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Received: 16 May 2016 Revised: 18 November 2016 Accepted: 28 December 2016

DOI 10.1002/acs.2757

RESEARCH ARTICLE

A robust optimal design for strictly positive realness in recursive

parameter adaptation

Hui Xiao1Ioan D. Landau2Xu Chen1

1Department of Mechanical Engineering,

University of Connecticut, USA

2Control System Department of GIPSA Lab, France

Correspondence

Xu Chen, 191 Auditorium Road U3139, University

of Connecticut, Storrs, CT, 06269-3139, USA.

Email: xchen@engr.uconn.edu

Summary

This paper provides an optimization-based approach to assure the strict positive

real (SPR) condition in a set of recursive parameter adaptation algorithms (PAA).

The developed algorithms and tools enable a multiobjective formulation of the SPR

problem, creating new controls of the stability and parameter convergence in PAAs.

In addition to assuring the SPR condition for global stability in PAAs, we pro-

vide an algorithmic solution for uniform convergence under performance constraints

in PAAs. Several new aspects of parameter convergence were observed with the

adoption of the algorithm in a narrow-band identification problem. The proposed

algorithm is verified in simulation and experiments on a precision motion control

platform in advanced manufacturing.

KEYWORDS

adaptive control, strictly positive real, system identification

1INTRODUCTION

The strictly positive real (SPR) condition of a transfer func-

tion has substantial importance in adaptive control and system

identification.1,2Let Pbe the set of finite-degree polynomi-

als with real coefficients. An essential problem in recursive

parameter adaptation algorithms (PAA) is: given an uncertain

A()∈P(denotes sin continuous-time problems and zor

z−1in discrete-time problems), design a polynomial C()∈

Psuch that the transfer function

C()

A()−(1)

is SPR for all possible values of A(). Here, (∈[0,

1]) is a fixed scalar that depends on the adaptation

algorithm. For instance, in the identification of a model

G(z−1)=B(z−1)/A(z−1) using the output error method with

a fixed compensator, C(z−1)/A(z−1)−1/2 being SPR is cru-

cial to assure the stability of the PAA when using a decreasing

adaptation gain.3The same problem occurs in the more gen-

eral pseudolinear regression algorithm, where the importance

of the SPR condition has been remarked in Section 8.6 of

Goodwin and Sin.1One can find various additional important

applications of the problem in, eg, Dasgupta and Bhagwat,4

Ljung,5Ioannou and Tao,6Landau et al,7and the references

therein.

Under slow adaptation and for special systems with small

gains at frequencies where the regressor has low-spectral

energy, the SPR condition can be relaxed locally by the aver-

aging theory.3The global SPR condition, on the other hand,

is a strong requirement and is not easy to guarantee for an

uncertain A(). Conventionally, one has to guess or apply

another parameter adaptation algorithm to obtain a C()that

is hopefully close to A(). Alternative approaches that use

(1) complex polynomial analysis,4,6,8-16

(2) geometrical design,14,15 and

(3) linear matrix inequalities17,18

have also been investigated. To be more specific, previ-

ous studies by Dasgupta and Bhagwat,4by Marquez and

Damaren,13 and by Wang and Yu16 characterized the SPR

condition and discussed the case when A() belongs to a

set of stable and known polynomials. Two previous studies

by Tesi et al14,15 analyzed the situation when the uncer-

tainty in A() comes from its root locations or bounded

uncertain frequency responses. A general classification was

discussed in references8-12,17,18,where A() is assumed to

lie in a known polytope, with bounded coefficients in the

polynomial. Among the existing results, most discussed

Int J Adapt Control Signal Process 2017; 1–12 wileyonlinelibrary.com/journal/acs Copyright © 2017 John Wiley & Sons, Ltd. 1

2XIAO ET AL.

the case where =0; Anderson and Landau10 and Fu

and Dasgupta17 investigated the more difficult situation

where >0. References4,6,9-13,16-18 mainly analyzed the

continuous-time version of the problem. The discrete-time

robust SPR problem has different characteristics compared

with the continuous-time version.10 Within this category,

Anderson et al8provided conditions for the existence of a

solution; Tesi et al15 showed a geometrical design approach

for systems with disk uncertainties; later in the previous study

by Fu and Dasgupta,17 linear matrix inequalities (LMIs) are

formed to analyze the general SPR condition for an uncertain

transfer function G()−.

The most natural (and recommended in the related text

books1,2,7) way of designing C() is to make it “close” to

A(), such that C()/A()−is approximately 1 −(recall

that ≤1). In fact, such a condition has substantial influ-

ence on the profile of the parameter convergence in adap-

tation algorithms.19 (Sections 2 and 5) This aspect, however, has

been largely discredited in previous literature results. Step-

ping out of the traditional regime of single-objective SPR

design (ie, the goal is exclusively focused on stability of

PAAs), this paper provides a solution to the missing piece

of coupling robust SPRness with the performance of the

PAA. The solution mapping is constructed under a unified

framework using frequency-domain characterization, convex

optimization, and time-domain verification. Using semidef-

inite programing, we provide a design approach that not

only assures the robust SPR condition but also is capa-

ble of finding the optimal C() that is “closest” to A().

