Controller optimization via BOD

BOD - The Digital Magnitude Optimum

BOD Toolbox Version 3.1

Applying MATLAB there is a toolbox BOD version 3.0 available. Download may be done regarding following several m-files und demos, or packed as ZipFile. Please note: if you select several functions only, do not forget to download directory private in either case:

File name Function

bod
bod_2 Calculation of discontinuous controllers using the Discontinuous Magnitude Optimum (BOD) - starting via a plant given in z-domain, typically applying function trans of the BOD toolbox. LTI syntax optionally; menu-driven optionally.
based on reconfigurations of the generel equation system; application of fsolve only in case of PID controller and delayed input signals; bod_2: old fsolve syntax

bodg Supporting file for bod.m; calling makes sense by bod.m only.

bod_gen
Application of the Discontinuous Magnitude Optimum (BOD) to calculate the coefficients of a discontinuous controller RZ (and optionally pre-filter FZ if applicable); without any internal pole compensation - if desired: previous external realization necessary!
=> direct application of the generel BOD equation system, generel use of fsolve; also Lead/Lag controller of 1. and 2. order

bod_gen_eq Supporting file for bod_gen.m; calling makes sense by bod_gen.m only.

bod_prefi_opt Optimization of the pre-filter, hybrid optimization according to the Digital Magnitude Optimum (BOD), i.e. controller setting causes behaviour for delayed inputs, setting of the new pre-filter causes behaviour for undelayed inputs.

bod_rst Application of the Discontinuous Magnitude Optimum (BOD) to calculate directly the coefficients of a RST control structure.
For RST control structure see: Landau/Zito "Digital Control Systems", Springer 2010.

bod_rst_eq Supporting file for bod_rst.m; calling makes sense by bod_rst.m only.

eez Calculation of discontinuous controllers using the method of Finite Time Settling (EEZ=EndlicheEinstellZeit) - starting via a plant given in z-domain, typically applying function trans of the BOD toolbox; LTI syntax optionally; menu driven optionally.

gmt Calculation of weighted mean value dead times in case of application of bus systems within control loops.

guete Calculation of step response quality criteria using SIMULINK simulation results of a closed loop.

kaskade Optimization of a cascade control structure using discontinuous controllers; optionally 2 or 3 cascaded control loops.

kmatrix Calculation weighting factors for the BOD optimization equation system.

simu Step response of a single control loop - simulation and quality criteria.

trans All-purpose z-transform of transfer functions starting from Laplace into z-domain: G(s)=za(s)/ne(s) - whole-number dead times separately! LTI syntax optionally; menu driven input optionally; output vectors of same order; real zeros in numerator;
Upgrading of c2d/c2dm by consideration of the coupling of controller and plant (T/H) as well as of the measuring method (A/M).

urlaz Conversion of controller parameters inside Laplace(s) and z-domain(z); conversion based on approximations; optionally menu driven; applicable to impart quasi-continuous knowledge of controllers designed by means of the Discontinuous Magnitude Optimum (BOD).

zust_au1
zust_au2 State control structure 1: coefficient number 3; automated; here >>>plant<<< only; application of the Discontinuous Magnitude Optimum (BOD) to calculate coefficients of a discontinuous state control structure; variants 1 and 2 differ in fsolve calling.

zust_aug1
zust_aug2 Supporting files for zust_au1.m resp. zust_au2.m - here >>>control structure<<< in z-domain;
calling makes sense via zust_au1 resp. zust_au2 only.

zust_aup1
zust_aup2 State control structure 1; coefficient number 3; here: parameter variations: motor, spring and load time constant;
variants 1 and 2 differ in fsolve calling.

zust_aut1
zust_aut2 State control structure 1; coefficient number 3; here: parameter variations: sampling time and first order time delay model of the closed current control loop;
variants 1 and 2 differ in fsolve calling.

zust_bran_2 State control structure 2: coefficient number 4; automated; here >>>plant<<< only; application of the Discontinuous Magnitude Optimum (BOD) to calculate coefficients of a discontinuous state control structure.

zust_bran_2gl Supporting file for zust_bran_2.m - here >>>control structure<<< in z-domain; calling makes sense via zust_bran_2 only.

