# Neural-Fuzzy Digital Strategy of Continuous-Time Nonlinear Systems Using Adaptive Prediction and Random-Local-Optimization Design

Authors: Zhi-Ren Tsai
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A tracking problem, time-delay, uncertainty and stability analysis of a predictive control system are considered. The predictive control design is based on the input and output of neural plant model (NPM), and a recursive fuzzy predictive tracker has scaling factors which limit the value zone of measured data and cause the tuned parameters to converge to obtain a robust control performance. To improve the further control performance, the proposed random-local-optimization design (RLO) for a model/controller uses offline initialization to obtain a near global optimal model/controller. Other issues are the considerations of modeling error, input-delay, sampling distortion, cost, greater flexibility, and highly reliable digital products of the model-based controller for the continuous-time (CT) nonlinear system. They are solved by a recommended two-stage control design with the first-stage (offline) RLO and second-stage (online) adaptive steps. A theorizing method is then put forward to replace the sensitivity calculation, which reduces the calculation of Jacobin matrices of the back-propagation (BP) method. Finally, the feedforward input of reference signals helps the digital fuzzy controller improve the control performance, and the technique works to control the CT systems precisely.


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## Body

## 1. Introduction

During the past decade, many fuzzy theories [1–7] and delay analysis [8–13] have attracted great attention from both the academic and industrial communities, and there have been many successful applications. Despite this success, it has become evident that many basic and important issues remain to be further addressed. These stability analysis and systematic designs are among the most important issues for robust control systems [1], and there has been significant research on these issues (see [2, 14–16]). However, the problems of modeling error and prediction are still worth to be solved further. Therefore, the two-stage control method in this paper is proposed to suppress the modeling error to guarantee the stability of predictive control system in the presence of this modeling error.In addition, neural-network- (NN-) based modeling has become an active research field because of its unique merits in solving complex nonlinear system identification and control problems (see [15]). Neural networks (NNs) or NARMAX/NARX neural networks [17] are composed of simple elements operating in parallel, inspired by biological nervous systems. A neural network can be trained to represent a particular function by adjusting the weights between elements. Due to discrete-time (DT) controllers (microcontrollers) being cheaper and more flexible than continuous-time (CT) controllers, the DT control problem for CT plant is worth studying. In modern control engineering, controllers are commonly implemented directly by the hardware or software of digital computers. However, one important issue has to be faced; that is, the proposed design (DT-CT design) in this paper effects a new type of application.The study of CT control of CT time-delay systems has received considerable attention in recent years since delay is a major cause of poor performance in many important engineering systems [18, 19]. As is known, the delay control problem is an important and complex factor in the stability performance of CT nonlinear systems. In general, a delay signal happens in a signal’s long-distance or heat translations. However, the amount of delay has different impacts on the various approaches [1, 3]. In this study, the direction of CT time-delay control systems needs to involve the DT control problem. Moreover, the recommended adaptive NN-model-based design method has not yet been developed to adjust the parameters of a discrete-time (DT) adaptive fuzzy controller such that the original continuous-time (CT) system, with time delays and uncertainties, is uniformly ultimately bounded (UUB) stable in Section 2.Based on the timer of the micro-controller, the effect of delay in neural system identification can be approximated by many tape-delay terms. This reduces the difficulty of delay identification. The DT NARMAX model is generally sufficient to approximate an unknown, nonlinear, dynamical, and delayed CT system by selecting an appropriate sampling time. Moreover, the measured modeling error between the model and the physical system is considered in the theorems by Lyapunov functions. In Remark1, this kind of modeling error is used to estimate the proposed corner of sampling period (CSP) which decides this kind of sampling time to support the intersample method.The feedforward term in [20] is derived indirectly by assuming some constraints. Due to the overfitting problem and the local optimal problem of NN, the method [20] is not suitable for real applications because of the need for lengthy convergence time. These neural techniques [20, 21] have usually been demonstrated under nonlinear control due to their powerful nonlinear modeling capability [22] and adaptability. However, they must exhibit the optimal problems of falling into the local minimum easily by using the back-propagation (BP) [23] method. To alleviate the requirements for accurate modeling of the plant, the proposed NARMAX plant and control models are trained by initially using novel offline methods with the RLO algorithm to improve this drawback. It not only guarantees the gradient decent method against the local optimal solution but also speeds up the convergence of the PSO [24].Conventional optimization methods assume that all parameters and goals of a model are precisely known [25, 26]. However, in many practical problems incomplete and unreliable information exists [27]. Therefore, the two-stage scheme is proposed to treat this unknown system as reliability problem.Inspired by the DT neural controller of [20] only for a DT system, a digital fuzzy control design for a CT system is proposed and an approximate inverse of the delayed plant dynamics is used to act as the NARMAX fuzzy controller. The adaptive controller and NARMAX models are easier to converge than [20, 21] by the recommended two-stage scheme. Moreover, this paper concludes with a simulation example and experimental data to demonstrate these techniques.The remainder of this paper is organized as follows. In Section2, the system description and its problem formulation are provided. In Section 3, robustness controllers and stability analysis in the presence of modeling errors for a series of control design algorithms are introduced. In Section 4, the effectiveness of the proposed approach is illustrated by a numerical example with three cases. Finally, the conclusions are drawn in Section 5.


