A control-theoretic study on iterative solutions to nonlinear equations forapplications in embedded systems✩
Ying Yang a,1, Steven X. Ding b
a State Key Lab for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, PR Chinab Institute for Automatic Control and Complex Systems, University of Duisburg-Essen, 47057 Duisburg, Germany
a r t i c l e i n f o
Article history:Received 22 November 2010Received in revised form5 August 2011Accepted 18 August 2011Available online 3 March 2012
Keywords:Numerical analysisStability theoryFixed point iterationNewton’s methodsObserver designLMI technique
a b s t r a c t
In this paper, the fixed point iteration and Newton’s methods for iteratively solving nonlinear equationsare studied in the control theoretical framework. This work is motivated by the ever increasing demandsfor integrating iterative solutions of nonlinear functions into embedded control systems. The use of thewell-established control theoreticalmethods for our application purpose is inspired by the recent control-theoretical study on numerical analysis. Our study consists of two parts. In the first part, the existing fixedpoint iteration and Newton’s methods are analysed using the stability theory for the sector-boundedLure’s systems. The second part is devoted to the modified iteration methods and the integration ofsensor signals into the iterative computations. The major results achieved in our study are, besides someacademic examples, applied to the iterative computation of the air path model embedded in the enginecontrol systems.
Iterative computation is one of the standard techniques forsolving nonlinear equations (Quarteroni, Sacco, & Saleri, 2000;Stoer & Bulirsch, 2002). It is a powerful mathematical tool widelyused in engineering applications (Eich-Soellner & Führer, 1998;Hoffmann, Marx, & Vogt, 2005). Thanks to the rapid developmentof microprocessor technology, more and more iterative solutionsof nonlinear equations are implemented on electrical controlunits (ECUs) in real-time embedded systems. For instance, forthe real-time control and on-board-diagnosis (OBD) of an internalcombustion engine numerous iterative computation blocks areintegrated into the ECU (Kiencke & Nielsen, 2005).
Nonlinear equations like
x = ϕ(x) and f (x) = 0 (1)
✩ This work is supported by the National Natural Science Foundation of Chinaunder Grant 60874011, 61174052, 90916003. The material in this paper was notpresented at any conference. This paper was recommended for publication inrevised form by Associate Editor Yoshito Ohta under the direction of Editor RobertoTempo.
E-mail addresses: [email protected] (Y. Yang), [email protected](S.X. Ding).1 Tel.: +86 10 62751815; fax: +86 10 62764044. She has completed this study
are most typical forms met in engineering applications (Eich-Soellner & Führer, 1998; Hoffmann et al., 2005). The so-called fixedpoint iteration described by
x(k + 1) = ϕ(x(k))
and Newton’s methods with the general iterative form
x(k + 1) = x(k)− Ψ (x(k))f (x(k))
are standard algorithms for solving (1) iteratively, where k standsfor the iterative number and Ψ (x(k)) is some matrix (Quarteroniet al., 2000; Stoer & Bulirsch, 2002). Under certain conditions, theiteration will converge to the solution of the equations x∗, i.e.
limk→∞
x(k) = x∗, x∗= ϕ(x∗) or f (x∗) = 0.
For the real-time applications in embedded control systems, thenonlinear equations in (1) build typically sub-models embedded ina complex functional block and thus are often a function of systeminputs and parameters (see also the example in Section 5). In thiscontext, the nonlinear equations in (1) can be extended to
x = ϕ(x, p) and f (x, p) = 0 (2)
with p being a vector representing these (time-varying) systeminputs and parameters. The iterative computation will then betriggered by each update of p. For applications in embeddedcontrol systems, such computation often demands for high real-time ability. Although ECUs of the new generation are becoming
584 Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594
more powerful, the low cost requirement on the one hand and everincreasing demands for the high system performance on the otherhand call for more attention to the real-time implementation ofiterative algorithms.
Our study presented in this paper is mainly motivated bythe real-time implementation of control and OBD algorithms onthe ECU for the internal combustion engine control (Weinhold,2007). In this industrial application, we have been confrontedwith the following two problems: (a) the convergence rate ofthe existing iterative solutions for some nonlinear equations doesnot satisfy the real-time requirement (b) the quantisation errorsdue to the use of look-up tables instead of analytical functionsmay considerably affect the computation performance. As a result,poor control performance and, in the worst case, instability can beobserved. As investigating solutions for these problems, we noticethe recent efforts on applying the modern control theoreticalmethods to numerical analysis (Bhaya & Kaszkurewicz, 2006,2007; Kashima & Yamamoto, 2007; Schaerer & Kaszkurewicz,2001; Söderlind, 2002). In particular, encouraged by the resulton the Newton’s method reported in Kashima and Yamamoto(2007), we focus our study on (a) the convergence conditionsof the fixed point iteration and Newton’s methods for nonlinearfunctions satisfying the so-called sector conditions (Khalil, 2002)(b) the robustness issue with respect to computation errorse.g. caused by the quantisation. In this study, vector-valuednonlinear equations are considered, and the well-establishednonlinear control theory (Khalil, 2002) and the LMI (linear matrixinequality) technique (Boyd, Ghaoui, & Feron, 1994) are applied asthe main analysis and design tool. This effort allows us to expressthe convergence conditions in terms of some LMIs, which can thenbe checked using some standard software.
Driven by the demands for high system performance andreliability, a trend can be observed in the area of embedded controlsystems that the number of the integrated sensors is continuouslyincreasing. The intuitive idea of our further study on solvingnonlinear equations is to integrate those sensor signals, which areavailable in the embedded system, into the iterative computation.This work has been strongly motivated by the moderate and, insome cases, poor performance delivered by the standard iterativemethods as they were applied in the engine control and OBDalgorithms running on the ECU. It is reasonable to expect that theadditional information provided by the sensors will improve theconvergence rate and enhance the robustness. For our purpose,an observer-like iterative method is proposed, which modifies theexisting methods and allows us to integrate the available sensorsignals so that both the convergence rate and robustness areimproved. From the control-theoretical viewpoint, this work is acombination of our study on the existing iterative solutions andthe well-established observer design technique.
By testing our new approach to the practical real-timeimplementation problems, we notice that the sensor noisesmay have a strong influence on the computation performance.Considering that e.g. in automotive systems the sensors embeddedin the control loops and in OBD are often low-cost products, thelast topic in our study is dedicated to the robustness analysis withrespect to the measurement noises.
