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EGGN 307 Introduction to Feedback Control Systems Transfer Functions (Lectures 13-15) Professor Kevin L. Moore Spring 2010 http://engineering.mines.edu/course/eggn307a Previously 3.0 Laplace Transform Modeling 3.1 Review of Complex Numbers 3.2 Laplace Transforms 3.3 Inverse Laplace and LODE Solutions 3.4 Transfer Functions Recall: Laplace differentiation theorem (1) Colorado School of Mines The differentiation theorem Higher order derivatives Due to Katie Johnson or Tyrone Vincent or someone 3 Differentiation Theorem (revisited) Colorado School of Mines d n 1 dn dt n f t s n F s s n 1 f 0 s n 2 d dt f 0 n 1 f 0 dt Differentiation Theorem when initial conditions are zero Due to Katie Johnson or Tyrone Vincent or someone 4 Introduction to Transfer Functions Colorado School of Mines Consider a system with input r(t), output x(t), and zero initial conditions: The Laplace Transform of the output is related to the Laplace Transform of the input by a function of s Due to Katie Johnson or Tyrone Vincent or someone 5 Definition of a Transfer Function Colorado School of Mines Definition Due to Katie Johnson or Tyrone Vincent or someone 6 •Transfer Function of a General LODE •Impulse Response of a General LODE Colorado School of Mines 3.0 Laplace Transform Modeling 3.1 Review of Complex Numbers 3.2 Laplace Transforms 3.3 Inverse Laplace and LODE Solutions 3.4 Transfer Functions 3.5 Block Diagrams 13 Block Diagram Components Colorado School of Mines Block diagrams are graphical representations of algebraic relationships. System Summing Junction C (s) R1 ( s ) Y (s) R(s) G(s) R2 ( s ) C ( s ) G ( s ) R( s ) Y ( s) R1 ( s) R2 ( s) Due to Katie Johnson or Tyrone Vincent or someone 14 Basic Block Diagram Simplifications (1) Colorado School of Mines Cascade Systems R(s) X (s ) C (s) G1 ( s ) G2 ( s ) X ( s) G1 ( s) R( s) C ( s) G2 ( s) X ( s) C ( s) G2 ( s)G1 ( s) R( s) R(s) C (s) G2 ( s)G1 ( s) Due to Katie Johnson or Tyrone Vincent or someone 15 Basic Block Diagram Simplifications (2) Colorado School of Mines Systems In Parallel G1 ( s ) R(s) G2 ( s ) C ( s) G1 ( s) R( s) G2 ( s) R( s) Due to Katie Johnson or Tyrone Vincent or someone 16 Basic Block Diagram Simplifications (3) Colorado School of Mines Feedback Connections R(s) E (s) C (s) G(s) H (s ) C ( s) G( s) E ( s) E (s) R(s) H (s)C (s) Due to Katie Johnson or Tyrone Vincent or someone 17 Basic Block Diagram Simplifications (4) Colorado School of Mines G( s ) C( s) R( s ) 1 G( s )H ( s ) R(s) G (s) C (s) 1 G (s) H (s) Due to Katie Johnson or Tyrone Vincent or someone 18 Example 1 Colorado School of Mines Simplify the following block diagram. All variables are functions of s. R C G1 G2 H1 H2 Due to Katie Johnson or Tyrone Vincent or someone 19 Example 1 (2) Colorado School of Mines R G1 G2 C H1 H 2 Due to Katie Johnson or Tyrone Vincent or someone 20 Example 1 (3) Colorado School of Mines R G2 C G1 1 G2 ( H1 H 2 ) R G2G1 C 1 G2 ( H1 H 2 ) Due to Katie Johnson or Tyrone Vincent or someone 21 Other Manipulations (1) Colorado School of Mines Sometimes additional manipulations