Persistency of Excitation and Performance of DeterministicLearning

YUAN Chengzhi, WANG Cong

College of Automation and the Center for Control and Optimization, South China University of Technology, Guangzhou 510640, P. R.China

E-mail: chengzhiy06@hotmail.com, wangcong@scut.edu.cn

Abstract: Recently, a deterministic learning theory was proposed for locally-accurate identification of nonlinear systems. In thispaper, we investigate the performance of deterministic learning, including the learning speed and learning accuracy. By analyzingthe convergence properties of a class of linear time-varying (LTV) systems, explicit relations between the persistency of excitation(PE) condition (especially the level of excitation) and the convergence properties of the LTV systems are derived. It is shown thatthe learning speed increases with the level of excitation and decreases with the upper bound of PE. The learning accuracy alsoincreases with the level of excitation, in particular, when the level of excitation is large enough, locally-accurate learning can beachieved to the desired accuracy, whereas low level of PE may result in the deterioration of the learning performance. This paperreveals that the performance analysis of deterministic learning can be established on the basis of classical results on stability andconvergence of adaptive control. Simulation studies on the Moore-Greitzer model, a well-known axial flow compressor model,are included to illustrate the effectiveness of the results.

Key Words: Persistency of excitation (PE), deterministic learning, linear time-varying (LTV) systems, adaptive control.

1 Introduction

Adaptation and learning are two closely related topics that

had been extensively studied in the 1960s. At that time,

adaptive control and learning control were competing terms

having similar but somewhat undeveloped meanings (see,

[1, 2] and the references therein). Since 1970s, there has

been many fundamental developments of adaptive control,

see the surveys and books including [2–6] and the references

therein. Adaptive control has as a key feature the ability

to adapt to, or ‘learn’ the unknown parameters during on-

line adjustment of controller parameters in order to achieve

stability and control performance. However, the learning

ability of adaptive control is actually very limited. In the

process whereby an adaptive control algorithm adjusts on-

line the controller parameters so that closed-loop stability is

maintained, it is not required that the parameters converge to

their true values. The adaptive controllers need to recalcu-

late the controller parameters even for repeating exactly the

same control task [7].

Learning is clearly a very desirable characteristic of ad-

vanced control systems, however, learning from a dynam-

ical closed-loop control process is a very difficult problem.

This problem is related to closed-loop identification, or more

generally, learning in a nonstationary (dynamic) environ-

ment [1], and has remained unresolved for a long period of

time. To achieve accurate identification of system dynamic-

s, it is required that the persistent excitation (PE) condition

be satisfied [8], and exponential stability of a class of linear

time-varing (LTV) systems arising in adaptive identification

and control be established [5, 6]. The PE condition is one

of the most important concepts in identification and adap-

tive control. The past decades have witnessed a considerable

progress on the investigation concerning the PE condition,

such as the PE of neural networks [9–11], the PE in fuzzy

systems [12], and the PE in passive learning control [13].

This work was supported by the National Basic Research Program

(973) of China under Grant No. 2007CB311005.

For general nonlinear systems, however, the PE condition is

very difficult to characterize and usually cannot be verified apriori [8, 14].

Recently, a deterministic learning (or dynamic learning) ap-

proach was proposed for identification and adaptive control

of nonlinear systems [15, 16]. By using the localized radi-

al basis function network (RBFN), a partial PE condition,

i.e., the PE condition of a certain regression subvector con-

structed out of the radial basis functions (RBFs) along a re-

current trajectory is proven to be satisfied. This partial PE

condition leads to exponential stability of the identification

error system which is in the form of the LTV systems arising

in identification and adaptive control. Consequently, accu-

rate NN identification of the nonlinear dynamics (including

closed-loop dynamics) is achieved within a local region a-

long the recurrent trajectory. This approach is referred to as

“deterministic learning” since it is developed not by using s-

tatistical principles (e.g. [17]), but by utilizing deterministic

algorithms from adaptive control (e.g. [5, 6]). The determin-

istic learning approach provides an effective solution to the

problem of learning in dynamic environments, and is use-

ful in many applications such as dynamical pattern recogni-

tion [18], learning and control of robotics [19], and oscilla-

tion faults diagnosis [20].

It has been shown that the nature of deterministic learning is

related to the PE condition and the exponential convergence

of the class of LTV systems. Concerning the stability of the

class of LTV systems, many significant results have been

achieved (see, e.g., [5, 6, 21–23]). By employing the con-

cept of uniform complete observability (UCO), exponential

convergence of the LTV system was established under the

satisfaction of the PE condition [4, 6, 21]. Interpretations of

the relationship among PE, UCO and exponential stability of

the LTV system are nicely summarized in [21]. In [5], com-

prehensive discussions on the PE condition and parameter

convergence were provided. [24, 25] placed more emphasis

on the issues of robustness and effects of disturbances on

Proceedings of the 30th Chinese Control ConferenceJuly 22-24, 2011, Yantai, China

2764

adaptive algorithms. By extending the definition of the PE

condition, and using modern tools which are ”integral” ver-

sions of classical Lyapunov theorems, stability and conver-

gence of a class of nonlinear time-varying (NLTV) systems

arising in nonlinear adaptive identification were investigat-

ed [22]. These results contribute greatly to the development

of adaptive control, as well as to the establishment of deter-

ministic learning.