## 2. System Description and Problem Formulation

First, the conventional PWM buck converter by using AM-OTS-DS [28] methodology is modeling to the following equivalent circuit plant:(1a)[i˙L(t)v˙C(t)]=[-1L[RL+R-·RCR-+RC]-R-L·(R-+RC)R-C·(R-+RC)-1C·(R-+RC)][iL(t)vC(t)]+[-1L[RMiL(t)-Vin-VD]0]u(t)+[-VDL0],vo(t)=[R-·RCR-+RCR-R-+RC][iL(t)vC(t)].
In this paper, the robustness of the above control system is emphasized, so uncertainty Δ and delay τ are added to the original control system (1a)
(1b)[i˙L(t)v˙C(t)]=[-1L[RL+R·RCR+RC]-RL·(R+RC)RC·(R+RC)-1C·(R+RC)][iL(t)vC(t)]+[-1L[RMiL(t)-Vin-VD]0]u(t)+[-VDL0],vo(t)=[R·RCR+RCRR+RC][iL(t)vC(t)]+ζ·vo(t-τ),where R=R-+Δ; R-=6; Vin=30 is a DC voltage source; C=202.5×10-6; Δ=0.52sin(2π·t/3); ζ=0.1; τ=10-2(1+sin(10t)); L=98.58×10-6; RL=48.5×10-3, and RC=162×10-3 are the parasitic resistances of the inductor and capacitor, respectively. The element RM=0.27 is the static drain to source resistance of the power MOSFET, and VD=0.82 is the forward voltage of the power diode. u(t) is duty ratio of conventional PWM buck converter. The state x(t)=x=[iL(t),vC(t)]T and the output y(t)=vo(t) of this power system are defined.The nonlinear, uncertain, hotter circuit’s components, time-delay, and digital control problems of PWM buck converter CT system renders a tracking control problem difficult to analyze. A simulation system in (1a) and (1b) is built with uncertainty. In this study, it is assumed that the parameters of the circuit’s components are not ideal, and the capacity of the digital controller is limited by using a lower-cost chip. Here, the sampling period Ts=10-4 is designed for this power system (1a) and (1b). Hence, the delay τ is very large for this system.Then, consider a general nonlinear system with delays described as follows:(2)P:x˙(t)=f(x,u,t,τ,Δ),y(t)=g(x),
where the bounded uncertainties Δ(t) create the dynamic quality of the system parameters which refer to electrical elements of the power system; the zero-order-hold control input u(t)=u(k·Ts)=u(k)=uk, t=k·Ts, Ts is the appropriate sample time of DT-CT design, k is the index of the discrete result uk of u(t) referring to the NN model of (1a) and (1b); τ is the time delay; g(·) is the relational function of the state x(t) and system output y(t).Remark 1.
With the understanding that normal physical systems are low-pass filter and smooth nonlinearity, the upper bound, UBSP (shown in Figure1), of the test sampling period Ts is decided based on a test intersample method with the plot of logarithm x-axis for Ts and logarithm y-axis for ∑k|emod(k)|, where emod(k)=y^k-yk. In Figure 1, we can find an obvious corner of sampling period (CSP) easily to support the intersample method.Figure 1
UBSP of (1a) and (1b) is decided based on CSP.


