Prg_SFunctionEst Class Reference

#include <Prg_SFunctionEst.h>

Inheritance diagram for Prg_SFunctionEst:

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Collaboration diagram for Prg_SFunctionEst:

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List of all members.

Detailed Description

Estimation of parameters and initial states for a model given as MEX S-function.

The estimation problem is formulated at the discrete time points $t^{0,l}<t^{1,l}<\ldots<t^{K_l}$ with $l=1,\ldots,N_{ex}$ a number of experiments. The data of all experiments is concatenated into combined vectors of time points, model inputs, and reference outputs, i.e. $K_l+1=k_{0,l+1}$ . In order to distinguish multiple experiments, it is assumed that $t^{K_l}>t^{0,l+1}$ ; for instance each experiment could start with $t^{0,l}=0$ . Parameter values are constrained to be the same for all experiments, whereas initial states are individual for each experiment. Note that this is not a restriction as generally states can be initialized with parameters and as parameters can be modeled as constant states.

In the following all vector operations are defined element wise. The treated estimation problem reads

$\begin{array}{l} \displaystyle{} J\ =\ \sum_{l=1}^{N_{ex}} \sum_{k_l=k_{l,0}}^{K_l} \sum_{i=1}^{dim(I)} \left\{ \left[\frac{ys^{k_l}(I)-ys^{k_l}_{ref}}{y_{nominal}(I)}\right]^2 \right\}_i \quad\to\quad \min, \quad I = \mbox{find}(y_{active}), \end{array}$

subject to the model given with the S-function methods mdlDerivatives $f$ and mdlOutputs $g$

$\begin{array}{l} \displaystyle \dot{x}(t) = \frac{f[p,\ x_{nominal}\,x(t),\ u(t)]} {x_{nominal}}, \\[3ex] \displaystyle y(t) = \frac{g[p,\ x_{nominal}\,x(t),\ u(t)]} {y_{nominal}} \end{array}$

with piecewise constant or linear interpolation of the inputs

$\begin{array}{ll} u_i(t)\ = & \left\{\begin{array}{ll} us_i^{k_l}, & i \in \mbox{find}(u_{order} = 0) \\[1ex] \displaystyle \frac{t^{k_l+1}-t}{t^{k_l+1}-t^{k_l}}\ us^{k_l} + \frac{t-t^{k_l}}{t^{k_l+1}-t^{k_l}}\ us^{k_l+1}, & \mbox{else} \end{array}\right. \\[5ex] & \quad t\in[t^{k_l},t^{k_l+1}), \quad k_l=k_{l,0},\ldots,K_l-1, \quad l=1,\ldots,N_{ex}, \end{array}$

and subject to the constraints

$\begin{array}{rcccll} \displaystyle \left\{ \frac{p_{min}}{p_{nominal}} \right. &\le& \displaystyle \frac{p}{p_{nominal}} &\le& \left. \displaystyle \frac{p_{max}}{p_{nominal}} \right\}_i, \quad & i \in \mbox{find}(p_{active}), \\[3ex] \displaystyle \left\{ \frac{x^0_{min}}{x_{nominal}} \right. &\le& x(t^{0,l}) &\le& \displaystyle \left. \frac{x^0_{max}}{x_{nominal}} \right\}_i, \quad & i \in \mbox{find}(x^0_{active}), \quad l=1,\ldots,N_{ex}, \\[3ex] \displaystyle \left\{ \frac{der\_x^0_{min}}{x_{nominal}} \right. &\le& \dot{x}(t^{0,l}) &\le& \displaystyle \left. \frac{der\_x^0_{max}}{x_{nominal}} \right\}_i, \quad & i \in \mbox{find}(x^0_{active}), \quad l=1,\ldots,N_{ex}, \\[3ex] && \{ x(t^{0,l}) &=& \displaystyle \frac{x0s^l}{x_{nominal}} \}_i, \quad & i \notin \mbox{find}(x^0_{active}), \quad l=1,\ldots,N_{ex}, \\[3ex] \displaystyle \frac{x_{min}}{x_{nominal}} &\le& x(t^{k_l}) &\le& \displaystyle \frac{x_{max}}{x_{nominal}}, \quad & k_l=k_{l,0},\ldots,K_l, \quad l=1,\ldots,N_{ex}. \end{array}$

