Multi-Objective Robust Control of Rotor/Active Magnetic Bearing Systems

Multi-Objective Robust Control of Rotor/Active Magnetic Bearing Systems İbrahim Sina Kuseyri Ph.D. Dissertation June 13, 211 İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 1 / 51

Outline 1 Introduction Overview Applications 2 System Dynamics Magnetic Bearings Rotordynamics 3 Robust Control Controller Design Model Uncertainty Robust Stability and Performance Numerical Results and Simulations 4 Multi-Objective LPV Control Linear Parametrically Varying (LPV) Systems Mixed Performance Specifications Numerical Results and Simulations İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 2 / 51

Overview Radial electromagnetic bearing 5 1 15 2 25 5 1 15 2 25 3 35 İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 4 / 51

Overview Radial electromagnetic bearing 5 1 15 2 25 5 1 15 2 25 3 35 Horizontal rotor with active magnetic bearings (AMBs) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 4 / 51

Advantages of rotor/amb systems No mechanical wear and friction. No lubrication therefore non-polluting. High circumferential speeds possible (more than 3 m/s). Operation in severe and demanding environments. Easily adjustable bearing characteristics (stiffness, damping). Online balancing and unbalance compensation. Online system parameter identification possible. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 5 / 51

Applications Satellite flywheels Turbomachinery High-speed milling and grinding spindles Electric motors Turbomolecular pumps Blood pumps Computer hard disk drives, x-ray devices,... İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 6 / 51

Electromagnetic Bearings The AMB model considered is based on the zero leakage assumption: Magnetic flux in a high permeability magnetic structure with small air gaps is confined to the iron and gap volumes. In the configuration above, the forces in orthogonal directions are almost decoupled and can be calculated separately. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 8 / 51

Electromagnetic bearings Two opposing electromagnets at orthogonal directions cause the force ( ( ) 2 ( ) ) 2 i+ i F r = F + F = k M s r s + r on the rotor. The magnetic bearing constant k M is k M := µ A A nc 2 cos α M 4 with α M denoting the angle between a pole and magnet centerline. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 9 / 51

Electromagnetic bearings The non-linearities of the magnetic force are generally reduced by adding a high bias current i to the control currents i c in each control axis. Linearization in one axis around the operating point leads to F r = Fr OP + F r i (i c i c OP ) + F r OP r (r r OP ). OP At i c OP = and r OP =, the linearized magnetic bearing force of the bearing for small currents and small displacements is given by F r,lin = k i i c k s r with the actuator gain k i and the open loop negative stiffness k s defined as k i := 4k M i s 2 and k s := 4k M i 2 s 3 İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 1 / 51

Rotordynamics Equations of motion for a rigid rotor may be derived from F = Ṗ = d dt (M r v), and M = Ḣ = d dt (Iω). f a1 f b1 bearing A a f a4 b bearing B f b4 m ub,s z, ξ φ m ub,c CG d 2 m ub,c θ f a3 f a2 y, η ψ x, ζ f b3 f b2 İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 11 / 51

Rotordynamics The equations of motion for the four degrees of freedom are ẍ = 1 [f A,x + f B,x + M r g + m ub,s Ω 2 d cos (Ωt + ϕ s )], M r 2 2 ÿ = 1 [f A,y + f B,y + M r g + m ub,s Ω 2 d sin (Ωt + ϕ s )], M r 2 2 ψ = 1 (a + b) [ ΩI p θ + a( fa,y ) + b(f I B,y ) + m r 2 ub,c Ω 2 d sin (Ωt + ϕ c )], θ = 1 (a + b) [ΩI p ψ + a(f I A,x ) + b( f B,x ) m r 2 ub,c Ω 2 d cos(ωt + ϕ c )]. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 12 / 51

