Plan of the Lecture Review: control, feedback, etc. Todays topic: - PowerPoint PPT Presentation

Plan of the Lecture ◮ Review: control, feedback, etc. ◮ Today’s topic: state-space models of systems; linearization

Plan of the Lecture ◮ Review: control, feedback, etc. ◮ Today’s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest. Once we have mastered this framework, we can proceed to analysis and then to design .

Plan of the Lecture ◮ Review: control, feedback, etc. ◮ Today’s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest. Once we have mastered this framework, we can proceed to analysis and then to design . Reading: FPE, Sections 1.1, 1.2, 2.1–2.4, 7.2, 9.2.1. Chapter 2 has lots of cool examples of system models!!

Notation Reminder We will be looking at dynamic systems whose evolution in time is described by differential equations with external inputs . We will not write the time variable t explicitly, so we use x instead of x ( t ) x ′ ( t ) or d x ˙ instead of x d t x ′′ ( t ) or d 2 x ¨ instead of x d t 2 etc.

Example 1: Mass-Spring System x m u

Example 1: Mass-Spring System x m u Newton’s second law (translational motion): F = ma ���� total force

Example 1: Mass-Spring System x m u Newton’s second law (translational motion): F = ma = spring force + friction + external force ���� total force

Example 1: Mass-Spring System x m u Newton’s second law (translational motion): F = ma = spring force + friction + external force ���� total force spring force = − kx (Hooke’s law) friction force = − ρ ˙ x (Stokes’ law — linear drag, only an approximation!!)

Example 1: Mass-Spring System x m u Newton’s second law (translational motion): F = ma = spring force + friction + external force ���� total force spring force = − kx (Hooke’s law) friction force = − ρ ˙ x (Stokes’ law — linear drag, only an approximation!!) F = − kx − ρ ˙ x + u

Example 1: Mass-Spring System x m u Newton’s second law (translational motion): F = ma = spring force + friction + external force ���� total force spring force = − kx (Hooke’s law) friction force = − ρ ˙ x (Stokes’ law — linear drag, only an approximation!!) m ¨ x = − kx − ρ ˙ x + u

Example 1: Mass-Spring System x m u Newton’s second law (translational motion): F = ma = spring force + friction + external force ���� total force spring force = − kx (Hooke’s law) friction force = − ρ ˙ x (Stokes’ law — linear drag, only an approximation!!) m ¨ x = − kx − ρ ˙ x + u Move x, ˙ x, ¨ x to the LHS, u to the RHS: m ¨ x + ρ ˙ x + kx = u

Example 1: Mass-Spring System x m u Newton’s second law (translational motion): F = ma = spring force + friction + external force ���� total force spring force = − kx (Hooke’s law) friction force = − ρ ˙ x (Stokes’ law — linear drag, only an approximation!!) m ¨ x = − kx − ρ ˙ x + u Move x, ˙ x, ¨ x to the LHS, u to the RHS: x + ρ x + k mx = u ¨ m ˙ m

Example 1: Mass-Spring System x m u Newton’s second law (translational motion): F = ma = spring force + friction + external force ���� total force spring force = − kx (Hooke’s law) friction force = − ρ ˙ x (Stokes’ law — linear drag, only an approximation!!) m ¨ x = − kx − ρ ˙ x + u Move x, ˙ x, ¨ x to the LHS, u to the RHS: x + ρ x + k mx = u ¨ m ˙ 2nd-order linear ODE m

Example 1: Mass-Spring System x m u x + ρ x + k mx = u ¨ m ˙ 2nd-order linear ODE m

Example 1: Mass-Spring System x m u x + ρ x + k mx = u ¨ m ˙ 2nd-order linear ODE m Canonical form: convert to a system of 1st-order ODEs

Example 1: Mass-Spring System x m u x + ρ x + k mx = u ¨ m ˙ 2nd-order linear ODE m Canonical form: convert to a system of 1st-order ODEs x = v ˙ (definition of velocity) v = − ρ mv − k mx + 1 ˙ mu

Example 1: Mass-Spring System x m u State-space model: express in matrix form � ˙ � 0 � � � x � � 0 � x 1 = + u m − ρ 1 − k v ˙ v m m

Example 1: Mass-Spring System x m u State-space model: express in matrix form � ˙ � 0 � � � x � � 0 � x 1 = + u m − ρ 1 − k v ˙ v m m Important: start reviewing your linear algebra now !! ◮ matrix-vector multiplication; eigenvalues and eigenvectors; etc.

