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n a recent note Ortega and Espinosa SI presented a globally stable controller for torque regulation of a complete induction motor model with partial state feedback, i.e., no assumption of flux measurement. The result was established under the assumptions that both the desired and load torques are constant, that the former does not exceed certain bounds which depend on the systems natural damping, and that the motor parameters are known. In the present contributions we extend these results in several directions. First, by “adding mechanical damping” to the closedloop system we relax the upper bound condition on the desired torque. Second, we use a new controller structure that allows us to treat the case of timevarying desired torque. Finally, a new estimator is proposed to handle timevarying (linearly parameterized)unknown loads
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 38, 1221 [231 1241 [251 NO 11, NOVEMBER 1993 REFERENCES V L Kharitonov, “Asymptotic stability of an equilibrium position of a family of systems of linear differential equations,” Difjerential’nye Uracneniya, vol 14, pp 2086-2088, 1978 B R Barmish, “Invariance of the strict Hunvitz property for polynomials with perturbed coefficients,” IEEE Trans Automat Contr., AC-28, pp 935-937, 1984 N K Bose and E Zeheb, “Kharitonov’s theorem and stability test of multidimensional digital filters,” IEE Proc G, vol 133, pp 187-190, 1986 A C Bartlett, C V Hollot, and H Lin, “Root locations of an entire polytope of polynomials: It suffices to check the edges,” Math Contr Signals Syst., vol 1, pp 61-71, 1988 B R Barmish, “A generalization of Kharitonov’s four-polynomials concept for robust stability problems with linearly dependent coefficients,” IEEE Trans Automat Contr., vol 34, pp 157-165, 1989 H Chapellat and S P Bhattacharyya, “A generalization of Kharitonov’s theorem: Robust stability of interval plants,” IEEE Trans Automat Contr., vol 34, pp 306-311, 1989 Y C Soh, “Stability of an entire polynomials,” Inf J Contr., vol 49, pp 993-999, 1989 M Fu and B R Barmish, “Polytopes of polynomials with zero in a prescribed set,” IEEE Trans Automat Contr., vol 34, pp 544-546, 1989 I R Petersen, “A new extension to Kharitonov’s theorem,” IEEE Trans Automat Contr., vol 35, pp 825-828, 1990 A Cavallo, G Celentano, and G de Maria, “Robust stability analysis of polynomials with linearly dependent coefficient perturbations,” IEEE Trans Automat Contr., vol 36, pp 380-384, 1991 I R Petersen, “A class of stability regions for which a Kharitonov-like theorem holds,” IEEE Trans Automat Contr., vol 34, pp 1111-1115, 1989 Y C Soh, “Strict Hunvitz property of polynomials under coefficient perturbation,” IEEE Trans Automat Contr., vol 34, pp 629-632, 1989 Y K Foo and Y C Soh, “A generalization of strong Kharitonov’s theorem to polytopes of polynomials,” IEEE Trans Automat Contr., vol 35, pp 936-939, 1990 -, “Kharitonov’s regions: It suffices to check a subset of vertex polynomials,” IEEE Trans Automat Contr., vol 36, pp 1102-1105, 1991 M B Argoun, “Frequency domain conditions for the stability of perturbed polynomials,” IEEE Trans Automat Contr., vol 32, pp 913-916, 1987 S Dasgupta, P J Parker, B D Anderson, F J Kraus, and M Mansour, “Frequency domain conditions for the robust stability of linear and nonlinear dynamical systems,” IEEE Trans Circ Syst., vol 38, pp 389-397, 1991 B D Anderson, E I Jury, and Mansour, “On robust Hunvitz polynomials,” IEEE Trans Automat Confr., vol 32, pp 909-913, 1987 M B Argoun, “On the stability of low-order perturbed polynomials,” IEEE Trans Automat Contr., vol 35, pp 180-182, 1990 F Kraus, B D Anderson, and M Mansour, “Robust stability of polynomials with multilinear parameter dependence,” Int J Contr., vol 50, pp 1745-1762, 1989 B R Barmish and Z C Shi, “Robust stability of a class of polynomials with coefficients depending multilinearly on perturbations,” IEEE Trans Automat Contr., vol 35, pp 1040-1043, 1990 J Ackermann, H Z Hu, and D Kaesbauer, “Robustness analysis: A case study,” IEEE Trans Automat Contr., vol 35, pp 