Feedback Linearization and Model Reference Adaptive Control of a Magnetic Levitation System

Transcription

1 eedback Linearization and Model Reference Adaptive Control of a Magnetic Levitation Syste Luiz H. S. Torres, Leizer Schnitan, Carlos A. V. V. Júnior, J. A. M. elippe de Souza Centro de Capacitação Tecnológica e Autoação Industrial (CTAI), Universidade ederal da Bahia, Rua Aristides Novis, nº, Escola Politécnica, º andar,.-63, Salvador, Bahia, Brasil. luizhenrique@ieee.org; leizer@ufba.br; carlos@trecon.co.br. Electroechanical Engineering Dept, University of Beira Interior, Covilhã, Portugal. felippe@ubi.pt Abstract: The ai of this paper is to cobine two techniques to control a nonlinear Magnetic Levitation Syste (MLS). irstly, a feedback linearization technique (here, exact linearization with state feedback) is applied to obtain a linear syste. Secondly, the linearization is ade via direct cancellation of nonlinear functions, which represent the phenoenological odel of the syste. inally, to deal with the presence of uncertainty in the syste odel, an adaptive controller is used. The controller is based on odel reference adaptive control to estiate the functions that contain the nonlinearities of the syste. The exact linearization and the adaptive controller were ipleented in a siulated environent (Matlab Siulink ). The linear adaptive controller structure guarantees the paraeters adaptation and the overall stability of the syste. The results show that the controller output signal tracks a reference input signal with a sall error. Keywords: Adaptive Control; Direct Approach; Exact Linearization; Magnetic Levitation; Model Reference.. Introduction In recent years, due to coputational developents that have enabled ore coplex applications of nonlinear probles, the area of nonlinear control systes has been the subject of any studies (Soltanpour and Shafie, ). The present paper shows a cobination of an adaptive controller and a feedback linearization technique to control a Magnetic Levitation Syste (MLS). This syste was chosen since it has nonlinear dynaics and a didactic kit of the physical syste is available to continue with future work. The MLS used is anufactured by ECP Educational Control Products ( and will be described in ore detail in section II. Here one desires to control a agnet displaceent over a glass stick as a result of the application of an electrical current on a coil (ECP, 999). The relationship between the electrical flow and the agnetic disc oveent is given by a second order nonlinear ordinary differential equation. This nonlinear relationship belongs to a class of engineering systes of the type x t, x Ax Bx Gxu. Several nonlinear control strategies can be used to control the disc position, such as, for exaple: fuzzy, neural network, adaptive control, feedback linearization (Khalil, 996; Abdel- Hady and Abuelenin, 8; Torres et al., a). Here both the exact feedback linearization technique and adaptive control will be used. Exact feedback linearization can enable a transforation fro a nonlinear syste to a linear through the addition of nonlinear copensators. Thus, this transforation allows designing a linear controller for the syste linearized. However, the exact linearization technique with state feedback requires a atheatic odel that represents the dynaics of the real plant (Slotine, 99). urtherore, the uncertainties in the phenoenological odel can coproise better results. To deal with soe uncertainties in the syste's odel, an adaptive controller is used. The controller is based on direct odel reference approach (Narendra and Valavani, 978; Ioannou and Sun, 996) to provide a control law that will be used to the syste after the linearization. The ai of this paper is to use the cobination of two techniques to control the MLS: exact linearization with state feedback and Model Reference Adaptive Control (MRAC). These two techniques cobined will enable a linear controller structure to deal with soe uncertainties in the syste odel to MSL s control disk position (a typical nonlinear dynaical syste). Studies in Inforatics and Control, Vol., No., March 67