are needed to apply basic simplifications Goal: move the pick-off point so that the dotted box has a single input and a single output. R C G1 G2 Due to Katie Johnson or Tyrone Vincent or someone 22 Other Manipulations (2) Colorado School of Mines Pick off points can be moved around blocks 1 G1 G1 G1 G1 G1 G1 Due to Katie Johnson or Tyrone Vincent or someone 23 Other Manipulations (3) Colorado School of Mines R C G1 G2 Equivalent Diagram 1 G1 R C G1 G2 Due to Katie Johnson or Tyrone Vincent or someone 24 Example 2 Colorado School of Mines Due to Katie Johnson or Tyrone Vincent or someone 25 Example 2 (2) Colorado School of Mines Due to Katie Johnson or Tyrone Vincent or someone 26 Block diagrams just represent algebra Colorado School of Mines If you ever get stuck – just convert back to algebraic relationships and solve by hand Add variables at inputs to blocks R A C G1 C G1 A A B G2 A R G2 B B G1 A Due to Katie Johnson or Tyrone Vincent or someone 27 Block diagrams just represent algebra (2) Colorado School of Mines C G1 A A A R G2G1 A A R G2 B 1 A R B G1 A 1 G2G1 1 1 C G1 R R 1 G2G1 1 G2G1 G1 1 C R 1 G2G1 Due to Katie Johnson or Tyrone Vincent or someone 28 Colorado School of Mines 3.6 Transfer Functions of Physical Systems Mesh and Nodal Equations 29 Electrical impedance Colorado School of Mines To work with algebraic relationships, we take the Laplace Transform of the defining equations with zero initial conditions – Recall: Transfer Function zero initial conditions resistor capacitor inductor dv di v iR C i vL dt dt V ( s) RI ( s) CsV ( s ) I ( s ) V ( s ) LsI ( s ) 1 V (s) I ( s) Cs Impedance: Ratio of Laplace Transform of across variable to Laplace Transform of through variable Due to Katie Johnson or Tyrone Vincent or someone 30 Finding Transfer Functions using mesh or nodal equations Colorado School of Mines Vout s Example: Find Vin s R Vin C L Vout Steps 1. Convert elements to impedances 2. Either • apply Kirchoff’s Current Law (KCL) at every node, or • apply Kirchoff’s Voltage Law (KVL) around every mesh 3. Solve resulting system of equations for output variable Due to Katie Johnson or Tyrone Vincent or someone 31 Mesh equations from KVL Colorado School of Mines R I1 ( s ) I 2 (s) Vin (s ) 1 sL Vout ( s ) sLI 2 ( s ) sC 1 Vin ( s) RI1 ( s) ( I1 ( s) I 2 ( s)) sC 1 0 ( I 2 ( s) I1 ( s)) LsI 2 ( s) sC Two equations: need to solve for I 2 ( s ) Due to Katie Johnson or Tyrone Vincent or someone 32 Cramer’s Rule Colorado School of Mines Cramer’s Rule is a tool that can be used to solve algebraic equations. b1 a11 a12 x1 If b a a22 x2 2 21 determinant Then b1 a12 a11 b1 b2 a22 a21 b2 x1 x2 a11 a12 a11 a12 a21 a22 a21 a22 Due to Katie Johnson or Tyrone Vincent or someone 33 Mesh equations in Matrix-Vector form Colorado School of Mines The mesh equations can be re-written in matrix- vector form 1 Vin ( s) RI1 ( s) ( I1 ( s) I 2 ( s)) sC 1 0 ( I 2 ( s ) I1 ( s )) LsI 2 ( s ) sC Vin ( s ) R 1 1 I ( s) sC sC 1 1 1 0 sC Ls I 2 ( s ) sC Use Cramer’s Rule Due to Katie Johnson or Tyrone Vincent or someone 34 Mesh equations in Matrix-Vector form (2) Colorado School