The paper is organised as follows. In Section 2, the neededpreliminaries in numerical analysis and stability theory are brieflypresented, based on which the main problems to be addressedin this paper are formulated. Section 3 is devoted to the control-theoretical study on the existing fixed point iterative solution andNewton’smethodswith a focus on the convergence and robustnessissues. In Section 4, we shall first propose a unified approach formodifying the existing iterative algorithms and integrating sensorsignals aiming at improving the convergence rate. It is followedby an analysis of the influence of the measurement noises on
the computation performance. The last part in this section dealswith the optimisation of the iterative schemes with respect to theconvergence rate and robustness. To illustrate the major results,some (academic) examples are included in each section, and anapplication example from the real-time implementation of the airpath model on the ECU is presented in Section 5.Notation. The notation adopted throughout this paper is fairlystandard. Rn denotes the n-dimensional Euclidean space andRn×m the set of all n×m real matrices. The superscript ‘‘T ’’ standsfor the transpose of a matrix. ‘‘I ’’ and ‘‘0’’ denote the identity andzero matrix with appropriate dimension, respectively. ‘‘P > 0 (≥0)’’ means matrix P is positive definite (semi-definite). σ (·), σ (·)denote the maximum and minimum singular value of a matrixrespectively. ∥·∥ stands for the Euclidean norm. For vector x ∈ Rn,∥x∥ =
√xT x. We use E(·) to denote the expectation of a statistical
or stochastic variable.
2. Preliminaries, basic ideas and problem formulation
2.1. Fixed point iteration and Newton’s methods
Let the function ϕ : Rn→ Rn have a fixed point x∗
: ϕ(x∗) =
x∗∈ Rn. The fixed point iteration algorithm is given e.g. by Stoer
and Bulirsch (2002)
x(k + 1) = ϕ(x(k)) ∈ Rn. (3)
The following definition and theorem are standard results innumerical analysis (Quarteroni et al., 2000; Stoer & Bulirsch, 2002).
Definition 1. ϕ : D ⊆ Rn→ Rn is called contractive on a set
Do ⊂ D if there exists a constant α < 1 such that for all x, y ∈ Do
∥ϕ(x)− ϕ(y)∥ ≤ α∥x − y∥. (4)
Theorem 1 (Eich-Soellner and Führer (1998), Banach’s Fixed PointTheorem). Let ϕ : D ⊆ Rn
→ Rn be contractive in a closed set Do ⊂
D and suppose ϕ (Do) ⊂ Do. Then ϕ has a unique fixed point x∗∈ Do.
Moreover, for any x(0) = x0 ∈ Do, the iteration (3) converges to x∗.The distance to the solution is bounded by
∥x∗− x(k)∥ ≤
αk
1 − α∥x(1)− x(0)∥. (5)
It is worthmentioning that under the same conditions given in theabove theorem it is easy to prove that
∥x∗− x(k)∥ ≤ αk
∥x∗− x(0)∥. (6)
Let the function f : Rn→ Rn have a unique solution x∗
∈ Rn:
f (x∗) = 0. The standard Newton’smethod is an iterative algorithmdescribed by
x(k + 1) = x(k)− (Df (x(k)))−1f (x(k)) (7)
with Df (x(k)) ∈ Rn×n as the Jacobian matrix of f (x(k)). Thefollowing theorem describes the major properties of Newton’smethod (Stoer & Bulirsch, 2002).
Theorem 2. Given f : D ⊆ Rn→ Rn and the convex set Do ⊆ D,
let f be differentiable for all x ∈ Do and continuous for all x ∈ D. Forxo ∈ Do let positive constants r, α, β, γ be given with the followingproperties:
Sr(xo) = {x|∥x − xo∥ < r} ⊆ Do,
h = αβγ /2 < 1, r =α
1 − h,
and let f (x) have the following properties:
Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594 585
(a) ∥Df (x)− Df (y)∥ ≤ γ ∥x − y∥ for all x, y ∈ Do,(b) (Df (x))−1 exists and satisfies ∥(Df (x))−1
∥ ≤ β for all x ∈ Do,(c) ∥(Df (x(0)))−1f (x(0))∥ ≤ α.
Then x(k), k = 0, 1, . . . , is well defined and satisfies x(k) ∈
Sr(xo), limk→∞ x(k) = x∗, f (x∗) = 0, and
∀k ≥ 0, ∥x(k)− x∗∥ ≤ α
h2k−1
1 − h2k. (8)
It is evident that computing (Df (x(k)))−1 in each iteration needsconsiderable computational effort and has to be avoided for a real-time application. The so-called simplified Newtonmethod (Stoer &Bulirsch, 2002) is amostly used solution in this case, which is givenby
x(k + 1) = x(k)− (Df (x(0)))−1f (x(k)), (9)
where (Df (x(0)))−1 is a constant matrix.
2.2. Sector-bounded nonlinearities and Lure’s systems
In their pioneering study, Kashima and Yamamoto (2007) haveinvestigated the convergence condition of Newton’s method forscalar nonlinear equations satisfying the so-called sector condi-tions. It is demonstrated that replacing the Lipschitz (contractive)condition given in Theorem 1 by the sector boundedness mayrelax the convergence conditions. Motivated by this result, weshall study the convergence conditions of the fixed point iterationand Newton’smethods for vector-valued nonlinear equationswithsector-bounded nonlinearities.
The following definition of the sector condition is givenby Khalil (2002), which is widely used in the study on systempassivity, nonlinear control (Khalil, 2002), observer (Arcak &Kokotovic, 2001) and filter design (Wang, Liu, & Liu, 2008).
Definition 2. Function ψ : Rn→ Rn is said to belong to the
sector [K1, K2] on D with K = K T= K2 − K1 > 0, K1, K2 ∈ Rn×n,
and denoted by ψ ∈ [K1, K2] if
(ψ(x)− K1x)T (ψ(x)− K2x) ≤ 0 for all x ∈ D. (10)
The sector boundedness on nonlinearities is quite general thatalso includes the widely used Lipschitz conditions as a specialcase (Khalil, 2002). One of the applications of the sector conditionis the stability study on the so-called Lure’s system, which is ingeneral described by
y(z) = G(z)u(z) ∈ Rn, u(k) = −ψ(y(k)) ∈ Rn (11)
where G(z) is the n × n transfer function matrix of an LTI (lineartime invariant) system andψ(y(k)) a nonlinearmapping satisfyingthe sector condition ψ ∈ [K1, K2]. It is well-known that by meansof a loop transformation as sketched in Fig. 1 (Khalil, 2002), wehave
System (13) satisfying (12) is called passive and the nonlinearityψ(y(k)) is said belong to [0,∞] and denoted by ψ ∈ [0,∞]
(Khalil, 2002).
