() Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 1 / 52

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1 Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécanique. A.GLUMINEAU, IRCCyN Institut de Recherche en Communications et Cybernétique de Nantes, Nantes, France, from joint works with J. DE LEON (Universidad Autonoma de Nuevo Leon, Mexico), F. PLESTAN, D. TRAORE, M. HAMIDA, R. BOISLIVEAU (IRCCyN). IFAC WC 2008, IFAC PPPSC 2012, Groupe inter GDR MACS/SEEDS : CSE () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 1 / 52

2 Outline Introduction 1 Introduction () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 2 / 52

3 Outline Introduction 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 2 / 52

4 Outline Introduction 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 2 / 52

5 Outline Introduction 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case 4 Sensorless control results 4.a IPMSM case Benchmark trajectories Simulation results 4.b IM case Benchmark trajectories () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 2 / 52

6 Outline Introduction 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case 4 Sensorless control results 4.a IPMSM case Benchmark trajectories Simulation results 4.b IM case Benchmark trajectories 5 Conclusions () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 2 / 52

7 Introduction () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 3 / 52

8 Introduction () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 3 / 52

9 Introduction Classical control scheme x: Fluxes, currents and speed u: Voltages T l : Load torque (disturbance) y: Currents, speed and load torque () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 3 / 52

10 Introduction Classical control scheme x: Fluxes, currents and speed u: Voltages T l : Load torque (disturbance) y: Currents, speed and load torque - Cost, - Breakdowns of sensor maintenance cost, - Fault tolerant control. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 3 / 52

11 Introduction Classical control scheme x: Fluxes, currents and speed u: Voltages T l : Load torque (disturbance) y: Currents, speed and load torque - Cost, - Breakdowns of sensor maintenance cost, - Fault tolerant control. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 3 / 52

12 Introduction Classical control scheme Software sensor x: Fluxes, currents and speed u: Voltages T l : Load torque (disturbance) y: Currents, speed and load torque - Cost, - Breakdowns of sensor maintenance cost, - Fault tolerant control. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 3 / 52

13 Introduction Classical control scheme Software sensor x: Fluxes, currents and speed u: Voltages T l : Load torque (disturbance) y: Currents, speed and load torque - Cost, - Breakdowns of sensor maintenance cost, - Fault tolerant control. Control without mechanical sensor + low speed problems () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 3 / 52

14 Introduction Classical control scheme Software sensor x: Fluxes, currents and speed u: Voltages T l : Load torque (disturbance) y: Currents, speed and load torque - Cost, - Breakdowns of sensor maintenance cost, - Fault tolerant control. Control without mechanical sensor + low speed problems Loss of observability, () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 3 / 52

15 Introduction Classical control scheme Software sensor x: Fluxes, currents and speed u: Voltages T l : Load torque (disturbance) y: Currents, speed and load torque - Cost, - Breakdowns of sensor maintenance cost, - Fault tolerant control. Control without mechanical sensor + low speed problems Loss of observability, Robustness. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 3 / 52

16 Introduction Industrial sensorless inverter experimental results Figure : 1. Test on the sensorless benchmark. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 4 / 52

17 Outline Observer without mechanical sensor 2.a Observer design for non linear systems 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case 4 Sensorless control results 4.a IPMSM case Benchmark trajectories Simulation results 4.b IM case Benchmark trajectories 5 Conclusions () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 5 / 52

18 Observer without mechanical sensor Observer design for non linear systems (1) 2.a Observer design for non linear systems () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 6 / 52

19 Observer without mechanical sensor Observer design for non linear systems (1) 2.a Observer design for non linear systems No general constructive algorithm for a non linear system: x = f ( x) + g( x)u, y = h( x). (1) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 6 / 52

20 Observer without mechanical sensor Observer design for non linear systems (1) 2.a Observer design for non linear systems No general constructive algorithm for a non linear system: x = f ( x) + g( x)u, y = h( x). (1) A first solution: to find a generalized transformation X = φ( x, u, u,, u (q 1) ) such that the nonlinear system (1) is equivalent to a linear one modulo an input-output (NL) injection: Ẋ = AX + ϕ(y, u, u,, u (q) ), ỹ = CX. (2) Hypothesis: A et C are in observability canonical form. Then a observer (Luenberger Like) can be: ˆX = A ˆX + ϕ(y, u, u,, u (q) ) + K C (X ˆX) Exponentiel convergence tuned by gain K, Observation of system (1) given by ˆ x = φ 1 ( ˆX, u, u,, u (q 1) )., See Souleiman, Glumineau, Guay, Respondek. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 6 / 52

21 Observer without mechanical sensor Observer design for non linear systems (2) 2.a Observer design for non linear systems () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 7 / 52

22 Observer without mechanical sensor Observer design for non linear systems (2) 2.a Observer design for non linear systems A second solution: to find a state transformation X = φ( x, u) such that the nonlinear system (1) is equivalent to an affine one modulo an input-output (NL) injection: Ẋ = A(y, u)x + ϕ(y, u) ỹ = CX. (3) with some observability hypothesis (see Besançon and al.), an Kalman Like observer can be designed: ˆX = A(y, u) ˆX + ϕ(y, u) S 1 C T (C ˆX ỹ) Ṡ = θs A(y, u) T S SA(y, u) + C T C ŷ = C ˆX There exists θ > 0 such that system (4) is an observer for (3) and ˆX X λ exp( τt), τ > 0. Observation of system (1) is given by ˆ x = φ 1 ( ˆX, u, u,, u (q 1) ). See Souleiman, Glumineau, Besancon (4) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 7 / 52

