Neural Network Augmented Predictive PID Controller Mohammad H. Moradi, Hemen Showkati Electrical Engineering Department, Bu Ali Sina University, Hamedan, Iran [email protected], [email protected], Corresponding Author: M.H. Moradi

Abstract-This paper is concerned with the design of a new PID tuning method with the capability of prediction. This predictive PID controller has similar performance in comparison with Generalized Predictive Controller (GPC). The predictive PID method has an extra simplicity for implementing in industrial applications because this scheme could be implemented on existing conventional PID hardware currently in use in industrial applications. A PID type structure is defined which includes prediction of output and the recalculation of new set point using the future set point data. The optimal values of the PID gains are calculated in two stages. First in off-line stage optimal values of PID gains are derived using optimal signal matching with GPC, then in online stage a neural network is used to fine tune the calculated parameters in off-line stage. Simulation results of the method show similar performance in comparison with GPC algorithm.

I. INTRODUCTION Proportional-Integral-Derivative (PID) controllers are control algorithms commonly found in industrial applications. Because of their robust performance, simplicity of structure and easy comprehension in principle, PID controllers are extensively used in process industries. [15], [5].The three parameters of PID controllers must be tuned to the process to obtain a satisfactory closed loop performance. Conventional PID controller despite the simplicity of implementation, lacks the sufficient degree of flexibility needed to handle complex processes such as systems with long time delay and non-minimum phase systems. Over the years numerous techniques have been suggested for tuning of PID parameters. Some well known techniques can be found in [18], [1]. On the other hand, the Generalized Predictive Controller (GPC) was introduced by [4]. And this has become one of the most popular Model Predictive Control (MPC) algorithms both in industry and academia. It has been successfully implemented in many industrial applications [4] showing good performance and certain degree of robustness. It can handle many different control problems for a wide range of plants with a reasonable number of design variables, which have to be specified by the user depending upon prior knowledge of plant and control objectives. GPC is an optimal method which can deal with unstable and nonminimum phase plants.

Because of hardware and software restrictions and practical implementation issues for using GPC and other advanced control techniques many researchers have attempted to restrict these advanced methods to retrieve a PID controller with similar features. [13] introduced an Internal Model Control (IMC) based controller design for first order process model and [3] extended IMC-PID controller to cover the second order process model. The limitation of these methods is that tuning rules are derived for the delay free system. In [8] a mathematically equivalent PID control law to GPC was proposed. This was done by equating the discrete PID control law with linear form of GPC. Although there is no restriction on choice of GPC tuning parameters, the process model order is restricted to maximum of two. In [16] a least square algorithm was used to compute the closest equivalent PID controller to an IMC design but still ineffective for time-delay and unstable systems. In [7] a model-based PID controller was used by utilizing multiple Distributed Control System (DCS) PID blocks to implement a model based predictive control strategy. [6] have introduced a predictive PID algorithm which has similar features to model-based predictive control (MPC). Corresponding to a prediction horizon of size M, a bank of M parallel conventional PID controllers is defined. Three term PID gains are calculated using optimal signal matching with GPC. In [9] a two layer auto-tuning method was proposed in which the lower layer is a conventional PID controller and upper level is identification and tuning modules. In [10] by use of Radial Basis Function neural network (RBFNN) an adaptive predictive PID controller was introduced which has high degree of robustness. Based on Lagrange Particle Swarm Optimization [14] introduced a robust PID tuning strategy. Although it is difficult to guarantee it’s effectiveness in theoretical way because of stochastic approach of particle swarm optimization. This paper presents a new PID type controller based on the GPC approach. The proposed PID control design is used and applied to low order, high order and non-minimum phase processes. In the proposed method no Pade approximation is made and the new PID control design is applicable to both time-delay and unstable processes. A new fine tuning method is also proposed which is a model free algorithm based on neural networks. This module gives the overall control system an extra flexibility to handle certain

