Optimal control problem for a model given as MEX S-function treated with multi-stage control vector parameterization. More...

#include <Prg_SFunctionOpt.h>

Inheritance diagram for Prg_SFunctionOpt:

[legend]

Public Member Functions
	Prg_SFunctionOpt ()
	constructor

	~Prg_SFunctionOpt ()
	destructor

const char *	name ()
	name SFunctionOpt

Access methods for program specific members (If prefix: prg_)
int	multistage () const
	indicate if problem is treated with one stage per time interval

void	set_multistage (int val)
	set multistage flag

Access methods for scaling of time
int	mdl_t_scale_idx () const
	optional index of a model input used for scaling of time (default: -1)

void	set_mdl_t_scale_idx (int val)
	set index of model input used for scaling of time

const VECP	taus () const
	vector of start time points in each sample period

void	set_taus (const VECP n_taus)
	set vector of start times

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_x0 () const
	initial states

const IVECP	mdl_x0_active () const
	free initial states (default: 0)

const VECP	mdl_x0_min () const
	lower bounds for initial states

const VECP	mdl_x0_max () const
	upper bounds for initial states

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_u0 () const
	model inputs at initial time

const VECP	mdl_u0_min () const
	lower bounds for model inputs at initial time

const VECP	mdl_u0_max () const
	upper bounds for model inputs at initial time

const VECP	mdl_y0 () const
	model outputs at initial time

const VECP	mdl_y0_min () const
	lower bounds for model outputs at initial time

const VECP	mdl_y0_max () const
	upper bounds for model outputs at initial time

const VECP	mdl_y0_weight1 () const
	weight for linear objective term at initial time (default: 0)

const VECP	mdl_y0_weight2 () const
	weight for quadratic objective term at initial time (default: 0)

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

const IVECP	mdl_u_active () const
	indicate optimized inputs

const IVECP	mdl_u_integer () const
	indicate integers

const IVECP	mdl_u0_nfixed () const
	numbers of fixed control inputs at begin of time horizon (default: 1)

const IVECP	mdl_u_decimation () const
	decimation for optimized model inputs (default: 1)

const VECP	mdl_u_nominal () const
	nominal input values (for scaling)

const IVECP	mdl_u_periodic () const
	first and last value are equal (default: false)

const VECP	mdl_u_min () const
	lower bounds for optimized model inputs

const VECP	mdl_u_max () const
	upper bounds for optimized model inputs

const VECP	mdl_u_ref () const
	reference values for optimized model inputs (default: 0)

const VECP	mdl_u_weight1 () const
	weight for linear objective term (default: 0)

const VECP	mdl_u_weight2 () const
	weight for quadratic objective term (default: 0)

const VECP	mdl_der_u_min () const
	lower bounds for rates of change of optimized model inputs

const VECP	mdl_der_u_max () const
	upper bounds for rates of change of optimized model inputs

const VECP	mdl_der_u_ref () const
	reference values for rates of change of optimized inputs (default: 0)

const VECP	mdl_der_u_weight1 () const
	weight for linear objective term (default: 0)

const VECP	mdl_der_u_weight2 () const
	weight for quadratic objective term (default: 0)

const VECP	mdl_der_u_soft_min () const
	soft lower bounds for rates of change of optimized model inputs

const VECP	mdl_der_u_soft_max () const
	soft upper bounds for rates of change of optimized model inputs

const VECP	mdl_der_u_soft_weight1 () const
	weight for linear objective term (default: 0)

const VECP	mdl_der_u_soft_weight2 () const
	weight for quadratic objective term (default: 0)

const IVECP	mdl_x_integer () const
	indicate integer valued states

const VECP	mdl_x_nominal () const
	nominal state values (for scaling)

const IVECP	mdl_x_periodic () const
	first and last value are equal (default: false)