## 3. Stability Analysis for Two-Stage Control Scheme

### 3.1. Stability Analysis with Neural-Fuzzy Design

In this paper, an adaptive prediction control structure is proposed, as shown in Figure2, where the FRP controller CF is designed as follows:
(3)u(z(k·T))=uk=CF(zk,(WC)k,Ts)+s·CX(ek)=uF+s·uX,
where the switch index
(4)s={0,e~k+1≥0,1,e~k+1<0,uk(z^)=CF(z^k,(WC)k,Ts),
where uP=uF+uX, y^(k+1)1=P^(uP(k+1)), y^(k+1)2=P^(uF(k+1)), e^(k+1)1=rk+1-y^(k+1)1, e^(k+1)2=rk+1-y^(k+1)2, |e^(k+1)1|-|e^(k+1)2|=e~k+1, as shown in Figure 2. The feedforward terms are reference signals [rk,rk-1,…,rk-p], and recursive terms are control signals [uk-1,uk-2,…,uk-q].Two-stage adaptive prediction structure of DT-CT control system. (a) Two-stage block diagram and (b) online adaptive prediction block diagram.
(a)
(b)The offline training input of controller is(5)z^(k)=z^k=[yk,yk-1,…,yk-p,uk-1,uk-2,…,uk-q],z(k)=zk=[rk,rk-1,…,rk-p,uk-1,uk-2,…,uk-q],yk=y(k), rk=r(k). The controller has two working phases: z(k) is the data vector of the testing phase, and z^(k) is the data vector of the training phase. The fuzzy controller is as follows: (6a)uF(z(k·T))=uF(k)=∑l=1Mh-l[∏i=1nexp(-((zi-z-il)/σil)2)]∑l=1M[∏i=1nexp(-((zi-z-il)/σil)2)]=CF(rk,rk-1,…,uk-1,uk-2,…,(WC)k,Ts),(6b)uF(z^(k))=CF(yk,yk-1,…,uk-1,uk-2,…,(WC)k,Ts),where feedforward terms are reference signals [rk,rk-1,…,rk-p], and recursive terms are control signals [uk-1,uk-2,…,uk-q]. Assume that Y=[yk,yk-1,…,yk-p]T, U=[uk,uk-1,…,uk-q] are measured to train model/controller. M is fixed;
(7)zi=ci0yk+ci1yk-1+⋯+cipyk-p+c-i1uk-1+c-i2uk-2+⋯+c-iquk-q.
The tuned parameter vector of controller is as follows:
(8)(WC)k=[(c-i1,c-i2,…,c-iq)(z-11,z-12,…,z-il,…,z-nM),(σ11,σ12,…,σil,…,σnM),(h-1,h-2,…,h-l,…,h-M),(ci0,ci1,…,cip),(c-i1,c-i2,…,c-iq)].The proposed digital fuzzy controlleruk has feedforward terms and recursive structure. It can use a NARMAX neural model or inverse of the plant dynamics to aid control precision in the face of a delayed plant. Owing to the adaptive fuzzy controller, the uncertainty of the plant can be suppressed. Although the structure of the fuzzy controller is chosen, the parameter vector (WC)k is not specified; therefore, the fuzzy controller has not been designed. γ·Ts is the tape-delay time, γ is a positive integer. The idea of the inverse-model-based fuzzy controller is proposed by the following simplified relation:
(9)Ifyk=P^(uk),uk=P^-1(rk)=CF(rk),thenyk=rk,
where P^(·) is the adaptive NARMAX neural model of plant; CF(·) is the adaptive NARMAX fuzzy controller; rk is the desired output. According to the idea of (9), the recursive structure P^(·) can be designed with tape delays as follows:
(10)yk≈y^k=P^(y^k-1,y^k-2,…,y^k-n,uk,uk-1,…,uk-p,(WP)k,Ts),
but the weights and biases (WP)k of this model are trained by the way of the feedforward structure neural network (NN) as follows:
(11)yk≈y^k=P^(yk-1,yk-2,…,yk-n,uk,uk-1,…,uk-p,(WP)k,Ts).
Due to the parameters of the recursive structure of the plant model, convergence may be much stricter, where n,p+1 are the amount of tape delays of y^,u, respectively. The plant output is compared with the desired output to create a system error signal ek=rk-yk. The errors e^k=rk-y^k and ek are used by the adaptation algorithm to update the parameters of P^ and CF. Next, the performance index is designed, as follows:
(12)Jk=12ekTek=12(rk-yk)T(rk-yk)=12(yk-rk)T(yk-rk),
is a cost function to be minimized by the proposed algorithm. Then, the BP algorithm adapts the control parameters (WC)k. That is, the change in control parameters (ΔWC)k is calculated as
(13)(ΔWC)kT=-ηC(k)dJkd(WC)k=-ηC(k)d(yk-rk)T(yk-rk)2·d(WC)k=-ηC(k)(yk-rk)dykd(WC)k=-ηC(k)(yk-rk)dykdukdukd(WC)k,
where the small positive ηC(k) can be selected as a stable learning rate via the following theorems.Theorem 2.
If the amount of neurons of the neural model is sufficient and the appropriate sampling timeTs is selected to let ∥y-k-yk∥≤ε- and the following condition
(14)0<ηP(k)<2∥dy^k/d(WP)k∥2≤η-P
be satisfied, where y-k is an output of the optimal model, then the trajectories y^k of converging to plant output yk is a uniformly ultimately bounded (UUB) approximation on the bounded error y^k-yk.Proof of Theorems.
