Volume 49, Number, 008 89 Predictive Cascade Control of DC Motor Alexandru MORAR Abstract: The paper deals with the predictive cascade control of an electrical drive intended for positioning applications. The rule for choosing the objective functions of the two loops of the cascade structure is given for dynamical separation of the loops. The method is applied to a DC motor employed as an actuator of a flexible arm joint. A number of simulations are presented to illustrate the performance of the predictive cascade structure. Keywords: predictive control, cascade structure, electric drive, DC motor.. INTRODUCTION Electrical drives represent an important field of the industrial activity and must be considered in terms of improvement of performances. To increase the performances for this type of applications, it becomes necessary to adapt the control strategies to more complex structures and more severely objectives concerning the productivity: increasing the rapidity, better accuracy and complete reliability. Most drive systems for servo applications use cascade control structures developed with classical PID controllers for the velocity and positions loops. For improving the performance it is necessary to use an advanced control technology and according to this feature, predictive control is a well-adapted strategy which enables to consider explicitly the already known behavior of the trajectory to be followed in the future. In recent years several applications of predictive motion control were reported in [], [], or [3]. In this paper, an alternative predictive cascade structure will be proposed for velocity and position control of a servo application. Both control loops included UPS (Unified Predicted Control) algorithm designed on the basis of an objective function that contains prediction error weighted with two polynomials P and M. By minimizing the objective function, the desired closed-loop transfer function can be defined by means of P and M. Using the method mentioned above, the desired speed of the closed-loop system can be adjusted. This control approach was used to accommodate the position of a DC motor employed as an actuator of the joint of a flexible robot arm. Finally a simulation was carried out to demonstrate the improvement of performances.. PREDICTIVE CASCADE STRUCTURE A cascade structure is necessary to control the two variables (velocity and position) of a motor drive application. It is also necessary to define two models corresponding to the plant of the inner and external loops. For both loops UPC algorithms have been used that were obtained by minimization the unified criterion function from [4]. The advantage of such structure is that the behaviors of the two loops are known, defined by a pole-placement method.. Unified prediction controller Most single-input single-output plants, when considering operation around a particular set-point and after linearization, can be described by an ARX model of the form: A(q - ) y(k) = B(q - )u(k-)+e(k) () 008 Mediamira Science Publisher. All rights reserved.
90 ACTA ELECTROTEHNICA where u(k), y(k) are the control and output sequences of the plant, e(k) is a zero mean white noise and A, B are polynomials in the backward shift operator q -. The UPC algorithm consists of applying a control sequence that minimizes the multistage objective function: H p J = [ Pyˆ ( k+ i) Mw( k+ i)] i= Hs H p u k i i= + λ ( + ) + () with the assumption: Δu (k + i - ) = 0 for i > H c (3) where ŷ(k+i) is an i step ahead prediction of the system output on data up to time t = kt (T sampling period ); H s and H p are the minimum and the maximum prediction horizons, H c is the control horizon, w(k+i) is a future set-point or reference trajectory, λ is a weighting factor, P is a polynomial in q - and M is a scalar equal to P(). In this case, the future set point will be known. Imposing a reference trajectory is a way of defining the desired closed-loop response and this goal can be achieved by defining P and M in the criterion function (). The objective of prediction control is to compute the future control