This leads to superior parameter convergence where the

single-objective SPR design is inefficient, or infeasible, to

assure fast convergence with small transient variations (cf

Section 5).

Along the optimal designs, a second contribution of the

paper is the achievement of additional optimal properties in

the compensator design. This enables to investigate several

new issues. For instance, in output-error–based adaptation

algorithms, it is favorable for the compensator to have min-

imum order and/or small gain in the high-frequency region.

Such properties are particularly beneficial for high-order sys-

tems and situations where the signal-to-noise ratio is small.

Finally, a major result of the paper is the implication of dif-

ferent design options in practical applications. We implement

different SPR designs experimentally on a precision motion

control platform in advanced manufacturing, and show that

the proposed design and added optimization help improve the

parameter convergence.

The remainder of the paper is organized as follows:

Section 2 reviews SPR transfer functions and formally defines

the problem. In Section 3.1, the basic SPR problem is

addressed. Section 3.2 discusses the introduction of opti-

mal properties to the compensator. Additional extensions are

discussed in Section 4. Section 5 provides several design

examples. Section 6 concludes the paper.

The common polytopic uncertainty8-12,17,18 is adopted here.

We will be focusing on the discrete-time version of the

problem, partially because of its fundamental role in system

identification and adaptive control, and partially because of

the fact that results in the more explored continuous-time

robust SPR problem do not necessarily generalize to

discrete-time systems.10,17 *

2BACKGROUND AND PRELIMINARIES

A proper and rational discrete-time transfer function G(z−1)

is SPR if (1) G(z−1) does not possess any pole outside of or

on the unit circle in the complex plane, and (2) ∀||<,

G(e−j)+G(ej)=2Re{G(e−j)} >0, ie, the real part of

G(e−j) is positive.

From the above definition, if G(z−1) is SPR, then (1)

G(z−1) is stable; (2) the phase response of G(z−1), after nor-

malization to [ −,], lies inside the region (−

2,

2)(see, eg

Anderson et al8and Wang and Yu16); and (3) the Nyquist plot of

G(z−1) lies in the closed right-half complex plane.20

The SPR condition has strong implications in adaptive con-

trol and system identification. Consider recursive PAAs in the

general form of

(k+1)=

(k)+F(k)(k)(k+1),(2a)

F(k+1)−1=F(k)−1+2(k)(k)T(k),0<

2(k)<2;F(0)0,

(2b)

(k+1)=H(z−1)T(k)∗−

(k+1)+w(k+1),(2c)

where

is an estimate of *, the true parameter vector;

(k) is the regressor; and F(k) is the adaptation gain matrix.

H(z−1) is a discrete transfer operator (z−1is regarded as a

1-step delay operator) with monic and coprime numerator and

denominator polynomials. w(k+ 1) is the image of the dis-

turbance under the adaptation law. It is assumed to be either

a sequence of independently distributed normal random vari-

ables, or be stochastically independent of the regressor vector

(k).

(k)is assumed to belong infinitely often to the domain

for which the stationary processes (k,

)and v(k+1,

)can

be defined. Then the PAA converges to a convergent domain

with probability 1 as follows:

Prob lim

t→∞

(k)∈Dc=1,Dc∶

∶T(k)[∗−

]=0,

if there exists 2with 2(k)≤2<2, such that H(z−1)− 2

2is

SPR.3,19

The structure in Equations 2A-C generalizes various forms

of adaptation algorithms. As a particular example, consider

the pseudolinear regression algorithm applied to systems with

the model

y(k)= B0(z−1)

F0(z−1)u(k)+e(k),

*An extension of the algorithm is discussed in Section 6, so that the

continuous-time problem can be similarly addressed.

XIAO ET AL. 3

where e(k) is a white noise that is independent of u(k).

Then applying the theory in Chapter 4.5 of Ljung and

Soderstrom19 gives that the eigenvalues of the con-

vergence matrix associated to the ordinary differential

equation are −1and −1/F0(k), where k’s are the

nonzero poles of B0(z−1)/F0(z−1). For us to assure

negative eigenvalues for convergence, a necessary and suffi-

cient condition for parameter convergence is thus 1/F0(z−1)

must be SPR. Filtering the adaptation error through an finite

impulse response (FIR) filter T(z−1) relaxes the condition to

T(z−1)/F0(z−1)−1/2 being SPR, in which case the eigenval-

ues of the associated convergence matrix change to −1and

−[T(k)/F0(k)−1/2].

The robust SPR problem to be solved is as follows:

Problem 1. Given ∈[0,1] and a monic†schur polynomial

A(z−1)=1+a1z−1+a2z−2+···+anz−n,(3)

with nunknown but bounded real coefficients

ai≤ai≤ai,i=1,2,…,n,(4)

find a real-coefficient polynomial C(z−1) such that C(z−1)

A(z−1)−

is SPR.

Remark 1. By the definition of SPRness, C(z−1)/A(z−1)−

is stable. Hence, roots of A(z−1)=0, although uncertain,

are all stable. This implies that the box region of coeffi-

cients in Equation 4 must be a subset of the stability region

of C(z−1)/A(z−1)−(or equivalently, that of 1/A(z−1)). For

example, when A(z)=1+a1z−1+a2z−2, the stability region

(obtained by, eg, bilinear transformation z=(1 + s)/(1 −s)

and Routh test) is a reverse triangle defined by 1 −a1+a2>0,

1+a1+a2>0,1 −a2>0. The rectangle defined by a1≤

a1≤a1and a2≤a2≤a2must stay inside the triangle.

