In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives.
| Published in | Automation, Control and Intelligent Systems (Volume 12, Issue 1) |
| DOI | 10.11648/j.acis.20241201.11 |
| Page(s) | 1-14 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Pi-State Controller, Actuator Saturation, Anti-Windup Methods, Continuous-Time and Discrete-Time Controllers, Controlled System with Dead Time Elements
2.1. State Equations of the Controlled System
,(1)
denotes the 
-dimensional state vector, 
the 
-dimensional manipulated variable vector, 
the 
-dimensional dynamics matrix and 
the (
)-di mensional control input matrix. Disturbance variables are not considered without any generality restriction. Eq. (1) is supplemented by the output equation
(2)
-dimensional vector 
of the control quantities and the (
)-dimensional output matrix 
. The possibility of the manipulated variables affecting directly the control quantities is disregarded.
and the reference variables in the vector 
, then the vectorial differential equation of the controller integrators is as follows, provided they operate continuously in time,
.(3)
(4)
and 
(
) indicate the sampling time instants 
respectively 
of the state variables and the time instants at which the manipulated variables take effect. 
is the (
) -dimensional transition matrix for which
(5)
is the discrete-time control input matrix with 
.(6)
is the sampling time. The output equation is in the discrete-time case
.(7)
.(8)2.2. Calculating the Corrected Reference Variables
. To determine the value of in the limiting case, the control law is specified firstly for the unlimited case and then again for the case of an active manipulated variable limitation. If the matrix of the feedback coefficients of the system state variables to the manipulated variable vector is denoted by 
, the matrix of the feedback coefficients of the output variables of the controller integral-action components to the manipulated variable vector by 
and the matrix of the amplification factors for the reference variables, the so-called pre-filter matrix, by 
, then the control law in the unlimited case is as follows
.(9) in brackets in Eq. (9) only applies to the discrete-time case. requested by the controller, only a manipulated variable vector modified by the saturation can act on the system. If this is designated as 
, the control law
(10)
is used instead of 
and the corrected reference variable vector 
is used instead of 
. The two relationships (9) and (10) can now be interpreted in such a way that they apply at the same time. Eq. (9) generates the manipulated variable vector requested by the controller, while Eq. (10) describes with which corrected reference variable vector the realizable manipulated variable vector can be generated. If both equations are subtracted from each other and the resulting difference is solved for , the result is as follows
.(11) must be modified in order to obtain a realizable reference variable vector. The corrected reference variables are then fed to the setpoint inputs of the controller integrators. This means that instead of Eq. (3), the following applies for continuous-time control
,(12)
(13)
for the already known P-state controller, the following results for continuous-time controllers
,(14)
,(15)
,(16)
,(17)
,(18)
.(19)
describes a diagonal matrix in which the main diagonal contains those control eigenvalues that have been added to the eigenvalues which result from the controlled system without controller integrators. 
is the corresponding diagonal matrix for discrete-time systems. It should be emphasized in both cases that the control eigenvalues generated by of the non-actuator-saturated P-state-controlled system are not changed by applying Eqs. (9) and (14) to (16) or (17) to (19). It should also be noted that Eqs. (14) to (16) or (17) to (19) lead to control eigenvalues that cannot be controlled via (see section 4). Finally, it should be noted that the calculation rules (16) and (19) for the pre-filter matrix are the same as the corresponding calculation rules for pure P-state controllers.
.(20) instead of , to form an overall system, the following results
(21)
,(22)
,(23)
.(24)
(25) instead of , to form an overall system, taking into account Eq. (22), gives the result
(26)
,(27)
.(28)
and the transition matrix 
are lower block triangular matrices. Their eigenvalues are identical in their entirety with the eigenvalues of the matrix blocks on the main block diagonal. The eigenvalues of are therefore composed of the eigenvalues of and the control eigenvalues on the main diagonal of caused by the controller integrators. For discrete-time systems, the eigenvalues of the transition matrix are composed of the eigenvalues of and the elements on the main diagonal of , i.e. the control eigenvalues caused by the discrete-time controller integrators. and can be assumed to be stable, the dynamic matrix and the transition matrix each describe a stable system, provided the associated controlled system is stable. This is remarkable, especially as the controller integrators included in the model – considered on their own – show critically stable behavior. The reference variable correction according to Eq. (11), in conjunction with the controller equations (14) to (16) respectively (17) to (19), has thus succeeded in forming a stable system from a critically stable system – assuming a stable controlled system but an unknown controller matrix . Exactly this is extremely advantageous for the applicability of Lyapunov's direct method in stability analysis and controller synthesis for linear systems with manipulated variable limits (see section 4).