2.3. Basic ideas and problem formulation
Our first task is to study the convergence conditions of thefixed point iteration and Newton’s methods for vector-valued
Fig. 1. A loop transformation.
nonlinear equations satisfying the sector condition. Inspired by thework in Kashima and Yamamoto (2007), where scalar nonlinearequations are dealt with, we shall address this topic in the stabilityframework using the LMI technique. The major objective of thisstudy is to derive some new convergence conditions, which mayrelax the existing ones, and to build the basis for investigatingpossible modifications of the existing methods, the integration ofthe sensor signal into the iterative computation and the robustnessissues.
As previously mentioned, in real-time embedded systems withlow-cost ECUs, there may exist computation errors at each iterate.These can be caused by the quantisation error εx(k + 1) = x(k +
1)− xo(k + 1), where xo(k + 1) represents the ideal computationbeing free of the round-off error, or by the approximation error ofthe nonlinear function εϕ(k) = ϕ(x(k)) − ϕo(x(k)) with ϕo(x(k))denoting the original function and ϕ(x(k)) its approximation bye.g. look-up tables. In practice, such computation errors mayremarkably affect the computation performance. Similar to thestudy in Kashima and Yamamoto (2007), we model these errorsin terms of an additional term ε(k) ∈ Rn in the iteration as followsx(k + 1) = ϕ(x(k))+ ε(k), fixed point iteration,
It is our task to find themaximum influence of ε(k) on limk ∥x(k)−x∗
∥ and to minimise it.Suppose that the sensors, being available in the embedded
control system and used for our purpose of improving the iterativecomputation, is modelled by
y = Cx∗+ η, y ∈ Rm, m < n, (16)
where y represents the vector of the measurements, η themeasurement noise. It is assumed that
Eη = 0, E(ηTη) = E∥η∥2≤ δ2η, δη > 0, (17)
matrix C ∈ Rm×n is known. In real applications, a sensor modelmay be nonlinear. In that case, a linearisedmodel can be used. Also,advanced nonlinear observer methods (Arcak & Kokotovic, 2001)are helpful.
Recall that in embedded systems iterations are often triggeredby each update of the system inputs and parameters. According to(2), the solution x∗ is in fact a function of p. As a result, (16) can berewritten into
y = Cx∗(p)+ η. (18)
For instance, the sampling time of today’s ECU for the moderninternal combustion engine control system is 10 ms, which means
586 Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594
e.g. the angle of the throttlemay change every 10ms. It is required,in turn, that all those iterative computations of the nonlinearfunctions parameterised by the angle of the throttle (see alsothe example in Section 5) should be completed within a coupleof milliseconds and before a new sampling. Thus, although thesensor signals are time sequences, y is constant during the wholeiteration running between two sampling instants. Also, x∗(p) is afunction of the system parameters, but remains constant betweentwo sampling instants. In the context of iterative computationsof nonlinear functions, which depend on system inputs andparameters and are implemented in an embedded system, theintegration of sensor signals modelled by (18) will enhance thereal-time ability of the iteration schemes.
The basic idea of integrating the sensor signals into the itera-tive computation is to add an additional term r(k) = L(y − Cx(k))∈ Rn in the iterative computations. Such an iterative computationsystem is called observer-like iteration due to its similar structurewith an observer. Inspired by this structure, we will propose a uni-fied form for the iterative computation, in which the sensor signalsand further possiblemodifications can be included in a generalisedform. Our task consists in (a) analysing the convergence rate androbustness of the unified iteration solution and (b) developing anapproach to optimise the iterative computation by finding matrixL ∈ Rn×m and the possible modifications.
3. Study on the existing iterative solutions
In this section, we will study the fixed point iteration andNewton’s methods, and discuss about the issues related to theneeded computation efforts and robustness.
3.1. On the fixed point iteration
For our purpose, we rewrite (3) into
x(k + 1) = x(k)− (x(k)− ϕ(x(k))). (19)
Let e(k) = x(k)− x∗ be the iteration error. It holds
e(k + 1) = e(k)− ψ(e(k)) (20)
with a new function ψ(e(k)) ∈ Rn defined by
ψ(e(k)) = e(k)− (ϕ(x(k))− ϕ(x∗))
= e(k)− (ϕ(e(k))− ϕ(x∗)), ϕ(e(k)) = ϕ(x(k)).(21)
Note that ψ(0) = ϕ(x∗) − ϕ(x∗) = 0. We now present and provethe first result of this work, whose proof plays an essential role inthe subsequent study.
Theorem 3. Consider the iterative algorithm (3) and the associatedEq. (20). Suppose that ϕ : Rn
→ Rn has a unique fixed point x∗∈
D ⊆ Rn and ψ ∈ [K1, K2] on Do, where Do = {x − x∗, x ∈ D}. Ifthere exist ν > 0 and P = PT > 0 such that the following linearmatrix inequality holds νI − P −K P − K T
1 P−K −2I P
P − PK1 P −P
< 0, (22)
then for an arbitrary (start) point with x(0) ∈ D the iterationconverges to the fixed point x∗. Moreover, it holds
∥e(k)∥ < τσ k∥e(0)∥, k = 1, . . . , (23)
where 0 < σ =
1 −
νσ (P)
1/2< 1, τ =
σ (P)σ (P) .
Proof. The proof consists of three steps. In the first step, a looptransformation, as shown in Fig. 1, is carried out. To this end, write(20) into the form given in (11)
e(z) = (zI − I)−1u(z), u(k) = −ψ(e(k)).