2 In section the MLS is briefly explained.the proposal of the exact linearization with state feedback and its application over the MLS are presented in section 3. In section, the adaptive controller structure with a linear control law is presented. The siulation and analysis results are discussed in section 5. inally, soe rearks about the controller perforance are presented in section 6. It is iportant to ention that a first version of this paper was published in (Torres et al., b).. The Model Magnetic levitation syste In this paper the MLS ade by ECP was used and is shown in igure. It coprises two agnetic discs, a glass stick, two laser sensors and two coils. The sensors are used to obtain the syste response associated with the disc positions. The syste input is given by the application of an electrical current to the coils. The physical syste counicates with a coputer via Digital Signal Processing (DSP) and a black box is responsible for the electrical current drivers and the energy supply. This MLS can be classified according to two odes, SISO (Single Input Single Outpu or MIMO (Multiple Input Multiple Outpu, which depends on the desired syste configuration. In the SISO ode only one disc is used whereas in the MIMO ode two discs are used. Here the MLS was configured to operate as SISO and in the repulsive ode over the disc. In other words, only the botto coil was used in this work. igure. Magnetic Levitation Syste ade by ECP (ECP, 999). The MLS anual (ECP, 999) shows the atheatic odel, based on the physical laws, which allows us to obtain its differential equation odel. The developent of the atheatic odel is beyond the scope of this paper. Through the balance of forces, the equation is given by (Laithwaite, 965) y c y where: y - agnetic disc position g, () y - first derivative of the agnetic disc position y - second derivative of the agnetic disc position c - air viscosity coefficient - agnetic disc ass - agnetic force applied to the agnetic disc. The agnetic force can be written in the following way (ECP, 999) i =, () a(y +b) where: i - electrical current applied on the coil; a and b - are constants related with the coil properties. By substituting () in (), a nonlinear relationship between the agnetic disc position and electrical current applied to the coil gives y g c y a( y b) Paraeter estiation i (3) There are five paraeters in (3): g, c,, a, and b. The paraeters g = 9.8 [/s ], =. [Kg] and c =.5 [Ns/] are considered as constant and known values (ECP, 999). The paraeters a and b are constants related with agnetic coil properties and ust be estiated. In (Silva, 9), the least square and Monte Carlo ethods were used to estiate a and b. Accordingly with (Silva, 9), based on a cost function, the Monte Carlo ethod presented the best values for these paraeters. The values are a =.95 and b = 6.8. These values will be used in this paper Studies in Inforatics and Control, Vol., No., March

3 3. eedback Linearization Exact linearization with state feedback eedback linearization can be applied to a certain class of nonlinear systes, including the MLS studied, and enables to transfor the original syste odels into equivalent odels of a sipler for. The control schee uses the exact linearization with state feedback based on the cancellation of nonlinear functions. However, to enable the application of the technique, the syste dynaic ust be represented by (Guadarbassi and Savaresi, ) X ( X ) G( X ) u, () where the functions (X) and G(X) represent the nonlinearities of the states, u is the control syste input and X is the state vector. urtherore, two conditions ust be satisfied. The first one is that the syste ust be controllable. or this first condition the atrix fored by vectorial fields in (5) ust has order n, where n is the syste order (Khalil, 996) n [ ad G ad G... ad G], (5) where ad n G is the notation of Lie s bracket (Slotine, 99). The second one is that the syste should be involutive. It eans that the distribution expressed in (6) also should be involutive (Guadarbassi and Savaresi, ) n D = span ad G ad G... ad G, (6) where D is the involutive distribution of G(X) expanded in Taylor s series (represented here by the notation span{.}) on an equilibriu state X (Na et al., 993) The order of D is given by n. In order to the distribution in (6) to be involutive, it is necessary that the order n of the expression in (7) be equal to diension of D in (6) n [ ad G,ad G], (7) Once the conditions are satisfied it is possible to deterine a diffeoorphis Z T(X ). After this, the dynaic of the syste given by () can be transfored into the for Z EZ ( Z)[ u f ( Z)], (8) where (Z) and (Z) represent the state feedback, u f is the feedback signal, and E and are, in this work, considered as constant with known values. A feedback signal u f for the nonlinear syste is chosen in the for in (9) u f ( Z) ( Z) u, (9) Thus, the linear syste can be written as Z EZ u () where u is the control signal (control law) for the syste after linearization. The deterination of u will be discussed in next section. Linearization of the MLS The odel of the MLS ade by ECP was presented in (3) and the two conditions for application of the exact linearization were presented in the last subsection. The feedback signal u f and the variables of states can be set as u f i x y, () x y The dynaic of the syste given by (3) can be rewritten in the for given in () x x x c g x a( x b) u f., () The functions (X) and G (X ) that contain the nonlinearities of the syste can be set as follows in (3) and () x ( X ) c g x (3) ( ). ( ) X a x b G () The transforation Z T(X ) can be set in the for given by (5) (Khalil, 996) Z Z T ( X ), (5) Z The functions (Z) and (Z) can be calculated in the for given by )( Z ) ( Z) ( ga caz b, (6) ) ( Z) a( Z b (7) inally, the feedback signal u f can be rewritten as Studies in Inforatics and Control, Vol., No., March 69

