One concept that has been dened in that context is robustly convergent identication al- gorithms(tuned or untuned), see (Helmicki et al., 1991)

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Identiability Implies Robust Identiability

Lennart Ljung, Torkel Glad and Torbjorn Andersson

email: ljung@isy.liu.se torkel@isy.liu.se toand@isy.liu.se

Department of Electrical Engineering Linkoping University

S-581 83 Linkoping, Sweden Keywords: identication, bounded disturbances, ro-

bustly convergent, global identiability, robust global identiability.

1 Identication and robustly convergent algorithms

There has been a recent interest in identication from a deterministic perspective, likeH¹- techniques (Helmicki et al., 1991), (Helmicki et al., 1992), (Gu and Khar- gonekar, 1992), L¹ - techniques (Makila, 1991), (Chen et al., 1992) etc. One concept that has been dened in that context is robustly convergent identication algorithms(tuned or untuned), see (Helmicki et al., 1991).

The concept aims at capturing the following property:

Suppose a true system G⁰ ² ^G generates exact input- output data

Z⁰^N=^fy⁰(t) u(t) t= 1::: N^g (1) Here ^G is a set of systems. Suppose also that the data are subjected to errorsv(t):

y(t) =y⁰(t) +v(t) (2) Z^N=^fy(t) u(t) t= 1::: N^g (3) Let^Abe an identication algorithm, i.e a mapping from Z^N to a space of models:

G^N=^A(Z^N) (4) This algorithm is then said to be robustly convergent (Helmicki et al., 1991) if

lim^!0_Nlim

!1

jjG^N^;G⁰^jj= 0 (5)

jv(t)^j ⁸t (6) for allG⁰²^G. The algorithm is \untuned" if it does not use knowledge of either^G or.

In (5)^jj^jjdenotes a suitable norm, for linear systems, it is typically chosen as theH¹ - norm (sup_!^jG^t(e^i!)^; G⁰(e^i!)^j).

Now traditional identication theory has paid substan- tial attraction to (5). In, e.g. (Ljung and Yuan, 1985)

(Thm 3.3), (5) is established to hold (with probability 1) without the limit , provided (6) is replaced by an assumption that^fv(t)^gis a stochastic process with a co- variance function that decays suciently fast. In (Ljung and Yuan, 1985) a least squares algorithm based on a FIR model is used, and similar results are obtained for ARX - models in (Ljung and Wahlberg, 1992).

Khargonekar and Akcay have established (6) and (5) for ARX - models, see (Akcay, 1992). We shall in the next section show how this can be proved within a \classical identication framework".

Another and major part of this contribution is however that the linear regression (ARX or FIR case) also sets the pattern for a very general \robust identiability" problem in the spirit of (6) and (5). We could formulate this as follows: Let ^M be a certain model structure, i.e. a smoothly parametrized set of models

M=^fM()^j²D^M^g (7) Let a system = ^M(⁰) generate exact data Z⁰^N and let it be disturbance corrupted as in (1), (2) and (3).

We would then say that ^Mis globally identiable at ⁰ if there exists an inputu(t), such that the value ⁰ can uniquely (globally) be recovered from the identication in Z⁰^N. We could say that it is robustly globally identiable if there exists an identication algorithm^J and aN such

that ^N =^J(Z^N) (8)

and lim

!0

j^_N^;⁰^j= 0 (9) More precise denitions will be given in section 3.

A conceptually interesting result is that

Global identiability ⁾ Robust global identiability In section 4 it is described how (10) can be established.(10)

2 The least squares algorithm is globally convergent

It is here proved for both the continuous- and discrete case simultaneously that the least squares algorithm is

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robustly convergent for ARX-models. Let the letter denote either the dierential operatorp= _ddt or the shift operatorq (qy(t) =y(t+ 1)). Consider the ARX-model

A()y(t) =B()u(t) (11) where

A() =ⁿ+a¹ⁿ^;1++an

B() =b¹ⁿ^;1++bn (12) Let

'(t) = ^;ⁿ^;1y(t) ^;ⁿ^;2y(t)::: ^;y(t)

ⁿ^;1u(t)::: u(t)]^T

= a¹ a²::: an b¹::: bn]^T: The model (11) can then be written

ⁿy(t) =^T'(t) (13) The least squares solution to (11)-(13) can be written

^N=

"

N1

N

X

k⁼¹'(k)'^T(k)

#

;1

| {z }

R^;1^N

N1

N

X

k⁼¹'(k)ⁿy(k) provided the inverseR^;1_N exists. The conditions on^fu(t)^g such that RN is uniformly positive denite are called persistence of excitation (p.e.)(Astrom and Wittenmark, 1989), (Ljung, 1987).