In this paper, we further investigate the performance of deter-

ministic learning, including the learning speed and learning

accuracy. This is a very important issue for both theoret-

ical and practical reasons. The speed and accuracy of de-

terministic learning is studied by analyzing the convergence

properties of the class of LTV systems. Specifically, explicit

relations between the PE condition (especially the level of

excitation) and the convergence properties of the LTV sys-

tems are derived. The convergence rate and residual error of

the perturbed LTV system are subsequently obtained. It is

thus shown that the learning speed increases with the level

of excitation and decreases with the upper bound of PE. The

learning accuracy also increases with the level of excitation,

in particular, when the level of excitation is large enough,

locally-accurate learning can be achieved to the desired ac-

curacy, whereas low level of PE may result in the deteriora-

tion of the learning performance. The attraction of this paper

lies in that it reveals that the performance analysis of deter-

ministic learning can be conducted via convergence analysis

of the class of LTV systems arising in adaptive control, and

thus classical results on stability and convergence of adap-

tive control play a significant role in the establishment and

development of the new dynamic learning methodology.

The rest of the paper is organized as follows: Preliminary

results and problem formulation are contained in Section I-

I. Convergence properties of the perturbed LTV system and

performance of deterministic learning are analyzed in Sec-

tion III. Section IV is devoted to the simulation studies on

the Moore-Greitzer model. Section V concludes the paper.

2 Preliminaries and Problem Formulation

2.1 Localized RBF Networks and PEThe RBF networks can be described by fnn(Z) =∑N

i=1 wisi(Z) = WTS(Z) [26], where Z ∈ ΩZ ⊂ Rq

is the input vector, W = [w1, · · · , wN ]T ∈ RN is the

weight vector, N is the NN node number, and S(Z) =[s1(‖Z − ξ1‖), · · · , sN (‖Z − ξN‖)]T , with si(·) being a

radial basis function, and ξi (i = 1, · · · , N ) being distinct

points in state space. The Gaussian function si(‖Z − ξi‖)= exp

[−(Z−ξi)

T (Z−ξi)η2i

]is one of the most commonly used

radial basis functions, where ξi = [ξi1, ξi2, · · · , ξiq]T is the

center of the receptive field and ηi is the width of the re-

ceptive field. The Gaussian function belongs to the class of

RBFs in the sense that si(‖Z − ξi‖) → 0 as ‖Z‖ → ∞.

It has been shown in [11, 26] that for any continuous func-

tion f (Z) : ΩZ → R where ΩZ ⊂ Rq is a compact

set, and for the NN approximator, where the node num-

ber N is sufficiently large, there exists an ideal constan-

t weight vector W ∗, such that for each ε∗ > 0, f (Z) =W ∗TS (Z) + ε, ∀Z ∈ ΩZ , where |ε| < ε∗ is the approx-

imation error. Moreover, for any bounded trajectory Zζ(t)

within the compact set ΩZ , f(Z) can be approximated by

using a limited number of neurons located in a local re-

gion along the trajectory: f(Z) = W ∗Tζ Sζ(Z) + εζ , where

Sζ(Z) = [sj1(Z), · · · , sjζ (Z)]T ∈ RNζ , with Nζ < N ,

|sji | > ι (ji = j1, · · · , jζ), ι > 0 is a small positive con-

stant, W ∗ζ = [w∗

j1, · · · , w∗

jζ]T , and εζ is the approximation

error, with εζ = O(ε) = O(ε∗).Based on the previous results on the PE property of RBF

networks [9, 10, 14], it is shown in [16] that for a localized

RBF network WTS(Z) whose centers are placed on a reg-

ular lattice, almost any recurrent trajectory Z(t) can lead to

the satisfaction of the PE condition of the regressor subvec-

tor Sζ(Z).

2.2 Deterministic Learning and Problem FormulationIn deterministic learning, identification of system dynamic-

s of general nonlinear systems is achieved according to the

following elements: (i) employment of localized RBF net-

works; (ii) satisfaction of a partial PE condition; (iii) expo-

nential stability of the adaptive system along the periodic or

recurrent orbit; (iv) locally-accurate NN approximation of

the unknown system dynamics [16].