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

const VECP	mdl_y_bias () const
	bias for outputs

const VECP	mdl_y_nominal () const
	nominal output values (for scaling)

const VECP	mdl_y_min () const
	lower bounds for model outputs

const VECP	mdl_y_max () const
	upper bounds for model outputs

const VECP	mdl_y_ref () const
	reference values for model outputs (default: 0)

const VECP	mdl_y_weight1 () const
	weight for linear objective term (default: 0)

const VECP	mdl_y_weight2 () const
	weight for quadratic objective term (default: 0)

const VECP	mdl_y_soft_min () const
	soft lower bounds for model outputs

const VECP	mdl_y_soft_max () const
	soft upper bounds for model outputs

const VECP	mdl_y_soft_weight1 () const
	weight for linear objective term (default: 0)

const VECP	mdl_y_soft_weight2 () const
	weight for quadratic objective term (default: 0)

const VECP	mdl_uf () const
	model inputs at final time

const VECP	mdl_uf_min () const
	lower bounds for model inputs at final time

const VECP	mdl_uf_max () const
	upper bounds for model inputs at final time

const VECP	mdl_yf () const
	model outputs at final time

const VECP	mdl_yf_min () const
	lower bounds for model outputs at final time

const VECP	mdl_yf_max () const
	upper bounds for model outputs at final time

const VECP	mdl_yf_weight1 () const
	weight for linear objective term (default: 0)

const VECP	mdl_yf_weight2 () const
	weight for quadratic objective term (default: 0)

const VECP	mdl_yf_soft_min () const
	soft lower bounds for model outputs at final time

const VECP	mdl_yf_soft_max () const
	soft upper bounds for model outputs at final time

const VECP	mdl_yf_soft_weight1 () const
	weight for linear objective term (default: 0)

const VECP	mdl_yf_soft_weight2 () const
	weight for quadratic objective term (default: 0)

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_x0 (const VECP v)
	set initial states and copy them to all mdl_xs

void	set_mdl_x0_active (const IVECP v)
	set free initial states

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_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_u0 (const VECP v)
	set initial values for model inputs and copy them to all mdl_us

void	set_mdl_u0_min (const VECP v)
	set lower bounds for initial model inputs

void	set_mdl_u0_max (const VECP v)
	set upper bounds for initial model inputs

void	set_mdl_y0_min (const VECP v)
	set lower bounds for initial model outputs

void	set_mdl_y0_max (const VECP v)
	set upper bounds for initial model outputs

void	set_mdl_y0_weight1 (const VECP v)
	set linear weight

void	set_mdl_y0_weight2 (const VECP v)
	set quadratic weight

void	set_mdl_u_order (const IVECP v)
	set interpolation order

void	set_mdl_u_active (const IVECP v)
	set optimized inputs

void	set_mdl_u_integer (const IVECP v)
	set integers

void	set_mdl_u0_nfixed (const IVECP v)
	set numbers of fixed control inputs

void	set_mdl_u_decimation (const IVECP v)
	set decimation for optimized model inputs

void	set_mdl_u_nominal (const VECP v)
	set nominal inputs

void	set_mdl_u_periodic (const IVECP v)
	set periodic congtrols

void	set_mdl_u_min (const VECP v)
	set lower bounds for model inputs

void	set_mdl_u_max (const VECP v)
	set upper bounds for model inputs

void	set_mdl_u_ref (const VECP v)
	set reference model inputs

void	set_mdl_u_weight1 (const VECP v)
	set linear weight

void	set_mdl_u_weight2 (const VECP v)
	set quadratic weight

void	set_mdl_der_u_min (const VECP v)
	set lower bounds for rates of change of model inputs

void	set_mdl_der_u_max (const VECP v)
	set upper bounds for rates of change of model inputs

void	set_mdl_der_u_ref (const VECP v)
	set reference rates of change of model inputs

void	set_mdl_der_u_weight1 (const VECP v)
	set linear weight

void	set_mdl_der_u_weight2 (const VECP v)
	set quadratic weight

void	set_mdl_der_u_soft_min (const VECP v)
	set soft lower bounds for rates of change of model inputs

void	set_mdl_der_u_soft_max (const VECP v)
	set soft upper bounds for rates of change of model inputs

void	set_mdl_der_u_soft_weight1 (const VECP v)
	set linear weight

void	set_mdl_der_u_soft_weight2 (const VECP v)
	set quadratic weight

void	set_mdl_x_integer (const IVECP v)
	set integer valued states

void	set_mdl_x_nominal (const VECP v)
	set nominal states

void	set_mdl_x_periodic (const IVECP v)
	set periodic states

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_order (const IVECP v)
	set interpolation order