23 Observer without mechanical sensor Observer design for non linear systems (2) 2.a Observer design for non linear systems A second solution: to find a state transformation X = φ( x, u) such that the nonlinear system (1) is equivalent to an affine one modulo an input-output (NL) injection: Ẋ = A(y, u)x + ϕ(y, u) ỹ = CX. (3) with some observability hypothesis (see Besançon and al.), an Kalman Like observer can be designed: ˆX = A(y, u) ˆX + ϕ(y, u) S 1 C T (C ˆX ỹ) Ṡ = θs A(y, u) T S SA(y, u) + C T C ŷ = C ˆX There exists θ > 0 such that system (4) is an observer for (3) and ˆX X λ exp( τt), τ > 0. Observation of system (1) is given by ˆ x = φ 1 ( ˆX, u, u,, u (q 1) ). See Souleiman, Glumineau, Besancon Otherwise... Other observers can be designed with Kalman Like High gain observer, Interconnected observers,... (see Besançon, Hammouri...) (4) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 7 / 52

24 Sensorless problem Observer without mechanical sensor 2.a Observer design for non linear systems Means () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 8 / 52

25 Sensorless problem Observer without mechanical sensor 2.a Observer design for non linear systems Means Design of an adaptive interconnected high-gain observers for the estimation of the magnetic(fluxes), mechanical (speed, load torque) variables, stator resistance value,... () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 8 / 52

26 Observer without mechanical sensor 2.a Observer design for non linear systems Sensorless problem Means Design of an adaptive interconnected high-gain observers for the estimation of the magnetic(fluxes), mechanical (speed, load torque) variables, stator resistance value,... Design of High Order Sliding Mode Controllers to achieve speed and flux tracking (IM) or speed and optimum torque (IPMSM) with robust performances () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 8 / 52

27 Observer without mechanical sensor 2.a Observer design for non linear systems Sensorless problem Means Design of an adaptive interconnected high-gain observers for the estimation of the magnetic(fluxes), mechanical (speed, load torque) variables, stator resistance value,... Design of High Order Sliding Mode Controllers to achieve speed and flux tracking (IM) or speed and optimum torque (IPMSM) with robust performances Analysis of the stability of the Schemes. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 8 / 52

28 Observer without mechanical sensor 2.a Observer design for non linear systems Sensorless problem Means Design of an adaptive interconnected high-gain observers for the estimation of the magnetic(fluxes), mechanical (speed, load torque) variables, stator resistance value,... Design of High Order Sliding Mode Controllers to achieve speed and flux tracking (IM) or speed and optimum torque (IPMSM) with robust performances Analysis of the stability of the Schemes. Validation on Sensorless Control Benchmarks from french research group of the CNRS Control of Electric Systems, () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 8 / 52

29 Outline Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case 4 Sensorless control results 4.a IPMSM case Benchmark trajectories Simulation results 4.b IM case Benchmark trajectories 5 Conclusions () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 9 / 52

30 Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Nonlinear model of the IPM synchronous motor in the dq frame From the PARK et CONCORDIA transformations: { ẋ = f (x) + g(x)u Σ NL : y = h(x) (5) with x = ( i d i q Ω θ ) T, u = ( ud, u q ) T, y = ( h1, h 2 ) T = ( id, i q ) T f (x) = p R s i L d + p Lq Ωi d L q d Rs i L q p L d Ωi q L q d p 1 φ L q f Ω s J (L d L q)i d i q fv J Ω + p J φ f i q 1 J T l Ω 1 0 and g(x) = 0 L d 1 L q with i d, i q, u d, u q: dq currents, voltages and Ω, T l, R s, θ: motor speed, load torque, stator resistance, rotor position. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 10 / 52

31 Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Nonlinear model + uncertainties of the IPMSM in the synchronous frame (d-q) or i d = k 4 i d + k 5 Ωi q + k 6 v d i q = k 9 i q + k 8 Ωi d + k 7 Ω + k 10 v q Ω = k 1 i d i q + k 3 Ω + k 2 i q θ = Ω (6) with Parameter uncertainties k 1 = k 01 + δk 1 = p J (L d L q), k 2 = k 02 + δk 2 = pφ f J, k 3 = k 03 + δk 3 = fv J, k 4 = k 04 + δk 4 = Rs, k 5 = k 05 + δk 5 = p Lq, k 6 = k 06 + δk 6 = 1, L d L d L d k 7 = k 07 + δk 7 = pφ f L q, k 8 = k 08 + δk 8 = pl d L q, k 9 = k 09 + δk 9 = Rs L q, k 10 = k δk 10 = 1 L q. (7) k 0i (1 i 10) the nominal value of the concerned parameter, δk i the uncertainty on this parameter such that δk i δk 0i < k 0i with δk 0i a known positive bound. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 11 / 52

32 Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Interconnected Model The IPMSM model (1) can be written in an interconnected form: Σ 1 { Ẋ1 = A 1 (y)x 1 + g 1 (X 2, u) y 1 = C 1 X 1 (8) Σ 2 { Ẋ2 = A 2 (y)x 2 + g 2 (X 1, X 2, u) + ΦT l y 2 = C 2 X 2 (9) T l and R s are assumed to be piecewise functions of time. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 12 / 52

