Hopfield-based DNN Approximator

respectively. Solve the differential equation (3-5), we obtain

( ) (

0 ⁰,

)

3.1.2 Hopfield-based DNN Approximator

A DNN approximator for continuous functions can be defined as

( ) (

0 ⁰,

)

difficult to be determined and might not be unique. The modeling error χ~ is defined as _i

i

In this paper, a Hopfield-based dynamic neural network is adopted as the approximator.

It is known as a special case of DNN with a_i =1/(R_iC_i) and b_i =1/C_i, where R_i >0 and

>0

Ci representing the resistance and capacitance at the ith neuron, respectively [25],[29].

The sigmoid functionσ(χ)=[σ(χ₁ )σ(χ₂ )_Lσ(χ_n)]^T is defined by a hyperbolic tangent

3.2 Problem Formulation

Let S ⊂Rⁿ be an open set, D_S ⊂ be and compact set. Consider the nth-order S nonlinear dynamic system of the form

d is the output of the system, and d∈R is a bounded external disturbance. We consider only the nonlinear systems which can be represented in (3-14). In order for (3-14) to be controllable, it is required that g ≠0. Without losing generality, we assume that 0< g <∞. The control objective is to force the system output y to follow a given bounded reference signal y_r ∈C^h, h≥n. The reference signal vector y_r and the error vector e are defined as

with e= y_r −x= y_r −y.

If the functions f(x) and g are known and the system is free of external disturbance, the ideal controller can be designed as

[

^{x kT}c^e

]

n r

id f y

u = g1 − ( )+ ^{( )} +

(3-16) where k_c =

[

k_n k_n−₁Lk₁

]

^T. Applying (3-16) to (3-14), we have the following error dynamics system

) 0

1 ( 1 )

( +k e ⁻ + +k e=

e ^{n n} L _n . (3-17) If ki, i=1, 2, …, n are chosen so that all roots of the polynomial H(s)∆sⁿ +k1sⁿ⁻¹ +L+k_n lie strictly in the open left half of the complex plane, then ^lim_t→∞ ^{e t}

( )

⁼⁰ can be implied for any initial conditions. However, since the system dynamics may be unknown or perturbed, the ideal feedback controller u _id in (3-16) cannot be implemented.

3.3 Design of DACHDNN

To solve this problem, a new direct adaptive control scheme using Hopfield neural networks for SISO nonlinear systems is proposed. In the DACHDNN, a Hopfield-based DNN is used to estimate the ideal controller u_id . The direct adaptive Hopfield-based DNN controller takes the following form

s HDNN

d u u

u = + (3-18) where u_HDNN is the Hopfield-based DNN controller used to approximate the ideal controller u in (3-16); id u is the compensation controller employed to compensate the effects of _s external disturbance and the approximation error introduced by the Hopfield-based DNN approximation (described later). The overall DACHDNN is shown in Fig. 3-2, wherein the adaptive laws are described later. Substituting (3-18) into (3-14) and using (3-16) yield

d u

u u

g _{c ideal HDNN s c}

ce B B

A

e& = + ( − − )−

d u

u

g _{c s c}

ce B B

A + − −

= (~ ) (3-19)

x

Fig. 3-2 The Block diagram of the DACHDNN

Note that the ideal controller u_id is a scalar, and thus the Hopfield-based DNN used to approximate u_id contains only a single neuron. The output of such a Hopfield-based DNN can be express as Hopfield-based DNN containing only a single neuron.

(⋅)

Fig. 3-3 The electric circuit of the Hopfield-based DNN containing only a single neuron

Substituting (3-20) into (3-19) yields

where ∆ is the approximation error. In order to derive the one of the main theorems in this chapter, the following assumption and lemma is required.

Assumption: Let d g and Θ, respectively. If the adaptive laws are designed as

( )

satisfies the following Riccati-like equation

1 0

⎥⎥

Following the preceding consideration, we have the following theorem.

Theorem 3-1: Suppose the Assumption (3-22) holds. Consider the plant (3-14) with the control law (3-18). The Hopfield-based DNN controller u_HDNN is given by (3-20) with the adaptive laws (3-23) and (3-24). The compensation controller u_s is given as

B^T_cPe scheme guarantees the following properties:

i)

∫

^{≤ + +Θ Θ Θ+}

∫

ii) The tracking error e can be expressed in terms of the lumped uncertainty as

) the minimum eigenvalue of P.