The problem is treated as multistage problem with $K_{N_{ex}}$ stages and one sample period per stage per default (multistage=1 or multistage=2). Discrete-time state variables with unknown initial value are introduced for the estimated parameters p, in order to preserve a sparse structure. Additional $K_{N_{ex}}$ junction conditions (equality constraints) are introduced for the estimated parameters and $K_{N_{ex}}-N_{ex}+1$ junction conditions are introduced for the state variables x. Alternatively the problem can be treated without stages hiding model states from the optimizer (multistage=0).

The states of all stages are either initialized with the initial states $x0s^l$

$\begin{array}{rcll} x_{initial}(t^{k_l}) &=& \displaystyle \frac{x0s^l}{x_{nominal}}, & k_l=k_{l,0},\ldots,K_l, \quad l=1,\ldots,N_{ex} \end{array}$

or with individually given initial guesses $xs$

$\begin{array}{rcll} x_{initial}(t^{k_l}) &=& \displaystyle \frac{xs^{k_l}}{x_{nominal}}, & k_l=k_{l,0},\ldots,K_l, \quad l=1,\ldots,N_{ex}. \end{array}$

All but the initial states of the experiments may be initialized with the results of an initial-value simulation for given initial states and inputs (multistage=0 or multistage=1). With multistage=2, individual initial states are used for each stage, resulting in the multiple shooting method. This might be useful if no sensible initial guesses can be given for unknown parameters, so that a simulation fails.

states and inputs. If the problem is treated as multistage problem, then not performing the simulation results in the multiple shooting method. This might be useful if no sensible initial guesses can be given for unknown parameters, so that a simulation fails.

Model inputs, states and outputs can be accessed through

$\begin{array}{l} us^{k_l} = u_{nominal}\,u(t^{k_l}), \\[1ex] xs^{k_l} = x_{nominal}\,x(t^{k_l}), \\[1ex] ys^{k_l} = y_{nominal}\,y(t^{k_l}), \quad k_l=k_{l,0},\ldots,K_l, \quad l=1,\ldots,N_{ex}. \end{array}$

In addition to the estimated parameters and initial states, the measurement matrix M is calculated in each optimization iteration. It is defined as

$M = \left[\begin{array}{cc} M_p^0, & M_{x^0}^0 \\[2ex] M_p^1, & M_{x^0}^1 \\[2ex] \vdots, & \vdots \\[2ex] M_p^{K_{N_{ex}}}, & M_{x^0}^{K_{N_{ex}}} \end{array}\right]$

with the sub-matrices

$\begin{array}{r} \displaystyle M_p^{k,l} = \frac{d\,\displaystyle\frac{ys^{k,l}} {y_{nominal}}(I_y)} {d\,\displaystyle\frac{p}{p_{nominal}}(I_p)}, \quad M_{x^0}^{k,l} = \frac{d\,\displaystyle\frac{ys^{k,l}}{y_{nominal}}(I_y)} {d\,\displaystyle\frac{x0s^l}{x_{nominal}}(I_{x^0})}, \quad k_l=k_{l,0},\ldots,K_l, \quad l=1,\ldots,N_{ex}, \\[7ex] I_y = \mbox{find}(y_{active}),\ I_p = \mbox{find}(p_{active}), \ I_{x^0} = \mbox{find}(x^0_{active}). \end{array}$

Note that the sub-matrices for estimated initial states of all experiments are stored in compressed form in one column, even though individual initial states are being estimated for each experiment.