In practical applications, is usually strictly positive.1,2In

this case, the problem can be normalized as follows:

Lemma 1. For >0, there exists a polynomial C(z−1)such

that C(z−1)

A(z−1)−is SPR, if and only if there exists a polynomial

C′(z−1) such that C′(z−1)

A(z−1)−1

2is SPR.

Proof. Under the assumption that >0, we have C(z−1)

A(z−1)−

=2C′(z−1)

A(z−1)−1

2where C′(z−1)=C(z−1)/(2). As scal-

ing a transfer function by a positive number does not change

the SPR property, C(z−1)/A(z−1)−is SPR if and only if

C′(z−1)/A(z−1)−1/2 is SPR.

For the above normalized problem, it is clear that letting

C′(z−1)/A(z−1)≈1 is a feasible solution. This is the suggested

way of designing the compensator C′(z−1) in text books of

system identification and adaptive control1-3,19 and is also

†That is, the leading coefficient of the polynomial is equal to 1.

important for the parameter convergence, on the basis of the

previous eigenvalue analysis.

3PROPOSED ROBUST AND OPTIMAL SPR

SOLUTIONS

The SPR definition itself is specified at an infinite amount

of frequencies. The celebrated positive-real lemma translates

the infinite dimensional problem to a single LMI.7,21 We adapt

the form that is most relevant to the focused problem in

this paper:

Lemma 2. (positive-real lemma) A square discrete-time sys-

tem Cp(zI −Ap)−1Bp+Dp,withApBpCpand Dpbeing

state-space matrices of proper dimensions, is SPR if and only

if there exists a positive definite matrix P=PT0 such that

the following matrix inequality holds

P−AT

pPApCT

p−AT

pPBp

Cp−BT

pPApDT

p+Dp−BT

pPBp0.(5)

We propose to leverage the capability of Lemma 2 to

solve Problem 1 (objective of Section 3.1) and then develop

approaches to embed optimal properties to improve parameter

convergence (Section 3.2).

For the first task, the overarching obstacle is the infinite

number of possible realizations of A(z−1). Parameteriza-

tion of this uncertainty will be conducted to yield a solv-

able Equation 5. This involves (1) characterization of the

(infinite-choice) parameter uncertainty in Equation 3 by a

finite amount of transfer functions and (2) parameteriza-

tion of state-space system matrices in Equation 5 such that

the inequality is convex with respect to the decision vari-

ables. This section first addresses part (1) in the next several

paragraphs, then discusses part (2) in Section 3.1.

Rewrite Equation 3 as

A(z−1)=1+z−1,z−2,…,z−n[a1,a2,…,an]T.(6)

By using the concept of convex hulls, the uncertain

[a1,a2,…,an]Tcan be characterized by the extreme edge

vectors that are defined by lower and upper bounds of ai’s:

[a1,a2,…,an]T=

2n

j=1

jbj,1,bj,2,…,bj,nT,

j≥0,

2n

j=1

j=1,

where bj,i=aior aiin Equation 4. There are 2nedge vec-

tors bj,1,bj,2,…,bj,nT. This number can be reduced if some

parameters are known a priori. Applying the above result to

Equation 6 yields

A(z−1)=

2n

j=1

jAj(z−1),

j≥0,

2n

j=1

j=1,(7)

Aj(z−1)≜1+bj,1z−1+bj,2z−2+· · ·+bj,nz−n,bj,i∈ai,ai.(8)

4XIAO ET AL.

Note that the roots of A(z−1)=0 are stable by assumption.‡

Roots of Aj(z−1)=0 thus must also be stable ∀j=1,2,…,2n.

With Equation 7, instead of an uncertain polynomial with

infinite choices of coefficients, we now have a convex com-

bination of a finite number of fixed edge polynomials. More-

over, we have the following result:

Proposition 1. If >0, A(z−1) is given by Equation 7 and

Aj(z−1) is defined by Equation 8, then C(z−1)

A(z−1)−is SPR if and

only if C(z−1)

Aj(z−1)−is SPR ∀j=1,2,…,2n.

Before proving the result, we remark that if

C(z−1)/A(z−1)−is SPR, then C(z−1) cannot have zeros

on the unit circle. Otherwise, there exists 0∈[0, 2)such

that Ce−j0=0, yielding Ce−j0∕Ae−j0−≤0,

which contradicts the SPR definition. (By a residual-theory

analysis, all zeros of C(z−1) further must be inside the unit

circle.26)

Proof of Proposition 1. The necessity part of the proof is

readily obtained because C(z−1)/Aj(z−1) is a particular case

of C(z−1)/A(z−1). For the sufficiency part, ∀j=1,2,…,2n,

if C(z−1)

Aj(z−1)−is SPR, then by definition,

ReCe−j

Aje−j−>0⇐⇒

Ce−j

Aje−j−+Cej

Ajej−>0,

(9)

where C(ej)=C(e−j)is the complex conjugate of C(e−j).