Consider a general nonlinear dynamical system:

x = F (x; p), x(t0) = x0 (1)

where x = [x1, · · · , xn]T ∈ Rn is the state of the sys-

tem, which is measurable, p is a system parameter vector,

F (x; p) = [f1(x; p), · · · , fn(x; p)]T is a smooth but un-

known nonlinear vector field.

Assumption 1. The state x remains uniformly bounded, i.e.,x(t) ∈ Ω ⊂ Rn, ∀t ≥ t0, where Ω is a compact set.Moreover, the system trajectory starting from x0, denotedas ϕζ(x0), is a recurrent trajectory.

The recurrent trajectory represents a large class of trajecto-

ries generated from nonlinear dynamical systems, includ-

ing not only periodic trajectories, but also quasi-periodic,

almost-periodic and even some chaotic trajectories. Rough-

ly, a recurrent trajectory is characterized as follows: given

ν > 0, there exists a finite T (ν) > 0 such that the trajectory

returns to the ν-neighborhood of any point on the trajectory

within a time not greater than T (ν). Please refer to [27] for

a rigorous definition of recurrent trajectory.

Consider the following dynamical RBF network for identifi-

cation of the unknown system dynamics F (x; p).

˙xi = −ai(xi − xi) + WTi Si(x), i = 1, · · · , n (2)

where x = [x1, · · · , xn]T is the state vector, x is the state of

system (1), ai > 0 is the design constant, and WTi Si(x) is a

RBF network used to approximate the unknown nonlinearity

fi(x; p) of (1). The weight estimates Wi in (2) are updated

by using the following Lyapunov-based learning law

˙Wi = −ΓiSi(x)xi (3)

where Γi = ΓTi > 0.

Define xi = xi − xi and Wi = Wi − W ∗i . According to

the properties of localized RBF networks, the identification

2765

error system can be derived as follows [16].[˙xi

˙Wζi

]=

[ −ai Sζi(x)T

−ΓζiSζi(x) 0

][xi

Wζi

]+

[−ε′ζi0

](4)

and˙Wζi =

˙Wζi = −ΓζiSζi(x)xi (5)

where subscripts (·)ζi and (·)ζi stand for the regions close to

and away from the trajectory, respectively, Sζi(ϕζ) is a sub-

vector of Si(x), Wζi is the corresponding weight subvector,

and |ε′ζi| is close to ε∗i where ε∗i is the ideal approximation

error (as given in Section 2.1). The following assumption is

given with regarding Sζi(x) in (4).

Assumption 2. There exists a constant k1 > 0 such that forall t ≥ 0, the following bound is satisfied,

max

{‖Sζi(x)‖,

∥∥∥∥dSζi(x)

dt

∥∥∥∥}

≤ k1 (6)

As analyzed in [16], PE of Sζi(x) is guaranteed under As-

sumption 1. Thereby, exponential stability of the LTV sys-

tem (4) is derived. Consequently, a locally-accurate NN ap-

proximation for the unknown fi(x; p) is obtained along the

trajectory ϕζ(x0) as

fi(x; p) = WTζiSζi(x) + εζi1 (7)

or by the entire RBF network as

fi(x; p) = WTi Si(x) + εi1 (8)

where εζi1 = O(ε∗i ) and εi1 = O(ε∗i ) are the practical ap-

proximation errors.

It is obvious that the above results are obtained on the basis

of qualitative analysis. The purpose of this paper is to fur-

ther analyze quantitatively the performance of deterministic

learning. Specifically, our objective is to derive explicit ex-

pression of the solution of system (4) in the following form

‖χζi(t)‖ ≤ me−λ(t−t0)‖χζi(t0)‖+ b, ∀t ≥ t0 (9)

where χζi(t) �[xi W

Tζi

]T, λ and b represent respective-

ly the convergence rate and the bound of the residual error

of system (4). On such a stage, performance of determinis-

tic learning, including learning speed and learning accuracy,

will be revealed in terms of λ, b, and εi1 , respectively. Re-

lationship between the PE level of Sζi(x) and the learning

performance will also be established.

3 Main Results

In this section, we firstly derive exponential convergence rate

and bound for the residual error of the perturbed LTV sys-

tem (4), and then analyze the performance of deterministic

learning.

3.1 Convergence Properties of the Perturbed LTV Sys-tem

Consider the nominal part of (4) as described below. We

have the following result.[˙xi

˙Wζi

]=

[ −ai STζi(x)

−ΓζiSζi(x) 0

] [xi

Wζi

]

� Aζi(x)χζi(t)

(10)

Lemma 1. Consider the LTV system (10), Under Assump-tions 1 and 2, the solution of system (10) satisfies

‖χζi(t)‖ ≤ me−λ(t−t0)‖χζi(t0)‖ (11)

with m and λ being as (12) and (13), where λmin(P ) =min {1,Γ−1

ζi }, λmax(P ) = max {1,Γ−1ζi }, N is the dimen-

sion of vector Sζi(x), and m′′, n′′ > 0 are two positive con-stants.

