Workshop in Celebration of the Life, Mathematics and Memories …gilliam/ttu/cib_mem_webpage/Chris... · 2011. 11. 26. · Properties of Dynamical Systems Workshop in Memories of

Workshop in Celebrationof the Life, Mathematics and Memories of

Christopher I. Byrnes

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“For a researcher, it is important to be able totellWhat Is Easy, What Is Hard ”

— Chris Byrnes

Workshop in Memories of Chris Byrnes

Some Examples of Observability forDynamical Systems

Wei KangDepartment of Applied Mathematics

Naval Postgraduate SchoolMonterey, California, USA

NPS - Wei Kang

Properties of Dynamical Systems


Consider a dynamical system

ut = f(u, ux, uxx, · · · , µ)

Some properties of interest

• Observability

• Reachability and controllability

• Robustness

They have a wide spectrum of applications− control system theory− Inverse problem− data assimilation− · · · · · ·

NPS - Wei Kang

Observability


Issues to be considered:

• Observability of state and control functions

• Partial observability of complex systems

• User-knowledge

• Heterogeneous sensor information

• Unknown input or unknown parameters

NPS - Wei Kang

Observability


Control system :

x = f(t, x, u, µ), −control system

y = y(t, x, u, µ), −system output

z = z(t, x, u, µ), −variable to be estimated

(x(·), u(·), µ) ∈ C −constraint or user-knowledge

Examples of user-knowledge

E(x(t0), x(tf )) ≤ 0, end point condition

s(x, u) ≤ 0, state-control constraints

s(x(t1)) = 0, known event at time t1

µmin ≤ µ ≤ µmax model uncertainties

s(x, µ) = 0, DAE (differential-algebraic equations, µ is a variable)

Total Variation of u ≤ Vmax, non-sensor information

NPS - Wei Kang

Observability


Definition

Given a trajectory (x(t), u(t), µ), t ∈ [t0, t1] and an output errorbound ǫ > 0. Define ρ(ǫ)

ρ(ǫ) = max(x(t),u(t),µ) ||z(t, x(t), u(t), µ)− z(t, x(t), u(t), µ)||Zsubject to

||y(t, x(t), u(t), µ)− y(t, x(t), u(t), µ)||Y ≤ ǫ

˙x = f(t, x, u, µ),

(x(·), u(·), µ) ∈ C

The number ρ(ǫ) is called the ambiguity in the estimation of z alongthe trajectory (x(t), u(t), µ). The ratio ρ/ǫ is also a measure ofobservability.

NPS - Wei Kang

Observability


Remarks

• The ratio ρ/ǫ measures the sensitivity of z relative to the erroror noise in y. A small value of ρ/ǫ implies strong observabilityof z.

• For linear systems, ǫ2/ρ2 equals the smallest eigenvalue ofobservability gramian

• The definition is applicable with general metrics, || · ||Lp , || · ||∞,......

NPS - Wei Kang

Observability


Computational algorithms : The concept of observability isnumerically implementable

• gradient projection method

• pseudospectral or collocation method

• empirical covariance matrix (empirical observability gramian -Krener)

• ......

NPS - Wei Kang

Observability


Consider a system

∂u(x, t)

∂t+ u(x, t)

∂u(x, t)

∂x− κ

∂2u(x, t)

∂x2= 0,

u(0, t) = 0

u(2π, t) = 0

yi(tj) = u(λi, tj),i = 1, 2, · · · , Nsj = 0, 1, 2, · · · , Nt

λi − sensor location , Ns − number of sensors ,

time interval/Nt − sensor sampling rate

0

1

2

3

4

5

0 1 2 3 4 5 6 7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

x

NPS - Wei Kang

Observability


Define

||y||2Y =

Ns∑

i=1

Nt∑

j=1

yi(tj)2

u(x, 0) ∈W = {α0 +6

∑

k=1

αk cos(kx) + βk sin(kx)}

Sensor locations:

equally spaced:[

λ1 λ2 · · · λ7

]

=

[

2π

822π

8· · · 7

2π

8

]

improved locations:[

λ1 p ∗ λ2 p ∗ λ3 p ∗ λ4 λ5 λ6 λ7

]

, p = 0.7

optimal locations:[

0.75 1.06 1.77 2.08 3.98 4.83 5.43]

NPS - Wei Kang

Observability


Empirical covariance matrix method : Suppose the metrics of yand u0 = u(x, 0) are defined by inner products

||y||Y =√< y, y >Y , ||u0||W =

√< u0, u0 >W

Let {v1, v2, · · · , vnz} be a basis of W . Define

∆i(t) =1

2ρ(y(t, u0 + ρvi)− y(t, u0 − ρvi))

GY = (< ∆i,∆j >Y )nz

i,j=1, GW = (< vi, vj >W )nz

i,j=1

Then ǫ(ρ)2/ρ2 is approximately the smallest eigenvalue, λmin, ofGY relative to GW and

ρ2/ǫ2 ≈ 1

λmin

NPS - Wei Kang

Observability


Discretization

ui(t) = U(xi, t), for i = 1, 2, · · · , N − 1

N = 50

Discretized Model

u1(t) = −u1(t)u2(t)− f1(t)

2∆x+ κ

u2(t) + f1(t)− 2u1(t)

∆x2

u2(t) = −u2(t)u3(t)− u1(t)

2∆x+ κ

u3(t) + u1(t)− 2u2(t)

∆x2...

uN−1(t) = −uN−1f2(t)−NN−2(t)

2∆x+ κ

f2(t) + uN−2(t)− 2uN−1(t)

∆x2

NPS - Wei Kang

Observability


Remark : The optimal sensor locations are computed usinggradient projection method that minimizes the value of ǫ forρ = 0.01.

