Frequency domain accuracy of approximate sampled-data models

Frequency domain accuracy of approximate sampled-data models

Eduardo A. Yucra and Juan I. Yuz

Department of Electronic Engineering

Universidad T´ecnica Federico Santa Mar´ıa, Valpara´ıso, Chile.

Email: eyucra@elo.utfsm.cl, juan.yuz@usm.cl

Abstract: Accurate sampled-data models are required when a digital device interacts with a continuous-time system. Even though exact sampled models can be obtained for linear systems, simple approximate models are sometimes preferred in applications to reduce computational load or to preserve the role of physical parameters. In this paper we quantify the frequency domain relative error of three kinds of approximate sampled models: (i) simple models obtained by derivative approximations , (ii) models obtained including the asymptotic sampling zeros, and (iii) models obtained by a truncated Taylor expansion of the system state equations. In particular, we characterize the bandwidth where each of the proposed models provides the highest accuracy.

1. INTRODUCTION

Most real systems and processes evolve in continuous- time, however, nowadays most control strategies are imple- mented using digital devices. As a consequence, discrete- time representations are required to be, both, simple and accurate in order to achieve good performance of control strategies or identification algorithms. For linear systems, exact sampled-data models can be obtained, for example, assuming a zero-order hold (ZOH) input (see, e.g., ˚Astr¨om and Wittenmark (1997)). One of the consequences of the sampling process is the presence of, so called,sampling ze- rosthat have no counterpart in the underlying continuous- time model (˚Astr¨om et al., 1984). Similar results have been obtained for sampled stochastic systems (Wahlberg, 1988;

Yuz and Goodwin, 2005a). In the nonlinear case, the pres- ence ofsampling zero dynamicshas also been characterized based on normal forms (Monaco and Normand-Cyrot, 1988; Kazantzis and Kravaris, 1997; Yuz and Goodwin, 2005b; Goodwin et al., 2010).

Approximate sampled models are commonly used in appli- cations. However, accuracy is a key issue for sampled data control. In fact, there are cases where a controller based on an approximate model that stabilizes the nominal control loop for every sampling period ∆>0, may fail to stabilize the true continuous-time model no matter how small the sampling period is chosen (Neˇsi´c and Teel, 2004).

In this paper we quantify the accuracy of approximate sampled-data models for linear deterministic systems in terms of the associated relative error in the frequency domain. The paper extends the results in Goodwin et al.

(2008), Ag¨uero et al. (2009) by characterizing the band- width where different approximate models provide the highest accuracy. Our main result shows that this band- width depends only on the continuous-time system poles and zeros, and is independent of the sampling period. The results show that the inclusion of sampling zeros are crucial to achieve low error at high frequencies. We include in the

analysis a class of models obtained by a truncated Taylor series expansion that was not considered in Goodwin et al.

(2008).

2. BACKGROUND ON SAMPLING OF LINEAR SYSTEMS

We consider a linear system given by the transfer function:

G(s) =F(s) E(s) =

Qm

k=1(s−σ_k) Qn

i=1(s−pi) (1) Whereσ_k andp_i are the continuous time zeros and poles, respectively. We assume a zero-order hold (ZOH) input, i.e., u(t) = u(k∆) = uk;k∆ ≤ t < (k+ 1)∆, where ∆ is the sampling period. If the system output is sampled instantaneouslyyk =y(k∆), then the exact sampled-data model (ESD) is given by (˚Astr¨om et al., 1984).

G^ESD_d (z) =F_d(z) Ed(z)

=(z−1) z Z

L⁻¹

G(s) s

_t=k∆

(2) An equivalent expression related to the aliasing effect is given by:

G^ESD_d (e^s∆) = (1−e^−s∆)

∞

X

`=−∞

G(s+ 2πj`/∆) s∆ + 2πj` (3) It is well known that the continuous-time poles p_i are mapped to z = e^pi∆ in the exact sampled model (2).

However, the mapping of the system zeros is much more involved (Blachuta, 1999). The sampled-data model (2) has, in general, relative degree one. This means that there are sampling zeros in the discrete model which have no continuous-time counterpart, these zeros can be characterized for fast sampling rates (˚Astr¨om et al., 1984):

G^ESD_d (z) −−−→^∆≈0 ∆^rBr(z) (z−1)^m

r! (z−1)ⁿ (4) wherer=n−m is the system relative degree andBr(z) are the Euler-Fr¨obenius polynomials (Weller et al., 2001).

Copyright by the 8711

Br(z) =b^r₁z^r−1+b^r₂z^r−2+. . .+b^r_r (5) b^r_k =

k

X

`=1

(−1)<sup>k−`</sup>`^r r+ 1

k−`

(6) These polynomials satisfy several properties:

(P1) From (5) and (6) it follows thatBr(1) =r!