As all roots of Aj(z−1)=0 are inside the unit circle,

Aj(e−j)≠0∀. Multiplying Aj(e−j)Aj(ej)=

Aj(e−j)2>0 on both sides of Equation 9 yields that

Equation 9 is equivalent to

Ce−jAjej+CejAje−j−2Aje−jAjej>0.

(10)

Division by C(e−j)C(ej)( ≠0) yields

Ajej

Cej+Aje−j

Ce−j−2Aje−jAjej

Ce−jCej>0.(11)

As

2

1

2−Aje−j

Ce−j

2

=21

2−Ajej

Cej1

2−Aje−j

Ce−j

=1

2−Aje−j

Ce−j+Ajej

Cej+2Aje−j

Ce−jAjej

Cej,

Equation 11 is equivalent to

‡For us to obtain the robust stability condition of such “interval polynomi-

als” with bounded coefficients, the Kharitonov’s Theorem22,23 provides a

complete solution to the continuous-time domain problem and has motivated

various research towards its discrete-time equivalences (see, eg, Bartlett et

al24 and Hollot and Bartlett25 and the references contained therein.)

1

2−Aje−j

Ce−j<1

2.

Recall that stability of C(z−1)/Aj(z−1)−is already assured.

Therefore, C(z−1)/Aj(z−1)−is SPR if and only if ∀∈

[0, 2), Aj(e−j)/C(e−j) lies inside a disk of radius 1

2cen-

tered at 1

2,0in the complex plane. Considering the convex

combination Equation 7, we have

1

2−Ae−j

Ce−j

(7)

=

1

2−2n

j=1jAje−j

Ce−j

=

2n

j=1j1

2−Aje−j

Ce−j

≤

2n

j=1j

1

2−Aje−j

Ce−j

(11)

<

2n

j=1j

1

2=1

2,(12)

where Equation 12 comes from the triangle equality. There-

fore A(e−j)∕C(e−j)lies in the aforementioned disk, and

C(z−1)

A(z−1)−is thus SPR.

3.1 Design for robust SPR

Given the SPR problem and the convex hull formulation of the

uncertain A(z−1), Lemma 1 leads to investigation of the SPR

condition for each edge transfer function C(z−1)/Aj(z−1)−.

Let G(z−1)=C(z−1)/K(z−1)−,whereK(z−1)represents

an edge polynomial Aj(z−1). Define

C(z−1)=c0+c1z−1+···+clz−l,(13)

K(z−1)=1+k1z−1+···+knz−n.(14)

The order of C(z−1) is a design parameter here. Depending on

the values of land n, different situations exist for the design

of Equation 5:

Case 1: If l≥n, simplification yields

C(z−1)

K(z−1)−=(c0−)+(c1−c0k1)zl−1+···+(cn−c0kn)zl−n+cn+1zl−n−1+···+cl

zl+k1zl−1+···+knzl−n,

which has the following state-space realization:

Ap=0l−1,1Il−1,l−1

0∗l×l

,Bp=0l−1,1

1,(15)

∗=01,l−n−1,−kn,…,−k1

Cp=cl,…,cn+1,cn,…,c1

−c001,l−n,kn,…,k1,

Dp=c0−.

(16)

Here, the controllable canonical form is proposed, so that

when we form Equation 5, the matrix on the left-hand side

is affine in the decision variables [c0,c1,…,cl], leading to

Equation 5 being a convex inequality constraint.

XIAO ET AL. 5

Case 2: If n>l, similar procedure gives the controllable

canonical form

Ap=

010⋱

⋮⋱⋱ 0

0··· 01

−kn··· −k2−k1

,Bp=

0

⋮

0

1n×1

,(17)

Cp=−c0kn…kl+1,kl…,k1

+01,n−l,cl,…,c1

Dp=c0−. (18)

Equation 15 or 17 can now be applied to construct (5). Such

constructions are repeated for each edge transfer function. We

can now formulate the feasibility problem:

find c0,…cl∈Rand Pj=PT

j0 (19)

subject to

Pj−AT

p,jPjAp,jCT

p,j−AT

p,jPjBp,j

Cp,j−BT

p,jPjAp,jDT

p,j+Dp,j−BT

p,jPjBp,j0,j=1,2,…2n,

(20)

where for each j,(Ap,j,Bp,j,Cp,j,Dp,j) is defined by (15) or

(17), with K(z−1)=Aj(z−1).

By construction, Cp,jand Dp,jdepend affinely on ciand

Pj, which renders Problem (19)-(20) to a convex semidefi-

nite programming problem, and can be solved by efficient

interior-point methods in convex optimization tools.§

3.2 Design for optimal properties

The last subsection provides solutions to obtain a feasible

solution, i.e., one set of parameters for C(z−1) will be obtained

if the robust SPR problem is solvable. Many feasible solutions

usually exist in practice. From the viewpoint of implications

in adaptive control and system identification, another main

design aspect is to obtain coefficients c0,…,clin (19) with

designer-assigned optimal properties (one of which is to keep

C(z−1)/A(z−1) close to 1). In this section, together with the

“close-to-1” condition, we provide a few examples to obtain

the optimal compensator C(z−1). The discussions contain

three thrusts, which can be combined by a weighted sum, to

satisfy multiple design objectives. All the results in this sub-

section are placed on top of the baseline SPR constraint in

Section 3.1.

