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Slope Restrictions. Multipliers. Stability

Inequalities

Vladimir Rasvan Dan Popescu Daniela Danciu

Department of Automatic Control, University of Craiova, A.I.Cuza, 13Craiova, RO-200585 Romania (e-mail:

{vrasvan,dpopescu,daniela}@automation.ucv.ro).

Abstract: It is considered an overview of the frequency domain inequalities for the absolute stabilityof the systems with monotone and slope restricted nonlinearities. It appears that the same type ofmultiplier is associated with different augmentations of the state space and this fact explains variousadditional assumptions accompanying the stability inequalities. These inequalities are applied to thePIO II problem in aircraft dynamics where the feedback structure of the absolute stability contains thesaturation nonlinearity which is both non-decreasing and slope restricted.

Keywords: Absolute stability, Multipliers, Slope restrictions, Frequency domain

1. THE STARTING POINTS OF THE PROBLEM

We shall start from the standard system of ordinary differentialequations

x = Axb(cx) (1)

where the state vector x has dimension n, : R R is a scalarcontinuous function and the constant coefficients A, b, c haveappropriate dimensions.

A. We assume that is a sector restricted nonlinear functioni.e. that it is subject to

()

, (0) = 0 (2)

Obviously (2) defines an entire class of functions; since eachfunction of this class defines a system (1) when considered inits equations, one may say that (1) - (2) define an entire class ofnonlinear systems. Since(0) = 0 these systems havex 0 (theequilibrium at the origin) as solution. Asymptotic stability ofthis equilibrium is a standard problem of the Liapunov theory.Less standard is the requirement that global asymptotic stability

should hold for all nonlinear functions subject to (2). Thisis some kind of robustness of the stability and, following analmost 70 years tradition, Bulgakov (1942), absolute stability.

We mention here one of the most recent applications of absolutestability is the so-called PIO II problem in aircraft dynamics -the P(ilot) I(n-the-loop) O(scillations) of the second category,defined by the activation of the position and rate limiters; thismeans that in the feedback structure composed of the airframeand the pilot dynamics, a nonlinearity of the saturation typeoccurs (Fig. 1)

The saturation function is of sector restricted type. On theother hand system (1) may be viewed as describing a feedback

structure composed of a linear and a nonlinear block as follows This work was supported by CNCSIS-UESFISCSU project number PN II -IDEI 95/2007

Hc(s)r

-

yHa(s)

1s

-

H Hauc uu.

Fig. 1. System with rate limiter

x = Ax + b1(t) , 1 = cx ;

2 = (2) ; 2 = 1 , 1 = 2(3)

(see also Fig.2)

L

N

u V

-

Fig. 2. Absolute stability feedback structure.

A straightforward approach to take in the PIO II problem isthat of the absolute stability - Rasvan and Danciu (2010). Buta problem occurs from the beginning- that of the sharpness ofthe results. Saturation is a specific sector restricted nonlinearitywhile the absolute stability techniques are valid for an entireclass for nonlinearities, hence the stability conditions will beonly sufficient i.e. lacking enough sharpness.

B. The sharpness problem of the absolute stability approacheshas been considered from its early days. The so-called Aizer-man conjecture, Aizerman (1949), says that the maximal ab-

solute stability sector (2) coincides with the Hurwitz sector -the linear stability sector corresponding to () = h. Thisconjecture is valid for first order systems (n = 1), also for


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second order systems (n = 2) except some limit situations asthat described by a counter example due to Krasovskii (1952)but was clearly disproved for third order systems (n = 3) bya celebrated counter example due to V. A. Pliss around 1957,Pliss (1958). Approximately around the same date Kalman(1957) started another conjecture: the Hurwitz stability sectorcoincides with the maximal sector of absolute stability for theso-called shape restricted nonlinear functions satisfying

() (4)

Slope restrictions clearly make the class of nonlinearities morenarrow hence the result of Barabanov (1988) proving the va-lidity of this conjecture for third order systems was somehowexpected; at the same time i.e. in the same paper a procedurefor constructing fourth order counterexamples to Kalman con-

jecture was proposed. Both conjectures disproved, there is stillroom for applications of the problem they generated: to test thesharpness of any method applied in absolute stability by com-parison between the Hurwitz sector and the nonlinear stabilitysector provided by that method. The aim of this paper is tocontribute to these methodological aspects in the case of theslope restrictions where several absolute stability criteria havebeen worked out. All of them have in common the so-calledtechnique of the augmented state space Barabanov (2000). Ourapproach will be however an engineering one, based on fre-quency domain inequalities, frequency domain characteristicsand frequency domain stability multipliers. The hyperstabilitytheory, Popov (1973), is furnishing a philosophy as follows: tocope with the usual point of view of the control engineer wholikes to have at his disposal a wide range of elements capableof being combined in various ways to form control systems ascomplex as desired, but who does not like to burden his creative

imagination with instability problems Popov (1973), page 5.