(P2) The coefficients can be computed recursively, i.e., b^r₁=b^r_r= 1, for allr≥1, and

b^r_k =kb^r−1_k + (r−k+ 1)b^r−1_k−1 (7) fork= 2, . . . , r−1.

(P3) From the symmetry of the coefficients it follows that Br(z) =z^r−1Br(z⁻¹) (8) (P4) Ifris an even number,Br(−1) = 0.

(P5) Their roots are always negative real and satisfy an interlacing property: every root of the polynomial B_r+1(z) lies between every two adjacent roots of Br(z), for allr≥2.

(P6) The following recursive relation holds:

B_r+1(z) =z(1−z)B_r⁰(z) + (rz+ 1)B_r(z) (9) for allr≥1, and whereBr0 =^dB_dz^r.

Additionally, we will use the following property:

Lemma 1. LetBr(z) be the polynomial of Euler-Fr¨obenius of orderr, then the derivative atz= 1 is given by:

B⁰_r(1) = r!(r−1)

2 (10)

Proof. If we differentiate equation (8) we obtain:

B⁰_r(z) = (r−1)z^r−2Br(z⁻¹)−z^r−3B_r⁰(z⁻¹) (11) The result is obtained evaluatingz= 1 and using (P1). 2

3. APPROXIMATE SAMPLED-DATA MODELS In this section we present different approximate sampled models. In the next sections we compare the proposed models with ESD model (2).

(1) Simple Derivate Replacement(SDR) model: ob- tained using the Euler approximation for time deriva- tives, i.e., we replaces= ^z−1_∆ in (1):

G^SDR_d (z) =G(^z−1_∆ ) (12) This model does not include any sampling zero and has the same relative degree as the continuous- time system. We also consider a Tustin Derivate Replacement(TDR) model

G^{T DR}_d (z) =G _∆^{2 z−1}_z+1

(13) This model has relative degree equal to 0, but does not include any of the sampling zeros in (4).

(2) Asymptotic Sampling Zeros (ASZ) model: Ob- tained including the asymptotic sampling zeros in the SDR model (12).

G^ASZ_d (z) = Br(z)

r! G^SDR_d (z) (14) The scaling factor 1/r! ensuresG^ASZ_d (1) =G(0).

From (P4), if r is an even number, then the rel- ative error corresponding to the ASZ model grows to infinity as we approach the Nyquist frequency _∆^π (see Section 4). Thus, we introduce a Corrected

Asymptotic Sampling Zeros (CSZ) model that includes a refined approximation of the asymptotic sampling zero atz=−1. The model is given by

G^CSZ_d (z) =K(z)

K(1)G^ASZ_d (z) (15) where

K(z) =(z+ 1 +c∆)

(z+ 1) (16)

When r is an odd number, then c∆ = 0 and, as a consequence G^CSZ_d (z) = G^ASZ_d (z). When r is an even number, then c∆ is chosen such that an approximation of the order of ∆ of the sampling zero aroundz=−1 is obtained (Blachuta, 1999), i.e.,

c∆= ∆ r+ 1

( _n X

i=1

pi−

m

X

k=1

σk

)

(17) (3) Different-order Taylor Expansion(DTE) model:

This model was proposed in (Yuz and Goodwin, 2005b) for nonlinear systems expressed in the, so called, normal form (Isidori, 1995). A state-space representation of the linear system given in (1) in normal form is given by













˙ x1

...

˙ x_r−1

˙ x_r

˙ η













=













x2

... xr

Q₁₁ξ+Q₁₂η+u Q21ξ+Q22η













(18)

whereξ= [x₁, . . . , x_r]^T,η= [x_r+1, . . . , x_n]^T, and the system output is y=x1. Explicit expressions forQij

above are given in (A.12)-(A.15). An approximate sampled-data model is then obtained performing a Taylor series expansion of each state up to an order such that the input explicitly appears:











q x1

.. . q xr−1

q xr

q η











=

















x1+ ∆x2+. . .+∆^r

r!(Q11ξ+Q12η+u) ..

.

xr−1+ ∆xr+∆²

2 (Q11ξ+Q12η+u) xr+ ∆(Q11ξ+Q12η+u)

ηk+ ∆(Q21ξ+Q22η)

















(19)

whereqis the forward shift operator. The associated transfer functionG^{DT E}_d (z) can be obtained from (19).

A consequence of Lemma 2 (below) for systems with relative degreeran even number, is that we have to introduce a Corrected Different-order Tay- lor Expansion (CTE) model in order to keep the associated relative error bounded near the Nyquist frequency (similarly as we modified ASZ models to obtain CSZ models):

G^{CT E}_d =K(z)

K(1)G^{DT E}_d (z) (20) where K(z) is defined in (16)-(17).