By the above described loop transformation, we have
G :
e(k + 1) = Ae(k)+ u(k), A := (I − K1)
y(k) = Ke(k)+ u(k), u(k) = −ψ(y(k)), (24)
ψ(y(k)) = ψ(ς(k))− K1ς(k), ς(k) = K−1(y(k)− u(k))
with ψ(y) ∈ [0,∞) on Do. Next, define V (k) = eT (k)Pe(k), P >0,1V (k) = V (k + 1)− V (k). It holds
Note that in the above equation and also in the remaining ofthis proof, we drop variable k for simplifying the notation, if noconfusion is caused. Since (22) is, by Schur complement (Boyd et al.,1994), equivalent to
Adding and subtracting 2ψT (y)Ke and 2ψT (y)ψ(y) to and from theright side of the above inequality lead to
1V < −νeT e − eT (PA − K)T (2I − P)−1(PA − K)e
−2eT (PA − K)T ψ(y)− ψT (y)(2I − P)ψ(y)
−2ψT (y)(Ke − ψ(y)) = −νeT e − zT z − 2ψT (y)y,
where z = (2I − P)−1/2(PA − K)e + (2I − P)1/2ψ(y). Sinceψ(y) ∈ [0,+∞) means ψT (y)y ≥ 0 for all e ∈ Do (Khalil, 2002),we have
1V < −νeT e < 0 (27)
for e(k) = 0. Following the Lyapunov stability theory (Khalil,2002), system (24) and equivalently system (20) are asymptoticallystable at origin, i.e. limk→∞ e(k) = 0 ⇐⇒ limk→∞ x(k) = x∗. Inthe third step, the inequality (23) will be proven. It follows from(27) that V (k + 1) < V (k) − νeT (k)e(k) ≤ σ 2V (k). Note thatP > νI ⇒ 0 < ν
Theorem 3 provides us with an alternative convergencecondition different from the contractive condition (4). Moreover, itdemonstrates that the convergence rate of the iterative algorithmunder the sector condition (10) is determined by σ and analogousto (6). In the following corollary and examplewe are going to checkif the sector condition (10) relaxes the contractive condition (4).
Corollary 1. Suppose that ϕ : Rn→ Rn has a unique fixed point
x∗∈ D ⊆ Rn and the contractive condition (4) holds for all x, y ⊆ D.
Then the convergence conditions given in Theorem 3 are satisfied.
Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594 587
Proof. It follows from the contractive condition (4) that
i.e. ψ(x) ∈ [K1, K2] on Do = {x − x∗, x ∈ D} with K1 = (1 −
α)I, K2 = (1 + α)I and so K = 2αI. To check if the LMI condition(22) is satisfied, we study (26), which is, by Schur complementlemma, equivalent to (22)ATPA − P + νI ATP − K
PA − K P − 2I
< 0
⇐⇒
α2P − P + νI α(P − 2I)α(P − 2I) P − 2I
< 0
⇐⇒ (2α2+ ν)I < P < 2I. (30)
Since P > 0, ν > 0 satisfying (30) exist, we can conclude that boththe sector condition ψ ∈ [K1, K2] and the LMI (22) are satisfied.The corollary is thus proven. �
Corollary 1 illustrates that the result with the sector conditiongiven in Theorem 3 includes the one satisfying the contractivecondition (4). The following example adopted from Kashima andYamamoto (2007)will show that the convergence conditions givenin Theorem 3 are less strict than the contractive condition (4).
Example 1. Consider
x = ϕ(x), ϕ(x) =2x3 + x2 − 13x2 + 2x + 1
,
for which we have solution x∗= −1. It is clear that the contractive
condition (4) is not satisfied e.g. by checking |ϕ(1) − ϕ(0)| =
1 +13 > 1. On the other hand,
ψ(e) = e − (ϕ(x)− ϕ(−1)) =e3 − 2e2 + 2e3e2 − 4e + 2
satisfies the sector condition
(ψ(e)− εe)(ψ(e)− (2 − ε)e) ≤ 0, ∀e ∈ R
i.e. ψ ∈ [ε, 2 − ε] on R, where 0 < ε ≪ 1 is a constant enoughclose to zero. With K1 = ε, K = 2 − 2ε, it is straightforward tocheck that for β > ε and 0 < β − ε ≪ 1 being enough close tozero
P =2 − ε − β
1 − ε= 1 +
1 − β
1 − ε< 2
and a small enough ν > 0 solve the LMI (22). Thus, it can be con-cluded that starting from any point in R, the iteration error willconverge to zero. Fig. 2 shows the results with three different ini-tial starts: x(0) = 0, x(0) = 1, x(0) = −2. All three iterationsconverge to −1.
Fig. 2. Iterative computation results in Example 1.
(31) is identical with (20). Thus, we have the following theoremwhose proof is identical with the one of Theorem 3 and thereforeomitted.
Theorem 4. Consider the iterative algorithm (7) and the associatedequation (31). Suppose that f : Rn
→ Rn has a unique solutionx∗
∈ D ⊆ Rn and ψ ∈ [K1, K2] on Do, where Do = {x − x∗, x ∈ D}.If there exist ν > 0 and P = PT > 0 such that the following linearmatrix inequality holds νI − P −K P − K T
1 P−K −2I P
P − PK1 P −P
< 0, (32)
then for an arbitrary (start) point x(0) with x(0) ∈ D the iterationconverges to the solution x∗.
Next, we demonstrate the relations between the convergenceconditions given in Theorems 2 and 4.
Corollary 2. Consider the iterative algorithm (7) and the associatedequation (31). Suppose that the conditions in Theorem 2 are satisfied.Then there exist a neighbourhood of x∗,D∗
o ⊆ Do, and associated withit D∗
= {x−x∗, x ∈ D∗o}, ν > 0 and P = PT > 0 so that ψ ∈ [K1, K2]
on D∗ and the LMI (32) is satisfied.
Proof. In Stoer and Bulirsch (2002), Section 5.3, it is proven thatconditions (a) and (b) in Theorem 2 lead to
∥D−1f (y)(f (x)− f (y)− Df (y)(x − y))∥ ≤βγ
2∥x − y∥2 (33)
for all x, y ∈ Do, where β, γ > 0. Let 0 < α < 1 such that D∗o ⊆ Do
be a neighbourhood of x∗
D∗
o =
x| ∥x − x∗
∥ ≤α
ω, ω =
βγ
2
.
Now, setting x = x∗, y = e + x∗∈ D∗
o in (33) yields
∥D−1f (e + x∗)f (e + x∗)− e∥ ≤βγ
2∥e∥2
≤ α∥e∥. (34)
Note that (34) can be rewritten as
∥ψ(e)− e∥ ≤ α∥e∥, e ∈ D∗,
which is of the same form like (29). Thus, using the results given inthe proof of Corollary 1 we can conclude that ψ ∈ [K1, K2] on D∗
and the LMI (32) is satisfied. �
588 Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594
Remember that the real-time ability is themajor concern in ourstudy on the application of the iterative solutions to the embeddedsystems. Among themodifiedNewton’smethods (Quarteroni et al.,2000), the one given in (9) is,widely used in the embedded systemapplications. Based on Theorem 4, the following convergenceconditions can be derived.
Corollary 3. Consider the iterative algorithm (9). Suppose that f :
Rn→ Rn has a unique solution x∗
∈ D ⊆ Rn and f ∈ [K1, K2]
on Do, where f (x) = f (x + x∗),Do = {x − x∗, x ∈ D}. LetB = (Df (x(0)))−1. If there exist ν > 0, P = PT > 0 such thatthe following linear matrix inequality holds νI − P −K P − K T
ψ(y(k)) = f (ς(k))− K1ς(k), ς(k) = K−1(y(k)− u(k)).
