Discrete Time Linear Quadratic Optimal Control for an Electrical Servo Drive System

Transcription

1 Discrete ime Linear Quaratic Optimal Control or an Electrical Servo Drive System CORNELIU BOAN, FLORIN OSAFI an ALEXANDRU ONEA Dept. o Automatic Control an Inustrial Inormatics Gh. Asachi echnical University o Iasi B. D. Mangeron 53A ROMANIA Abstract: - he iscrete-time linear quaratic optimal problem with inite inal time or a servo-rive system with a brushless or a c motor is presente. he propose algorithm allows simpliying the optimal control law implementation. he optimal control ensures a goo behavior o the system an the ecrease o the energy consumption. Key-Wors:- Optimal control, Brushless, DC machines, Eiciency, Servo rives. Introuction he optimal control o the electrical rive systems allows to obtain several avantages: the reuction o the energy consumption an, in some cases, the reuction o the rate power o the rive motor an thus the ecrease o the imensions an the weight o the system; in the same time, a goo behavior in the transient perio can be achieve. hese avantages are very important nowaays, when the aspects connecte with the prouction quality, the cost o the plants an the energy losses are consiere with priority. he paper reers to a linear servo-rive system controlle on the basis o the linear quaratic optimal (LQ) problem []. In orer to obtain a ast response, a inite inal time is impose. A servo-rive system with a c or brushless motor is consiere. A linear moel or the system is aopte: x(t) = A x(t) + B u(t) + W m(t) () c c c 3 where x(t) =θ [ ω i] is the state vector ( enotes the transposition), u(t) is the control variable, m(t) is the isturbance (loa torque), θ an ω are the angular position an spee, respectively, an a A = a a, B =, W = w () c 3 c c c a3 a 33 b 3 he elements o the matrices in () epen on the parameters o the servo-rive system (incluing motor, converter, gear an transucers) an these are: ρ Cm Ce R a = r, a =, a 3 =, a3 =, a33 =, J J L L where r is the gear constant, ρ is the viscous riction coeicient, J is the inertia, an R, L are the resistance an inuctance o the controlle circuit o the motor. For a.c. motor, i an u are the rotor current an voltage, respectively. In the case o a brushless motor (a surace mounte permanent rotor is consiere), in the q reerence rame, i=i q an u=u q represent the q-axis current an voltage, respectively. It is suppose that u is chosen so that i is very small []. Since the optimal controller must be implemente in a iscrete manner, it is useul to treat the problem in iscrete case. With the notations A c A c t A c t c c (3) A= e, B= e tb, W= e tw ( is the sampling perio), the iscrete moel o the servo-rive system is x(k + ) = Ax(k) + Bu(k) + Wm(k) (4) where x(k) = x(k), k, an similar notations are use or all other variables. he aim o the optimal control is to obtain a convenient transient response, a small error or the inal position an a reuce energy consumption.

