Vai's Uni Notes

State-Space Models

4 min read

How a (possibly nonlinear) physical model is packaged as a state equation plus an output equation. Symbols follow Prof. Sanner’s notation: physical state 𝐳, physical input 𝐰, physical output 𝛈.

State of a System

  • The state of a system is a collection of quantities 𝑧1,𝑧2,…,𝑧𝑛 such that knowledge of these quantities at a single time 𝑡0, together with knowledge of the inputs 𝐰(𝑡), 𝑡≥𝑡0, is sufficient to predict the outputs 𝛈(𝑡) ∀𝑡≥𝑡0
  • Block diagram: input 𝐰(𝑡)∈ℝ𝑚 → [system, with internal state 𝐳(𝑡)] → output 𝛈(𝑡)∈ℝ𝑝
  • The state defines a mapping:
    • {𝐳(𝑡0),𝐰(𝑡),𝑡≥𝑡0}→𝛈(𝑡),𝑡≥𝑡0
    • A feasible mapping must make this constructive
  • State vector of the system:
    • 𝐳(𝑡)=[𝑧1(𝑡)𝑧2(𝑡)⋮𝑧𝑛(𝑡)]∈ℝ𝑛

State Equation

Implicit Form

  • Assume physical modeling specifies the rate of change of the chosen states 𝑧𝑖(𝑡) via implicit DEs of the form:
    • 0=𝑓𝑘(̇𝑧1(𝑡),̇𝑧2(𝑡),…,̇𝑧𝑛(𝑡),𝑧1(𝑡),…,𝑧𝑛(𝑡),𝑤1(𝑡),…,𝑤𝑚(𝑡),𝑡),𝑘=1,…,𝑛
    • (𝑛 equations)
  • i.e. the rate of change of each 𝑧𝑘(𝑡) potentially depends on the instantaneous values of the 𝑛 states, the 𝑛 state derivatives, the 𝑚 inputs, and time 𝑡 itself (2𝑛+𝑚+1 variables)
  • More compactly:
    • 0=𝑓𝑘(̇𝐳(𝑡),𝐳(𝑡),𝐰(𝑡),𝑡),𝑘=1,…,𝑛
  • Even more compactly, stack them into a vector of functions:
    • 𝐟(̇𝐳,𝐳,𝐰,𝑡)=[𝑓1(̇𝐳,𝐳,𝐰,𝑡)𝑓2(̇𝐳,𝐳,𝐰,𝑡)⋮𝑓𝑛(̇𝐳,𝐳,𝐰,𝑡)]

State Equation

  • 𝟎=𝐟(̇𝐳(𝑡),𝐳(𝑡),𝐰(𝑡),𝑡)
  • is the state equation for the system

Explicit Form

  • Common alternate form is the explicit form:
    • ̇𝐳(𝑡)=𝐟(𝐳(𝑡),𝐰(𝑡),𝑡)
  • But this can be an unnecessarily restrictive assumption in some cases

Linear State Equations

  • The state equation is linear iff:
    • 𝐟(̇𝐳,𝐳,𝐰,𝑡)=𝐸(𝑡)⏟𝑛×𝑛̇𝐳+𝐹(𝑡)⏟𝑛×𝑛𝐳+𝐺(𝑡)⏟𝑛×𝑚𝐰

Standard Form

  • If the state eq’n is linear, with invertible matrix 𝐸, then
    • 𝟎=𝐸(𝑡)̇𝐳+𝐹(𝑡)𝐳+𝐺(𝑡)𝐰
  • can be re-written as the “standard form” for a linear state eq’n:
    • ̇𝐳=𝐴(𝑡)𝐳+𝐵(𝑡)𝐰𝐴(𝑡)=−𝐸(𝑡)−1𝐹(𝑡)𝐵(𝑡)=−𝐸(𝑡)−1𝐺(𝑡)

NOTE

  • 𝐸(𝑡) will be invertible in all examples considered in this class

Output Equation

  • Block diagram: 𝐰 (𝑚) → [system, 𝐳 (𝑛)] → 𝛈 (𝑝)
  • Full model:
    • 𝟎=𝐟(̇𝐳,𝐳,𝐰,𝑡)state eq'n𝛈=𝐡(𝐳,𝐰,𝑡)output eq'n
  • Like the state eq’n, the output eq’n is linear iff:
    • 𝐡(𝐳,𝐰,𝑡)=𝐶(𝑡)⏟𝑝×𝑛𝐳+𝐷(𝑡)⏟𝑝×𝑚𝐰
  • The system is linear if both the state and output eq’ns are

Example: Simple Pendulum

  • Simple pendulum with applied torque at the pivot (from Prof. Sanner's notes)

Simple Pendulum Model

  • Take states:
    • 𝑧1=𝜃
    • 𝑧2=̇𝜃=𝜔
  • Physics:
    • 𝑚𝑙2̇𝜔=−𝑚𝑔𝑙sin𝜃−𝑏(𝜔)+𝜏𝑚̇𝜃=𝜔
    • 𝑏(𝜔): friction, drag
    • 𝜏𝑚: applied torque (servo motor) → this is the input, 𝑤
  • Equivalently:
    • ̇𝑧1=𝑧2𝑚𝑙2̇𝑧2=−𝑚𝑔𝑙sin𝑧1−𝑏(𝑧2)+𝑤
  • Rewrite in implicit form:
    • [00]=[̇𝑧1−𝑧2𝑚𝑙2̇𝑧2+𝑚𝑔𝑙sin𝑧1+𝑏(𝑧2)−𝑤]=[𝑓1(̇𝐳,𝐳,𝑤,𝑡)𝑓2(̇𝐳,𝐳,𝑤,𝑡)]=𝐟(̇𝐳,𝐳,𝑤,𝑡)
  • Note that this is not linear, b/c of the sin𝑧1 term, and b/c 𝑏(𝑧2) is not explicitly specified
  • However, if we assume sin𝑧1≈𝑧1, 𝑏(𝜔)≈𝛽𝜔 (linear friction), then the state eq’n is linear with:
    • [00]=[100𝑚𝑙2]⏟𝐸[̇𝑧1̇𝑧2]+[0𝑚𝑔𝑙−1𝛽]⏟𝐹[𝑧1𝑧2]+[0−1]⏟𝐺𝑤
    • Note: 𝐸,𝐹,𝐺 constant ⇒ eq’n is also time invariant
  • Output: suppose we want to control the angle 𝜃=𝑧1. Then
    • 𝜂=ℎ(𝐳,𝑤,𝑡)=𝑧1=[10]⏟𝐶𝐳+0⏟𝐷𝑤
    • (linear and time-invariant here)
  • Why is sin𝑧≈𝑧 justified? See Linearization

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