void	set_mdl_y_bias (const VECP v)
	set output bias

void	set_mdl_y_nominal (const VECP v)
	set nominal outputs

void	set_mdl_y_min (const VECP v)
	set lower bounds for model outputs

void	set_mdl_y_max (const VECP v)
	set upper bounds for model outputs

void	set_mdl_y_ref (const VECP v)
	set reference model outputs

void	set_mdl_y_weight1 (const VECP v)
	set linear weight

void	set_mdl_y_weight2 (const VECP v)
	set quadratic weight

void	set_mdl_y_soft_min (const VECP v)
	set soft lower bounds for model outputs

void	set_mdl_y_soft_max (const VECP v)
	set soft upper bounds for model outputs

void	set_mdl_y_soft_weight1 (const VECP v)
	set linear weight

void	set_mdl_y_soft_weight2 (const VECP v)
	set quadratic weight

void	set_mdl_uf_min (const VECP v)
	set lower bounds for final model inputs

void	set_mdl_uf_max (const VECP v)
	set upper bounds for final model inputs

void	set_mdl_yf_min (const VECP v)
	set lower bounds for final model outputs

void	set_mdl_yf_max (const VECP v)
	set upper bounds for final model outputs

void	set_mdl_yf_weight1 (const VECP v)
	set linear weight

void	set_mdl_yf_weight2 (const VECP v)
	set quadratic weight

void	set_mdl_yf_soft_min (const VECP v)
	set soft lower bounds for model outputs at final time

void	set_mdl_yf_soft_max (const VECP v)
	set soft upper bounds for model outputs at final time

void	set_mdl_yf_soft_weight1 (const VECP v)
	set linear weight

void	set_mdl_yf_soft_weight2 (const VECP v)
	set quadratic weight

void	set_mdl_us (const MATP v)
	set model inputs

void	set_mdl_xs (const MATP v)
	set model states

Public Member Functions inherited from Prg_SFunction
	Prg_SFunction ()
	constructor

	~Prg_SFunction ()
	destructor

const char *	mdl_name () const
	S-function name.

void	set_mdl_name (const char *str)
	set S-function name

const char *	mdl_path () const
	S-function path, including name, used for dynamic loading. More...

void	set_mdl_path (const char *str)
	set S-function path

const char *	mdl_args () const
	String representation of S-function arguments.

void	set_mdl_args (const char *str)
	set S-function arguments

const VECP	mdl_p () const
	parameters

void	set_mdl_p (const VECP value)
	set parameters

const VECP	mdl_x0 () const
	initial states

void	set_mdl_x0 (const VECP value)
	set initial states

Public Member Functions inherited from Omu_Program
	Omu_Program ()
	constructor

virtual	~Omu_Program ()
	destructor

virtual void	update (int kk, const adoublev &x, const adoublev &u, adoublev &f, adouble &f0, adoublev &c)
	High-level update for the optimization criterion, constraints, and discrete state equations.

virtual void	consistic (int kk, double t, const adoublev &x, const adoublev &u, adoublev &xt)
	High-level consistic (consistent initial conditions) routine for the initialization of continuous-time states from optimization variables x and u. More...

Protected Member Functions
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. More...

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. More...

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. More...

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)

Overloaded gradient routines
These routines call the S-function method mdlJacobian if available; otherwise they direct calls to the according methods by Omu_Program.
void	update_grds (int kk, const Omu_StateVec &x, const Omu_Vec &u, const Omu_StateVec &xf, Omu_DependentVec &f, Omu_Dependent &f0, Omu_DependentVec &c)
	Overloaded update routine for obtaining gradients.

void	continuous_grds (int kk, double t, const Omu_StateVec &x, const Omu_Vec &u, const Omu_StateVec &dx, Omu_DependentVec &F)
	Overloaded continuous routine for obtaining gradients.