degree of model-plant mismatch. This method can be applied in processes which some extra software capability is available. Predictive PID algorithm with fine tuning module is suitable, when high degree of accuracy is needed so for typical industrial processes fine tuning module can be optional. In configuration of this paper, the system transient behavior is directly managed by the predicted trajectories. The actual closed-loop is a combination of the PID response and the GPC output predicted system trajectories. Note that this approach considers also similar assumptions to that of GPC approach. Another important feature of this scheme is related to hardware implementation and current operator’s practice. If there exists a PID controller regulating the system, changing to a MBPC strategy implies the removal of all the components currently in use. Such changes involve the approval of the operator and training in the use of new tuning rules of new control algorithm. The approach of this paper is based on existing PID controller configuration and requires only little extra additional hardware. Because for most of typical industrial applications the calculations can be done off-line and be tested on a computer away from the actual system and then downloaded to the appropriate storage device. And more ever there are no changes from the point of view of operator’s training; the operator can continue to manage the plant in the same manner they used before. The main idea is based on calculating an equivalent set of PID parameters from a GPC control law derived using a general process model. In this way an optimal controller is developed which has a simple and desirable structure but yields the level of performance expected from GPC. The paper has been organized as follows: section 2 describes the structure of predictive PID controller and calculates the control law. Section 3 presents the optimal value for predictive PID controller. In section 4 the stability issues are discussed. A comparison between the proposed method and GPC is presented in section 5. Finally, conclusions close the paper.

equation (1). For M>0 the proposed controller has predictive capability similar to MBPC where M is prediction horizon of PID controller. The horizon, M will be selected to find the best approximation to GPC solution. Using (1), equation (2) can be decomposed into M control signal as follows: u (k ) = u~ (k ) + u~ (k + 1) + ... + u~ (k + M ) (3) In our previous work, we have shown that control signal increment can be written as follows: (4) ∆u ( k ) = − K α∆uˆ (k ) + F f y0 (k ) + G g ∆u0 ( k )

{

where M

α= i =0

(1)

where kP , k I , and k D are the proportional, integral, and derivates gains, respectively. A type of predictive PID controller is defined as follows: k

( k P e( k + i ) + k I i =0

Gi′ i =0

{

}

Shifting the control signal for d step ahead gives: ∆u (k ) = − K α d ∆uˆ (k ) + F fd y0 (k ) + G gd ∆u0 (k )

{

}

(6)

Where coefficients matrices are: d +M

αd =

d +M

Gi F fd = i =d

d +M

Fi i =d

Gi′

Ggd = i =d

III. OPTIMAL VALUES OF PREDICTIVE PID GAINS A. Off-line Calculation of Optimal Values To obtain optimal values of the gains, the generalized predictive control (GPC) algorithm is used. For process control, default setting of output cost horizon {N1 : N 2 } = {1 : N } and control cost horizon N u = 1 can be used in GPC to give reasonable performance [4]. GPC consists of applying a control sequence that minimizes the following cost function: [ y (k + d + i ) − w(k + d + i )]2 + λ∆u (k )

(7)

i =1

k

M

i =0

J (1, N ) =

j =1

u (k ) =

M

Fi Gg =

And Gi , Fi and Gi′ are matrices defined in [2] used for ith step ahead prediction of output. For systems with time delay, d, the output of the process will not be affected by ∆u (k ) until the time instant (k+d+1), the previous outputs will be a part of free response. We have shown that control signal increment for systems with time-delay can be written as follows [11]: ∆u (k − d ) = − K α∆uˆ (k − d ) + F f y0 (k ) + Gg ∆u0 (k ) (5)

A discrete PID controller has the following form:

e( j ) + k D [e(k ) − e(k − 1)]

M

Ff =

Gi

N

II. PID TYPE PREDICTOR

u~ (k ) = k P e(k ) + k I

}

e( j + i ) + k D [e(k + i ) − e(k + i − 1)]) j =1

(2) The controller consists of M parallel PID controllers. For M=0, the controller is identical to the conventional PID in

The minimum of J (assuming there are no constraints on the control signal) is found using the usual gradient analysis, which leads to [4]: ∆u ( k ) = (G T G + λI ) −1 G T [ w − Fy0 ( k ) + G ′∆u0 (k )] (8) Which can be summarized (assuming the future set point w(t+i)=0): y (k ) y (k ) = −K0 0 ∆u ( k ) = − K GPC [ − F − G ′] 0 (9) ∆u 0 ( k ) ∆u 0 ( k ) Where: K 0 = − K GPC [ − F

