Nonlinear Multivariable Decoupling PID ControlUsing Neural Networks

Lianfei Zhai , Tianyou Chai and Yujing ShiResearch Center of Automation

Northeastern UniversityShenyang, 110004, P. R. ChinaE-mail: zlffly@yahoo.com.cn

Abstract-A nonlinear multivariable decoupling PIDcontroller is derived from generalized minimum variancecontrol law, which consists of a PID controller with decouplingdesign and a feedforward compensator for the unmodeleddynamics. Then a nonlinear multivariable decoupling PIDcontrol algorithm using neural networks is proposed. By usingneural networks online estimating and compensating theunmodeld dynamics, the proposed algorithm has the adaptivecapability to the variations of both parameters and structure ofthe process. Under some assumptions, it is proved that all thesignals in the closed-loop system are globally bounded and thetracking error can be made less than any specified constantover a compact set by properly choosing the structures andparameters of neural networks. Simulation results show theeffectiveness and robustness of the proposed algorithm.

I. INTRODUCTION

In many complex industrial processes, the couplingamong control loops often invalidates conventional singleloop controllers. How to achieve decoupling control of suchprocesses has become a considerable topic in the field ofcontrol engineering. Decoupling control was initiallydeveloped for deterministic linear systems. Typicalapproaches of decoupling control for deterministic linearsystems include [2], [4], [10] and [11]. For linear systemswith unknown parameters, many adaptive decouplingcontrol algorithms have been proposed and developed [9],

For decoupling control problems of the nonlinearmultivariable processes, the development and controlapplications of the neural networks[61 brought chances todeal with them. In [8], neural networks were used as acompensator and the sufficient and necessary condition ofdecoupling was given. Combing multivariable generalizedpredictive control with neural networks, a novel robustdecoupling method was proposed in paper [7]. In [16], astructure of nonlinear decoupling control system waspresented, which combined the generalized minimumvariance control based decoupling design with theunmodeled dynamics online feedforward compensating byneural networks. Based on [16], a switching mechanism wasemployed to deal with the complex industrial processes withmultiple operating points in [15].

Although the development of decoupling control theory is

considerable, the number of applications of these controllersin the process control is still discouraging. One main reasonis the domination of PID controllers because of their simplestructures. As a result, [1] derived SISO self-tuning PIDcontroller by orientating the self-tuning control law of thegeneralized minimum variance to have a PID-like structure,and later [12] and [13] extended it into multivariable case.In [17], a self-tuning feedforward PID controller waspresented to deal with linear multivariable processes withnon-minimum phase. Although these control algorithmshave PID structures, they are not real PID controllers andcannot be easily implemented on the standard modules ofDCS/PLC.

In this paper, consulting the reference [17], a nonlinearmultivariable decoupling PID controller is derived from thecontroller proposed by [16] to deal with the nonlinearmultivariable processes. Such controller consists of anumerical multivariable PID controller with decouplingdesign and a feedforward compensator for the unmodeleddynamics. Then a control algorithm using neural networks issummarized. Through estimating and compensating theunmodeld dynamics online by neural networks, theproposed algorithm has the adaptive capability to thevariations of both parameters and structure of the process.Then global stability and convergence analysis are givenunder some assumptions. Simulation results show theeffectiveness and robustness of the proposed algorithm.Since the PID controller is the standard form that is widelyused in industrial process control, making it more favorableto be used in industrial.

II. NONLINEAR MULTIVARIABLE DECOUPLING PID CONTROLUSING NEURAL NETWORKS

A. Statement ofthe ProblemThe process to be controlled is assumed to be an n-input

and n-output deterministic nonlinear system described bythe following discrete-time model:

y(t+k) = f[y(t),.* ,y(t- n +±1),u(t),...,u(t- nU)] (1)where YU[Y y]TERn and [Ul[ ...Un,u]TeRn are

system input and output vectors, respectively; n, and nu

0-7803-9422-4/05/$20.00 C2005 IEEE1843

are the system orders; k is the system time delay which isknown; m represents the number of multiple operatingpoints; f() = [f1(.), ,fs (.)]T is a vector-valued nonlinearfunction that is assumed unknown and continuouslydifferentiable.Assume that equation (1) can be divided into an