Let u~(t) = 1A()u(t) (14) so that

y(t) =B()~u(t) We then have

'(t) =

2

6

4

0 ^;b¹ ^;b² ^;bn 0 0 0 0 ^;b¹ ^;bn^;1 ^;bn 0 ... ... ...

0 0 ^;b¹ ^;b² ^;bn

1 a¹ a² an 0 0 0 1 a¹ an^;1 an 0

0 0... 1 a¹ an

3

7

5

| {z }

A

2

6

4

²ⁿ^;1u~(t)

²ⁿ^;2u~(t) ......

......

u~(t)

3

7

5

Let '~(t) = ²ⁿ^;1u~(t)::: u~(t)]^T so '(t) =^A'~(t)

The Sylvester matrix ^A is non-singular if A() and B() have no common factors. Hence

RN¹I if R~N = 1N

N

X

t^;1'~(t)~'^T(t)²I (15) The condition (15) is known as ~u(t) is p.e. of order 2n (sometimes the denition is applied to the limit asN ^!

1, but we should here let it hold uniformly inN N⁰ - which is the same for large enoughN⁰). In view of (14) u~(t) is p.e. of order 2nif u(t) is, provided A() has no zeros on the unit circle (no zeros on the imaginary axis).

There are many ways to generate sequences^fu(t)^gthat are p.e. as required (Astrom and Wittenmark, 1989), (Ljung, 1987). Now suppose that we have a disturbance

A()y(t) =B()u(t) +v(t)

where^jv(t)^j< 0 <¹. in the discrete case and in addition^jv⁽^j⁾(t)^j< cj 0cj<¹for the continuous case. Let

y(t) =yu(t) +yv(t) where

A()yu(t) =B()u(t) A()yv(t) =v(t)

IfA() has no roots on the unit circle (no roots on the imaginary axis) we have

jyv(t)^j³max^jv(t)^j³ (16) Let '(t) ='u(t) +'v(t)

where

'u(t) = ^;ⁿ^;1yu(t)^;yu(t)ⁿ^;1u(t)u(t)]^T 'v(t) = ^;ⁿ^;1yv(t)^;yv(t)00]^T

Then R_N =R_uN +R_vN

with obvious notation, and for (15) and (16) we have that RN¹I

for small enough. Now, with

~N= ^N^;⁰ we have

~N=R^;1_N N¹

N

X

t⁼¹'(t)v(t)

(3)

and

jj~N^jj 1

¹max^fj'(t)^jgC

We conclude that the least squares algorithm is robustly convergent for ARX-models providedA() has no roots on the unit circle (no roots on the imaginary axis), A() andB() have no common roots andu(t) is p.e. of order 2n. If there are less parameters thanninB() the p.e. condition on u(t) can be weakened. Suppose that B(q) =b¹qⁿ^;1+:::+bmqⁿ^;^m(B(p) =b¹p^m+:::+bm) then it can be shown, analogously, thatu(t) has to be p.e.

of ordern+m. The continuous case and the discrete case have to be treated separately here.

3 Problem statement

In section 1 some denitions for discrete time systems have been stated. The denitions will now be carried over to the continuous time case. Let ^M be a model structure (continuous or discrete time) which is a smoothly parametrized set of models

M=^fM()^j²^D^M^g (17) Let a system = ^M(⁰) generate exact input-output data

Z⁰=^fy⁰(t) u(t) t= 1 ::: N or t²ab]^g (18) where t = 1::: N for the discrete time case and t ²

ab] for the continuous time case. Let the disturbance corrupted data be written

Z=^fyv(t) u(t) t= 1 ::: N or t²ab]^g (19) Herey_v(t) depends on the disturbancev(t) with^jv(t)^j

⁸ t 0 < ¹ in the discrete case and in the continuous casev(t) also has to be band limited^jv⁽ⁱ⁾(t)^j<

ci 0ci <¹.

