Multi-Objective Robust Control of Rotor/Active Magnetic Bearing Systems
|
|
- Jody Lambert
- 6 years ago
- Views:
Transcription
1 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, / 51
2 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, / 51
3 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, / 51
4 Overview Radial electromagnetic bearing İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
5 Overview Radial electromagnetic bearing Horizontal rotor with active magnetic bearings (AMBs) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
6 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, / 51
7 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, / 51
8 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, / 51
9 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, / 51
10 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, / 51
11 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 İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
12 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, / 51
13 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, / 51
14 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, / 51
15 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, / 51
16 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, / 51
17 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, / 51
18 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, / 51
19 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, / 51
20 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, / 51
21 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, / 51
22 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, / 51
23 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, / 51
24 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, / 51
25 Model uncertainty Uncertainty in Rotor/AMB Models İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
26 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, / 51
27 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, / 51
28 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, / 51
29 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, / 51
30 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, / 51
31 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, / 51
32 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. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
33 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, / 51
34 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, / 51
35 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 m M D 77.1 kg L D.5 m s m I r kg m 2 d S.1 m i 3. A I p 2.41 kg m 2 d D.5 m k M N m 2 /A 2 a.58 m s A.73 m k s N/m b.58 m s B.73 m k i N/A İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
36 Numerical Results - Weighting functions W u = ( 38 s + 12 ) I 4 W e = s + 5 ( ) s +.5 I 4 s Gain [db] 15 1 Wu Gain [db] 6 4 We Frequency [rad/s] Frequency [rad/s] İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
37 Results with the H controllers for the nominal system Maximum operation speed = 3 rpm ( 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 K Frequency (rad/sec) Frequency (rad/sec) 4 Singular Values 4 Singular Values 2 2 Singular Values (db) Closed loop SVs with K1 Singular Values (db) Closed loop SVs with K Frequency (rad/sec) Frequency (rad/sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
38 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 Table: H performance with K 2 for different design parameters Maximum speed (rpm) Maximum mass center displacement (m) γ İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
39 Critical speeds (eigenfrequencies) Pole Zero Map 6 4 x: Openloop eigenfrequencies at standstill (rad/s) Imaginary Axis (x2) 65.8 (x2) 65.8 (x2) 117 (x2) Real Axis Closedloop Phaseshift for journal displacements(unbalance channel) 1 XA YA XB 5 YB 5 Phase shift with K Frequency (Speed) [rad/s] Closedloop Phaseshift for journal displacements(unbalance channel) XA 2 YA XB 4 YB Frequency[rad/s] İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
40 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 to and from 21.5 to 62.7 respectively. Singular Values Singular Values Singular Values (db) Closed loop SVs with K1r Singular Values (db) Closed loop SVs with K2r Frequency (rad/sec) Frequency (rad/sec) Closedloop Phaseshift for journal displacements(unbalance channel) 5 XA YA XB YB Frequency[rad/s] Closedloop Phaseshift for journal displacements(unbalance channel) XA 2 YA XB YB Frequency[rad/s] İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
41 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 mu Maximum rotor speed (RPM) Uncertainty in bearing stiffness (%) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
42 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 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) Frequency (rad/sec) Frequency (rad/sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