Protected Member Functions inherited from Prg_SFunction
void	read_mx_args (VECP p)
	read _mx_args into vector p

void	write_mx_args (VECP p)
	write vector p to _mx_args

bool	setContinuousTask (bool val)
	enable continuous sample time; return true if a continuous sample time exists and has been enabled

bool	setSampleHit (bool val)
	enable hit for all discrete sample times; return true if a discrete sample time exists and has been enabled

Protected Attributes
Omu_VariableVec	_mdl_x0
	initial states for optimization

IVECP	_mdl_x0_active
	free initial states (default: 0)

VECP	_mdl_der_x0_min
	minimum for time derivative of x0

VECP	_mdl_der_x0_max
	maximum for time derivative of x0

Omu_VariableVec	_mdl_u0
	initial inputs

Omu_OptVarVec	_mdl_y0
	model outputs at initial time

Omu_OptVarVec	_mdl_u
	model inputs

Omu_OptVarVec	_mdl_der_u
	rates of change of inputs

Omu_OptVarVec	_mdl_der_u_soft
	relaxed rates of change of inputs

Omu_VariableVec	_mdl_x
	state bounds

Omu_OptVarVec	_mdl_y
	model outputs

Omu_OptVarVec	_mdl_y_soft
	attributes for relaxed output constraints

Omu_OptVarVec	_mdl_uf
	model inputs at final time

Omu_OptVarVec	_mdl_yf
	model outputs at final time

Omu_OptVarVec	_mdl_yf_soft
	relaxed output constraints at final time

IVECP	_mdl_u_order
	interpolation order (default: 1 (linear))

IVECP	_mdl_y_order
	interpolation order (default: 1 (linear))

VECP	_taus
	scaled time communicated to outside as prg_ts

int	_t_scale_idx
	index into mdl_u vector for variable used for scaling of time

int	_t_active
	time is being scaled

int	_t_scale_i
	index of internal optimization variable for scaling of time

double	_t_scale_nominal
	nominal value for time scaling (default: 1)

double	_t_nominal
	nominal time (used internally for scaling)

VECP	_mdl_u_nominal
	nominal inputs (for scaling)

VECP	_mdl_x_nominal
	nominal states (for scaling)

VECP	_mdl_y_nominal
	nominal outputs (for scaling)

VECP	_mdl_y_bias
	bias correction (offset) for outputs

int	_nx
	number of states for optimizer

int	_nu
	number of optimized control inputs

int	_nc
	number of constrained outputs

int	_nc0
	number of constrained/used outputs at initial time

int	_ncf
	number of constrained/used outputs at final time

int	_ns
	number of slack variables for soft constraints

int	_nsc
	number of soft constraints

int	_nsu
	number of slack variables for soft cns on inputs

int	_nsuc
	number of soft constraints on inputs

int	_nsf
	number of slacks for soft constraints at final time

int	_nscf
	number of soft constraints at final time

int	_multistage
	treat as multistage problem

int	_sps
	Number of sample periods per stage (default: 1). More...

MATP	_mdl_us
	given model inputs (controls and disturbances)

MATP	_mdl_xs
	given and calculated model states

MATP	_mdl_ys
	calculated model outputs

IVECP	_mdl_u0_nfixed
	Numbers of fixed control inputs at begin of time horizon (default: 0). More...

IVECP	_mdl_u_decimation
	Decimation factor, i.e. More...

IVECP	_mdl_u_periodic
	Periodic controls, i.e. More...

IVECP	_mdl_x_periodic
	Periodic states, i.e. More...

Protected Attributes inherited from Prg_SFunction
char *	_mdl_name
	S-function name.

char *	_mdl_path
	full S-function path

char *	_mdl_args
	S-function parameters.

SimStruct *	_SS
	pointer to SimStruct

mxArray **	_mx_args
	S-function parameters after parsing.

int	_mdl_nargs
	number of S-function arguments

double	_t0_setup_model
	time used for initialization of model

int	_mdl_np
	number of model parameters

int	_mdl_nd
	number of discrete-time states

int	_mdl_nx
	number of model states (incl. discrete)

int	_mdl_nu
	number of model inputs

int	_mdl_ny
	number of model outputs

VECP	_mdl_p
	parameters

VECP	_mdl_x0
	initial states

bool	_mdl_needs_setup
	indicate that setup_model needs to be called as _mdl_name, _mdl_path, or _mdl_args changed

Detailed Description

Optimal control problem for a model given as MEX S-function treated with multi-stage control vector parameterization.

