NEURAL CONTROL OF NONLINEAR SYSTEMS: A REFERENCE GOVERNOR APPROACH

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1 NEURAL CONTROL OF NONLINEAR SYSTEMS: A REFERENCE GOVERNOR APPROACH L. Schnitman Institto Tecnológico e Aeronática S.J. os Campos, SP Brazil leizer@ele.ita.cta.br J.A.M. Felippe e Soza Universiae a Beira Interior Covilhã Portgal felippe@emnet.bi.pt T. Yoneyama Institto Tecnológico e Aeronática S.J. os Campos, SP Brazil taashi@ele.ita.cta.br ABSTRACT This wor presents the esign an implementation of a controller for nonlinear, nstable an constraine systems. For instance, a magnetic levitation system is selecte to highlights the controller properties especially with respect to stability an constraints satisfaction. The control action is base on the reference governor (RG) approach that ses a Lyapnov's concepts of energy to prevent constraints violation both on the state an on the maniplate variable even ring large changes in the reference signal. To esign the RG an inner loop controller is propose. The RG receives the system states an the esire reference to compte an spply the reference signal to the inner loop controller. This procere garantees the stability while avoiing constraints violation. The paper escribes in fll etail the inner loop controller an RG esign, bt in spite of the sccessfl reslts, the RG approach is not able to treat ncertainties. Here the athors propose to replace the whole linearization an control action for a single Artificial Neral Networ (ANN). The ANN is traine sing the system states an esire reference as inpt an the noal control action as the otpt. Training ata is generate thoght simlation base on the action of the esigne inner loop controller with the RG. The major objective in the se of the ANN is may also be able to treat ncertainties an allows straightforwar implementation of training techniqes to frther provie aaptation capabilities. KEY WORDS Constraine control, Nonlinear control, Neral control INTRODUCTION The control of nonlinear systems has attracte wiesprea attention in the recent years (e.g. [8], [9], [], [4], [5], [6]). Classical linearization methos wors well when the moel is accrate, the reference signals are well conitione an have low amplite. However, this may not be the case for inpts sch as large steps or when ncertainties are present. The problem of the larger inpt steps can be treate sing the reference governor (RG) approach [] - [7], [0], [], [3], which garantees the constraint satisfaction for a general class of inpt commans while stability is assre by a Lyapnov base esign techniqe. Bt the problem of ncertainties may still be critical. This paper proposes to replace the inner loop controller as well as the RG by a single ANN, which is traine with ata pairs generate from the simlation of the previos controller actions. Thereafter the neral controller is able to on-line aaptation by means of

2 learning techniqes an may be robst against a class of ncertainties. Section presents the inner loop an RG esign theory. In Section 3 the stie plant is presente. Section 4 is evote to apply the controller theory to the magnetic levitation system an the esire control action is shown. Section 5 presents the propose neral control an its simlation reslts. The conclsions are presente in Section 6. Controller esign theory Let the nonlinear system in the form: & f ( ) + g( ) y h( ) Using the eact feebac linearization theory [9], [4], [5], [6], if Eqation () can be written as [ α( ) ] () & A + Bβ ( ) () where - A an B are n n matri - The pair (A,B) is controlable - α: R n R p is efine in a omain D R n - β: R n R p p is efine in a omain D R n - β is nonsinglar D an β - is its inverse then one can se the following control law α ( ) + β ( ) v (3) Using Eqations () an (3) we can obtain the linear epression of the form & A + Bv (4) For the linear system we simply se a state feebac for obtaining pole placement. The bloc iagram of the controller scheme is shown in Figre. r v v r -. α() + β().v Nonlinear Plant α() K β () Figre : Controller scheme 3 A magnetic levitation system Consier a magnetic levitation system as shown in Figre. y i( F m g ( Figre : Magnetic levitation system This constraine nonlinear system can be escribe by i & g (5) m where m is the mass of the ball, > 0 is a constant parameter, g is the gravitational acceleration an i F (6) is the electromagnet force controlle by i( Note that the system is clearly open-loop nstable. Moreover, classical controllers that o not tae into accont the constraints, an the satration crrent i may fail in the sense of letting the ball fall or having the ball magnetically attache to the base of the coil. If we efine, & an i, then the system can be rewritten as & (7) & g m y h( ) [ 0] An the objective is to esign the controller in orer to garantee the stability while avoiing constraints violation. 4 Practical controller esign 4. Inpt feebac linearization An inner loop controller shol be responsible to provie local stability, withot consiering the constraints. Here we se the eact feebac linearization theory in orer to obtain an aeqate local performance. To linearize the system sing the propose metho, we mst tae the erivatives of the otpt y ntil we fin the inpt signal, i.e.: y& & && y & g m (8) Let & y v ths,