Proof. The proof is decomposed into three steps.

Step 1: Firstly, consider the following linear time-invariant

(LTI) system:

xLTI(t) = −aixLTI(t) + u(t)

y′(t) = xLTI(t), xLTI(t0) = 0(14)

Obviously, the transfer function of the LTI system is sta-

ble, minimum phase and rational. We choose here u(t) =STζi(x)c, where c is a constant vector. According to the anal-

ysis in [15], Assumption 1 leads to the satisfaction of the PE

condition of Sζi(x), i.e., there exist constants α1, α2, T0 >0, such that

α1 ≤∫ t0+T0

t0

‖Sζi(x(τ))T c‖2dτ ≤ α2, ∀t0 ≥ 0, ∀c ∈ RN

(15)

where α1 is referred to as the level of PE, and α2 as the upper

bound of PE. Then, under Assumption 2, and according to

the Lemma 2.6.7 in [6], we have:

α′1‖c‖2 ≤

∫ t0+δ′

t0

y′(τ)2dτ ≤ α′2‖c‖2 (16)

where α′1 = α1

2a2i, α′

2 = α2

a2i, δ′ = T0.

Step 2: In this part, we need to prove system (10) with the

output defined as

y(t) = [1, 0, . . . , 0]χζi(t) � Cχζi(t) (17)

is uniform complete observable (UCO).

In order to apply the invariant theory of UCO under output

injection [4, 6], we define the following matrixes.

A(t) =

[ −ai STζi(x)

0 0

], K(t) =

[0

−ΓζiSζi(x)

](18)

Thus, according to (15),

∫ t0+T0

t0

‖K(τ)‖2dτ

=

∫ t0+T0

t0

‖ΓζiSζi(x(τ))‖2dτ

≤∫ t0+T0

t0

‖Γζi‖2‖Sζi(x(τ))‖2dτ

= ‖Γζi‖2tr[∫ t0+T0

t0

Sζi(x(τ))Sζi(x(τ))T dτ

]

≤ N‖Γζi‖2α2 � kT0

(19)

2766

m =

√λmax(P )/λmin(P )

1− (aiα1)/[λmax(P )(ai + ‖Γζi‖α2(m′′ + n′′)√2N)2]

(12)

λ =1

2(m′′ + n′′)T0· ln

(1

1− α1/[λmax(P )(√ai + (‖Γζi‖α2(m′′ + n′′)

√2N)/

√ai)2]

)(13)

As a result, system (10) with (17) is UCO is equivalent to

system (20) is UCO.[˙xi

˙Wζi

]=

[ −ai STζi(x)

0 0

] [xi

Wζi

]= A(t)χζi(t)

y′′(t) = Cχζi(t)

(20)

The output of system (20) can be expressed as:

y′′(t) = xi(t) = e−ai(t−t0)xi(t0)

+

∫ t

t0

e−ai(t−τ)Sζi(x(τ))T Wζi(t0)dτ

� y′′1 (t) + y′′2 (t)(21)

As for y′′2 (t), by taking u(t) of (14) as Sζi(x(τ))T Wζi(t0),

it follows

α′1‖Wζi(t0)‖2 ≤

∫ t0+T0

t0

y′′2 (τ)2dτ ≤ α′

2‖Wζi(t0)‖2

(22)

On the other hand, since −ai is stable and [A(t), C] is ob-

servable, there exist γ1, γ2 > 0 and γ3(m′′T0) which in-

crease along with m′′T0 , such that∫ ∞

t0+m′′T0

y′′1 (τ)2dτ ≤ γ1‖xi(t0)‖2e−γ2m

′′T0 (23)

∫ t0+m′′T0

t0

y′′1 (τ)2dτ ≥ γ3(m

′′T0)‖xi(t0)‖2 (24)

for all t0, xi(t0) and an integer m′′ > 0 to be defined later.

Now, we choose δ′′ = (m′′ + n′′)T0. In light of inequality

(a+ b)2 ≥ 0.5a2 − b2, and following the proof of Theorem

2.6.5 in [6], it yields

∫ t0+δ′′

t0

y′′(τ)2dτ

=

∫ t0+m′′T0

t0

(y′′1 (τ) + y′′2 (τ))2dτ

+

∫ t0+δ′′

t0+m′′T0

(y′′1 (τ) + y′′2 (τ))2dτ

=[0.5γ3(m

′′T0)− γ1e−γ2m

′′T0

]‖xi(t0)‖2

+ [0.5n′′α′1 −m′′α′

2] ‖Wζi(t0)‖2

(25)

Let m′′, n′′ satisfy

0.5γ3(m′′T0)− γ1e

−γ2m′′T0 ≥ 0.25γ3(m

′′T0)

