Mathematical Models of Systems
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1 Mathematical Models of Systems Transfer Function and Transfer functions of Complex Systems M. Bakošová and M. Fikar Department of Information Engineering and Process Control Institute of Information Engineering, Automation and Mathematics FCFT STU in Bratislava Process Control MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
2 Contents Mathematical Models of Systems System Quantities of Systems Types of Systems Approches to Modelling and Types of Systems Input-Output Model of the Dynamic System Input-Output Differential Equations Transfer functions Conversion of the Differential Equation to the Transfer Function Transformation of the Transfer Function to the Differential Equation Transfer Functions of Complex Systems Serial Connection Parallel Connection Feedback Connection with Negative Feedback Feedback Connection with Positive Feedback MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
3 Mathematical Models of Systems System System System Real object Model MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
4 Mathematical Models of Systems Quantities of Systems Quantities of systems input control inputs u(t) disturbances r(t) state x(t) output y(t) MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
5 Mathematical Models of Systems Types of Systems Types of Systems Systems dynamic, static single-variable, multi-variable deterministic, static continunous-time, dicrete-time linear, non-linear with lumped parameters, with disributed parameters with constant coefficients, with varying coefficients MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
6 Mathematical Models of Systems Approches to Modelling and Types of Systems Approaches to modelling deterministic experimental-statistic Models according to modelling approach theoretical, mathematical empirical theoretical-empirical MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
7 Mathematical Models of Systems Approches to Modelling and Types of Systems Models according to the mathematical description algebraic equations differential equations - ordinary, partial difference equations rational functions probabilistic relations Models according to the variables state-space models input-output models (input-output) differential equations transfer function MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
8 Input-Output Differential Equations Input-output differential equation a n y (n) (t)+a n 1 y (n 1) (t)+ +a 1 y (t)+a 0 y(t) = b m u (m) (t)+b m 1 u (m 1) (t)+ +b 1 u (t)+b 0 u(t) initial conditions y(0), y (0),, y (n 1) (0) MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
9 Transfer functions Definition of the transfer function Prenos G(s) = Y(s) U(s) Grafical representation of the system described by the transfer function U Y G Calculation of the Laplace transform of the system output, when the system is described by the transfer function Y(s) = G(s)U(s) MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
10 Conversion of the Differential Equation to the Transfer Function Conversion of the Differential Equation to the Transfer Function Differential equation (DE) a n y (n) (t)+a n 1 y (n 1) (t)+ +a 1 y (t)+a 0 y(t) = b m u (m) (t)+b m 1 u (m 1) (t)+ +b 1 u (t)+b 0 u(t) initial conditions y (n 1) (0) = = y (0) = y(0) = 0. MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
11 Conversion of the Differential Equation to the Transfer Function Laplace transform of the DE a n s n Y(s)+a n 1 s n 1 Y(s)+ +a 1 sy(s)+a 0 Y(s) = b m s m U(s)+b m 1 s m 1 U(s)+ +b 1 su(s)+b 0 U(s) mathematical treatment transfer function G(s) (a n s n + a n 1 s n 1 + +a 1 s + a 0 )Y(s) = (b m s m + b m 1 s m 1 + +b 1 s + b 0 )U(s) G(s) = Y(s) U(s) = b ms m + b m 1 s m 1 + +b 1 s + b 0 a n s n + a n 1 s n 1 + +a 1 s + a 0 = B(s) A(s) MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
12 Conversion of the Differential Equation to the Transfer Function Minimal realization of the transfer function The order of the system n Physical feasibility of the transfer function m n Gain Z = b 0 a 0 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
13 Transformation of the Transfer Function to the Differential Equation Transformation of the Transfer Function to the Differential Equation G(s) = Y(s) U(s) = b ms m + b m 1 s m 1 + +b 1 s + b 0 a n s n + a n 1 s n 1 + +a 1 s + a 0 mathematical treatment (a n s n + a n 1 s n 1 + +a 1 s + a 0 )Y(s) = (b m s m + b m 1 s m 1 + +b 1 s + b 0 )U(s) = B(s) A(s) MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