Rotor/AMB model in state-space The equations of motion for the electromechanical system in the state-space form are ( ) I ẋ r = x A G (Ω) r + B wr w + B ur u + ḡ, A S where x r := (x y ψ θ ẋ ẏ ψ θ ) T, u = (i ca,x i ca,y i cb,x i cb,y ) T, w = ( 1 2 m ub,sd 1 2 m ub,cd) T. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 13 / 51

Rotor/AMB model in state-space The equations of motion for the electromechanical system in the state-space form are ( ) I ẋ r = x A G (Ω) r + B wr w + B ur u + ḡ, A S where x r := (x y ψ θ ẋ ẏ ψ θ ) T, u = (i ca,x i ca,y i cb,x i cb,y ) T, w = ( 1 2 m ub,sd 1 2 m ub,cd) T. Control objective is to stabilize the system and to minimize the rotor displacements (whirl) with acceptable control effort. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 13 / 51

Controller design d i y m K u + G v + n w { n d i u e } z u P G + + v y m y K İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 15 / 51

Controller design Exogenous Input w u P z y Performance Output Manipulated Input Measurement (Feedback) K İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 16 / 51

Controller design Exogenous Input w u P z y Performance Output Manipulated Input Measurement (Feedback) K Q: How to choose K? İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 16 / 51

Controller design Exogenous Input w u P z y Performance Output Manipulated Input Measurement (Feedback) K Q: How to choose K? A: Minimize the size (e.g. H or H 2 -norm) of the closed-loop transfer function M from w to z. w M z İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 16 / 51

H 2 and H -norms The definitions are M := sup σ ( M(jω) ) ( Note : σ(m) := λ max (M M) ) ω 1 M 2 := Trace ( M(jω) 2π M(jω) ) dω İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 17 / 51

H 2 and H -norms The definitions are M := sup σ ( M(jω) ) ( Note : σ(m) := λ max (M M) ) ω 1 M 2 := Trace ( M(jω) 2π M(jω) ) dω For SISO LTI systems, M = sup ω M(jω) = peak of the Bode plot M 2 = M(jω) 2 dω area under the Bode plot 1 2π İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 17 / 51

Frequency Weighting Can fine-tune the solution by using frequency weights on w and z. ũ d i do ẽ W u W i W o W e d i d o r i r + u + + W i r i y m r K + G + v + e y m + n W + n ñ W db W db W db ω c log ω ω l ω u log ω ω c log ω İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 18 / 51

Model uncertainty Uncertainty in Rotor/AMB Models İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 19 / 51

Model uncertainty Uncertainty in Rotor/AMB Models Model Parameter Uncertainty (such as AMB stiffness k s ) Neglected High Frequency Dynamics (high frequency flexible modes of the rotor) Nonlinearities (such as hysteresis effects in AMB) Neglected Dynamics (such as vibrations of rotor blades) Setup Variations (e.g., a controller for an AMB milling spindle should function with tools of different mass) Changing System Dynamics (gyroscopic effects change the location of the poles at different operating speeds) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 19 / 51

Closed-loop rotor/amb system with uncertainty p q W p p q W q w W w w u P z y Wz z P K σ ( Wp 1 (jω) (jω) Wq 1 (jω) ) = σ ( ) (jω) 1 ω Re [ ] δks I := ΩI İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 2 / 51

Closed-loop rotor/amb system with uncertainty Overall system in the state-space form p q w W p W w p w u P q z y W q Wz z ẋ = Ax + B p p + B w w + B u u q = C q x + D qw w z = C z x + D zu u P y = C y x + D yw w p = q K İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 21 / 51

Robust stability and performance w p M q z w N z İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 22 / 51

Robust stability and performance w p M q z w N z Nominal Stability (NS) M is internally stable Nominal Performance NS, and σ ( M(jω) ) < γ ω R e Robust Stability (RS) NS, and N to be stable : σ ( (jω) ) 1 ω R e Robust Performance RS, and σ ( N(jω) ) < γ : σ ( (jω) ) 1 ω R e İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 22 / 51