33 Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Interconnected Model The IPMSM model (1) can be written in an interconnected form: Σ 1 { Ẋ1 = A 1 (y)x 1 + g 1 (X 2, u) y 1 = C 1 X 1 (8) Σ 2 { Ẋ2 = A 2 (y)x 2 + g 2 (X 1, X 2, u) + ΦT l y 2 = C 2 X 2 (9) T l and R s are assumed to be piecewise functions of time. where [ ] [ ] [ iq idω X 1 =, X R 2 =, A 1 ( ) = s 0 iq L q 0 0 ], A 2 ( ) = [ 0 p Lq L d i q 0 fv J ] g 1 ( ) = [ p L d L q Ωid p φ f L q Ω + 1 L q u q 0 ], g 2 ( ) = Rs L d i d + 1 L d u d p J Φ f i qi d + p J (L d L q)i q, [ Φ = 0 1 J ], C 1 = C 2 = [1 0]. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 12 / 52

34 Design (1) Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Objectives Design two adaptive interconnected observers for Σ 1 and Σ 2 to estimate the speed the load torque and the stator resistance. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 13 / 52

35 Design (1) Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Objectives Design two adaptive interconnected observers for Σ 1 and Σ 2 to estimate the speed the load torque and the stator resistance. Assumption 1 (u, X 2 ) and (u, X 1 ) are respectively regularly persistent inputs for Σ 1 and Σ 2. (i.e. the inputs do not force the system to stay in an observability loss area) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 13 / 52

36 Design (1) Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Objectives Design two adaptive interconnected observers for Σ 1 and Σ 2 to estimate the speed the load torque and the stator resistance. Assumption 1 (u, X 2 ) and (u, X 1 ) are respectively regularly persistent inputs for Σ 1 and Σ 2. (i.e. the inputs do not force the system to stay in an observability loss area) Remark 1. X 1 and X 2 are respectively considered as inputs for subsystem Σ 2 and Σ 1. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 13 / 52

37 Design (1) Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Objectives Design two adaptive interconnected observers for Σ 1 and Σ 2 to estimate the speed the load torque and the stator resistance. Assumption 1 (u, X 2 ) and (u, X 1 ) are respectively regularly persistent inputs for Σ 1 and Σ 2. (i.e. the inputs do not force the system to stay in an observability loss area) Remark 1. X 1 and X 2 are respectively considered as inputs for subsystem Σ 2 and Σ 1. Remark 2. To design robust adaptive observers, it will be necessary to verify (easy to do): A 1 (y) is globally Lipschitz w.r.t.x 1. A 2 (y) is globally Lipschitz w.r.t.x 1. g 1 (X 2 ) is globally Lipschitz w.r.t.x 2 uniformly w.r.t.(u, y). g 2 (X 1, X 2 ) is globally Lipschitz w.r.t.x 1, X 2 uniformly w.r.t.(u, y). () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 13 / 52

38 Design (2) Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Observers Then adaptive interconnected observers of Σ 1 and Σ 2 are given by: (10) Ż 1 = Ṡ 1 = A 1 (y)z 1 + g 1 (y 2, Z 2, u) + S 1 1 CT 1 (y 1 ŷ 1 ) ρ 1 S 1 A T 1 (y)s 1 S 1 A 1 (y) + C1 T C 1 ŷ 1 = C 1 Z 1 Ż 2 = A 2 (y)z 2 + g 2 (y, Z 1, u) + ΦˆT l + KC1 T (y 1 ŷ 1 ) +(ϖλs 1 θ ΛT C2 T + ΓS 1 x C2 T )(y 2 ŷ 2 ) ˆT l = ϖs 1 θ ΛT C2 T (y 2 ŷ 2 ) + B(Z 1 )(y 1 ŷ 1 ) Ṡ x = ρ x S x A T 2 (Z 1, y)s x S x A 2 (Z 1, y) + C2 T C 2 Ṡ θ = ρ θ S θ + Λ T C2 T C 2Λ Λ = (A 2 (y) ΓSx 1 C2 T C 2)Λ + Φ ŷ 2 = C 2 Z 2 ] T ] T with Z 1 = [îq ˆRs et Z2 = [îd ˆΩ are the estimated states of X1 and X 2. S 1 1 CT 1 is the gain of the observer (10), ϖλs 1 θ ΛT C2 T + ΓS 1 x C2 T and KCT 1 are the gains of the observer (11). (11) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 14 / 52

39 Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Observer Stability analysis under parameters uncertainties, [HAMIDA, 12], [Besançon, 06] Observer Stability Let us define: ɛ 1 = X 1 Z 1, ɛ 2 = X 2 Z 2, ɛ 3 = T l ˆT l. We take the following Lyapunov candidate function with V 1 = ɛ T 1 S 1ɛ 1, V 2 = ɛ T 2 S 2ɛ 2 et V 3 = ɛ T 3 S 3ɛ 3. The time derivative of V o is: V o = V 1 + V 2 + V 3 V o δ(v 1 + V 2 + V 3 ) + µ( V 1 + V 2 ). δv o + µψ V o, (12) where ψ > 0, such that ψ V 1 + V 2 + V 3 > V 1 + V 2 and δ = min(δ 1, δ 2, δ 3 ) with δ 1 = ρ 1 λψ 1 > 0, δ 2 = ρ 2 λ 2 ψ 2 λ 1 ψ 1 λ 3 > 0, δ 3 = ρ n λ 2 ψ 2 > 0 and µ an uncertainties parameter. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 15 / 52

40 Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Discussion () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 16 / 52

41 Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Discussion No parameters variation (i.e. nominal case) µ = 0, by choosing ρ 1, ρ 2 and ρ 3 such that δ 1 > 0, δ 2 > 0 and δ 3 > 0 implies V o δv o, () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 16 / 52