43 Simulations Simulation Environment in SIMULINK İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
44 Simulations Simulation Environment in SIMULINK (Rotor/AMB) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
45 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 Sensor Noise Unbalance Force (Newtons) Volts Time (sec) Time (msec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
46 Simulations Rotor displacement in Bearing A (x direction) 1 Rotor displacement in Bearing A (y direction) X A (Volts) 2 Y A (Volts) Time (sec) Time (sec) Control current for Bearing A (x axis) 3 Control current for Bearing A (y axis) 2 ic,ax (Amperes) 1 ic,ay (Amperes) Time (sec) 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, / 51
47 Simulations Mass center displacement (eccentricity) due to unbalance of the rotor is assumed to be m in the simulations. Peak value of the vibration (except the transient) is less than.1 V, corresponding to 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) Time (sec) Time (sec) 4 4 Control current for Bearing A (x axis) Control current for Bearing A (y axis) ic,ax (Amperes) 1 ic,ay (Amperes) Time (sec) Time (sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
48 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, / 51
49 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, / 51
50 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, / 51
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, / 51
52 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, / 51
53 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, / 51
54 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. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
55 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, / 51
56 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, / 51
57 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, / 51
58 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, / 51
59 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, / 51
60 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, / 51
61 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 Singular Values (db) 1 15 LPV Closed loop SVs at 3 RPM Singular Values (db) 5 1 LPV Closed loop SVs at 6 RPM Frequency (rad/sec) Frequency (rad/sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
62 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. İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
63 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 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 y ) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
64 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 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, / 51
65 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 Acceleration Feedback X = X Y = Y + ΩY No Acc. Feedback X = X Y = Y 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 Acceleration Feedback X = X Y = Y + ΩY No Acc. Feedback X = X Y = Y No Acc. Feedback İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
66 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 at 6 rpm. Singular Values Singular Values 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 Frequency (rad/sec) Frequency (rad/sec) İ. Sina Kuseyri (B.U. Mech.E.) Robust Control of Rotor/AMB Systems June 13, / 51
67 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, / 51
68 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 ic,ay (Amperes) 1 1 Y A Time (sec) 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, / 51
69 ic,ay (Amperes) 1 1 Y A Time (sec) Time (sec) Figure: Control current and rotor displacement with LPV L 2 control ic,ay Y A Time (sec) 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, / 51
70 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, / 51
Lecture 6. Chapter 8: Robust Stability and Performance Analysis for MIMO Systems. Eugenio Schuster.
Lecture 6 Chapter 8: Robust Stability and Performance Analysis for MIMO Systems Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 6 p. 1/73 6.1 General
More informationControl of Chatter using Active Magnetic Bearings
Control of Chatter using Active Magnetic Bearings Carl R. Knospe University of Virginia Opportunity Chatter is a machining process instability that inhibits higher metal removal rates (MRR) and accelerates
More informationDesign of Robust AMB Controllers for Rotors Subjected to Varying and Uncertain Seal Forces
Downloaded from orbit.dtu.dk on: Nov 9, 218 Design of Robust AMB Controllers for Rotors Subjected to Varying and Uncertain Seal Forces Lauridsen, Jonas Skjødt; Santos, Ilmar Published in: Mechanical Engineering
More informationApplication of Neuro Fuzzy Reduced Order Observer in Magnetic Bearing Systems
Application of Neuro Fuzzy Reduced Order Observer in Magnetic Bearing Systems M. A., Eltantawie, Member, IAENG Abstract Adaptive Neuro-Fuzzy Inference System (ANFIS) is used to design fuzzy reduced order
More informationDenis ARZELIER arzelier