Robust stability - Structured singular value Transfer matrix of the closed-loop uncertain system in LFT form is N = M zw + M zp (I M qp ) 1 M qw. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 23 / 51

Robust stability - Structured singular value Transfer matrix of the closed-loop uncertain system in LFT form is N = M zw + M zp (I M qp ) 1 M qw. For robust stability ( I M qp (s) (s) ) 1 should have no poles in C + for all with σ( ) 1. Meaning that = det ( I M qp (jω) ), with σ( ) 1, ω R e. Therefore, robust stability holds if and only if inf {σ( ) : det( I M qp (jω) ) =, ω R e } > 1. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 23 / 51

Robust stability - Structured singular value Transfer matrix of the closed-loop uncertain system in LFT form is N = M zw + M zp (I M qp ) 1 M qw. For robust stability ( I M qp (s) (s) ) 1 should have no poles in C + for all with σ( ) 1. Meaning that = det ( I M qp (jω) ), with σ( ) 1, ω R e. Therefore, robust stability holds if and only if inf {σ( ) : det( I M qp (jω) ) =, ω R e } > 1. Inversion leads to the definition µ (M) := 1 inf {σ( ) : det ( I M qp (jω) ) = } < 1 ω R e. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 23 / 51

Numerical Results - System Data Section A-A ds dd g A displacement sensors touch-down bearing A magnetic bearing A touch-down bearing B magnetic bearing B a b sa sb A LD LS Symbol Value Unit Symbol Value Unit Symbol Value Unit M S 85.9 kg L S 1.5 m s 2. 1 3 m M D 77.1 kg L D.5 m s 1.5 1 3 m I r 17.28 kg m 2 d S.1 m i 3. A I p 2.41 kg m 2 d D.5 m k M 7.8455 1 5 N m 2 /A 2 a.58 m s A.73 m k s 3.535 1 5 N/m b.58 m s B.73 m k i 235.4 N/A İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 24 / 51

Numerical Results - Weighting functions W u = ( 38 s + 12 ) I 4 W e = s + 5 ( ) s +.5 I 4 s +.1 35 12 3 1 25 2 8 Gain [db] 15 1 Wu Gain [db] 6 4 We 5 2 5 1 2 1 1 2 1 4 1 6 Frequency [rad/s] 1 2 1 1 2 1 4 1 6 Frequency [rad/s] İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 25 / 51

Results with the H controllers for the nominal system Maximum operation speed = 3 rpm ( 314.2 rad/s) Singular Values Singular Values 2 2 Singular Values (db) 2 4 Singular values of controller K1 Singular Values (db) 2 4 Singular values of controller K2 6 6 8 8 1 1 2 1 1 2 1 4 1 6 Frequency (rad/sec) 1 1 2 1 1 2 1 4 1 6 Frequency (rad/sec) 4 Singular Values 4 Singular Values 2 2 Singular Values (db) 2 4 6 Closed loop SVs with K1 Singular Values (db) 2 4 6 Closed loop SVs with K2 8 8 1 1 12 12 14 1 2 1 1 2 1 4 1 6 Frequency (rad/sec) 14 1 2 1 1 2 1 4 1 6 Frequency (rad/sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 26 / 51

Results with the H controllers for the nominal system Table: H performance with K 1 for different design parameters Maximum speed (rpm) Maximum mass center displacement (m) γ 15.25 1 3 7.96 3.25 1 3 97.6 6.25 1 3 99.81 15.5 1 3 89.57 3.5 1 3 99.24 6.5 1 3 1.7 Table: H performance with K 2 for different design parameters Maximum speed (rpm) Maximum mass center displacement (m) γ 15.25 1 3 11.41 3.25 1 3 15.42 6.25 1 3 31.77 15.5 1 3 12.62 3.5 1 3 21.5 6.5 1 3 52.1 İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 27 / 51