The optimization time horizon \([t_0,t_f]\) is split into \(k=0,...,K\) stages with time points \(t_0=t^0<t^1<\ldots<t^K=t_f\). Each stage may be further subdivided into \(sps\) sample periods per stage. This leads to the overall number of \(KK=sps\,K\) sample periods with the sample time points \(t^{kk}, kk=0,...,KK\) (sample time points within a stage are for instance useful to better treat path constraints). The model time may be parameterized by defining a piecewise constant model input as scaling factor. Sought control trajectories are described piecewise as constant or linear functions of control parameters in each stage.

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

\[ \begin{array}{l} \displaystyle J\ =\ \sum_{i=1}^{n_y} \left\{ y_{0\_weight1}\left[y(t_0)-\frac{y_{ref}}{y_{nominal}}\right] \ +\ y_{0\_weight2}\left[y(t_0)-\frac{y_{ref}}{y_{nominal}}\right]^2 \right\}_i \\[4ex] \displaystyle \qquad \ + \ \sum_{kk=0}^{KK} \sum_{i=1}^{n_u} \Delta t_{u,i}^{kk} \left\{ u_{weight1}\left[u(t^{kk})-\frac{u_{ref}}{u_{nominal}}\right] \ +\ u_{weight2}\left[u(t^{kk})-\frac{u_{ref}}{u_{nominal}}\right]^2 \right\}_i \\[4ex] \displaystyle \qquad \ + \ \sum_{k=0}^{K-1} (t^{k+1}-t^{k}) \sum_{i=1}^{n_u} \left\{ {der\_u}_{weight1}\left[du^k -\frac{{der\_u}_{ref}}{u_{nominal}}\right] \ +\ {der\_u}_{weight2}\left[du^k -\frac{{der\_u}_{ref}}{u_{nominal}}\right]^2 \right\}_i \\[4ex] \displaystyle \qquad \ + \ \sum_{kk=0}^{KK} \Delta t_{y,i}^{kk} \sum_{i=1}^{n_y} \left\{ y_{weight1}\left[y(t^{kk})-\frac{y_{ref}}{y_{nominal}}\right] \ +\ y_{weight2}\left[y(t^{kk})-\frac{y_{ref}}{y_{nominal}}\right]^2 \right\}_i \\[4ex] \displaystyle \qquad \ + \ \sum_{i=1}^{n_y} \left\{ y_{f\_weight1}\left[y(t_f)-\frac{y_{ref}}{y_{nominal}}\right] \ +\ y_{f\_weight2}\left[y(t_f)-\frac{y_{ref}}{y_{nominal}}\right]^2 \right\}_i \\[4ex] \displaystyle \qquad \ + \ \sum_{kk=0}^{KK} \Delta t_{y,i}^{kk} \sum_{i=1}^{n_y} \left\{ y_{soft\_weight1}\,s^{kk} + y_{soft\_weight2}\,s^{kk}s^{kk} \right\}_i \\[4ex] \displaystyle \qquad \ + \ \sum_{i=1}^{n_y} \left\{ y_{f\_soft\_weight1}\,s_f + y_{f\_soft\_weight2}\,s_fs_f \right\}_i \\[4ex] \displaystyle \qquad \ \to\quad \min \end{array} \]

with

\[ \begin{array}{l} \displaystyle \Delta t_{u,i}^{kk} = \left\{\begin{array}{ll} \Delta t_0^{kk}, & u_{order,i} = 0, \\[1ex] \Delta t^{kk}, & u_{order,i} = 1, \end{array}\right. \\[5ex] \displaystyle \Delta t_{y,i}^{kk} = \left\{\begin{array}{ll} \Delta t_0^{kk}, & y_{order,i} = 0, \\[1ex] \Delta t^{kk}, & y_{order,i} = 1, \end{array}\right. \\[5ex] \displaystyle \Delta t_0^{kk} = \left\{\begin{array}{ll} t_{scale}^{kk+1}(t^{kk+1} - t^{kk}), & kk < KK, \\[1ex] 0, & kk = KK, \end{array}\right. \\[5ex] \displaystyle \Delta t^{kk} = \frac{1}{2} \left\{\begin{array}{ll} t_{scale}^{kk+1}(t^{kk+1} - t^{kk}), & kk=0, \\[1ex] t_{scale}^{kk}(t^{kk} - t^{kk-1}), & kk=KK, \\[1ex] t_{scale}^{kk+1}(t^{kk+1} - t^{kk}) + t_{scale}^{kk}(t^{kk} - t^{kk-1}), & \mbox{else}, \end{array}\right. \\[5ex] \displaystyle t_{scale} = \left\{\begin{array}{ll} u_{t\_scale\_idx}, & t\_scale\_idx \ge 0, \\ 1, & \mbox{else}, \end{array}\right. \end{array} \]

