State space analysis mechanical

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This document discusses state space analysis for a mechanical mass-damper-spring system. The system is modeled as a set of first-order differential equations describing how the mass, damping, and spring interact over time. State space analysis can be used to understand the behavior and stability of the system by analyzing the system's response to different inputs.

STATE SPACE ANALYSIS
MECHANICAL
(Mass-Damper –Spring)
1

Introduction As systems become more complex, representing them with differential equations or transfer functions becomes cumbersome. This is even more true if the system has multiple inputs and outputs. This document introduces the state space method which largely alleviates this problem. The state space representation of a system replaces an nth order differential equation with a single first order matrix differential equation. The state space representation of a system is given by two equations : The first equation is called the state equation, the second equation is called the output equation. For an nth order system (i.e., it can be represented by an nth order differential equation) with r inputs and m outputs the size of each of the matrices is as follows: Several features:The state equation has a single first order derivative of the state vector on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.The output equation has the output on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side. q is nx1 (n rows by 1 column)q is called the state vector, it is a function of timeA is nxn; A is the state matrix, a constantB is nxr; B is the input matrix, a constant u is rx1; u is the input, a function of time C is mxn; C is the output matrix, a constant D is mxr; D is the direct transition matrix, a constant y is mx1; y is the output, a function of time Derivation of of State Space Model (Electrical) To develop a state space system for an electrical system, they choosing the voltage across capacitors, and current through inductors as state variables. Recall that so if we can write equations for the voltage across an inductor, it becomes a state equation when we divide by the inductance (i.e., if we have an equation for einductor and divide by L, it becomes an equation for diinductor/dt which is one of our state variable). Likewise if we can write an equation for the current through the capacitor and divide by the capacitance it becomes a state equation for ecapacitor There are three energy storage elements, so we expect three state equations. Try choosing i1, i2 and e1 as state variables. Now we want equations for their derivatives. The voltage across the inductor L2 is e1 (which is one of our state variables)so our first state variable equation is This equation has our input (ia) and two state variable (iL2 and iL1) and the current through the capacitor. So from this we can get our second state equation Our third, and final, state equation we get by writing an equation for the voltage across L1 (which is e2) in terms of our other state variables references: http://lpsa.swarthmore.edu/Representations/SysRepSS.html https://en.wikipedia.org/wiki/State-space_representation

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