2. ABOUT THE STABILITY MULTIPLIERS

We shall follow here the way of Krasovskii (1978): startingfrom the basic structure of Fig. 2 we perform equivalencetransformations of it. Consider the equivalent structure of Fig.3

1

0

B [M

0B V

Fig. 3. Series augmented system.

IfB0 is a linear proper (causal) block, its inverse is also a proper

i.e. causal block. This is the case, for instance, with the blockdescribing the so-called Brockett Willems multiplier, whosetransfer function is

BW(s) =p

1

js +j

s + j +j, j 0 ,

j > 0 , j 0 (5)

But B0 may be even improper: in fact the oldest stabilitymultiplier - the Popov multiplier - is improper since it reads

P(s) = +s (6)

being thus a PD multiplier. Obviously B10 is in this case astrictly proper linear block. While the theory of the absolutestability based on non-causal multipliers has attracted someresearchers in the 70ies of the 20th century, only a few of thecriteria obtained a broader use. The oldest of these criteria isthat due to Yakubovich (1962): it corresponds to = 0 and hasthe form

1

1

+e ()

+ 2e ()+

+32e(1 + ())(1 + ()) 0

(7)

with (s) = c(sIA)1b and for some real numbers i, 1 0,3 0.

If additionally we take = 0 i.e. the admissible functions arealso non-decreasing then (7) becomes:

e (1 + 2+ 32)() +

1

+ 3

2 > 0 (8)

and one may recognize the multiplier

Z(s) = 1 + 2s 3s2 (9)

The next criterion was concerned with slope restrictions only,see Barabanov and Yakubovich (1979), i.e. only the restrictions(4) are taken into account. The frequencydomain criterion takesthe form

e

(1 +())(1 + ())

()

0 (10)

and if= 0 then (10) becomes

e

1 + ()

()

0 (11)

which suggest the PI multiplier

Z(s) =

s=

1

1

s

(12)

This multiplier is causal. Several years later the same case wasconsidered in Singh (1984)with a multi- variable counterpart inHaddad and Kapila (1995), Haddad (1997); the slope restric-tions were (0, ) and the frequency domain inequality was ofthe type (11) with changed in . Since the sign of is notspecified we have obviously the same inequality.

Moreover, if we take 1 = 0 in (8) and divide the inequality

by 2

> 0 we rediscover (11). It appears that from the pointof view of the frequency domain inequality all criteria areidentical. There exist however several differences connected


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with proof techniques and they introduce additional assump-tions which have corresponding effects on the practical stabilityconditions.

3. THE AUGMENTED SYSTEMS

We mention from the beginning that a good overall reference is

Barabanov (2000). Here we focus on the systems associatedto the cases described in the previous section. In the caseof Yakubovich (1965a,b) (in fact this was mentioned even inYakubovich (1962)) the augmented system was defined by thestate variables

z = x , = (cx) (13)

what sends to the (n + 1)-dimensional system

z =Az + b(14)

=(cz)c(Az + b)

Obviously this system has the prime integral

(t) +(cz(t)) const (15)

hence its dimension may be reduced by one; moreover if thesolutions of (14) are viewed on the invariant set +(cz) 0- suggested by (13) - then z(t) x(t) provided z(0) = x(0).This extended system was considered in Barbalat and Halanay(1974) for the case of several nonlinear functions. Consider nowthe approach of Barabanov and Yakubovich (1979); here thenew state variables are

z = Axb(c

x) , = (c

x) (16)and unlike (13) here z = x. From here the following is obtained

z =Az + b(t)(17)

= (t) , (t) = (cx(t))cz

If det A = 0 then we may compute cx = cA1(z b) toobtain the(n + 1)-dimensional system

z = Az + b(18)

=(cA1(zb))cz

with the prime integral

(t) +(cA1(z(t)b(t))) const (19)

The third approach of Singh (1984), Haddad and Kapila (1995)is based on differentiating the initial system; this means

z = Axb(cx) , = cx (20)

hence

z =Azb()cz(21)

= cz

with the prime integral

(t) cA1z(t) cA1b((t)) const (22)

Not only that (15) are the simplest in defining the new statevariables but also the return to the basic system ((13) viathe associated prime integral generating a family of invariantsets is much simpler. This suggests, especially when thinking

to the assumption det A = 0 that slope restrictions are takeninto account in a more natural way if considered together withthe sector restrictions.

4. SOME APPLICATIONS

Several applications with purely mathematical character maybe found in Barbalat and Halanay (1970, 1971, 1974), Rasvan(2007).

A. We consider first the previously mentioned celebrated coun-terexample of Pliss. In this case the transfer function of thelinear part is

(s) =1

s + 1 + a+

s1s2 + 1

,a > 0 (23)

which is irreducible and has two poles on R. It is quite easilychecked that the Hurwitz sector is given by 0 < h < (1 + a)/a.Consequently the maximal achievable result for the absolutestability sector is subject to > 0 and < (1 + a)/a.