File name	Function
bod bod_2	Calculation of discontinuous controllers using the Discontinuous Magnitude Optimum (BOD) - starting via a plant given in z-domain, typically applying function trans of the BOD toolbox. LTI syntax optionally; menu-driven optionally. based on reconfigurations of the generel equation system; application of fsolve only in case of PID controller and delayed input signals; bod_2: old fsolve syntax
bodg	Supporting file for bod.m; calling makes sense by bod.m only.
bod_gen	Application of the Discontinuous Magnitude Optimum (BOD) to calculate the coefficients of a discontinuous controller RZ (and optionally pre-filter FZ if applicable); without any internal pole compensation - if desired: previous external realization necessary! => direct application of the generel BOD equation system, generel use of fsolve; also Lead/Lag controller of 1. and 2. order
bod_gen_eq	Supporting file for bod_gen.m; calling makes sense by bod_gen.m only.
bod_prefi_opt	Optimization of the pre-filter, hybrid optimization according to the Digital Magnitude Optimum (BOD), i.e. controller setting causes behaviour for delayed inputs, setting of the new pre-filter causes behaviour for undelayed inputs.
bod_rst	Application of the Discontinuous Magnitude Optimum (BOD) to calculate directly the coefficients of a RST control structure. For RST control structure see: Landau/Zito "Digital Control Systems", Springer 2010.
bod_rst_eq	Supporting file for bod_rst.m; calling makes sense by bod_rst.m only.

eez	Calculation of discontinuous controllers using the method of Finite Time Settling (EEZ=EndlicheEinstellZeit) - starting via a plant given in z-domain, typically applying function trans of the BOD toolbox; LTI syntax optionally; menu driven optionally.
gmt	Calculation of weighted mean value dead times in case of application of bus systems within control loops.
guete	Calculation of step response quality criteria using SIMULINK simulation results of a closed loop.
kaskade	Optimization of a cascade control structure using discontinuous controllers; optionally 2 or 3 cascaded control loops.
kmatrix	Calculation weighting factors for the BOD optimization equation system.
simu	Step response of a single control loop - simulation and quality criteria.
trans	All-purpose z-transform of transfer functions starting from Laplace into z-domain: G(s)=za(s)/ne(s) - whole-number dead times separately! LTI syntax optionally; menu driven input optionally; output vectors of same order; real zeros in numerator; Upgrading of c2d/c2dm by consideration of the coupling of controller and plant (T/H) as well as of the measuring method (A/M).
urlaz	Conversion of controller parameters inside Laplace(s) and z-domain(z); conversion based on approximations; optionally menu driven; applicable to impart quasi-continuous knowledge of controllers designed by means of the Discontinuous Magnitude Optimum (BOD).
zust_au1 zust_au2	State control structure 1: coefficient number 3; automated; here >>>plant<<< only; application of the Discontinuous Magnitude Optimum (BOD) to calculate coefficients of a discontinuous state control structure; variants 1 and 2 differ in fsolve calling.
zust_aug1 zust_aug2	Supporting files for zust_au1.m resp. zust_au2.m - here >>>control structure<<< in z-domain; calling makes sense via zust_au1 resp. zust_au2 only.
zust_aup1 zust_aup2	State control structure 1; coefficient number 3; here: parameter variations: motor, spring and load time constant; variants 1 and 2 differ in fsolve calling.
zust_aut1 zust_aut2	State control structure 1; coefficient number 3; here: parameter variations: sampling time and first order time delay model of the closed current control loop; variants 1 and 2 differ in fsolve calling.
zust_bran_2	State control structure 2: coefficient number 4; automated; here >>>plant<<< only; application of the Discontinuous Magnitude Optimum (BOD) to calculate coefficients of a discontinuous state control structure.
zust_bran_2gl	Supporting file for zust_bran_2.m - here >>>control structure<<< in z-domain; calling makes sense via zust_bran_2 only.