First, consider the following ideal Lyapunov candidate [29] of the model part,
(15)V1k=12(y^k-yk)T(y^k-yk)=12∥y^k-y-k+y-k-yk∥2=12∥y^k-y-k∥2+εk=V2k+εk,
where V2k=(1/2)∥y^k-y-k∥2 is an actual Lyapunov candidate of reachable and assumptive trajectory y-k; the bounded approximation error
(16)εk=12∥y-k-yk∥2+(y^k-y-k)T(y-k-yk)
and the number of neurons of the neural model are sufficient, and the appropriate sampling time Ts is selected to let y-k≈yk. The next task is to train the model such that V2k is minimized,
(17)(ΔWP)kηP(k)=-dV2kd(WP)k=-(y^k-y-k)d(y^k-y-k)d(WP)k=-(y^k-y-k)dy^kd(WP)k,
where (ΔWP)k=WP(k+1)-WP(k); WP(k)=(WP)k.
Then, the following Lyapunov candidate of the controller part is designed:(18)V3(k)=V3k=12(y^k-rk)T(y^k-rk)=12∥y^k-rk∥2,
Thus, the change in the Lyapunov function is obtained by
(19)V3(k+1)-V3(k)=12(∥y^(k+1)-r(k+1)∥2-∥y^(k)-r(k)∥2).
Finally, the update law of the control parameters of the controller is obtained as follows:
(20)(ΔWC)kTηC(k)≈-dV3(k)d(WC)k=-(y^k-rk)dy^kd(WC)k.
This study develops some convergence theorems to select appropriate stable learning rates. First, the difference of modeling error eP(k)=y^k-y-k can be represented by
(21)eP(k+1)=eP(k)-[deP(k)d(WP)k]T[ηP(k)eP(k)deP(k)d(WP)k]=eP(k)(1-[deP(k)d(WP)k]TηP(k)deP(k)d(WP)k)=eP(k)(1-[dy^kd(WP)k]TηP(k)dy^kd(WP)k);
thus, the change in the Lyapunov function is obtained by
(22)V2(k+1)-V2(k)=12(∥eP(k+1)∥2-∥eP(k)∥2)=12(∥eP(k)(1-[dy^kd(WP)k]TηP(k)dy^kd(WP)k)∥2-∥eP(k)∥2∥eP(k)(1-[dy^kd(WP)k]TηP(k)dy^kd(WP)k)∥2)=12∥eP(k)∥2[(1-[dy^kd(WP)k]TηP(k)dy^kd(WP)k)2-1].
Hence, if -1<(1-[dy^k/d(WP)k]TηP(k)(dy^k/d(WP)k))<1 and ∥y-k-yk∥≤ε-, then V2(k+1)<V2(k), that is, V2(k)→0 or y^k→y-k makes the UUB approximation of this model on the bounded y^k-yk. The proof is thereby completed.Furthermore, the following theorem for the convergence of the controller is obtained by the same procedure as the above proof.Theorem 3.
If Theorem2 in (14) is satisfied, the function dy^k/d(WC)k in (20) is computed to let the following condition
(23)0<ηC(k)<2∥dy^k/d(WC)k∥2≤η-C,
be satisfied. Then, the nonlinear systems (1a) and (1b) are UUB stable, and the tracking errors e(k)=rk-yk are bounded via the controller.Hence, the dynamic response of the systemP can be controlled using CF, as shown in Figure 2. This CF needs the plant model P^ to adjust control parameters via sensitivity function ∂y^k/∂uk-i.The digital feedback controller includes a delay blockD, as shown in Figure 2. Here, the error e~k+1 is used to estimate uX, and the proposed predictor of the delayed system can let us cancel some complex computations, such as ∂y^k/∂uk≈(y^k-y^k-1)/Δuk=Δy^k/Δuk=Δy^k/uX(k), of sensitivity function ∂y^k/∂uk-i in the BP algorithm. Hence, the following theorem is proposed to update the control parameters of FRP under the assumption of providing a model which applies a lower prediction error and a more correct uX. The prediction error e^k+1=rk+1-y^k+1, being bounded due to the previous e^k=rk-y^k, is bounded at any time. Hence, the prediction error e^k+1 will be bounded by using Theorems 2 and 3. Furthermore, the following theorem is obtained for the convergence of the adaptive prediction controller by the same procedure as Theorem 2.Theorem 4.
If Theorem2 in (14) is satisfied, the predictive function duk+1/d(WC)k+1 is computed to let the following condition
(24)0<ηX(k+1)<2∥duk+1/d(WC)k+1∥2≤η-X
be satisfied, then the nonlinear systems (1a) and (1b) are UUB stable, and the tracking errors ek=rk-yk are bounded via the predictive controller uk+1=uF(k+1)+s·uX(k+1).The tracking error isek=rk-yk and ek=[e1k,e2k,e3k,…]T, but the parameters of adaptive control CF are updated by using the predictive offset
(25)Δuk+1=uX(k+1)=CX(ek)=[K1(e1k),K2(e2k),K3(e3k),…]T,
so its previous term is
(26)Δuk=uX(k)=CX(ek-1)=[K1(e1(k-1)),K2(e2(k-1)),K3(e3(k-1)),…]T,
of control input uk, where K1(·),K2(·),K3(·),… are defined by the user, and the predictive controller is uk+1=uF(k+1)+s·uX(k+1), so its previous term is uk=uF(k)+s·uX(k).