COURSE ON LMI OPTIMIZATION WITH APPLICATIONS IN CONTROL PART II.2 LMIs IN SYSTEMS CONTROL STATE-SPACE METHODS PERFORMANCE ANALYSIS and SYNTHESIS Denis ARZELIER www.laas.fr/ arzelier arzelier@laas.fr 15
More informationIntroduction to Control (034040) lecture no. 2
Introduction to Control (034040) lecture no. 2 Leonid Mirkin Faculty of Mechanical Engineering Technion IIT Setup: Abstract control problem to begin with y P(s) u where P is a plant u is a control signal
More informationSpacecraft Attitude Control with RWs via LPV Control Theory: Comparison of Two Different Methods in One Framework
Trans. JSASS Aerospace Tech. Japan Vol. 4, No. ists3, pp. Pd_5-Pd_, 6 Spacecraft Attitude Control with RWs via LPV Control Theory: Comparison of Two Different Methods in One Framework y Takahiro SASAKI,),
More information8 A First Glimpse on Design with LMIs
8 A First Glimpse on Design with LMIs 8.1 Conceptual Design Problem Given a linear time invariant system design a linear time invariant controller or filter so as to guarantee some closed loop indices
More informationThe Q-parametrization (Youla) Lecture 13: Synthesis by Convex Optimization. Lecture 13: Synthesis by Convex Optimization. Example: Spring-mass System
The Q-parametrization (Youla) Lecture 3: Synthesis by Convex Optimization controlled variables z Plant distubances w Example: Spring-mass system measurements y Controller control inputs u Idea for lecture
More informationCourse Outline. FRTN10 Multivariable Control, Lecture 13. General idea for Lectures Lecture 13 Outline. Example 1 (Doyle Stein, 1979)
Course Outline FRTN Multivariable Control, Lecture Automatic Control LTH, 6 L-L Specifications, models and loop-shaping by hand L6-L8 Limitations on achievable performance L9-L Controller optimization:
More informationAn LMI Approach to the Control of a Compact Disc Player. Marco Dettori SC Solutions Inc. Santa Clara, California
An LMI Approach to the Control of a Compact Disc Player Marco Dettori SC Solutions Inc. Santa Clara, California IEEE SCV Control Systems Society Santa Clara University March 15, 2001 Overview of my Ph.D.
More informationState Regulator. Advanced Control. design of controllers using pole placement and LQ design rules
Advanced Control State Regulator Scope design of controllers using pole placement and LQ design rules Keywords pole placement, optimal control, LQ regulator, weighting matrixes Prerequisites Contact state
More informationMIMO analysis: loop-at-a-time
MIMO robustness MIMO analysis: loop-at-a-time y 1 y 2 P (s) + + K 2 (s) r 1 r 2 K 1 (s) Plant: P (s) = 1 s 2 + α 2 s α 2 α(s + 1) α(s + 1) s α 2. (take α = 10 in the following numerical analysis) Controller:
More informationFunnel control in mechatronics: An overview
Funnel control in mechatronics: An overview Position funnel control of stiff industrial servo-systems C.M. Hackl 1, A.G. Hofmann 2 and R.M. Kennel 1 1 Institute for Electrical Drive Systems and Power Electronics
More informationExam. 135 minutes, 15 minutes reading time
Exam August 6, 208 Control Systems II (5-0590-00) Dr. Jacopo Tani Exam Exam Duration: 35 minutes, 5 minutes reading time Number of Problems: 35 Number of Points: 47 Permitted aids: 0 pages (5 sheets) A4.
More informationLinear Matrix Inequality (LMI)
Linear Matrix Inequality (LMI) A linear matrix inequality is an expression of the form where F (x) F 0 + x 1 F 1 + + x m F m > 0 (1) x = (x 1,, x m ) R m, F 0,, F m are real symmetric matrices, and the
More informationFRTN10 Multivariable Control, Lecture 13. Course outline. The Q-parametrization (Youla) Example: Spring-mass System
FRTN Multivariable Control, Lecture 3 Anders Robertsson Automatic Control LTH, Lund University Course outline The Q-parametrization (Youla) L-L5 Purpose, models and loop-shaping by hand L6-L8 Limitations
More informationUncertainty and Robustness for SISO Systems
Uncertainty and Robustness for SISO Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Outline Nature of uncertainty (models and signals). Physical sources of model uncertainty. Mathematical
More informationCollocated versus non-collocated control [H04Q7]
Collocated versus non-collocated control [H04Q7] Jan Swevers September 2008 0-0 Contents Some concepts of structural dynamics Collocated versus non-collocated control Summary This lecture is based on parts
More informationFEEDBACK CONTROL SYSTEMS
FEEDBAC CONTROL SYSTEMS. Control System Design. Open and Closed-Loop Control Systems 3. Why Closed-Loop Control? 4. Case Study --- Speed Control of a DC Motor 5. Steady-State Errors in Unity Feedback Control
More informationMechatronics Modeling and Analysis of Dynamic Systems Case-Study Exercise
Mechatronics Modeling and Analysis of Dynamic Systems Case-Study Exercise Goal: This exercise is designed to take a real-world problem and apply the modeling and analysis concepts discussed in class. As
More informationÜbersetzungshilfe / Translation aid (English) To be returned at the end of the exam!
Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3. 8. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid
More informationOutline. Classical Control. Lecture 1
Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction
More informationPositioning Servo Design Example
Positioning Servo Design Example 1 Goal. The goal in this design example is to design a control system that will be used in a pick-and-place robot to move the link of a robot between two positions. Usually
More informationD(s) G(s) A control system design definition
R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure
More informationDynamics of the synchronous machine
ELEC0047 - Power system dynamics, control and stability Dynamics of the synchronous machine Thierry Van Cutsem t.vancutsem@ulg.ac.be www.montefiore.ulg.ac.be/~vct October 2018 1 / 38 Time constants and
More informationI. D. Landau, A. Karimi: A Course on Adaptive Control Adaptive Control. Part 9: Adaptive Control with Multiple Models and Switching
I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 1 Adaptive Control Part 9: Adaptive Control with Multiple Models and Switching I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 2 Outline
More informationProblem Set 4 Solution 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback
More informationECSE 4962 Control Systems Design. A Brief Tutorial on Control Design
ECSE 4962 Control Systems Design A Brief Tutorial on Control Design Instructor: Professor John T. Wen TA: Ben Potsaid http://www.cat.rpi.edu/~wen/ecse4962s04/ Don t Wait Until The Last Minute! You got
More informationLecture 8. Applications
Lecture 8. Applications Ivan Papusha CDS270 2: Mathematical Methods in Control and System Engineering May 8, 205 / 3 Logistics hw7 due this Wed, May 20 do an easy problem or CYOA hw8 (design problem) will
More informationDESIGN OF ROBUST CONTROL SYSTEM FOR THE PMS MOTOR
Journal of ELECTRICAL ENGINEERING, VOL 58, NO 6, 2007, 326 333 DESIGN OF ROBUST CONTROL SYSTEM FOR THE PMS MOTOR Ahmed Azaiz Youcef Ramdani Abdelkader Meroufel The field orientation control (FOC) consists
More informationDesign and Experimental Evaluation of the Flywheel Energy Storage System
Research article Design and Experimental Evaluation of the Flywheel Energy Storage System Jun-Ho Lee *, Byeong-Song Lee * * orea Railroad Research Institute #176, Cheoldo bangmulgwan-ro, Uiwang, Gyeonggi-Do,
More informationTwo-Mass, Three-Spring Dynamic System Investigation Case Study
Two-ass, Three-Spring Dynamic System Investigation Case Study easurements, Calculations, anufacturer's Specifications odel Parameter Identification Which Parameters to Identify? What Tests to Perform?
