Parameter Estimation in System Identification

1. System Identification = Model structure selection or define + Model parameter estimation

2. Parameter estimation is a mapping: Z----> sigma*

3. Essence of Model is its prediction ability.

4. Two criteria:

Scalar norm or function of errors

Data are not related to Predictive error.

pem and lsqcurvefit

they are really sensitive to initial condition and parameter values.

this really haunting on me several days.

Something Help Kalman

Hinfinity Filter is a minmax filter, which minimise worsts estimation. Meanwhile, Kalman minimise average estimation error.

Kalman Smoother is useful, when you want to estimate the state trajectory after obtaining the whole signal serials.

Steady State Kalman filter have constant K and P.

State Space Realization of IIR

cited from [https://ccrma.stanford.edu/~jos/fp/State_Space_Realization.html]

IIR filters have an extensively used matrix representation called state space form (or ``state space realizations''). They are especially convenient for representing filters with multiple inputs andmultiple outputs (MIMO filters). An order digital filter with inputs and outputs can be written in state-space form as follows:

			(F.4)

where is the length state vector at discrete time , is a vector of inputs, and the output vector. is the state transition matrix, and it determines thedynamics of the system (its poles, or resonant modes).

aij

I saw "Introduction to linear dynamical systems" on YouTube, It is much better than what my teacher taught me, that is a little bothered me.

matrix is about linear algebra.

y=Ax

1) y is measurement, x is something we do not know.

2) y is output, x is input.

3) x is signal, y is transformation output.

aij is is weight on xi to produce yj, ith row concerns with ith output, jth column concerns with jth input.

1) aij = 0, then yj does not depend on xi

2) some aij is bigger in i row, y is more depend on xi

3) some aij is bigger in j column, xi more affect yj

4) lower triangular are zeros, yi only depends on x1....xi

5) diagnal, yi only depends on xi.

6) sparsity means relationship between x and y is not so complex.

System Identification

yeah, i am here, jumped too fast!

some commands of matlab:

lsqnonlin--------non-linear least square optimization

lsqcurvefit------use lsqnonlin to fit a curve

pem------------prediction error model

idgrey----------grey box model

idnlgrey--------grey box for non linear model.

ss --------------state space model

sim------------model simulation

bode-----------bode graph

step------------step response

impulse---------impulse response

c2d-------------convert continuous model to discrete model

system identification method:

state space

poles and zeros

frequency method i.e., bode graph

PID control

optimization method:

newton

gauss-newton

steepest gradient

levenberg marquardt

small bugs haunting me

C++:

; is important in class{};

string is in using namespace std;

fabs not abs

-> . :: thing

c++ call matlab

long time from c++, suffering from small bugs,

yes, c++ is a bug itself.

before i start, i would like to log this bugs:

cout &it&it fixed &it&it setprecision(5) &it&it x

yes, if x is float, and you want to output it to file or console, then you get nothing.

same with textread in matlab, but you get something instead of nothing.

ok, let us start.

C++ call matlab

remember, what we are using are some libs of matlab, which is mainly the "engine" lib.

1) so add includes, add lib search path. add libs themselves.

2) coding

#include " engine.h"

14 Challenges of 21st Century[From USA NAE]

This is cited from USA National Academy of Engineering.

1. Make solar energy affordable
2. Provide energy from fusion
3. Develop carbon sequestration methods
4. Manage the nitrogen cycle
5. Provide access to clean water
6. Restore and improve urban infrastructure
7. Advance health informatics
8. Engineer better medicines
9. Reverse-engineer the brain
10. Prevent nuclear terror
11. Secure cyberspace
12. Enhance virtual reality
13. Advance personalized learning
14. Engineer the tools for scientific discovery

dynamic system

Dynamic system follows a fixed rule, which can be formulated by:

y(t+1) = A[y(t),y(t-1)...y(t-M)] + K[u(t)]

So the rule and initial value determine the whole time serials.