0.5n′′α′1 −m′′α′

2 ≥ α′1

(26)

Then, we have∫ t0+δ′′

t0

y′′(τ)2dτ ≥ min{α′1, 0.25γ3(m

′′T0)}‖χζi(t0)‖2

� α′′1‖χζi(t0)‖2

(27)

Similarly, applying inequality (a+b)2 ≤ 2a2+2b2, we have∫ t0+δ′′

t0

y′′(τ)2dτ ≤ max{2(m′′ + n′′)α′2, 2γ1}‖χζi(t0)‖2

� α′′2‖χζi(t0)‖2

(28)

Consequently, we achieve the following results for this step:

α′′1‖χζi(t0)‖2 ≤

∫ t0+δ′′

t0

y′′(τ)2dτ ≤ α′′2‖χζi(t0)‖2 (29)

where

α′′1 = min {α′

1, 0.25γ3(m′′T0)} =

α1

2a2i

α′′2 = max {2(m′′ + n′′)α′

2, 2γ1} = 2(m′′ + n′′)α2

a2iδ′′ = (m′′ + n′′)T0

(30)

with m′′, n′′ > 0 large enough.

Coming back to system (10) with (17), from (19), we have∫ t0+δ′′

t0

‖K(τ)‖2dτ ≤ (m′′+n′′)N‖Γζi‖2α2 � k′T0(31)

According to the Lemma 2.5.2 in [6], ∀t ≥ t0, we have:

β1‖χζi(t0)‖2 ≤∫ t0+δ

t0

y(τ)2dτ ≤ β2‖χζi(t0)‖2 (32)

where

β1 =α′′1

(1 +√k′T0

α′′2 )

2

=α1

2[ai + ‖Γζi‖α2(m′′ + n′′)√2N ]2

β2 = α′′2e

k′T0

α′′2

δ = δ′′ = (m′′ + n′′)T0

(33)

Step 3: Choosing the following Lyapunov function candi-

date:

V =1

2x2i +

1

2WT

ζiΓ−1ζi Wζi =

1

2χTζi(t)

[1 00 Γ−1

ζi

]χζi(t)

� 1

2χTζi(t)Pχζi(t)

(34)

2767

Obviously, we have

b1‖χζi(t)‖2 ≤ V ≤ b2‖χζi(t)‖2 (35)

where b1 � λmin(P ), b2 � λmax(P ).Evaluating its time derivative along the trajectories of system

(10) yields:

V = xi˙xi + WT

ζiΓ−1ζi

˙Wζi

= xi(−aixi + WTζiSζi(x)) + WT

ζiΓ−1ζi (−ΓζiSζi(x)xi)

= −aix2i ≤ 0

(36)

Integrating both sides of (36), and based on the results from

Step 2, we have∫ t0+δ

t0

V dτ = −ai

∫ t0+δ

t0

x2i dτ = −ai

∫ t0+δ

t0

y(τ)2dτ

≤ −aiβ1‖χζi(t)‖2 � −b3‖χζi(t)‖2(37)

The lemma is proved due to the satisfaction of V in (34) to

(37), and according to Lemma 1.5.2 in [6].

Based on Lemma 1, subsequently the perturbed LTV system

(4) is studied.

Theorem 1. Consider the perturbed LTV system (4). UnderAssumptions 1 and 2, the state error xi and parameter errorWζi converge exponentially to the residual set

Re ={xi, Wζi

∣∣∣‖[xi WTζi]

T ‖ ≤ b =m

λε′∗ζi

}(38)

with the rate of convergence as λ, where m and λ are givenin Theorem 1, and ‖ε′ζi(x)‖ ≤ ε

′∗ζi.

Proof. Lemma 1 establishes that the state transition matrix

Φ(t, t0) of the system (4) satisfies

‖Φ(t, t0)‖ ≤ me−λ(t−t0) (39)

Under Assumptions 1 and 2, the solution χζi(t) of (4) can

be expressed as

χζi(t) = Φ(t, t0)χζi(t0) +

∫ t

t0

Φ(t, τ)gζi(t;x)dτ (40)

where gζi(t;x) =[−ε′ζi, 0

]T. Therefore,

‖χζi(t)‖ ≤ ‖Φ(t, t0)‖‖χζi(t0)‖

+

∫ t

t0

‖Φ(t, τ)‖‖gζi(t;x)‖dτ(41)

Using (39) and ‖ε′ζi(x)‖ ≤ ε′∗ζi, we have

‖χζi(t)‖ ≤ me−λ(t−t0)‖χζi(t0)‖+mε′∗ζi

∫ t

t0

e−λ(t−τ)dτ

≤ me−λ(t−t0)‖χζi(t0)‖+ m

λε′∗ζi

(42)

This ends the proof.