subject to the model given with the S-function methods mdlDerivatives \(f\), mdlOutputs \(g\) and mdlUpdate \(h,\ t\in[t_0,t_f]\), as well as with parameterized time \(\tau\)

\[ \begin{array}{l} \displaystyle \quad\ \frac{d\tau(t)}{dt} = t_{scale}(t), \\[3ex] \displaystyle x(\tau(t)) = \left[\begin{array}{ll} x_d^{kk}, & t\in[t^{kk},t^{kk+1}),\ kk=0,\ldots,KK \\ x_c(\tau(t)) \end{array}\right], \\[3ex] \displaystyle x_d^{kk} = \frac{h[x_{nominal}\,x(\tau(t^-)),\ u_{nominal}\,u(\tau(t^-))]} {x_{nominal_d}},\quad t=t^{kk},\ kk=1,\ldots,KK,\\[3ex] \displaystyle \frac{dx_c(\tau(t))}{d\tau(t)} = \frac{f[x_{nominal}\,x(\tau(t)),\ u_{nominal}\,u(\tau(t))]} {x_{nominal_c}}, \\[3ex] \displaystyle y(\tau(t)) = \frac{g[x_{nominal}\,x(\tau(t)),\ u_{nominal}\,u(\tau(t))]}{y_{nominal}} \ +\ \frac{y_{bias}}{y_{nominal}}, \end{array} \]

with piecewise constant or linear approximation of \(u(t)\) either using optimized control parameters \(du^k\) or given inputs \(us\)

\[ \begin{array}{ll} \left\{ u(t) = u(t^{k-1}) + \Delta u(t^{k-1}) \right\}_i, & i \in \mbox{find}(u_{active}\ \mbox{and}\ u_{order}=0), \\[1ex] & t\in[t^{k},t^{k+1}),\ k=1,\ldots,K-1, \\[1ex] \ \ \Delta u(t^{k-1}) = \left\{\begin{array}{ll} (t^{k}-t^{k-1})du^{k-1}, & i = t\_scale\_idx \\[1ex] (t^{k}-t^{k-1})du^{k-1}t_{scale}^k, & \mbox{else} \end{array}\right. & k=1,\ldots,K-1, \\[3ex] \left\{ \frac{du(\tau(t))}{d\tau(t)} = du^{k} \right\}_i, & i \in \mbox{find}(u_{active}\ \mbox{and}\ u_{order}=1), \\[1ex] & t\in[t^{k},t^{k+1}),\ k=0,\ldots,K-1, \\[3ex] \left\{ u(t) = \displaystyle \frac{us^{kk}}{u_{nominal}} \right\}_i, & i \in \mbox{find}(\mbox{not}\ u_{active}\ \mbox{and}\ u_{order}=0),\\[3ex] \left\{ u(t) = \displaystyle \frac{t^{kk+1}-t}{t^{kk+1}-t^{kk}}\ \frac{us^{kk}}{u_{nominal}} + \frac{t-t^{kk}}{t^{kk+1}-t^{kk}} \ \frac{us^{kk+1}}{u_{nominal}} \right\}_i, & i \in \mbox{find}(\mbox{not}\ u_{active}\ \mbox{and}\ u_{order}=1), \\[3ex] & t\in[t^{kk},t^{kk+1}),\ kk=0,\ldots,KK-1, \end{array} \]