is mapped to the state vector 
for continuous-time control using the transformation rules
(29)
.(30)
by the right-hand side of Eq. (21) and finally replacing by Eq. (29) solved for then leads to the transformed differential state equation
(31)
(32)
.(33)
and 
the Eqs. (14) to (16) are taken into account, then after a few reforming steps the block diagonal matrix
(34) the result
(35) transformed state variables cannot be controlled from , neither directly because of the zero matrix in , nor indirectly via the upper 
state variables because of the zero matrix in the lower block row of . This means that the control eigenvalues contained in cannot be controlled via . As these eigenvalues can be considered to be stable, the subsystem described by the lower block row in and is both stable and not controllable and therefore does not need to be considered further in the stability investigations.
,(36)
(37)
(38)
,(39)
.(40)
, which has the eigenvalues contained in that can be assumed as stable. Due to its stability and non-control lability, this subsystem can also be disregarded in the further stability investigations. as the input variable. It is therefore sufficient to find a stabilizing control law for the controlled system using a P-state controller with the feedback matrix . The stability of the resulting PI-state control by means of Eqs. (14) to (16) respectively (17) to (19) is then automatically ensured on the basis of the above explanations. In order to achieve a comparable statement for discrete-time systems, a complex modal transformation of the extended controlled system was carried out in 
.(41)
denotes the deviation of the state vector 
from its stationary position, which is denoted below by 
. The following therefore applies
.(42))-dimensional matrix 
contained in Eq. (41) must be a positive definite matrix yet to be determined. This means that 
holds for 
and 
only for 
. If it is now possible to ensure that 
constantly decreases for and approaches the steady state, i.e. 
, for 
then the stability of the system under consideration is proven 
decreases with a continuous-time system description, is derived with respect to time and then the sign of 
is examined. This results in
.(43)
itself is obtained from the controlled system state differential equation (1) with instead of . Here, is also divided into a stationary component 
and the deviation
(44)
, the following follows from the controlled system state differential equation
.
must be the zero vector in the steady state and according to Eq. (1) is identical with 
in the steady state, 
holds. Overall, this gives the state differential equation
(45)
.(46) to decrease for 
, must be negative. To ensure this even with saturated manipulated variables, the expression 
is considered separately from the expression 
. Because the former expression depends quadratically on 
, it is generally to be expected that reaches values which, due to the limitation of 
, lead to such a large amount of that this term determines the sign of . The matrix term 
is therefore assigned a positive definite or at most a positive semi-definite matrix 
respectively 
by
(47)
.(48)
respectively 
is positive definite, the matrix is also positive definite if the controlled system is stable, i.e. if only has eigenvalues with a negative real part can be any -column matrix without violating the positive semi-definiteness of . Only if is to be positive definite, the number of rows of must be at least as large as the number of columns of and, in addition, 
must apply in Eq. (47) or (48), then it can be enforced that the right-hand side of Eq. (46) does not become positive, provided the controlled system is asymptotically stable or critically stable. To do this, 
is set in the way
(49)
is a positive definite (
) -dimensional diagonal matrix with otherwise arbitrary diagonal elements and 
is an arbitrary matrix of suitable dimension, provided that Eq. (47) is used as the basis for . When using Eq. (48), 
must be selected. The chosen approach based on Eq. (47) is oriented towards the so-called Kalman-Yakubovich-Popov equations . However, it is often sufficient to work with and the somewhat more simply structured formulas, using either Eq. (47) or Eq. (48) as a basis.
.(50)
.(51) is now chosen so that 
is positive definite – for which the diagonal elements of only have to be chosen sufficiently large – then 
is a positive semi-definite matrix term 
. Furthermore, 
is also at least positive semi-definite. This ensures that does not become positive. With 
, quadratic and positive definite and positive definite in any case, is then always negative for 
and zero for 
itself. is negative definite in this case, while is positive definite. Since these properties apply to the entire state space, the stability condition of Lyapunov's direct method is fulfilled and thus the system under consideration is stable results in
.(52) to be smaller than specified by the controller. However, this can be modeled according to Eq. (49) by a corresponding increase in the relevant diagonal element of . However, since the diagonal elements of can be chosen to be arbitrarily large as long as they are only positive and fulfill the condition 
, this has no negative influence on the definiteness of 
and thus on the stability conclusion.
and must be positive definite if is positive definite and is negative definite. This is possible if the controlled system is stable. However, if the controller integrators had been included in the system model without splitting off the uncontrollable, stable subsystem as described above, then the simultaneous specification of and as positive definite matrices would not have been possible due to the then critically stable system.
