Standard Forms of System of Equations

# Standard Forms of System of Equations - ME375 Handouts...

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Unformatted text preview: ME375 Handouts Standard Forms for System Models • State Space Model Representation – State Variables – Example • Input/Output Model Representation – General Form – Example • Comments on the Difference between State Space and Input/Output Model Representations School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 1 State Space Model Representation • State Variables The smallest set of variables {q1, q2, …, qn} such that the knowledge of these variables at time t = t0 , together with the knowledge of the input for t ≥ t0 completely determines the behavior (the values of the state variables) of the completely variables) system for time t ≥ t0 . Example: x K EOM: M f(t) B Q: What information about the mass do we need to know to be able to solve for x(t) for t ≥ t0 ? for School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 2 1 ME375 Handouts State Space Model Representation • State Space Representation – Two parts: • A set of first order ODEs that represents the derivative of each state variable qi as an algebraic function of the set of state variables {qi} and the inputs {ui}. ⎧ q1 = f 1 (q1 , q 2 , q 3 , … , q n , u1 , u 2 , u 3 , … , u m ) ⎪ q = f (q , q , q , … , q , u , u , u , … , u ) ⎪2 2 1 2 3 n 1 2 3 m ⎨ ⎪ ⎪ q n = f n (q1 , q 2 , q 3 , … , q n , u1 , u 2 , u 3 , … , u m ) ⎩ • A set of equations that represents the output variables as algebraic algebraic functions of the set of state variables {qi} and the inputs {ui}. ⎧ y1 = g 1 ( q 1 , q 2 , q 3 , … , q n , u 1 , u 2 , u 3 , … , u m ) ⎪ ⎪ y 2 = g 2 ( q1 , q 2 , q 3 , … , q n , u 1 , u 2 , u 3 , … , u m ) ⎨ ⎪ ⎪ y = g (q , q , q , … , q , u , u , u , … , u ) 1 2 3 1 2 3 p n m ⎩p School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 3 State Space Model Representation • Example EOM M x + B x + K x = f (t ) K M B x f(t) State Variables: Output: State Space Representation: School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 4 2 ME375 Handouts State Space Model Representation • Obtaining State Space Representation – Identify State Variables • Rule of Thumb: – Nth order ODE requires N state variables. – Position and velocity are natural state variables for translational translational mechanical systems. – Eliminate all algebraic equations written in the modeling process. process. – Express the resulting differential equations in terms of state variables and variables inputs in coupled first order ODEs. – Express the output variables as algebraic functions of the state variables and inputs. – For linear systems, put the equations in matrix form. x = A⋅ x State Variables in vector form +B⋅ u Inputs in vector form y Outputs in vector form = C ⋅x + D⋅ u School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 5 State Space Model Representation • Exercise Represent the 2 DOF suspension system in a state space representation. Let the system output be the position of mass M1. M 1 x1 + B1 x1 − B1 x2 + K1 x1 − K1 x2 = 0 M 2 x2 − B1 x1 + B1 x2 − K1 x1 + ( K1 + K 2 ) x2 = K 2 xr State Variables: Output: State Space Representation: K2 xr M1 g K1 M2 B1 x2 x1 ⎡ q1 ⎤ ⎡ ⎢q ⎥ ⎢ ⎢ 2⎥ = ⎢ ⎢ q3 ⎥ ⎢ ⎢⎥⎢ ⎣ q4 ⎦ ⎣ x A ⎤ ⎡ q1 ⎤ ⎡ ⎥ ⎢q ⎥ ⎢ ⎥⎢ 2⎥ + ⎢ ⎥ ⎢ q3 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎦ ⎣ q4 ⎦ ⎣ x B ⎤ ⎥ ⎥ xr ⎥ ⎥ ⎦ , y =[ C ⎡ q1 ⎤ ⎢q ⎥ ] ⎢ q2 ⎥ ⎢ 3⎥ ⎢⎥ ⎣ q4 ⎦ x School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 6 3 ME375 Handouts Input/Output Representation • Input/Output Model Uses one nth order ODE to represent the relationship between the input variable, u(t), and the output variable, y(t), of a system. For linear time-invariant (LTI) systems, it can be represented by : time- an y ( n ) + (n) where y =( + a2 y + a1 y + a0 y = bmu ( m ) + dt d ) + b2u + b1u + b0u (t ) n y – To solve an input/output differential equation, we need to know – To obtain I/O models: • Identify input/output variables. • Derive equations of motion. • Combine equations of motion by eliminating all variables except the input and output variables and their derivatives. School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 7 Input/Output Representation • Example Vibration Absorber EOM: z2 M2 K2 z1 M1 K1 B1 M 1 z1 + B1 z1 + ( K1 + K 2 ) z1 − K 2 z2 = f (t ) M 2 z2 + K 2 z2 − K 2 z1 = 0 – Find input/output representation between input f(t) and output z2. f(t) School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 8 4 ME375 Handouts Input/Output Representation • Vibration Absorber z2 M2 K2 z1 M1 K1 B1 f(t) Q: Find input/output representation between input f(t) and output z1. Find Q: What if another damper is added between masses M1 and M2 ? What School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 9 Comments on Input/Output and State Space Models • State Space Models: – consider the internal behavior of a system – can easily incorporate complicated output variables – have significant computation advantage for computer simulation – can represent multi-input multi-output (MIMO) systems and multimultinonlinear systems School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 10 5 ME375 Handouts Comments on Input/Output and State Space Models • Input/Output Models: – are conceptually simple – are easily converted to frequency domain transfer functions that are more intuitive to practicing engineers – are difficult to solve in the time domain (solution: Laplace transformation) School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 11 6 ...
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