More informationROBUST STABILITY AND PERFORMANCE ANALYSIS* [8 # ]
ROBUST STABILITY AND PERFORMANCE ANALYSIS* [8 # ] General control configuration with uncertainty [8.1] For our robustness analysis we use a system representation in which the uncertain perturbations are
More informationAutonomous Helicopter Landing A Nonlinear Output Regulation Perspective
Autonomous Helicopter Landing A Nonlinear Output Regulation Perspective Andrea Serrani Department of Electrical and Computer Engineering Collaborative Center for Control Sciences The Ohio State University
More informationAcceleration Feedback
Acceleration Feedback Mechanical Engineer Modeling & Simulation Electro- Mechanics Electrical- Electronics Engineer Sensors Actuators Computer Systems Engineer Embedded Control Controls Engineer Mechatronic
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 14: LMIs for Robust Control in the LF Framework ypes of Uncertainty In this Lecture, we will cover Unstructured,
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect
More informationTHE SLIDING MODE VARIABLE STRUCTURE CONTROL BASED ON COMPOSITE REACHING LAW OF ACTIVE MAGNETIC BEARING. Received November 2007; accepted January 2008
ICIC Express Letters ICIC International c 2008 ISSN 1881-803X Volume 2, Number 1, March 2008 pp. 59 63 THE SLIDING MODE VARIABLE STRUCTURE CONTROL BASED ON COMPOSITE REACHING LAW OF ACTIVE MAGNETIC BEARING
More informationRobust fuzzy control of an active magnetic bearing subject to voltage saturation
University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 2010 Robust fuzzy control of an active magnetic bearing subject to voltage
More informationFEL3210 Multivariable Feedback Control
FEL3210 Multivariable Feedback Control Lecture 5: Uncertainty and Robustness in SISO Systems [Ch.7-(8)] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 5:Uncertainty and Robustness () FEL3210 MIMO
More informationTowards Rotordynamic Analysis with COMSOL Multiphysics
Towards Rotordynamic Analysis with COMSOL Multiphysics Martin Karlsson *1, and Jean-Claude Luneno 1 1 ÅF Sound & Vibration *Corresponding author: SE-169 99 Stockholm, martin.r.karlsson@afconsult.com Abstract:
More informationRobust Multi-Objective Control for Linear Systems
Robust Multi-Objective Control for Linear Systems Elements of theory and ROMULOC toolbox Dimitri PEAUCELLE & Denis ARZELIER LAAS-CNRS, Toulouse, FRANCE Part of the OLOCEP project (includes GloptiPoly)
More informationAn Adaptive Control System for an Accelerating Rotor Supported by Active Magnetic Bearings under Unbalance Disturbances.
An Adaptive Control System for an Accelerating Rotor Supported by Active Magnetic Bearings under Unbalance Disturbances by Xianglin Wang A thesis submitted to the Graduate Faculty of Auburn University
More informationChapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 7 Interconnected
More information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 24: H2 Synthesis Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology May 4, 2011 E. Frazzoli (MIT) Lecture 24: H 2 Synthesis May
More informationMultiobjective Optimization Applied to Robust H 2 /H State-feedback Control Synthesis
Multiobjective Optimization Applied to Robust H 2 /H State-feedback Control Synthesis Eduardo N. Gonçalves, Reinaldo M. Palhares, and Ricardo H. C. Takahashi Abstract This paper presents an algorithm for
More informationFinite element analysis of rotating structures
Finite element analysis of rotating structures Dr. Louis Komzsik Chief Numerical Analyst Siemens PLM Software Why do rotor dynamics with FEM? Very complex structures with millions of degrees of freedom
More informationStructured LPV Control of Wind Turbines
fda@es.aau.dk Department of Electronic Systems, November 29, 211 Agenda Motivation Main challenges for the application of wind turbine control: Known parameter-dependencies (gain-scheduling); Unknown parameter
More informationIntroduction to Feedback Control
Introduction to Feedback Control Control System Design Why Control? Open-Loop vs Closed-Loop (Feedback) Why Use Feedback Control? Closed-Loop Control System Structure Elements of a Feedback Control System
More informationAdvanced Digital Controls
Advanced Digital Controls University of California, Los Angeles Department of Mechanical and Aerospace Engineering Report Author David Luong Winter 8 Table of Contents Abstract 4 Introduction..5 Physical
More information9. Two-Degrees-of-Freedom Design
9. Two-Degrees-of-Freedom Design In some feedback schemes we have additional degrees-offreedom outside the feedback path. For example, feed forwarding known disturbance signals or reference signals. In
More informationDesign Methods for Control Systems
Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term 2002-2003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9
More informationChapter 5 Design. D. J. Inman 1/51 Mechanical Engineering at Virginia Tech
Chapter 5 Design Acceptable vibration levels (ISO) Vibration isolation Vibration absorbers Effects of damping in absorbers Optimization Viscoelastic damping treatments Critical Speeds Design for vibration
More information6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski. Solutions to Problem Set 1 1. Massachusetts Institute of Technology
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Solutions to Problem Set 1 1 Problem 1.1T Consider the
More informationAA 242B / ME 242B: Mechanical Vibrations (Spring 2016)
AA 242B / ME 242B: Mechanical Vibrations (Spring 206) Solution of Homework #3 Control Tab Figure : Schematic for the control tab. Inadequacy of a static-test A static-test for measuring θ would ideally
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering Dynamics and Control II Fall 2007
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering.4 Dynamics and Control II Fall 7 Problem Set #9 Solution Posted: Sunday, Dec., 7. The.4 Tower system. The system parameters are
More informationModel Uncertainty and Robust Stability for Multivariable Systems
Model Uncertainty and Robust Stability for Multivariable Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Devron Profile Control Solutions Outline Representing model uncertainty.