C(z−1)/A(z−1)Being Close to 1. The intuition and

the importance of this objective has been discussed in

Section 2. The condition that the z-domain transfer function

C(z−1)/A(z−1) is close to 1 is mathematically equivalent to

minimizing the maximum value of |C(ej)/A(ej)−1|, i.e.,

§When the problem is formulated for computer solvers, the positive definite

constraint P=PT0istransformedtoP−I⪰0, where is a small positive

number chosen as the lower bound of all the eigenvalues of P.

min

c0,…cl∈R∶

C(z−1)

A(z−1)−1∞

(21a)

subject to ∶C(z−1)

Aj(z−1)−is SPR,j=1,…,2n.(21b)

Note that min ||G(z−1)||∞is the same as

min

≥0

s.t.G(e−j)≤,∀∈[0,2),

which is equivalent to, based on the Bounded-real lemma (see,

eg, Boyd27), another LMI:

min

≥0,P∶(22)

subject to ∶−

AT

rPAr−PA

T

rPBrCT

r

BT

rPArBT

rPBr−ID

T

r

CrDr−I⪰0

P=PT0,(23)

where (Ar,Br,Cr,Dr) is the state-space realization of G(z−1).

By proper state-space formulations and using the

bounded-real lemma, Problems 21A,B is transformed to the

semidefinite programing problems:

min

,Pr,Pj,ci

∶≥0 (24a)

subject to ∶−

AT

rPrAr−PrAT

rPrBrCT

r

BT

rPrArBT

rPrBr−ID

T

r

CrDr−I⪰0,(24b)

Pr=PT

r0 (24c)

Pj−AT

p,jPjAp,jCT

p,j−AT

p,jPjBp,j

Cp,j−BT

p,jPjAp,jDT

p,j+Dp,j−BT

p,jPjBp,j0 (24d)

Pj=PT

j0,j=1,2,···2n,(24e)

where Equations 24D,E is as defined in Section 3.1. Analo-

gous to the formulation for C(z−1)/K(z−1)−in Section 3.1,

for G(z−1)=C(z−1)∕A(z−1)−1,Ar,Br,Cr,andDrare defined

by

Ar=

01 0··· 0

⋮⋱⋱ ⋱

⋱⋱ ⋱ ⋮

⋱⋱ 0

··· 01

0··· 0−an··· −a1

l×l

,l≥n

010⋱

⋮⋱⋱ 0

0…01

−an··· −a2−a1

,l<n

,

Br=

0l−1,1

1,l≥n

0n−1,1

1,l<n

Cr=cl…c1−c001,l−nan…a1,l≥n

−c0an…a1+01,n−lcl…c1,l<n,

Dr=c0−1.

6XIAO ET AL.

Again, we proposed the controllable canonical form, so

that the left hand side of Equation 24B is affine in ,

Pr,andci. After adding the SPR constraint, the mini-

mization in Equation 24 remains a convex optimization

problem.

A candidate polynomial A(z−1) is needed in the above

optimization. Similar to general system identification, some

engineering judgment needs to be applied. For the specific

polytopic uncertainty (Equation 4), the geometric center of

the polytope can be used.

Minimum High-Frequency Gains. In the output-error

method with a fixed compensator3and filtered pseudolin-

ear regression, the output error is filtered through C(z−1)to

obtain the adaptation error, denoted as (k), for parameter

identification or adaptive control. Excessive high-frequency

amplifications in (k) reduces the signal-to-noise ratio and

increases the quantization error. It is therefore favorable to

limit the high-frequency magnitude of C(z−1) (and hence the

high-frequency energy in (k)).

Recall that the compensator is given by C(z−1)=c0+

c1z−1+···+clz−l, whose frequency response at =is

C(e−j)=C(z)|z=− 1. To minimize the high-frequency gain (at

the Nyquist frequency) of C(z−1), we can add the following

objective:

min C(e−j)=[1,−1,1,−1,…][

c0,c1,…,cl]T,(25)

subject to the feasibility constraint Equation 20. The cost

function Equation 25 is linear in ci’s, rendering the optimiza-

tion to remain convex.

Analogous procedure can be performed to minimize/

maximize the low-frequency (DC) gain of C(z−1), with the

objective of min/max C(1)=[1,1,…,1][

c0,c1,…,cl]T.

Minimum-Order Compensator. Theorderofthecom-

pensator is directly related to the required computation com-

plexity in the related system identification or adaptive control

problems. The common practice in system identification is

to apply l=nin Equation 14. With the proposed optimiza-

tion framework, it is readily available to find the compensator

C(z−1) with the minimum number of coefficients (and in the

meantime satisfying the discussed optimal properties). This

is achieved through the optimization formulation, by starting

the feasibility problem Equation 19 with l=n, and iteratively

reducing luntil Equation 19 becomes infeasible as shown in

Algorithm 1.