The DTE model satisfies the following property:

Lemma 2. Consider the model (19), where the output is y = x1. Then, the zeros of the associated discrete time transfer functionG^{DT E}_d (z) are given by

(1) An Euler approximation of theintrinsic zeros of the system, (i.e., they appear atz= 1 + ∆σ_k), and (2) The asymptotic sampling zeros corresponding to a

system of relative degreer, (i.e., the roots ofB_r(z)).

Thus, the associated transfer function has the form G^{DT E}_d (z) = Br(z)F(^z−1_∆ )

r! ˜E(z) (21) Proof. It follows from (Yuz and Goodwin, 2005b, Theo- rem 2), where it is shown that the zero dynamics of the approximate sampled model are given by(i) the intrinsic zero dynamics discretized using Euler approximation, and (ii) the sampling zero dynamics, whose eigenvalues are equal to the asymptotic sampling zeros described by the Euler-Fr¨obenius polynomials. 2

The mapping of the continuous-time poles to the roots of ˜E(z) of the DTE model (21) is not straightforward to obtain (see Appendix A). This is in contrast to the exact sampled-data model (2).

4. RELATIVE ERRORS

In this section we consider the following relative error measures in the frequency domain:

Rⁱ₁(ω) =

G^ESD_d (e^jω∆)−Gⁱ_d(e^jω∆) G^ESD_d (e^jω∆)

(22) Rⁱ₂(ω) =

G^ESD_d (e^jω∆)−Gⁱ_d(e^jω∆) Gⁱ_d(e^jω∆)

(23) where the superscript i∈ {SDR, TDR, ASZ, CSZ, DTE, CTE}andω∈[0,_∆^π].

We use the, so called, big-O notation for the asymptotic behavior of a functionf(∆) as the sampling period ∆ goes to zero, i.e.,f(∆)∈ O(∆ⁿ) if and only if∃c >0 such that 0≤f(∆)≤c∆ⁿ for all ∆<∆o.

Theorem 3. The relative error performance of the approx- imate sampled-data models are as follows:

Relative error r: odd r: even kR^SDR₁ (ω)k_∞ O(1) O(_∆¹) kR^{T DR}₁ (ω)k∞ O(1) O(1) kR^ASZ₁ (ω)k_∞ O(∆) O(1) kR₁^CSZ(ω)k∞ O(∆) O(∆) kR₁^{DT E}(ω)k∞ O(∆) O(1) kR^{CT E}₁ (ω)k∞ O(∆) O(∆) Relative error r: odd r: even kR^SDR₂ (ω)k∞ O(1) O(1) kR^{T DR}₂ (ω)k∞ ∞ ∞ kR^ASZ₂ (ω)k∞ O(∆) ∞ kR₂^CSZ(ω)k_∞ O(∆) O(∆) kR₂^{DT E}(ω)k∞ O(∆) ∞ kR^{CT E}₂ (ω)k_∞ O(∆) O(∆) wherekR(ω)k_∞= sup_ω∈[0,π

∆]|R(ω)|

Proof. See Appendix A. 2

5. BANDWIDTH OF ACCURACY

Theorem 3 gives the maximum magnitude of the error as a function of the sampling period. However, no insights are given into the accuracy of the approximate models at different frequencies. In this section we present the main result of the paper: we characterize the bandwidth where each approximate sampled-data model provides higher accuracy.

Theorem 4. Assume that G(s) has relative degree r >1, and has no pure imaginary poles or zeros. Then,

(1) For low frequencies, SDR models are more accurate than ASZ models, i.e., when ω→0,

R^SDR_k (ω)≤R^ASZ_k (ω) (24) where k∈ {1,2}.

(2) The frequencies at which SDR and ASZ models have approximately the same relative error are given by the solutions of the following equation

n

X

i=1

ω² ω²+|p_i|² −

m

X

k=1

ω²

ω²+|σ_k|² = r+ 1

2 (25)

The solutionsω`of (25) satisfy ˇ

p rr+ 1

2n ≤ω_`≤

r(n+m+ 1)bp²+ ˇσ²

r−1 (26)

where ˇp, p, ˇb σ and σb denote the minimum and max- imum magnitude of the system poles, and the mini- mum and maximum magnitude of the system zeros, respectively.

Proof. See Appendix B. 2

The previous theorem can be extended to compare SDR and CSZ models, obtaining the same bounds that appear in (26). Moreover, the assumptions in Theorem 4 can be relaxed: If the relative degree ofG(s) is r= 1 then SDR and ASZ models are the same. On the other hand, if the continuous-time system includes pure imaginary poles (zeros), the relative error goes to one (infinity) at those frequencies.

Equation (25) can be numerically solved for any given system, however, the bounds in (26) are, from our point of view, more insightful. Indeed, the bandwidth defined by (26) depends on the continuous-time poles and zeros of the system, but is independent of the sampling period.

This result provides a clear guideline to which approximate models are better to use: If one will use an approximate model up to the continuous-time system bandwidth, then SDR models may provide enough accuracy. However, if one requires a model more accurate for higher frequencies (and, in particular, near the Nyquist frequency), then the sampled modelhas to incorporate sampling zeros.