approximated linear model (with lower order) for thecontroller design plus a higher order nonlinear term asexpressed in the following:

y(t +k) = -A(z )(t) +B(z1)u(t) +B(z-')u(t) +vx(t)] (2)

where A (z. 1), BR(z-') andB=(z-) are polynomial matrices interms of the unit back shift operator z-' with A(z-1) andR(z-§1) being diagonal, B(z-1) is a polynomial matrix withzero diagonal elements, let A(z-1) =I+Azk(z1) andB(z') = B(z-) + B(z-') having the following format

A(z-I ) = I+ A z-i +...*+An Z_n,B(z-')=BO +Blz1 ++Bnb Z-nb

where I is the identitymatrix, na and nb are the ordersof A,(z-1) and B(z-1) respectively. The nonlinear termrepresents the unmodeled dynamics

v[x(t)] = f(E) + A(z- )y(t) - B(z 1 )u(t) -B (z 1 )u(t) (3)Qualitatively, the objective of decoupling control is to

determine an input control law so that all the signals in theclosed-loop system remain bounded, whilst the outputvector y is confmed in some pre-specified range, and theinfluence of the couples is suppressed as small as possible.

B. Multivariable PID ControllerThe numerical multivariable PID controller can be written

into the following equation

u(t) = Kpe(t) + K1, e(i) + KD[e(t) - e(t -1)] (4)i=O

where e(t) = w(t) - y(t), w(t) is the reference input vector,Kp is the proportional gain of PID controller, K, andKDdenote the integral gain and derivative gain respectively. Inaddition, a velocity-type form of the PID controller isobtained

u(t) = u(t - 1) + Kp [e(t) - e(t - 1)]+ K,e(t) + KD [e(t) - 2e(t - 1) + e(t - 2)]

Above form of PD controller has relatively independentgains, which is usually designed as a standard PID moduleof DCS/PLC and widely used in industrial process control.Equation can be written into the following form

H(z-' )u(t) = [K,, + K, + KD ]e(t) (6)+ [-KP - 2KD ]e(t - 1) + KDe(t - 1)

where H(z )=(1-z I.

C. Nonlinear Multivariable Decoupling PIDController

Introduce the following performance index[161J =11 P(z- )y(t + k) - R(z-' )w(t) (7)

+ S(z-7 )u(t) + Q(z- )u(t) + K(z- )v[x(t)] 112where P(z-1), Q(z 1), R(z 1) and K(z ') are the diagonalweighting polynomial matrices about z-' , S(z-') is aweighting polynomial matrix with zero diagonal elements.Introduce Diophantine equation

P(z'- ) = F(z l )A(z l ) +z kG(z-l) (8)where G(z )=G0 +Gz-I+G2Gz2. In order to get the PIDcontrol law, it is required that the order of G(z-1) isng <2 . Since ng =na -k , if the order of A(z-§) isna ' (2 + k), the least rank resolution of the Diophantineequation can be obtained, otherwise the general solution ofDiophantine equation will be received. It can be proved thatthe optimal controller that minimizes (7) is described by[F(z ')B(z') + Q(z1)]u(t)= R(z' )w(t) - G(z- )y(t) - [F(z- )B (z- ) + S(z' )]u(t) (9)- [F(z-') + K(z- )]v[x(t)]

Let F(z 1)B(z-')+Q(z-1)=2A-H(z-1) and R(z-)=G(z'),a nonlinear multivariable decoupling PID controller can beobtainedH(z')u(t) = XG(z- )e(t) - X[F(zJ' )B(z-' ) + S(z-')]u(t) (10)

- X[F(z-') + K(z-1)]v[x(t)]where A=diag{21,..., An} is a constant diagonal matrix.Equation (10) can be transform into the following formH(z ')u(t) = AG(z- )e(t) - H(z-1 )u(t) - K(z-')v[x(t)] (11)

whereH(z-') = [F(z' )B (z-1) + S(z-')]K(z-1) = X[F(z1 )+ K(z-')]