Proof:

i) Define the Lyapunov function candidate as

Θ

η . Differentiating (3-31) with respect to time and using (3-21) yield

Θ Substituting (3-28) into (3-32), we have

+ Θ

By using the Riccati-like equation (3-25), (3-36) can be rewritten as + Θ

(3-38) For the condition

⎟⎟ and the second line of (3-40) can be rewritten as

0 Integrating both sides of the inequality (3-43) yields

∫

Substituting (3-31) into (3-44), we can prove (3-29).

ii) From (3-44) and since 0

0 ≥

∫

^{t Te Qedt , we have}

µ ρ² ) 2

0 ( 2 ) (

2V t ≤ V +g , 0≤ t <∞ (3-45) From (3-31), it is obvious that e Pe^T ≤2 , for any V V . Because P is a positive definite symmetric matrix, we have

Pe e e e P e

P) = ( ) ^T ≤ ^T

( ² _min

min λ

λ (3-46) Thus, we obtain

µ ρ

λ_min(P)e ² ≤ Pee^T ≤2V(t)≤2V(0)+g^{2 2} (3-47) from (3-45)-(3-46). Therefore, from (3-47), we can easily obtain (3-30), which explicitly describe the bound of tracking error e . If initial state V(0)=0, tracking error e can be made arbitrarily small by choosing adequate ρ. Equation (3-30) is very crucial to show that the proposed DACHDNN will provide the closed-loop stability rigorously in the Lyapunov sense under the Assumption (3-22). Q.E.D.

Remark: Equation (3-30) shows the relations among e , ρ, and λ_min(P). For more insight of (3-30), we first choose ρ² =δ in (3-25) to simplify the analysis. Thus, from (3-25), we can see that λmin(P) is fully affected by the choice of λmin(Q) in the way that a larger

)

min(Q

λ leads to a larger λ_min(P), and vice versa. Now, one can easily observe form (3-30) that the norm of tracking error can be attenuated to any desired small level by choosing ρ and λ_min(Q) as small as possible. However, this may lead to a large control signal which is usually undesirable in practical systems.

3. 4 Simulation Results

In this section, two examples are presented to illustrate the effectiveness of the proposed DACHDNN. It should be emphasized that the development of the DACHDNN does not need to know the exact dynamics of the controlled system.

Example 3-1: Chaotic dynamic systems are known for their complex, unpredictable behavior and extreme sensitivity to initial conditions as well as parameter variations. Consider a second-order chaotic dynamic system, the well known Duffing’s equation, which describes a special nonlinear circuit or a pendulum moving in a viscous medium under control [65]:

2

1 x

x& =

u wt q x p x p x p

x&₂ =− &− ₁ − ₂ ³ + cos( )+ x

y= (3-48)

Fig. 3-4 The Phase plane of uncontrolled chaotic system

where p , p₁, p₂, q and w are real constants. Depending on the choices of these constants, the solutions of system (3-49) may display complex phenomena, including various periodic orbits behaviors and some chaotic behaviors [66]. Fig. 3-4 shows the complex open-loop system behaviors simulated with u=0, 4p=0. , p₁ =−1.1, p₂ =1.0, w=1.8,

95 .

=1

q , and

[

^x₁ ^x₂

] [ ]

^{T = 0 0T}. Assume the system is free of external disturbance in this example. The reference signal is y_r(t)=sin(0.5t)+cos(t).Some initial parameter settings of DACHDNN are chosen as

[

^{x x}2⁰

]

^T

^[

^{0.5 0}

^]

^T

0

1 = , 0u_HNN,0 = , 0ξ⁰_W = , ξ⁰_Θ =

[ ]

0 0 ^T,Wˆ⁰ =0, and Θˆ⁰ =

[ ]

11 ^T. These initial settings are chosen through some trials to achieve favorable transient control performance. The learning rates of weights adaption are selected as

5 .

=7

=β_Θ

β_W ; the slope of tanh(⋅ at the origin are selected as ) κ =1; g_L =0.1 and 5

.

=0

δ for the compensation controller. The resistance and capacitance are chosen asR= 5Ω andC=0.005F. Solving the Riccati-like equation (3-25) for a choice of Q 10= I,

[ ]

^T

c = 21

k , we have ⎥

⎦

⎢ ⎤

⎣

=⎡

5 5

5

P 15 . The simulation results for are shown in Figs. 3-5, where

the tracking responses of state x₁ and x₂ are shown in Figs. 3-5(a) and 3-5(b), respectively, the associated control inputs are shown Fig. 3-5(c), and the trained weightings are shown in Fig. 3-5(d). From the simulation results, we can see that the proposed DACHDNN can achieve favorable tracking performances without external disturbance.

yr

x1

time (sec) (a)

x1

yr

x1

time (sec) (a)

x1

y&r

x2

time (sec) (b)

x2

y&r

x2

time (sec) (b)

x2

time (sec) (c)

ud

time (sec) (c)

ud

θ11

Fig. 3-5 Simulation results of Example 3-1

Example 3-2:

Consider the following nonlinear dynamic system described as [58, 67]

2

P . The simulation results for

are shown in Figs. 3-6, where the tracking responses of state x₁, x₂ and x are shown in ₃ Figs. 3-6(a), 3-6(b), and 3-6(c), respectively, the associated control inputs are shown in Fig.