Here as elsewhere we shall follow the philosophy of someparsimony principle: to use as few free parameters as necessaryi. e. the stability multiplier should be as simple as possible.We are guided by our experience which tells that more freeparameters are in use, more difficult is to manipulated them ina reasonable way.

Application of the Popov criterion requires to take in (7) 3 = 0to find

1

+e(1 + )() =

1

+

1 + a +2

(1 + a)2 +2+

1 +2

2 1 0

(24)The only choice for is = 1 to find

< 1/2 < (1 + a)/a

Next, the case when only slope restrictions are taken intoaccount does not apply for the Yakubovich criterion since itwould require in (7) both 1 = 2 = 0, the inequality thuslacking any free parameter. Consequently we shall considerboth sector and slope restrictions:

= = 0 , = , = 2/1 , = 3/1

Then (7) reads

1

+e(1 + +2)() 0 (25)

that is

1

+

(1 + a)(1 +2) +2

(1 + a)2 +2

1 +2(+)

12 0

and a necessary choice is += 1 hence = 1< 0.Further an elementary computation shows that by choosing > 2/a the inequality (25) holds provided 1/ a/(1 +


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a) > 0 what recovers the Hurwitz sector for all non-decreasingnonlinear functions of this sector.

B. The next application accounts for a preliminary computationfor PIO II prevention in the short period longitudinal motionof the so-called ADMIRE standard model. With the specificnotations we have

MqMMc = 0(26)

c = e(k kqc)

with - the incidence angle and c - the control deflectionangle. The function () is the saturation function

() =

VL , ||> L

VL

L , || L

(27)

In order to estimate the Hurwitz sector we take () = with > 0. The characteristic equation of the linear system thusobtained is

D() 3 + (eMq)2 + (e(Mkq Mq)M)+

+e(MkM) = 0(28)

Since k 0, kq 0,Mq < 0 we shall haveAq =MkqMq > 0from the first Stodola inequality. The Hurwitz sector will bedefined by e> +, where + is the positive root of thetrinomial

Aq 2 (M+AqMq +A) +MMq = 0 (29)

We rotate the sector by introducing

() = +() (30)

where e+ = +. We obtain in this way a feedback structureas in Fig.2 with the nonlinear function subject to

+ 0 (34)

where we denoted p1 = + Mq > 0, 20 = Aq+ M> 0.

This choice and the fact that > 0 gives

e (1 + )() =2 +Ap1

20

2 + p21> 0 , (35)

Since the entire Hurwitz sector has been recovered, we deducethat (26) is absolutely stable in the sector (+,). But thissector is violated by the specific nonlinear function (27).We are thus stressed to find an invariant set of the state spacewhere this sector is not violated. From the graphical condition|| 0the largest possible such that

supc>0

{x R3 : V(x) < c}

{x R3 : (k+ kq+c)2 < (VL/+)

2}

(37)

Paradoxically, we need here a Liapunov function while inaircraft dynamics and PIO analysis all available data are ex-pressed in the frequency domain. Fortunately we may use theYakubovich Kalman Popov lemma to associate to the frequency

domain inequality (33) - and (35) - a Liapunov function of theform

V(x) = xHx + cx

0()d (38)

where > 0 is that of (34) while H is a result of solving someLinear Matrix Inequalities.

C. Another application is the analysis of the PIO II pronenessfor the roll attitude of the lateral directional motion of a genericaircraft, see Klyde et al. (1995). The mathematical model reads

+1

TR= Laa

(39)a = e(k kpa)

where is the bank angle and a the aileron deflection angle;() is again given by (27). The transfer function of the linearpart is

(s) = e

1

s+La

k+ kps

s2(s + 1/TR)

(40)

hence it is in the critical case of the double pole. The Popovcriterion holds for the infinite parameter i.e. for

m () > 0 T2R

2 + 1 +LaTR(kp kTR)

T2R2 + 1

> 0 (41)


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which holds for

1 +LaTR(kp kTR) > 0 (42)

5. CONCLUSIONS AND PERSPECTIVES

Throughout this paper there were presented two kinds of prob-lems - theoretical and applied. The theoretical issues accountfor the specific features of the stability multipliers which areconnected with monotonicity and slope restrictions of the non-linear functions. The fact the same structure of the frequencydomain inequality is obtained under various proof assumptionsis a sound explanation for the additional assumptions accompa-nying the frequency domain inequalities.

At the same time this analysis shows the role of the non-causalstability multipliers which are still not enough investigated. Onthe other hand it is an interesting coincidence that such animportant application as PIO II may be embedded within theabsolute stability problem and that the saturation function is

both non-decreasing and slope-restricted. Moreover, since theaircraft dynamics databases are expressed in frequency domain,this is a rather important field of applications of the frequencydomain inequalities. Once more the almost 70 year old fieldof the absolute stability is rewarding for both theoretical andapplied research.

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