More informationDISTURBANCE ATTENUATION IN A MAGNETIC LEVITATION SYSTEM WITH ACCELERATION FEEDBACK
DISTURBANCE ATTENUATION IN A MAGNETIC LEVITATION SYSTEM WITH ACCELERATION FEEDBACK Feng Tian Department of Mechanical Engineering Marquette University Milwaukee, WI 53233 USA Email: feng.tian@mu.edu Kevin
More informationIntroduction to Controls
EE 474 Review Exam 1 Name Answer each of the questions. Show your work. Note were essay-type answers are requested. Answer with complete sentences. Incomplete sentences will count heavily against the grade.
More informationAppendix A Solving Linear Matrix Inequality (LMI) Problems
Appendix A Solving Linear Matrix Inequality (LMI) Problems In this section, we present a brief introduction about linear matrix inequalities which have been used extensively to solve the FDI problems described
More informationRobust Multivariable Control
Lecture 2 Anders Helmersson anders.helmersson@liu.se ISY/Reglerteknik Linköpings universitet Today s topics Today s topics Norms Today s topics Norms Representation of dynamic systems Today s topics Norms
More informationRichiami di Controlli Automatici
Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici
More informationLPV MODELING AND CONTROL OF A 2-DOF ROBOTIC MANIPULATOR BASED ON DESCRIPTOR REPRESENTATION
Copyright c 9 by ABCM January 4-8, 1, Foz do Iguaçu, PR, Brazil LPV MODELING AND CONTROL OF A -DOF ROBOTIC MANIPULATOR BASED ON DESCRIPTOR REPRESENTATION Houssem Halalchi, houssem.halalchi@unistra.fr Edouard
More informationGain-Scheduling Approaches for Active Damping of a Milling Spindle with Speed-Dependent Dynamics
Gain-Scheduling Approaches for Active Damping of a Milling Spindle with Speed-Dependent Dynamics Simon Kern, Andreas Schwung and Rainer Nordmann Abstract Chatter vibrations in high-speed-milling operations
More informationIntroduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31
Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured
More informationA brief introduction to robust H control
A brief introduction to robust H control Jean-Marc Biannic System Control and Flight Dynamics Department ONERA, Toulouse. http://www.onera.fr/staff/jean-marc-biannic/ http://jm.biannic.free.fr/ European
More informationA Novel Adaptive Estimation of Stator and Rotor Resistance for Induction Motor Drives
A Novel Adaptive Estimation of Stator and Rotor Resistance for Induction Motor Drives Nagaraja Yadav Ponagani Asst.Professsor, Department of Electrical & Electronics Engineering Dhurva Institute of Engineering
More informationAdaptive Control of Variable-Speed Variable-Pitch Wind Turbines Using RBF Neural Network
Schulich School of Engineering Department of Mechanical and Manufacturing Engineering Adaptive Control of Variable-Speed Variable-Pitch Wind Turbines Using RBF Neural Network By: Hamidreza Jafarnejadsani,
More informationMotion System Classes. Motion System Classes K. Craig 1
Motion System Classes Motion System Classes K. Craig 1 Mechatronic System Design Integration and Assessment Early in the Design Process TIMING BELT MOTOR SPINDLE CARRIAGE ELECTRONICS FRAME PIPETTE Fast
More informationPositioning Control of One Link Arm with Parametric Uncertainty using Quantitative Feedback Theory
Memoirs of the Faculty of Engineering, Okayama University, Vol. 43, pp. 39-48, January 2009 Positioning Control of One Link Arm with Parametric Uncertainty using Quantitative Feedback Theory Takayuki KUWASHIMA,
More informationLecture 1: Introduction to System Modeling and Control. Introduction Basic Definitions Different Model Types System Identification