352-356, 1990 K H We1 and R K Yedavalli, “Invariance of strict Hunvitz property for uncertain polynomials with dependent coefficients,” IEEE Trans Automat Contr., vol 32, pp 907-909, 1987 L R Pujara, “On the stability of uncertain polynomials with dependent coefficients,” IEEE Trans Automat Contr., vol 35, pp 756-759, 1990 N K Bose, “A system theoretic approach to stability of sets of polynomials,” Contemporary Math., vol 47, pp, 25-34, 1985 N K Bose and Y Q Shi, “A simple general proof of Kharitonov’s generalized stability criterion,” IEEE Trans Circults Syst., vol 34, pp 1233-1237, 1987 1675 [26] K S Yeung and S S Wang, “A simple proof of Kharitonov’s theorem,” IEEE Trans Automat Contr., vol 32, pp 822-823, 1987 [27] H Chapellat and S P Bhattacharyya, “An alternative proof of Kharitonov’s theorem,” IEEE Trans Automat Contr., vol 34, pp 448-450, 1989 [28] F R Gantmacher, Theory of Matrices, vol 11, New York Chelsea, 1964 Nonlinear Control of Induction Motors: Torque Tracking with Unknown Load Disturbance Romeo Ortega, Carlos Canudas, and Seleme I Seleme Abstrecf-In a recent note Ortega and Espinosa [SI presented a globally stable controller for torque regulation of a complete induction motor model with partial state feedback, i.e., no assumption of flux measurement The result was established under the assumptions that both the desired and load torques are constant, that the former does not exceed certain bounds which depend on the systems natural damping, and that the motor parameters are known In the present contributions we extend these results in several directions First, by “adding mechanical damping” to the closed-loop system we relax the upper bound condition on the desired torque Second, we use a new controller structure that allows us to treat the case of time-varying desired torque Finally, a new estimator is proposed to handle time-varying (linearly parameterized) unknown loads I PROBLEM FORMULATION We consider in this note the classical dq model [lo] of the induction motor (1.1) (1.2) with generated torque Manuscript received March 14, 1992; revised August 21, 1992 R Ortega is with Genie Informatique, Universite de Technologie de Compiegne, BP 649-60206, Cedex, France He was a Visiting Professor at the Department of Electrical Engineering, McGill University, Montreal, Canada, when this work was completed C Canudas is with the Laboratoire D’Automatique de Grenoble, ENSIEG, BP 46, 38402, Saint Martin D’Heres, France S I Seleme is on leave from Faculdad de Engenharia de Joinville, UDESC, Brazil and is currently with the Laboratoire D’Automatique, de Grenoble, ENSIEG, BP 46, 38402, Saint Martin D’Heres, France His work was supported in part by CNPq/CEFI, Brazil-France IEEE Log Number 9208709 0018-9286/93$03.00 1993 IEEE 1676 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 38, NO 11, NOVEMBER 1993 TABLE I where we have defined the signal vectors u l , u ,and u w,-primary frequency, are the control inputs, x is the state vector and y , is a load torque All symbols are explained in Table I It has been shown in [8] that a more convenient form for (1.11, which reveals the work less forces acting on the motor, can be obtained directly from the total energy function and Lagrange's equations as (1.6) The matrices D , R , C , and M are given by D diag{D,, c J } [o, o,o, o E - I R S x ,R L e ~E I ~R ~ I ~ diag{R,, 16) E I R j X , (1.7) dX) + L [ ~ , x 2L , , ~ - ~( L, , ~ + , L , , ~ , ) 0, , 0i7 with 0, the x identity matrix and matrix List of Symbols p = ( d / d i ) = Derivative opertator R , = Stator Resistance R, = Rotor Resistance L , = Stator Inductance L , = Rotor Inductance L,, = Mutual Inductance J = Rotor Inertia b = Motor Damping l ' = Number of Pole Pairs x, = ; = I , = d-component of stator current x2 = I' = I , = q-component of stator current x j L- I , - d-component of rotor current xq = i; = I, = q-component of rotor current xj = w, = Rotor angular speed U , = Vj = d-component of stator voltage u = V; = q-component of stator voltage u = w, = Primary frequency y = Generated torque y, = Load torque wr = ci3 - l'x5 = Slip frequency = Observed value of I,, i = 1; ,4 J, the x antisymmetric (1.8) Notice that the matrix C is skew-symmetric The control problem can now be formulated as follows Consider the induction motor model (1.3), (1.6) with states x, control variable U , disturbance and regulated signal y Assume (see point 1) of the discussion below) A.l) Stator currents x , , x and rotor speed x are available for measurement A.2) Motor parameters are exactly known A.3) Load torque can be linearly parameterized as Discussion: 1) The following comments regarding the assumptions are in order: A.l) is the only realistic situation for practical applications A.2) is a very strong assumption since it is well known that, e.g., the rotor resistance changes considerably in operation Our contention is that the fact that our scheme does not rely on nonlinearity cancellations makes this assumption less stringent (see [81) Further, in the known load case, we are able to establish exponential stability of the scheme Thus, a robustness margin to parameter variations is expected As is well known in adaptive control, this property still holds under suitable excitations assumptions A.3) is a more realistic assumption than constant load torque It is well known that bearings and viscous forces vary linearly with speed, while large fluid systems as pumps and fans have loads proportional to the square of the speed Thus, we propose a torque load of the form y, = (0, + O,x:)sgn(xi) + ozxs (1.12) Clearly the assumption of bounded x for all bounded generated torques y restricts the values of Other prior knowledge can be used to select the vector A.4) we believe is a reasonable, pratical assumption 2) A brief review of the literature follows The problem of torque regulation assuming full state measurement was studied y1 = O W t ) (1.9) using linearization techniques by [3] for a model neglecting the mechanical dynamics, that is, x s = const [7] proposes an adapwhere E IRq is an unknown constant vector and contains tive version of the feedback linearization scheme of [5] to measurable signals Further, and I#J are such that for all address the speed control problem with measurable state, unbounded y the solution of (1.2) yields a bounded x , known parameters and constant load torque In [ l l ] sliding A.4) Desired torque yd is a differentiable function with known mode techniques are used for partial state-feedback velocity first derivative control [4] established local stability of a scheme designed using Under these conditions, design a control law that will ensure backstepping, which is recent Lyapunov-based stabilization technique, for the velocity control problem with flux observer [8] (1.10) lim ( y - y d ) = provided the first solution to the torque regulation problem with partial state feedback and unknown constant load for constant with all internal signals bounded Further, we want our con- desired torques which satisfy an upperbound determined by the troller to attain asymptotically field orientation [6], i.e., motor mechanical damping An adaptive scheme to handle un(1.11) certain rotor resistance is also presented in that note, but lim ( L , , X , + L , x , ) = r+= requires measurement of rotor signals T o the best of our I#J _ 1677 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 38, NO 11, NOVEMBER 1993 knowledge, the problem of combining parameter adaptation and K, is a nonlinear gain and K,, is a time-varying gain chosen as flux observation remains open Torque control is also studied in [ l ] using some ideas of energy minimization and simple linear P D control laws 3) Our interest in this paper for torque control is motivated by the unquestionable potential of A C drives in robotics applications This potential stems from the fact that A C drives supersede D C drives in nowadays technology due to its simpler construction, reliability and lower cost The first proof of stability of robot motion controllers with induction motors was recently reported in [2] It is not clear to the authors how this problem can be addressed with a controller that regulates the motor speed instead of the