An interesting question at this point is how to obtain a model accurate as SDR models for low frequencies, but, at the same time, accurate as ASZ (or CSZ) models for frequencies close to the Nyquist frequency _∆^π. In fact, the next result shows that CTE models provide such desired behavior.

Theorem 5. For a sampling period sufficiently small, the relative error associated with CTE models behaves as

10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³

−80

−70

−60

−50

−40

−30

−20

−10 0 10 20

Magnitude (dB)

System: R2_CSZ Frequency (rad/sec): 2.77 Magnitude (dB): −17.9

System: R2_CSZ Frequency (rad/sec): 2.75 Magnitude (dB): −37.9 Bode Diagram

Frequency (rad/sec) R2_SDR

R2_ASZ R2_CSZ

Fig. 1. Relative errorsR^SDR₂ (ω) ,R^ASZ₂ (ω) andR^CSZ₂ (ω) for two different sampling periods ∆ ∈ {0.1,0.01}

(Example 6).

the relative error corresponding to SDR models for low frequencies and as the relative error corresponding to CSZ models near the Nyquist frequency, i.e.,

R^{CT E}_k (ω)−−−→^ω→0 R^SDR_k (ω) (27) R^{CT E}_k (ω) ^ω→

π

−−−−→∆ R^CSZ_k (ω) (28) wherek∈ {1,2}.

Proof. See Appendix C. 2 6. EXAMPLES

In this section we present examples to illustrate the bounds presented in Theorem 4. Additionally, we analyze the accuracy of DTE and CTE models.

Example 6. ConsiderG(s) = _(s+1)(s+2)² . Then:

G^ESD_d (z) = b1(∆)z+b0(∆)

(z−e^−∆)(z−e^−2∆) (29) G^SDR_d (z) = 2

(^z−1_∆ + 1)(^z−1_∆ + 2) (30) G^ASZ_d (z) = (z+ 1)

(^z−1_∆ + 1)(^z−1_∆ + 2) (31) G^CSZ_d (z) = 2(z+ 1−∆)

(^z−1_∆ + 1)(^z−1_∆ + 2)(2−∆) (32) The coefficients b1(∆) and bo(∆) in can be explicitly obtained.

Figure 1 shows the relative errorsR^SDR₂ (ω),R^ASZ₂ (ω) and R^CSZ₂ (ω) for two different sampling periods, ∆ = 0.1 and

∆ = 0.01. The bounds predicted in Theorem 4 are given by 0.8165≤ω`≤3.4641. Figure 1 shows that the frequency where relative errors intersect is within the band given in (6) and this does not change if the sampling frequency is increased . 2

Example 7. We consider:

G(s) = p1ω_n²

(s+p1)(s²+ 2ξωns+ω_n²) (33) wherep1= 1, ξ= 0.1, ωn= 4.

10⁻¹ 10⁰ 10¹ 10²

−70

−60

−50

−40

−30

−20

−10 0

Magnitude (dB)

Bode Diagram

Frequency (rad/sec) R2_SDR

R2_CSZ R2_CTE

Fig. 2. Relative errorsR^SDR₂ (ω),R^CSZ₂ (ω) andR^{CT E}₂ (ω), for a sampling period ∆ = 0.01 (Example 7).

We compare the associated SDR and CSZ models with CTE models. In this case, G^CSZ_d (z) = G^ASZ_d (z) and G^{DT E}_d (z) = G^{CT E}_d (z). Figure 2 shows that CTE models provide the same relative error than SDR models for low frequencies. For higher frequencies, CTE models provide higher accuracy than SDR models, because they include the asymptotic sampling zeros. Moreover, we see that CTE models provides the highest accuracy for low and high frequencies, and it is only slightly outperformed by CSZ models in a band aroundω= 1. 2

7. CONCLUSION

In this paper we have characterized the accuracy of differ- ent approximate sampled-data models for linear systems in terms of frequency-domain relative error. We have shown that a simple derivative replacement model using Euler approximation provides a smaller relative error within the continuous-time system bandwidth. However, if one is required to have a more accurate model for higher frequencies (in particular, near the Nyquist frequency), then sampling zeros must be included in the model. Addi- tionally, the results show that a particular model obtained by different-order Taylor expansion of the state equations provides an accurate description for all frequencies. The insights provided by the relative error also presented in this paper can be taken into account for the case of approximate sampled data models for nonlinear systems.

Appendix A. PROOF OF THEOREM 3

The H_∞-norm of Rⁱ₁(ω) and Rⁱ₂(ω) for SDR, ASZ, and CSZ models was characterized in Goodwin et al. (2008).

Thus, the proof only considers theH_∞-norm analysis for TDR, DTE, and CTE models.