42 Observer without mechanical sensor 2.b Interior Permanent Magnet Synchronous Motor case (IPMSM) Discussion No parameters variation (i.e. nominal case) µ = 0, by choosing ρ 1, ρ 2 and ρ 3 such that δ 1 > 0, δ 2 > 0 and δ 3 > 0 implies V o δv o, Parameters variation µ 0 then V o (1 ς)δv o, ɛ ɛ such that B(0, ɛ ) is a ball which the radius depends of the uncertainties parameter µ and of the gains of the observer. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 16 / 52

43 Outline Observer without mechanical sensor 2.c Induction Motor case (IM) 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case 4 Sensorless control results 4.a IPMSM case Benchmark trajectories Simulation results 4.b IM case Benchmark trajectories 5 Conclusions () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 17 / 52

44 Observer without mechanical sensor 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Nonlinear model of the induction motor (dq frame) From Park and Concorda transformations, one gets ẋ = f (x) + g(x)u y = h(x) (13) with x = ( ) i sd i sq φ rd φ rq Ω T l R T s, u = ( ) u sd u T sq, y = ( y 1 baφ rd + bpωφ rq γi sd + ω si sq m 1 0 ) y T 2 = ( i sd ) i T sq baφ rq bpωφ rd γi sq ω si sd 0 m 1 aφ rd + (ω s pω)φ rq + am sr i sd 0 0 f (x) = aφ rq (ω s pω)φ rd + am sr i sq and g(x) = 0 0. m(φ rd i sq φ rqi sd ) cω 1 J l with i sd, i sq, φ rd, φ rq, u sd, u sq: d q currents, flux, voltage, Ω, T l, ω s, R s: motor speed, load torque, stator pulsation, stator resistance. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 18 / 52

45 Observer without mechanical sensor 2.c Induction Motor case (IM) Interconnected observers: principle [Besançon, 98], [Besançon, 06] () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 19 / 52

46 Observer without mechanical sensor 2.c Induction Motor case (IM) Interconnected observers: principle [Besançon, 98], [Besançon, 06] A class of nonlinear systems may be seen as the interconnection between several subsystems, such that each of these subsystems satisfies some required properties to build an observer. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 19 / 52

47 Observer without mechanical sensor 2.c Induction Motor case (IM) Interconnected observers: principle [Besançon, 98], [Besançon, 06] A class of nonlinear systems may be seen as the interconnection between several subsystems, such that each of these subsystems satisfies some required properties to build an observer. The main idea is to design adaptive interconnected observers for the whole system, from the separate synthesis of an observer for each subsystem, assuming that, for each of these separate observer, the state of the other subsystem is available. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 19 / 52

48 Observer without mechanical sensor 2.c Induction Motor case (IM) Design (1) The IM model (13) can be written in an interconnection form { X Σ 1 : 1 = A 1 (X 2, y)x 1 + g 1 (u, y, X 2, X 1 ) + ΦT l (14) y 1 = C 1 X 1 { X Σ 2 : 2 = A 2 (X 1 )X 2 + g 2 (u, y, X 1, X 2 ) (15) y 2 = C 2 X 2 X 1 = ( i sd, Ω, R s ) T, X2 = ( i sq, φ rd, φ rq ) T, u = ( usd, u sq ) T, y = ( isd, i sq ) T, C 1 = C 2 = ( ). T l is the unknown load torque supposed a piecewise constant function. R s is the unknown stator resistance value supposed constant. 0 bpφ rq m 1 i sd A 1 ( ) = mφ rq c 0, A 2 ( ) = γ 1 bpω ab 0 a pω, pω a g 1 ( ) = γ 1i sd + abφ rd + m 1 u sd + ω si sq mφ rd i sq, 0 g 2 ( ) = m 1R si sq ω si sd + m 1 u sq ω sφ rq + am sr i sd ω sφ rd + am sr i sq, Φ = 0 1 J 0 () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 20 / 52

49 Design (2) Observer without mechanical sensor 2.c Induction Motor case (IM) Objectives Design of two adaptive interconnected high-gains observers for Σ 1 and Σ 2 in order to estimate mechanical variables (speed, load torque) and stator resistance value and magnetic variables (fluxes) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 21 / 52

50 Design (2) Observer without mechanical sensor 2.c Induction Motor case (IM) Objectives Design of two adaptive interconnected high-gains observers for Σ 1 and Σ 2 in order to estimate mechanical variables (speed, load torque) and stator resistance value and magnetic variables (fluxes) Assumption 1 1. X 1 and X 2 are respectively considered as inputs for subsystem Σ 2 and Σ 1, 2. (u, X 2 ) and (u, X 1 ) are respectively regularly persistent inputs for Σ 1 and Σ 2. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 21 / 52

51 Design (2) Observer without mechanical sensor 2.c Induction Motor case (IM) Objectives Design of two adaptive interconnected high-gains observers for Σ 1 and Σ 2 in order to estimate mechanical variables (speed, load torque) and stator resistance value and magnetic variables (fluxes) Assumption 1 1. X 1 and X 2 are respectively considered as inputs for subsystem Σ 2 and Σ 1, 2. (u, X 2 ) and (u, X 1 ) are respectively regularly persistent inputs for Σ 1 and Σ 2. Assumption 2 g 1 (u, y, X 2, X 1 ) is globally Lipschitz with respect to X 2, X 1 and uniformly w.r.t. (u, y). () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 21 / 52

