On line tuning of a neural PID controller based on variable structure RBF network

This paper presents the use of a variable structure radial basis function (RBF) network for identification in PID control scheme. The parameters of PID control are online tuned by a sequential learning RBF network, whose hidden units and connecting parameters are adapted online. The RBFnetworkbased PID controller simplifies modeling procedure by learning inputoutput samples while keep the advantages of traditional PID controller simultaneously. Simulation results of ship course control simulation demonstrate the applicability and effectiveness of the intelligent PID control strategy

Based on Variable Structure RBF Network

Jianchuan Yin, Gexin Bi, and Fang Dong College of Navigation, Dalian Maritime University,

1 Linghai Road, Dalian 116026, China yinjianchuan@gmail.com, bigexin@gmail.com, dfhhxy@sohu.com

Abstract This paper presents the use of a variable structure radial

basis function (RBF) network for identification in PID control scheme The parameters of PID control are on-line tuned by a sequential learn-ing RBF network, whose hidden units and connectlearn-ing parameters are adapted on-line The RBF-network-based PID controller simplifies mod-eling procedure by learning input-output samples while keep the advan-tages of traditional PID controller simultaneously Simulation results of ship course control simulation demonstrate the applicability and effec-tiveness of the intelligent PID control strategy

Keywords: Radial basis function network, Variable structure, PID

control

1 Introduction

The design of ship motion controller presents challenges because ship’s motion

is a complex nonlinear system with time-varying dynamics [1] The dynamics of ship motion varies in case of any changes in sailing conditions such as speed, loading conditions, trim, etc Similar changes may also be caused by environ-mental disturbances, such as waves, wind, current, etc So it is hard to obtain

a suitable physically founded ship motion model due to the complexity of the time-varying dynamics and the underlying processes

In recent years, neural networks have been used to model unknown nonlinear systems for a broad variety of control problems [2] Whereas, most satisfactory performances of neural-network-based intelligent controller are achieved when applied to systems with static dynamics When applied to systems with non-stationary dynamics, a neural network with static structure is less satisfactory [3] in representing time-varying dynamics So it is desirable to develop variable structure neural networks whose structure and connecting weights can adapt to changes of system dynamics

Sequential learning algorithms are designed for on-line constructing variable structure network They are featured by low computational burden and par-simonious network structure Radial basis function (RBF) neural network has gained much popularity due to its features such like simple topological struc-ture and best approximation [4] These feastruc-tures enables RBF network suitable

W Yu, H He, and N Zhang (Eds.): ISNN 2009, Part II, LNCS 5552, pp 1094–1104, 2009 c

 Springer-Verlag Berlin Heidelberg 2009

for sequential learning scheme The most widely used sequential learning algo-rithm are resource allocation network (RAN) [5], RAN with extended kalman filter (RANEKF) [6], Minimal-RAN (MRAN) [7] and generalized growing and pruning RBF (GGAP-RBF) [8]

In this paper, we introduce a sequential learning algorithm for RBF network referred to as dynamic orthogonal structure adaptation (DOSA) algorithm The algorithm can achieve compact network structure by employing a small num-ber of parameters It takes advantage of a sliding data window for monitoring system dynamics, and improve the well-known idea of error reduction ratio as contribution criteria for network pruning First-order derivative information is also included in the network to trace the trend of system dynamics

Conventional PID control is still the most widely used control scheme in in-dustrial applications attribute to its merits such as strong robustness, under-standability and simple structure But it may become laborious when applied

to practical systems which are usually both nonlinear and nonstationary By in-corporating the neural network in the PID control scheme, resulting intelligent adaptive PID controller simplifies the modeling procedure by merely learning input-output samples as well as keep the advantages of conventional PID con-troller simultaneously

In this study, we apply the PID controller with variable structure RBF net-work in ship course control simulation The RBF netnet-work was on-line constructed

by DOSA algorithm Simulation of ship course control was finally conducted to demonstrate the applicability and effectiveness of the RBF-network-based PID controller

2.1 Algorithm Description

By combining a sequential learning mode with the subset selection scheme of orthogonal least squares (OLS) algorithm [9], we introduce a learning algorithm referred to as dynamic orthogonal structure adaptation (DOSA) algorithm The network structure becomes variable by adding the newly received observation as hidden unit directly, while pruning units which contribute little to output over a number of observations The contribution of each hidden unit is measured by its normalized error reduction ratio, which generalized from error reduction ratio in orthogonal least squares (OLS) algorithm [9]

We employ in this algorithm a sliding window which is a first-in-first-out sequence When a new observation is received, the sliding window is updated by incorporating the new observation and discarding the foremost one

window = [(x1, y1), (x2, y2), · · · , (x N , y N)] (1)

where N is the width of the sliding data window The data in the sliding window are used to represent the dynamics of system with input X ∈ R n×N and output

Y ∈ R N ×m n and m are dimensions of the input and output, respectively.