(53)
decreases at each sampling step as long as 
is not a zero vector. The aim is therefore to ensure that 
is negative definite in the entire state space. For this purpose, 
and are replaced in Eq. (53) according to the right-hand side of Eq. (41), from which
(54)
(55)
.(56)
(57)
is negative definite or at least negative semi-definite. This is ensured by
(58)
(59) as a matrix with at least as many rows as columns at maximum rank when applying Eq. (58) and aiming for positive definiteness of 
as in the continuous-time system description. If only has stable eigenvalues, then the positive definiteness of follows from the positive definiteness of respectively 
according to Eq. (41). If, on the other hand, has stable and/or critically stable eigenvalues, can be chosen to be positive definite, but respectively can then at best be positive semi-definite (see example from section 6). to be negative (semi-)definite, the sum of all three summands must be negative (semi-)definite in addition to the first summand of Eq. (57), which can now be represented in the form 
respectively 
. To achieve this, starting from the second summand, the control law
(60)
(61) in the case of Eq. (59). Eq. (60), after insertion into Eq. (57) and taking into account Eqs. (58) and (61) yields the result
.(62) is then simply specified as the zero matrix and is replaced by . If is now chosen so that the bracket expression 
is positive definite, then the entire matrix term in the last row of Eq. (62) is positive semi-definite due to 
. Again, this can always be achieved with sufficiently large amounts 
(
) of . Furthermore, because the matrix term in the second last row of Eq. (62) is at least positive semi-definite, cannot become positive. With , quadratic, positive definite specification of and 
, a stable discrete-time controlled system according to Eq. (4) can therefore always be stabilized using the control law (60) via a PI-state controller. This also applies in particular when manipulated variable constraints occur, because when limiting 
(), only the element of needs to be increased in thought until corresponds to the relevant limiting value using Eq. (60)., which was determined on the basis of the state vector 
, is then set to a newly introduced vector 
, using the difference equation
.(63)
is now used instead of as the manipulated variable vector acting on the controlled system, the dead time is taken into account in the model. If and are then combined to form the overall state vector
(64),(65)
(66)
.(67)
(68)
.(69)
(70)
and then for the remaining terms negative definiteness or negative semi-definiteness must be ensured. With the approaches
,(71)
(72)
instead of 
and at the same time 
, this is achieved in the same way as previously described in section 4 for discrete-time systems without dead times in the manipulated variable paths. If you now specify 
as a symmetrical matrix in block matrix notation
,(73) as block diagonal matrix
(74)
in block matrix notation
,(75)
,(76)
,(77)
.(78)
,(79)
(80)
and 
of . The specification of therefore determines . Because is positive definite if respectively with stable transition matrix is positive definite as upper block triangular matrix with stable matrix fulfills this condition, the positive definiteness of no longer needs to be proven separately. If we also evaluate Eq. (61) for according to Eq. (73) instead of , according to Eq. (66) instead of , 
according to Eq. (67) instead of , according to Eq. (74) instead of and according to Eq. (75) instead of , then the following results with 
instead of , taking into account Eq. (79):
.(81) the controller matrix of the system with dead time is associated with the controller matrix of the system without dead time via the relationship
.(82)) is requested for the system with dead time, which produces the same control behavior as the associated dead time-free system, only delayed by one sampling interval. Interestingly, this controller setting also fulfills the requirement for stability in the case of manipulated variable constraints, provided that the dead time-free system (with the controller matrix ) fulfills this requirement. Furthermore, when applying Eq. (82), the pre-filter matrix ensuring stationary accuracy in the command behavior, which is denoted below as 
to distinguish it from from Eq. (19), is identical to , i.e. it holds
.(83) is first written according to Eq. (19) with 
instead of , instead of , instead of and instead of . It reads
.(84)
(85)
(86)
by 
to obtain the unit matrix. If is then multiplied from the left by according to Eq. (69) and from the right by according to Eq. (67), the following results
,
. The renewed inversion and negation according to Eq. (84) finally leads to the statement of Eq. (83).
from Eq. (82) is inserted as the controller matrix 
in Eq. (17). The extended system matrices 
, 
and 
from Eqs. (66), (67) respectively (69) can be used. The pre-filter matrix 
can remain unchanged due to Eq. (83) and the statements in section 3.