3-6(d), and the trained weightings are shown in Fig. 3-6(e). From Fig. 3-6(a), we can observe that the output of the system well tracks the reference signal throughout the whole control process, even with the external disturbance occurring in the middle time (t ≥10). This fact shows the strong disturbance-tolerance ability of the proposed system.

yr

x1

time (sec) (a)

x1

yr

x1

time (sec) (a)

x1

y&r

x2

time (sec) (b)

x2

y&r

x2

time (sec) (b)

x2

y&&r

x3

time (sec) (c)

x3

y&&r

x3

time (sec) (c)

x3

ud

time (sec) (d)

ud

time (sec) (d)

t i m e (

θ11

θ12

θ13

time (sec) (e)

weightings

W t i m e ( t i m e (

θ11

θ12

θ13

time (sec) (e)

weightings

W

Fig. 3-6 Simulation results of Example 3-2

3.5 Performance analysis of Hopfield-based DNNs with and without the self-feedback loop

The performance of Hopfield-based DNNs with and without the self-feedback loop will be compared in this section. Hopfield networks are sometimes composed of neurons without self-feedback loops in some applications, such as pattern recognition [68]. This is to minimize the number of potential stable states so as to increase the recognition rate [68]. However, is it true that a Hopfield-based DNN composed of neurons without self-feedback loops performs better in the control problem of SISO affine nonlinear systems? We will try to answer this question by the following discussions and simulation results.

Because the proposed Hopfield-based DNN contains only a single neuron for SISO affine nonlinear systems, we can simply set W =0 (and hence W^* =Wˆ =W~ =0) when a neuron without self-feedback loop is used. Thus, repeating the discussions with W =0 in sections 3.1 and 3.3, we have the following theorem:

Theorem 3-2: Suppose the required assumption holds. Consider the plant (3-14) with the control law (3-18), where the Hopfield-based DNN controller u_HDNN is given as

0 1

0

1 1 ˆ

1 ˆ

Θ

−

−

Θ + − Θ

Θ

= ^Tξ ^RCt _HDNN ^{RCt T}ξ

HDNN e u e C

u C . (3-50)

with the adaptive law (3-24). The compensation controller u_s is given as (3-28). Then, the overall control schemes guarantees that

i)

∫

^{≤ +Θ Θ Θ+}

∫

Θ T t

t T T g d

d 0

2 2 2 0

0 0

0 0 2 2

~

~ 2

1 2

1 ρ ε τ

τ e Pe β^&

Qe

e (3-51)

for 0≤ t <∞.

ii) The tracking error e can be expressed in terms of the lumped uncertainty as

) ( 2

min 2 2 0

e P

λ

µ ρ g V +

≤ . (3-30) Proof: Theorem 3-2 can be easily proven by following the proof of Theorem 3-1 under the

premise that W =W^* =Wˆ =W~ =0. Q.E.D.

From Theorem 3-2, we ascertain that the convergence performance of the Hopfield-based DNN without the self-feedback loop can still be guaranteed.

Next, simulations for the Hopfield-based DNN without the self-feedback loop are performed. For Example 3-1, the tracking responses of state x₁ and x₂ are shown in Figs.

3-7(a) and 3-7(b), respectively. The norms of error vectors, e , for the cases of Hopfield-based DNN with and without the self-feedback loop are shown in Fig. 3-8(c). For example 3-2, the tracking responses of state x₁, x₂, and x are shown in Figs. 3-8(a), ₃ 3-8(b), and 3-8(c), respectively; Fig. 3-8(d) shows the e for both cases. From the simulation results, we can see that as we expect, a Hopfield-based DNN without the self-feedback loop can also result in acceptable tracking performance. However, form Figs.

3-8(c) and 3-8(d), it can be easily observed that a Hopfield-based DNN with the self-feedback loop perform better in the tracking control problem of SISO nonlinear systems. This fact is totally opposite to the knowledge that a Hopfield network without the self-feedback loop can be used to increase recognition rate in pattern recognition. [68].