Denition 3.1 A model structure^Mis said to be globally identiable atwith respect to^D^Mif there exists an input signalusuch that the output signal !y( u)⁶= and

y!( u)y!( u) ²^D^M⁾= (20)

2

Denition 3.2 A model structure^Mis robustly globally identiable if there exists an identication algorithm

J and aZ such that

^=^J(Z) (21)

and lim

!0

j^^;⁰^j= 0 (22)

2

The model structure considered here are all those systems that can be described in the following form

x_ =f(xu)

y=h(xu) (23)

with the statesx²Rⁿ, outputy²R^m, inputu²R^kand parameters²R^d. The functions f andhare assumed to be polynomial in the variablesxuand .

The disturbed system is considered to be z_=fv(zuv)

y_v=h_v(zuv) (24) with z ² Rⁿ yv ² R^m u ² R^k and ²R^d. The disturbancev²R^l is bounded above and band limited, i.e.

jv_i⁽^j⁾^j < cj 0cj <¹ j = 01:::n+d+ 1 i = 1:::l. It is assumed that the signalsuv ²Cⁿ⁺^d⁺¹ab], whereCⁿab] denotes the space ofntimes continuously dierentiable functions with t ² ab]. The functions f_v and h_v are polynomials in the variables z u v such that f_v(zu0) = f(zu) and h_v(zu0) = h(zu).

Given initial statesx(a) of the system (23) there exists a solution to (23) in some intervalt²ab] (Birkho and Rota, 1969), (Arnold, 1987).

Theorem 3.1 If a solution exists to the system (23) with t²ab] then the solution for the system (24) also exists witht²ab] if is suciently small

Proof. The proof is done by using similar methods as for continuation of solutions in (Birkho and Rota, 1969)

2The parameter space^D^M for this model structure is here assumed to be equal to R^d. However it may be a smaller set if the physics is taken into account. Finally, a dierential polynomialF(uu:::y_ y:::_ ) will here be denotedF(uyp) wherep= _ddt.

4 Global identiability implies robust global identiability

It is shown in (Glad and Ljung, 1990a), (Glad and Ljung, 1990b) that a system (23) is globally identiable if and only if there can be found dierential equations in the form #i(uyp) +i$i(uyp) = 0 (25) where i = 1::: d by dierentiating, adding, scaling and multiplying the equations in (23). An algorithm for

nding these polynomials is given in (Glad and Ljung, 1990a), (Glad and Ljung, 1990b). It can be shown by using the result in (Glad, 1988) that there are no higher derivatives thann+din the polynomials #i(uyp) and

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$i(uyp). Replacingy withyv of the disturbed system (24) yields

#i(uyvp)+i$i(uyvp)+%i(uyvvzp) = 0 (26) where %i(uyvvzp) is a polynomial in the parameter

and the variables uyv v and their derivatives. The expression for %i() is given in the proof of theorem 4.1 below.

The identication algorithm can now be dened.

Denition 4.1 Let the identication algorithm^J be

^i =^;

Rb

a#i(uyvp)$i(uyvp)dt

Rb

a$²_i(uyvp)dt ⁽²⁷⁾

2

Here it is assumed that ^R_a^b$²_idt ⁶= 0, i.e., the signal u is p.e. (Glad and Ljung, 1990a), (Glad and Ljung, 1990b). Using equation (25) in (27) gives that ^ = in the undisturbed case (23).

Theorem 4.1 Assume the model (23) to be globally identiable, as in (Glad and Ljung, 1990a), (Glad and Ljung, 1990b). Let the initial conditionx(a) and the con- troluonab] be chosen such that (23) has a solution on

ab] and that the excitation conditions giving global identiability, (25), (see (Glad and Ljung, 1990a), (Glad and Ljung, 1990b)) are satised. Then (22) is satised, i.e.

the model is robustly globally identiable.