### 3.2. Two-Stage Scheme in the Presence of Stability

Figure2 shows a block-diagram of an adaptive recursive control system. The system to be controlled is labeled as the plant P, which is subject to modeling errors, uncertainties, and delays. Due to gradient-descent-based training algorithms, let the model/controller converge to some local minimums which destroy the control system stability by enlarging the modeling error. Hence, we proposed the two-stage training algorithm to reduce this modeling error as follows.In the first stage, the measured data is used to train the global optimal NARMAX plant and the fuzzy controller by the training-data-shuffle method. This method shuffles the training data to avoid most of the local optimal solutions obtained by the offline training procedure in the next section. The measured data is divided into a training data and other testing data. This testing data is not used for training the NN. However, the final performance of the NN is decided by the testing data and the training data.In the second stage, the global optimal NARMAX plant model and fuzzy controller is adapted. The two stages are divided into the following five steps.Step 1.
First, the reference signalr(k)=rk is designed. By the white noise of input u(k)=uk for plant, output data y(k)=yk is collected, and a training-data-shuffle method is used to shuffle the input/output pairs’ data. These shuffled data are ready to train the NARMAX model/controller. Here, the following reasonable conditions need to be taken into account:
(27)maxk(rk)≈maxk(yk),mink(rk)≈mink(yk),maxk(uk)≤uU,mink(uk)≥uL
need be satisfied, where uU is the upper bound of uk, and uL is the lower bound of uk. According to (27), much of the excessive control effort uk can be avoided. If (27) is satisfied, then go to Step 2.Step 2.
The feedforward structure modelP^ is trained/tested offline
(28)y^(k)=P^((k-py),(WP)k,TsSuuk,Suuk-1,…,Suu(k-pu),Syyk-1,Syyk-2,…,Syy(k-py),(WP)k,Ts)(1Sy),
via the shuffled input/output pairs’ data. After system identification, P^ is performed, and the digital fuzzy controller CF for the CT system can be built by using this inverse NARMAX plant model P^-1 in the next step.Step 3.
In practice, according to the exchanged output/input pairs’ data from Step2, the offline stage to train/test the fuzzy controller can be passed through
(29)uk=CF(Suuk-1,Suuk-2,…,Suu(k-cu),Syyk,Syyk-1,…,Syy(k-cy),(WC)k,Ts)(1Su).
If (28) and (29) work, go to Step 4.Step 4.
Update the online weights and biasesWP of the recursive structure model P^:
(30)y^(k)=P^((k-py),(WP)k,TsSuuk,Suuk-1,…,Suu(k-pu),Syy^k-1,Syy^k-2,…,Syy^(k-py),(WP)k,Ts)(1Sy),
to approximate the CT nonlinear system by using Theorem 2. Due to the adaption laws for (28) and (30), an exchange for both of them can be designed to switch into the system, as a switching in Figure 2, when (30)’s absolute approximation error is too big. If (28) and (30) work, go to Step 5.Step 5.
Adapt the digital fuzzy controller for the modeling error and tracking error by using Theorems2 and 3. Finally, update the online parameters of the fuzzy controller CF(31)uk=CF((k-cy),(WC)k,TsSuuk-1,Suuk-2,…,Suu(k-cu),Syrk,Syrk-1,…,Syr(k-cy),(WC)k,Ts)(1Su),
to minimize the tracking error, and finish the above two stages: the offline stage and online stage.To make sure of the robustness of the control system, the convergence to the global optimal solution of parameters of the model/controller has to be guaranteed. Hence, some random initial weights and biases of the model are designed by particle swarm optimization (PSO) [24] with the parameters of the controller first. The PSO algorithm consists of the velocity
(32)vi(j+1)=vi(j)+γ1i·(pi-x-i(j))+γ2i·(G-x-i(j)),
and position
(33)x-i(j+1)=x-i(j)+vi(j+1),
where i=1,2,…,H is the particle index; j=1,2,…,N is the iteration index; vi is the velocity of ith particle; x-i is the position of ith particle; pi is the best position found by ith particle (personal best); G is the best position found by the swarm (global best, best of personal best); γ1i,γ2i are the random numbers on the interval [0,1] applied to the ith particle.The PSO supplies random initial parameters, hence, it is an initial parameters’ conductor. These initial parameters are then converged locally by the BP method and the best solution for the initial model/controller is chosen. Finally, the global optimal solution of parameters can be found every time. Hence, this idea has been named the random-local-optimization (RLO) algorithm. The RLO algorithm is a composite of the BP algorithm and a random initialization procedure of evaluating fitness value1/(Ξ+0.01), where Ξ=ρ·Ξ1+(1-ρ)·Ξ2, ρ∈[0,1]. The total of absolute training error Ξ1 is obtained by BP via the training data, and Ξ2 is the total of absolute testing error of the model/controller output via the testing data input. In this paper, offline RLO is used as a learning algorithm for the feedforward structure model (28) due to the online tuning parameters of the recursive structure of the plant model being not converged. After the offline training stage, in order to tune the online parameters of the plant model (30) recursively, dy^k/d(WP)k of (14) needs to be calculated as follows:
(34)dy^kd(WP)k=∂y^k∂(WP)k+∑i=0pu∂y^k∂uk-iduk-id(WP)k+∑i=1py∂y^k∂y^k-idy^k-id(WP)k.
Similarly, in order to tune the online parameters of the controller (31) recursively, and dy^k/d(WC)k of (24) needs to be calculated as follows:
(35)dy^kd(WC)k=∑i=0pu∂y^k∂uk-iduk-id(WC)k+∑i=1py∂y^k∂y^k-idy^k-id(WC)k,
where
(36)dukd(WC)k=∂uk∂(WC)k+∑i=1cu∂uk∂uk-iduk-id(WC)k.
Hence, the following algorithm adapts a NARMAX neural controller for a NARMAX neural model of plant.Step 1.
Back propagate throughCF to form ∂uk/∂uk-i and ∂uk/∂(WC)k in (36). If update duk/d(WC)k of (36) and shift duk-i/d(WC)k down in (35), then go to Step 2.Step 2.
Back propagate throughP^ to form ∂y^k/∂uk-i and ∂y^k/∂y^k-i in (35). If update dy^k/d(WC)k of (35) and shift dy^k-i/d(WC)k down in (35), then go to Step 3.Step 3.
Compute(ΔWC)kT=-ηC(k)(y^k-rk)(dy^k/d(WC)k). If we update weights
(37)(WC)k+1=(WC)k+(ΔWC)k,(WP)k+1=(WP)k+(ΔWP)k,
then go to Step 1.To clarify this method, in [20], a robust and adaptive method was used to allow learning to occur online, tuning performance as the system runs. But, [20] did not consider the prediction, modeling error, global optimal initialization of control parameters, the problem of lengthy convergence time of online control, delayed terms, uncertainties in plant, and DT-CT problems. Moreover, the choice method of initial parameters of the online controller still lacks the ability to overcome the overfitting problem of the controller. Hence, the offline stage is proposed for an RLO learning algorithm to choose the initial weights and biases of the online neural controller in the simulation example of the power plant, as shown in the following case study.