Lecture 1: Introduction to System Modeling and Control Introduction Basic Definitions Different Model Types System Identification What is Mathematical Model? A set of mathematical equations (e.g., differential
More informationMEMS Gyroscope Control Systems for Direct Angle Measurements
MEMS Gyroscope Control Systems for Direct Angle Measurements Chien-Yu Chi Mechanical Engineering National Chiao Tung University Hsin-Chu, Taiwan (R.O.C.) 3 Email: chienyu.me93g@nctu.edu.tw Tsung-Lin Chen
More informationUncertainty Analysis for Linear Parameter Varying Systems
Uncertainty Analysis for Linear Parameter Varying Systems Peter Seiler Department of Aerospace Engineering and Mechanics University of Minnesota Joint work with: H. Pfifer, T. Péni (Sztaki), S. Wang, G.
More informationMTNS 06, Kyoto (July, 2006) Shinji Hara The University of Tokyo, Japan
MTNS 06, Kyoto (July, 2006) Shinji Hara The University of Tokyo, Japan Outline Motivation & Background: H2 Tracking Performance Limits: new paradigm Explicit analytical solutions with examples H2 Regulation
More informationA Comparative Study on Automatic Flight Control for small UAV
Proceedings of the 5 th International Conference of Control, Dynamic Systems, and Robotics (CDSR'18) Niagara Falls, Canada June 7 9, 18 Paper No. 13 DOI: 1.11159/cdsr18.13 A Comparative Study on Automatic
More information2.010 Fall 2000 Solution of Homework Assignment 1
2. Fall 2 Solution of Homework Assignment. Compact Disk Player. This is essentially a reprise of Problems and 2 from the Fall 999 2.3 Homework Assignment 7. t is included here to encourage you to review
More information6.241 Dynamic Systems and Control
6.41 Dynamic Systems and Control Lecture 5: H Synthesis Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology May 11, 011 E. Frazzoli (MIT) Lecture 5: H Synthesis May 11, 011
More informationAutomatic Control Systems. -Lecture Note 15-
-Lecture Note 15- Modeling of Physical Systems 5 1/52 AC Motors AC Motors Classification i) Induction Motor (Asynchronous Motor) ii) Synchronous Motor 2/52 Advantages of AC Motors i) Cost-effective ii)
More informationRegulating Web Tension in Tape Systems with Time-varying Radii
Regulating Web Tension in Tape Systems with Time-varying Radii Hua Zhong and Lucy Y. Pao Abstract A tape system is time-varying as tape winds from one reel to the other. The variations in reel radii consist
More informationControl System Design
ELEC4410 Control System Design Lecture 19: Feedback from Estimated States and Discrete-Time Control Design Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science
More informationROBUST CONTROL OF MULTI-AXIS SHAKING SYSTEM USING µ-synthesis
ROBUST CONTROL OF MULTI-AXIS SHAKING SYSTEM USING µ-synthesis Y. Uchiyama, M. Mukai, M. Fujita IMV CORPORATION, Itami 664-847, Japan Department of Electrical and Electronic Engineering, Kanazawa University,
More informationRobust Controller Design for Cancelling Biodynamic Feedthrough
Robust Controller Design for Cancelling Biodynamic Feedthrough M. R. Sirouspour and S. E. Salcudean University of British Columbia Vancouver, BC, Canada shahins@ece.ubc.ca, tims@ece.ubc.ca Abstract: There
More informationControl System Design
ELEC ENG 4CL4: Control System Design Notes for Lecture #11 Wednesday, January 28, 2004 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Relative Stability: Stability
More informationENGG4420 LECTURE 7. CHAPTER 1 BY RADU MURESAN Page 1. September :29 PM
CHAPTER 1 BY RADU MURESAN Page 1 ENGG4420 LECTURE 7 September 21 10 2:29 PM MODELS OF ELECTRIC CIRCUITS Electric circuits contain sources of electric voltage and current and other electronic elements such