When the minimum order of C(z−1) is still high, a related

design is to obtain sparse (having large amounts of zeros)

coefficients in C(z−1). In that case, if a feasible order l

is firstly assigned, we can apply the 1-norm approximation for

cardinality minimization (see, eg, Boyd and Vandenberghe28)

and add the cost function minci,Pi[c0,c1,…,cl]T1

to Equation 19, which shall provide a sparse

[c0,c1,…,cl].

4EXTENSIONS

The proposed algorithms can be readily extended to accom-

modate additional SPR design constraints. For instance, con-

sider again the general pseudolinear regression algorithm

described before Problem 1. By introducing filtering in PAAs,

the convergence condition can be relaxed to T(z−1)

F0(z−1)S(z−1)−

1

2being SPR, where Tand Sare filters of the adapta-

tion error and regressor vectors, respectively (Ljung and

Soderstrom19, Section 4.5).¶Additional convergence require-

ments can occur in more involved PAAs. For instance, for

pseudolinear regression with filtered adaptation error applied

to a Box-Jenkins model

y(k)= B0(z−1)

F0(z−1)u(k)+ C0(z−1)

D0(z−1)e(k),

both T(z−1)D0(z−1)/C0(z−1)/F0(z−1)−1/2 and T(z−1)/

C0(z−1)−1/2 must be SPR.19

To address the general SPR and magnitude constraints, we

provide next the formulations to solve

min

c0,…cl∈RR(z−1)∞≜

W(z−1)

V(z−1)C(z−1)−∞

.(26)

Here, we illustrate the addressing of the magnitude constraint

by the bounded-real lemma. The case for SPR constraint

is analogous by the adoption of the positive-real lemma.

The formulation extends Equation 21A by combining gen-

eral polynomials W(z−1), V(z−1)withC(z−1). This way, we

can constrain C(z−1) to have an arbitrary desired (if feasible)

frequency response.

For brevity, it is assumed that W(z−1), V(z−1), and C(z−1)

have the same order, ie, W(z−1)=w0+w1z−1+···+wlz−l,

V(z−1)=1+v1z−1+···+vlz−l,andC(z−1)=c0+c1z−1+

···+clz−l. If not, one can classify different situations in a

¶For the most common algorithms that filter the regressor vector, S(z−1)has

the form of 1/L(z−1) (see, eg, Landau et al3,Chapter 5 ).

XIAO ET AL. 7

way similar to that in Section 3, or simply constrain the

coefficients of the excessive high-order terms to be 0.

Equation 26 can be transformed to a tractable optimization

problem in a form similar to Equation 19. Before that can be

performed, we need the following system construction: notice

that H(z−1)≜W(z−1)C(z−1)=h0+h1z−1+···+h2lz−2lis

given by the convolution

h0

h1

⋮

⋮

h2l

=

w00··· ··· 0

w1w0⋱⋱ ⋮

⋮⋱⋱⋱ ⋮

⋮⋱⋱⋱ 0

wlwl−1··· w1w0

0wl⋱⋱ w1

⋮⋱⋱⋱ ⋮

⋮⋱⋱⋱wl−1

0··· ··· 0wl

(2l+1)×(l+1)

c0

c1

⋮

⋮

cl

,

from which R(z−1)=H(z−1)/V(z−1)−has the following

state-space realization:

Ar=02l−1,1I2l−1,2l−1

0∗2l×2l

,Br=02l−1,1

1

∗=01,l−1,−vl,−vl−1,…,−v1,Dr=h0−

(27)

Cr=[h2l,h2l−1,…,h1]−h001,l,vl,…,v1.

By using the bounded-real lemma, Equation 26 can be

achieved if and only if the following problem can be solved:

min

,Pr,ci

∶≥0 (28)

subject to ∶−

AT

rPrAr−PrAT

rPrBrCT

r

BT

rPrArBT

rPrBr−ID

T

r

CrDr−I⪰0

Pr=PT

r0.(29)

Again, we applied the controllable canonical form in

Equation 27, so that the left hand side of Equation 29 is affine

in ,Pr,andci. After adding the SPR constraint (Equation 19),

the minimization in Equation 28 remains a convex problem.

5RESULTS AND ANALYSIS

Consider the rejection of disturbances with band-limited

spectra that are modeled by

w(k)=

nd

i=1

sin(ik+i)+(k),(30)

where ndis the number of peaks in the disturbance spectra,

iand iare the frequency and initial phase of each har-

monic component, and (k) is a zero-mean white stochastic

noise. Note that under the internal model principle, passing

nd

i=1sin(ik+i)through B(z−1)

A(z−1)=nd

i=1

1−2cosiz−1+z−2

1−2cos iz−1+2z−2

yields a null output. When the disturbance frequencies are

unknown, i’s can be online identified via adaptively mini-

mizing a filtered output:

TABLE 1 Parameter adaptation algorithms summary for the case with

nd=1

Definition Equation

Regressor (k−1) =w(k−1) −e(k−1)

a posteriori output error e(k)=(k−1)T

(k)+w(k)+w(k−2)−2e(k−2)

a priori output error eo

(k)=(k−1)T

(k−1)+w(k)+w(k−2)−2e(k−2)

a posteriori adaptation error (k)=Cz−1e(k)≜(1+cz−1+2z−2)e(k)

a priori adaptation error 0(k)=e0(k)+2e(k−2) + e(k−1)c.