## 3.1. Stability Analysis with Neural-Fuzzy Design

In this paper, an adaptive prediction control structure is proposed, as shown in Figure2, where the FRP controller CF is designed as follows:
(3)u(z(k·T))=uk=CF(zk,(WC)k,Ts)+s·CX(ek)=uF+s·uX,
where the switch index
(4)s={0,e~k+1≥0,1,e~k+1<0,uk(z^)=CF(z^k,(WC)k,Ts),
where uP=uF+uX, y^(k+1)1=P^(uP(k+1)), y^(k+1)2=P^(uF(k+1)), e^(k+1)1=rk+1-y^(k+1)1, e^(k+1)2=rk+1-y^(k+1)2, |e^(k+1)1|-|e^(k+1)2|=e~k+1, as shown in Figure 2. The feedforward terms are reference signals [rk,rk-1,…,rk-p], and recursive terms are control signals [uk-1,uk-2,…,uk-q].Two-stage adaptive prediction structure of DT-CT control system. (a) Two-stage block diagram and (b) online adaptive prediction block diagram.
(a)
(b)The offline training input of controller is(5)z^(k)=z^k=[yk,yk-1,…,yk-p,uk-1,uk-2,…,uk-q],z(k)=zk=[rk,rk-1,…,rk-p,uk-1,uk-2,…,uk-q],yk=y(k), rk=r(k). The controller has two working phases: z(k) is the data vector of the testing phase, and z^(k) is the data vector of the training phase. The fuzzy controller is as follows: (6a)uF(z(k·T))=uF(k)=∑l=1Mh-l[∏i=1nexp(-((zi-z-il)/σil)2)]∑l=1M[∏i=1nexp(-((zi-z-il)/σil)2)]=CF(rk,rk-1,…,uk-1,uk-2,…,(WC)k,Ts),(6b)uF(z^(k))=CF(yk,yk-1,…,uk-1,uk-2,…,(WC)k,Ts),where feedforward terms are reference signals [rk,rk-1,…,rk-p], and recursive terms are control signals [uk-1,uk-2,…,uk-q]. Assume that Y=[yk,yk-1,…,yk-p]T, U=[uk,uk-1,…,uk-q] are measured to train model/controller. M is fixed;
(7)zi=ci0yk+ci1yk-1+⋯+cipyk-p+c-i1uk-1+c-i2uk-2+⋯+c-iquk-q.
The tuned parameter vector of controller is as follows:
(8)(WC)k=[(c-i1,c-i2,…,c-iq)(z-11,z-12,…,z-il,…,z-nM),(σ11,σ12,…,σil,…,σnM),(h-1,h-2,…,h-l,…,h-M),(ci0,ci1,…,cip),(c-i1,c-i2,…,c-iq)].The proposed digital fuzzy controlleruk has feedforward terms and recursive structure. It can use a NARMAX neural model or inverse of the plant dynamics to aid control precision in the face of a delayed plant. Owing to the adaptive fuzzy controller, the uncertainty of the plant can be suppressed. Although the structure of the fuzzy controller is chosen, the parameter vector (WC)k is not specified; therefore, the fuzzy controller has not been designed. γ·Ts is the tape-delay time, γ is a positive integer. The idea of the inverse-model-based fuzzy controller is proposed by the following simplified relation:
(9)Ifyk=P^(uk),uk=P^-1(rk)=CF(rk),thenyk=rk,
where P^(·) is the adaptive NARMAX neural model of plant; CF(·) is the adaptive NARMAX fuzzy controller; rk is the desired output. According to the idea of (9), the recursive structure P^(·) can be designed with tape delays as follows:
(10)yk≈y^k=P^(y^k-1,y^k-2,…,y^k-n,uk,uk-1,…,uk-p,(WP)k,Ts),
but the weights and biases (WP)k of this model are trained by the way of the feedforward structure neural network (NN) as follows:
(11)yk≈y^k=P^(yk-1,yk-2,…,yk-n,uk,uk-1,…,uk-p,(WP)k,Ts).
Due to the parameters of the recursive structure of the plant model, convergence may be much stricter, where n,p+1 are the amount of tape delays of y^,u, respectively. The plant output is compared with the desired output to create a system error signal ek=rk-yk. The errors e^k=rk-y^k and ek are used by the adaptation algorithm to update the parameters of P^ and CF. Next, the performance index is designed, as follows:
(12)Jk=12ekTek=12(rk-yk)T(rk-yk)=12(yk-rk)T(yk-rk),
is a cost function to be minimized by the proposed algorithm. Then, the BP algorithm adapts the control parameters (WC)k. That is, the change in control parameters (ΔWC)k is calculated as
(13)(ΔWC)kT=-ηC(k)dJkd(WC)k=-ηC(k)d(yk-rk)T(yk-rk)2·d(WC)k=-ηC(k)(yk-rk)dykd(WC)k=-ηC(k)(yk-rk)dykdukdukd(WC)k,
where the small positive ηC(k) can be selected as a stable learning rate via the following theorems.Theorem 2.
If the amount of neurons of the neural model is sufficient and the appropriate sampling timeTs is selected to let ∥y-k-yk∥≤ε- and the following condition
(14)0<ηP(k)<2∥dy^k/d(WP)k∥2≤η-P
be satisfied, where y-k is an output of the optimal model, then the trajectories y^k of converging to plant output yk is a uniformly ultimately bounded (UUB) approximation on the bounded error y^k-yk.Proof of Theorems.
First, consider the following ideal Lyapunov candidate [29] of the model part,
(15)V1k=12(y^k-yk)T(y^k-yk)=12∥y^k-y-k+y-k-yk∥2=12∥y^k-y-k∥2+εk=V2k+εk,
where V2k=(1/2)∥y^k-y-k∥2 is an actual Lyapunov candidate of reachable and assumptive trajectory y-k; the bounded approximation error
(16)εk=12∥y-k-yk∥2+(y^k-y-k)T(y-k-yk)
and the number of neurons of the neural model are sufficient, and the appropriate sampling time Ts is selected to let y-k≈yk. The next task is to train the model such that V2k is minimized,
(17)(ΔWP)kηP(k)=-dV2kd(WP)k=-(y^k-y-k)d(y^k-y-k)d(WP)k=-(y^k-y-k)dy^kd(WP)k,
where (ΔWP)k=WP(k+1)-WP(k); WP(k)=(WP)k.
Then, the following Lyapunov candidate of the controller part is designed:(18)V3(k)=V3k=12(y^k-rk)T(y^k-rk)=12∥y^k-rk∥2,
Thus, the change in the Lyapunov function is obtained by
(19)V3(k+1)-V3(k)=12(∥y^(k+1)-r(k+1)∥2-∥y^(k)-r(k)∥2).
Finally, the update law of the control parameters of the controller is obtained as follows:
(20)(ΔWC)kTηC(k)≈-dV3(k)d(WC)k=-(y^k-rk)dy^kd(WC)k.
This study develops some convergence theorems to select appropriate stable learning rates. First, the difference of modeling error eP(k)=y^k-y-k can be represented by
(21)eP(k+1)=eP(k)-[deP(k)d(WP)k]T[ηP(k)eP(k)deP(k)d(WP)k]=eP(k)(1-[deP(k)d(WP)k]TηP(k)deP(k)d(WP)k)=eP(k)(1-[dy^kd(WP)k]TηP(k)dy^kd(WP)k);
thus, the change in the Lyapunov function is obtained by
(22)V2(k+1)-V2(k)=12(∥eP(k+1)∥2-∥eP(k)∥2)=12(∥eP(k)(1-[dy^kd(WP)k]TηP(k)dy^kd(WP)k)∥2-∥eP(k)∥2∥eP(k)(1-[dy^kd(WP)k]TηP(k)dy^kd(WP)k)∥2)=12∥eP(k)∥2[(1-[dy^kd(WP)k]TηP(k)dy^kd(WP)k)2-1].
Hence, if -1<(1-[dy^k/d(WP)k]TηP(k)(dy^k/d(WP)k))<1 and ∥y-k-yk∥≤ε-, then V2(k+1)<V2(k), that is, V2(k)→0 or y^k→y-k makes the UUB approximation of this model on the bounded y^k-yk. The proof is thereby completed.Furthermore, the following theorem for the convergence of the controller is obtained by the same procedure as the above proof.Theorem 3.
If Theorem2 in (14) is satisfied, the function dy^k/d(WC)k in (20) is computed to let the following condition
(23)0<ηC(k)<2∥dy^k/d(WC)k∥2≤η-C,
be satisfied. Then, the nonlinear systems (1a) and (1b) are UUB stable, and the tracking errors e(k)=rk-yk are bounded via the controller.Hence, the dynamic response of the systemP can be controlled using CF, as shown in Figure 2. This CF needs the plant model P^ to adjust control parameters via sensitivity function ∂y^k/∂uk-i.The digital feedback controller includes a delay blockD, as shown in Figure 2. Here, the error e~k+1 is used to estimate uX, and the proposed predictor of the delayed system can let us cancel some complex computations, such as ∂y^k/∂uk≈(y^k-y^k-1)/Δuk=Δy^k/Δuk=Δy^k/uX(k), of sensitivity function ∂y^k/∂uk-i in the BP algorithm. Hence, the following theorem is proposed to update the control parameters of FRP under the assumption of providing a model which applies a lower prediction error and a more correct uX. The prediction error e^k+1=rk+1-y^k+1, being bounded due to the previous e^k=rk-y^k, is bounded at any time. Hence, the prediction error e^k+1 will be bounded by using Theorems 2 and 3. Furthermore, the following theorem is obtained for the convergence of the adaptive prediction controller by the same procedure as Theorem 2.Theorem 4.
If Theorem2 in (14) is satisfied, the predictive function duk+1/d(WC)k+1 is computed to let the following condition
(24)0<ηX(k+1)<2∥duk+1/d(WC)k+1∥2≤η-X
be satisfied, then the nonlinear systems (1a) and (1b) are UUB stable, and the tracking errors ek=rk-yk are bounded via the predictive controller uk+1=uF(k+1)+s·uX(k+1).The tracking error isek=rk-yk and ek=[e1k,e2k,e3k,…]T, but the parameters of adaptive control CF are updated by using the predictive offset
(25)Δuk+1=uX(k+1)=CX(ek)=[K1(e1k),K2(e2k),K3(e3k),…]T,
so its previous term is
(26)Δuk=uX(k)=CX(ek-1)=[K1(e1(k-1)),K2(e2(k-1)),K3(e3(k-1)),…]T,
of control input uk, where K1(·),K2(·),K3(·),… are defined by the user, and the predictive controller is uk+1=uF(k+1)+s·uX(k+1), so its previous term is uk=uF(k)+s·uX(k).