General n -Dimensional State-Space Model     x 1 u 1 . .  ∈ R n  ∈ R m  .   .  state x = input u = . .   x n u m

General n -Dimensional State-Space Model     x 1 u 1 . .  ∈ R n  ∈ R m  .   .  state x = input u = . .   x n u m           x 1 ˙ x 1 u 1 A B . . .  =  + . . . . . .         n × n n × m ˙ x n x n u m matrix matrix

General n -Dimensional State-Space Model     x 1 u 1 . .  ∈ R n  ∈ R m  .   .  state x = input u = . .   x n u m           x 1 ˙ x 1 u 1 A B . . .  =  + . . . . . .         n × n n × m ˙ x n x n u m matrix matrix x = Ax + Bu ˙

Partial Measurements     x 1 u 1 . .   ∈ R n     ∈ R m . . state x = input u = . .   x n u m

Partial Measurements     x 1 u 1 . .    ∈ R n   ∈ R m  . . state x = input u = . .   x n u m   y 1 .  ∈ R p  .  output y = y = Cx C – p × n matrix .  y p

Partial Measurements     x 1 u 1 . .   ∈ R n    ∈ R m  . . state x = input u = . .   x n u m   y 1 .  ∈ R p  .  output y = y = Cx C – p × n matrix .  y p x = Ax + Bu ˙ y = Cx

Partial Measurements     x 1 u 1 . .    ∈ R n   ∈ R m  . . state x = input u = . .   x n u m   y 1 .  ∈ R p  .  output y = y = Cx C – p × n matrix .  y p x = Ax + Bu ˙ y = Cx Example: if we only care about (or can only measure) x 1 , then   x 1 x 2   � �   y = x 1 = 1 0 . . . 0 .  .  .   x n

State-Space Models: Bottom Line x = Ax + Bu ˙ y = Cx

State-Space Models: Bottom Line x = Ax + Bu ˙ y = Cx State-space models are useful and convenient for writing down system models for different types of systems, in a unified manner.

State-Space Models: Bottom Line x = Ax + Bu ˙ y = Cx State-space models are useful and convenient for writing down system models for different types of systems, in a unified manner. When working with state-space models, what are states and what are inputs ?

State-Space Models: Bottom Line x = Ax + Bu ˙ y = Cx State-space models are useful and convenient for writing down system models for different types of systems, in a unified manner. When working with state-space models, what are states and what are inputs ? — match against ˙ x = Ax + Bu

Example 2: RL Circuit V R + − + + I V S V L − −

Example 2: RL Circuit V R + − + + I V S V L − − − V S + V R + V L = 0 Kirchhoff’s voltage law V R = RI Ohm’s law V L = L ˙ I Faraday’s law − V S + RI + L ˙ I = 0

Example 2: RL Circuit V R + − + + I V S V L − − − V S + V R + V L = 0 Kirchhoff’s voltage law V R = RI Ohm’s law V L = L ˙ I Faraday’s law − V S + RI + L ˙ I = 0 I = − R LI + 1 ˙ LV S (1st-order system)

Example 2: RL Circuit V R + − + + I V S V L − − − V S + V R + V L = 0 Kirchhoff’s voltage law V R = RI Ohm’s law V L = L ˙ I Faraday’s law − V S + RI + L ˙ I = 0 I = − R LI + 1 ˙ LV S (1st-order system) I – state, V S – input

Example 2: RL Circuit V R + − + + I V S V L − − − V S + V R + V L = 0 Kirchhoff’s voltage law V R = RI Ohm’s law V L = L ˙ I Faraday’s law − V S + RI + L ˙ I = 0 I = − R LI + 1 ˙ LV S (1st-order system) I – state, V S – input Q: How should we change the circuit in order to implement a 2nd-order system ?

Example 2: RL Circuit V R + − + + I V S V L − − − V S + V R + V L = 0 Kirchhoff’s voltage law V R = RI Ohm’s law V L = L ˙ I Faraday’s law − V S + RI + L ˙ I = 0 I = − R LI + 1 ˙ LV S (1st-order system) I – state, V S – input Q: How should we change the circuit in order to implement a 2nd-order system ? A: Add a capacitor.

Example 3: Pendulum θ ` θ T e external m g sin θ torque m g

Example 3: Pendulum Newton’s 2nd law (rotational motion): θ ` θ T e external m g sin θ torque m g

Example 3: Pendulum Newton’s 2nd law (rotational motion): θ ` = θ T J α T e ���� ���� ���� total moment angular torque of inertia acceleration external m g sin θ torque m g