sequence u(k), u(k+),,u(k + H c - ) in such a way that the optimal i step ahead prediction ŷ(k+i) is driven close to w(k+i). This is accomplished by minimizing the objective function J which leads to the polynomial form of the control law: R(q - )u(k)=t(q - )w(k+h p )-S(q - )y(k) (4) where the expressions of the polynomials R,T and S are given in [4]. Knowing the plant model () and the controller (4) the following closed loop w c UPC transfer function can be calculated: q B( q ) T( q ) yk ( ) = Aq ( ) Rq ( ) + q Bq ( ) Sq ( ) wk ( + H ). p. Cascade structure design (5) The design of a cascade structure implies a dynamical separation of the control loops with the inner loops established to be faster. These are possible by an adequate choosing of the polynomials P(q - ) and the scalars M from the objective functions J. Thus, if a second order trajectory is used transforming the continuous model to the discrete domain using a zero-order hold circuit yields for P: P(q - )=+α q - +α q - (6) with ζωnt α = e cosω T ζ, α = e ζωnt. n (7) The dynamical separation is achieved by prescribing the settling time t s of the closedloop system assuming the damping ratio ζ = 0,707: t = 4/ ζ w ω = 4/ t ζ (8) s n n s w = u u Inner UPC System Inner loop External loop Fig.. Predictive cascade system. By using the above methods, the imposed speed of the closed-loop system can be adjusted by using a single parameter: the settling time of the closed loop step response: It is common practice that M is given by: M = P() = +α +α (9) Considering the cascade structure from Fig., the design of the two control loops means to compute two UPC controllers so that to obtain the desired performances: rapidity, accuracy and, of course, dynamical y y External System
Volume 49, Number, 008 9 separation of the loops. Design algorithm of the two predictive controllers implies the following steps: o Definition of the models The models are defined in an ARX form as it follows: - Inner plant model A( q ) y( k) = B( q ) u( k ) + e( k). (0) - External plant model: A ( q ) y ( k) = B ( q ) y ( k) + e( k). () e e - Inner closed-loop model: q B( q ) T( q ) y( k) = A( q ) R( q ) + q B( q ) S( q ) w ( k+ Hp ). () - Model used by external UPC controller obtained from () and (): A ( q ) A( q ) y( k) = B( q ) w( k) + ( ( ) ) e k Ae q (3) where: A (q - )=A e (q - )[A (q - )R (q - )+ + q - B (q - )S (q - )] (4) B (q - )=B e (q - )q - B (q - )T (q - ). o Definition of the objective functions The criteria of both loops are the classical ones of the form () with the assumption (3). In order to obtain the dynamical separation of the lops, for the inner one it will be chosen a setting time lower than the external loop that means a natural frequency greater with a factor equal with to 5. The minimization of J j, j=, gives the optimal control signals u and u c. 3 o Polynomial controllers The minimization of the two previous objective functions gives the equivalent polynomial representation (4) of the predictive controllers. The algorithm implies then to compute only two difference equations during the real time loop, one for the inner loop with the R, S, T inner polynomial controller, the other for the external loop with the R, S, T external polynomial controller. 3. SIMULATION EXAMPLE In this section, some simulation results are presented to illustrate the properties and the performances of the predictive cascade structure proposed in Section. The UPC algorithms have been designed, implemented and tested within a flexible experimental environment [4] developed in MATLAB- SIMULINK. The electrical drive considered was a DC permanent magnet torque motor used as actuator for a joint of a robot arm. The transfer of a single- joint arm in a cascade control system represented in Fig., is taken from a model presented by Fu et al in [5]. The aim of the reductive cascade structure is to servo the motor do that the actual angular displacement of the joint will w w = u Position Controller uc - Speed k Controller a + + Las+Ra + - - T r J s Ωm Ω Θ N s E b k b y k Ω y k p Fig.. Predictive cascade structure for a DC motor.