Some simple quality measures are directly calculated in order to support a first interpretation of estimation results. These are the precision matrix

$P = (M^TM)^{-1},$

and the covariance matrix

$COV = s_R^2 P, \qquad s_R^2=\frac{J}{n}, \qquad n=\mbox{dim}(I_y)(K+1) - \mbox{dim}(I_p) - N_{ex}\mbox{dim}(I_{x^0}) - 1,$

assuming that the residual variance $s_R^2$ is equal to the measurement variance. Last but not least confidence intervals for estimated parameters and initial states $px^0$ are obtained applying the t-Test:

$px^0_{confidence}\ =\ |px^0_{estimated} - px^0_{actual}|\ = \ px_{nomainal} t_{\alpha/2,n}\sqrt{\mbox{diag(COV)}}, \qquad \alpha=0.05.$

Public Member Functions

Prg_SFunctionEst ()

constructor

~Prg_SFunctionEst ()

destructor

char * name ()

name SFunctionEst

Access methods for program specific members (If prefix: prg_)

int nex () const

number of experiments used for estimation

void set_nex (int val)

set number of experiments

int multistage () const

indicate if problem is treated with one stage per time interval

void set_multistage (int val)

set multistage flag

const MATP ys_ref () const

reference values for active model outputs (size: KK+1 . ny)

void set_ys_ref (const MATP val)

set reference outputs

const MATP M () const

measurement matrix M=dy/d(p,x0) (size: (KK+1)*ny . np+nx0)

const MATP P () const

precision matrix P=(M^TM)^(-1) (size: np+nx0 . np+nx0)

const MATP COV () const

covariance matrix (size: np+nx0 . np+nx0)

Read methods for model specific members (no If prefix).

Note that the prefix mdl_ is omitted in the detailed mathematical problem description.

const VECP mdl_p () const

model parameters

const VECP mdl_p_min () const

lower bounds for model parameters (note: default is 0)

const VECP mdl_p_max () const

upper bounds for model parameters

const IVECP mdl_p_active () const

indicate estimated parameters

const VECP mdl_p_confidence () const

confidence intervals of estimated parameters

const VECP mdl_p_nominal () const

nominal parameter values (for scaling)

const VECP mdl_x0 () const

model initial states (read only, note: write initial states of experiments via mdl_x0s)

const VECP mdl_x0_min () const

lower bounds for initial states

const VECP mdl_x0_max () const

upper bounds for initial states

const IVECP mdl_x0_active () const

indicate estimated initial states

const VECP mdl_x0_confidence () const

confidence intervals of estimated initial states, averaged over multiple experiments

const VECP mdl_der_x0_min () const

minimum for time derivative of x0

const VECP mdl_der_x0_max () const

maximum for time derivative of x0

const VECP mdl_x_nominal () const

nominal state values (for scaling)

const IVECP mdl_u_order () const

const VECP mdl_x_min () const

lower bounds on states at stage boundaries (default: -Inf)

const VECP mdl_x_max () const

upper bounds on states at stage boundaries (default: Inf)

const IVECP mdl_y_active () const

indicate measured outputs

const VECP mdl_y_nominal () const

nominal output values (for scaling)

const MATP mdl_x0s () const

initial states for each experiment (size: nex . mdl_nx)

const MATP mdl_us () const

model inputs (size: KK+1 . mdl_nu)

const MATP mdl_xs () const

model states (size: KK+1 . mdl_nx)

const MATP mdl_ys () const

model outputs (read only, size: KK+1 . mdl_ny)

Write methods for model specific members (no If prefix).

void set_mdl_p (const VECP v)

set values of model parameter

void set_mdl_p_min (const VECP v)

set lower bounds for model parameters

void set_mdl_p_max (const VECP v)

set upper bounds for model parameters

void set_mdl_p_active (const IVECP v)

set estimated parameters

void set_mdl_p_nominal (const VECP v)

set nominal parameters

void set_mdl_x0_min (const VECP v)

set lower bounds for initial states

void set_mdl_x0_max (const VECP v)

set upper bounds for initial states

void set_mdl_x0_active (const IVECP v)

set estimated initial states

void set_mdl_der_x0_min (const VECP v)

set minimum for dx0dt

void set_mdl_der_x0_max (const VECP v)

set maximum for dx0dt

void set_mdl_x_nominal (const VECP v)

set nominal states

void set_mdl_u_order (const IVECP v)

set interpolation order for inputs

void set_mdl_x_min (const VECP v)

set lower bounds on states

void set_mdl_x_max (const VECP v)

set upper bounds on states

void set_mdl_y_active (const IVECP v)

set measured outputs

void set_mdl_y_nominal (const VECP v)

set nominal outputs

void set_mdl_x0s (const MATP v)

set initial states

void set_mdl_us (const MATP v)

set model inputs

void set_mdl_xs (const MATP v)

set model states

Protected Member Functions

void write_active_mx_args (VECP p)

write active parameters that are packed in p to _mx_args

Implementation of predefined methods.

