State-space models 2 2 nd order ODEs

Transcription

1 1 State-space models 2 2 nd order ODEs J A Rossiter

2 Introduction This resource focuses on derivations of state space model equivalents for systems described by ODEs. Here we consider 2 nd order ODEs (see separate resources for detailed derivation). The state-space model is defined in terms of the derivatives of the states. If you know the derivatives of all the states, then you can capture the system behaviour. States relate to dynamic variables such as splacement, height, tension, temperature, etc. 2

3 States of a 2 nd order ODE 3 1. The first step in forming a state space model is to define the states. 2. There should be enough independent states to capture the entire system dynamics for low order systems this selection is usually obvious. 3. However, when presented with a high order fferential equation, the user may have no access (or information) relating to the definition of the original underlying states, and thus an arbitrary definition can be used.

4 Modelling a mass-spring-damper Force balance can be used to determine the overall model of behaviour. f 1 Bv ˆ f f f 2 kx dv 3 M f 1 f 2 f 3 4 dv f M Bv ˆ kx States are v and x

5 States for mass-spring-damper dv f M Bv ˆ kx It is clear that the model contains states velocity v and splacement x so these are a logical choice. 5 For a state space model, find the derivatives of each state and stack into a single vector.

6 State-space model Write the derivatives of v and x and stack in a vector. dv dv f kx M f kx Bv ˆ dx dx M v v Bv ˆ 6 dv dx Bˆ k v 1 M M M f x A z B z In compact form Az Bf

7 Resistor-inductor-capacitor in series Consider the following circuit and use Kirchhoff s voltage law to derive an appropriate model. 7 v1 ir1 R1 dq v2 1 v3 q C v v1 v2 v3 v R dq States are i and q 1 C q

8 State-space model Write the derivatives of i and q and stack in a vector. q q v ir v ir C C dq i q dq R 1 i 1 C v q A z B z In compact form Az Bv 8

9 The model can be summarised with some simple equations. Here it may be less obvious which states to choice due to the possibilities of two angles/velocities, current, back emf and torque. DC servo v ki Bw ˆ kw J dw ir 9

10 DC servo choice of state The states of interest must be defined with an equation inclung their derivative. Other states could be viewed as outputs (possible measurements) and in this case will be linearly dependent on the selected states. Two obvious choices are w and i: 10 v kw ir; ki Bw ˆ J dw

11 State-space model of DC servo Express the derivatives of the selected states in a column vector. dw v kw ir; ki Bw ˆ J v k R dw k Bˆ w i ; i w J J dw Bˆ k w J J v k R i 1 0 z Az Bv z B A 11

12 DC servo with splacement Should the user be interested in the angular position as well as the angular velocity, then an adtional state is needed. 12 d w dw J Bw ki i kw ; ˆ R; v v i w R k J k J B d dw ˆ

13 Pendulum A pendulum of length l with end mass m is able to swing freely (assume some friction constant k). A model can be derived using force balance in the tangential rection (small angles). ml mg length - l kl mass - m 13 Choose states:, w

14 Pendulum state space model State derivatives: dw d ml mg klw w length - l mass - m 14 dw d kl mg w 0 ml ml A z B f z Az Bf

15 Summary 15 Illustrated state-space model derivation for several common 2 nd order systems. It is noted that the selection of states is important, but this choice is often obvious from the underlying component and balance equations as only certain states will have their derivatives defined explicitly. The compact form has vectors of states/inputs and matrices of coefficients. x Ax Bu

16 Anthony Rossiter Department of Automatic Control and Systems Engineering University of Sheffield University of Sheffield This work is licensed under the Creative Commons Attribution 2.0 UK: England & Wales icence. To view a copy of this licence, visit or send a letter to: Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA. It should be noted that some of the materials contained within this resource are subject to third party rights and any copyright notices must remain with these materials in the event of reuse or repurposing. If there are third party images within the resource please do not remove or alter any of the copyright notices or website details shown below the image. (Please list details of the third party rights contained within this work. If you include your institutions logo on the cover please include reference to the fact that it is a trade mark and all copyright in that image is reserved.)