3 m ( g v) (9) Replacing in Eqation (9) in Eqation (7) one obtains & & v (0) which is a linear system. Comparing Eqations (9) an (3) we obtain m m α ( ) g an β ( ) () 4. State feebac With the linear system, se a state feebac to pole placement. Let ς A then, Eqation (0) becomes an it can be rewritten as an v r - () & (3) & r ς A & 0 0 (4) + r & ς A which is clearly linear an stable for A an ς > 0. From Eqation (9) one obtains the final control law of the form m g + + ς (5) A In orer to avoi the steay state error, let A an for the sae of the simplicity let also ς. Note that A is relate to nll steay state error with respect to the reference signal, ς is relate to the amping coefficient while the pair A, ς >0 garantee the stability for the close loop system. 4.3 The reference governor esign The controller escribe in the last two sbsections wors well when the constraints are not violate. However, when the constraints are violate the controller may fail. The main iea of the RG is to garantee the constraint satisfaction while the local controller assres stability. It is assme that there is a previos controller esigne so as to provie an aeqate performance aron the efine set points bt withot consiering the constraints. To avoi the constraint violation, the inpt to this close loop system shol be spplie throgh the RG. The RG receives the esire reference signal r ( an the state variables an then tracs the reference inpt as close as possible to the esire r ( bt sbject to constraint satisfaction while garantee that r ( when t. Figre 3 illstrates the RG scheme. r Reference Governor r v v r -. α() + β().v Nonlinear Plant α() K β() Figre 3: Reference governor scheme The first step in the esign of the RG is the selection of a fnction in the sense of energy. Ths, let an note that V r ( r ) + 0 (6) & ς 0. (7) V r The constraints are represente as q (r) V r () s.t. ( - 0 q (r) V r () s.t. ( - 0 i i( i y (8) Therefore, for the propose case, one gets q (r) (r - ) (9) q (r) (r ) The violation of the constraints can now be etecte by efining C V r () - q (0) ( r ) + ( r C V r () - q ) ( r ) + ( r ) An sing the following criteria: a) For positive step inpts, the constraint is violate if C < 0, i.e. > r ( ) b) For negative step inpts, the constraint is violate if C < 0, i.e. ()

4 < r () ( ) c m g + (30) Therefore, one can se the following control law to avoi constraints violation: a) For positive step inpts, [r (, r ] b) For positive step inpts, [r (, r ] c) Compte Eqation (5) an apply i(. 4.4 Crrent satration analysis Consier now the crrent i( in the case of the initial conition with nll spee 0, ths Eqations () an () becomes: a) For positive step inpts, the constraint is violate if C < 0, i.e. > r + (3) b) For negative step inpts, the constraint is violate if C < 0, i.e. + < r (4) 4.4. C violation With 0, which is the worst case with respect to the imm positive amplite of the inpt step (i.e. initial an r ), C is violate if ( ) + r t > (5) an then the Eqation (5) becomes: m g (6) c Consiering, at the eqilibrim with, the reqire control effort is an m g (7) c t ( 8) Consiering, at the eqilibrim with, the reqire control effort is m g (3) an c t (3) Therefore, the reqire crrent can be always spplie withot satration. 4.5 Simlation reslts For the magnetic levitation system, let consier the following parameters m 0.05; g 9.8; (33) [, ] [,3] Initial conitions [, ] i [., 0] Desire references [.,.9] The inner loop controller propose in Eqation (5) garantees the tracing of the reference signal bt it oes not tae into accont the constraints satisfaction. Figre 4 shows the esire reference r ( an the evoltion of which garantee that the constraints are satisfye as shown in Figre 5. These inpt/otpt ata pairs represents the esire control action an they are se to train the propose Artificial Neral Networ (ANN) Simlation time (s) Figre 4: Reference evoltion Therefore, the reqire crrent can be always spplie withot satration C violation Withot RG W ith RG With 0, which is the worst case with respect to the imm negative amplite of the inpt step (i.e. initial an r ), C is violate if ( ) + r t < (9) an then the Eqation (5) becomes: ( Simlation time (s) Figre 5: Simlation reslts

