

Re (Po (s ) 1a.)

= mm = eE (:;) Re (T e P K ()/) S a. Re (Po (s )1a. )

\

1

o.~

0.1 ·0-'

-.~ . -

!\ext, let PK(s)T PPK = MT M, M E R2X2. Another application of Cauchy-Schwarz inequality gl yes

,

Figure 1. Pa(s) is the size of the smallest ellipsoid touching the the line with normal direction Q

It is easy to find explicit formulae for Pars) and p( s) in the ellipsoid case. To each complex vector U E {;rrtX 1, assign a real m x 2-matrix 11 = [Re U lm u]. We then have the following result, which motivates the new notation.

142

with equality when Po(s)M- 1 is parallel to aM T and in particular when a. = 0'( s). D

Minimization over w

Algorithm

The following theorem gives both lower and upper !:'ounds on the stability margin and is the basis for the algorithm to calculate Pstab:

We are now ready to formulate the algorithm for computing Pstab

THEOREM 5-Lower and Upper Bounds Let e be an arbitrary convex parameter set, then

p(jw)

(6)

sup Po.(jw) 0.#0

•

Choose as initial frequencies W2, . . . WN-l the zeros of Repo(jw) and Impo(jw). Then Po satisfies (7).

•

At each iteration step, add the frequency (Wl+WI-d/2 obtained from the minimizing 1 of the lower bound.

•

Stop when lower and upper bounds are sufficiently close.

Furthermore, if N

L arg[PO(jwl)/PO(jWI-dl = 2mr

(7)

2

The algorithm has been implemented in Matlab. A typical calculation time on Sun 3/50 is ten seconds. The calculation time is relative insensitive to the number of uncertain parameters.

then minsupmin{pu(jWI_d , po.(jWI)} I

-:::

0#0

(8) -:::

Pstab

-:::

minp(jwL).

4.

Proof: Suppose that P < p(jw). Then 0 rf. p(jw, pe). Convexity of p(jw, pe) implies that o rf. Re [p(jw, pe)/Cl'l for some complex Cl' i- 0, so p < Pa(jw). Hence Po.(jw) -::: sUPa#O Pa(w)

Example

As an example we will study the robustness of an inverted pendulum with uncertain length and mass , see figure 4.

and since the converse inequality is obvious from definitions, the desired equality follows. Suppose for the second statement that

P < min sup min{Pa(jwl_d, Pa(jWI)} I

a#O

u

m

Then 0 rf. conv{p(jwl_l , pe),p(jwl, pe)} for all j, so the function

Figure 3. The inverted pendulum with uncertain mass and length.

If the system is linearized around the upright position the transfer function from force to pendulum angle is

is continuous in pe. It takes only integer values, so it must be constant and by our assumption the value is 27rn. Hence pC, pe) is Hurwitz and P < Pstab· This proves the lower bound on Pstab. The upper bound is obvious from definitions so the proof is complete. 0

b = l/ml

b y= - - - u , s2 - a

a

= g/l

The control objective is to stabilize the pendulum around the upright position. The following regulator gives good nominal performance:

Discontinuities of p(jw) In (Barmish et.al 1990) it was observed that the stability margin can be a discontinuous function of the problem data. The example in the next section is one example where p(jw) is discontinuous w. The discontinuity can however only arise when the polynomials Pk all have equal argument. We have the following result,

s

+ 0.7

u=-(s+2)2Y The nominal values after identification is ao = 1 and bo = 8. The covariance of the estimates are

P

Corollary. If PK(jW) has full rank, then Pa(jw), p(jw) and Cl'(jw) are all continuous at w. Proof: This follows from the expressions in Theorem 4. o 143

= E(

(

a-a o) b-b o

(a-a o) T ) = b-b o

(

0.2 0.2

0.2 ) 10

6.

References

B'-\R~llSH,

B.R., P.P. KHARGONEKAR, Z.C. SHl, and R. TE!\lPO (1990): "Robust ness \ Iargin need not be a continuous function of the problem data ," System and Control Letters, 91 - 98 .

11 -

Ol -

0-·--- - -- . - - 41

1 O~

- ---

I:

-'- I-~

--

l:S

Figure 4_ p(jw) fo r Example 2. !\ot e that the functi o n is di s co ntinu o us at th e critical frequency. The lo we r and upp e r bound s from theorem 5 are also shown

p(jw) is shown in Fig. 4. !\" ote that because of the discontinuity the lower and upper bounds in theorem 5 will be unequal. The \latlab algorithm gi\'es pstab

= 0.91 ,

DO'r'LE, J.C. (1982): "Analysis of Feedback Systems with Structured Uncertainties," IEE- D, 2-12- 250. O~R / Honeywell

IIl i\llI CllSEN, D., and A.J. PRITCHARD (1988): ":'\ew Robustness Results for Linear Systems under Real Perturbations," 27th CDC, :\ustin, Texas, pp. 1375- 1378. HJAL ~ IAllSSO N

(1990): "On estimation of model quality in system identification," Licentiate Thesis, Linkoping, Sweden.

the critical frequency is

(1987): System Identification : Theory for the User, Prentice-Hall, Englewood Cliffs, NJ ..

LJU'\'G Wc

= l.10,

For clarity this example only had two uncertain parameters. The computation time increases relatively slowly with the number of parameters, around 100 parameters are \\'ell within the limits of the existing software.

5.

J .C.

(1984): "Lecture Notes for Workshop on Advances in :\1u ltivariable Control," Minneapolis, Minnesota.

DOYLE,

Conclusions

\Ve have defined and calcu lated a structured stability margin that gives interesting information in connection with parameter identification. This stability margin uses information on parameter covarian ces. It is not always necessary to be able to maximize a robustness measure over all controllers. It is very seldom that stabiJjty robustness is the only design crit eria. A robustness measure is mostly helpful as a piece of information to judge a controller design. A good robustness measure can signal a potentially bad design but should not be used as the only design criteria. We have not studied how to minimize our robustness measure over all controll ers. This is an area for future research. We belie\"e that identification based structured stabiJjty margins has a great potential as robustness measures for adaptive robust control. This is another area for future research. The matlab code to calculate the structured stability margin is available by Email from the authors. 144