See also:
Omu_Program

void setup_model ()

load S-function

void setup_stages (IVECP ks, VECP ts)

Setup stages and time sampling.

void setup (int k, Omu_VariableVec &x, Omu_VariableVec &u, Omu_VariableVec &c)

Allocate states x, control parameters u, and constraints c for stage k.

void setup_struct (int k, const Omu_VariableVec &x, const Omu_VariableVec &u, Omu_DependentVec &xt, Omu_DependentVec &F, Omu_DependentVec &f, Omu_Dependent &f0, Omu_DependentVec &c)

void init_simulation (int k, Omu_VariableVec &x, Omu_VariableVec &u)

Initialize problem variables using simulation.

void update (int kk, const Omu_StateVec &x, const Omu_Vec &u, const Omu_StateVec &xf, Omu_DependentVec &f, Omu_Dependent &f0, Omu_DependentVec &c)

void consistic (int kk, double t, const Omu_StateVec &x, const Omu_Vec &u, Omu_DependentVec &xt)

void continuous (int kk, double t, const Omu_StateVec &x, const Omu_Vec &u, const Omu_StateVec &dx, Omu_DependentVec &F)

Protected Attributes

Omu_VariableVec _mdl_p

model parameters (note: default min is 0)

Omu_VariableVec _mdl_x0

initial states

Omu_VariableVec _mdl_x

state bounds

IVECP _mdl_p_active

indicate estimated parameters

VECP _mdl_p_confidence

confidence intervals for est parameters

IVECP _mdl_x0_active

indicate estimated states

VECP _mdl_x0_confidence

confidence intervals for est x0

VECP _mdl_der_x0_min

minimum for time derivative of x0

VECP _mdl_der_x0_max

maximum for time derivative of x0

IVECP _mdl_u_order

interpolation order (default: 1 (linear))

IVECP _mdl_y_active

indicate measured outputs

VECP _mdl_p_nominal

nominal parameter values (for scaling)

VECP _mdl_x_nominal

nominal state values (for scaling)

VECP _mdl_y_nominal

nominal output values (for scaling)

int _nx

number of states for optimizer

int _np

number of estimated parameters

int _nx0

number of estimated initial states

int _ny

number of reference outputs for estimation

int _multistage

treat as multistage problem

int _nex

number of experiments used for estimation

MATP _mdl_x0s

initial states for each experiment

IVECP _exs

index identifying experiment in estimation data

MATP _mdl_us

given model inputs

MATP _mdl_xs

calculated model states

MATP _mdl_ys

calculated model outputs

MATP _ys_ref

reference values for active outputs

double _ssr

sum of squared output residuals

MATP _M2

measurement matrix M=dy/d(p,x0)

MATP _P2

precision matrix P=(M^TM)^(-1)

MATP _COV

covariance matrix COV=

MATP _dydx

help variable dy/dx

MATP _dxdpx0

help variable dx/d(p,x0)

MATP _dydpx0

help variable dy/d(p,x0)

MATP _dxfdx

help variable dxf/dx

MATP _dxfdpx0

help variable dxf/d(p,x0)

Member Function Documentation

void Prg_SFunctionEst::setup_stages	(	IVECP	ks,
		VECP	ts
	)			`[protected, virtual]`

Setup stages and time sampling.

The default implementation sets up an optimization problem without stages.

Reimplemented from Omu_Program.

void Prg_SFunctionEst::setup	(	int	k,
		Omu_VariableVec &	x,
		Omu_VariableVec &	u,
		Omu_VariableVec &	c
	)			`[protected, virtual]`

Allocate states x, control parameters u, and constraints c for stage k.

Additionally setup bounds and initial values.

Implements Omu_Program.

void Prg_SFunctionEst::init_simulation	(	int	k,
		Omu_VariableVec &	x,
		Omu_VariableVec &	u
	)			`[protected, virtual]`

Initialize problem variables using simulation.