0 1 2 3 4 5 6

Observability :

Equally spaced Improved location Optimal location

ρ/ǫ 12.92 6.30 1.75

NPS - Wei Kang

Observability


Error variance for two hundred data sets :

Observability The variance of The variance of

ρ/ǫ ||u(0)− u(0)|| ||u(·)− u(·)||Equally spaced 12.92 0.0087 0.0135

Improved location 6.30 0.0042 0.0066

Optimal location 1.75 0.0023 0.0036

Error variance as a function of x

0 1 2 3 4 5 6 70

1

2

3

x 10−4

NPS - Wei Kang

Observability - Partial Observability of Networked System s


A platoon of networked vehicles

Vehicles are treated as point mass

xi1 = xi2 yi1 = yi2

xi2 = ui yi2 = vi

Partial information about control inputs

u2 = a1(x21 − x11 − d1) + a2(x22 − x12)

u3 = b1(x31 −x11 + x21

2− d2) + b2(x32 −

x12 + x222

)

NPS - Wei Kang



Information

Sensor information: output =[

x21 y21 x31 y31

]T

User-knowledge: Variation of u1 ≤ Vmax

Variable to be estimated: z =[

x11 x12

]T

Question: what is the ambiguity in the estimation of the states ofVehicle 1 with unknown input?

NPS - Wei Kang



ρ(ǫ) = max(x,u1)

||x11(t)− x11(t)||L2

subject to

||x21(t)− x21(t)||L∞ ≤ ǫ1

||x31(t)− x31(t)||L∞ ≤ ǫ2

˙x = f(x)

V (u1) ≤ Vmax

System configuration

t0 tf d1 d2 a1 a2 b1 b2 (x0

11, x0

12) (x0

21, x0

22) (x0

31, x0

32)

0 20 −2 −2 −1 −2 −3 −7 (0, 4) (d1, 4) (d2, 4)

u1 = sin(tf − t0)t

π

NPS - Wei Kang



Result of computation using N = 18 LGL nodes

Vmax ǫ ρx11ρx11

/||x11||L2ρx12

ρx12/||x12||L2

3 10−2 1.2257 2.8× 10−3 0.5901 1.16× 10−2

The worst estimation

0 2 4 6 8 10 12 14 16 18 20−100

0

100

200

300

time (second)

x (m

)1

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

time (second)

x (m

)2

To summarize, a networked system with unknown input becomespractically observable by using user-knowledge. In addition, ρ is ameasure of partial observability using local information.

NPS - Wei Kang

Input-Output Gain


System model

x = f(t, x, w, µ)

z = z(t, x, w, µ)

w ∈ U

The Lp-gain from w to z along x∗(t):

γ(σ) = max

||w||Lp ≤ σ,

µmin ≤ µ ≤ µmax

||z(t, x, w, µ)− z(t, x∗, 0, µ)||Lp

σ

Remark: Without parameter, the maximum value of||z(t, x, w)− z(t, x∗, 0)||Lp is the ambiguity in the estimation of zunder the observation of w with an error bound σ.

NPS - Wei Kang

Reachability


Ambiguity in Control: Given x0 and x1 in ℜnx . Define

ρc(x0, x1)2 = min

(x,u)||x(t1)− x1||2X

subject to

x = f(x, u)

x(t0) = x0

(x(·), u(·)) ∈ C

Cost of control: Given x0 and x1, define

W (x0, x1) = min(x,u)

limψ→∞

(||u(t)− u∗(t)||U + ψ||x(t1)− x1||X)

subject to

x = f(x, u)

x(t0) = x0,

(x(·), u(·)) ∈ C

NPS - Wei Kang

Reachability


For linear system x = Ax+Bu:

max(x0,x1)

ρc(x0, x1) =

0 if (A,B) is controllable

∞ if (A,B) is uncontrollable

DefineW = max

x1∈Bǫ(x0)W (x0, x1)

Then (ǫ/W )2 equals the smallest eigenvalue of the controllabilitygramian.

NPS - Wei Kang

Conclusions


• Properties on observability, robustness, and reachability aredefined under the same framework: dynamic optimization.

• System properties are quantitatively defined.

• Constraints are accommodated.

• The concepts can be numerically implemented.

• Limitations and challenges: computational approach is local,lacks mathematically proved observability, difficulty insearching for optimal solutions, ......

• Future work: applications; theoretical foundation; estimationmethod, computational algorithms and numerical analysis, ......

NPS - Wei Kang


I will never forget youeven for an intervalShort as those for the convergenceto zero dynamics

NPS - Wei Kang

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Workshop in Celebration of the Life, Mathematics and Memories …gilliam/ttu/cib_mem_webpage/Chris... · 2011. 11. 26. · Properties of Dynamical Systems Workshop in Memories of