torque 4) In many applications speed regulation is required The design methodology of [8] also applies cerbatim for this problem Actually, as will become clear from the developments below, most of the difficulty in our study comes from the nonlinear dependence of torque uis a cis the state variables, a problem that is clearly absent in speed control Kp(xdj) with < E L$c = - R, (2.8) < R, The feedforward terms -yl(t),-y2(t)are given by and the slip frequency U, is (2.12) Finally, the estimate of the load torque is obtained as 11 MAINRESULT 6% jL= Proposition 2.1: Consider the induction motor model (1.21, (1.6) under assumptions A.l)-A.4) and the bounded desired generated torque y , with known derivative yd Let the controller be defined as follows: -x:j 2€ where (2.13) is updated with = -g+(xj - x d ) , 6(0) = Bo E IRq, g > (2.14) Under these conditions, (2.15) lim ( y - y d ) = f + X (2.2) with all internal signals bounded Further, the field orientation objective (1.11) is attained and the observer states asymptotically converge to their true values 111 PROOFOF MAINRESULT First, we proceed to define the error equations T o this end, let n e = x -xd (3.1) where where xd5 is the controller state which satisfies xd' p [x,,, X d , Xd3, Xd4, Xdjl E IR5 (3.2) plays the role of desired values for x , see [8] for further explanations We will choose with xl xd, = p X& = - = const (3.3) P > a desired value for the stator current d component and 9, are the estimates of the currents x and x4 P, respectively, and they are obtained from the nonlinear observer -1 Lsr p Y d (3.6) Notice that if x = x d we have y = y d Further, the choices of x d and x d ensure LsrXd2 + Lrx,, = (3.7) which reflects our objective of attaining field orientation We will also define a state observation error - A I k I - I and a parameter estimation error ijL (2.7) 6- (3.8) (3.9) In terms of the error signals (3.1) we can write (1.6) as De + Ce + ( R + K ) e = (3.10) IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 38, NO 11, NOVEMBER 1993 1678 where K a diag{K K p , 0, 0, Kpb) pi perturbation vector with components ?= U1 E I R x and IC, is a + ws(LSxd2 + LsrXd4) + uxd5(Lsx2+ L s r x ) - whose derivative, taking into account the skew-symmetry of CA - F and C, looks like + K ) e - f T R , i + e T S ( x d , ) f : = -zTQz and the matrix where we have defined z [eT,iTIT H, + K,e, R,P (3.11) = -eT(R QA[R+K -ST/2 Noting that R definite iff -JXdsU - - ubx,, L'L, (LsX2 f LsrX4)PC'+ ( L s x l + LsrX3)- PGYd + KpbeS - ueTO (3.15) where we have used, where convenient, (3.3143.6) Now, if we replace the control law in the expressions above we get -Uxd5Lsri4 (3.16) CXd5LsrI3 (3.17) *2 = (3.18) UL (3.19) sr The closed-loop system equations can be written in compact form as De x, EP~ i =, 1; ,4 y ~ y ? , (from(l.3)) D e i + (CA - F ) J = - R e f (3.21) e = -g+e, (3.22) where, to get (3.21) we have used the fact that the first four equations of (1.6) can be written using the notation (2.6), (2.7) as D,I I:[ + [ C A ( u , x 5+) R,]I = M I 0 0 0 0 xd5J2 0 0 (3.23) E I R x (3.24) The system is fully described by the state equations (2.4), (3.20)-(3.22) T o study the stability of (3.20)-(3.221, consider the quadratic function 11 H, L - I T D p i + -eTDe 2 + - c0 %2g (3.25) (3.28) - (from A.3) xs xdi E P ~ (since e5 EY_) = e , ~ E (from P (3.20) ~ and (3.21) - respectively) e,i+o + Ce = - ( R + K ) e + S ( x d ) i +[O, , , , ~ $ ~where the last implication fol!ows (3.20) > It is easy to check that (2.81, (2.9) ensures (3.28) NO?, from and H , I -A,,,,,(Q)ll~11? we conclude that' e,I and are bounded, and further e and I are square integrable Convergence to zero of the torque tracking and observation errors can be established with the following chain of implications (Zdenotes the set of essentially bounded functions): *3 = *4 = + K)-'S H, x d , ,e, E Z ~ i =, 1; ,4 = (3.27) Re we see that this matrix is positive Re - - S T ( R (3.14) l/ls + K > 0, -s'21 (3.26) ast+m from square integrability and uniform continuity of e and I Internal stabilitv follows from boundedness of the state vector IV SIMULATION