4 ( ga caz )( Z b) a( Z b u (8) u f ) The application of the feedback signal u f over the syste given by () will cancel the nonlinearities and the syste will be transfored into a linear syste given by (). In the sae way but using (5) (Slotine, 99) x Z. (9) x A block diagra was ipleented in Matlab Siulink which siulates the exact linearization technique applied in the MLS (see igure ). The block diagra in igure 3 ay explain the general idea of a syste using MRAC. MRAC applied to the MSL after the linearization In this work MRAC classical technique using the stability theory fro the input-output view is applied to the MSL after the exact linearization. Once the dynaics are now linear, the control proble will be forulated as odel-following. The derivation of the MRAC will follow the 3 steps below (Aströ and Wittenark, 8) igure. Block diagra for the exact linearization in the MLS. igure 3. General idea of a syste using MRAC.. Adaptive Controller Model Reference Direct Approach The Model Reference Adaptive Control (MRAC) is one of ain techniques in adaptive control. The changes in the controller paraeters are provided by the adjustent echanis with the objective to iniize the error between the syste under control and a odel reference output (that is the desired response). MRAC uses integral laws to adaptation which can introduce a slow and oscillatory transitory, however in steady state there is a soft control signal (Narendra and Valavani, 978; Ioannou and Sun, 996). Step : ind a controller structure that adits perfect output tracking; Step : Derive an error odel of the for T G ( { ( ( )} () where G ( p ) is a Strictly Positive Real (SPR) transfer function in p, is the process paraeters (or the true controller paraeters), and is the controller paraeters (or the adjustable controller paraeter). Step 3: Use the paraeter adjustent law ( () 7 Studies in Inforatics and Control, Vol., No., March

5 where is the adaptation gain, an auxiliary vector of filtered signals and the error signal. All these paraeters and variables in () and () will be defined in the next subsection. The controller structure The desired response given by a continuoustie odel reference signal could be written as follows A y( Br( () where A and B are polynoials to obtain the odel transfer function in the for B G ( s) (3) A The desired response (in this case here is the height value) and the input reference signal r( is bounded. The continuous-tie plant (the MSL after exact linearization) could be described as follows Ay( b Bu( () It is assued that the polynoials A and B do not have coon factors and the polynoial B is onic with all zeros in the left half-plane. The paraeter b is called the high-frequency gain. The variable y( is the easured output and it is also bounded. The plant is a iniuphase and, in this work, it is assued the sign of b is known. If the paraeter b is not known, it ust be estiated. A linear controller can be written as Ru( Sy( Tr( (5) where R, S and T are polynoials. Since the polynoial B is stable, the corresponding poles can be canceled by the controller. In other words, R RB. The closed-loop syste when (5) is applied becoes ( AR bs) y btr (6) where T will be chosen as T t A, and A is a stable onic polynoial and R and S satisfy AR bs A A (7) One way to achieve the perfect odelfollowing is set A y ( btr( (8) and the error equation (the difference between the plant response and the desired response given by the reference odel) is set as e y( y ( (9) Let introduce the polynoials P, P and P : P P P n n, P p p... n, and k k P p p... k, where n = degree ( A ) = degree ( P ), and k = degree ( R ) = degree ( P ). It is assued that degree ( P ) > degree ( P ) and the polynoial P is a stable onic polynoial. ro (7) and after soe algebraic anipulations, let define the filtered error e f as bq R S T e f u y r( (3) A A P P P where Q is also a polynoial whose degree is not greater than degree A A ) such that ( bq is SPR. This filtered error e f will be A A necessary to obtain the error signal in the for of (). One can writes P R as R R P P R P (3) P P P P P The filtered error then becoes bq R P S T e f u y r( (3) A A P P P P Let l and be the degrees of the polynoials S and T, respectively. Let introduce a vector of true controller paraeters T ( r... rk s... sl t... t ) (33) where ri are the coefficients of the polynoial R P. Also define the auxiliary vector of filtered input, output and input reference signals k T p ( u... u p r... r) l p y... y (3) The filtered error in (3) can then be rewritten as bq e f u T (35) A A P Studies in Inforatics and Control, Vol., No., March 7

