Review: control, feedback, etc. Today s topic: state-space models of systems; linearization

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1 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 11, 12, 21 24, 72, 921 Chapter 2 has lots of cool examples of system models!!

2 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 etc x instead of x(t) ẋ instead of x (t) or dx dt ẍ instead of x (t) or d2 x dt 2

3 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 friction force = ρẋ (Hooke s law) mẍ = kx ρẋ + u Move x, ẋ, ẍ to the LHS, u to the RHS: (Stokes law linear drag, only an approximation!!) ẍ + ρ mẋ + k m x = u m 2nd-order linear ODE

4 Example 1: Mass-Spring System x m u ẍ + ρ mẋ + k m x = u m 2nd-order linear ODE Canonical form: convert to a system of 1st-order ODEs ẋ = v (definition of velocity) v = ρ m v k m x + 1 m u

5 Example 1: Mass-Spring System x m u State-space model: express in matrix form (ẋ ) ( 0 1 = v k m ρ m ) ( ) ( ) x u v m Important: start reviewing your linear algebra now!! matrix-vector multiplication; eigenvalues and eigenvectors; etc

6 General n-dimensional State-Space Model state x = ẋ 1 ẋ n = x 1 x n R n input u = A n n matrix x 1 x n + B u 1 u m n m matrix R m u 1 u m ẋ = Ax + Bu

7 Partial Measurements state x = output y = x 1 x n y 1 y p R n input u = u 1 u m R m R p y = Cx C p n matrix ẋ = Ax + Bu y = Cx Example: if we only care about (or can only measure) x 1, then x 1 y = x 1 = ( ) x 2 x n

8 State-Space Models: Bottom Line ẋ = 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 ẋ = Ax + Bu

9 Example 2: RL Circuit V S + + V R I + V L V S + V R + V L = 0 V R = RI V L = LI V S + RI + L I = 0 Kirchhoff s voltage law Ohm s law Faraday s law I = R L I + 1 L V 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

10 Example 3: Pendulum ` mg sin mg T e external torque Newton s 2nd law (rotational motion): T }{{} total torque = J }{{} moment of inertia α }{{} angular acceleration = pendulum torque + external torque pendulum torque = mg sin θ }{{} }{{} l force lever arm moment of inertia J = ml 2 mgl sin θ + T e = ml 2 θ θ = g l sin θ + 1 ml 2T e (nonlinear equation)

11 Example 3: Pendulum θ = g l sin θ + 1 ml 2T e For small θ, use the approximation sin θ θ sin Θ Θ ( θ1 θ 2 ) = ( 0 1 g l 0 (nonlinear equation) θ = g l θ + 1 ml 2 T e State-space form: θ 1 = θ, θ 2 = θ θ 2 = g l θ + 1 ml 2 T e = g l θ ml 2 T e ) ( ) θ1 θ 2 + ( 0 1 ml 2 ) T e

12 Linearization Taylor series expansion: f(x) = f(x 0 ) + f (x 0 )(x x 0 ) f (x 0 )(x x 0 ) 2 + f(x 0 ) + f (x 0 )(x x 0 ) linear approximation around x = x 0 Control systems are generally nonlinear: ẋ = f(x, u) x = x 1 x n u = u 1 u m f = f 1 f n nonlinear state-space model Assume x = 0, u = 0 is an equilibrium point: f(0, 0) = 0 This means that, when the system is at rest and no control is applied, the system does not move

13 Linearization Linear approx around (x, u) = (0, 0) to all components of f: For each i = 1,, n, ẋ 1 = f 1 (x, u),, ẋ n = f n (x, u) f i (x, u) = f i (0, 0) }{{} =0 + f i x 1 (0, 0)x f i x n (0, 0)x n + f i u 1 (0, 0)u f i u m (0, 0)u m Linearized state-space model: ẋ = Ax + Bu, where A ij = f i x j x=0 u=0, B ik = f i u k x=0 u=0 Important: since we have ignored the higher-order terms, this linear system is only an approximation that holds only for small deviations from equilibrium

14 Example 3: Pendulum, Revisited Original nonlinear state-space model: θ 1 = f 1 (θ 1, θ 2, T e ) = θ 2 already linear θ 2 = f 2 (θ 1, θ 2, T e ) = g l sin θ ml 2 T e Linear approx of f 2 around equilibrium (θ 1, θ 2, T e ) = (0, 0, 0): f 2 = g θ 1 l cos θ 1 f 2 = g θ 1 l 0 f 2 = 0 θ 2 f 2 = 0 θ 2 0 Linearized state-space model of the pendulum: θ 1 = θ 2 θ 2 = g l θ ml 2T e f 2 = 1 T e ml 2 f 2 = 1 T e ml 2 0 valid for small deviations from equ

15 General Linearization Procedure Start from nonlinear state-space model ẋ = f(x, u) Find equilibrium point (x 0, u 0 ) such that f(x 0, u 0 ) = 0 Note: different systems may have different equilibria, not necessarily (0, 0), so we need to shift variables: x = x x 0 u = u u 0 f(x, u) = f(x + x 0, u + u 0 ) = f(x, u) Note that the transformation is invertible: x = x + x 0, u = u + u 0

16 General Linearization Procedure Pass to shifted variables x = x x 0, u = u u 0 ẋ = ẋ (x 0 does not depend on t) = f(x, u) = f(x, u) equivalent to original system The transformed system is in equilibrium at (0, 0): f(0, 0) = f(x 0, u 0 ) = 0 Now linearize: ẋ = Ax + Bu, where A ij = f i f i, B x j ik = x=x0 u k u=u 0 x=x0 u=u 0

17 General Linearization Procedure Why do we require that f(x 0, u 0 ) = 0 in equilibrium? This requires some thought Indeed, we may talk about a linear approximation of any smooth function f at any point x 0 : f(x) f(x 0 )+f (x 0 )(x x 0 ) f(x 0 ) does not have to be 0 The key is that we want to approximate a given nonlinear system ẋ = f(x, u) by a linear system ẋ = Ax + Bu (may have to shift coordinates: x x x 0, u u u 0 ) Any linear system must have an equilibrium point at (x, u) = (0, 0): f(x, u) = Ax + Bu f(0, 0) = A0 + B0 = 0

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