9 ACTA ELECTROTEHNICA track a desired displacement specified by a pre-planned trajectory. In Fig., the following variables were used: Θ angular displacement of the arm; E b back electromotive force; T r - load torque; Ω m - angular velocity of the motor; Ω - angular velocity of the arm; k a - motor torque constant; k b - motor voltage constant; R a - armature resistance; L a armature inductance; J m motor inertia; J o load inertia; N gear ratio; J total inertia rated to the motor shaft; k Ω velocity factor; k p position sensor gain; In order to obtain a good tracking of the position profile the predictive cascade system from Fig. was designed with the method from Section and the following parameters of the DC motor: k a =0.7 N m/a; R a =3.0 Ω; L a =3.0 mh ; J m =8.4686 0-5 N m sec /rad; J 0 = 0.9 N m sec /rad; N=30; J0 4 J = Jm + =.7 0 Nmsec / rad ; N k p =.6667 V/rad; k Ω =0.038 V/(rad/sec). Applying the design procedure from Section, the first inner loop was designed and then the external loop. - Inner loop design The inner plant model was obtained from the block scheme of Fig. with the above DC motor parameters, and after discretization it can be represented by the linear discrete-time equation (0) with the polynomials: A (q - ) = - 0.7705q - ; B (q - ) = 0.0053q -. Fixing for the inner loop the settling time t s = 0. sec. and taking the sampling period T = 0.0 sec. the weighting polynomials are obtained: P(q-) = -.347q - + 0.4493q -, M = 0.46. Minimizing J with the tuning parameters Hs =, H p =3, H c = and λ = 0, the following polynomials of the UPC controller resulted: S (q - ) = -0.930q + 0.3565q + 0.0463q -3 T (q - ) =.4967 +.399q - +.95q - R (q - )=4.9436 8.455q - + 6.757q - 0.0004q -3. - External loop design The closed-loop transfer function of the inner loop using can now be calculated, using () and taking into account the external plant, the polynomials of the model (3) used by the external UPC controller result: A (q - ) = - 0.9q - +.087q - -0.64q -3-0.044q -4 B (q - ) = 0.303q - 0.35q - +0.035q -3 + 0.094q -4. In order to obtain a slower external loop and thus dynamical separation of the loops, it was chosen the settling time t s =0.5 sec. and so the weighting polynomials for objectives function J are: P (q - ) = -.8403q - +0.85q - ; M = 0.08. With tuning parameters H s =, H p = 3, H c = and λ = 0 by minimizing J the following polynomials of the UPC controller were obtained: S (q - ) = -0.797q - -0.0986q - -0.8788q - + 0.8735q -4 +0.0977q -5 T (q - ) = 0.076+0.00q - +0.0048q - R (q - ) = 7.4457-9.5946q - +0.8475q - - 6.865q -5 +0.6377q -4 +0.7q -5. Using the above information the predictive cascade structure was simulated. The control system output and the send point are given in Fig. 3 and Fig. 4. As it can be seen from the figures the controlled outputs track very well the reference trajectories, for a reference composed by ramps and plateaus and for a sinusoidal reference, respectively.
Volume 49, Number, 008 93 y - dot, w line Fig. 3. Control system response to ramp reference. y dot, w - line Fig. 4. - Control system response to sinusoidal reference 4. CONCLUSIONS A new predictive cascade control structure of an electric drive for positioning application was proposed. The advantage of the structure is that the asymptotic behavior of the inner loop is well known, defined by the reference model of the UPC algorithm through the polynomials P and M, achieving thus the dynamical separation of the two loops. The predictive cascade structure was used in a servo application of a DC motor and tested by simulation. The results obtained show the good tracking of the position profile and superior performances regarding rapidity and accuracy. R E F E R E N C E S. Boucher P., Dumer D., Daűmuller S., Predictive cascade control of machine tools motor drives. Proc. of the EPE 9, vol., Florence, 0-5 (99).. Boucher P., Dumer D., Multirate polynomial predictive cascade control. Proc. of the nd IEEE Conference on Control Applications, vol., Vancouver, 63-68 (993). 3. Giarré L., Implicit adaptive predictive control : A planar -link manipulator case. Proc. of the 4 th
94 ACTA ELECTROTEHNICA International Symposium on Automatic Control and Computer Science, Iaşi, 54-58, (993). 4. Soeterboek R., Predictive Control. A Unified Approach, Prentice Hall, Englewood Cliffs, New York, 99. 5. Fu K. S., Gonzales R. C., Lee C.S.C., Robotics : Control, Sensing, Vision and Intelligence, McGraw Hill, New York, 993. Alexandru MORAR Petru Maior University of Targu Mures Faculty of Electrical Engineering Nicolae Iorga St. 540088 Targu Mures e-mail: morar@upm.ro