The remaining proof of LMI (35) can be done along with the linesin the proof of Theorem 3 and thus omitted.
3.3. Computation issues
In our previous study, no modification has been made on theiterative algorithms and thus, in the real-time application, noadditional on-line computation is needed. On the other hand,different (off-line) computational efforts for checking the proposedconvergence conditions are expected. Below, we will analysethe needed (off-line) computational costs briefly and proposean algorithm for a systematic checking procedure. It followsfrom Theorems 3 and 4 and Corollary 3 that the computationsintroduced above for checking the convergence conditions consistof two steps: (a) checking the sector condition (10) (b) solving anLMI.
When a scalar nonlinear function is addressed, the sectorcondition (10) can be checked analytically or graphically as shownin Example 1. The latter is preferred in engineering applications,since the computation range is, in general, physically well definedand considerably limited. In case of dealing with a n-dimensionalvector of nonlinear functions, Khalil (2002) suggests to setK1, K2 tobe n-dimensional diagonal matrices if the nonlinear functions aredecoupled. Thus, the problem is reduced to check n scalar sectorconditions. In some cases, choosing K1, K2 suitably may result indecoupled nonlinear functions, see e.g. Examples 2 and 3 in thispaper.
In our study, the LMIs given in Theorems 3 and 4 and Corollary 3are solved using the MATLAB LMI toolbox (Gahinet, Nemirovski,Laub, andChilali, 1995). It should be pointed out that the solvabilityof the LMIs given in (22), (32) and (35) depends on K1 and K2(K).For instance, in Example 1, althoughψ ∈ [0,∞] onR, the LMI (22)is solvable only for ψ ∈ [ε, 2 − ε] on R. In other words, it is oftennecessary to determine suitable K1 and K2(K). For this purpose, wepropose the following algorithm: (a) solvingνI − P −K P − Q T
−K −2I BTPP − Q PB −P
< 0 (37)
for P > 0, ν > 0, K = K T > 0 and Q , where B = I for the fixedpoint and Newton’s methods and B = (Df (x(0)))−1 for themodified Newton’s method (9). (b) setting
K1 =
P−1Q , for the iterations (3), (7)Df (x(0))P−1Q , for the iteration (9)
(38)
and K2 = K + K1. The LMI (37) and Eq. (38) follow directly from(22) and (35) and can be solved using e.g. the above-mentionedMATLAB LMI toolbox.
3.4. Convergence error
In this subsection, we analyse the convergence error caused bythe computation error ε(k) in each iterate. Let e(k) = x(k) − x∗.It follows from (14) that for both the fixed point iteration andNewton’s method
e(k + 1) = e(k)− ψ(e(k))+ ε(k) (39)
ψ(e(k)) =
e(k)−
ϕ(e(k))− ϕ(x∗)
,
for the fixed point iterationϕ(e(k)), for the Newton’s method withϕ(e(k)) = ϕ(x(k)+ x∗).
We suppose that ψ ∈ [K1, K2] on D, and the convergence condi-tions in Theorems 3 and 4 are satisfied respectively for the fixedpoint iteration and Newton’ method, which ensure x(k) convergesto x∗ if ε(k) = 0. Next, we analyse how large the convergenceerror will become for the bounded computation error ε(k). LetV (k) = eT (k)Pe(k), P > 0. It holds
1V = V (k + 1)− V (k) = (e − ψ(e))TP(e − ψ(e))− eTPe + εTPε + 2(e − ψ(e))TPε H⇒
Recalling that under convergence conditions given in Theorem3 orTheorem 4 we have
V (k)+ (e − ψ(e))TP(e − ψ(e))− eTPe < σ 2V (k)
with σ = (1 −ν
σ (P) )1/2 < 1, it yields
V (k + 1) < σ 2V (k)+ γ1δεV (k)+ σ (P)δ2ε ,
γ1 = 2σ (P)σ (P−1/2)(σ (N)+ σ (M − I)). (40)
Let δ > 0 satisfy
ξ = σ 2+ γ1/δ + σ (P)/δ2 ≤ 1. (41)
As a result, if V (k) ≥ δ2δ2ε , we have V (k+ 1) < ξV (k). That means(a) if x(0) is selected so that V (0) ≥ δ2δ2ε ,
limk
V (k) ≤ δ2δ2ε H⇒ limk
∥e(k)∥ ≤ σ (P−1/2)δδε, (42)
(b) if x(0) is selected so that V (0) < δ2δ2ε ,
V (k) ≤ δ2δ2ε H⇒ ∥e(k)∥ ≤ σ (P−1/2)δδε. (43)
In summary, we have proven the following theorem.
Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594 589
Theorem 5. Consider (39)with bounded ε(k)whose boundedness isgiven in (15). Suppose that the convergence conditions in Theorems 3and 4 are satisfied. Then,
limk
∥e(k)∥ ≤ ϑδε, ϑ = σ (P−1/2)δ > 0 (44)
with δ satisfying (41).
(44) gives an estimation of the upper bound of the convergenceerror. Recall that σ in (41) is a key parameter which decides theconvergence rate of the iteration (see also Theorem 3). If σ in (41)is very close to 1, δ and so ϑ will become very large to ensure (41).In other words, if the error-free iteration (i.e. ε(k) = 0) converges(very) slowly to its solution x∗, the convergence error caused byε(k)may become (very) large.
4. A unified observer-like iterative solution
We now study improving the real-time ability of the iterativesolutions by modifying the algorithms and embedding the sensorsignals into the iterative computation.
4.1. A unified form of the observer-like solutions
For iteratively solving nonlinear equations f (x) = 0 and x =
ϕ(x), we propose the following two algorithms:(a) modified Newton’s method:
where B ∈ Rn×n and L ∈ Rn×m are design parameter matrices tobe determined.
The introduction of the term −Bf (x(k)) in (45) is a naturalextension of (9), while adding−B(x(k)−ϕ(x(k))) in (46) is inspiredby the study on the fixed point iteration in Section 3.1. It is worthnoting that these two additional terms act as a feedback of theestimation error of the nonlinear equations in (1) at each iterate:
It is known from the control theory that suitably selecting Bmay improve the convergence rate. Both (45) and (46) are ofthe observer structure (Chen, 1984), whose core is the feedbackof signal r(k) = L(y − Cx(k)) under use of the sensor modely = Cx∗
+ η. It is well known that by a suitable selection ofmatrix L, which is also called observer matrix, feeding back r(k)will significantly improve the convergence rate of the iteration. Onthe other hand, a most important pre-condition for achieving niceconvergence performance is the observability which depends onthe output matrix C and the system matrix. The introduction ofmatrix B, which is a part of the system matrix, also serves for thispurpose.