52 Design (2) Observer without mechanical sensor 2.c Induction Motor case (IM) Objectives Design of two adaptive interconnected high-gains observers for Σ 1 and Σ 2 in order to estimate mechanical variables (speed, load torque) and stator resistance value and magnetic variables (fluxes) Assumption 1 1. X 1 and X 2 are respectively considered as inputs for subsystem Σ 2 and Σ 1, 2. (u, X 2 ) and (u, X 1 ) are respectively regularly persistent inputs for Σ 1 and Σ 2. Assumption 2 g 1 (u, y, X 2, X 1 ) is globally Lipschitz with respect to X 2, X 1 and uniformly w.r.t. (u, y). Remark It is easy to verify that A 1 (X 2, y) is globally Lipschitz w.r.t. X 2, A 2 (X 1 ) is globally Lipschitz w.r.t. X 1 and g 2 (u, y, X 2, X 1 ) is globally Lipschitz w.r.t. X 2, X 1 and uniformly w.r.t. (u, y) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 21 / 52

53 Design (3) Observer without mechanical sensor 2.c Induction Motor case (IM) Observers Then, the adaptive interconnected observers of Σ 1 and Σ 2 read as O 1 : Ż 1 = A 1 (Z 2, y)z 1 + g 1 (u, y, Z 2, Z 1 ) + ΦˆT l +(ϖλs 1 3 ΛT C1 T + ΓS 1 1 CT 1 )(y 1 ŷ 1 ) + KC2 T (y 2 ŷ 2 ) ˆT l = ϖs 1 3 ΛT C1 T (y 1 ŷ 1 ) + B 1 (Z 2 )(y 2 ŷ 2 ) + B 2 (Z 2 )(y 1 ŷ 1 ) Ṡ 1 = θ 1 S 1 A T 1 (Z 2, y)s 1 S 1 A 1 (Z 2, y) + C1 T C 1 Ṡ 3 = θ 3 S 3 + Λ T C1 T C 1Λ Λ = (A 1 (Z 2, y) ΓS 1 1 CT 1 C 1)Λ + Φ ŷ 1 = C 1 Z 1 Ż 2 = A 2 (Z 1 )Z 2 + g 2 (u, y, Z 1, Z 2 ) + S 1 2 CT 2 (y 2 ŷ 2 ) O 2 : S 2 = θ 2 S 2 A T 2 (Z 1)S 2 S 2 A 2 (Z 1 ) + C2 T C 2 ŷ 2 = C 2 Z 2 () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 22 / 52

54 Design (4) Observer without mechanical sensor 2.c Induction Motor case (IM) Definitions Z 1 = ( î sd, ˆΩ, ˆR ) T s, Z2 = ( ) T î sq, ˆφ rd, ˆφ rq are the estimated states. θ1, θ 2, θ 3 are positive constants and S 1, S 2 are symmetric positive definite matrices [Besançon, 96], S 3 (0) > 0. B 1 (Z 2 ) = km ˆφ rd, B 2 (Z 2 ) = km ˆφ rq, K = k c1 0 0 k c2 0 0, Γ = , where k, α k c1, k c2, α and ϖ are positive constants. Note that (ϖλs 1 3 ΛT C1 T + ΓS 1 1 CT 1 ) and KCT 2 are the gains of observer (O 1) and S 1 2 CT 2 gain of observer (O 2 ). is the () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 23 / 52

55 Observer without mechanical sensor 2.c Induction Motor case (IM) Observer Stability analysis under parameters uncertainties (and tuning), [Traore, 07], [Besançon, 06] Observer Stability and Tuning Let us define V o = V 1 + V 2 + V 3 a candidate Lyapunov function where V 1 = ɛ T 1 S 1ɛ 1, V 2 = ɛ T 2 S 2ɛ 2 and V 3 = ɛ T 3 S 3ɛ 3. with ɛ 1, ɛ 2 and ɛ 3 are the errors between the state and its estimation. The time derivative of V o is: V o δ(v 1 + V 2 + V 3 ) + µ( V 1 + V 2 ) where ψ > 0, such that ψ V 1 + V 2 + V 3 > V 1 + V 2. δv o + µψ V o. (16) δ = min(δ 1, δ 2, δ 3 ) and µ = max(µ 6, µ 7 ) parameters uncertainties, δ 1 = θ 1 2k 12 2k 1 k 16 µϕ 1 µ 9 ϕ 3 > 0, δ 3 = θ 3 + 2k 10 µ 8 ϕ 2 µ 9 ϕ 3 > 0, and ϕ i (i = 1, 2, 3) ]0, 1[ () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 24 / 52

56 Observer without mechanical sensor 2.c Induction Motor case (IM) Discussion () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 25 / 52

57 Observer without mechanical sensor 2.c Induction Motor case (IM) Discussion Known parameters µ = 0, to choose θ 1, θ 2 and θ 3 such that δ 1 > 0, δ 2 > 0 and δ 3 > 0 then V o δv o, () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 25 / 52

58 Observer without mechanical sensor 2.c Induction Motor case (IM) Discussion Known parameters µ = 0, to choose θ 1, θ 2 and θ 3 such that δ 1 > 0, δ 2 > 0 and δ 3 > 0 then V o δv o, Parameters variation µ 0 then V o (1 ς)δv o, ɛ µψ ςδ, 1 > ς > 0 () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 25 / 52

59 Outline High Order Sliding Mode Control (HOSMC) 3.a HOSMC design 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case 4 Sensorless control results 4.a IPMSM case Benchmark trajectories Simulation results 4.b IM case Benchmark trajectories 5 Conclusions () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 26 / 52