Substituting (11) into (2) yields[H(Z1AZ-1 + -kXff(Z-1)(-1 )]y(t + k)

= XB(z )G(z ')wQ) +[H(z ')B(z 1) B(z1 )H(z- )]u(t) (12)+ [H(z-') - B(z- )K(z- )]v[x(t)]Equation (2) can be regarded as representation of theclosed-loop system. From equation (12), it can be seen thatif matrices X , H(z ) and K(z') are chosen asfollows:

H(z1 ) = B(z )K(z') (13)H(Z 1)B(z-1) =B(z-1)H(z1) (14)

det{H(z-1)A(z1)+ Z -k28..(Zl1 )G(z1)} .0, IzI > 1 (15)

then the closed-loop system will be stable, the decouplingcontrol effect and the tracking errors can be eliminated.The nonlinear multivariable decoupling PID controller

(11) can be expressed byH(z- )u(t) = AGoe(t) + kG1e(t -1) + XG2e(t - 2) (16)

- H(z- )u(t) - k(Z- )v[x(t)]According to PID controller equation (6), the proportional

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gain Kp, the integral gain K, and the derivative gain KDin the nonlinear multivariable decoupling PID controller canbe selected as

Kp =-A[2G2 +G1] (17)K, =A[GO +G1 +G2] (18)KD= AG2 (19)

D. Estimation of the Unmodeled Dynamics UsingNeural NetworksAs for the estimation of the unmodeled dynamics v[x(t)],

a Back-Propagation (BP) neural network is employed, asshown in Fig. 1, which has one hidden layer and the linearoutput. It has n(ny + nu + 1) inputs and n outputs

v(t) = NN[x(t), W(t)] (20)

where NN denotes the format of the neural network, x(t)is the input of neural network and W(t) is the connectionweighting matrix. The cost function E is defined as:

I P

E = -E {v[x(j)] v(j)} 2 (21)2P j=1

where P is the number of input and output data.

E.Nonlinear Mutiaial Decoplin Contro

u V

.. . .

Irj- e Te _ i

AlgorithmThe nonlinear multivariable decoupling PID control

algorithm can be divided into two parts: 1) offline controllerdesign and 2) adaptive compensating algorithm for theunmodeled dynamics. The algorithm of offline controllerdesign can be summarized as follows:Step 1. Choose the system orders na and nbStep2. If the process is known, obtain A(z-1), B(z-1) by using

Taylor's expansion at its equilibrium; otherwise theyare substituted by their estimnations A(z1) B(z1)

Step3. Properly choose X, k(z-1) and H-(z-1) so as tosatisfy (13)-(15);

Step4. Establish neural network NN to approximnate theunmodeled dynamics v[x(t)] , train it in batch.

The decoupling control algorithm with an adaptivecompensator for the unmodeled dynamics can besummarized as follows:Step1. Read input-output data to construct vector x(t);Step2. Get the estimation v(t) by using the neural network

NN, where u(t) is substituted by u(t-l);Step3. Calculate the control input u(t) using equation (11);Step4. Implement u(t) on the process (1), and then get the

new output y(t + 1) of the process;Step5. Obtain the tutorial signal from equation (3);Step6. Train the neural networks NN once;Step7. Let t = t +1, and return to step 1.

III. GLOBAL STABILITY AND CONVERGENCE ANALYSIS

In the proposed algorithm, the unmodeled dynamicsv[x(t)] is substituted by v(t), the real control inputsimplemented on the plant is

H(z 1)u(t) = AG(z§ )e(t) - H(z- )u(t) - K(z-' );(t) (22)There are some assumptions made:Assumption 1. w(t) is a bounded deterministic sequence.Assumption 2. v[x(t)] is globally bounded, jvIx(t)0<4A,

where the boundary A is known.Assumption 3. The choices of A, P(z-1) and S(z ') are such

that det{H(z§ )A(z-' ) + z-kB(z-I )AG(z-' )} 0, for jZj >1.Assumption 4. The neural network identification obtain

the estimation v(t) such that the cost function is madeacceptable small, regardless of the estimation algorithmsused.Lemma. If the nonlinear decoupling PID controller (22) is