It is evident that limk→∞ x(k) in (45) or (46) respectively solvesf (x) = 0 or x = ϕ(x) if x(k) converges to x∗.
4.2. Convergence study
For our purpose, we first assume η = 0 and let e(k) = x(k)−x∗.It follows respectively from (45) and (46) thate(k + 1) = (I − LC)e(k)− Bf (e(k)+ x∗)for the modified Newton’s iteration (45),
e(k + 1) = (I − B − LC)e(k)− B(x∗− ϕ(e(k)+ x∗))
for the modified fixed point iteration (46),
which is, to simplify the notation, unified written as
e(k + 1) = ALe(k)− Bψ(e(k)), (47)AL = I − LC, ψ(e(k)) = f (e(k)+ x∗)for the modified Newton’s iteration (45)
AL = I − B − LC, ψ(e(k)) = x∗− ϕ(e(k)+ x∗)
for the modified fixed point iteration (46).
We now introduce a theorem that provides us with the sufficientconditions for the convergence of (45) and (46).
Theorem 6. Consider the iterative algorithm (45) or (46). Supposethat they have a unique solution x∗
∈ D ⊆ Rn and ψ ∈ [K1, K2] onDo,Do = {x − x∗, x ∈ D}. If there exist ν > 0, P = PT > 0 and Q , Rsuch that the following linear matrix inequality holds νI − P −K P − K TQ T
− CTRT
−K −2I Q T
P − Q K − RC Q −P
< 0, (48)
where K = K1 for the modified Newton’s method (45) and K = I+K1for the modified fixed point iteration (46), then for B = P−1Q , L =
P−1R and x(0) ∈ D the iteration converges to x∗ with
∥e(k)∥ < τσ k∥e(0)∥, τ =
σ (P)σ (P)
, σ =
1 −
ν
σ (P)
1/2
. (49)
The proof of this theorem can be done along with the lines in theproof of Theorem 3 and is thus omitted.
In order to demonstrate the application of the above result, wegive the following example which shows the convergence of theobserver-like Newton’s method.
Example 2. Consider the iteration algorithm (45) with
f (x) =
x31 + 2x21 + 2x1 + 1
2x21 + 2x1 + 1
x32 + x1x22 + x2 + x1x22 + 1
.It holds ψ = f (e + x∗) ∈ [K1, K2] on R2 with
x∗=
−11
, K1 =
0 0
−1 0
, K2 =
4 0
−1 3
.
Using the MATLAB LMI toolbox, it can be found out that (48) isinfeasible for C = 0. That means we cannot find B to ensure theconvergence of the iteration (45). Now, assume that we have anembedded sensor signal y = Cx∗ with C = [2 1]. It can be verifiedthat (48) is feasible with
B =
0.0906 −0.1951
−0.1795 0.3818
, L =
0.32360.3483
.
Fig. 3 shows the computation results of x(k) starting from the initialvalue x(0) = [1 − 1]T .
4.3. Convergence error caused by measurement noise
Considering that the influence of ε(k) on the convergenceerror limk ∥e(k)∥ can be estimated along with the discussion inSection 3.4, we shall, in this subsection, focus on the analysis ofthe influence of the measurement noise η on the convergenceperformance for both iteration algorithm (45) and (46). To this end,consider the unified form of the iteration errors
e(k + 1) = ALe(k)− Bψ(e(k))+ Lη. (50)
590 Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594
Fig. 3. Computation results of x(k)with observer-like Newton’s method.
Recall that if the convergence conditions given in Theorem 6 aresatisfied, then we have
(ALe − Bψ(e))TP(ALe − Bψ(e))− eTPe < −ν ∥e∥2
⇒ (ALe − Bψ(e))TP(ALe − Bψ(e)) < (σ (P)− ν)∥e∥2.
As a result, setting ζ (> 0) such that ζ (σ (P)− ν)− ν < 0 leads to
V (k + 1) < V (k)+ (ζ (σ (P)− ν)− ν)∥e∥2
+ ηT LT1ζP + I
Lη. (53)
Note that for νI − P < 0, ζ (σ (P) − ν) − ν > −σ (P) results in1 > 1 −
ν−ζ (σ (P)−ν)σ (P) > 0. Let
ϖ =
1 −
ν − ζ (σ (P)− ν)
σ (P)
1/2
< 1, (54)
λ = σ
LT
1ζP + I
L, (55)
we have finally
V (k + 1) < ϖ 2V (k)+ ληTη
H⇒ V (k + 1) < ϖ 2kV (0)+λ(1 −ϖ 2k)
1 −ϖ 2ηTη,
limk→∞
(eT (k)e(k)) <λ
σ(P)(1 −ϖ 2)ηTη
H⇒ limk→∞
E(eT (k)e(k)) <λ
σ(P)(1 −ϖ 2)δ2η. (56)
The main result from the above discussion can be summarised inthe following theorem.
Theorem 7. Consider (50)with measurement noise η whose bound-edness is given in (17). Suppose that the convergence conditionsin Theorem 6 are satisfied. Then,
limk→∞
E∥e(k)∥2 <λ
σ(P)(1 −ϖ 2)δ2η (57)
withϖ,λ satisfying (54)–(55).
Wewould like to remark thatϖ will (very) close to 1 if σ in (49)(see Theorem 6) is (very) close to 1. Moreover, in this case ζ willbecome (very) small,which leads to a (very) largeλ. In otherwords,if the noise-free iteration (i.e. η = 0) converges (very) slowly to itssolution x∗, the convergence error caused by η will become (very)large. Also, due to the term λ = σ (LT ( 1
ζP + I)L) a large σ (LTPL)
will cause large convergence error.