## 3.2. Two-Stage Scheme in the Presence of Stability

Figure2 shows a block-diagram of an adaptive recursive control system. The system to be controlled is labeled as the plant P, which is subject to modeling errors, uncertainties, and delays. Due to gradient-descent-based training algorithms, let the model/controller converge to some local minimums which destroy the control system stability by enlarging the modeling error. Hence, we proposed the two-stage training algorithm to reduce this modeling error as follows.In the first stage, the measured data is used to train the global optimal NARMAX plant and the fuzzy controller by the training-data-shuffle method. This method shuffles the training data to avoid most of the local optimal solutions obtained by the offline training procedure in the next section. The measured data is divided into a training data and other testing data. This testing data is not used for training the NN. However, the final performance of the NN is decided by the testing data and the training data.In the second stage, the global optimal NARMAX plant model and fuzzy controller is adapted. The two stages are divided into the following five steps.Step 1.
First, the reference signalr(k)=rk is designed. By the white noise of input u(k)=uk for plant, output data y(k)=yk is collected, and a training-data-shuffle method is used to shuffle the input/output pairs’ data. These shuffled data are ready to train the NARMAX model/controller. Here, the following reasonable conditions need to be taken into account:
(27)maxk(rk)≈maxk(yk),mink(rk)≈mink(yk),maxk(uk)≤uU,mink(uk)≥uL
need be satisfied, where uU is the upper bound of uk, and uL is the lower bound of uk. According to (27), much of the excessive control effort uk can be avoided. If (27) is satisfied, then go to Step 2.Step 2.
The feedforward structure modelP^ is trained/tested offline
(28)y^(k)=P^((k-py),(WP)k,TsSuuk,Suuk-1,…,Suu(k-pu),Syyk-1,Syyk-2,…,Syy(k-py),(WP)k,Ts)(1Sy),
via the shuffled input/output pairs’ data. After system identification, P^ is performed, and the digital fuzzy controller CF for the CT system can be built by using this inverse NARMAX plant model P^-1 in the next step.Step 3.
In practice, according to the exchanged output/input pairs’ data from Step2, the offline stage to train/test the fuzzy controller can be passed through
(29)uk=CF(Suuk-1,Suuk-2,…,Suu(k-cu),Syyk,Syyk-1,…,Syy(k-cy),(WC)k,Ts)(1Su).
If (28) and (29) work, go to Step 4.Step 4.
Update the online weights and biasesWP of the recursive structure model P^:
(30)y^(k)=P^((k-py),(WP)k,TsSuuk,Suuk-1,…,Suu(k-pu),Syy^k-1,Syy^k-2,…,Syy^(k-py),(WP)k,Ts)(1Sy),
to approximate the CT nonlinear system by using Theorem 2. Due to the adaption laws for (28) and (30), an exchange for both of them can be designed to switch into the system, as a switching in Figure 2, when (30)’s absolute approximation error is too big. If (28) and (30) work, go to Step 5.Step 5.
Adapt the digital fuzzy controller for the modeling error and tracking error by using Theorems2 and 3. Finally, update the online parameters of the fuzzy controller CF(31)uk=CF((k-cy),(WC)k,TsSuuk-1,Suuk-2,…,Suu(k-cu),Syrk,Syrk-1,…,Syr(k-cy),(WC)k,Ts)(1Su),
to minimize the tracking error, and finish the above two stages: the offline stage and online stage.To make sure of the robustness of the control system, the convergence to the global optimal solution of parameters of the model/controller has to be guaranteed. Hence, some random initial weights and biases of the model are designed by particle swarm optimization (PSO) [24] with the parameters of the controller first. The PSO algorithm consists of the velocity
(32)vi(j+1)=vi(j)+γ1i·(pi-x-i(j))+γ2i·(G-x-i(j)),
and position
(33)x-i(j+1)=x-i(j)+vi(j+1),
where i=1,2,…,H is the particle index; j=1,2,…,N is the iteration index; vi is the velocity of ith particle; x-i is the position of ith particle; pi is the best position found by ith particle (personal best); G is the best position found by the swarm (global best, best of personal best); γ1i,γ2i are the random numbers on the interval [0,1] applied to the ith particle.The PSO supplies random initial parameters, hence, it is an initial parameters’ conductor. These initial parameters are then converged locally by the BP method and the best solution for the initial model/controller is chosen. Finally, the global optimal solution of parameters can be found every time. Hence, this idea has been named the random-local-optimization (RLO) algorithm. The RLO algorithm is a composite of the BP algorithm and a random initialization procedure of evaluating fitness value1/(Ξ+0.01), where Ξ=ρ·Ξ1+(1-ρ)·Ξ2, ρ∈[0,1]. The total of absolute training error Ξ1 is obtained by BP via the training data, and Ξ2 is the total of absolute testing error of the model/controller output via the testing data input. In this paper, offline RLO is used as a learning algorithm for the feedforward structure model (28) due to the online tuning parameters of the recursive structure of the plant model being not converged. After the offline training stage, in order to tune the online parameters of the plant model (30) recursively, dy^k/d(WP)k of (14) needs to be calculated as follows:
(34)dy^kd(WP)k=∂y^k∂(WP)k+∑i=0pu∂y^k∂uk-iduk-id(WP)k+∑i=1py∂y^k∂y^k-idy^k-id(WP)k.
Similarly, in order to tune the online parameters of the controller (31) recursively, and dy^k/d(WC)k of (24) needs to be calculated as follows:
(35)dy^kd(WC)k=∑i=0pu∂y^k∂uk-iduk-id(WC)k+∑i=1py∂y^k∂y^k-idy^k-id(WC)k,
where
(36)dukd(WC)k=∂uk∂(WC)k+∑i=1cu∂uk∂uk-iduk-id(WC)k.
Hence, the following algorithm adapts a NARMAX neural controller for a NARMAX neural model of plant.Step 1.
Back propagate throughCF to form ∂uk/∂uk-i and ∂uk/∂(WC)k in (36). If update duk/d(WC)k of (36) and shift duk-i/d(WC)k down in (35), then go to Step 2.Step 2.
Back propagate throughP^ to form ∂y^k/∂uk-i and ∂y^k/∂y^k-i in (35). If update dy^k/d(WC)k of (35) and shift dy^k-i/d(WC)k down in (35), then go to Step 3.Step 3.
Compute(ΔWC)kT=-ηC(k)(y^k-rk)(dy^k/d(WC)k). If we update weights
(37)(WC)k+1=(WC)k+(ΔWC)k,(WP)k+1=(WP)k+(ΔWP)k,
then go to Step 1.To clarify this method, in [20], a robust and adaptive method was used to allow learning to occur online, tuning performance as the system runs. But, [20] did not consider the prediction, modeling error, global optimal initialization of control parameters, the problem of lengthy convergence time of online control, delayed terms, uncertainties in plant, and DT-CT problems. Moreover, the choice method of initial parameters of the online controller still lacks the ability to overcome the overfitting problem of the controller. Hence, the offline stage is proposed for an RLO learning algorithm to choose the initial weights and biases of the online neural controller in the simulation example of the power plant, as shown in the following case study.