Applying MATLAB and SIMULINK directory DEMO offers following m- and mdl-Files containing examples for scripts to call above listed functions resp. Simulink simulation structures to check the results:

File name Function Simulation

demo_1 Demo for use of trans.m and bod.m - see also DEMO_BOD (LTI model definition), the demo contains two menu-driven examples;
example 1: undelayed input signals without pole compensation;
example 2: delayed input signals including pre-filter demo_1s

demo_2 Demo for use of zust_au.m / zust_aug.m - application of BOD for state control structures;
example: two mass system with reference in journal "Automatisierungstechnik";
application of Digital Magnitude Optimum results in three controller parameters. demo_2s

DEMO_BOD Application of function bod.m (Digital Magnitude Optimum) and optionally input / output of LTI models and / or optionally menu-driven or vector control;
example 1: two time-delay elements
controller variants: 5 regarding undelayed, 3 regarding delayed input signals;
example 2: demo applying a damped controller derivative part DEMO_BODs

demo_BOD_rst Application of function bod_rst.m, i.e. application of the Digital Magnitude Optimum (BOD) to a RST control structure by Landau/Zito (see bod_rst.m), plant features a large dead time. demo_BOD_rst_s

DEMO_EEZ Application of function eez.m (Finite Time Settling) and optionally input / output of LTI models and / or optionally menu-driven or vector control;
example 1: two time-delay elements
controller variants for undelayed input signals: 4 regarding complete pole compensation, 3 regarding ramp input, 2 regarding partial pole compensation;
example 2: one time-delay element and mean value measurement
controller variants for delayed input signals: 9 controller and pre-filter resulting from possible combinations of {0, 1, 2} suboptimal rise time steps;
example 3: more complex example, according reference in journal "Automatisierungstechnik". DEMO_EEZs

DEMO_GUETE Application example for guete.m: calculation of quality criteria based on a single step responses of a closed control loop; computation each with undelayed and delayed input signal, latter with pre-filter; determination of quality criteria's of two set-point and one disturbance step responses. DEMO_GUETEs

DEMO_KASKADE1 Application of Kaskade.m to three cascaded control loops; calculation of 4 controller variants regarding undelayed input signals:
BOD/BOD/BOD; FTS/FTS/FTS; FTS/BOD/BOD; FTS/FTS/BOD
"BOD" - Discontinuous Magnitude Optimum; "FTS" - Finite Time Settling
no consideration of controller output limitations: for practical use see method: "limitation backward calculation" DEMO_KASKADEs

DEMO_KASKADE2
DEMO_KASKADE2a Application of Kaskade.m to three cascaded control loops; calculation of 4 controller variants regarding undelayed input signals:
BOD/BOD/BOD; FTS/FTS/FTS resp. FTS/FTS; FTS/BOD/BOD; FTS/FTS/BOD;
"BOD" - Discontinuous Magnitude Optimum; "FTS" - Finite Time Settling;
>>>DIFFERENCE to DEMO_KASKADE1: partially mean value measuring<<<
>>>DEMO_KASKADE2a - one example features two control loops only:<<< DEMO_KASKADEs
DEMO_KASKADE2s

DEMO_KASKADE3
DEMO_KASKADE3b Application of Kaskade.m to three cascaded control loops; calculation of 4 controller variants regarding undelayed input signals:
BOD/BOD/BOD; FTS/FTS resp. FTS/FTS/BOD; FTS/FTS/FTS; FTS/BOD/BOD;
"BOD" - Discontinuous Magnitude Optimum; "FTS" - Finite Time Settling;
>>>DIFFERENCE to DEMO_KASKADE1: delayed input signals for middle control loop<<<
>>>DEMO_KASKADE3: one example features two control loops only<<< DEMO_KASKADEs

DEMO_SIMU Use of function simu.m for a fast check of discontinuous controller parameters without Simulink;
z-transform and controller parameter computation for one plant as well as subsequent simulation and quality criteria computations regarding undelayed and delayed step responses. DEMO_SIMUs

regber1 Controller parameter computation - comparison of different variants 1:
use of two time-delay elements as plant, z-transform of the plant; computation of 4 controllers: applying Magnitude Optimum, quasi-continuous design, Digital Magnitude Optimum (BOD) and Finite Time Settling (EEZ); simulation of plant and control loop step responses;
user inputs may vary BOD and EEZ controller parameters; use of numerator and denominator polynomial representations of plants and controllers. regber1s

regber2 Controller parameter computation - comparison of different variants 2:
use of two time-delay elements as plant, z-transform of the plant; computation of 7 controllers: applying Digital Magnitude Optimum (PI-type and PID-type controllers each with and without pole compensation) and Finite Time Settling (minimum form, one and two suboptimal rise time steps); simulation of plant and control loop step responses;
controller computation is based on control vectors and therefore may not be varied; use of LTI representations of plants and controllers. regber2s