More informationH-infinity Model Reference Controller Design for Magnetic Levitation System
H.I. Ali Control and Systems Engineering Department, University of Technology Baghdad, Iraq 6043@uotechnology.edu.iq H-infinity Model Reference Controller Design for Magnetic Levitation System Abstract-
More informationSubspace identification and robust control of an AMB system
21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 WeC184 Subspace identification and robust control of an AMB system HMNK Balini, Ivo Houtzager, Jasper Witte and
More informationCo-simulation of Magnetic Bearing System based on Adams and MATLAB
6th International Conference on Machinery, Materials, Environment, Biotechnology and Computer (MMEBC 016) Co-simulation of Magnetic Bearing System based on Adams and MATLAB Sijia Liu1, a, Jingbo Shi, b
More informationAdvanced Control Theory
State Feedback Control Design chibum@seoultech.ac.kr Outline State feedback control design Benefits of CCF 2 Conceptual steps in controller design We begin by considering the regulation problem the task
More informationChapter 6 Feedback Control Designs
Chapter 6 Feedback Control Designs A. Schirrer, M. Kozek, F. Demourant and G. Ferreres 6.1 Introduction A. Schirrer and M. Kozek 6.1.1 General Properties of Feedback Control The general concept of feedback
More informationControl for. Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e
Control for Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e Motion Systems m F Introduction Timedomain tuning Frequency domain & stability Filters Feedforward Servo-oriented
More informationTHE REACTION WHEEL PENDULUM
THE REACTION WHEEL PENDULUM By Ana Navarro Yu-Han Sun Final Report for ECE 486, Control Systems, Fall 2013 TA: Dan Soberal 16 December 2013 Thursday 3-6pm Contents 1. Introduction... 1 1.1 Sensors (Encoders)...
More informationExperimental Evaluation on H DIA Control of Magnetic Bearings with Rotor Unbalance
Experimental Evaluation on H DIA Control of Magnetic Bearings with Rotor Unbalance Hiroki Seto and Toru Namerikawa Division of Electrical Engineering and Computer Science Kanazawa University Kakuma-cho,
More informationTheory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 2 Simpul Rotors Lecture - 2 Jeffcott Rotor Model In the
More informationState Feedback Controller for Position Control of a Flexible Link
Laboratory 12 Control Systems Laboratory ECE3557 Laboratory 12 State Feedback Controller for Position Control of a Flexible Link 12.1 Objective The objective of this laboratory is to design a full state
More informationLecture 7 (Weeks 13-14)
Lecture 7 (Weeks 13-14) Introduction to Multivariable Control (SP - Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 13-14) p.
More informationRandom Eigenvalue Problems in Structural Dynamics: An Experimental Investigation
Random Eigenvalue Problems in Structural Dynamics: An Experimental Investigation S. Adhikari, A. Srikantha Phani and D. A. Pape School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:
More informationNonlinear Rolling Element Bearings in MADYN 2000 Version 4.3
- 1 - Nonlinear Rolling Element Bearings in MADYN 2000 Version 4.3 In version 4.3 nonlinear rolling element bearings can be considered for transient analyses. The nonlinear forces are calculated with a
More informationDynamics of Structures
Dynamics of Structures Elements of structural dynamics Roberto Tomasi 11.05.2017 Roberto Tomasi Dynamics of Structures 11.05.2017 1 / 22 Overview 1 SDOF system SDOF system Equation of motion Response spectrum
More information