Critical speeds (eigenfrequencies) Pole Zero Map 6 4 x: Openloop eigenfrequencies at standstill (rad/s) Imaginary Axis 2 2 117 (x2) 65.8 (x2) 65.8 (x2) 117 (x2) 4 6 25 2 15 1 5 5 1 15 2 25 Real Axis Closedloop Phaseshift for journal displacements(unbalance channel) 1 XA YA XB 5 YB 5 Phase shift with K1 1 15 12 2 1 1 1 1 2 1 3 1 4 Frequency (Speed) [rad/s] Closedloop Phaseshift for journal displacements(unbalance channel) XA 2 YA XB 4 YB 6 8 1 12 14 16 18 15 2 1 1 1 1 2 1 3 1 4 Frequency[rad/s] İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 28 / 51

Results with the reduced order H controllers The H norm of the closed-loop system at 3 rpm with the reduced ordered controllers K 1r and K 2r (4 states are eliminated) increases from 99.24 to 529.55 and from 21.5 to 62.7 respectively. Singular Values Singular Values 6 4 4 2 2 Singular Values (db) 2 4 6 Closed loop SVs with K1r Singular Values (db) 2 4 6 Closed loop SVs with K2r 8 8 1 1 12 12 14 1 2 1 1 2 1 4 1 6 Frequency (rad/sec) 14 1 2 1 1 2 1 4 1 6 Frequency (rad/sec) Closedloop Phaseshift for journal displacements(unbalance channel) 5 XA YA XB YB 5 1 15 17 2 1 2 1 1 1 1 1 1 2 1 3 1 4 Frequency[rad/s] Closedloop Phaseshift for journal displacements(unbalance channel) XA 2 YA XB YB 4 6 8 1 12 14 16 18 185 2 1 2 1 1 1 1 1 1 2 1 3 1 4 Frequency[rad/s] İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 29 / 51

Robust stability of the uncertain closed-loop system Keeping the uncertainty on the bearing stiffness constant (25%), robust stability of the closed-loop system is tested for several maximum operating speeds with µ-analysis. Moreover, keeping the operation speed constant (3 rpm), robust stability is tested for uncertainty in bearing stiffness. mu 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.9 mu.9.8.7.6.5.4.3.2.1.8 3 35 4 45 5 55 6 Maximum rotor speed (RPM) 5 1 15 2 25 Uncertainty in bearing stiffness (%) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 3 / 51

Results with the robust H controller Singular values of the controller and the closed-loop system for a maximum operating speed of 485 rpm are shown below. H performance γ of the system for Ω max = 485 rpm is 47.86. Order of the controller K 3 (twelve) can not be reduced since it leads to the instability of the closed-loop system. Singular Values Singular Values 2 2 Closed loop SVs with K3 Singular Values (db) 2 4 Singular values of controller K3 Singular Values (db) 2 4 6 6 8 8 1 1 2 1 1 2 1 4 1 6 Frequency (rad/sec) 1 1 2 1 1 2 1 4 1 6 Frequency (rad/sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 31 / 51

Simulations Simulation Environment in SIMULINK İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 32 / 51

Simulations Simulation Environment in SIMULINK (Rotor/AMB) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 33 / 51

Simulations We analyze the H performance of the closed-loop system using the controller K 2 in the simulations. Disturbance acting on the system, i.e., unbalance force and sensor/electronic noise, are shown below. 1 8 6.4.3 Sensor Noise Unbalance Force (Newtons) 4 2 2 4 6 8 Volts.2.1.1.2 1.1.2.3.4.5 Time (sec).3 1 2 3 4 5 6 Time (msec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 34 / 51

Simulations 6 2 5 Rotor displacement in Bearing A (x direction) 1 Rotor displacement in Bearing A (y direction) 4 3 1 X A (Volts) 2 Y A (Volts) 2 1 3 4 1 5 2.1.2.3.4.5 Time (sec) 6.1.2.3.4.5 Time (sec) 2 4 1 Control current for Bearing A (x axis) 3 Control current for Bearing A (y axis) 2 ic,ax (Amperes) 1 ic,ay (Amperes) 1 2 3 1 4.1.2.3.4.5 Time (sec) 2.1.2.3.4.5 Time (sec) Rotor position and control currents during start-up İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 35 / 51