Parameter adaptation

(k)=

(k−1)+F(k−1)(−(k−1))0(k)

1+(k−1)TF(k−1)(k−1)

Adaptation gain F(k)= 1

(k)F(k−1)− F(k−1)(k−1)T(k−1)F(k−1)

(k)+T(k−1)F(k−1)(k−1)

e(k)=

B(z−1)

A(z−1)w(k)≜

Ad(z−1)

Ad(−1z−1)w(k)

=

nd

i=1

1−2cos iz−1+z−2

1−2cos iz−1+2z−2w(k).

(31)

Such a technique is key in a number of applications in pre-

cision engineering systems.29,30 A more detailed discussion

of the importance and state-of-the-art research activities on

rejecting such narrow-band disturbances is available in a

recent benchmark problem on adaptive regulation.31

We discuss parameter identification on Equation 31 using

the output error method with a fixed compensator C(z−1). For

the case with nd=1, denote =a1=−2cos(1). The PAA is

summarized in Table 1. The full-order version of the formu-

las follows those defined in the previous study by Chen and

Tomizuka.29 Under standard PAA formulations, convergence

of parameters require C(z−1)/A(z−1)−1/2 to be SPR (see, eg,

Landau et al3, Chapter 3 and Ljung and Soderstrom19, Section 4.5).‖

The proposed algorithms were experimentally verified on

a wafer scanner testbed. The hardware setup has been dis-

cussed in the previous study.30 We provide next the robust and

multiobjective SPR design results.

5.1 SPR condition

It is easy to check that A(z−1) is firstly stable ∀i, satisfying

the first condition for SPR transfer functions. Let =0.98,

andthesamplingperiodinthesystembeTs=1/2500 sec-

ond. Let the disturbance frequency vary from 5 to 120 Hz.

Figure 1A demonstrates the frequency responses of a set of

possible 1/A(z−1)’s uniformly sampled from the uncertainty

region. One can observe that 1/A(z−1) contains significant

uncertainties; and in a large frequency region, the phase

responses of 1/A(z−1)isbelow−90◦,ie,Re(1/A(e−j)) ≤0.

Therefore, 1/A(z−1) is not SPR, not to say 1/A(z−1)−1/2.

Under such conditions, without a compensator, the PAA is not

asymptotically stable. In fact, one can find examples where

the probability that the parameter estimate converges to the

desired value is zero.19

The 2 edge polynomials in this case are A1(z−1)=1−

2a1z−1+2z−2and A2(z−1)=1−2a1z−1+2z−2,

‖Note that w(k) is persistently exciting for the specified model.

8XIAO ET AL.

FIGURE 1 Frequency responses in the design example [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2 Parameter convergence (experimental results) under feasibility solution. The adaptation was turned on at 50 samples and initialized at 48 Hz. As

the output error method with a fixed compensator converges locally,7a recursive least squares was used in the first 250 time steps, which provides biased

parameter estimation when the noise term in the excitation signal is not white (in the experiments, the environmental noise is small; yet still there isasmall

bias for some of the frequency estimations) [Colour figure can be viewed at wileyonlinelibrary.com]

with a1=cos (2Ts×5)and a1=cos (2Ts×120).For-

mulating and solving (via the cvx computation tool32)the

feasibility design in Section 3, with l=n=2, we obtain

Figure 1B. One observes that ∀,C(z−1) is capable of pro-

viding robust compensation such that C(e−j)/A(e−j)−1/2

stays strictly in the open right-half complex plane (phase∈

−

2,

2). Combined with the condition that A(z−1)is

always stable, this indicates the success of the robust SPR

design.

Notice however in Figure 1B that large gain variations

exist in C(z−1)/A(z−1)−1/2 and that C(z−1)/A(z−1)isfar

away from the unity function (particularly at low frequen-

cies). Indeed, applying the PAA under the feasibility solution

gives Figure 2, which plots the convergence under 6 dif-

ferent A(z−1)’s using the same C(z−1) compensator. The

parameters were seen to all converge to the true values,

which verified the benefit of the robust SPR design. Yet,

meanwhile, the convergence was highly unbalanced. With

C(e−j)/A(e−j)−1/2 having large variations at different ’s,

the eigenvalues of the corresponding convergence matrices

(recall Section 2) contain large variations leading to large

condition numbers of the matrix. Correspondingly, the con-

vergence at low frequency was significantly more noisy than

that at higher frequencies.

5.2 Design for optimal properties

Adding the objective of min C(z−1)∕A∗(z−1)−1∞with

A*(z−1) being the center of the polytope, we obtain the solid

line in Figure 3A. Compared to the dashed solution from

the previous feasibility design, C(z−1) has reduced magni-

tude response. In fact, the resulting DC gain of −30dB

is seen to match the inverse of the center of A(z−1)in

Figure 1A.