## 4. Cases Study

First, the conventional PWM buck converter, by using AM-OTS-DS [28] methodology, is modeled to the equivalent circuit plant. In this paper, the robustness of this control system is emphasized, so uncertainty and delay are considered in (1b). The state is x(t)=x=[iL(t),vC(t)]T, and the output is y(t)=vo(t) in this power system.Referring to Figure2, and the above sections about two-stage control structure, it can be seen how to model the CT plant dynamics from Figure 2(a) by considering the modeling error, and how to use the DT neural model of plant to adapt a DT fuzzy controller and predict the DT compensation of total control input uk from Figure 2(b).To compare with other methods, the following cases are introduced.Case 1.
This case is in [28], and its digital controller is a kind of fuzzy controller with integral term shown in (38).Case 2.
This case is in [30], and its controller is a kind of single neuron shown in (39) without multilayer structure, and this CT controller is discretized to DT controller to compare with Case 3.Case 3.
This is the control method presented here, and the proposed neural-model-based fuzzy controller is adaptive, predictive, and globally optimal.The detail designs of the Cases1–3 are as follows.Case1 is an LMI control method of original example for this power plant. The control parameters of Case 1 are solved by the LMI toolbox as follows.K
11
=
K
21
=
[
0.0476,0.9348
]
T and K12=K22=-426.4969 are referred to [28], and this T-S fuzzy controller of Case 1 is designed as
(38)uk=-h1k·(K11xk+K12∑kekTs)-h2k·(K21xk+K22∑kekTs),
where ∑kekTs is integral term.Case2 is an adaptive control method. This neural controller is designed as
(39)uk=a·(1-e-b·net)1+e-b·net,net=I-ϕ,
where I=ek is a net input of single neuron.And the design parametersP2=1200 and learning rate η=1 for this neuron of Case 2 are referred to [30].Case3 is also an adaptive control method, but its NARMAX fuzzy control design method with neural-model-based prediction is very different from Case 2. The predictive controller of Case 3 is uX=K1ek, where K1=0.1.First, the sampling time of the model/controller is set forTs=10-4 sec to simulate the proposed adaptive prediction control system, the initial state x(0)=[0,0]T, and the reference signal rk=12sin(30k·Ts). Case 3 uses NN structure 5-8-1 of the NARMAX plant model, it has 5 inputs, [uk,uk-1,uk-3,y^k-1,y^k-2], 8 tansig(·) neurons in the hidden layer, and 1 purelin(·) neuron in the output y^k layer.The adaptive weights and biases,WP, are trained as follows by selecting a suitable scaling factor Su=1, Sy=1/24 of the plant model. The summation WPS of adaptive plant model parameters WP is updated as shown in Figure 3(a). The summation WCS of adaptive fuzzy control parameters WC is updated as shown in Figure 3(b). The tracking control performances ek=rk-yk and ∑k|ek| of Case 3 are shown in Figures 3(c) and 5. The bounded modeling error of Case 3 is shown in Figure 3(d). The trajectory of predictive input signal uX of Case 3 is shown in Figure 3(e). Note that this predictive control input signal is just the minor portion of total control signal. It works to control the CT system precisely. To clarify this difference between predictive control input and total control input, the figure of total control signal is shown in Figure 3(f).(a) The learning curve of the summation ofWP, (b) the learning curve of the summation of WC, (c) the tracking control performance, (d) the modeling error y^k-yk=yNk-yk, (e) the trajectories of predictive input signal uX, and (f) total input signal u of Case 3.
(a)
(b)
(c)
(d)
(e)
(f)And the tracking control performance of Case2 is shown in Figures 4(a) and 4(c), and the controller of Case 2 includes the adaptive parameters ϕ,a,b whose learning curves are shown in Figure 4(b). From Figures 3(d) and 5(d), the offline training scheme of Case 3 does let the neural model be optimized globally to make the adaptive laws of Case 3 converge faster than that of Case 2. Therefore, the more precise result from Case 3 is made as shown in Figure 5.(a) The tracking control performance and (b) the parametersϕ,a,b whose learning curves of Case 2.
(a)
(b)(a)(b) Comparison of the output trajectories for Cases1 to 3, and (b) is the detail view of (a). (c) Comparisons of ek=rk-yk and (d) ∑k|ek| of the control performances for Cases 1 to 3.
(a)
(b)
(c)
(d)Finally, the control performances of Cases1–3 are compared, as shown in Figure 5. Figure 3 shows the precise fuzzy control performance of Case 3. Figure 4 shows the digital adaptive neural control performance of Case 2. Figure 5 shows the LMI control performance of Case 1. From the control results of Cases 1 and 3 shown in Figure 5(d), the proposed adaptive and predictive laws do let the proposed controllers adapt and predict themself to overcome these problems about robustness as the delay, uncertainty, and nonlinearity.It is clear that the two-stage scheme, Case3, for the proposed adaptive controller and predictive controller has excellent tracking performance when compared with Cases 1 and 2.


## 5. Conclusion

The proposed two-stage adaptive prediction control converges very fast, works highly effective, and precise. It simplifies the complex model-based adaptive control design, and works for nonlinear delayed plants with uncertainty. The proposed recursive and feedforward control scheme is partitioned into two stages that can be independently optimized. First, an offline neural model of continuous-time (CT) nonlinear power plant is made. Second, a constrained offline digital fuzzy controller is generated; then, an adaptive plant model is made, and an adaptive NARMAX prediction tracker is generated. Finally, all processes may continue concurrently, and robustness and adaptive prediction design with DT-CT problems for a power plant are solved. Although this power system is only a simulation, the control strategy can be extended to LED dimmer systems and time-delay robot systems based on visual servo, and is within my plans of future research.



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*Source: 836414-2013-04-22.xml*