respectively the actual value 
of the electric torque are used directly as input and output variables of the subordinate current control loop instead of the corresponding torque-forming current (setpoint) components. The time constant of the closed current control loop is 
. The dead time element comes from the modeling of the computing time of the signal processor used for control. The difference between the electric torque and the load torque 
results in the acceleration torque, which leads to the speed (angular velocity) 
when integrated via the moment of inertia 
. and neglecting the load torque, the discrete-time state equations of the dead-time-free system are approximately as follows 
,
,
,
is understood to be the actual torque value and the state variable 
, which is also the controlled variable 
, is understood to be the actual speed value. In contrast, the output variable of the dead time element from Figure 2 serves as the manipulated variable 
in the dead time-free system. For 
, it holds 
. For the relevant system matrices, this results in the values
,
,
. has an eigenvalue at 
, which is why the controlled system is not asymptotically stable. It can therefore be assumed that respectively will not be a positive definite matrix term. The calculation of 
according to Eq. (58) or (59) shows this immediately. Because with the symmetrical matrix
you get
respectively can at best be positive semi-definite, and only if the secondary diagonal elements are zero. From this follows directly
. and thus use Eq. (48) as a basis, it applies that
.
can be positive semi-definite. For the matrix , it follows under the above conditions
.
and 
. Furthermore, the positive semi-definiteness of must also be fulfilled, which leads in the example to the condition
. In the next step, the P-state controller matrix is calculated by means of Eq. (61). Because was selected, the result is
.
holds and 
is chosen without restricting the generality, then 
and 
, for example, fulfill the above conditions. For and , it follows from this, but for general 
,
,
.
.
assigned to it is added, it follows from Eqs. (17) and (18)
,
.
is specified at time 
with vanishing initial state variable values. The step height was intentionally chosen so large that the manipulated variable of the speed controller, i.e. the torque setpoint , reaches the limit 
at the start of the transient response. In addition to the uncorrected speed setpoint 
, the actual speed value and the unlimited torque setpoint , Figure 3 also shows the corrected speed setpoint 
, the saturated torque setpoint 
and the output variable 
of the speed controller integrator. With regard to the latter, it should be noted that Figure 3 is based on the fact that the control difference is first multiplied by 
and only then integrated and fed to the manipulated variable determination with a positive sign. In addition to the aforementioned value for the weighting factor, Figure 3 also shows curves for other values of the weighting factor in order to demonstrate its influence on the control quality. All diagrams are based on the moment of inertia 
and the sampling time 
. decreases. At 
, however, there appears a clear torque ripple. But, at , the previously mentioned sufficient stability condition is no longer fulfilled. Since this is only a sufficient condition, stable operation cannot be excluded, which is also shown in principle in Figure 3d.| [1] | Hippe, P. Windup in Control. London: Springer; 2010. |
| [2] | Adamy, J. Nonlinear Systems and Controls. Berlin, Heidelberg: Springer Vieweg; 2022. |
| [3] | Tarbouriech, S., Garcia, G., Goes da Silva Jr., J. M., Queinnec, I. Stability and Stabilization of Linear Systems with Saturating Actuators. London: Springer; 2011. |
| [4] | Vidyasagar, M. Nonlinear Systems Analysis. 2nd edition. Philadelphia: SIAM; 2002 (unabridged republication). |
| [5] | Lerch, S., Dehnert, R., Damaszek, M., Tibken, B. Anti Windup PID Control of Discrete Systems Subject to Actuator Magnitude and Rate Saturation: An Iterative LMI Approach. Proceedings of the 25th International Conference on System Theory, Control and Computing (ICSTCC). Iasi, Romania, 2021, pp. 413–418. |
| [6] | Chen, Y., Yang, M., Liu, K., Long, J., Xu, D., Blaabjerg, F. Reversed Structure Based PI-Lead Controller Antiwindup Design and Self-Commissioning Strategy for Servo Drive Systems. IEEE Transactions on Industrial Electronics. 2022, 69 (7), pp. 6586–6599. |
| [7] | Wang, K., Wu, F., Sun, X.-M. Switchung Anti-windup Control for Aircraft Engines. IEEE Transactions on Industrial Electronics. 2023, 70 (2), pp. 1830–1840. |
| [8] |
Nuss, U. Stabilitätsverhalten von zweistufig entworfenen zeitdiskreten PI-Zustandsreglern bei Stellgrößenbegrenzungen [Stability properties of two-stage designed discrete-time PI state controllers considering the limitation of input variables]. at – Automatisierungstechnik. 2017, 65 (10), pp. 705 – 717.