2 he ollowing iscrete criterion is introuce or this purpose: s s J = ( θ(k ) θ ) + ( ω(k ) ω) (5) + [q ( θ(k) θ ) + q ( ω(k) ω ) + q3i(k) + pu(k) ] k = where k correspons to the inal time, s, s, q, q, q 3, p> are the weight coeicients an x [ ] = θ ω i is the esire state vector. Usually, ω = an i is ree. he irst an the secon terms in (5) penalize the inal errors o the position an o the spee, respectively. he terms containe in the sum penalize the transient errors an the great values o the current an o the control voltage; note that the presence o the term q3i(k) in the criterion ensures a small value or the issipate energy in the wening on the optimization interval [, k ]. he criterion (5) can be expresse in the general orm J= [ x(k ) x] S[ x(k ) x] x (k)qx(k) u (k)pu(k) k= (6) where S=iag(s,s, ), Q=iag(q, q, q 3 ) an P=p. he problem is to in the optimal control u(x(t)) which minimizes the criterion (6) having in view the equation (4) o the servo rive system. A similar problem but or the continuous case was stuie in [3]. here are two type o optimal control problems in connection with the inal state: the ixe enpoint problem an the ree en-point problem, respectively. In the irst one, the inal value o the state vector is impose, or instance x(k ) = x. For the other type o problems, the inal vector x(k ) is ree, but the minimization o the criterion (6) ensures that x(k ) is in the proximity o the esire value x. he ixe en-point problem ensures that the esire vector is achieve at the inal moment k an this act is very important. We have to remark that usually the optimal control law o not ensures that the system stops in x or k>k. In many applications it is suicient to arrive at a ixe moment k in a neighborhoo o x. I x must be achieve, we can change the control law (not obligatory optimal) in the last part o the traectory, ater k. he main avantage o the ree en-point problem is the act that the control variable u(k) is smaller than in the ixe en-point problem. he present paper reers to a ree en-point optimal control problem.. Main Results Using the Hamilton conitions [], the optimal control is u(k) P B (k ) = λ +, (7) n where λ(k) is the state vector that in the classical proceures is aopte as λ (k) = R(k)x(k) + v(k), (8) where the component v(k) epens on the exogenous variables x an m(t). In orer to avoi the on-line computation in inverse time the time-variant matrix R(k) as solution to a Riccati ierence matrix equation, instea o (8) one proposes to choose λ(k) as λ (k) = Rx(k) + v(k) (9) where R is a constant matrix an v(k) = v(k) + v(k) ; the corrective component v(k) ensures that the solution is the same as one obtaine i the orm (8) is aopte. Using the Hamilton equations an transversallity conitions, one obtains that R satisies the Riccati matrix algebraic equation or the iscrete case [4]: + = = () R A R(I NR) A Q, N BP B (I-the ientity matrix). In orer to establish the vector v(k), some supplementary issues must be solve, since the transversallity conitions oer only the inal value ( ) ( ) v k = (S R)x k Sx () his inal value can not be use in the real time computing an have to be replace with x(), that is beorehan known. For this purpose, the vector z(k) = x (k) v (k) is introuce. One can veriy that z(k) satisies the equation z(k + ) = Gz(k) + r(k), ()

3 where r(k) r(k) = r(k) = ENZ E Q x A REWm(k) = Z Wm(k) with Z = A (I REN) an E = (I + NR) an G G G EA ENZ = G = Z his matrix an its inverse G G A E A N G = = G Z (3) (4) (5) () = h(k) = Z r (k + ). Remark. he computation o the vector g impose to beorehan know the vector r(k), epening on exogenous variables. his act implies to know at least the shape o the loa torque m(k) an to estimate its amplitue [5] in the beginning o the optimization perio. he simplest an most requently met case or the servo-rive systems is m(k)=const. In this case, r an r have constant values in (9) an (), an the vector g in (9) is constant. Finally, rom (7), (9) an (), the optimal control can be written as * u (k) u (k) u c(k) = + () he eeback component u(k) = P BREAx(k) (3) have some useul eatures, as G G G =, with G A similar eature is obtaine or G -. From () yiels k = i i i= G = G G G (6) z(k) = G z(k ) (G ) r(k + ) (7) One obtains rom () an (7) or k= = +, (8) x(k ) M [x() g] where M= G + G (S R) an i= g= G r(i ) + G r (i ) + G Sx From () an (7) we also obtain k k v(k) Z (S R)M (x() g) Sx h(k) with (9) = +,() is ientical with one obtaine in the similar problem, but with ininite inal time. he corrective component u (k) = P B (A ) v(k) + Qx (4) c [ ] (with v(k) given by ()) epens on the initial state x() an on the exogenous variables x an m. Remark. he above relations are rather complicate, but the most part o the computation is perorme o-line, in the stage o the controller esign, when are establishe all the constant matrices. In the real time computing, only two variable elements must be compute: the vector h(k) given by () an the matrix k k Γ k = Z. he both elements can be recurrently compute: Γ =Γ Γ = h(k) h(k ) Z Z r k k Z, Z, + = = = = h() Z r, k,,...,k (5) he terms epening on x an m rom u c (k) must be compute inepenently o the use metho. he speciic term or the propose algorithm is the term epening on x() o the corrective component u c (k). his term ensures the ientity o the optimal control () with one