60 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design Hypothesis and Definition HOSM (1) Consider the uncertain nonlinear system ẋ = f (x) + g(x)v (17) y = σ(x, t) (18) with x the state variable, v the input control and σ(x, t) a smooth output function (sliding variable) defined to satisfy the control objective (tracking trajectory,...). H1. The relative degree r d of (17) with respect to s is assumed to be constant and known, and the associated zero dynamics are stable. After a possible preliminary I/0 linearization feedback (computed with the nominal parameters) applied to the nonlinear system (17) there exists χ and Γ such that σ (r d ) = χ( ) + Γ( )v (19) H2. Functions χ( ) and Γ( ) are such that there exist K m IR +, K M IR +, C 0 IR + with χ C 0, 0 < K m < Γ < K M. Furthermore, control input v is bounded. (20) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 27 / 52

61 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design HOSM controller design (1) The design of the controller is obtained in two steps () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 28 / 52

62 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design HOSM controller design (1) The design of the controller is obtained in two steps Design of a switching variable S(σ, σ,...) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 28 / 52

63 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design HOSM controller design (1) The design of the controller is obtained in two steps Design of a switching variable S(σ, σ,...) Design of a control to maintain, from t = 0, the system trajectories on the corresponding switching surface which ensures the establishment of a r th order sliding mode, in spite of uncertainties in a fixed finite time t F. = for t > t F, σ = 0, σ = 0,..., σ (r) = 0, with r the relative degree r d. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 28 / 52

64 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design HOSM controller design (2) Switching variable. S denotes the switching variable defined as S = σ (r 1) F (r 1) [ (t) + λ r 2 σ (r 2) F (r 2) (t) ] + + λ 0 [σ(x, t) F(t)] (21) with () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 29 / 52

65 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design HOSM controller design (2) Switching variable. S denotes the switching variable defined as with S = σ (r 1) F (r 1) (t) + λ r 2 [ σ (r 2) F (r 2) (t) ] + + λ 0 [σ(x, t) F(t)] (21) λ r 2,, λ 0 defined such that P(z) = z (r 1) + λ r 2 z (r 2) + + λ 0 is a Hurwitz polynomial in the complex variable z, () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 29 / 52

66 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design HOSM controller design (2) Switching variable. S denotes the switching variable defined as with S = σ (r 1) F (r 1) (t) + λ r 2 [ σ (r 2) F (r 2) (t) ] + + λ 0 [σ(x, t) F(t)] (21) λ r 2,, λ 0 defined such that P(z) = z (r 1) + λ r 2 z (r 2) + + λ 0 is a Hurwitz polynomial in the complex variable z, F(t) is a C r function defined such that and σ (k) (x(0), 0) F (k) (0) = 0 σ (k) (x(t F ), t F ) F (k) (t F ) = 0 (0 k r 1), t F the desired convergence time. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 29 / 52

67 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design HOSM controller design (3) More precisely, the problem consists in finding the function F(t) such that from initial and final conditions σ(x(0), 0) = F(0), σ(x(t F ), t F ) = F(t F ) = 0, σ(x(0), 0) = F(0), σ(x(t F ), t F ) = F(t F ) = 0,. σ (r 1) (x(0), 0) = F (r 1) (0), σ (r 1) (x(t F ), t F ) = F (r 1) (t F ) = 0 (22) A solution for F(t) reads as (1 j r) F(t) = Ke Ft T σ (r j) (0) (23) with F a 2r 2r-dimensional Hurwitz matrix (strictly negative eigenvalues), T a 2r 1-dimensional vector and j such that σ (r j) (0) 0 and bounded. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 30 / 52

68 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design HOSM controller design (4) Then, the switching variable S (7) reads as S = σ (r 1) KF r 1 e Ft T σ (r j) [ (0) + λ r 2 σ (r 2) KF r 2 e Ft T σ (r j) (0) ] [ + + λ 0 σ(x, t) Ke Ft T σ (r j) (0) ]. (24) H3. There exists a finite positive constant Θ IR + such that KF r e Ft T σ (r j )(0) λ r 2 [ σ (r 1) KF r 1 e Ft T σ (r j) (0) ] λ 0 [ σ(x, t) KFe Ft T σ (r j) (0) ] < Θ (25) Equation S = 0 describes the desired dynamics which satisfy the finite time stabilization of vector [σ (r 1) σ (r 2) σ] T to zero. Then, the switching manifold on which system is forced to slide on, via a discontinuous control v, is defined as S = {x S = 0} (26) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 31 / 52

69 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design HOSM controller design (5) The discontinuous control law v must force the system trajectories to slide on S, to reach in finite time the origin and to maintain the system at the origin. Theorem Consider nonlinear system (17) with a relative degree r d with respect to σ(x, t). Suppose that it is minimum phase and that hypotheses H 1 and H 2 are fulfilled. Let r be the sliding mode order and 0 < t F < the desired convergence time. Define S IR by (24) with K unique solution of (23) and that assumption H3 is fulfilled. The control input v defined by with v = α sign(s) (27) α C 0 + Θ + η K m, (28) C 0, K m defined by (20), Θ defined by (25), η > 0, leads to the establishment of a r th order sliding mode with respect to σ. The convergence time is t F. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 32 / 52

70 High Order Sliding Mode Control (HOSMC) 3.a HOSMC design SM1 and this HOSM Figure : 2. Comparaison of SM of order 1 and this HOSM design. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 33 / 52

71 Outline High Order Sliding Mode Control (HOSMC) 3.b IPMSM case 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case 4 Sensorless control results 4.a IPMSM case Benchmark trajectories Simulation results 4.b IM case Benchmark trajectories 5 Conclusions () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 34 / 52

