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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 provide 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. Copyright
Received: 16 May 2016 Revised: 18 November 2016 Accepted: 28 December 2016
DOI 10.1002/acs.2757
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
Xu Chen, 191 Auditorium Road U3139, University
of Connecticut, Storrs, CT, 06269-3139, USA.
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.
adaptive control, strictly positive real, system identification
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
z1in discrete-time problems), design a polynomial C()∈
Psuch that the transfer function
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(z1)=B(z1)/A(z1) using the output error method with
a fixed compensator, C(z1)/A(z1)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
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 Copyright © 2017 John Wiley & Sons, Ltd. 1
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 *
A proper and rational discrete-time transfer function G(z1)
is SPR if (1) G(z1) does not possess any pole outside of or
on the unit circle in the complex plane, and (2) ||<,
G(ej)+G(ej)=2Re{G(ej)} >0, ie, the real part of
G(ej) is positive.
From the above definition, if G(z1) is SPR, then (1)
G(z1) is stable; (2) the phase response of G(z1), after nor-
malization to [ ,], lies inside the region (−
2)(see, eg
Anderson et al8and Wang and Yu16); and (3) the Nyquist plot of
G(z1) 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
is an estimate of *, the true parameter vector;
(k) is the regressor; and F(k) is the adaptation gain matrix.
H(z1) is a discrete transfer operator (z1is 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)is assumed to belong infinitely often to the domain
for which the stationary processes (k,
)and v(k+1,
be defined. Then the PAA converges to a convergent domain
with probability 1 as follows:
Prob lim
if there exists 2with 2(k)2<2, such that H(z1)− 2
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(z1)
*An extension of the algorithm is discussed in Section 6, so that the
continuous-time problem can be similarly addressed.
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(z1)/F0(z1). For us to assure
negative eigenvalues for convergence, a necessary and suffi-
cient condition for parameter convergence is thus 1/F0(z1)
must be SPR. Filtering the adaptation error through an finite
impulse response (FIR) filter T(z1) relaxes the condition to
T(z1)/F0(z1)1/2 being SPR, in which case the eigenval-
ues of the associated convergence matrix change to 1and
The robust SPR problem to be solved is as follows:
Problem 1. Given [0,1] and a monicschur polynomial
with nunknown but bounded real coefficients
find a real-coefficient polynomial C(z1) such that C(z1)
is SPR.
Remark 1. By the definition of SPRness, C(z1)/A(z1)
is stable. Hence, roots of A(z1)=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(z1)/A(z1)(or equivalently, that of 1/A(z1)). For
example, when A(z)=1+a1z1+a2z2, 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
a1a1and a2a2a2must 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(z1)such
that C(z1)
A(z1)is SPR, if and only if there exists a polynomial
C(z1) such that C(z1)
2is SPR.
Proof. Under the assumption that >0, we have C(z1)
2where C(z1)=C(z1)/(2). As scal-
ing a transfer function by a positive number does not change
the SPR property, C(z1)/A(z1)is SPR if and only if
C(z1)/A(z1)1/2 is SPR.
For the above normalized problem, it is clear that letting
C(z1)/A(z1)1 is a feasible solution. This is the suggested
way of designing the compensator C(z1) 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.
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
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(z1). 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
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:
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
Aj(z1)1+bj,1z1+bj,2z2+· · ·+bj,nzn,bj,iai,ai.(8)
Note that the roots of A(z1)=0 are stable by assumption.
Roots of Aj(z1)=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(z1) is given by Equation 7 and
Aj(z1) is defined by Equation 8, then C(z1)
A(z1)is SPR if and
only if C(z1)
Aj(z1)is SPR j=1,2,,2n.
Before proving the result, we remark that if
C(z1)/A(z1)is SPR, then C(z1) cannot have zeros
on the unit circle. Otherwise, there exists 0[0, 2)such
that Cej0=0, yielding Cej0Aej00,
which contradicts the SPR definition. (By a residual-theory
analysis, all zeros of C(z1) further must be inside the unit
Proof of Proposition 1. The necessity part of the proof is
readily obtained because C(z1)/Aj(z1) is a particular case
of C(z1)/A(z1). For the sufficiency part, j=1,2,,2n,
if C(z1)
Aj(z1)is SPR, then by definition,
where C(ej)=C(ej)is the complex conjugate of C(ej).
As all roots of Aj(z1)=0 are inside the unit circle,
Aj(ej)0. Multiplying Aj(ej)Aj(ej)=
Aj(ej)2>0 on both sides of Equation 9 yields that
Equation 9 is equivalent to
Division by C(ej)C(ej)( 0) yields
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.)
