# Linear Systems Theory

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1 ME 3253 Linear Systems Theory Review

2 Class Overview and Introduction 1. How to build dynamic system model for physical system? 2. How to analyze the dynamic system? -- Time domain -- Frequency domain (Laplace transformation, etc) Future: how to control a dynamic system?

3 Ordinary Differential Equations (ODE) Relation (function) involves independent variable and its derivatives (could be higher order). Comparing to algebra equation 1. Time involved 2. History (initial value) matters 3. Solution is a signal over time Dynamic Systems

4 Scope of Systems Linear ODE Linear Dynamic Systems x (3) + sin(2t) && x+ cos( t) x& + e t x + sin( t) + 1 = 0 Linear Time-Invariant Systems x (3) + 2&& x+ 3x& + 5x+ sin( t) + 1 = 0

5 Linear Ordinary Differential Equations Coefficients of variable and its derivative only depends on self-variable or time Linear: 1. Additivity 2. Scalability Linear Dynamic Systems

6 Solution of Linear Systems Solution depends on: System Dynamics (Equation) x (3) + sin(2t) && x+ cos( t) x& + e Initial Condition & x&= 1, x(0) = 2 Input (If we have) t x+ sin( t) + 1= 0 Block diagram (physical meaning): Input Signal System dynamics Output Signal sin( t ) + 1 with initial value x(t)

7 Representation of System Dynamics How to represent system dynamics? LTI System (ODE) Transfer function (Laplace transformation) State space X & = AX + Bu Analysis Analysis Classical control theory Modern control theory

8 Uniform and Standard Representation Linear ODE can be transformed into a system of differential equation of order 1 X& = AX + y= CX Bu

9 Laplace Transformation Closely related to frequency analysis.

10 Example of Mechanical Systems 1. Dynamic equation (ODE) 2. Laplace domain 2.1 Input, output signal 2.2 Transfer function 3. Laplace transformation of input signal 4. Laplace transformation of output signal 5. Inverse Laplace transformation Output signal

11 Laplace Transformation We have a time domain function: s L st f ( t ) ( ) 0 Complex variable in Laplace domain Stands for the Laplace Transform = L[] e [] dt 0 st F( s) = L[ f ( t)] = e f ( t) dt = = f t = for t< 0 Notations: Small case f, x, y, etc : time domain Capital case F, X, Y, etc : Laplace domain or frequency domain f(t) is called inverse Laplace transform of F(s) 0

12 Differentiation Theorem Differentiation Theorem df L[ ] sf( s) f (0) dt 2 d f L[ ] dt 2 2 s F( s) sf (0) f & (0)

13 ODE: Frequency Domain && x + 2x& + 5x= u( t) = sin( t) 1 2 s X ( s) + 2sX ( s) + 5X ( s) = U ( s) X 1 ( s) = U ( s) 2 s + 2s+ 5 Input Signal System dynamics Output Signal u (t) with initial value x(t)

14 Solution of Linear Systems Linear Time-Invariant Systems && x + 2 x& + 5x+ sin( t) + 1= 0 Block diagram (physical meaning): Input Signal u( t) = sin( t) 1 System dynamics with initial value Output Signal x(t) && x + 2& x+ 5x= u & x + 2x& + 5x= u& + u

15 Solution of Linear Systems & x + 2x& + 5x= u& + u Perform Laplace transformation to both sides: Differential equation becomes algebraic equation. Laplace: st F( s) = L[ f ( t)] = e f ( t) dt = = 0 Inverse Laplace: f ( t) = 1 2 πj β+ j β j F( s) e st ds

16 Existence and Property Existence of Laplace Transform: f ( t) continuous, of exponential order as t goes to infinity Laplace Transform is a Linear Operator La [ f ( t) + a f ( t)] = al[ f ( t)] + a L[ f ( t)]

17 Multiplication by exponential function L [ f ( t)] = F( s) αt L [ e f ( t )] = F ( s+ + α )

18 Integration Theorem L [ f ( t)] = F( s) t L [ f ( τ ) d τ ] = 0 F( s) s Compare with differentiation theorem f ( t) = t

19 Translated Function L [ f ( t)1( t)] = F( s) ( ) αs t α ] = e F( ) L[ f ( t α)1 s α 0

20 Laplace Transform Property lim f ( t) Final Value Theorem: if exists t lim f ( t ) = lim sf ( s ) t s 0 Initial Value Theorem: f (0) = lim sf( s) s

21 Pulse function f f ( t) ( t) = = A,0 t0 0, t> t 0 t t 0

22 Inverse Laplace Transform Interesting Case: Complex Conjugate Poles Trick: remain in second order form 3 Example X ( s) = 2 s( s + 2s + 2) Poles: p = p2,3 = 1 ± 1 0 Complex numbers involved (can use previous method, tedious) j Use either book method Or brute force method 3 3 t jt jt 3 jt jt x( t) = e ( e + e ) + j( e e ) t 3 t = e cos( t) e sin( t) 2 2 2

23 Don t have to go through that.. Trick: Expansion format retain the quadratic factor relating to the complex conjugate poles Pole equation: p 1 = 0 p = ± 2,3 1 j s s + s + = 2 ( ) = 0 s 2 2 s + 2s + 2 = ( s + 1) + 1 X ( s) a = + s bs+ c ( s+ 1) + 1 Note: numerator must be bs+ c

24 L bs + c bs + b b + c [ ] = L [ ] ( s + 1) + 1 ( s + 1) s = bl [ ] + ( c b) L [ ] 2 2 ( s + 1) + 1 ( s + 1) + 1 t t = be cos( t) + ( c be ) sin( t) Can find that (use brute force method, compare the coefficients of like power of s) a= b= c= 3 2

25 Properties: 1. Stability (real part of components) 2. Natural frequency (imaginary part of components) 3. Initial/End values

26 Mechanical Systems Mass: inertia elements Moment of Inertia :analogous to mass, rotating wheel of your car Spring elements: translational spring and torsional spring linear assumption Damper elements: convert mechanical energy into heat energy dissipation (velocity dependent) Modeling characterize how the system move Newton s Law: F=ma

27 Mechanical Systems Forced response: behavior under an external force Natural response (free response): behavior under initial condition only, without external force Example: brake system Example: simple spring mass system under initial condition natural frequency why natural? How to measure moment of inertia of complicated structure? Use natural frequency! J && θ + k θ = 0 ω Natural frequency can be measured, and torsional spring k can be measured. k ω = J J

28 Mechanical Systems Friction Static friction: not moving Sliding friction: translational motion (sliding friction coefficient smaller than static friction coefficient) Rolling friction: friction force magnitude unknown. Rolling without sliding: x= Rθ

29 Electrical Systems Complex impedances: directly write system equations in Laplace domain E( s) = Z( s) I( s) e= ir E = IR Z = R di e= L E = LsI Z = Ls dt e= idt E = I Z = C Cs Cs Advantage: can be used in parallel and series circuit modeling, just like resistances Only suitable for Transfer Function Derivation (why??)

30 How to Solve Electrical systems 1. Loop law (Voltage equations) 2. Node law (Current equations) 3. Basic element definitions In Laplace domain: a set of linear equations: Laplace transformation of input signals. Solve to get Laplace transformation of output signal. Inverse-Laplace transformation. Initial values?

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