Simulations Mass center displacement (eccentricity) due to unbalance of the rotor is assumed to be.25 1 3 m in the simulations. Peak value of the vibration (except the transient) is less than.1 V, corresponding to 14 1 6 m. Therefore, the H controller K 2 reduces the unbalance whirl amplitude of the rotor more than 95%..5 Rotor displacement in Bearing A (x direction) 1 Rotor displacement in Bearing A (y direction).5 X A (Volts).5 1 Y A (Volts).5 1 1.5 1.5 2.1.2.3.4.5 Time (sec) 2.1.2.3.4.5 Time (sec) 4 4 Control current for Bearing A (x axis) Control current for Bearing A (y axis) 3 3 2 2 ic,ax (Amperes) 1 ic,ay (Amperes) 1 1 1 2 2 3.1.2.3.4.5 Time (sec) 3.1.2.3.4.5 Time (sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 36 / 51

LPV Systems ẋ A(ρ) B w (ρ) B u (ρ) x z = C z (ρ) D zw (ρ) D zu (ρ) w y C y (ρ) D yw (ρ) D yu (ρ) u Parameters ρ(t) are measured in real-time with sensors for control. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 38 / 51

LPV Systems ẋ A(ρ) B w (ρ) B u (ρ) x z = C z (ρ) D zw (ρ) D zu (ρ) w y C y (ρ) D yw (ρ) D yu (ρ) u Parameters ρ(t) are measured in real-time with sensors for control. Hence controller is also parameter-dependent, using the available real-time information of the parameter variation. w u P ρ y z ρ K ρ İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 38 / 51

Mixed Performance Specifications Suppose a specific control task leads to the generalized LPV plant ẋ A(ρ) B 1 (ρ) B 2 (ρ) z 1 z 2 = C 1 (ρ) D 11 (ρ) D 12 (ρ) x C 2 (ρ) D 21 (ρ) D 22 (ρ) w u y C(ρ) D(ρ) Using an LPV controller, K(ρ, ρ), the closed-loop system can be described in the form ẋ cl A B ( ) z 1 = C 1 D 1 xcl w z 2 C 2 D 2 İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 39 / 51

Mixed Performance Specifications ẋ cl A B ( ) z 1 = C 1 D 1 xcl w z 2 C 2 D 2 L 2 gain of the w z 1 channel is defined as α opt := inf sup K K w 2 z 1 2 w 2 where K := {set of all stabilizing controllers}. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 4 / 51

Mixed Performance Specifications ẋ cl A B ( ) z 1 = C 1 D 1 xcl w z 2 C 2 D 2 L 2 gain of the w z 1 channel is defined as α opt := inf sup K K w 2 z 1 2 w 2 where K := {set of all stabilizing controllers}. To quantify the gain of the channel w z 2 we use the induced norm Remark: z 2 := β opt := inf sup K K w 2 z 2 w 2 z(t)t z(t) dt <, z := ess sup t R z(t) <. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 4 / 51

Mixed Performance Specifications We can use a single Lyapunov function to achieve both of the control objectives (though conservatively) and the problem can be defined as minimizing an upper bound β m under the constraint α < α m. This leads to defining the mixed objective functional I ( K(X) ) := inf {β m a function X(ρ) satisfying α < α m and β < β m } from the solution of the following infinite dimensional LMIs for all (ρ, ρ): X + A T X + X A X B C T X = X T 1, B T X I D1 T, C 2 X 1 C 2 β m I, C 1 D 1 α 2 m I İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 41 / 51