−G ′]

K GPC = (G T G + λI ) −1 G T

y0 ( k ) = [ y ( k )

y (k − 1) ... y (k − na )]

B. Fine Tune Module In section A. off-line stage of finding optimal value of predictive PID controller using signal matching have been discussed. In this section on-line fine tuning module will be presented. In off-line calculation of predictive PID optimal gains because of approximations used in derivation of the parameters there is always some error between two control signal increments i.e. ∆u PID − ∆uGPC ≠ 0 . So it can not

T

∆u 0 ( k ) = [∆u (k − 1) ∆u (k − 2) ... ∆u ( k − nb )]

T

To compute the optimal values of predictive PID gains with N u = 1 [uˆ ( k ) = u ( k )] , the PID control signals should be made the same as GPC controller. This means using equation (4) and (9) and solving the following optimal problem: MinK∈K s ,M J ( K PPID , K 0 )

guarantee precise approximation of GPC in presence of model-plant mismatch. So on-line fine tune module will be used to increase the accuracy of approximation. In section 3 the goal was to produce a control signal similar to that of GPC. In this stage we will use actual plant output to retune the parameters. The proposed method uses neural network optimization approach. The cost function J is introduced to be minimized by training of a neural network in order to fine tune predictive PID parameters. The optimization problem can be formulated as follows [12]: NN : min F{K * − K nn ( w,...)}

PID

K

S PID

= Set of stability gains for PID

Where K PPID is predictive PID gains and,

[

Z=

]

− (1 + K PPID α ) −1 K PPID F f

J ( K PPID , K 0 ) =

G g Z ( K PPID )

+ K 0 Z (K 0 )

2

y0 (k ) is dependent on the control gains used. Write ∆u0 (k )

Z (k ) = Z ( k 0 ) + ∆Z Inserting N u = 1 in equation (4), then the optimization problem will be: J ( K PPID , K 0 ) =

[

]

− (1 + K PPIDα ) −1 K PPID F f + K0Z (K0 )

2 −1

[ [F [F [F

≤ (−(1 + K PPIDα ) K PPID F f −1

+ − (1 + K PPIDα ) K PPID −1

≤ − (1 + K PPIDα ) K PPID −1

f

f

+ − (1 + K PPIDα ) K PPID

] G ]∆Z G ]− K G ] ∆Z

Gg − K 0 ) Z ( K 0 ) g

2

0

g

f

2

g

2

Z (K0 )

2

[

]

Gg + K 0

2

2

[

This is found as − (1 + K PPIDα ) −1 K PPID F f

]

Gg = K 0

its assumed that it is possible to find suitable gain K PPID close to K 0 so that ∆Z 2 is suitably small. The solution for K PPID can be found in terms of K 0 as:

[

K 0 = (1 + K PPIDα ) −1 K PPID F f

Gg

]

10)

→ K 0 = (1 + K PPIDα ) −1 K PPID S 0 K 0 (1 + K PPIDα ) = K PPID S 0

(11)

→ K PPID ( S 0 − αK 0 ) = K 0

[

Where S 0 = F f

Gg

]

A unique solution to equation (10) always exists and takes the form:

[

]

−1

2

Where K * denotes the PID controller parameters as target and K nn stands for the predicted values by neural network. Parameters also depend on desired closed loop control system characteristics. The controller parameters K * are solution to the following closed loop control optimization: min F ′{ y * − y p (u ,...)}; u = C ( K ,...) K

Thus: A minimum norm solution is sought from: − (1 + K PPIDα ) −1 K PPID F f

w

G g ( Z ( K 0 ) + ∆Z )

K PPID = K 0 ( S 0 − αK 0 )T ( S 0 − αK 0 )( S 0 − αK 0 )T (12) Remark 1: For second order system one level of PID (M=0) is enough to achieve the GPC performance. For higher order systems, M will be selected to find the best approximation to GPC solution.