RESULTS The performance of the control scheme of Proposition 2.1 was investigated by simulations The numerical values of the fourpole squirrel-cage induction motor used in [8] were chosen, that is R, = 687C12,R , = 842R, L , = 84mH, L , = 85.2mH, L,, = 81.3mH, J = 03Kgm2 and b = 0.01Kgm2s-' We present here simulations of torque sinusoidal change with load torque as given in (1.12) The motor is initially in stand-still with zero iFitial conditions The initial conditions of the observer are I ( ) = [21.87,- 11, 85, 0, - 11.29IT, and the estimator initial values are zero The values of the torque load parameters are = [2.75,0.15,0.003]T.At time t = the load torque parameters are changed to = [5.5, 0.25, 0.004IT Fig shows the response of the generated and the desired torque Fig shows the rotor currents and its observed values transients The rotor speed and it corresponding reference are shown in Fig Load torque and its estimated value are illustrated in Fig Notice that good load estimation is achieved in spite of the rapid changes of the actual load torque during the rotor speed zero crossover and at t = 0.6 Finally, the appied voltages at the stator end are presented in Fig 'Notice that these two conditions insure exponential convergence when the load is known IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 38, NO 11, NOVEMBER 1993 torque (N.m) 1679 speed (rads) 20 I time (s) time (s) Fig Rotor speed, x s , and its reference, xSd, Fig I Torque response with respect to a reference signal current (A) torque (N.m) 201 10 0.2 current (A) 0.4 0.8 (a) i time (s) -30 0.2 time (s) (b) 0.4 0.6 0.8 time (s) Fig Load torque, y,, and its estimated value, j L Fig Transient of the rotor current compo-nents on the (d-q-axes frame and its pbserved values: (a) id, = x and id, = P,, (b) i,, = P, and i,, = x, and i, = 2, V CONCLUDING REMARKS 1) As discussed in [8] a key step in the derivation of the control law and the state observer is the selection of a suitable representation to obtain the skew-symmetric property of C and CA - F As explained in that note, see also [9], this is tantamount to identifying the workless forces of the system Also the definition of the controller dynamics x d s follows directly from the design methodology of [8] 2) Three are the modifications of the controller of [8] introduced here First, the selection of the desired values for the current allows us to solve the output regulation problem posed in that paper for time varying desired torques Notice that this choice is more consistent with the field orientation philosophy [6] since here we require the q coordinate of the rotor flux to be zero all the time Second, the inclusion of the term KpbeS in (2.4) allows us to inject mechanical damping to the closed loop hence relaxing the magnitude condition on the desired torque of [8] Finally, by allowing the load torque to be time varying, though linearly parameterized, i.e., (I 9), we considerably extend the realm of application of our control scheme ) The proposed control law is very simple to implement and tune The controller is always well defined, even in startup This in contrast with most existing schemes where the control calculation may cross through singularities during the transients 1680 IEEE TRANSAClTONS ON AUTOMATIC CONTROL, VOL 38, NO 11, NOVEMBER 1993 voltage (V) 60 On Invariant Polyhedra of Continuous-TimeLinear Systems 40 E B Castelan” and J C Hennet 20 Abstract-This note presents some conditions of existence of positively invariant polyhedra for linear continuous-time systems These conditions are first described algebraically, then interpreted on the basis of the system eigenstructure Then, a simple state-feedback placement method is described for solving some linear regulation problems under constraints 0 0.2 0.4 0.6 0.8 i1 time (s) voltage (V) “2 2o01 -400 // 0.2 0.4 0.6 0.8 time (s) (b) Fig The input voltages with respect to the (d-q)-axes frame: (a) U I = Vd,, (b) U Z VqS 4) The main open problem that remains to be solved is the case when we combine motor parameter estimation with state ObSeNerS 5) As pointed out in [8], the