Theorem 1 provides explicit rate of exponential convergence

and bounds of residual error for system (4). Now, we are able

to give further quantitative analysis of issues concerning the

performance of deterministic learning.

3.2 Performance of Deterministic Learning3.2.1 Learning Speed

In [16], preliminary relationship between the recurrent tra-

jectories generated by the nonlinear system (1) and the PE

condition of Sζi(x) was presented, which states that

α1 = θ2τ0, α2 = s∗T0 (43)

where α1 is the level of PE, α2 is the upper bound of PE,

τ0 is the minimum amount of time that ϕζ stays within a

small neighborhood of the involved neural center, and T0

is the period by which the trajectory passes through each

center of the RBF network, θ and s∗ are constants given in

Proposition 2.1 of [16]. Moreover, it has been shown via

simulation experiments that a larger τ0 and a small T0 will

make the learning proceed at a higher speed.

From Lemma 1 and Theorem 1, the following result on

learning speed can be obtained.

Corollary 1. Consider the adaptive system consisting of thenonlinear dynamical system (1), the dynamical RBF network(2) and the learning law (3). Under Assumptions 1 and 2,we have: the learning speed λ (as given in Equation (13))increases with α1 and decreases with α2. Furthermore, λ isalso proportional to τ0 and inverse proportional to T0.

Proof. In light of Lemma 1 and Theorem 1, evaluating the

partial derivative of λ in Equation (13) with respect to α1,

α2, it can be easily obtained that dλdα1

> 0, dλdα2

< 0. This

means that λ ∝ α1, λ ∝ 1α2

. Further, from (43), α1 ∝ τ0and α2 ∝ T0. It is thus clear that λ ∝ τ0 and λ ∝ 1

T0. This

ends the proof.

Remark 1. Corollary 1 shows that the higher the PE level ofSζi(x) is, the faster the learning speed will be. Furthermore,the faster the trajectory moves, as well as that the longer theperiod of the trajectory is, the slower the learning speed willbe.

3.2.2 Learning Accuracy

The learning accuracy discussed in this part is related to both

the convergence bound b and the practical NN approximation

error εi1 (as in (8)). In the following, we further study the

relationship between the PE level and the learning accuracy,

and analyze the practical NN approximation error.

Corollary 2. Consider the adaptive system consisting of thenonlinear dynamical system (1), the dynamical RBF network(2) and the learning law (3). Under Assumptions 1 and 2, wehave: (i) there exist a positive constant α∗

1 such that ∀α1 ∈[0, α∗

1), b ∝ 1α1

and ∂b∂α1

∝ α1; (ii) there exist constants 0 <α∗1 < α∗

1 < α∗1 such that ∀α′

1 ∈ [0, α∗1] and ∀α′′

1 ∈ [α∗1, α

∗1),

b(α′1) � b(α′′

1 ); and (iii) when α1 ∈ [α∗1, α

∗1), the practical

approximation error εi1 (as given in (8)) is of the order of ε∗i ,where ε∗i is the ideal approximation error.

Proof. (i) Substituting (12) and (13) into (38), and de-

note C = 2(m′′ + n′′)T0

√λmax(P )/λmin(P ) and K =

λmax(P )(√ai + (‖Γζi‖α2(m

′′ + n′′)√2N)/

√ai)

2. Then,

we have:

b =Cε

′∗ζi√

1− α1/K ln(

11−α1/K

) (44)

2768

Evaluating the partial derivative of b with respect to α1,

yields

∂b

∂α1= − Cε

′∗ζi[√

1− α1/K ln(

11−α1/K

)]2

·∂[√

1− α1/K ln(

11−α1/K

)]∂α1

= −Cε

′∗ζi

[1− 1

2 ln(

11−α1/K

)]K (1− α1/K)

32

[ln(

11−α1/K

)]2

(45)

Let ∂b∂α1

< 0, we have

1− ln

√1

1− α1/K> 0

α1 < K(1− e−2) � α∗1

(46)

Further, it can be easily obtained that b∣∣α1=0

= ∞, and

b∣∣α1=α∗

1= 1

2Ceε′∗ζi � b(α∗

1). Then, it can be concluded

that for all α1 ∈ [0, α∗1), b is monotone decreasing with α1

from ∞ to b(α∗1).

Now, we examine the speed of this monotone decreasing

variation of b. Evaluating the second derivative of b with

respect to α1, yields

∂2b

∂α21

=Cε

′∗ζi (1− α1/K)

12 ln

(1

1−α1/K

)K2 (1− α1/K)

3[ln

(1

1−α1/K

)]4·{3

4

[ln

(1

1− α1/K

)]2− 2 ln

(1

1− α1/K

)+ 2

}

(47)

According to the condition (46), it can be derived that ∂2b∂α2

1>

0, ∀α1 ∈ [0, α∗1). Moreover, calculating ∂b

∂α1at α1 = 0

and α1 = α∗1, respectively, yields ∂b

∂α1

∣∣∣α1=0

= −∞ and

∂b∂α1

∣∣∣α1=α∗

1

= 0. This means that ∀α1 ∈ [0, α∗1), b is mono-

tone decreasing from ∞ to b(α∗1) with the decreasing speed

increasing monotonously from −∞ to 0.