LL_ExampleZanasi Lead Lag controller of 1. and 2. order based on BOD, i.e. about 63� phase margin - applying bod_gen;
Simulation of an example by Zanasi LLC_ExampleZanasis

LL_ExampleYeung Lead Lag controller of 1. and 2. order based on BOD, i.e. about 63� phase margin - applying bod_gen;
Simulation of an example by Yeung LLC_ExampleYeungs
bod_prefi_opt_check01_par Application of bod_prefi_opt.m to example: PID controller and plant with two first-order time-delay elements;
...RA... test with reduced accuracy bod_prefi_opt_check01_sim
bod_prefi_opt_check01_RA_sim

bod_prefi_opt_check02_par Application of bod_prefi_opt.m to example: PI, PID, and PID2 controller and non-minimum-phase (2 zeros) plant with 5th order time-delay element and dead time; ...RA... test with reduced accuracy bod_prefi_opt_check02_sim
bod_prefi_opt_check02_RA_sim

bod_prefi_opt_check03_par Application of bod_prefi_opt.m to example: PI, PID, and PID2 controller and plant with four first-order time-delay elements with similar time constants; ...RA... test with reduced accuracy bod_prefi_opt_check03_sim
bod_prefi_opt_check03_RA_sim

bod_prefi_opt_check04_par Application of bod_prefi_opt.m to example: PID controller and non-minimum-phase (1 zero) plant with one second-order time-delay element and one fourth-order time-delay element; test with reduced accuracy in parallel bod_prefi_opt_check04_sim

bod_prefi_opt_check05_par Application of bod_prefi_opt.m to example: PID2 controller and plant with five first-order time-delay elements; test with reduced accuracy in parallel bod_prefi_opt_check05_sim