Lemma 4.1

#i(uy_vp) +_i$i(uy_vp) + %i(uy_vvzp) = 0 where%i() satises

j%i(uyvvzp)^jC for suciently small

Proof: Since the right hand sides of equation (23) and (24) are polynomials in the variables ux respectively uvzanduv²Cⁿ⁺^d⁺¹the solutionsx(t) andz(t) are n+dtimes continuously dierentiable functions (Arnold, 1987). It also follows that z(t) and its n+d derivatives converges uniformly to x(t) and the corresponding derivatives (Birkho and Rota, 1969). The output sig- nals yv y and their derivatives up to order n+d can now be obtained from (23) and (24). Furthermorey⁽v^j⁾(t) converges uniformly to y⁽^j⁾(t) j = 0:::n+d. This follows from the fact that fvhv are polynomial in the variableszuv and converges tofh, ^jv_i⁽^j⁾^j< cj and uv² Cⁿ⁺^d⁺¹. Polynomials in yv(t) and its derivatives will also converge uniformly to the corresponding polynomials iny(t) and its derivatives.

Introduce the notion

2

6

4

A¹(uxp) ...An(uxp) An⁺¹(uyx) ...An⁺m(uyx)

3

7

5

=

2

6

4

x_¹^;f¹(xu) ..._

xn^;fn(xu) y¹^;h¹(xu) ...ym^;hm(xu)

3

7

5

: (28)

for the system (23). The polynomials (25) can then be expressed as

#i(uyp) +_i$i(uyp) =

X

ij Qij(uyxp)A⁽_i^j⁾(uyx) (29) where the sum is nite andQij are some polynomials in the parameterand the variablesuyxand their derivatives. Similarly introduce the notionAvi(uyzp) for the system (24). Changing y and x to yv and z in the expression (29) gives

#i(uyvp) +i$i(uyvp) =

X

ij Qij(uyvzp)A⁽_i^j⁾(uyvz) =

X

ij Qij(uyvzp)A⁽_vi^j⁾(uyvzv)+

| {z }

=0

X

ij Qij(uyvzp)A⁽_i^j⁾(uyvz)^;A⁽_vi^j⁾(uyvzv)

| {z }

; i(uy^vvzp⁾

which can be compared with (26). Now (30)

j%i(uyvvzp)^j

X

ij

jQij(uyvzp)^j

cîj⁰^jjv^jj+cîj¹^jjv_^jj+:::+cîj_n⁺_d^jjv⁽ⁿ⁺^d⁾^jjC for some constants 0cîj_k <¹ ²

Proof of Theorem 4.1

If the system (23) is globally identiable then there are expressions of the form (26). The identication algorithm

J gives

^i=^;

Rb

a#i(uyvp)$i(uyvp)dt

Rb

a $²_i(uyvp)dt ⁼

i+

Rb

a %i(uyvvzp)$i(uyvp)dt

Rb

a$²_i(uyvp)dt ⁽³¹⁾

(5)

We will show that

Rb

a %i(uyvvzp)$i(uyvp)dt

Rb

a$²_i(uyvp)dt ^!⁰ ^!0 (32) It follows from Lemma 4.1 that

%i(uyvvzp)^!0 ^!0

uniformly. Using the result in the rst part of the proof of Lemma 4.1 it can be shown that

$i(uyvp)^!$i(uyp) ^!0

#i(uy_vp)^!#i(uyp) ^!0 uniformly. Uniform convergence also implies that

Z b

a $²_i(uy_vp)^!

Z b

a $²_i(uyp) ^!0

Z b

a #i(uyvp)%i(uyvvzp)^!0 ^!0 Since $²_i(uyvp) is continuous and converges uniformly to $²_i(uyp) and ^R_a^b$²_i(uyp) ⁶= 0 then

Rb

a $²_i(uyvp)⁶= 0 ifsuciently small.

Consequently (32) holds so that ^^!⁰ ^!0 ²

5 Conclusions

We have studied a class of systems which can be written in state space form _x =f(ux) y=h(ux) where f and hare polynomials inu x and. We have shown that if such a state space system is globally identiable its disturbed counterpart is robustly globally identiable.

We also remark that similar techniques can be used to establish the result for nonlinear discrete time state space systems.

6 Acknowledgement

This work has been supported by the Swedish Research Council for Engineering Sciences (TFR).

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