A key observation is that, for any fixed frequency Gⁱ_d(e^jω∆) −−−−→^∆→0 G(jω) This implies that, for any fixed ω, the relative errors (22) and (23) go to zero. Thus, the analysis is restricted to the behavior of the relative error at the Nyquist frequencyωN = _∆^π ⇔z=−1.

LetG(s) be the linear system in (1) represented as G(s) = s^m+bm−1s^m−1+· · ·+b1s+b0

sⁿ+a_n−1sⁿ⁻¹+· · ·+a1s+a0

(A.1) where b_m−1 = −Pm

k=1σk and a_n−1 = −Pn

i=1pi. The transfer function (A.1) can be expanded as

G(s) =_s¹r + Θ_sr+1¹ +O(s^−r−2) (A.2) where Θ =b_m−1−a_n−1=X

pi−X

σk (A.3) The ESD model for an r-th order integrator is given by the right hand side of (4) (˚Astr¨om et al., 1984). Thus,

G^ESD_d (z) = ∆^rBr(z)

r!(z−1)^r + Θ ∆^r+1Br+1(z)

(r+ 1)!(z−1)^r+1 +O(∆^r+2) (A.4) Moreover,G(s) can be expressed in state-space form

˙

x=Ax+Bu (A.5)

y =Cx (A.6)

where A=

0_n−1×1 I_n−1

−a₀ . . . −a_n−1

, B=

0_n−1×1 1

(A.7) C= (b_o b₁ · · · b_m−1 1 0· · · 0) (A.8) There exists a transformation_ξ

η

= Φxsuch that (A.5)- (A.8) is rewritten in the normal form (Isidori, 1995)

ξ˙=A_rξ+B_rCA^rΦ⁻¹_ξ

η

+B_ru (A.9)

˙

η=Q21ξ+Q22η (A.10)

where Ar=

0r−1×1 Ir−1

0 01×r−1

, Br=

0r−1×1

1

(A.11) Q21=

0_1×r−1 0_m−1×r−1 1 0_r−1×1

(A.12) Q₂₂=

0_m−1×1 I_m−1

−b0 −b1 · · · −bm−1

(A.13) and

Φ =









 C

... CA^r−1 Im 0_m×r











(A.14)

Note that, comparing (A.9)-(A.13) with (18) Q11ξ+Q12η+u=CA^rΦ⁻¹

ξ η

+u (A.15)

Then the DTE model (19) can be expressed as

q−1

∆ ξ=A^δ_rξ+B_r^δ[CA^rΦ⁻¹

ξ η

+u] (A.16)

q−1

∆ η=Q21ξ+Q22η (A.17)

where

A^δ_r= e^Ar∆−I

∆ =Ar+O(∆) =













0 1 · · · _(r−1)!^∆r−2 ... . .. . .. ... ... . .. . .. 1 0 · · · 0











 (A.18) B_r^δ=

R∆

0 e^ArνB_rdν

∆ =B_r+O(∆) =

∆^r−1 r! · · · 1

^T (A.19)

Then, the denominator of the transfer function (21) asso- ciated with the DTE model, is given by

E(z) = det˜

z−1

∆ −A_d

(A.20) where, from (A.16)-(A.17),

Ad=

A^δ_r 0r×m

0m×r 0m×m

+

B_r^δCA^rΦ⁻¹ Q₂₁ Q₂₂

(A.21) The characteristic polynomial (A.20) can be expanded as E(z) =˜ _∆¹n{(z−1)ⁿ−tr(Ad)(z−1)ⁿ⁻¹∆+O(∆²)} (A.22) Note that, from (A.9)-(A.10) ,

ΦAΦ⁻¹=

A_r 0_r×m 0_m×r 0_m×m

+

BrCA^rΦ⁻¹ Q21 Q22

(A.23) From (A.8), (A.18), (A.19) and (A.21), we obtain

Ad=ΦAΦ⁻¹+O(∆) (A.24)

⇒tr(Ad) =−a_n−1+O(∆) (A.25) Then

E(z) =˜

z−1

∆

n

1 +^an−1_z−1^∆+O(∆)

(A.26) The numerator of (21) can be expressed as

F˜(z) = ^Br(r)(z−1)_r!∆m ^m

1 + ^bm−1_z−1^∆+O(∆²)

(A.27) Equation (A.26) and (A.27) yields the following expansion

G^{DT E}_d (z) = ∆^rBr(z) r!(z−1)^r

(1 +b_m−1z−1^∆ +O(∆²) 1 +an−1 ∆

z−1+O(∆²) )

(A.28)

= ^∆_r!(z−1)^rBr(z)r + Θ^∆_r!(z−1)^r+1Brr+1^(z)+O(∆^r+2) (A.29) From (A.4) and (A.29), the absolute error between DTE and ESD models is given by

G^ESD_d −G^{DT E}_d = Θ^∆r+1(r+1)!(z−1)^{{Br+1−(r+1)Br+1 r}} +O(∆^r+2) Then

G^ESD_d −G^{DT E}_d

G^ESD_d =(r−1)(z−1)B^{∆Θ(Br+1−(r+1)B}r+∆ΘB^r_r+1⁾ +O(∆²) (A.30)

G^ESD_d −G^{DT E}_d

G^{DT E}_d =(r+1)((z−1)B^{∆Θ(Br+1−(r+1)B}_r+∆ΘB^r)_r)+O(∆²) (A.31) whereB_r(z) =B_r. Then, replacingω= _∆^π (z=−1) gives the result for DTE models.