design technique used here applies directly to the fixed frame motor model of 141, 151, [7] [31 [41 [51 [61 REFERENCES C Canudas and S Seleme, “An energy minimization approach to induction motor control,” LAG Int Rep., 1992 C Canudas, R Ortega, and S Seleme, “Robot motion control using AC drives,” in h o c IEEE ICRA Conf, Atlanta, GA, USA, 1993, May 3-7 A Deluca, “Design of an exact nonlinear controller for induction motors,” IEEE Trans Automat Contr., vol 34, no 12 pp 1304-1307, 1989 I Kanellakopoulos, Adaptive control of nonlinear systems, Ph.D dissertation, Univ of Illinois, Aug VILV-ENG-91-2233, DC-134, 1991 Z Krzseminski, “Nonlinear control of induction motor,” in Proc 10th IFAC World Congress, Munich, 1987, pp 349-354 W Leonhard, “Microcomputer control of high-performance dynamic AC drives: A survey,’’ Automatica, vol 22, no 1, pp 1-19, 1986 R Marino, S Peresada, and P Valigi, “Adaptive partial feedback linerization of induction motors,” IEEE Trans Automat Contr., vol 38, no 2, pp 208-221, 1993 R Ortega and G Espinosa, “A controller design methodology for systems with physical structures: Application to Induction Motors,” in Proc IEEE CDC, Brighton, UK, 1991; also in Automatica, vol 29, no 3, pp 621-633, 1993 R Ortega and M Spong, “Adaptive motion control of rigid robots: A tutorial,” Automatica, vol 25, no 6, pp 877-888, 1989 S Seely, Electromechanical Energv Concersion New York McGraw-Hill, 1962 V J Utkin, “Sliding mode control design principles and applications to electric drives,” IEEE Trans Ind Electr., vol 40, no 1, pp 23-36, 1993 I PRELIMINARY RESULTS ON POSITIVE INVARIANCE AND POLYHEDRAL SETS Any locally stable time invariant dynamical system admits some domains in its state-space from which any state-vector trajectory cannot escape These domains are called positively invariant sets of the system If a system is subject to constraints on its state vector and can be controlled, the purpose of a regulation law can be to stabilize it while maintaining its statevector in a positively invariant set included in the admissible domain Under a state feedback regulation law, this design technique can also be used to satisfy constraints on the control vector, possibly by transferring these constraints onto the statespace The existence and characterization of positively invariant sets of dynamical systems is therefore a basic issue for many constrained regulation problems T o analyze the desired properties of a closed-loop time invariant linear system under a linear state feedback, it suffices to study the “autonomous” model: i ( t ) =Ax(t), x(t) E SH”, A E % “ * “ , t (1) Definition 1: Positive Inuariance A nonempty set Cl is a positively invariant set of system (1) if and only if for any initial state xo E R,the complete trajectory of the state vector, x(t), remains in R Or, equivalently, fi has the property e A t R E Cl V t Definition is general and the set Cl can for example be a bounded polyhedron, a cone or a vectorial subspace In the last case, positive invariance is equivalent to the well-known property of A-invariance of subspaces [lo] Definition 2: Convex Polyhedron Any nonempty convex polyhedron of tli” can be characterized by a matrix Q E !Ti‘*“ and a vector p E 9tr, r E M - (0}, n E M- (O} It is defined by: R[Q,p ] = {X E 91”; QX p } By convention, equalities and inequalities between vectors and between matrices are componentwise Without loss of generality, it can be assumed that the set of inequality constraints defining R[Q,p ] is nonredundant Let Q, be the ith row-vector of matrix Q , and p , the ith component of vector p Then (see [8]), there exists a one-to-one corresponManuscript received March 14, 1992 This work was supported in part by CAPES, Brazil E B Castelan is on leave from LCMI/EEL/UFSC, Florian6polis, Brazil He is now with the Laboratoire d‘Automatique et d’halyse des SystSmes du CNRS, 7, avenue du Colonel Roche, 31077 Toulouse France J C Hennet is with Laboratoire d’Automatique et d’Analyse des Systkmes du CNRS, 7, avenue du Colonel Roche, 31077 Toulouse France IEEE Log Number 9208710 0018-9286/93$03.00 1993 IEEE