yr

x1

time (sec) (a)

x1

yr

x1

time (sec) (a)

x1

y&r

x2

time (sec) (b)

x2

y&r

x2

time (sec) (b)

x2

time (sec) (c)

with self-feedback loop without self-feedback loop

e

time (sec) (c)

with self-feedback loop without self-feedback loop with self-feedback loop without self-feedback loop

e

Fig. 3-7 Simulation results of Example 3-1 using Hopfield-based DNN without the feedback loop

yr

x1

time (sec) (a)

x1

yr

x1

time (sec) (a)

x1

y&r

x2

time (sec) (b)

x2

y&r

x2

time (sec) (b)

x2

y&&r

x3

time (sec) (c)

x3 y&&^r

x3

time (sec) (c)

x3

e

time (sec) (d)

with self-feedback loop without self-feedback loop

e

time (sec) (d)

with self-feedback loop without self-feedback loop with self-feedback loop without self-feedback loop

Fig. 3-8 Simulation results of Example 3-2 using Hopfield-based DNN without the feedback loop

Chapter 4

Conclusions and Future Works

For decades, many researchers and designers, from such broad areas as aircraft and spacecraft control, robotics, process control, and biomedical engineering, have shown an active interest in the control problem of nonlinear systems. Among these research efforts, adaptive fuzzy control and adaptive NN control have been shown to be powerful and effective methodologies for nonlinear control. However, in the control design, the structure determination is a difficult task for both FSs and NNs. More specifically, choosing the number of fuzzy rules, inherently involving fuzzy partitioning of input and output spaces, can greatly affect the approximation capability of fuzzy systems; similarly, the number of neurons can be a decisive factor to the performance of NNs.

In Chapter 2, the proposed self-structuring fuzzy system (SFS) can construct a compact fuzzy rule base by automatic rule generation and pruning. The problems of determining the fuzzy partitions of input spaces and the number of fuzzy rules are solved simultaneously. The provided systematic method can cope with the tradeoff between the approximation accuracy and computational load of FS. New rules are generated according to the newly added membership functions to adjust the improper fuzzy clustering of the input spaces.

Insignificant rules with negligible contribution toward the output of FS will be removed after a short period. Further, a robust adaptive self-structuring fuzzy control (RASFC) scheme for the uncertain or ill-defined nonlinear nonaffine systems is proposed. Some adaptive laws for on-line tuning the parameters of fuzzy rules are derived in the Lyapunov sense to realize favorable fuzzy approximation. As shown in Chapter 2, the RASFC can achieve a L2 tracking performance with arbitrarily attenuation level. This L2 tracking performance can provide a clear expression of tracking error in terms of the sum of lumped uncertainty and external disturbance, which has not been shown in previous works. Several examples are illustrated to show that the RASFC can achieve favorable tracking performance in the presence of external disturbance, yet heavy computational burden is relieved.

In Chapter 3, we propose a direct adaptive control scheme using Hopfield-based dynamic neural networks for SISO nonlinear systems. A simple Hopfield-based DNN is used to approximate the ideal controller and the synaptic weights Hopfield-based DNN are on-line

tuned by adaptive laws. A compensation controller is merged into control law to suppress the effect of modeling error and external disturbance. By Lyapunov stability analysis, we prove that the closed-loop system is stable, and the tracking error can be attenuated to a desired level.

Note that no strong assumptions and prior knowledge of the controlled plant are needed in the development of DACHDNN. Simulation results demonstrate the effectiveness and robustness of the proposed DACHDNN in the presence of external disturbance. The case of Hopfield-based neural network without the self-feedback loop is also studied. We show that this case has inferior results than those of Hopfield neural network with the self-feedback loop.

The most important is, for SISO affine nonlinear systems, we propose an adaptive control scheme which results in a Hopfield-based DNN containing only one neuron but still maintain good tracking performance. The parsimonious structure of the Hopfield-based DNN solve the structuring problem of NNs, and the simple Hopfield circuit makes the DACHDNN much easier to implement and more reliable in practical purposes.

Although we have basically solved the control problem of nonlinear systems by the fuzzy and NN control schemes with automatic structuring processes, some underlying details need to be examined to make the solutions more perfect and practical. The first is the universal approximation property of the SFS. It has been proven by many researchers that fuzzy systems can approximate any nonlinear function to any desired accuracy because of the universal approximation theorem. However, the validness of the universal approximation property for a fuzzy system with variable number of rules, such as the proposed SFS, is still left to be explored. Although the research results in our work and many other literatures have provided strong collateral evidences, a direct and rigorous proof of the universal approximation theorem for fuzzy systems with variable structure is indispensable. This will be one of our future works. The implementation of the proposed RASFC scheme in a real hardware platform is also a problem. Although the concepts of rule pruning and growing are quite intuitive and simple; however, it is not an easy task to realize them in hardware.

On the other hand, in Chapter 3, the parsimonious structure makes the proposed DACHDNN scheme has the best chance to be realized in hardware for real world applications.

However, the proposed DACHDNN scheme Chapter 3 is now only applicable to SISO nonlinear systems. In the future, we will work on extending the research results to the MIMO nonlinear systems.

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