For the case of the CTE and DTE models, we proceed similarly. Full details can be found in the associated Technical Report (Yucra and Yuz, 2010). 2

Appendix B. PROOF OF THEOREM 4

We first obtain an approximation for ESD, SDR, and ASZ models. We consider the ESD model expressed as in (3). When ∆ → 0, the terms in the infinite sum (3) are negligible except for`= 0. Thus, we use the following approximation

G^ESD_d (e^s∆)≈(1−e^−s∆)G(s) s∆

=G(s)

1−¹₂s∆ +O(∆²)

(B.1) Moreover, if we now examine the other terms in the infinite sum (3), and we replaceG(s) using (1), we have that

(1−e^−s∆) s∆ + 2πj`

Qm

k=1(s+ 2πj`/∆−σ_k) Qn

i=1(s+ 2πj`/∆−pi)

= ∆^r 1−e^−s∆

s∆ + 2πj`

Qm

k=1(2πj`+ ∆(s−σ_k)) Qn

i=1(2πj`+ ∆(s−pi)) =O(∆^r+1) (B.2)

ω₁= s

ˆ

p²(1 +n+m)−σˇ²(m+n−1) +p ˇ

σ⁴(n+m−1)²+ 2(n²−6nm+m²−1)ˇσ²pˆ²+ (1 +n+m)²pˆ⁴

2(r−1) (B.20)

Thus, all such terms are of the order of ∆^r+1, and do not modify the expansion in (B.1). Now, SDR models can be expressed as follows

G^SDR_d (e^s∆) =G

e^s∆−1

∆

=G(s+^s2₂^∆+O(∆²)) (B.3) If we now replaceG(s) using (1), we obtain

G^SDR_d (e^s∆) = Qm

k=1(s+¹₂s²∆ +O(∆²)−σk) Qn

i=1(s+¹₂s²∆ +O(∆²)−pi)

=G(s) ( Qm

k=1(1 +¹_2s−σ^s2∆

k+O(∆²)) Qn

i=1(1 +¹_2s−p^s2∆

i +O(∆²)) )

(B.4) Then We then obtain a Taylor series expansion for the expression in curly brackets

G^SDR_d (e^s∆) =G(s)

1 +¹₂s∆ψ(s) +O(∆²) (B.5) where ψ(s) =

m

X

k=1

s s−σ_k −

n

X

i=1

s

s−p_i (B.6) We now obtain an approximation for the ASZ model. A Taylor series expansion of Br(e^jω∆), at ∆ = 0 is given by

Br(e^s∆) =Br(1) +B_r⁰(1)s∆ +O(s²∆²)

=r! + ^r!(r−1)₂ s∆ +O(s²∆²) (B.7) where we have used the fact thatB_r(1) =r! and Lemma 1.

Using (14), (B.5), and (B.7), we obtain G^ASZ_d (e^s∆) =G(s)

1 +¹₂[ψ(s) +r−1]s∆ +O(∆²) (B.8) From (B.5) and (B.8), the relative errors are expanded as Rⁱ₁(ω) = ¹₂|ω∆(1 +ψ(jω))|+O(ω²∆²)i∈ {SDR, ASZ}

(B.9) Rⁱ₂(ω) = ¹₂|ω∆(1 +ψ(jω))|+O(ω²∆²)i∈ {SDR, ASZ}

(B.10) Then as ω→0, the terms O(ω²∆²) can be neglected for any sampling period ∆. From (B.9) and (B.10) gives part 1) of the theorem.

To show part 2) we consider the condition

R^SDR_k (ω) =R^ASZ_k (ω) (B.11) for k ∈ {1,2}. Using (B.9), it can be shown that (B.11) leads (fork= 1 and fork= 2}) to

ψ(jω) +r =

ψ(jω) + 1

+O(∆) (B.12)

If we neglect terms of the order of ∆ and expand, we obtain Ren

ψ(jω)o

=

m

X

k=1

ω(ω−Im{σk}) ω²+|σ_k|² −

n

X

i=1

ω(ω−Im{pi}) ω²+|p_i|²

=−r+ 1

2 (B.13)

Noting that complex poles and zeros appear in complex conjugate pairs, we obtain equation (25).