applied to the system (2), the input-output dynamics aredescribed by[H(z ')A(z') + Zkj(Z-1)XG(z-1)]y(t + k) (23)= B(z' )XG(z-')w(t) + H(z- )v[x(t)] -.i(z- )k(z-' v()[A(z- )H(z-1 ) + z-k G(z-l )B(z-' )]u(t) 24= A(z 1)G(z-1)w(t) _zZkXG(z ')v[x(t)]-A(z' )k(z-' )v(t) (2)and

det{H(z- )A(z-') + z- B(z- )G(z- )}= det{A(z1 )H(z-1) + z-kxG(z1 )B(z' )}

(25)

where H(z- ) = H(z-') +H(z-'),B(z-)H(z1 ) =H(z-' )B(z-'), det B(z') =det B(z-') .O

Proof. Left-multiplying equation (4) by [RG(z-')], left-multiplying equation (22) by A(z-), according to the inter-changeablity of matrices, equation (24) can be got. LetH(z- ) = H(z-')+ H(z-'). Introduce matrices B(z-') andH(z- ) satisfying

B(z- )H(z-')= H(z-')B(z-'), det{B(z1)} = det{B(z')}Left-multiplying equation (22) by B(z-') and combining

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equation (4), equation (23) can be obtained.Introduce matrices A(z-1) and R(z-') satisfying

A(z-' )B(z- ) = B(z-' )A(z 1), det{B(z1 )} = det{B(z' )}z

It can be obtained that det{B(z1 )} = det{B(z- )} = det{B(z1 )},then

det{H(z-1)A(z-§) + zikB(z-l)XG(z-l)}= detO(z-)B(z-)Br(z-)A(z-') +z-kh(z-')XG(z-1)}= det{B(z-)[H(z-1)ff(z- )A(z-) + Z-kXG(z-)]}= det (z-')[H(z-')B[(z-')A(z-)B(z-1) + z-kXG(z-l)B(z-')]-1 (z-')}= det I(z-)A(zl) +z-kXG(z-1)(z1)}= det{[H(z-l)+zzkXG(z-l)A-l(zl)B(z-l)A(z-l)}= det{H(z-')A(z1')+ z-kXG(z-')B(z-1)} . o

Theorem. For the process (2) with the control input (22),together with the parameters of the controller selected by(13)-(15), all the signals in the closed-loop switching systemdescribed above are bounded, and the tracking error can bemade less than any specified constant 6 over a compact setby properly choosing the structures and parameters ofneuralnetworks, that is iily(t + k) - w(t)I <c.

Proof. The unmodeled dynamics estimation errorproduced by neural networks can be described by

£(t) = V[X(t)] - (t) (26)Substitute (26) into (23) and (24) respectively, the followingequations can be obtained[A(z-')H(z- ) + z-kXG(z-')B(z- )]u(t)= MA(z- )G(z-1)w(t) _ [z-kXG(z-l) + A(z-' )k(z-1)]v[x(t)] (27)± A(z-')K-(z1')s(t)[H(z-')A(z-' ) + z-kj(z-1 )XG(z- )]y(t + k)= B(§'( )XG(z- )w(t) + [H(z§) -B(z-' )K(z 1)]v[x(t)] (28)+ (z1)K(z -1)(t)From (25), (27), (28) and Assumption 1-3, there existconstants C1, C2, C3 and C4 satisfying

Jy(t + k)I < C1 + C2 maxCr(T)j (29)Iu(t)l < C3 + C4 * maXl((T)l

From the structure of x(t), it can be got thatIx(t)I ' C5 + C6 maXI|e(T)l0OS,<t

(30)

(31)Since the neural networks are regarded as universal

approximators[5], the estimation error Iv[x(t)]-v(t)l can bemade less than any specified 4 over a compact set byproperly choosing their structures and parameters, that is

k£(t)I < 4 (32)According to equations (31) and (32), it can be get that

Ix(t) < C7 + C8 .4 < C9 (33)where, C7, C8 and Cg constants.