Where C is the predictive PID controller structure. The network here can be trained either in simulation or in real process. But in presence of model-plant mismatch it's better to train the network with real process data. Initial values for this optimization problem are of great importance. In conventional PID controller augmented with neural networks, Zeigler-Nichols values are used as initial values [17]. Here we use values calculated in section 3 derived using signal matching optimization, as initial values. Architecture used in this section is shown in fig (1). Depending on process dynamics H delayed error signals are needed to capture process dynamics. The neural network is trained by back propagating the error between plant output and neural network output as well as the error in the predicted proportional gain and the target proportional gain. [17] have shown that by considering the neural network structure with one parameter fixed, and determining other parameters directly by neural network as the outputs of the hidden layer greater efficiency will be achieved. Once the neural network is trained there will be no need to YGPC . By use of this method certain degree of model-plant mismatch can be handled since there is no assumption to model of process in and fine tuning is done by real process data. The results demonstrate that for typical industrial applications such as industrial boilers off-line optimizations give satisfactory performance although on-line fine tuning

module using neural networks increases accuracy of approximation in comparison with Generalized Predictive Control. The stability analysis of method is presented in section 4. IV. STABILITY STUDY FOR PROPOSED METHOD To study the stability for proposed method the closed loop transfer function of system is calculated. Writing control law equation (4) in matrix form with set point gives: (13) ∆u ( k ) = − Kˆ α∆u (k ) + F f θ (k ) + G g ∆U (k )

{

}

Where f (i −2 )1 Fi = f (i −1)1

g i −3 Gi = g i −2 g i −1

f (i −2 ) 2 ... f (i −1) 2 ...

f (i )1

g (′i −2 )1 Gi′ = g (′i −1)1 g i′1

...

f (i ) 2

f (i −2 )( na +1) f (i −1)( na +1) f i ( na +1)

g (′i −2 ) 2 ... g (′i −2 ) nb g (′i −1) 2 ... g (′i −1) nb g i′2 ... g in′ b

V. CASE STUDIES In this section, the stability and performance of purposed method applied to industrial boiler will be discussed.

θ (k ) = [− e(k ) − e(k − 1) ... − e(k − na )]T ∆U ( k ) = [∆ (k − 1) ∆ (k − 2) ... ∆ (k − nb )]

T

M

α=

M

Gi

M

Ff =

i =0

Gi′

Fi Gg = i =0

i =0

e(k − i ) = w(k − 1) − y (k − 1)

i = (0,..., nb )

Rewriting control law in compact form and after some straightforward algebra gives: Ac ( z −1 ) y = w* − Bc ( z −1 ) ∆u (14) Where: T A ( z −1 ) = Kˆ F Z Z = 1 z −1 ... z − na c

f

[

G = 1 + Kˆ α *

[

[

y

y

Kˆ G g

]

Z y = 1 z −1 ... z − nb

]

−1

*

Bc ( z ) = G Z u

]

T

using the following procedure: Step 1: initialization 1. Find the system model and calculate the discrete polynomials. 2. Choose the value of prediction horizon, M, and formulate the future set point vectors. Step2:off-line calculation 1. Calculate the MPC gain. 2. Calculate optimal value of PID gains using equation (12). Setp3: On-line calculation 1. Train neural network using actual plant outputs. 2. Calculate the gain matrix Kˆ as the output of neural network. 3. Iterate over M to find the best approximation to GPC. 4. Check the closed loop stability using equation (17). Change M to achieve stability if necessary.

w* = k ref w kref = Kˆ F f

A CARIMA model for system is given by: ~ A( z −1 ) y (k ) = z − d B ( z −1 )∆u ( k − 1) = z − d −1 B( z −1 ) ∆u (k )

(15)

Where d ≥ 1 is the time delay of system and ~ −1 A( z ) = D ( z −1 ) A( z −1 ) = (1 − z −1 ) A( z −1 ) Inserting equation (15) gives in equation (14) the closed loop transfer function of system as: B ( z −1 ) z − d −1 y= w* ~ Ac ( z −1 ) B( z −1 ) z −d −1 + Bc ( z −1 ) A( z −1 ) (16) So the closed loop poles are the roots of characteristic equation: ~ Ac ( z −1 ) B( z −1 ) z − d −1 + Bc ( z −1 ) A( z −1 ) = 0 (17)