4 obtaine via classical proceures, base on relations (7) an (8). In this last case the time variant matrix R(k) have to be compute on-line in inverse time as a solution to a Riccati ierence matrix equation. his signiicant iiculty is avoie in the propose metho. Remark 3. Generally, the optimal control reers to a speciie time interval. Usually, the control law u * (k) must be change ater the inal time t = k so that to ensure the esire state x. In the stuie problem with ree en-point, usually x(k ) x. In orer to achieve the esire inal state, the control law is change, or instance in the orm u(k) =α( θ θ(k)) βω (k), k >, α >, β> (6) I it is necessary, a iscrete low pass ilter can be introuce in orer to avoi an abrupt change o u(k) at the moment k. 3. Simulation Results he above results were veriie by simulation. he applications are or R=3.Ω, L=.6H, Ce=.58Vs/ra, Cm=.58Nm/A, J=.8Nms /ra, r=/5, =.3s, t =.9s, θ =ra. Some simulations were perorme or ierent weight matrices in (6). Figure shows the system behavior in the case S=iag(6, 5, ), Q=iag(,, 3.), P=.6. he behavior o the system or the same parameters o the optimal controller in the case o the change o the optimal control law at the moment t (see Remark 3) is presente in the igure. One can remark that the motor runs as a brake in the last part o the optimization interval. Remark 4: It is well-know the result obtaine or the ixe en-point problem or a simple irst orer servo-rive system, where the current i(k) is consiere as control variable: the open loop optimal control i(k) has a linear variation an the spee ω(k) has a parabolic variation. he igures presente above inicate that these shapes are on the whole respecte also or the close loop optimal system in the more realistic case, when the voltage u(k) is consiere as control variable. Moreover, this aspect is vali also or the ree enpoint optimal control problem. It was stuie the eect o some ierent moiications in the control law or in the unctioning conitions: A signiicant increase o the sampling perio (up to times) only or the corrective component u c (k) o not yiel a signiicant change o the traectory. his act allows reucing o the on-line computing time. he increase o the loa torque leas to a small increase o the maximum value o the current (or instance, or the change o the loa torque rom.5 Nm up to.5 Nm, the maximum value o the current increases only with.6 A). his act shows that the most part o the electromagnetic torque o the motor ensures the acceleration o the system. he ecrease o the impose inal time leas to a signiicant increase o the current (or example, or the ecrease o the inal time t rom.9s to.5s the maximum value o the current increase by.7 times). he change (or an inaccurate estimation) o the loa torque m o not prouce a great variation rom the reerence curves o ω(k) an θ(k). his act is illustrate in the ig. 3 or the change o the loa torque rom.5 Nm up to.5 Nm at the moment k /. he energy consumption was compute in each case an compare with one obtaine in the usual cascae control case. A signiicant ecrease o the energy losses (up to 3%, epening on the weight matrices) is obtaine in the optimal control case. time[s] Fig.. he behavior o the optimal servo rive system on the optimization interval [, t ]

5 time[s] Fig.. he behavior o the optimal servo rive system or t< t an or t > t (t =.9 s) Annual Conerence o the IEEE Inustrial Electronics Society IECON 98, Aachen, 998, pp [3] C. Botan, F. Ostai, S. Dumbrava, A. Onea, Linear Quaratic Optimal Control o an Electrical Servo Drive System, Proceeings o the th International Power Electronics & Motion Control Conerence, EPE-PEMC, Cavtat & Dubrovnik,, pp. 437&CDROM. [4] C. Botan, A. Onea, Discrete-ime Optimal Control o an Electrical Drive System, Bul. Instit. Politehnic Iasi, om XLV(IL), Fasc. -4, A.C., 999, pp [5] S. Komaa, M. Ishia, K. Ohnishi,. Hori, Disturbance Observer-Base Motion Control o Direct Drive Robots, IEEE ransactions on Energy Conversion, 6(3), 99, pp time[s] Fig. 3. he eect o the change o the loa torque rom.5 Nm up to.5 Nm at the moment t=t / 4. Conclusions A new metho or the iscrete optimal control or a servo-rive system is presente. he propose algorithm can be easier implemente than in the classical proceure or the linear quaratic optimal control. he aopte criterion ensures a goo behavior or the system an a signiicant ecrease o the energy consumption. Reerences [] Brian D.O. Anerson, John B. Moore, Optimal Linear Quaratic Methos, Prentice Hall, Englewoo Clis, New Jersey, 99. [] K.S. Low, Y.Z. Deng, X.L. Guo, wo-egreeo-reeom Control o a PMSM Drive without Mechanical Sensor, Proceeings o the 4 th