For nonlinear systems involving variants of local mean and trend, the dynam-ics of system can be easily tracked by applying a suitable difference on the signal

[3] Here we use the first order differences of the sliding window input X  ∈ R n×N

and output Y  ∈ R N ×mas input and output to the network, respectively

X  = [x1− x0, , x N − x N −1] =

⎛

. .

⎞

⎟

⎠ ∈ R n×N(2)

is the difference of sliding data window input and

Y  =[y1−y0, , y N −y N −1]T=

⎛

⎜

⎝

. .

⎞

⎟

⎠ ∈ R N ×m(3)

is the difference of desired output matrix of the sliding data window

The learning procedure begins with no hidden unit At each step, the differ-ence of new observation input is directly added into the hidden layer as a new hidden unit By Calculating the Gaussian functions of the Euclidean distance

between X  and the candidate hidden units, we have response matrix of the

hidden units to the input of sliding data window Φ ∈ R N ×M with

φ j,k= exp(− x j − c k 2

where c k are known as the k-th hidden units, σ a width constant and  ·  the

Euclidean norm

In order to avoid multicollinearity and evaluate the individual contribution of

each hidden units, we transform the set of basis vectors Φ into a set of orthogonal basis vectors using Gram-Schmidt method by decomposing Φ into Φ = W A The space spanned by the set of w k is the same space spanned by the set of φ k

The error reduction ratio of each vector w k is then calculated:

[err] ki= (wT k y i)2

According to vector space theory,M

θ k = 1 in single-output condition This explains whyM

k=1 [err] k = 1 in OLS algorithm under single output condi-tion Different from the square response matrix in OLS algorithm, the response matrix in DOSA algorithm is generally not square because the size of sliding data window input is generally not the same as the number of hidden units We can find thatM

k=1 [err] k > 1 when M > N andM

k=1 [err] k < 1 when M < N

For evaluating the contribution of hidden units to the trend of system dynamics

directly, the normalized error reduction ratio (nerr) is obtained by

[nerr] k = [err] k

M

k=1 [err] k

(6)

This transformation makesM

k=1 [nerr] k = 1 under any condition and enables

the direct application of nerr for evaluation purpose.

After the network growing process by adding the new observation, the pruning process is conducted by pruning units which contribute little to the output First,

we select hidden units whose sum of nerr value falls below an preset accuracy threshold ρ at each step Assume that [nerr] k1 = max{[nerr] k , 1 ≤ k ≤ M }.

If [nerr] k1 < ρ, then select [nerr] k2 = max{[nerr] k , 1 ≤ k ≤ M, k = k1} The

same selection is made for [nerr] k S = max{[nerr] k , 1 ≤ k ≤ M, k = k1, k =

k2, , k = k S−1 }.

The selection procedure continues until the sumk=k S+1

k=k1 [nerr] k ≥ ρ Select

k1, , k S and the corresponding hidden units are marked with S k={k1, , k S }

and considered as contributing little to the output at this step The selection

is made at each step When some hidden units are selected for consecutive M S

times, certain units will be pruned from network That is, remove the units in

the intersection of sets selected in the past M S observations

I = {S k

S k−1

.