4.4. Design of the observer-like solution
Remember that for the real-time applications in embeddedsystems the performance of an iterative computation will beevaluated in terms of (a) the convergence rate (b) the robustnessagainst computation errors and (c) in case of an integration ofsensor signals, the robustness against noises. In this context, wepropose an approach for the design of observer-like iterativesolutions (45) and (46). For our purpose, we first define a costfunction. It follows from (49) in Theorem 6 that the convergencerate is determined by σ = (1 −
νσ (P) )
1/2, and the smaller σ is, thehigher the convergence rate becomes. Moreover, the discussionsat the end of Sections 3.4 and 4.2 show that enhancing theconvergence rate by reducing σ will also reduce the convergenceerror. These facts motivate us to optimise the (observer-like)iteration by minimising σ , which can then be equivalentlyformulated as
minν,σ (P)
1 −
ν
σ (P)
, 1 −
ν
σ (P)> 0
⇐⇒ minν,σ (P)
(σ (P)− ν), σ (P)− ν > 0. (58)
Let σ (P)−ν < ς . Then, we are able to write optimisation problem(58) as
minν,σ (P)
ς subject to σ (P) < ς + ν, ς > 0 ⇐⇒ (59)
minν,P
ς subject to−(ς + ν)I P
P −(ς + ν)I
< 0, ς > 0. (60)
Recall that σ (LTPL) should be bounded in order to limit the in-fluence of the measurement noise. To this end, we introduceσ (LTPL) ≤ χ , for some given χ > 0. Together with the conver-gence conditions given in Theorem 6, we finally formulate our op-timisation problem as follows.
Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594 591
Theorem 8. Consider iterative computation (45) or (46)whose errorsystem is unified described by (50). Suppose the conditions givenin Theorem 6 are satisfied. If there exist ν > 0, P = PT > 0 and Q , Rsuch that the following linear matrix inequality optimisation problemhas a solution:
min ς subject to (61) νI − P −K P − K TQ T− CTRT
−K −2I Q T
P − Q K − RC Q −P
< 0, (62)
−(ς + ν)I P
P −(ς + ν)I
< 0, ς > 0, (63)
−χ I RT
R −P
< 0, (64)
for given χ > 0, where in (62)
K =
K1, for the mod. Newton’s method (45)I + K1, for the mod. fixed point iteration (46)
then for any x(0) ∈ D the iteration converges to x∗ with the minimumconvergence rate ς and the optimal solution for L, B is given by B =
P−1Q , L = P−1R.
Example 3. Consider the algorithm (3) and its modified form (46)with
ϕ(x) =
−x31 + 4x21 + 3x1 + 2
3x21 + 2x1 + 1
x1 + 0.5 sin(x2 + 1)
,for which we have the solution x∗
= [−1 − 1]T . We assume thata sensor signal y = Cx∗ is available with C = [10 5]. Moreover, weset χ = 1. Since
ϕ ∈ [K1, K2] on R2, K1 =
0 0
−1 −0.5
, K2 =
2 0
−1 0.5
,
we get the largest convergence rate with σmin = 0.5876 bysolving (61)–(64) using theMATLAB LMI toolbox. For a comparison,we consider the algorithm (3), i.e. L = 0, B = 0. It turns outσmin = 0.865, which indicates a lower convergence rate. It isclear from Fig. 4 that the algorithm (46) with B, L delivers a betterconvergence performance than the algorithm (3) regarding to thesame initial value x(0) = [1 2]T .
4.5. On the computation issue
In comparison with the standard fixed point and Newton’siteration algorithms (3) and (9), the realisation of the observer-like forms (45) and (46) needs the following additional on-linecomputations: L(y − Cx(k)) in (45) and (I − B)x(k)+ L(y − Cx(k))as well as Bϕ in (46). These computations will not significantlyincrease the on-line computation cost. For the needed off-linecomputations, the check of the sector boundedness requiresthe same computational efforts as mentioned in Section 3.3. InExamples 2, 3, K1, K2 are chosen in such a way: the off-diagonalitems are chosen to make the nonlinear function decoupled, thediagonal items are determined graphically as the lower and upperbounds of the sector which the nonlinearity belongs to. The LMI(48) and the optimisation problem in Theorem 8 are standardones (Scherer, Gahinet, & Chilali, 1997) and can be solved using theMATLAB LMI toolbox (Gahinet et al., 1995).
5. An application example
In this section, we describe the application of our major resultsto the real-time computation of the air path model of the SI (spark
Fig. 4. Computation of the modified fixed point iteration with and withoutfeedback of sensor data.
Fig. 5. Schematic description of the air path in the SI engines.
ignition) engines, which is integrated into the embedded enginecontrol systems (Guzzella & Onder, 2010; Weinhold, 2007).
5.1. A brief system description and problem formulation
Fig. 5 sketches the air path of the SI engine with
mLF = 3600
100pU(pU − pvDK )
cfrTU,
mDK = 3600A(αDK )100pvDK0.5
√RTvDK
Ψ
pS
pvDK
,
mZyl = 3600VZyl0.5 · 100pS
RTS
160
n · η(n).
Variable Meaning Unit
mLF air flow through the air filter kg/hmDK air flow entering the intake manifold kg/hmZyl air flow entering the cylinder kg/hpU ambient air pressure = 1013 hPapvDK pressure in manifold behind air filter hPapS pressure in the intake manifold hPan engine speed rpmαDK angle of the throttle o
The coefficients and physical constants are listed in Appendix.Ψ (
pSpvDK
), A(αDK ), η(n) represent three nonlinear functions, forwhich only look-up tables shown in Fig. 6 are available. αDK , nmaychange strongly depending on the driver. In the steady state, wehave
pS =A(αDK )
√RTS
√TvDKn · η(n)
pvDK · Ψ
pS
pvDK
, (65)
pvDK = pU −25cfrTU
pU
A(αDK )pvDK
√RTvDK
Ψ
pS
pvDK
2
. (66)
592 Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594
Fig. 6. Look-up tables for the three nonlinear functions.
For the control and OBD purposes, pS, pvDK are iterativelycomputed using (65)–(66). This is done at each sampling instantand depending on n, αDK . Our major task consists in a reliable andreal-time applicable iterative computation of pS, pvDK in the wholeoperating range defined by αDK , n. In addition, the computationshould be as robust as possible against uncertainties in the look-up tables for η(n), A(αDK ),Ψ (
pSpvDK
).
5.2. The iterative computations and the results
In order to reduce the coupling between the computations ofpS, pvDK ,we rewrite (65) into
pSpvDK
=A(αDK )
√RTS
√TvDKn · η(n)
Ψ
pS
pvDK
, (67)
and iteratively compute pSpvDK
instead of pS . For our purpose, thestandard fixed point iteration (13) has been first applied, whichleads to
We have noticed that in some operating ranges, the iterations donot converge. Fig. 7 shows a representative (unstable) iteration atthe operating point n = 2000, αDK = 8. The initial condition ispvDK (0) = 900, µ(0) = 700/900.
