Lecture Outline Systeem- en Regeltechniek II Previous lecture: representation of dynamic models as differential Lecture 2 – System Modeling and Analysis equations. Robert Babuˇ ska Today: Delft Center for Systems and Control • Transfer functions. Faculty of Mechanical Engineering Delft University of Technology • State-space models. The Netherlands e-mail: r.babuska@dcsc.tudelft.nl www.dcsc.tudelft.nl/ ˜ babuska tel: 015-27 85117 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 1 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 2 Example 3: Liquid Storage Tank Example 3 (cont’d): Liquid Storage Tank A A ˙ h ( t ) = Q i ( t ) − Q o ( t ) Q i � � Q o ( t ) = r 2 gh ( t ) = K h ( t ) , h ≥ 0 A ˙ � h ( t ) + K h ( t ) = Q i ( t ) h Q o • Nonlinear differential equation • Must be linearized for analysis or control design • Can be used to simulate the process Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 3 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 4
Laplace Transform – Definition Using Laplace Transform L Transform a signal from time domain to complex domain ( s -domain): Differentiation: f ( n ) ( t ) s n F ( s ) − → L f ( t ) − → F ( s ) Linear differential equation: a n y ( n ) ( t ) + a n − 1 y ( n − 1) ( t ) + · · · + a o y ( t ) = b m u ( m ) ( t ) + b m − 1 u ( m − 1) ( t ) + · · · + b o u ( t ) � ∞ f ( t ) e − st dt F ( s ) = 0 Linear algebraic equation: � a n s n + a n − 1 s n − 1 + · · · + a o � Y ( s ) = a number of useful properties � b m s m + b m − 1 s m − 1 + · · · + b o � U ( s ) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 5 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 6 Transfer Function Example 1 (revisited): Transfer Function m ¨ d ( t ) = F ( t ) − b ˙ d ( t ) � a n s n + a n − 1 s n − 1 + · · · + a o � Y ( s ) = ms 2 D ( s ) = F ( s ) − bsD ( s ) � b m s m + b m − 1 s m − 1 + · · · + b o � U ( s ) G ( s ) = D ( s ) 1 1 F ( s ) = ms 2 + bs = s ( ms + b ) U ( s ) = b m s m + b m − 1 s m − 1 + · · · + b o G ( s ) = Y ( s ) a n s n + a n − 1 s n − 1 + · · · + a o F D G s ( ) G ( s ) = B ( s ) . . . transfer function A ( s ) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 7 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 8
Example 2 (revisited): Transfer Function Example 3 (revisited) L di ( t ) dθ ( t ) dt + Ri ( t ) = V ( t ) − K t electrical part dt A ˙ � h ( t ) + K h ( t ) = Q i ( t ) J d 2 θ ( t ) dt 2 + b dθ ( t ) mechanical part = K t i ( t ) dt Can we use Laplace transform? ( Ls + R ) I ( s ) = V ( s ) − K t sθ ( s ) ( Js 2 + bs ) θ ( s ) = K t I ( s ) G ( s ) = θ ( s ) K t V ( s ) = s [( Ls + R )( Js + b ) + K 2 t ] Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 9 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 10 State-Space Models Linear State-Space Model Introduce state variable x ( t ) (vector) to parameterize the ‘memory’ x ( t ) = Ax ( t ) + Bu ( t ) ˙ of the system. y ( t ) = Cx ( t ) + Du ( t ) • The state contains all information needed to determine future A . . . state matrix behavior without reference to the derivatives of input and out- B . . . input matrix put variables. C . . . output matrix D . . . direct transmission matrix • The state is often determined from physical considerations Interpretation: (related to energy storage in the system). Derivative of each state is given by a linear combination of states plus a linear combination of inputs. Similarly for the output . . . • The dimension n of the state vector is the order of the system. Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 11 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 12
State-Space Model: Block Diagram Example 1 (revisited): State-Space Model m ¨ d ( t ) = F ( t ) − b ˙ Diff. equation for motion under friction: d ( t ) x ( t ) = Ax ( t ) + Bu ( t ) ˙ y ( t ) = Cx ( t ) + Du ( t ) Introduce velocity: v ( t ) = ˙ d ( t ) . ( ) ( ) u t y t ( ) ( ) x t x t Rewrite the above 2nd-order equation as a set of two 1st order DE: mv ( t ) + 1 v ( t ) = − b ˙ mF ( t ) ˙ d ( t ) = v ( t ) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 13 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 14 Example 1 (revisited): State-Space Model Example 2 (revisited): State-Space Model L di ( t ) dθ ( t ) dt + Ri ( t ) = V ( t ) − K t electrical part dt v ( t ) state: x ( t ) = , input: u ( t ) = F ( t ) , output: y ( t ) = d ( t ) J d 2 θ ( t ) dt 2 + b dθ ( t ) = K t i ( t ) mechanical part d ( t ) dt Introduce velocity: ω ( t ) = ˙ θ ( t ) − b 1 x 1 ˙ m 0 x 1 m = + u Rewrite the above equations as a set of three 1st order DE: x 2 ˙ 1 0 x 2 0 L ω ( t ) + 1 i ( t ) = − R Li ( t ) − K t ˙ LV ( t ) x 1 � � y = ω ( t ) = K t J i ( t ) − b 0 1 ˙ Jω ( t ) x 2 ˙ θ ( t ) = ω ( t ) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 15 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 16
Compare to the Input Output Model Example 2 (revisited): State-Space Model � T � state: x ( t ) = , input: u ( t ) = V ( t ) , output: G ( s ) = θ ( s ) i ( t ) ω ( t ) θ ( t ) K t V ( s ) = s [( Ls + R )( Js + b ) + K 2 y ( t ) = θ ( t ) t ] − R L − K t 1 � LJs 3 + ( RJ + Lb ) s 2 + ( Rb + K 2 � x 1 ˙ 0 x 1 θ ( s ) t ) s = K t V ( s ) L L K t − b = + u x 2 ˙ 0 x 2 0 J J Corresponds to the following differential equation: ˙ 0 1 0 0 x 2 x 3 ... θ ( t ) + ( RJ + Lb )¨ θ ( t ) + ( Rb + K 2 t ) ˙ LJ θ ( t ) = K t V ( t ) x 1 � � y = 0 0 1 x 2 Note: current i not in the model! Input-output models do not use internal variables, x 3 instead use higher derivatives of input and outputs. Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 17 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 18 The big picture Purpose of Analysis Analyze the available model in order to: Type of model First principles • Understand the behavior of the process under study. Simulation, prediction, Nonlinear differential eq. Physical world (process) better understanding • Define meaningful specification for the controlled system. Linearization • Give basis for control design choices (controller structure, pa- Data Basis for control - rameters). Linearized differential eq. oriented models We are mainly interested in: Analysis, control design Transfer function State-space model • Stability of the open-loop process. Design • Transient response (impulse, step, ramp). Influence a process, Controller modify behavior • Steady-state response (constant or sinusoidal input). Implementation Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 19 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 20
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