5 5 Neral control The propose ANN mst replace the whole control action shown in Figre 3 for a single ANN as shown in Figre 6. i( r ( Neral Controller Nonlinear Plant Figre 6: Neral controller ( Looing for robst operation in close-loop systems, before efining an training the neral strctre, it is interesting to analyze the states an for ifferent r an generate a larger grop of training ata pairs. Figres 7 an 8 show the simlation reslts sing the noal inner controller an the noal RG that characterize the esire soltion to be learne for the neral controller. In the first figre, r changes from r to r an bac to r, where r an r [,.4,.8,.,.6] for simlations s, s,..., s5 respectively. In the secon one, r changes from r to r an bac to r, where r [.4,.8,.,.6, 3] an r, for simlation s6, s7,..., s0 respectively. For both Figres 7 an 8 the larger ots inicate the sitations when the RG action was necessary to avoi constraints violation. otpt layer. The ANN is traine ner MATLAB simlations sing the Levenberg Marqart procere. The trajectories from Figre 7 were se. Each trajectory represents 30s of simlation an the ata pairs to ANN training are obtaine with a sample time of 0.s. The weights an bias are initialize ranomly an the ANN is traine for 500 epochs. The following tables present the ANN parameters after being trainne w b w b Finally, Figre 9 presents the simlation reslts with the conventional RG compare with the neral controller reslts s s s3 s4 s Conventional RG Neral controller X ( X Figre 7: Desire soltion Simlation time (s) X s6 s7 s X s8 Figre 8: Desire soltion 5. Neral strctre an training Practical simlation mae s to select an ANN with one hien layer an 6 hien nerons (tanh) an a linear s9 Figre 9: Neral controller reslts 6 CONCLUSIONS The control of constraine nonlinear systems may be tacle by the RG approach, which se two istinct blocs to compte separately the inner-loop controller an the RG control actions. Here, a single ANN controller is propose to replace both an sccessfl reslts are verifie with a large nmber of simlations. The robstness of the ANN in close-loop is verifie only when the training ata pairs consier ifferent trajectories as shown in Figres 7 an 8. The sccessfl reslts shows that for a noal cases a single ANN is able to replace both the esigne inner-loop controller

6 an the RG. If real cases are consiere an ncertainties cannot be neglecte, then with the se of the ANN we may also be able to treat ncertainties by proviing aaptation capabilities. ACKNOWLEDGEMENT The athors are inebte to FAPESP (Fnação e Amparo à Pesqisa o Estao e São Palo, Brazil), for spport ner grant 98/ References: [] A. Bempora, Control of constraine nonlinear systems via reference management. Proc. of the American Control Conference, 5 (997), pp [], Reference governor for constraine nonlinear systems. IEEE Trans. on Atomatic Control, 43 (998), pp [3] A. Bempora, A. Casavola an E. Mosca, Nonlinear control of constraine linear systems via preictive reference management. IEEE Trans, on Atomatic Control, 4 (997) pp [4] F. Blanchini an S. Miani, Constant signal tracing for state constraine systems. Proc. of the 36 th IEEE Conference on Decision an Control (997) pp [5] A. Casavola an E.Mosca, Reference governor for constraine ncertain linear Systems sbject to bone inpt istrbances. Proc. of the 35 th IEEE Conference on Decision an Control (996), pp [6] E.G. Gilbert an I. Kolmanovsy, Set-point control of nonlinear systems with states an control constraints: A lyapnov-fnction, referencegovernor approach. Proc. of the 38 th IEEE Conference on Decision an Control (999), pp [7] K. Hirata an M. Fjita, Set of amissible reference signals an control of systems with state an control constraints. Proc. of the 38 th IEEE Conference on Decision an Control (999), pp [8] F. Hoppensteat, Analysis an simlation of chaotic systems. Springer Verlag, 993. [9] H. K. Khalil, Nonlinear Systems. Prentice Hall n. E [0] J. McNamee an M. Pachter, Efficient nonlinear reference governor algorithms for constraine tracing control systems. Proc. of the American Control Conference (999) pp [] R.H. Miller, I. Kolmanovsy, E.G. Gilbert an D. Washabagh, Control of constraine nonlinear systems: A case sty. IEEE Control Systems Magazine, (000), pp [] H. Nijmeijer an A. Van Der Shaft, Nonlinear Dynamic Control. Springer Verlag 990. [3] J. Rossiter an B. Kovaritais, reference governors an preictive control. Proc. of the American Control Conference, (998), pp [4] J.-J. E. Slotine an W. Li., Applie Nonlinear Control, Prentice Hall, 99. [5] E.D. Sontag, Mathematical Control Theory. Springer verlag 990. [6] J. Zabczy, Mathematical Control Theory. Birhaser, 99.