72 High Order Sliding Mode Control (HOSMC) 3.b IPMSM case The control objective is double: Ω Ω ref i d i dref where i dref is computed by the Maximum-Torque-Per-Ampere method [Uddin et al. 2007] to increase the efficiency of the drive. HOSM controller design (1) The design of the controller is given by: The choice of sliding variables for system (2), to obtain trajectory tracking to Ω ref and i dref. The sliding variables are chosen as: [ ] [ ] σω Ω Ωref (t) σ = = (29) σ id i d i dref (t) The vector relative degree of σ = [2 1] T. Design of a control to maintain, from t = 0, the system trajectories on a switching manifold which ensures the establishment of a r th order sliding mode, in spite of uncertainties in a fixed finite time t F. = for t > t F, σ = 0, σ = 0,..., σ (r) = 0, with r the relative degree. As the discontinuous control introduced in the following, could imply a chattering phenomenon, the controller is chosen as a [3 2] order sliding mode. = The discontinuity control part will act on the first time derivative of the control inputs. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 35 / 52

73 High Order Sliding Mode Control (HOSMC) 3.b IPMSM case HOSM controller design (2) The second and first time derivatives of σ Ω and σ id are: σ (2) Ω = χ 1 + Γ 11 u d + Γ 12 u q, σ (1) i d = χ 2 + Γ 21 u q. (30) Applying the following static state feedback [ ] ud = Γ 1 u q 0 [ χ 0 + v] (31) denoting [ Γ110 Γ Γ 0 = 120 Γ ] [ ] χ10, χ 0 = χ 20 (32) with Γ 0, χ 0 the nominal values terms and v := [v d v q] T the new control vector, then [ σω (3) σ id (2) ] = [ ] [ χ1 Γ + 11 Γ 12 χ 2 Γ 21 0 ] [ vd vq ] (33) with v d, v q the new inputs defined later. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 36 / 52

74 High Order Sliding Mode Control (HOSMC) 3.b IPMSM case HOSM controller design (3) Switching vector Then, following [Plestan et al.2008], the switching vector S = [S Ω S id ] T is written as S Ω = σ (2) Ω F (2) 1 + λ 11 [ σ Ω F 1 ] + λ 12 [σ Ω F 1 ] S id = σ (1) F i 2 + λ 21 [σ id F 2 ] d (34) with F 1 (t) = K 1 F 1 e F 1t T 1 σ Ω (0) et F 2 (t) = K 2 e F 2t T 2 σ id (0). and λ 11 = 2ζ Ω ω nω, λ 12 = 2ω 2 nω, λ 21 = ω nd with F i being a 2r 2r-dimensional stable matrix, and T i being a 2r 1-dimensional vector. = Finite time convergence (t F ) to σ = 0, σ = 0,..., σ (r) = 0. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 37 / 52

75 High Order Sliding Mode Control (HOSMC) 3.b IPMSM case HOSM controller design (3) Discontinuous input From these switching surfaces (9), the discontinuous control input acting the new inputs is defined by: [ ] [ ] vd α1.sign(s = Ω ). (35) vq α 2.sign(S id ) where with Θ i such that ([Plestan et al, 2008], [Traore et al, 2008]): α i C 0i + Θ i + η i K mij (36) Θ > K ctf r e Ft σ (r j) c (0) λ r 2 [σ (r 1) c K ctf r 1 e Ft σ (r j) c (0)] λ 0 [ σ c(x, t) K ctfe Ft σ (r j) c (0)], (37) with η 1 := η Ω, η 2 := η d. For the IPMS motor, as the uncertainties are bounded, then there exist gains α Ω and α d such that Ṡ Ω S Ω η Ω S Ω and Ṡi d S id η d S id. Then, the finite time convergence of tracking errors is ensured. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 38 / 52

76 Outline High Order Sliding Mode Control (HOSMC) 3.c IM case 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case 4 Sensorless control results 4.a IPMSM case Benchmark trajectories Simulation results 4.b IM case Benchmark trajectories 5 Conclusions () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 39 / 52

77 Principle and objectives High Order Sliding Mode Control (HOSMC) 3.c IM case Fied Oriented Control strategy: FOC [Blaschke, 72] Make electromagnetic torque proportional to the stator current use the IM such as a Direct Motor. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 40 / 52

78 Principle and objectives High Order Sliding Mode Control (HOSMC) 3.c IM case Fied Oriented Control strategy: FOC [Blaschke, 72] Make electromagnetic torque proportional to the stator current use the IM such as a Direct Motor. Control law () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 40 / 52

79 Principle and objectives High Order Sliding Mode Control (HOSMC) 3.c IM case Fied Oriented Control strategy: FOC [Blaschke, 72] Make electromagnetic torque proportional to the stator current use the IM such as a Direct Motor. Control law High Order Sliding Mode controller to achieve speed tracking in finite time () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 40 / 52

80 High Order Sliding Mode Control (HOSMC) 3.c IM case Principle and objectives Fied Oriented Control strategy: FOC [Blaschke, 72] Make electromagnetic torque proportional to the stator current use the IM such as a Direct Motor. Control law High Order Sliding Mode controller to achieve speed tracking in finite time High Order Sliding Mode controller to achieve flux tracking in finite time () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 40 / 52