This result hasmotivatedus to check the sector conditions givenin Theorem 3 for the fixed point iteration. In Fig. 8, the values of thenonlinear functionsψ1(e1) = e1 − c1(αDK , n)(Ψ (e1 + µ∗)− Ψ (µ∗))
= µ− c1(αDK , n)Ψ (µ), ψ2(e2)
= e2 − pU + c2(αDK )
(e2 + p∗
vDK )2Ψ 2(e1 + µ∗)
−(p∗
vDKΨ (µ∗))2
= pvDK − pU + c2(αDK )((pvDKΨ (µ))2)
e1 = µ∗− µ, e2 = pvDK − p∗
vDK
are sketched for the (physically) realistic values of µ, pvDK ,representatively at the operating points αDK = 8, n = 1500, 2000
1.5
1
0.5
0
–0.50 5 10 15 20 25 30 35 40 45 50
Iteration numbr (k)
Iteration numbr (k)
1100
1000
900
800
7000 5 10 15 20 25 30 35 40 45 50
Fig. 7. The (unstable) computation result using the standard fixed point iteration.
Fig. 8. ψ1(e1), ψ2(e2) in the operating ranges.
respectively. It is evident that ψ1 ∈ [0, 4], ψ2 ∈ [0, 2], and thusthe known sufficient sector condition [0, 2] is not satisfied for ψ1.
Motivated by this observation, we have decided to applythe modified fixed point iteration given in (46), first withoutintegrating the sensor data (i.e. L = 0):µ(k + 1)
pvDK (k + 1)
=
µ(k)
pvDK (k)
− B
µ(k)− c1(αDK , n)Ψ (µ(k))
pvDK (k)− pU + c2(αDK )(pvDK (k)Ψ (µ(k)))2
.
Applying Theorem 6 to our system results in B = diag(0.39, 0.30).As shown in Fig. 9, themodified fixedpoint iterationwith the aboveB delivers a reliable and fast iterative computation of pvDK , µ(pS)over the operating range.
Having ensured the stable iterative computation of pvDK , µ(pS),we have further focused on enhancing the computation robustnessagainst uncertainties in η(n), A(αDK ) by means of the observer-like iterative solution studied in Section 4. To this end, themeasurement pS delivered by an embedded pressure sensor isintegrated into the iteration. Considering that µ = pS/pvDK isa nonlinear function, which can be approximated by a standardlinearisation around pvDK (k),
µ ≈ pS/pvDK (k)−pS
p2vDK (k)1(k), 1(k) = µ− pS/pvDK (k),
we have used y = pS/pvDK (k) as an approximatedmeasurement ofµ in order to simplify the design and analysis. On this assumption,applying Theorem6 results in B = diag(0.32, 0.30), L = [0.19 0]T .As a result, the iterative computations areµ(k + 1) = µ(k)− 0.32(µ(k)− c1(αDK , n)Ψ (µ(k)))
+ 0.19(ps/pvDK (k)− µ(k)),
Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594 593
Fig. 9. Iterative computations of µ, pvDK with different initial conditions. Theoperating point in this example: αDK = 8, n = 2000. The real solution: p∗
vDK =
887.5, p∗
S = 718.5, µ∗= 718.5/887.5 = 0.81.
Fig. 10. Robustness comparison. The operating point in this example: αDK =
8, n = 2000. The real solution: p∗
vDK = 887.5, p∗
S = 718.5, µ∗= 718.5/887.5
= 0.81 The final values of the iteration without measurement: pvDS = 882.9, µ =
0.720. The final values of the iteration with measurement: pvDS = 888.0, µ =
In our study, the values of η(n), A(αDK ) given by the look-uptables have been manipulated up to ±20% to simulate the naturaluncertainties in η(n), A(αDK ). It can be observed that also in thiscase all iterations are stable and converge to the values, whichmaybe different from the real solutions. In that case, the iterationswiththe integrated measurement (sensor signal pS) deliver values withsmaller convergence errors, although a linearised sensormodel hasbeen applied. In Fig. 10, an example of a such iteration is presented.
6. Conclusions
In this paper, we have studied the iterative solutions fornonlinear equations and their applications to embedded systemsin the control theoretical framework. The first part of this studydeals with the analysis of the existing fixed point iteration andNewton’s methods using the stability theory for the sector-bounded Lure’s systems. As a result of this study, sufficientconditions for the convergence of the iterations have been derived,which are expressed in terms of the sector-boundedness andsome LMIs. Moreover, estimations of the convergence rate and theboundedness of the convergence errors caused by uncertaintiesin the nonlinear equations or by the quantisation are provided.The second part of our study addresses the modified iterationmethods with a focus on (a) the feedback of the estimationerror of the nonlinear equations (b) the integration of sensor
signals into the iterative computations. This work is stronglymotivated by the ever increasing demands for integrating reliableand fast converging iterative solutions into the ECUs in theembedded control systems. To this end, the major focus of ourstudy is on the improvement of the convergence rate, and onenhancing the computation robustness against uncertainties in thenonlinear equations as well as the measurement noises. We havederived sufficient conditions for the convergence of the so-calledobserver-like iterative algorithms as well as the estimations of theconvergence rate and the boundedness of the convergence errors.
To illustrate the theoretical results, three academic examplesare included in the paper. Our final result is a successful applicationto the iterative computation of the air pathmodel embedded in theengine control systems.
Acknowledgements
The authors thank Mr. de Moll for his help in the applicationstudy described in Section 5. The authors are also grateful tothe reviewers for the constructive and valuable comments andsuggestions.
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594 Y. Yang, S.X. Ding / Automatica 48 (2012) 583–594
Ying Yang received her Ph.D. degree in control theoryfrom Peking University, China in 2002. From January2003 to November 2004, she worked as a PostdoctoralResearcher at Peking University. Since 2005, she has beenan associate professor at the Department of Mechanicsand Aerospace Engineering, College of Engineering, PekingUniversity. Her research interests include robust andoptimal control, nonlinear systems control, numericalanalysis, fault detection and fault tolerant systems.
Steven Ding received Ph.D. degree in electrical engineer-ing from the Gerhard-Mercator University of Duisburg,Germany, in 1992. From 1992 to 1994, he was an R&D en-gineer at Rheinmetall GmbH. From 1995 to 2001, he wasa professor of control engineering at the University of Ap-plied Science Lausitz in Senftenberg, Germany, and servedas vice president of this university during 1998–2000. Heis currently a full professor of control engineering andthe head of the Institute for Automatic Control and Com-plex Systems (AKS) at the University of Duisburg-Essen,Germany. His research interests are model-based and
data-driven fault diagnosis, fault tolerant systems, real-time control, and their ap-plication in industry with a focus on automotive systems and chemical processes.