After hidden units being added or pruned at each step, the weights between the hidden layer and output layer are adapted to the difference of the sliding data window output using the linear least mean squares estimation (LLSE) [10]:

When the difference of output y  N +1 is obtained according to input x  N +1, the output can be finally achieved by

2.2 Performance Results of DOSA Algorithm

In this section, DOSA algorithm is implemented to prediction of Mackey-Glass chaotic time series whose dynamics are both nonlinear and nonstationary [11] The Mackey-Glass series is governed by the following time-delay ordinary differential equation:

ds(t)

dt =−bs(t) + a s(t − τ )

with a = 0.2, b = 0.1 and τ = 17 Integrating the equation over the time interval [t, t + Δt] by the trapezoidal rule yields

x(t+Δt) =2− bΔt

2 + Δt x(t) + [

x(t + Δt − τ )

1 + x10(t + Δt − τ )+

x(t − τ )

1 + x10(t − τ )]

aΔt

2 + bΔt(11) Set Δt = 1, the time series is generated under the condition x(t − τ ) = 0.3 for

0≤ t ≤ τ(τ = 17) The series is predicted with μ = 50 sample steps ahead using

four past samples: s n−μ , s n−μ−6 , s n−μ−12 , s n−μ−18 Hence, the n-th input-output

data pair for the network to learn are expressed as

and

whereas the predicted value at time n is given by

where f (x n ) is the network prediction and the μ-step-ahead prediction error is

This experiment is designed to evaluate the on-line prediction capability of the RBF obtained by DOSA algorithm, so after learning at each step, the unlearned

next value in the series s n+1 is predicted using f (x n) and get the predicted value

z n+1:

and the prediction error was got

The exponentially weighted prediction error (EWPE) is chosen as a measure

of the performance of the network prediction The value of EWPE at time n can

be recursively computed as

where 0 < λ < 1 Here λ is chosen as λ = 0.95.

5000 data were generated from (11) for the prediction experiment of proposed DOSA algorithm The MRAN algorithm is a popular sequential learning algo-rithm, so the same experiment was performed by MRAN algorithm for

compar-ison The parameter values for DOSA algorithm are selected as follows: N = 10,

ρ = 0.00002, M S = 3 We notice that there are only three parameters, which facilitate its practical applications Parameter values for MRAN algorithm are

max min = 0.07, γ = 0.999, e min = 0.095, e  min = 0.078,

κ = 0.87, P0= R n = 1.0, Q = 0.0002, η = 0.02, δ = 0.01, M = 90.

The evolution of hidden units number, prediction error and value of lg(e W P E) with different algorithms are shown in Figs 1- 3.Figure 1 shows that the hidden units number of network generated by DOSA algorithm initiates with 0 and remains between 5−8 after 29-th step, while that of network generated by MRAN

algorithm remains at 16 which is much larger than that of DOSA algorithm

We can see from Figs 2 and 3 that the prediction error of DOSA algorithm is much smaller than that of MRAN algorithm It shows that the DOSA algorithm can quickly track the changes of system dynamics with parsimonious network structure, is an efficient modelling method to sequentially construct a variable network structure

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

2

4

6

8

10

12

14

16

18

Number of Observations

DOSA MRAN

Fig 1 Comparison of hidden units number curve

−0.2

0

0.2

0.4

0.6

Number of Observations

−0.5

0

0.5

1

Number of Observations

DOSA

MRAN

Fig 2 Comparison of prediction error curve

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

−3

−2.5

−2

−1.5

−1

−0.5

Number of Observations

Fig 3 Comparison of WPE value curve

3 The Neural PID Controller

In this study, sequential DOSA algorithm is used for RBF network training The

PID control parameters K p , K i and K d can be adjusted online through RBF network self-training And the best values of them can be obtained corresponding

to the outputs of RBF network with a certain optimal control law

The general configuration of the neural network PID controller is shown in Fig 4 In the control scheme, neural network act for system identification, here

Fig 4 Configuration of RBF network PID control scheme

we use the RBF network constructed by the DOSA algorithm Ship model is set

as the plant, and the PID control parameters are tuned on-line

The conventional increment PID algorithm is given by

Δu(k) = K p [e(k) − e(k − 1)] + K i e(k) + K d [e(k) − 2e(k − 1) + e(k − 2) (20)

where K p is the proportional coefficient, K i the integral coefficient, K d is the

differential coefficient, u is the control variable, r is the expected output value and y is the actual output value obtained during evaluation.

So we can calculate u(k) if we know u(k − 1), y(k), e(k), e(k − 1) and e(k − 2).