Recall that stability of C(z1)/Aj(z1)is already assured.
Therefore, C(z1)/Aj(z1)is SPR if and only if
[0, 2), Aj(ej)/C(ej) lies inside a disk of radius 1
tered at 1
2,0in the complex plane. Considering the convex
combination Equation 7, we have
where Equation 12 comes from the triangle equality. There-
fore A(ej)∕C(ej)lies in the aforementioned disk, and
A(z1)is thus SPR.
3.1 Design for robust SPR
Given the SPR problem and the convex hull formulation of the
uncertain A(z1), Lemma 1 leads to investigation of the SPR
condition for each edge transfer function C(z1)/Aj(z1).
Let G(z1)=C(z1)/K(z1),whereK(z1)represents
an edge polynomial Aj(z1). Define
The order of C(z1) is a design parameter here. Depending on
the values of land n, different situations exist for the design
of Equation 5:
Case 1: If ln, simplification yields
which has the following state-space realization:
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.
Case 2: If n>l, similar procedure gives the controllable
canonical form
0··· 01
kn··· −k2k1
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,clRand Pj=PT
j0 (19)
subject to
where for each j,(Ap,j,Bp,j,Cp,j,Dp,j) is defined by (15) or
(17), with K(z1)=Aj(z1).
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(z1) 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(z1)/A(z1) close to 1). In this section, together with the
“close-to-1” condition, we provide a few examples to obtain
the optimal compensator C(z1). 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(z1)/A(z1)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(z1)/A(z1) 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=PT0istransformedtoPI0, where is a small positive
number chosen as the lower bound of all the eigenvalues of P.
subject to C(z1)
Aj(z1)is SPR,j=1,,2n.(21b)
Note that min ||G(z1)||is the same as
which is equivalent to, based on the Bounded-real lemma (see,
eg, Boyd27), another LMI:
subject to ∶−
where (Ar,Br,Cr,Dr) is the state-space realization of G(z1).
By proper state-space formulations and using the
bounded-real lemma, Problems 21A,B is transformed to the
semidefinite programing problems:
0 (24a)
subject to ∶−
r0 (24c)
p,jPjBp,j0 (24d)
where Equations 24D,E is as defined in Section 3.1. Analo-
gous to the formulation for C(z1)/K(z1)in Section 3.1,
for G(z1)=C(z1)∕A(z1)−1,Ar,Br,Cr,andDrare defined
01 0··· 0
⋮⋱⋱ ⋱
⋱⋱ ⋱
⋱⋱ 0
··· 01
0··· 0an··· −a1
an··· −a2a1
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
A candidate polynomial A(z1) 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(z1)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(z1) (and hence the
high-frequency energy in (k)).
Recall that the compensator is given by C(z1)=c0+
c1z1··+clzl, whose frequency response at =is
C(ej)=C(z)|z=− 1. To minimize the high-frequency gain (at
the Nyquist frequency) of C(z1), we can add the following
min C(ej)=[1,1,1,1,][
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(z1), with the
objective of min/max C(1)=[1,1,,1][
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(z1) 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(z1) is still high, a related
design is to obtain sparse (having large amounts of zeros)
coefficients in C(z1). 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
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(z1)
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(z1)
F0(z1)u(k)+ C0(z1)
both T(z1)D0(z1)/C0(z1)/F0(z1)1/2 and T(z1)/
C0(z1)1/2 must be SPR.19
To address the general SPR and magnitude constraints, we
provide next the formulations to solve
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(z1), V(z1)withC(z1). This way, we
can constrain C(z1) to have an arbitrary desired (if feasible)
frequency response.
For brevity, it is assumed that W(z1), V(z1), and C(z1)
have the same order, ie, W(z1)=w0+w1z1··+wlzl,
···+clzl. If not, one can classify different situations in a
For the most common algorithms that filter the regressor vector, S(z1)has
the form of 1/L(z1) (see, eg, Landau et al3,Chapter 5 ).
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(z1)W(z1)C(z1)=h0+h1z1+···+h2lz2lis
given by the convolution
w00··· ··· 0
w1w0⋱⋱ ⋮
⋮⋱ 0
wlwl1··· w1w0
0wl⋱⋱ w1
0··· ··· 0wl
from which R(z1)=H(z1)/V(z1)has the following
state-space realization:
By using the bounded-real lemma, Equation 26 can be
achieved if and only if the following problem can be solved:
0 (28)
subject to ∶−
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.