81 Design (1) High Order Sliding Mode Control (HOSMC) 3.c IM case Stator pulsation ω s For FOC, one needs ω s = pω + a Msr φ rd i sq. One defines ω s = p ˆΩ + a Msr ˆφ rd i sq with ω s an estimated stator frequency, β 1 = (isq îsq) k ωs β 1 ˆφrd Msr σl sl r and k ωs > 0. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 41 / 52

82 High Order Sliding Mode Control (HOSMC) 3.c IM case HOSM controller design (6) Sliding vector s = [ ] [ σ1 ˆφ = rd φ ] σ 2 ˆΩ Ω (38) Relative degree of σ 1 and σ 2 =2 In order to limit chattering effect, design of a 3 rd order sliding mode controller The control input u is composed by a linearizing-decoupling term (in nominal case) and a term v defined such that [ ] [ ] v1 αφ sign(s v = = φ ) v2 α Ω sign(s Ω ) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 42 / 52

83 High Order Sliding Mode Control (HOSMC) 3.c IM case HOSM controller design (7) The switching vector reads as () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 43 / 52

84 High Order Sliding Mode Control (HOSMC) 3.c IM case HOSM controller design (7) The switching vector reads as For t t F. S φ = σ 1 χ φ, S Ω = σ 2 χ Ω where { χφ = K φ F 2 e Ft T σ 1 (0) 2ζ φ ω nφ ( σ 1 K φ Fe Ft T σ 1 (0)) ω 2 nφ (σ 1 K φ e Ft T σ 1 (0)) χ Ω = K Ω F 2 e Ft T σ 2 (0) 2ζ Ω ω nω ( σ 2 K Ω Fe Ft T σ 2 (0)) ω 2 nω (σ 2 K Ω e Ft T σ 2 (0)) () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 43 / 52

85 High Order Sliding Mode Control (HOSMC) 3.c IM case HOSM controller design (7) The switching vector reads as For t t F. S φ = σ 1 χ φ, S Ω = σ 2 χ Ω where { χφ = K φ F 2 e Ft T σ 1 (0) 2ζ φ ω nφ ( σ 1 K φ Fe Ft T σ 1 (0)) ω 2 nφ (σ 1 K φ e Ft T σ 1 (0)) χ Ω = K Ω F 2 e Ft T σ 2 (0) 2ζ Ω ω nω ( σ 2 K Ω Fe Ft T σ 2 (0)) ω 2 nω (σ 2 K Ω e Ft T σ 2 (0)) For t > t F. S φ = σ 1 + 2ζ φ ω nφ σ 1 + ω 2 nφ σ 1 S Ω = σ 2 + 2ζ Ω ω nω σ 2 + ω 2 nω σ 2 () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 43 / 52

86 High Order Sliding Mode Control (HOSMC) 3.c IM case HOSM controller design (7) The switching vector reads as For t t F. S φ = σ 1 χ φ, S Ω = σ 2 χ Ω where { χφ = K φ F 2 e Ft T σ 1 (0) 2ζ φ ω nφ ( σ 1 K φ Fe Ft T σ 1 (0)) ω 2 nφ (σ 1 K φ e Ft T σ 1 (0)) χ Ω = K Ω F 2 e Ft T σ 2 (0) 2ζ Ω ω nω ( σ 2 K Ω Fe Ft T σ 2 (0)) ω 2 nω (σ 2 K Ω e Ft T σ 2 (0)) For t > t F. S φ = σ 1 + 2ζ φ ω nφ σ 1 + ω 2 nφ σ 1 S Ω = σ 2 + 2ζ Ω ω nω σ 2 + ω 2 nω σ 2 Stability of Controller-Observer scheme Stability proof formally established in the paper. () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 43 / 52

87 Outline Sensorless control results 4.a IPMSM case 1 Introduction 2 Observer without mechanical sensor 2.a Observer design for non linear systems 2.b Interior Permanent Magnet Synchronous Motor synchronous motor case (IPMSM) Model in (d-q) axes Observer stability analysis 2.c Induction Motor case (IM) Induction motor model in a dq rotating frame Adaptive Interconnected observer design Stability analysis of observer 3 High Order Sliding Control (HOSMC) 3.a HOSMC design 3.b IPMSM case 3.c IM case 4 Sensorless control results 4.a IPMSM case Benchmark trajectories Simulation results 4.b IM case Benchmark trajectories 5 Conclusions () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 44 / 52

88 Sensorless control results 4.a IPMSM case Motor parameters Table : Motor parameters (nominal values) Current 6A Torque 5.3Nm Speed 3000 rpm ψ f Wb R s 3.25 Ω p 3 L d 18 mh L q 34 mh J kg.m 2 f v kg.m 2 s 1 () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 45 / 52

89 Sensorless control results 4.a IPMSM case Benchmark (defined within the framework of the CNRS french workgroup CSE) 350 a 6 b Speed (rad/s) Load torque (N.m) Time (s) Time (s) Figure : 3. Sensorless industriel benchmark. a. Speed reference, b. Load torque Electric System Control group SEEDS/MACS () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 46 / 52

90 Sensorless control results 4.a IPMSM case 350 a 10 b Speed (rad/s) Speed error (rad/s) Ω real Ω obs Time (s) Time (s) Figure : 4. a. Nominal case Observed speed, Measured speed b. Observation error () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 46 / 52

91 Sensorless control results 4.a IPMSM case a R s R s est b Resistance (ohm) Resistance error (ohm) Time (s) Time (s) Figure : 5. a. Nominal case Estimated resistance, Resistance b. Estimation error () Observation, Estimation et Commande robuste non linéaires d actionneurs électriques sans capteur mécaniqu 46 / 52