State Estimation - Very, Study notes of Control Systems Analysis

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Control Engineering

Ramkrishna Pasumarthy

Department of Electrical Engineering,

Indian Institute of Technology Madras

Module 12

Lecture 1

Controllability Matrix and Controllable Systems

Controllability Matrix

Consider the continuous time LTI system

˙x = Ax + Bu, x 2 Rn, u 2 Rk^ (1)

For the system in (1), the controllability Gramians is given, respectively, by

WC(t 0 , t 1 ) :=

∫ (^) t 1 t 0 e A(t 0 τ ) BB

e A′(t 0 τ ) dτ =

∫ (^) t 1 t 0 0 e At BB

e A′t dt

The controllability matrix of the system in (1) is defined as

C := [B AB A

B


A n 1 B]n(kn) (2)

Example: RC Circuit

u

R 1

C 1

x 1

R 2

C 2

x 2

Figure 1: Parallel RC cirucit

Consider the parallel RC circuit shown in figure 1. The controllability matrix for the

electrical circuit is given by

C = [B AB] =

[ 1 R 1 C 1 ^

R^21 C^21

R 2 C 2 ^

R^22 C^22

]

ControllabLe Systems

Controllable Systems

Consider the following continuous time LTV system

x^ ˙ = A(t)x + B(t)u, x 2 R

n , u 2 R

k (3)

Definition 7.2.

Given two times t 1 > t 0  0, the system in (3), or simply the pair (A(.), B(.)), is (completely

state- ) controllable on [t 0 , t 1 ] if C [t 0 , t 1 ] = Rn^ i.e. if every state can be transferred to the

origin.

Example: Pendulum on a Cart

Consider the dynamics of the cart-pendulum system discussed in Lecture 1. The

dynamics of the cart-pendulum system is

˙z =

d

dt

2

6 6 6 4

x

θ

x^ ˙

θ^ ˙

3

7 7 7 5

=

2 6 6 6 6 4 x^ ˙

θ^ ˙ Jtmlsθ θ˙^2 kJt ˙x+cmlcθ θ˙+m^2 l^2 gcθ sθ MtJtm^2 l^2 c^2 θ kmlcθ ˙xm^2 l^2 cθ sθ θ˙^2 cMt θ˙Mtmglsθ MtJtm^2 l^2 c^2 θ

3 7 7 7 7 5

2 6 6 6 6 4 0

0 Jt MtJtm^2 l^2 c^2 θ mlcθ MtJtm^2 l^2 c^2 θ

3 7 7 7 7 5

u

The equilibrium points of the system are [x 0 0 0]

T

i.e. the pendulum with upright

position and [x π 0 0]

T

, the pendulum with down position, 8 x 2 R.

Example: Pendulum on a Cart

Assume that the system stays within a small neighbourhood with the pendulum in the

upright position. Then

sin θ  θ, cos θ  1 , θ˙

 0

) The linearized dynamics leads to a set of linear state space equation

z^ ˙ =

2

6 6 6 4

0 0 1 0

0 0 0 1

0 m^2 l^2 g MtJtm^2 l^2 kJt MtJtm^2 l^2

cml MtJtm^2 l^2 0 Mtmgl MtJtm^2 l^2

kml MtJtm^2 l^2 cMt MtJtm^2 l^2

3

7 7 7 5

| {z } A

z +

2

6 6 6 4

0

0 Jt MtJtm^2 l^2 ml MtJtm^2 l^2

3

7 7 7 5

| {z } B

u

Overview

Summary: Module 12 Lecture 1

▶ Controllability Matrix

▶ Controllable Systems

Contents: Module 12 Lecture 2

▶ Controllability Tests

Pendulum on a Cart: Controllability Matrix

C = [B AB A

B A

B], where

B =

2

6 6 6 4

0

0 Jt MtJtm^2 l^2 ml MtJtm^2 l^2

3

7 7 7 5

, AB =

2 6 6 6 6 4 Jt JtMtl^2 m^2 lm l^2 m^2 JtMt

cl^2 m^2 +J^2 t k (l^2 m^2 JtMt)^2 lm(cMt+Jtk) (l^2 m^2 JtMt)^2

3 7 7 7 7 5

, A

B =

2 6 6 6 6 6 4

cl^2 m^2 +J^2 t k (l^2 m^2 JtMt)^2 lm(cMt+Jtk) (l^2 m^2 JtMt)^2

c^2 l^2 m^2 Mt+Jtl^2 m^2 ( 2 ckglmMt)+gl^5 m^5 +J^3 t k^2 (l^2 m^2 JtMt)^3 lm(c^2 M^2 t +JtMt(ckglmMt)+ckl^2 m^2 +gl^3 m^3 Mt+J^2 t k^2 ) (l^2 m^2 JtMt)^3

3 7 7 7 7 7 5 A

B =

2 6 6 6 6 6 6 4

c^2 l^2 m^2 Mt+Jtl^2 m^2 ( 2 ckglmMt)+gl^5 m^5 +J^3 t k^2 (l^2 m^2 JtMt)^3 lm(c^2 M^2 t +JtMt(ckglmMt)+ckl^2 m^2 +gl^3 m^3 Mt+J^2 t k^2 ) (l^2 m^2 JtMt)^3

2 Jtl^2 m^2 (c^2 kMtcglmM^2 t +gkl^3 m^3 )+cl^2 m^2 (c^2 M^2 t +ckl^2 m^2 + 2 gl^3 m^3 Mt)+J^2 t kl^2 m^2 ( 3 ck 2 glmMt)+J^4 t k^3 (l^2 m^2 JtMt)^4 lm(c^3 M^3 t + 2 c^2 kl^2 m^2 Mt+J^2 t kMt(ckglmMt)+cJt(ckM^2 t 2 glmM^3 t +2k^2 l^2 m^2 )+ 2 cgl^3 m^3 M^2 t +gkl^5 m^5 +J^3 t k^3 ) (l^2 m^2 JtMt)^4

3 7 7 7 7 7 7 5

Control Engineering Module 12 Lecture 1 Ramkrishna P.