Now we find upper and lower bounds for the roots of f(ω) =

n

X

i=1

ω² ω²+|p_i|² −

m

X

k=1

ω²

ω²+|σ_k|² −r+ 1

2 (B.14)

We find bounding functions ˇf(ω) and ˆf(ω) such that fˇ(ω)≤f(ω)≤fˆ(ω) (B.15) for all ω >0, such that ˆf(0) <0 and limω→∞fˇ(ω)>0.

This ensure, that the roots of f(ω) lie in the interval between the smallest positive root of ˇf(ω) and the largest positive root of ˆf(ω). We choose

fˇ(ω) =n_ω2^ω+ ˆ²p² −m_ω2^ω+ˇ²σ² −^r+1₂ (B.16) The positive real root of this equation isω=ω1, given in equation (B.20) at the top of the page . The frequencyω1

can be bounded as ω1≤ω2=

q(n+m+1) ˆp²+ˇσ²

r−1 (B.18)

whereω₂ its obtained adding the term

4ˇσ⁴(n+m) + 4(4nm+n+m+ 1)ˇσ²pˆ² (B.19) to the inner square root term inω₁in (B.20).

On the other hand, a majorizing function is given by fˆ(ω) =n^ω_pˇ2² −^r+1₂ (B.20) Note that ˆf(0) = −^r+1₂ < 0, and thus, the lower bound for the roots off(ω) is given by the only positive root of fˆ(ω), i.e.

ω=ω3= ˇp qr+1

2n (B.21)

Thus, the roots of f(ω) are bounded by (B.21) and (B.18). 2

Appendix C. PROOF OF THEOREM 5

To show that CTE relative errors behave as CSZ relatives errors, an approximation of ASZ model is first obtained.

From (14) and (A.2) we have that G^ASZ_d (z) = ∆^rBr(z)

r!(z−1)^r+ Θ ∆^r+1Br(z)

r!(z−1)^r+1+O(∆^r+2) (C.1) From (15) and (C.1), ifris an odd number, then

G^CSZ_d =G^ASZ_d = ∆^rB_r(z)

r!(z−1)^r + Θ∆^r+1B_r(z)

r!(z−1)^r+1 +O(∆^r+2) (C.2) Whenr is even number, the CSZ model takes the form

G^CSZ_d (z) = ^K(z)_K(1)G^ASZ_d (z) = _(z+1)(2+c^2(z+1+c∆)

∆)G^ASZ_d (z) (C.3) Replacing (C.1) into (C.3) and using (17), we obtain

G^CSZ_d (z) =^∆_r!(z−1)^rBr(z)_r + Θ^∆_r!(z−1)^r+1Br_r+1^(z)+

−Θ2(r+1)!(z+1)(z−1)^{∆r+1Br(z) r−1} +O(∆^r+2) (C.4) It can be shown that CSZ and CTE models are equal up to an error of the order of ∆^r+2. Then

|R_k^CSZ−R^{CT E}_k |=O(∆²) (k∈ {1,2}) (C.5) Moreover, from Theorem 3, Rⁱ_k = O(∆), for i ∈ {CSZ, CT E} and k = {1,2}. Thus, in (C.5) and for a small ∆, near the Nyquist frequencyR^CSZ_k =R^{CT E}_k . This corresponds to (28).

For low frequencies, CTE models can be expanded replac- ing ^z−1_∆ =s+O(s²∆) into (A.20). This yields

E(e˜ ^s∆) = det(sI−Ad+O(s²∆))

= det(sI−A) det(I+ (A−A_d)(s−A)⁻¹+O(s²∆))

=E(s) det(I−(A−Ad)(A⁻¹+sA⁻²+· · ·) +O(s²∆))

=E(s)[det(AdA⁻¹−s(A−Ad)A⁻²) +O(s²∆))]

=E(s)[det(A_dA⁻¹)

−tr((AA⁻¹_d )^A(A−Ad)A⁻²)s+O(s²∆)] (C.6) where (·)^Adenotes the adjugate matrix (Bernstein, 2009).

Simplifying the second term of the right side of (C.6) tr((AA⁻¹_d )^A(A−Ad)A⁻²) = det(A)tr(A^A_d)−det(Ad)tr(A^A)

=−^(r−1)∆₂ a_n−1 (C.7)

where the last equality follows from lengthy manipulation of the matrices involved (for full details see (Yucra and Yuz, 2010)). Then, using (C.6) and (C.7), yields

E(e˜ ^s∆) =E(s)(1 + ^(r−1)∆₂ a_n−1s+O(s²∆ + ∆²)) (C.8) where f(s,∆) ∈ O(s²∆ + ∆²) if and only if |f(s,∆)| <

C|s²∆ + ∆²|, in the vicinity of (s,∆) = (0,0). From (21) and (B.7) the following expansion is obtained

G^{DT E}_d (e^s∆) =G(s)