A performance function is given by

The three control coefficients are updated according to gradient-descent algo-rithm:

ΔK p=−η ∂E

∂K p

=−η ∂E

∂y

∂y

∂u

∂u

∂K p = ηe(k) ∂y

∂u (e(k) − e(k − 1)) (23)

ΔK i=−η ∂E

∂K i

=−η ∂E

∂y

∂y

∂u

∂u

∂K i = ηe(k) ∂y

ΔK d=−η ∂E

∂K d

=−η ∂E

∂y

∂y

∂u

∂u

∂K d = ηe(k) ∂y

∂u (e(k) − 2e(k − 1) + e(k − 2)) (25)

where ∂u ∂y is the Jacobian message of the controlled plant Here we employ ∂ ˆ ∂u y to approximate it and ∂ ˆ y

∂u is achieved from the RBF network identification The Jacobian matrix is

∂y(k)

∂u(k) ≈ ∂ ˆ y(k)

∂u(k) =

M j=1

w j h j c j − x1

b2

j

(26)

with x1= u(k) and j is the index of the hidden units.

4 Ship Course Control Simulation

The proposed neural network PID control strategy was experimented by Matlab simulation Ship motion is a typical nonlinear system with unstable dynamics,

so examine the performance of the control strategy by applying it in ship track-keeping control The simulation is based on ship ”Mariner” [12] via Abkowitz type mathematical model [13] expression The ship’s hydrodynamic coefficients are acquired from [13]

The objective of our simulation was to steer a ship on setting courses with small deviations as well as avoiding large control actions The desired course are set as−10 ◦during [0s, 300s], 10◦ during [301s, 600s], 20◦during [601s,900s]

and 0◦ during [901s,1200s] To make the simulation more realistic, influence

of wind, wave were both considered [14] Wind force is set to Beaufort scale

4 with speed ranging from [0m/s, 14m/s], speed and direction changes every

10 and 50 seconds, respectively Wave direction was set as 30◦ during [1s,600s]

and 60◦ during [601s,1200s] Influences of wind and wave on ship motion were

calculated according to [13] Here the ship motion control input-output data pair for the network to learn are expressed using nonlinear autoregressive model with exogenous inputs NARX system expression [15]:

ˆ

y(k) = f (y(k − 1), , y(k − n y ), u(k), , u(k − n u)) (27)

where y(·) and u(·) are discrete sequences of output and input n y and n u are

the maximum lags in the output and input, d is the delay of system, f (·) is the

unknown function

In the simulation, ship speed was set to 15 knot, rudder angle and rate were constrained to ±20 ◦ and ±5 ◦ /s separately The parameters were chosen as

fol-lows: n u = 1, n y = 2, η = 10, N = 10, M S = 3, ρ = 0.0005 DOSA algorithm was

implemented to construct RBF network on-line Simulation results are shown in Fig 5 The upper figure of Fig 5 shows the desired ship heading course and the actual heading course, the lower figure of Fig 5 shows the rudder actions during simulation For comparison, the conventional PID control is also implemented

0 200 400 600 800 1000 1200

−20

−10 0 10 20 30

time (s)

−20

−10 0 10 20

time (s)

Fig 5 Ship heading course and rudder angle (RBF-network-based PID control)

−20

−10 0 10 20 30

time (s)

−20

−10 0 10 20

time (s)

Fig 6 Ship heading course and rudder angle (PID control)

under the same condition and the results are shown in Fig 6 The parameters

of conventional PID controller are tuned as: K P = 8, K I = 0.01, K D= 80 Our control goal is to steer the ship continues to follow the desired courses with small tracking errors We can see from Figs 5 and 6 that, although both method can track the desired course well, the proposed RBF network-based PID control strategy uses much less rudder action, and the heading course curve is smooth either The effects of wind and wave are both eliminated to a considerably low level It indicates that the controller can react fast to the environmental changes with smooth rudder actions, it also shows that the RBF network which is on-line constructed by DOSA algorithm can react to the change of ship dynamics adaptively

5 Conclusion

With regard to the problems of controlling nonlinear system with unstable dy-namics, an RBF-network-based intelligent PID control algorithm is introduced The RBF network is on-line constructed by DOSA algorithm Simulation re-sults show that the proposed control strategy is featured by quick response, high stability and satisfactory anti-interference ability It is demonstrated that the proposed control scheme is a promising alternative to conventional autopilots

Acknowledgements

This work is supported by Application Fundamental Research Foundation of China Ministry of Communications under grant 200432922505

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