of Mines Vin ( s ) R 1 1 I ( s) sC sC 1 1 1 0 sC Ls I 2 ( s ) sC 1 R Vin ( s ) sC 1 0 1 sC sC Vin ( s ) I 2 ( s) R 1 1 R sC Ls sC sC 1 1 1 2 sC sC 1 1 Ls sC sC Due to Katie Johnson or Tyrone Vincent or someone 35 Mesh equations in Matrix-Vector form (3) Colorado School of Mines Vout ( s ) sLI 2 ( s ) sLVin ( s ) Vout ( s ) 2 s RLC Ls R Vout ( s ) sL 2 Vin ( s ) s RLC Ls R Due to Katie Johnson or Tyrone Vincent or someone 36 Mesh equations from circuit diagram Colorado School of Mines Mesh equations in matrix/vector form: sum of impedances on mesh 1 shared impedance Vin ( s ) R 1 1 I ( s) sC sC 1 1 1 0 sC Ls I 2 ( s ) sC sum of impedances on mesh 2 R I1 ( s ) I 2 (s) 1 Vin (s ) sL Vout (s ) sC Due to Katie Johnson or Tyrone Vincent or someone 37 Exercise: Find the mesh equations Colorado School of Mines 1 Vin I1 I1 I 2 1 I1 ( s ) I 2 (s) s 0 I1 I 2 3I 2 2I 3 I 2 Vin 1 3 1 s s I1 I 2 I3 I 2 0 2I 3 I 2 Vout I 3 I1 2 2s Vout I 3 ( s) Vin ( s ) 1 1 s 1 s 0 I1 ( s ) 0 1 I ( s ) s 23 2 1 s 2 0 0 2 2 2s I 3 ( s ) Due to Katie Johnson or Tyrone Vincent or someone 38 Mechanical impedance Colorado School of Mines Element Equation Impedance Admittance mass f M x X (s) F (s) 1 Ms2 F (s) X (s) Ms2 damper f Dx X (s) F (s) 1 Ds F (s) X (s) Ds spring f Kx X (s) F (s) K 1 F (s) X (s) K Admittance, or the inverse of impedance, is more commonly used in mechanical systems. Both admittances and impedances are commonly used in electrical systems. Due to Katie Johnson or Tyrone Vincent or someone 39 Nodal equations for current input Colorado School of Mines V1 R V2 1 I in (s ) Ls sC Vout ( s ) V2 ( s ) sum of admittances connected to node 1 negative of shared admittance I in ( s ) Ls R 1 1 R V1 ( s ) 1 0 1 1 R Cs R V2 ( s ) Due to Katie Johnson or Tyrone Vincent or someone 40 Nodal equations for mechanical systems Colorado School of Mines K1 K2 u M1 M2 x1 x2 Newton’s Laws at each node M 11 K1 x1 K 2 ( x1 x2 ) x M 2 2 K 2 ( x2 x1 ) u x Newton’s Laws with Admittances 0 M 1s 2 X 1 ( s) K1 X 1 ( s) K 2 ( X 1 ( s) X 2 ( s)) U ( s) M 2 s 2 X 2 ( s) K 2 ( X 2 ( s) X 1 ( s)) Due to Katie Johnson or Tyrone Vincent or someone 41 Nodal equations from mechanical diagram Colorado School of Mines The same pattern as electrical nodal equations! K1 K2 u M1 M2 x1 x2 Sum of admittances Negative of shared admittances 0 M 1s 2 K1 K 2 K 2 X 1( s) U ( s ) K2 M 2 s K 2 X 2 ( s ) 2 Due to Katie Johnson or Tyrone Vincent or someone 42 Colorado School of Mines •Motors 43 Electromechanical systems (1) Colorado School of Mines What happens when you run current through a wire in a magnetic field? f (out of screen) B i Lorentz force equation (SI units) f i B Key result: Magnitude of force is proportional to current and magnetic field f Ki Due to Katie Johnson or Tyrone Vincent or someone 44 Electromechanical Systems (2) Colorado School of Mines What happens when you move a wire through a magnetic field? x (out of screen) B Vb i Conservation of energy: iVb xf iVb xKi Key result: induced voltage is proportional to velocity Vb xK Due to Katie Johnson or Tyrone Vincent or someone 45 Recall - DC Motor Colorado School