(ii) Based on the results that ∀α1 ∈ [0, α∗1),

∂b∂α1

<

0, ∂2b∂α2

1> 0, b(0) = ∞, ∂b

∂α1

∣∣∣α1=0

= −∞, ∂b∂α1

∣∣∣α1=α∗

1

=

0, it can be concluded that ∀ε < 0, there exists a α1ε ∈[0, α∗

1) such that ∂b∂α1

∣∣∣α1=α1ε

= ε. Thereby, α∗1 and α∗

1 can

be obtained by choosing |ε| large enough and small enough,

respectively, such that b(α∗1) � b(α∗

1).(iii) When α1 ∈ [α∗

1, α∗1), it is clearly seen that

b(α1) = O(b(α∗1)) = O(ε

′∗ζi) = O(ε∗i ) (48)

Since ‖Wζi‖ ≤ b, and from (7) and (8), it can be obtained as

in [16] that

εζi1 = εζi − WTζiSζi(ϕζ) = O(ε

′∗ζi) = O(ε∗i ) (49)

Since the neural weights Wζi will only be slightly updated,

both Wζi and WTζiSζi(x) will remain to be very small [16].

Consequently, the practical approximation error εi1 satisfies

εi1 = WTζiSζi(ϕζ) + εζi1 = O(ε∗i ) (50)

Thus, it means that the approximation errors can be made

to the order of the ideal approximation error ε∗i . Thereby,

locally accurate approximation for fi(x; p) along the trajec-

tory ϕζ to the desired level ε∗i can be achieved by using RBF

network WTi Si(x). This ends the proof.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

200

400

600

800

1000

1200

α1

b/ε ζ

i*

(a)

0.5 1 1.5 2 2.5 3 3.5 4 4.58

10

12

14

16

18

20

22

24

26

α1

b/ε ζ

i*

(b)

Fig. 1: Relationship between PE level and learning accuracy

Remark 2. The relationship between b and α1 is shown inFig. 1. It is seen that b

ε′∗ζi

decrease with α1. When α1 is smal-

l, bε′∗ζi

is very large, which implies that low PE level α1 leads

to large convergence error b (as shown in Fig. 1(a)), and inturn, large approximation error εi1 . In this case, accuratelearning cannot be achieved. On the other hand, when α1

is large enough, bε′∗ζi

becomes small, and as shown in Fig.

1(b), b = O(ε′∗ζi) = O(ε∗i ) holds, which means that accurate

learning can be achieved. Moreover, the larger τ0 (whichmeans the slower the trajectory moves), and the shorter theperiod T0 of the trajectory will yield better learning accura-cy.

4 Simulation Studies on the Moore-Greitzer Mod-el

Modeling and control for axial flow compression systems

have received great attention in recent years (see [28] and

the references therein). Two of the important and challeng-

ing topics in this field are identification and detection of ro-

tating stall and surge in compressor, which can effectively

reduce engine performance, and consume a major fraction

of an engine development program [28].

In this section, the Moore-Greitzer model, a low-order non-

linear state-space model developed by Moore and Greitzer

[29] that captures the nonlinear dynamics of the compressor

system through its bifurcation characteristic [28], is prelimi-

narily taken as an example for simulation study.

According to [30], the Moore-Greitzer model provided by

[29] can be reduced to a set of three ordinary differential

equations as:

dΦ

dξ= −Ψ+Ψc(Φ)− 3ΦR+ p

dΨ

dξ=

1

β2(Φ− ΦT (Ψ)) + p

dR

dξ= σR(1− Φ2 −R)

(51)

2769

where ξ stands for the time nondimensionalized by the ro-

tor speed, Ψ is the pressure rise coefficient, Φ represents the

axial flow coefficient, R denotes the square of amplitude of

angular disturbance of axial-flow coefficient, and ΦT (Ψ) is

the inverse function of Ψ which satisfies Ψ = 1γ (ΦT + 1)2,

and where γ is proportional to the cross-sectional area of

the throttle, and is used to control the exit mass flow [30].

According to [29, 30], if the value of γ decreases, the sta-

bility margin of system (51) follows to decrease, which can

consequently lead to the occurrence of rotating stall. To il-

lustrate these variance more explicitly, a small disturbance

signal p = 110 (sin (ξ) + cos (ξ)) is added to the original dy-

namics with respect to Φ and Ψ, respectively (as denoted by

the last term of the corresponding equations in (51)). We

refer readers to [29] and [30] for definitions of the rest coef-

ficients and more specific descriptions. Details are omitted

here due to the space limitations.