XIAO ET AL. 9

FIGURE 3 Optimal strictly positive real design and corresponding parameter adaptation algorithms performance. A, Frequency responses of C(z−1). B,

Parameter convergence (experimental results) under optimization min||C(z−1)/A(z−1)−1||∞[Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 4 Zeros of optimal C*(z−1)frommin||C/A−1||∞and sampled

poles of the uncertain 1

A(z−1)[Colour figure can be viewed at

wileyonlinelibrary.com]

Figure 4 plots the zeros of the optimal C*(z−1) correspond-

ing to the solid line in Figure 3A and samples of the uncertain

poles of 1/A(z−1). Notice the nontrivial relationship between

locations of the zeros and the poles. One observes that for

robust SPRness, the roots of C(z−1)=0donotnecessar-

ily represent the geometric center of the sampled roots of

A(z−1)=0. This new insight is beyond the conventional

approach in system identification, where a rough estimate of

A(z−1) is assigned to C*(z−1), and neither robust SPR nor

optimality is assured.

Table 2 shows the coefficients of C(z−1) under different

design objectives. We explore first the optimal C(z−1)thathas

the minimum high-frequency magnitude. Applying the algo-

rithms in Section 3.2, we obtain the red dashed line marked

by plus signs in Figure 5. Noticing the deep notch in the solid

line of the magnitude responses, we can see that the optimiza-

tion indeed provides strong high-frequency gain attenuation

thanks to the cost function design in Equation 25. Figure 5

compares the online experimental parameter convergence

TABLE 2 Numerical values of the solved C(z−1)

Design objective Solved C(z−1)

D1 Feasibility 19.9469 −11.4016z−1−7.77161z−2

D2 min C(z−1)z=−19.95416 −0.705471z−1−8.69951z−2

D3 min C(z−1)∕A(z−1)−1∞1.31659 −1.95223z−1−0.666983z−2

D4 Feasibility + min order 25.8531 −25.0835z−1

D5 min C(z−1)z=−1+ min order 0.997506 −0.962617z−1

with that of the feasibility solution (copied and zoomed in). It

is seen that convergence of most of the parameters has been

improved. Interestingly, after adding the minimum-order cri-

teria, the combined minimum-order minimum-Nyquist-gain

solution (D5 in Table 2) provides more surprising improve-

ments as shown in Figure 6. The convergence is almost as

good as that in Figure 3B. This can be explained from the

frequency domain. One observes from Figure 6 that the bode

plots of the resulting optimal C(z−1)inD5andD3arequite

close to each other, leading to similar convergence perfor-

mances. The result is quite nice and suggests that the order of

the filter do not necessarily need to be the highest for good

performance in practice. Again, under reduced order designs,

the locations of the roots of C(z−1) are trivial. Designing

C(z−1) based on root locations seems not viable, especially

for high-order systems.

Figure 7 shows a further example under a different sampling

time. A second- and a fourth-order systems are considered.

From the left figure, one observes that besides the achieve-

ment of the robust SPR requirement, C(e−j)/A(e−j)−1/2

is significantly confined to be in a smaller region: the

phase of C(e−j)/A(e−j)−1/2 is close to 0◦at most fre-

quencies; and the magnitude response is condensed to be

within 0.2503 (−12.03 dB) and 2.6931 (8.605 dB). There-

fore, C(z−1)/A(z−1) is indeed closer to 1 from the optimal

design. Similar trend of response can be observed on the

right figure.

10 XIAO ET AL.

FIGURE 5 Parameter convergence under different optimal designs [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 6 Frequency responses and convergence of the optimal solutions [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 7 Bode plots for 1

A(z−1)and C(z−1)

A(z−1)−1

2in a second example with Ts=1/26400 sec: left–second-order case (under min||C/A−1||∞); right–forth-order

case (feasible solution) [Colour figure can be viewed at wileyonlinelibrary.com]

XIAO ET AL. 11

6CONCLUSION

In this paper, a convex-optimization approach is proposed to

address the design of robust strictly positive real transfer func-

tions; simulation and experimental results are provided that

suggest several new aspects of design in recursive parameter

adaptation algorithms. It is shown that a feasibility semidef-

inite programming formulation can be used to provide the

compensator that achieves the desired robust SPR condition.

Moreover, the important issue of maintaining the designed

transfer function to be close to 1 is addressed, by adding an

infinity-norm minimization in the optimization. Additional

concepts of cost function design are introduced, which lead

to solutions of several new problems. All the formulated opti-

mization problems can be efficiently solved by interior point

methods in convex optimization. The contribution of this

paper is important because (1) it provides a new approach of

designing for SPR condition, leading to more natural intuition

with greater performance, and (2) it also develops optimal

design methodologies for different practical design prefer-

ences. Experimental results show that the optimal design

could promote different performance gains during parameter

adaptation.

Although the focus has been placed on the discrete-time

SPR analysis, the presented work can be readily extended to

solve the continuous-time version of the problem by apply-

ing the continuous-time positive- and bounded-real lemmas

(see, eg, Boyd et al27). Note that in the continuous-time

case, for the SPR condition to hold, the relative degree

of G(s) must equal zero or one,8,20,33 ie, l+1≥

n≥lin Equation 14. Therefore, only Case 2 holds in the

continuous-time robust SPR formulation, hence simplifying

the formulation of matrix inequalities.

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How to cite this article: Xiao H, Landau ID,

Chen X. A robust optimal design for strictly positive

realness in recursive parameter adaptation. Int J Adapt

Control Signal Process. 2017. doi:10.1002/acs.2757