6 To obtain an error odel, one ust introduce a paraeterization of the controller. The control law u ( is given by T u( ( P ) (36) By using this control law and (35) the filtered error can be written as bq T T T T e f ( P ) (37) A A P Let introduce the signals and. After soe algebraic anipulation by using (37) these signals can be defined by P T u (38) Q bq ( y y ) (39) P A A e Q Q e y () P P f y P T u () bq e f (3) A A ( () T u( ( P ) (5) 5. Siulation and Analysis Results The Matlab Siulink was used to siulate the proposed controller for the MLS in the present work. The block diagra designed in Siulink is shown in igure. The odel paraeters of the MSL used here were presented in subsection.. igure. Block diagra in Siulink. The signal is called the augented error, and is called the error augentation. The error odel in (39) is the sae defined in (). It is also linear in the paraeters and satisfies the requireents of the step, and the paraeters will be updated by (). The stability of the closed-loop syste is obtained by considering that b Q ( A A ) is SPR and that signals in are bounded. inally, the equations needed to ipleent the MRAC syste can be suarized as follows: B y r () A Many different odel reference adaptive systes can be obtained by different choices of the design paraeters. or sake of siplicity, in this work the polynoials were chosen as follow: P (s) = A (s), P (s) = A (s), and Q(s) = A (s).a (s). The transfer functions of the reference odel was chosen as G ( s) s. n ; n s n.75 n (6) 7 Studies in Inforatics and Control, Vol., No., March

7 The polynoials A, R, S, and T were chosen as follow below and by using the solution given by the Diophantine equation (7) A () s = s+ a a = (7) Rs ( ) = s+ r r=.5 (8) Ss ( ) = ss+ s s=.3; s=.56 (9) Ts ( ) = ts+ t t=.8; t=.56 (5) The siulations were perfored regarding r( (reference) as a square wave signal with aplitude equal to and the adaption gain equal to.8. The value of b was chosen equal to.5. igure 5 shows the copared response between the odel reference output and the plant output. igure 6 and igure 7 show the error signal and the control effort, respectively. According to (3) the reference signal r is the electrical current applied on the coil i t (that is the anipulated variable). It could be observed in igure 5 that the odel reference output given by (3) was tracked by the plant output. The process variable y t that is the agnetic disc position could be observed in igure 5 with oscillations in transitory, but stable in steady state. The error e in steady state is little according to igure 6 and the control effort associated with the electrical current is bounded as shown in igure 7. igure 5. Coparison between the desired response (odel reference) and plant response. 6. Conclusions igure 6. Error signal. igure 7. Control effort. In this paper the cobination of two techniques to control a MLS were presented: exact linearization with state feedback and MRAC direct approach. It could be observed through the siulations and analysis results that the desired response (output signal of the odel reference) was tracked by the plant response. The error signal could be seen as bounded and near to zero and the control effort could be seen also as bounded. The siulation results show that the adaptive controller is able to control satisfactorily the agnetic disc position, in spite of the presence of odel uncertainties. or future work this adaptive controller should be ipleented in the real physical syste. Acknowledgeents The authors wish to acknowledge the support with infrastructure fro CTAI, and CAPES for the financial support. Studies in Inforatics and Control, Vol., No., March 73

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