https://doi.org/10.1515/auto-2016-0136 (in German) |
| [9] | Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German) |
| [10] | Aström, K., Wittenmark, B. Computer-Controlled Systems. 2nd edition. Englewood Cliffs: Prentice-Hall; 1990 |
| [11] | Quang, N.P., Dittrich, J.-A. Vector Control of Three-Phase AC Machines. Berlin, Heidelberg: Springer; 2008. |
| [12] | March, P., Turner, C.. Anti-Windup Compensator Designs for Nonsalient Permanent-Magnet Synchronous Motor Speed Regulators. IEEE Transactions on Industry Applications. 2009, 45 (5), pp. 1598–1609. |
| [13] |
Nuss, U. Ein einfacher Zustandsreglerentwurf im Zuge der Erweiterung der Systemstruktur um Reglerintegratoren und Rechentotzeiten [A simple state space controller design in the context of expanding the control structure by integrators and calculation dead time]. at – Automatisierungstechnik. 2016, 64 (1), pp. 29–40.
https://doi.org/10.1515/auto-2015-0058 (in German) |
| [14] | Nuss, U. Regelungstechnik einfach verpackt [control theory simply presented]. forschung im fokus. Offenburg: University of Applied Science Offenburg. 2016, pp. 19 – 22. (in German) |
| [15] | Grabmair, G., Gahleitner, R. PI-Zustandsregler – eine methodische Neubetrachtung [PI statespace control revisited]. at – Automatisierungstechnik. 2019, 67 (9), pp. 727–738. |
| [16] | Zurmühl, R., Falk, S. Matrizen und ihre Anwendungen [Matrices and their applications]. 5th edition. Berlin, Heidelberg, New York, Tokyo: Springer; 1984. (in German) |
| [17] | Nuss, U. Hochdynamische Regelung elektrischer Antriebe [Highly dynamic control of electrical drives]. 2nd edition. Berlin, Offenbach: VDE; 2017. (in German) |
APA Style
Nuss, U. (2024). New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation. Automation, Control and Intelligent Systems, 12(1), 1-14. https://doi.org/10.11648/j.acis.20241201.11
ACS Style
Nuss, U. New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation. Autom. Control Intell. Syst. 2024, 12(1), 1-14. doi: 10.11648/j.acis.20241201.11
AMA Style
Nuss U. New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation. Autom Control Intell Syst. 2024;12(1):1-14. doi: 10.11648/j.acis.20241201.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.acis.20241201.11,
author = {Uwe Nuss},
title = {New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation
},
journal = {Automation, Control and Intelligent Systems},
volume = {12},
number = {1},
pages = {1-14},
doi = {10.11648/j.acis.20241201.11},
url = {https://doi.org/10.11648/j.acis.20241201.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acis.20241201.11},
abstract = {In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives.
},
year = {2024}
}
TY - JOUR T1 - New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation AU - Uwe Nuss Y1 - 2024/04/12 PY - 2024 N1 - https://doi.org/10.11648/j.acis.20241201.11 DO - 10.11648/j.acis.20241201.11 T2 - Automation, Control and Intelligent Systems JF - Automation, Control and Intelligent Systems JO - Automation, Control and Intelligent Systems SP - 1 EP - 14 PB - Science Publishing Group SN - 2328-5591 UR - https://doi.org/10.11648/j.acis.20241201.11 AB - In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives. VL - 12 IS - 1 ER -
Department of Electrical Engineering, Medical Engineering and Computer Science, Offenburg University of Applied Science, Offenburg, Germany
Biography: Uwe Nuss teaches and researches since 2003 at Offenburg University of Applied Sciences. He is a lecturer in the fields of electrical drive technology, power electronics and control engineering at the Department of Electrical Engineering, Medical Engineering and Computer Science. He received his doctoral degree from Karlsruhe University (TH), Germany, in 1989, where he joined the electrotechnical institute and researched about inverter-based reactive power compensation. In 1994, he was habilitated at Karlsruhe University (TH), where he was awarded a teaching qualification on the subject of control of electrical drives. From 1994 to 2003, Prof. Nuss worked for a medium-sized company, where he was responsible for software development for the control of three-phase drives.
Research Fields: Uwe Nuss: Control of Electrical Drives, State Control Theory, Continuous-Time Systems, Discrete-Time Systems, Power Electronics.
Figure 1. Block diagram of discrete-time PI-state control with reference variable correction in the case of manipulated variable saturation
Figure 2. Continuous-time block diagram of the exemplary controlled system.
Figure 3. Transient response of the relevant variables of the speed control loop when a speed setpoint step is specified with a limited torque setpoint; a) r=10, b) r=5, c) r=0,5, d) r=0,05.
Information