Implement the control input (22) on the process (2), thenthe cl9sed-loop of the controlled process is expressed by

[H(z-I )A(z -1 kX z )G(z-l)]y(t + k)= XBf - )G(z-')w(t)+ [H(z-' )B(z- ) - B(z- )H(z- )]u(t) (34)+ [H(z -1) -B(z-1 )K(z -)]v[x(t)] + B(z-1 )K(z -')c(t)From equations (13)-(15) and (34), the state tracking errorof the system is

e = imjy(t + k) - w(t)l = fim|B(1)K(l)c(t)* < C10 *4 < a

where, C1o > 0 is a constant.o

IV. SIMULATIONS

To illustrate the effectiveness and robustness of theproposed decoupling control algorithm using neuralnetworks, a simulation study is described in this section.

Example 1. The process I is described byy1(t + 1) = 0.4sin[y1(t)]/[0.5 + y2(t - 1)]

- 0.45y1(t - 1) + 1. lul (t) + 0.8u2 (t) + 41 (t + 1)y2(t + 1) = cos[y22(t)] - 0.3y2(t -1) + 1.25u2(t) + O.lu2(t -1) (36)

+ 0.29cos[u2(t)] + 0.32u (t - 1) + 42(t + 1)where, R(t+l) and 42(t +1) are random disturbances.One equilibrium of the process to be controlled is Y1 = 0 ,

y2 =0.82272,u =0,u2 =0 , select the orders of A(z-'),B(z-') as na =2, nb =1, then the model for controllerdesign can be obtained by Taylor's expansion.

A(z-') I[ - 0.3024z-' + 0.45z-2 0L 1 + 1.0306z-' + 0.3z-

[z 0 1.25 +0.lz- ] B(Z) = [032z- 0.8]Let P(z-')=I, solve the Diophantine equation. Properlychoose X, K(z-') and H(z-') so as to satisfy (13)-(15).The control input is

H(z-1)u(t) = G(z-')e(t) - H(z-')u(t) - k(z-')i(t) (37)where H(z-1)=[l 1 0- ]X

G(z 1- 0.2854 +0.4247z1 06 2( )L ° ~ 0.3861 + 0. 1124z-l

H(z1)-[I .r7+ Z0 0.7273+0.7273z( -0.237+0.237zl o0K(z-) =-0.8579 +0.8579z- 0 0 1Z- L ° ~ - 0.2973 + 0.2973z- -

A group of BP neural network NN1 is used, where onlyone hidden layer is employed with 20 nodes, the learningrate = 5; learning rate increase = 1.05; learning rate decrease= 0.7; momentum constant = 0.05; sum-squared error goal =0.02; maximum error ratio = 1.64 are used in on-linetraining. The simulation result is shown in Fig. 2 and Fig.3. It can be seen that the performance of proposeddecoupling controller is very good in the most oftime.Example 2. For comparison, under the same simulation

conditions and with the same parameter selections, the

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outputs of the same system but controlled by the typical PIDcontroller is also given in Fig. 4 and Fig. 5, where the PIDcontroller has neither the decoupling compensator nor theunmodeled dynamics compensator, i.e. H(z-1) = K(z 1) = 0in (37). Obviously, the algorithm proposed in this paper issuperior to the typical one.

_ 40 X A.30Y V---,__

D ILO O10150ot -20()0 250 300 3S0 400

40

20

10 AO 5O 100 150 200 250 300 350 400

time

Fig. 2. Performance of the Proposed Decoupling PID Controller

4:;20

0

-200 50 100 150

Example 3. In order to study the robustness of thecontroller, the parameters of the process are changed but thecontroller is not changed, the simulation experiments aredescribed as follow. At t = 120, the process is changedfrom , to _2 which is described byy (t + 1) = 0.6 sin[y1(t)]/[O.6 + y2(t - 1)] (38

- 0.41y1(t -1) +1.3u1(t) + 0.5U2(t) + 41(t + 1)y2(t + 1) = cos[y22(t)] - 0.3y2(t - 1) + 1.25u2(t) + 0.3u2(t - 1)(39)

+ 0.3 Icos[u2(t)] + 0.2u1 (t - 1) + 42 (t + 1)Control the process ,2 by using the control input (37). Thesimulation result is shown in Fig. 6 and Fig. 7. It can be seenfrom Fig. 6, although the parameters of the process modelare varied, the system outputs can track the reference inputs.It shows that the proposed algorithm has the strongrobustness for the parameters variation.