and subject to the constraints at initial and final time

\[ \begin{array}{rcccll} \displaystyle && \{ x(t^0) &=& \displaystyle \frac{x^0}{x_{nominal}}\}_i, \quad & i\notin\mbox{find}(x_{0\_active}), \\[3ex] \displaystyle \left\{ \frac{x^0_{min}}{x_{nominal}} \right. &\le& x(t^{0}) &\le& \displaystyle \left. \frac{x^0_{max}}{x_{nominal}} \right\}_i, \quad & i\in\mbox{find}(x_{0\_active}), \\[3ex] \displaystyle \left\{ \frac{der\_x^0_{min}}{x_{nominal}} \right. &\le& \dot{x}(t^{0}) &\le& \displaystyle \left. \frac{der\_x^0_{max}}{x_{nominal}} \right\}_i, \quad & i\in\mbox{find}(x_{0\_active}), \\[3ex] \displaystyle \left\{ \frac{u^0_{min}}{u_{nominal}} \right. &\le& u(t^{0}) &\le& \displaystyle \left. \frac{u^0_{max}}{u_{nominal}} \right\}_i, \quad & i\in\mbox{find}(u_{active}\ \mbox{and}\ u_{0,nfixed}=0), \\[3ex] \displaystyle && \{ u(t^0) &=& \displaystyle \frac{us^0}{u_{nominal}} \}_i, \quad & i \in \mbox{find}(u_{0,nfixed}>1-u_{order}), \\[3ex] \displaystyle\left\{\frac{us^0+der\_u_{min}}{u_{nominal}}\right. &\le& u(t^0) &\le& \displaystyle\left.\frac{us^0+der\_u_{max}}{u_{nominal}}\right\}_i, \ & i \in \mbox{find}(u_{0,nfixed}=1\ \mbox{and}\ u_{order}=0), \\[3ex] \displaystyle \frac{y_{0,min}}{y_{nominal}} &\le& y(t^{0}) &\le& \displaystyle \frac{y_{0,max}}{y_{nominal}}, \\[3ex] \displaystyle \frac{u_{f\_min}}{u_{nominal}} &\le& u(t_f) &\le& \displaystyle \frac{u_{f\_max}}{u_{nominal}}, \\[3ex] \displaystyle \frac{y_{f\_min}}{y_{nominal}} &\le& y(t_f) &\le& \displaystyle \frac{y_{f\_max}}{y_{nominal}}, \\[3ex] \displaystyle \frac{y_{f\_soft\_min}}{y_{nominal}} - s_f &\le& y(t_f) &\le& \displaystyle \frac{y_{f\_soft\_max}}{y_{nominal}} + s_f, \\[3ex] \displaystyle && s_f &\ge& 0, \end{array} \]

as well as at all time points

\[ \begin{array}{rcccll} \displaystyle\left\{ \frac{u_{min}}{u_{nominal}} \right. &\le& u(t^{k}) &\le& \displaystyle\left. \frac{u_{max}}{u_{nominal}} \right\}_i, \quad & i \in \mbox{find}(k \ge u_{0,nfixed} + u_{order} - 1), \\[1ex] && && & k=0,\ldots,K, \\[2ex] \displaystyle \frac{{der\_u}_{min}}{u_{nominal}} &\le& du^{k} &\le& \displaystyle \frac{{der\_u}_{max}}{u_{nominal}}, \quad & k=0,\ldots,K-1, \\[3ex] \displaystyle \frac{x_{min}}{x_{nominal}} &\le& x(t^{k}) &\le& \displaystyle \frac{x_{max}}{x_{nominal}}, \quad & k=0,\ldots,K, \\[3ex] \displaystyle \frac{y_{min}}{y_{nominal}} &\le& y(t^{kk}) &\le& \displaystyle \displaystyle \frac{y_{max}}{y_{nominal}}, \quad & kk=0,\ldots,KK, \\[3ex] \displaystyle \frac{y_{soft\_min}}{y_{nominal}} - s^{kk} &\le& y(t^{kk}) &\le& \displaystyle \frac{y_{soft\_max}}{y_{nominal}} + s^{kk}, \\[3ex] \displaystyle && s^{kk} &\ge& 0, \quad & kk=0,\ldots,KK. \end{array} \]

Note that path constraints over continuous time intervals can be approximated by specifying \(sps>1\), leading to output constraints at interior sample time points.