Consider the rejection of disturbances with band-limited
spectra that are modeled by
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
i=1sin(ik+i)through B(z1)
12cos iz1+2z2
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
Definition Equation
Regressor (k1) =w(k1) e(k1)
a posteriori output error e(k)=(k1)T
a priori output error eo
a posteriori adaptation error (k)=Cz1e(k)(1+cz1+2z2)e(k)
a priori adaptation error 0(k)=e0(k)+2e(k2) + e(k1)c.
Parameter adaptation
Adaptation gain F(k)= 1
(k)F(k1)− F(k1)(k1)T(k1)F(k1)
12cos iz1+z2
12cos iz1+2z2w(k).
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(z1). 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(z1)/A(z1)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(z1) 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(z1)’s uniformly sampled from the uncertainty
region. One can observe that 1/A(z1) contains significant
uncertainties; and in a large frequency region, the phase
responses of 1/A(z1)isbelow90,ie,Re(1/A(ej)) 0.
Therefore, 1/A(z1) is not SPR, not to say 1/A(z1)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(z1)=1
2a1z1+2z2and A2(z1)=12a1z1+2z2,
Note that w(k) is persistently exciting for the specified model.
FIGURE 1 Frequency responses in the design example [Colour figure can be viewed at]
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]
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(z1) is capable of pro-
viding robust compensation such that C(ej)/A(ej)1/2
stays strictly in the open right-half complex plane (phase
2). Combined with the condition that A(z1)is
always stable, this indicates the success of the robust SPR
Notice however in Figure 1B that large gain variations
exist in C(z1)/A(z1)1/2 and that C(z1)/A(z1)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(z1)’s using the same C(z1) 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(ej)/A(ej)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(z1)∕A(z1)−1with
A*(z1) 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(z1) has reduced magni-
tude response. In fact, the resulting DC gain of 30dB
is seen to match the inverse of the center of A(z1)in
Figure 1A.
FIGURE 3 Optimal strictly positive real design and corresponding parameter adaptation algorithms performance. A, Frequency responses of C(z1). B,
Parameter convergence (experimental results) under optimization min||C(z1)/A(z1)1||[Colour figure can be viewed at]
FIGURE 4 Zeros of optimal C*(z1)frommin||C/A1||and sampled
poles of the uncertain 1
A(z1)[Colour figure can be viewed at]
Figure 4 plots the zeros of the optimal C*(z1) correspond-
ing to the solid line in Figure 3A and samples of the uncertain
poles of 1/A(z1). Notice the nontrivial relationship between
locations of the zeros and the poles. One observes that for
robust SPRness, the roots of C(z1)=0donotnecessar-
ily represent the geometric center of the sampled roots of
A(z1)=0. This new insight is beyond the conventional
approach in system identification, where a rough estimate of
A(z1) is assigned to C*(z1), and neither robust SPR nor
optimality is assured.
Table 2 shows the coefficients of C(z1) under different
design objectives. We explore first the optimal C(z1)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(z1)
Design objective Solved C(z1)
D1 Feasibility 19.9469 11.4016z17.77161z2
D2 min C(z1)z=−19.95416 0.705471z18.69951z2
D3 min C(z1)∕A(z1)−11.31659 1.95223z10.666983z2
D4 Feasibility + min order 25.8531 25.0835z1
D5 min C(z1)z=−1+ min order 0.997506 0.962617z1
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(z1)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(z1) are trivial. Designing
C(z1) 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(ej)/A(ej)1/2
is significantly confined to be in a smaller region: the
phase of C(ej)/A(ej)1/2 is close to 0at 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(z1)/A(z1) is indeed closer to 1 from the optimal
design. Similar trend of response can be observed on the
right figure.
FIGURE 5 Parameter convergence under different optimal designs [Colour figure can be viewed at]
FIGURE 6 Frequency responses and convergence of the optimal solutions [Colour figure can be viewed at]
FIGURE 7 Bode plots for 1
A(z1)and C(z1)
2in a second example with Ts=1/26400 sec: left–second-order case (under min||C/A1||); right–forth-order
case (feasible solution) [Colour figure can be viewed at]
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
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
nlin 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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... A positive real transfer function has non-negative real part on the closed complex right half-plane. It has a relative degree (that is, a pole-zero excess) of 0, +1, or −1 [1][2][3][4][5][6][7][8]. Several ways and methods of designing such transfer functions in circuitry synthesis problems are given in [3][4][5][6][7]. ...
... Several ways and methods of designing such transfer functions in circuitry synthesis problems are given in [3][4][5][6][7]. Their design in the context of recursive parameter adaptation is focused on in [8]. In [9], the global asymptotic stability property is studied for a composite system with an asymptotically hyperstable subsystem. ...
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