of Mines DC motors are used in many control applications (e.g., robots, disk drives) Torque on load K f i f - magneticflux Tm K1ia K1 K f i f ia Tm - torque Due to Katie Johnson or Tyrone Vincent or someone 46 Field Controlled DC motor – armature current fixed Colorado School of Mines Circuit Transducer Mechanical load Tm (t ) K1K f ia i f (t ) Tm K mi f (t ) I f ( s) 1 Tm ( s ) (s) 1 Km 2 V f ( s) ( L f s R f ) I f (s) Tm ( s ) Js bs Overall transfer function ( s) K m / JL f V f ( s) s( s b / J )(s R f / L f ) Due to Katie Johnson or Tyrone Vincent or someone 47 Armature controlled DC motor – field current fixed (1) Colorado School of Mines Vb K b K b Key relationships: Tm K1K t i f ia K mia Ra La Tm b Ia Va Vb J Armature Due to Katie Johnson or Tyrone Vincent or someone 48 Armature controlled DC motor – field current fixed (2) Colorado School of Mines Va Vb Note (from circuit diagram): I a Ra La s Thus, Tm K m I a Km Va Vb Ra La s Since net torque = Tm Td TL , we have TL 1 , Js b s Due to Katie Johnson or Tyrone Vincent or someone 49 Armature controlled DC motor – field current fixed (2) Colorado School of Mines Closed loop transfer function ( s) Km 1 1 ( Ra La s ) ( Js b ) Va ( s ) s (1 Kb K m 1 ( Ra La s ) ( Js b ) ) (s) Km Va ( s ) s (( Ra La s )( Js b) K b K m ) When inductance is negligible: ( s) Km Ra Va ( s ) s Js b K b Km Ra Due to Katie Johnson or Tyrone Vincent or someone 50 Example Colorado School of Mines Ra b 8 N m s rad Va J 7 kg m 2 m 500 Steady state Va 100 V motor load curve 50 m (s) Find the transfer function Va ( s ) Due to Katie Johnson or Tyrone Vincent or someone 51 Example (2) Colorado School of Mines Motor constant from stall torque Va Vb Va m 0 Vb kbm 0 ia Ra Ra K mVa m K mia @ m 0 Ra K m 100 500 Ra Km 5 Ra Due to Katie Johnson or Tyrone Vincent or someone 52 Example (3) Colorado School of Mines Motor constant from no-load velocity m 0 ia 0 Va Vb Vb K b Va K b @ m 0 100 K b 50 2 Kb Due to Katie Johnson or Tyrone Vincent or someone 53 Example (4) Colorado School of Mines Transfer function Km Ra 5 Kb 2 J 7 b8 (s) Km Ra Va ( s ) s Js b K b K m Ra 5 s7 s 18 Due to Katie Johnson or Tyrone Vincent or someone 54 Hydraulic Actuator Colorado School of Mines Used for large loads load x y Q drain ps supply P ps Q drain Nonlinear flow rate Q g ( x, P) Piston area A Pressure P Due to Katie Johnson or Tyrone Vincent or someone 55 Two dimensional nonlinear function Colorado School of Mines Operating point ( x0 , P0 ) Linearized flow g g Q x P x ( x , P ) P ( x , P ) 0 0 0 0 k xx k PP where g kx x ( x0 , P0 ) g kP P ( x0 , P0 ) Due to Katie Johnson or Tyrone Vincent or someone 56 Free Body Diagram Colorado School of Mines Piston pressure/force and flow/position relationships dy Q f PA dt A load y Y (s) f N ( s) f F ( s) y Q Y s Ps N s A P Q Due to Katie Johnson or Tyrone Vincent or someone 57 Hydraulic transfer function Colorado School of Mines Small signal relationships in Laplace domain Q( s) k xX ( s) k PP( s) 1 P ( s ) Y ( s ) N (s) A 1 Y ( s) Q( s) sA Q(s) Y (s) X (s) kx 1 sA P(s) 1 kp N ( s) A Due to Katie Johnson or Tyrone Vincent or someone 58