( 1−^(r−1)₂ s∆²+O(s²∆) 1−^(r−1)₂ s∆²+O(s²∆ + ∆²)

)

=G(s){1 +O(s²∆ + ∆²)} (C.9) Then, when the relative degree r is an odd number,

G^{CT E}_d (e^s∆) =G^{DT E}_d (e^s∆) =G(s){1 +O(s²∆ + ∆²)}

(C.10) When ris an even number, the correction term (16) has to be included, obtaining a similar expansion:

G^{CT E}_d (e^s∆) = K(e^s∆)

K(1) G^{DT E}_d (e^s∆)

= 1 +_es∆^c∆₊₁

1 + ^c₂^∆ G^{DT E}_d (e^s∆)

=G(s){1 +O(s∆²+s²∆)} (C.11) Equations (B.1) and (C.10) (or (C.11)) yield

G^ESD_d (e^s∆)−G^{CT E}_d (e^s∆) G^ESD

d (e^s∆) = ^∆₂s+O(s²∆ +s∆²) (C.12)

G^ESD_d (e^s∆)−G^{CT E}_d (e^s∆)

G^{CT E}_d (e^s∆) = ^∆₂s+O(s²∆ +s∆²) (C.13) Thus, the relative errors (22)-(23) can be expanded as

R^{CT E}_k (ω) = ^∆₂ω+O(ω²∆ +ω∆²), k∈ {1,2} (C.14) Note that ψ(jω) in (B.6) is of the order of ω. Thus, the expansion of the relative errors associated to SDR models ((B.9) and (B.10)) can be expressed as

R^SDR_k (ω) = ^∆₂ω+O(ω²∆ +ω∆²)), k∈ {1,2} (C.15) From (C.14) and (C.15) the error between the relative errors associated with CTE and CSZ models satisfy

R^{CT E}_k (ω)−R^CSZ_k (ω) =O(ω²∆ +ω∆²), k∈ {1,2} (C.16) For a sampling period sufficiently small, and for low frequencies, the terms of order ω²+ω∆ go to zero. This shows (27). 2

REFERENCES

Ag¨uero, J.C., Goodwin, G.C., S¨oderstr¨om, T., and Yuz, J.I. (2009). Sampled data errors in variables systems.

In 15th IFAC Symposium on System Identification - SYSID 2009. Saint Malo, France.

˚Astr¨om, K.J., Hagander, P., and Sternby, J. (1984). Zeros of sampled systems. Automatica, 20(1), 31–38.

˚Astr¨om, K.J. and Wittenmark, B. (1997). Computer Controlled Systems. Theory and Design. Prentice Hall, Englewood Cliffs, N. J., 3rd edition.

Bernstein, D.S. (2009). Matrix Mathematics: Theory, Facts, and Formulas. Princeton University Press, 2nd edition.

Blachuta, M.J. (1999). On zeros of pulse transfer func- tions. IEEE Transactions on Automatic Control, 44(6), 1229–1234.

Goodwin, G.C., Yuz, J.I., and Ag¨uero, J.C. (2008). Rela- tive error issues in sampled-data models. In17th IFAC World Congress. Seoul, Korea.

Goodwin, G.C., Yuz, J.I., Ag¨uero, J.C., and Cea, M.

(2010). Sampling and sampled-data models. In 2010 American Control Conference. Baltimore, USA.

Isidori, A. (1995). Nonlinear Control Systems. Springer, 3rd edition.

Kazantzis, N. and Kravaris, C. (1997). System-theoretic properties of sampled-data representations of nonlinear systems obtained via Taylor-Lie series. International Journal of Control, 67(9), 997–1020.

Monaco, S. and Normand-Cyrot, D. (1988). Zero dynamics of sampled nonlinear systems. Systems and Control Letters, 11, 229–234.

Neˇsi´c, D. and Teel, A.R. (2004). A framework for stabiliza- tion of nonlinear sampled-data systems based on their approximate discrete-time models. IEEE Transactions on Automatic Control, 49(7), 1103–1122.

Wahlberg, B. (1988). Limit results for sampled systems.

International Journal of Control, 48(3), 1267–1283.

Weller, S.R., Moran, W., Ninness, B., and Pollington, A.D. (2001). Sampling zeros and the Euler-Fr¨obenius polynomials.IEEE Transactions on Automatic Control, 46(2), 340–343.

Yucra, E.A. and Yuz, J.I. (2010). Frequency do- main accuracy of approximate sampled-data mod- els. Technical report, Electronics Dept., UTFSM.

http://altair.elo.utfsm.cl/public elo/.

Yuz, J.I. and Goodwin, G.C. (2005a). Generalised filters and stochastic sampling zeros. InJoint CDC-ECC’05.

Seville, Spain.

Yuz, J.I. and Goodwin, G.C. (2005b). On sampled-data models for nonlinear systems. IEEE Transactions on Automatic Control, 50(10), 1477–1489.