Equations (13-17) show that poles of closed loop system, which are the roots of characteristic equation, depend on PID gains. On the other hand, the PID gains are affected by selection of prediction horizon of proposed method (M). The neural Predictive PID controller can be implemented

A. Stability study The effect of M on PID parameters and variation of PID parameters on stability region for two types of systems has been considered; the systems are: • Unstable second order system • Non-minimum phase second order system The results showed that for unstable second order system the regions are not closed and bigger M causes bigger stability regions Fig (2). For non-minimum phase system the stability region is closed and bigger M causes the smaller stability region Fig (3). It should be considered that for non-minimum phase bigger M decreases the possibility of stability. And for unstable system increasing M increases the possibility of stability. B. Performance Study for Industrial boilers In this section loop one and loop two for De Mello Boiler model are considered. The loop one is with fuel as input and throttle pressure as output. And the loop two is control valve as input and mass flow as output. The systems are shown in table 1. GPC and Proposed predictive PID method were used to design controller for each system. For GPC, the prediction horizon of output N=25, the control horizon N u =1, =80 was assumed. The gains of two methods have been shown in table 1. The step response of closed loop system for loop one and loop two of boiler model has been shown in Fig (4). The results show that the method is applicable for industrial boiler with one and two level of PID for loop one and loop two, respectively.

VI. CONCLUSION A predictive PID controller was proposed which has important similar features to the model based predictive control. The controller reduces to the same structure as a PI or PID controller for first and second order systems, respectively. It was shown that optimal values of PID in the first stage can be found using signal matching optimization, then a fine tuning module was introduced which uses neural networks to fine tune the parameters derived from last stage. The proposed method was applied to two benchmark processes to illustrate the stability of proposed method. Performance of proposed method applied to two loops of industrial boilers was investigated showing good performance compared with GPC. One of the main advantages of proposed controller is that it can be used with systems of any order. Simulations reveal that this method shows good performance when applied to a wide range of industrial processes from first and second order ones to high order and non-minimum phase systems. Table1. GPC and predictive PID control design for industrial boiler.

kp 0 -50 -100 -100

5 kp

1

[.1658 .078 .o287 .1487]

-10 -5

Boiler model; transfer function between flow rate and Control Valve.

Back Propagation

Target Proportional gain −1

K

z −1

nn

p

ynn (k + 1)

-

K *P

-

-

+ y * ( k + 1) GPC

−1

Ki

Kd

u(k)

Process

0

kd

y(k+1)

Fig (1): Structure of online fine tuning module.

5

10

(b) Fig (2): The stability regions for unstable second order system over M, a) kd=constant, b) ki=constant.

2 1.5 1 0.5 kp 0 -0.5 -1 -1.5

Stability Regions M =0, 2, 10, 25

0

0.05 ki (a)

Kp

Predictive PID

10

0

0

Boiler model; transfer function between Throttle pressure and fuel/air.

50

Stability M=0, 5, 25

-5

description

0 ki

10

z 2 − 1.96 z + 0.96 z 2 − 1.93z + 0.93

[-.029 .1004 .0037]

-50

(a)

− 0.007 z + 0.007 z 2 − 1.98 z + 0.98

Predictive PID gains

z

50

g1 ( z ) =

[.374 -.129 .0005 -.11 .069] [.128 .25 .0005]

z

Stability Regions M=2, 25

g2 ( z) =

system

Level of PID (M) GPC gains

100

0.1

4 Stability Region M=0, 2, 10

3 2

REFERENCES

kp 1

[1]

0 [2]

-1 -2

[3]

0

2

4 kd

6

8

[4]

(b)

[5]

Fig (3): Stability regions for non-minimum phase second order system, a) kd=constant, b) ki=constant.

[6]

[7] [8]

T P 0.1

[9]

[10]

Proposed Method

0.06

[11]

GPC Method

0.02

[12] [13]

-0.02 0.8

1.2

1.6 Time

2 4 x 10

[14]

(a) [15]

CV 0.1

[16]

Propose

0.06

[18]

GPC

0.0 -0.02 0.8

[17]

1.2

1.6 Time 2

4

x 10

(b) Fig (4): Performance comparison of GPC and proposed Predictive PID for industrial boiler, a) loop one, b) loop two.

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