Tutorial 11 SS modelling 1

Tutorial 11: state-space modelling • State-space models are used for MIMO systems • Multiple input multiple output • Particularly useful for modelling systems with initial conditions • Also useful for computer analysis as is matrix based • State-space model comprises of four matrices A , B , C , D • A represents system characteristics • B represents system input • C represents system output • D represents initial conditions • Order of system dictates number of state variable & equations… • 2 nd order = two state variables & two state equations etc • State variables are assigned by the engineer • Show the link between current state & next state • State-space model is a matrix version of system equations • Mathematically identical Develop the state-space model for the following system; assume no initial conditions 30 𝑠𝑠 2 + 6𝑠𝑠 + 30 What do we know? • System shown in Laplace (s-domain) • Must map back to time domain • System is 2 nd order • Two state variables • Two state equations • No initial conditions but must include D • D =[0] Step 1: map to time domain 𝑠𝑠 2 𝑌𝑌 ( 𝑠𝑠 ) → 𝑑𝑑 2 𝑦𝑦 𝑑𝑑𝑡𝑡 2 ≡ ̈𝑦𝑦 𝑠𝑠𝑌𝑌 ( 𝑠𝑠 ) → 𝑑𝑑𝑦𝑦 𝑑𝑑𝑡𝑡 ≡ ̇𝑦𝑦 𝑌𝑌 𝑠𝑠 → 𝑦𝑦 𝑠𝑠 2 𝑌𝑌 𝑠𝑠 + 6𝑠𝑠𝑌𝑌 𝑠𝑠 + 30𝑌𝑌 𝑠𝑠 = 30𝑈𝑈 ( 𝑠𝑠 ) ̈𝑦𝑦 𝑡𝑡 + 6 ̇𝑦𝑦 𝑡𝑡 + 30𝑦𝑦 𝑡𝑡 = 30𝑢𝑢 ( 𝑡𝑡 )

Step 2: create two state variables 𝑥𝑥 1 𝑎𝑎𝑎𝑎𝑑𝑑 𝑥𝑥 2 Step 3 : Assign state variables In this example… ̈𝑦𝑦 𝑡𝑡 + 6 ̇𝑦𝑦 𝑡𝑡 + 30𝑦𝑦 𝑡𝑡 = 30𝑢𝑢 ( 𝑡𝑡 ) we can choose from ̈𝑦𝑦 , ̇𝑦𝑦 , 𝑦𝑦 Let us assign… 𝑥𝑥 1 ( 𝑡𝑡 ) = 𝑦𝑦 ( 𝑡𝑡 ), 𝑥𝑥 2 ( 𝑡𝑡 ) = ̇𝑦𝑦 (t) Step 4 : assign next states of 𝑥𝑥 1 𝑎𝑎𝑎𝑎𝑑𝑑𝑥𝑥 2 𝐼𝐼𝐼𝐼 𝑐𝑐𝑢𝑢𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑡𝑡 𝑠𝑠𝑡𝑡𝑎𝑎𝑡𝑡𝑐𝑐 𝑖𝑖𝑠𝑠 𝑥𝑥 1 𝑡𝑡 = 𝑦𝑦 𝑡𝑡 , 𝑡𝑡𝑡𝑐𝑐 𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝑠𝑠𝑡𝑡𝑎𝑎𝑡𝑡𝑐𝑐 𝑖𝑖𝑠𝑠 ̇ 𝑥𝑥 1 ( 𝑡𝑡 ) = ̇𝑦𝑦 ( 𝑡𝑡 ) This is the current state of the system 𝐼𝐼𝐼𝐼 𝑐𝑐𝑢𝑢𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑡𝑡 𝑠𝑠𝑡𝑡𝑎𝑎𝑡𝑡𝑐𝑐 𝑖𝑖𝑠𝑠 𝑥𝑥 2 ( 𝑡𝑡 ) = ̇𝑦𝑦 𝑡𝑡 , 𝑡𝑡𝑡𝑐𝑐 𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝑠𝑠𝑡𝑡𝑎𝑎𝑡𝑡𝑐𝑐 𝑖𝑖𝑠𝑠 ̇ 𝑥𝑥 2 ( 𝑡𝑡 ) = ̈𝑦𝑦 ( 𝑡𝑡 ) ̇ 𝑥𝑥 1 ( 𝑡𝑡 ) = ̇𝑦𝑦 ( 𝑡𝑡 ) ̇ 𝑥𝑥 2 ( 𝑡𝑡 ) = ̈𝑦𝑦 ( 𝑡𝑡 ) This is the next state of the system Step 5 : Establish the two state equations State equation 1: ̇ 𝑥𝑥 1 ( 𝑡𝑡 ) = ̇𝑦𝑦 ( 𝑡𝑡 ) 𝑡𝑜𝑜𝑜𝑜𝑐𝑐𝑜𝑜𝑐𝑐𝑐𝑐 , 𝑎𝑎𝑠𝑠 𝑥𝑥 2 ( 𝑡𝑡 ) = ̇𝑦𝑦 ̇ 𝒏𝒏 𝟏𝟏 ( 𝒏𝒏 ) = 𝒏𝒏 𝟐𝟐 ( 𝒏𝒏 ) State equation 2: Make ̈𝑦𝑦 ( 𝑡𝑡 ) the subject ̈𝑦𝑦 𝑡𝑡 = − 6 ̇𝑦𝑦 𝑡𝑡 − 30𝑦𝑦 𝑡𝑡 + 30𝑢𝑢 ( 𝑡𝑡 ) (don’t forget ̇ 𝑥𝑥 2 𝑡𝑡 ≡ ̈𝑦𝑦 𝑡𝑡 & 𝑥𝑥 2 ( 𝑡𝑡 ) ≡ ̇𝑦𝑦 (t) ) ↓ ̇ 𝑥𝑥 2 ( 𝑡𝑡 ) = − 6 𝑥𝑥 2 ( 𝑡𝑡 ) − 30 𝑥𝑥 1 ( 𝑡𝑡 ) + 30𝑢𝑢 ( 𝑡𝑡 ) Reorder ̇ 𝒏𝒏 𝟐𝟐 ( 𝒏𝒏 ) = −𝟑𝟑𝟑𝟑𝒏𝒏 𝟏𝟏 ( 𝒏𝒏 ) − 𝟔𝟔𝒏𝒏 𝟐𝟐 ( 𝒏𝒏 ) + 𝟑𝟑𝟑𝟑𝟑𝟑 ( 𝒏𝒏 ) We need to establish the equations for the next state, i.e., the equations for ̇ 𝑥𝑥 1 𝑡𝑡 & ̇𝑥𝑥 2 ( 𝑡𝑡 )