A problem specification may set state variables x and/or control parameters u prior to the evaluation of stage k. The default version causes an initial value simulation based on initial values given in setup.

Reimplemented from Omu_Program.

const VECP Prg_SFunctionEst::mdl_x_nominal ( ) const [inline]

nominal state values (for scaling)

interpolation order (0 (constant) or 1 (linear), default: 1)

References _mdl_x_nominal.

The documentation for this class was generated from the following file:

Prg_SFunctionEst.h


Public Member Functions
	Prg_SFunctionEst ()
	constructor
	~Prg_SFunctionEst ()
	destructor
char *	name ()
	name SFunctionEst
Access methods for program specific members (If prefix: prg_)
int	nex () const
	number of experiments used for estimation
void	set_nex (int val)
	set number of experiments
int	multistage () const
	indicate if problem is treated with one stage per time interval
void	set_multistage (int val)
	set multistage flag
const MATP	ys_ref () const
	reference values for active model outputs (size: KK+1 . ny)
void	set_ys_ref (const MATP val)
	set reference outputs
const MATP	M () const
	measurement matrix M=dy/d(p,x0) (size: (KK+1)*ny . np+nx0)
const MATP	P () const
	precision matrix P=(M^TM)^(-1) (size: np+nx0 . np+nx0)
const MATP	COV () const
	covariance matrix (size: np+nx0 . np+nx0)
Read methods for model specific members (no If prefix).
Note that the prefix mdl_ is omitted in the detailed mathematical problem description.
const VECP	mdl_p () const
	model parameters
const VECP	mdl_p_min () const
	lower bounds for model parameters (note: default is 0)
const VECP	mdl_p_max () const
	upper bounds for model parameters
const IVECP	mdl_p_active () const
	indicate estimated parameters
const VECP	mdl_p_confidence () const
	confidence intervals of estimated parameters
const VECP	mdl_p_nominal () const
	nominal parameter values (for scaling)
const VECP	mdl_x0 () const
	model initial states (read only, note: write initial states of experiments via mdl_x0s)
const VECP	mdl_x0_min () const
	lower bounds for initial states
const VECP	mdl_x0_max () const
	upper bounds for initial states
const IVECP	mdl_x0_active () const
	indicate estimated initial states
const VECP	mdl_x0_confidence () const
	confidence intervals of estimated initial states, averaged over multiple experiments
const VECP	mdl_der_x0_min () const
	minimum for time derivative of x0
const VECP	mdl_der_x0_max () const
	maximum for time derivative of x0
const VECP	mdl_x_nominal () const
	nominal state values (for scaling)
const IVECP	mdl_u_order () const
const VECP	mdl_x_min () const
	lower bounds on states at stage boundaries (default: -Inf)
const VECP	mdl_x_max () const
	upper bounds on states at stage boundaries (default: Inf)
const IVECP	mdl_y_active () const
	indicate measured outputs
const VECP	mdl_y_nominal () const
	nominal output values (for scaling)
const MATP	mdl_x0s () const
	initial states for each experiment (size: nex . mdl_nx)
const MATP	mdl_us () const
	model inputs (size: KK+1 . mdl_nu)
const MATP	mdl_xs () const
	model states (size: KK+1 . mdl_nx)
const MATP	mdl_ys () const
	model outputs (read only, size: KK+1 . mdl_ny)
Write methods for model specific members (no If prefix).
void	set_mdl_p (const VECP v)
	set values of model parameter
void	set_mdl_p_min (const VECP v)
	set lower bounds for model parameters
void	set_mdl_p_max (const VECP v)
	set upper bounds for model parameters
void	set_mdl_p_active (const IVECP v)
	set estimated parameters
void	set_mdl_p_nominal (const VECP v)
	set nominal parameters
void	set_mdl_x0_min (const VECP v)