Mixed Performance Specifications We can use a single Lyapunov function to achieve both of the control objectives (though conservatively) and the problem can be defined as minimizing an upper bound β m under the constraint α < α m. This leads to defining the mixed objective functional I ( K(X) ) := inf {β m a function X(ρ) satisfying α < α m and β < β m } from the solution of the following infinite dimensional LMIs for all (ρ, ρ): X + A T X + X A X B C T X = X T 1, B T X I D1 T, C 2 X 1 C 2 β m I, C 1 D 1 α 2 m I where X is defined to be X := m i=1 X ρ i ρ i İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 41 / 51

Controller Synthesis A full order controller K which satisfying the mixed objective functional I ( K(X) ) can be constructed, if there exist parameter-dependent functions X(ρ), Y(ρ) with X, Y, and E(ρ), F(ρ), G(ρ) with G(ρ) = D K (ρ), such that X + A T X + XA + FC + (FC) T XB 1 + FD (C 1 + D 12 GC) T (XB 1 + FD) T I (D 11 + D 12 GD) T C 1 + D 12 GC D 11 + D 12 GD α 2 mi, Ý + AY + YAT + B 2 E + (B 2 E) T B 1 + B 2 GD (C 1 Y + D 12 E) T (B 1 + B 2 GD) T I (D 11 + D 12 GD) T, C 1 Y + D 12 E D 11 + D 12 GD α 2 mi β mi C 2 Y + D 22 E C 2 + D 22 GC (C 2 Y + D 22 E) T Y I. (C 2 + D 22 GC) T I X Inequalities above consist of convex but infinite-dimensional optimization problem. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 42 / 51

LPV Model for Rotor/AMB Systems ẋ A(Ω) B 1 (Ω 2 ) B 2 z 1 z 2 = C 1 D 12 x C 2 w u y C D System has parameter dependence to Ω(t) due to gyroscopic effects and to Ω 2 (t) due to unbalance forces. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 43 / 51

LPV Model for Rotor/AMB Systems ẋ A(Ω) B 1 (Ω 2 ) B 2 z 1 z 2 = C 1 D 12 x C 2 w u y C D System has parameter dependence to Ω(t) due to gyroscopic effects and to Ω 2 (t) due to unbalance forces. Letting all of the parameter dependent functions to have an affine structure, (such as X(Ω) = X + ΩX 1 ) infinite-dimensional inequalities for controller synthesis become a series of LMIs with linear dependence on Ω and linear/quadratic/cubic dependence on Ω. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 43 / 51

LPV Model for Rotor/AMB Systems ẋ A(Ω) B 1 (Ω 2 ) B 2 z 1 z 2 = C 1 D 12 x C 2 w u y C D System has parameter dependence to Ω(t) due to gyroscopic effects and to Ω 2 (t) due to unbalance forces. Letting all of the parameter dependent functions to have an affine structure, (such as X(Ω) = X + ΩX 1 ) infinite-dimensional inequalities for controller synthesis become a series of LMIs with linear dependence on Ω and linear/quadratic/cubic dependence on Ω. Hence one only needs to check these matrix inequalities at the vertices of the polytope defined by P = [Ω min,ω max ] [ Ω min, Ω max ] İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 43 / 51

Numerical Results with LPV (L 2 ) Controllers LPV controller for the parameter (rotor speed) dependent rotor/amb system can be designed via semidefinite programming satisfying several LMIs at all the vertices of the convex hull. Singular values of the closed-loop system at two different speeds; 3 and 6 rpm are shown below: Singular Values Singular Values 5 5 5 Singular Values (db) 1 15 LPV Closed loop SVs at 3 RPM Singular Values (db) 5 1 LPV Closed loop SVs at 6 RPM 15 2 25 2 3 1 2 1 1 2 1 4 1 6 1 8 Frequency (rad/sec) 25 1 2 1 1 2 1 4 1 6 1 8 Frequency (rad/sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 44 / 51