14 Transformation of the Transfer Function to the Differential Equation the other mathematical treatment a n s n Y(s)+a n 1 s n 1 Y(s)+ +a 1 sy(s)+a 0 Y(s) = b m s m U(s)+b m 1 s m 1 U(s)+ +b 1 su(s)+b 0 U(s) inverse Laplace transform a n y (n) (t)+a n 1 y (n 1) (t)+ +a 1 y (t)+a 0 y(t) = b m u (m) (t)+b m 1 u (m 1) (t)+ +b 1 u (t)+b 0 u(t) initial conditions of DE y (n 1) (0) = = y (0) = y(0) = 0 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
15 Transformation of the Transfer Function to the Differential Equation Example 1 Finde the tranfer function of the dynamic system that is described by the DE 2y (t)+8y (t)+y(t) = 3u (t)+2u(t) y (0) = y (0) = y(0) = 0 Solution G(s) = 3s + 2 2s 3 + 8s MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
16 Transformation of the Transfer Function to the Differential Equation Example 2 Finde the DE, which describes the dynamic system described by the trnsfer function G(s) = 4s2 + 5s 5s 3 + 8s Solution 5y (t)+8y (t)+y(t) = 4u (t)+5u (t) y (0) = y (0) = y(0) = 0 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
17 Transformation of the Transfer Function to the Differential Equation Example 3 Find the transfer functions of the dynamic system, which is described by the state-space model in the form x 1 (t) = 5x 1(t)+4x 2 (t)+2u 1 (t)+u 2 (t), x 1 (0) = 0 x 2 (t) = 2x 1(t) 4x 2 (t)+3u 1 (t)+u 2 (t), x 2 (0) = 0 y 1 (t) = 3x 1 (t) y 2 (t) = 2x 2 (t) 6s + 60 G y1,u 1 = s 2 + 9s + 12, G 3s + 24 y 1,u 2 = s 2 + 9s s + 38 G y2,u 1 = s 2 + 9s + 12, G 2s + 14 y 2,u 2 = s 2 + 9s + 12 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
18 Transfer Functions of Complex Systems Serial Connection Serial connection of 2 systems U(s) Y1(s) Y(s) B1(s) A1(s) G1(s) B2(s) A2(s) G2(s) Deriving of the transfer function of 2 serially connected systems Y(s) = G 2 (s)y 1 (s) = G 2 (s)g 1 (s)u(s) MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
19 Transfer Functions of Complex Systems Transfer function of 2 serially connected systems Y(s) U(s) = G(s) = G 1(s)G 2 (s) Transfer function of n serially connected systems G(s) = n G i (s) i=1 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
20 Transfer Functions of Complex Systems Parallel Connection Parallel connection of 2 systems U(s) B1(s) A1(s) G1(s) Y1(s) Y(s) B2(s) A2(s) G2(s) Y2(s) Tranfer function of 2 system with parallel connection G(s) = G 1 (s)+g 2 (s) Tranfer function of n system with parallel connection G(s) = n G i (s) i=1 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
21 Transfer Functions of Complex Systems Feedback Connection with Negative Feedback Feedback connection with negative feedback U(s) E(s) B1(s) A1(s) Y(s) G1(s) B2(s) Y2(s) A2(s) G2(s) Deriving of the transfer function Y(s) = G 1 (s)e(s) = G 1 (s)(u(s) Y 2 (s)) = G 1 (s)u(s) G 1 (s)y 2 (s) = G 1 (s)u(s) G 1 (s)g 2 (s)y(s) MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
22 Transfer Functions of Complex Systems mathematical treatment of the equation Y(s) = G 1 (s)u(s) G 1 (s)g 2 (s)y(s) Transfer function Y(s)(1+G 1 (s)g 2 (s)) = G 1 (s)u(s) G(s) = Y(s) U(s) = G 1 (s) 1+G 1 (s)g 2 (s) MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
23 Transfer Functions of Complex Systems Feedback Connection with Positive Feedback Feedback connection with positive feedback U(s) E(s) B1(s) A1(s) Y(s) G1(s) B2(s) Y2(s) A2(s) G2(s) G(s) = G 1 (s) 1 G 1 (s)g 2 (s) MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
24 Transfer Functions of Complex Systems Example 4 Z G3 W G1 G2 G4 Y G5 G6 G yw (s) = Y W = G 1 G 2 G 4 1+G 1 G 2 G 5 + G 1 G 2 G 6 G yz (s) = Y Z = G 3 G 4 1+G 1 G 2 G 5 + G 1 G 2 G 6 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
25 Transfer Functions of Complex Systems Example 5 U G1 Y G2 G4 G3 G yu (s) = G 1 + G 2 1+G 1 G 3 G 4 + G 2 G 3 G 4 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
26 Transfer Functions of Complex Systems Example 6 G1 Z W G4 G2 Y G3 G y,w = G y,z = G 2 G 4 G 1 G 4 G 1 G 2 G 3 G 4 1+G 2 G 3 + G 2 G 4 G 1 G 4 G 1 G 2 G 3 G 4 1+G 2 G 3 1+G 2 G 3 + G 2 G 4 G 1 G 4 G 1 G 2 G 3 G 4 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
27 Transfer Functions of Complex Systems Example 7 Z W G1 G2 G3 G4 Y G y,w = G y,z = G 1 G 2 G 3 G 4 1+G 2 G 3 + G 1 G 2 G 3 G 4 G 3 G 4 1+G 2 G 3 + G 1 G 2 G 3 G 4 MB/MF (IRP) Mathematical Models of Systems WS 2017/ / 36
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