The actually optimized rates of changes \(du^k, k=0,\ldots,K-1\) for active inputs may be set to fixed values by specifying \(u_{0,nfixed}\) and \(u_{decimation}\) (defaults: 1), fixing initial values and holding optimized inputs constant over multiple stages, respectively

\[ du^k_i = \left\{\begin{array}{ll} \displaystyle 0 , \quad & i \in \mbox{find}(\mbox{mod}(k+1, u_{decimation}) \ne 0), \\[2ex] \displaystyle du^k_{initial,i}, \quad & i \in \mbox{find}(\mbox{mod}(k+1, u_{decimation}) = 0\ \mbox{and}\ k < u_{0,nfixed}+u_{order}-2), \\[2ex] \displaystyle \mbox{free} , \quad & \mbox{else}. \end{array}\right. \]

The initial guess is taken from given initial states and model inputs

\[ \begin{array}{rcll} x_{initial}(t^0) &=& \displaystyle \frac{x^0}{x_{nominal}}, \\[3ex] u_{initial}(t^0) &=& \displaystyle \frac{us^0}{u_{nominal}}, \\[3ex] du^k_{initial} &=& \displaystyle \left\{\frac{us^{sps\,(k+1)} - us^{sps\,k}} {(t^{sps\,(k+1)}-t^{sps\,k})\,u_{nominal}} \right\}_i, & i \in \mbox{find}(u_{active}), \\[1ex] &&& k=0,\ldots,K-1. \end{array} \]

The problem is treated as multistage problem with K stages per default. Consequently additional K junction conditions (equality constraints) are introduced for the state variables x and the control trajectories u. Alternatively the problem can be treated without stages applying pure control vector parameterization and hiding model states from the optimizer.

When treated as multistage problem, the additional optimization variables introduced for states are normally initialized with the results of an initial-value simulation for the given initial states and inputs (multistage=1). Alternatively, with multistage=2, the multistage problem may be treated with multiple shooting, i.e. all states are initialized with the initial states \(x^0\)

\[ \begin{array}{rcll} x_{initial}(t^k) &=& \displaystyle \frac{x^0}{x_{nominal}}, & k=1,\ldots,K \end{array} \]

or with an explicitly given initial guess for states \(xs\) in all stages

\[ \begin{array}{rcll} x_{initial}(t^k) &=& \displaystyle \frac{xs^{sps\,k}}{x_{nominal}}, & k=1,\ldots,K. \end{array} \]

The multiple shooting method is advantageous if the expected state trajectories are known (e.g. constant at a set point or following a ramp), but if an initial guess for the control trajectories causing those state trajectories is unknown.

Model inputs, states and outputs can be accessed through

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

Member Function Documentation

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

protectedvirtual

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.

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

protectedvirtual

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

Additionally setup bounds and initial values.

Implements Omu_Program.

void Prg_SFunctionOpt::setup_stages	(	IVECP	ks,
		VECP	ts
	)

protectedvirtual

Setup stages and time sampling.

The default implementation sets up an optimization problem without stages.

Reimplemented from Omu_Program.

Member Data Documentation

IVECP Prg_SFunctionOpt::_mdl_u0_nfixed

protected

Numbers of fixed control inputs at begin of time horizon (default: 0).

An active input is free for _mdl_u0_fixed=0, fixed at the initial time point for _mdl_u0_nfixed=1, fixed up to the last time point of the first stage for _mdl_u0_nfixed=2, and so on.

Referenced by mdl_u0_nfixed(), and set_mdl_u0_nfixed().

IVECP Prg_SFunctionOpt::_mdl_u_decimation

protected

Decimation factor, i.e.

~numbers of subsequent optimized inputs with equal values (default: 1). The controls of two subsequent stages are equal for decimation=2, three for decimation=3, and so on.

Referenced by mdl_u_decimation(), and set_mdl_u_decimation().

IVECP Prg_SFunctionOpt::_mdl_u_periodic

protected

Periodic controls, i.e.

~first and last value are equal

Referenced by mdl_u_periodic(), and set_mdl_u_periodic().

IVECP Prg_SFunctionOpt::_mdl_x_periodic

protected

Periodic states, i.e.

~first and last value are equal

Referenced by mdl_x_periodic(), and set_mdl_x_periodic().

int Prg_SFunctionOpt::_sps

protected

Number of sample periods per stage (default: 1).

The value can be increased to devide each control interval into multiple sample periods, e.g. for evaluating constraints and the objective within control intervals. Currently _sps>1 is only supported for multistage problems.

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

Prg_SFunctionOpt.h

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Detailed Description

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