Results with LPV (L 2 ) Controllers Controller to robustly stabilize the system with L 2 performance is synthesized inside a four-dimensional convex hull with the rotor speed range from rad/s to 614 rad/s (6 rpm), and angular acceleration range from -15 rad/s 2 to 15 rad/s 2. L 2 performance α of the closed-loop LPV system at the instantaneous speed 6 RPM is 56.31. Note that this performance is achieved with a controller of the form (ẋk u ) ( AK (Ω, = Ω) B K (Ω) C K (Ω) D K (Ω) )( xk If the matrix function X used for the stabilization of the closed-loop system is assumed to be constant (time-invariant), then the controller matrices will not depend on the angular acceleration of the rotor, and the controller will be of the form (ẋk u ) = ( AK (Ω) B K (Ω) C K (Ω) D K (Ω) y )( xk y ) ) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 45 / 51

Results with LPV (L 2 ) Controllers Comparing the L 2 performance of the controllers, it can be said that there is virtually no loss of performance if the controller is constructed without the information on angular acceleration of the rotor. Table: L 2 performance of LPV closed-loop systems at 3 RPM Structure of X and Y α Controller Form X = X + ΩX 1 Y = Y + ΩY 1 15.92 Acceleration Feedback X = X Y = Y + ΩY 1 19.13 No Acc. Feedback X = X Y = Y 27.56 No Acc. Feedback Table: L 2 performance of LPV closed-loop systems at 6 RPM Structure of X and Y α Controller Form X = X + ΩX 1 Y = Y + ΩY 1 56.31 Acceleration Feedback X = X Y = Y + ΩY 1 65.42 No Acc. Feedback X = X Y = Y 12.29 No Acc. Feedback İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 46 / 51

Numerical Results with Multi-objective LPV Controller A multi-objective LPV controller with mixed performance specification is synthesized within the same convex hull as the single objective LPV controller for a maximum operating speed of 6 rpm. Generalized L 2 L performance β m of the multi-objective LPV controller is found to be 364.4, with L 2 performance level α m of 72.12 at 6 rpm. Singular Values Singular Values 4 5 2 Singular Values (db) 2 4 SVs of Multi objective Controller at 6 RPM Singular Values (db) 5 1 Closed loop SVs of Multi objective LPV System at 6 RPM 15 6 8 2 1 1 2 1 1 2 1 4 1 6 Frequency (rad/sec) 25 1 2 1 1 2 1 4 1 6 1 8 Frequency (rad/sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 47 / 51

Simulations with Multi-objective LPV Controller Simulations for the LPV system are made using the LFR Toolbox from ONERA for MATLAB R -Simulink. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 48 / 51

Simulations with Multi-objective LPV Controller A pulse signal with 1 V amplitude and.25 seconds duration and is injected into the loop at.2 seconds of simulation time at the input of the controller. Control current and rotor position at bearing A in y-axis for LPV control with L 2 performance and with mixed performance is shown in the figures. 4.4 3.2 2 ic,ay (Amperes) 1 1 Y A.2.4.6 2.8 3 1 4.1.2.3.4.5 Time (sec) 1.2.1.2.3.4.5 Time (sec) Figure: Control current and rotor displacement with LPV L 2 control İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 49 / 51

.1.2.3.4.5 4.4 3.2 2 ic,ay (Amperes) 1 1 Y A.2.4.6 2.8 3 1 4 Time (sec) 1.2.1.2.3.4.5 Time (sec) Figure: Control current and rotor displacement with LPV L 2 control 2.5 2 1.5 1.4.2 ic,ay.5.5 1 1.5 2 Y A.2.4.6.8 2.5.1.2.3.4.5 Time (sec) 1.1.2.3.4.5 Time (sec) Figure: Control current and rotor displacement with LPV mixed control İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 5 / 51

Conclusion Comparing the results, it is clear that the peak values of both the control current and rotor position are suppressed in the closed-loop system with the multi-objective controller. Hence mixed control provides additional flexibility with respect to transients. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, 211 51 / 51