540

40

30

~2010

0 50 100 150 200 250 300 350 400-

1200 2S0R -,0o 350 400

30 1

G:20-

10-10 0 100 50 100 ? 50 200 2S0O 300 3S0 400

time

Fig. 3. Corresponding Control Inputs of Fig. 2

Ou- 00 1 1

;;20

0 50 100 150 200 250 300 350 400

Fig. 4. Performance of the Typical PD Controller

0 50 1 6010 0 200 250 300 350 400

40 i

30

20

-100

SO 100 ISO 200 250 300 350 400time

Fig. 6. Performance ofthe Proposed PID Controller for the Process withAbrupt Varied Parameters

60

40

-20

40

30

0 5o 100 150 200 250 300 350 400

SO

40

-20 0 50 100 150 200 250 300. 350O 400time

Fig. 7. Corresponding Control Inputs of Fig. 6

Example 4. To illustrate the adaptive capacity of theproposed algorithm to the structural variation of the process,a similar simulation is studied. At t = 120, the structure ofthe process El is changed into 3 which is described byyl(t + 1) = 0.4sin[yQ(t)]/[0.5 + y2(t - 1)]- 0.45 sin[y1(t - 1)]

+ 1. lul (t) + 0.8u2(t) + 41(t +1) (i40y2(t + 1) = sin[y22(t)] - 0.3y2(t - 1) + 1.25u2(t) + O.1u2(t - 1) (41)

+ 0.29cos[u,2(t)] + 0.32u1(t - 1) + 42 (t + 1)Implement the control input of (37) on the process 3 . Thesimulation result is shown in Fig. 8 and Fig. 9. From Fig. 8,it can be seen that the system outputs still track the referenceinputs, through the structure of the process model is vri"ed.It shows that the proposed algorithm also has the strong

1847

Fig. 5. Corresponding Control Inputs of Fig, 4

,_o V- --- oi,--

go, .

-4..~ 3

robustness for the structure variation.

40L'[7

5200

40

'I 3

1 20

10

process control. In addition, since the parameters of suchPID controller are chosen from the generalized minimumvariance control law, it is easier for field engineers andprocess operators to relate the parameters settings, as well asthe control input saturation can be avoided.

50 100 tS0 200" 250 300 3S0 400

100 ISO 200 250 300 350 400time

Fig. 8. Perfonnance ofthe Proposed PID Controller for the Process withAbrupt Varied Structure

Fig. 9. Conresponding Control Inputs of Fig. 8

From Fig. 6 and Fig. 8, it can be. shown that the systemoutputs can track the reference inputs although the structureand the parameters of the process model are changed.Because the unmodeld dynamics is estimated andcompensated online by neural networks, the proposednonlinear multivariable decoupling PID control algorithmusing neural network has strong robust performance for thevariations ofboth parameters and structure of the process.

V. CONCLUSIONS

Combining a PID controller with decoupling design and afeedforward compensator for the unmodeled dynamics, anonlinear multivariable decoupling PID controller is derived.Then a nonlinear multivariable decoupling PID algorithmusing neural networks is proposed. Through estimating andcompensating the unmodeld dynamics online by neuralnetworks, the proposed algorithm has the adaptive capabilityto the variations of both parameters and structure of theprocess. When the unmodeled dynamics is globally bounded,it is proved that all signals in the closed-loop system areglobally bounded and the tracking error can be made lessthan any specified constant over a compact set by properlychoosing the structures and parameters of neural networks.The simulation results show the effectiveness and strongrobustness of the proposed algorithm. Because the PIDcontroller can be realized on standard DCS/PLC modules,the algorithm is more favorable to be put into industrial

ACKNOWLEDGMENT

This work was supported in part by the Chinese NationalBasic Research Program (2002CB312201) and the ChineseNational Hi-Tech Development Program (2004AA412030).

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