	set lower bounds for initial states
void	set_mdl_x0_max (const VECP v)
	set upper bounds for initial states
void	set_mdl_x0_active (const IVECP v)
	set estimated initial states
void	set_mdl_der_x0_min (const VECP v)
	set minimum for dx0dt
void	set_mdl_der_x0_max (const VECP v)
	set maximum for dx0dt
void	set_mdl_x_nominal (const VECP v)
	set nominal states
void	set_mdl_u_order (const IVECP v)
	set interpolation order for inputs
void	set_mdl_x_min (const VECP v)
	set lower bounds on states
void	set_mdl_x_max (const VECP v)
	set upper bounds on states
void	set_mdl_y_active (const IVECP v)
	set measured outputs
void	set_mdl_y_nominal (const VECP v)
	set nominal outputs
void	set_mdl_x0s (const MATP v)
	set initial states
void	set_mdl_us (const MATP v)
	set model inputs
void	set_mdl_xs (const MATP v)
	set model states
Protected Member Functions
void	write_active_mx_args (VECP p)
	write active parameters that are packed in p to _mx_args
Implementation of predefined methods.
See also: Omu_Program
void	setup_model ()
	load S-function
void	setup_stages (IVECP ks, VECP ts)
	Setup stages and time sampling.
void	setup (int k, Omu_VariableVec &x, Omu_VariableVec &u, Omu_VariableVec &c)
	Allocate states x, control parameters u, and constraints c for stage k.
void	setup_struct (int k, const Omu_VariableVec &x, const Omu_VariableVec &u, Omu_DependentVec &xt, Omu_DependentVec &F, Omu_DependentVec &f, Omu_Dependent &f0, Omu_DependentVec &c)
void	init_simulation (int k, Omu_VariableVec &x, Omu_VariableVec &u)
	Initialize problem variables using simulation.
void	update (int kk, const Omu_StateVec &x, const Omu_Vec &u, const Omu_StateVec &xf, Omu_DependentVec &f, Omu_Dependent &f0, Omu_DependentVec &c)
void	consistic (int kk, double t, const Omu_StateVec &x, const Omu_Vec &u, Omu_DependentVec &xt)
void	continuous (int kk, double t, const Omu_StateVec &x, const Omu_Vec &u, const Omu_StateVec &dx, Omu_DependentVec &F)
Protected Attributes
Omu_VariableVec	_mdl_p
	model parameters (note: default min is 0)
Omu_VariableVec	_mdl_x0
	initial states
Omu_VariableVec	_mdl_x
	state bounds
IVECP	_mdl_p_active
	indicate estimated parameters
VECP	_mdl_p_confidence
	confidence intervals for est parameters
IVECP	_mdl_x0_active
	indicate estimated states
VECP	_mdl_x0_confidence
	confidence intervals for est x0
VECP	_mdl_der_x0_min
	minimum for time derivative of x0
VECP	_mdl_der_x0_max
	maximum for time derivative of x0
IVECP	_mdl_u_order
	interpolation order (default: 1 (linear))
IVECP	_mdl_y_active
	indicate measured outputs
VECP	_mdl_p_nominal
	nominal parameter values (for scaling)
VECP	_mdl_x_nominal
	nominal state values (for scaling)
VECP	_mdl_y_nominal
	nominal output values (for scaling)
int	_nx
	number of states for optimizer
int	_np
	number of estimated parameters
int	_nx0
	number of estimated initial states
int	_ny
	number of reference outputs for estimation
int	_multistage
	treat as multistage problem
int	_nex
	number of experiments used for estimation
MATP	_mdl_x0s
	initial states for each experiment
IVECP	_exs
	index identifying experiment in estimation data
MATP	_mdl_us
	given model inputs
MATP	_mdl_xs
	calculated model states
MATP	_mdl_ys
	calculated model outputs
MATP	_ys_ref
	reference values for active outputs
double	_ssr
	sum of squared output residuals
MATP	_M2
	measurement matrix M=dy/d(p,x0)
MATP	_P2
	precision matrix P=(M^TM)^(-1)
MATP	_COV
	covariance matrix COV=
MATP	_dydx
	help variable dy/dx
MATP	_dxdpx0
	help variable dx/d(p,x0)
MATP	_dydpx0
	help variable dy/d(p,x0)
MATP	_dxfdx
	help variable dxf/dx
MATP	_dxfdpx0
	help variable dxf/d(p,x0)