File name	Function	Simulation
demo_1	Demo for use of trans.m and bod.m - see also DEMO_BOD (LTI model definition), the demo contains two menu-driven examples; example 1: undelayed input signals without pole compensation; example 2: delayed input signals including pre-filter	demo_1s
demo_2	Demo for use of zust_au.m / zust_aug.m - application of BOD for state control structures; example: two mass system with reference in journal "Automatisierungstechnik"; application of Digital Magnitude Optimum results in three controller parameters.	demo_2s
DEMO_BOD	Application of function bod.m (Digital Magnitude Optimum) and optionally input / output of LTI models and / or optionally menu-driven or vector control; example 1: two time-delay elements controller variants: 5 regarding undelayed, 3 regarding delayed input signals; example 2: demo applying a damped controller derivative part	DEMO_BODs
demo_BOD_rst	Application of function bod_rst.m, i.e. application of the Digital Magnitude Optimum (BOD) to a RST control structure by Landau/Zito (see bod_rst.m), plant features a large dead time.	demo_BOD_rst_s
DEMO_EEZ	Application of function eez.m (Finite Time Settling) and optionally input / output of LTI models and / or optionally menu-driven or vector control; example 1: two time-delay elements controller variants for undelayed input signals: 4 regarding complete pole compensation, 3 regarding ramp input, 2 regarding partial pole compensation; example 2: one time-delay element and mean value measurement controller variants for delayed input signals: 9 controller and pre-filter resulting from possible combinations of {0, 1, 2} suboptimal rise time steps; example 3: more complex example, according reference in journal "Automatisierungstechnik".	DEMO_EEZs
DEMO_GUETE	Application example for guete.m: calculation of quality criteria based on a single step responses of a closed control loop; computation each with undelayed and delayed input signal, latter with pre-filter; determination of quality criteria's of two set-point and one disturbance step responses.	DEMO_GUETEs
DEMO_KASKADE1	Application of Kaskade.m to three cascaded control loops; calculation of 4 controller variants regarding undelayed input signals: BOD/BOD/BOD; FTS/FTS/FTS; FTS/BOD/BOD; FTS/FTS/BOD "BOD" - Discontinuous Magnitude Optimum; "FTS" - Finite Time Settling no consideration of controller output limitations: for practical use see method: "limitation backward calculation"	DEMO_KASKADEs
DEMO_KASKADE2 DEMO_KASKADE2a	Application of Kaskade.m to three cascaded control loops; calculation of 4 controller variants regarding undelayed input signals: BOD/BOD/BOD; FTS/FTS/FTS resp. FTS/FTS; FTS/BOD/BOD; FTS/FTS/BOD; "BOD" - Discontinuous Magnitude Optimum; "FTS" - Finite Time Settling; >>>DIFFERENCE to DEMO_KASKADE1: partially mean value measuring<<< >>>DEMO_KASKADE2a - one example features two control loops only:<<<	DEMO_KASKADEs DEMO_KASKADE2s
DEMO_KASKADE3 DEMO_KASKADE3b	Application of Kaskade.m to three cascaded control loops; calculation of 4 controller variants regarding undelayed input signals: BOD/BOD/BOD; FTS/FTS resp. FTS/FTS/BOD; FTS/FTS/FTS; FTS/BOD/BOD; "BOD" - Discontinuous Magnitude Optimum; "FTS" - Finite Time Settling; >>>DIFFERENCE to DEMO_KASKADE1: delayed input signals for middle control loop<<< >>>DEMO_KASKADE3: one example features two control loops only<<<	DEMO_KASKADEs
DEMO_SIMU	Use of function simu.m for a fast check of discontinuous controller parameters without Simulink; z-transform and controller parameter computation for one plant as well as subsequent simulation and quality criteria computations regarding undelayed and delayed step responses.	DEMO_SIMUs
regber1	Controller parameter computation - comparison of different variants 1: use of two time-delay elements as plant, z-transform of the plant; computation of 4 controllers: applying Magnitude Optimum, quasi-continuous design, Digital Magnitude Optimum (BOD) and Finite Time Settling (EEZ); simulation of plant and control loop step responses; user inputs may vary BOD and EEZ controller parameters; use of numerator and denominator polynomial representations of plants and controllers.	regber1s
regber2	Controller parameter computation - comparison of different variants 2: use of two time-delay elements as plant, z-transform of the plant; computation of 7 controllers: applying Digital Magnitude Optimum (PI-type and PID-type controllers each with and without pole compensation) and Finite Time Settling (minimum form, one and two suboptimal rise time steps); simulation of plant and control loop step responses; controller computation is based on control vectors and therefore may not be varied; use of LTI representations of plants and controllers.	regber2s
LL_ExampleZanasi	Lead Lag controller of 1. and 2. order based on BOD, i.e. about 63� phase margin - applying bod_gen; Simulation of an example by Zanasi	LLC_ExampleZanasis
LL_ExampleYeung	Lead Lag controller of 1. and 2. order based on BOD, i.e. about 63� phase margin - applying bod_gen; Simulation of an example by Yeung	LLC_ExampleYeungs
bod_prefi_opt_check01_par	Application of bod_prefi_opt.m to example: PID controller and plant with two first-order time-delay elements; ...RA... test with reduced accuracy	bod_prefi_opt_check01_sim bod_prefi_opt_check01_RA_sim
bod_prefi_opt_check02_par	Application of bod_prefi_opt.m to example: PI, PID, and PID2 controller and non-minimum-phase (2 zeros) plant with 5th order time-delay element and dead time; ...RA... test with reduced accuracy	bod_prefi_opt_check02_sim bod_prefi_opt_check02_RA_sim
bod_prefi_opt_check03_par	Application of bod_prefi_opt.m to example: PI, PID, and PID2 controller and plant with four first-order time-delay elements with similar time constants; ...RA... test with reduced accuracy	bod_prefi_opt_check03_sim bod_prefi_opt_check03_RA_sim
bod_prefi_opt_check04_par	Application of bod_prefi_opt.m to example: PID controller and non-minimum-phase (1 zero) plant with one second-order time-delay element and one fourth-order time-delay element; test with reduced accuracy in parallel	bod_prefi_opt_check04_sim
bod_prefi_opt_check05_par	Application of bod_prefi_opt.m to example: PID2 controller and plant with five first-order time-delay elements; test with reduced accuracy in parallel	bod_prefi_opt_check05_sim

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