Standard_Forms

Standard_Forms - Standard Forms for System Models Standard...

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Unformatted text preview: Standard Forms for System Models Standard Forms for System Models • State Space Model Representation 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 and Input/Output Model Representations School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 1 State Space Model Representation State Space Model Representation • State Variables The smallest set of variables {q1, q2, …, qn} such that the knowledge of these smallest set of variables such that the knowledge of these variables variables at time t = t0 , together with the knowledge of the input for t ≥ t0 completely completely determines the behavior (the values of the state variables) of the system for time t ≥ t0 . Example 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 for t ≥ t0 ? School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 2 State Space Model Representation State Space Model Representation • State Space Representation – Two parts: 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 ⎪q ⎪ &2 ⎨ ⎪ ⎪ qn ⎩& = f 1 (q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m ) = f 2 (q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m ) M = f n (q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m ) • A set of equations that represents the output variables as algebraic functions of the set of state variables {qi} and the inputs {ui}. ⎧ y1 ⎪ ⎪ y2 ⎨ ⎪ ⎪ ⎩ yp = g 1 ( q1 , q 2 , q 3 , K , q n , u 1 , u 2 , u 3 , K , u m ) = g 2 ( q1 , q 2 , q 3 , K , q n , u 1 , u 2 , u 3 , K , u m ) M = g p ( q1 , q 2 , q 3 , K , q n , u 1 , u 2 , u 3 , K , u m ) School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 3 State Space Model Representation State Space Model Representation • Example EOM && & M x + B x + K x = f (t ) x K M f(t) B State Variables: Output: State Space Representation: School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 4 State Space Model Representation 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 mechanical systems. – Eliminate all algebraic equations written in the modeling process. – Express the resulting differential equations in terms of state variables and inputs in coupled first order ODEs 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. linear systems put the equations in matrix form & x = A⋅ x { State Variables in vector form y { Outputs in vector form +B⋅ u { Inputs in vector form = C ⋅x + D⋅ u School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 5 State Space Model Representation 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&&1 + B1 x1 − B1 x2 + K1 x1 − K1 x2 = 0 x x1 M1 g K1 B1 x2 & & M 2 &&2 − B1 x1 + B1 x2 − K1 x1 + ( K1 + K 2 ) x2 = K 2 xr x M2 State Variables: K2 Output: xr State Space Representation: & ⎡ q1 ⎤ ⎡ ⎤ ⎡ q1 ⎤ ⎡ ⎤ ⎢q ⎥ ⎢ ⎥ ⎢q ⎥ ⎢ ⎥ &2 ⎢ ⎥=⎢ ⎥ ⎢ 2 ⎥ + ⎢ ⎥ xr & ⎢ q3 ⎥ ⎢ ⎥ ⎢ q3 ⎥ ⎢ ⎥ ⎢& ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎦ ⎣ q4 ⎦ { ⎣ q4 ⎦ ⎣ { 144 2444 { ⎣ ⎦ 4 3 & B x x A , School of Mechanical Engineering Purdue University ⎡ q1 ⎤ ⎢q ⎥ y =[ ⎢ q2 ⎥ 14 244 ⎢ 3 ⎥ 4 3 C ⎢⎥ ⎣ q4 ⎦ { x ME375 Standard Forms of Equation - 6 Input/Output Representation Input/Output Representation • Input/Output Model Uses one nth order ODE to represent the relationship between the input Uses one nth order ODE to represent the relationship between the input variable, 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 ) + L + a2 && + a1 y + a0 y = bmu ( m ) + L + b2u + b1u + b0u (t ) y where y ( n ) = ( d dt ) y n – To solve an input/output differential equation, we need to know – To obtain I/O models: I/O • Identify input/output variables. • Derive equations of motion. • Combine equations of motion by eliminating all variables except the Combine equations of motion by eliminating all variables except the input input and output variables and their derivatives. School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 7 Input/Output Representation Input/Output Representation • Example z2 Vibration Absorber EOM: M2 & M 1&&1 + B1 z1 + ( K1 + K 2 ) z1 − K 2 z2 = f (t ) z M 2 &&2 + K 2 z2 − K 2 z1 = 0 z – Find input/output representation between betwee input f(t) and output z2. K2 z1 M1 K1 School of Mechanical Engineering Purdue University f(t) B1 ME375 Standard Forms of Equation - 8 Input/Output Representation In • Vibration Absorber z2 M2 K2 z1 M1 K1 f(t) B1 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 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 ou (MIMO) nonlinear nonlinear systems School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 10 Comments on Input/Output and State Space Models Comments on Input/Output and State Space Models • Input/Output Models: – are conceptually simple – are easily converted to frequency domain transfer functions that that are more intuitive to practicing engineers – are difficult to solve in the time domain (solution: Laplace transformation) transformation) School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 11 ...
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