Assume that the state vector defined as [Ψ,Φ, J ]T � x are

available from measurement, and the system dynamics dΦdξ �

f1(x),dΨdξ � f2(x) and dJ

dξ � f3(x) are mostly unknown to

the identifier. The RBF network identifiers are obtained as in

(2) and (3) to learn the unknown system dynamics fi(x), i =1, 2, 3.

The relationship between the learning speed and α1, α2 as

presented by Corollary 1 can be illustrated by simulations in

[16]. Thus, the corresponding simulations are omitted here

due to limitation of space. The following simulations are

devoted to illustrating the identification results of rotating

stall associated with different PE levels.

According to [30], by setting different values of γ, we

can obtained different corresponding trajectories of stability

leading to the inception of rotating stall. For the concise-

ness of presentation, we only focus on the identification of

the system dynamic f1(x) undergoing a pattern of the incep-

tion of rotating stall which can be generated by by setting

γ = 1.02. Comparison experiments are given to illustrate

the results in Corollary 2.

We construct the RBF network with the centers μ evenly

placed on [0.4, 1.5]× [3, 4.4]× [−0.5, 0.5], and the neurons

constructed in the phase pace as 11×11×11. The rest param-

eters are fixed as a = 3.5, Γ = 3, the initial weights W (0) =0, and the initial conditions x(0) = [1.375, 3.75, 4]T and

x(0) = [0.2, 0.3, 0]T . Different PE levels will be generat-

ed and indicated by choosing different values of η, i.e., the

width of the receptive field (as described in section 2.1).

Figs. 2(a) and 2(b) respectively display the pattern of the

inception of rotating stall in phase space and states x of the

pattern.

Firstly, the value of η is respectively set as 0.15, 0.2 to ob-

tain two different but large enough PE levels. Accordingly,

compare the NN parameter convergence in Fig. 3(a) with

that in Fig. 4(a), and the function approximation result in

Fig. 3(b) with that in Fig. 4(b). It is shown that under large

enough PE levels, the learning results in both cases are sat-

isfactory, the approximation errors are both on the small lev-

els. Consequently, locally accurate learning can be achieved

to a desired accuracy level.

Secondly, let η = 0.08 and η = 0.4 to yield the correspond-

ing learning results presented by Figs. 5 and 6. It can be

clearly seen that approximation errors in Figs. 5(b) and 6(b)

are apparently larger than those in Figs. 3(b) and 4(b). As

the statement in Corollary 2, when the PE level cannot be

guaranteed to be large enough (i.e., α1 does not locate in

[α∗1, α

∗1)), approximation errors of the corresponding learn-

ing result may suddenly become very large.

From the preliminary experiments on the Moore-Greitzer

model presented above, it is seen that based on the deter-

ministic learning theory, with large enough PE levels being

guaranteed by appropriately constructing the RBFs, the non-

linear dynamics of the Moore-Greitzer model can be locally

accurately identified along recurrent pattern trajectories to

the desired error level. However, locally accurate learning

may be violated when the PE level can not be guaranteed

to be large enough. Furthermore, it is worth noticing that

the nonlinear dynamics of the compressor undergoing a tra-

jectory of the inception of rotating stall can be accurately

identified. From a perspective of practical applications, this

identification result provides a dispensable basis for further

detecting/predicting rotating stall and surge in compressors.

5 Conclusions

In this paper, performance analysis of deterministic learning

including the learning speed and accuracy has been investi-

gated. By analyzing the convergence properties of a class

of LTV systems arising in adaptive control, explicit rela-

tions between the PE condition (especially the level of ex-

citation) and the convergence properties of the LTV systems

have been derived. It has been shown that the learning speed

increases with the level of excitation and decreases with the

upper bound of PE; the learning accuracy also increases with

the level of excitation, and in particular, when the level of

excitation is large enough, locally-accurate learning can be

achieved to the desired accuracy, whereas low level of PE

may result in the deterioration of the learning performance.

As an application, it has been shown that local-accurate iden-

tification of the nonlinear dynamics of the Moore-Greitzer

model can be achieved along variant patterns, which is sub-

stantial for further research on the rapid detection of rotating

stall and surge of axial compression systems.

−0.50

0.51

1.5

2

3

4

50

1

2

3

4

Φ

Trajectory in phase space

Ψ

J

(a) System trajectory in phase s-

pace

0 50 100 150 200 250 300−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time(Seconds)

Periodic states

(b) System states: Φ : ”− ”; Ψ :”−−”; R : ” · · · ”

Fig. 2: Trajectory of the inception of rotating stall

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