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Robust Controller Tuning for Integral Dead Time Systems
Abstract
By using newly developed method of the performance portrait (PP) for achieving as fast as possible transients in the closed loops with the Integrator Plus Dead Time (IPDT) and Second Order Integrator Plus Dead Time (SOIPDT) plants this paper discusses optimality of the traditional controller tuning corresponding to the Multiple Real Dominant Pole (MRDP) and shows potential of the new method for arbitrarily shaping the close loop properties of time delayed systems.
Supplementary resource (1)
... In addition, the paper develops another new dimension of an intelligent approach to the design of optimal and robust controllers based on the performance portrait method (PPM). After several initial steps [36,37], the PPM was fully developed, especially in the works [38,39]. It could be simply stated that this contribution represents an extension of the work dealing with optimal and robust control design of systems approximated by integrator plus dead-time (IPDT) models using 2DoF PI controllers [38] to the case of controlling DIPDT models by 2DoF PID controllers. ...
... Especially in the field of vehicle motion control, it is certainly not possible to optimize their movement by minimizing the value of TV, but it must be adjusted for the values of useful interventions needed to accelerate and brake the vehicle. Thus, for a double integrator [37,38], the optimal shape of the control signal u(t), t ∈ [0 − , ∞) associated with an ideal monotonic setpoint changes and 1P disturbance responses of the output (given by the inverse plant dynamics) will be specified by two extreme points u m1 , u m2 occurring at times t 1 , t 2 ∈ (0, ∞), lying between the initial and final values u 0 and u ∞ and satisfying the condition: ...
... The performance portrait method (PPM) [28,[36][37][38] assumes verification of loop properties at all possible relevant controller settings. For the DIPDT models, such meaningful intervals of optimal controller parameters were analytically investigated by selecting different groupings of real dominant poles in [28]. ...
The article deals with a computer-supported design of optimal and robust proportional-integral-derivative controllers with two degrees of freedom (2DoF PID) for a double integrator plus dead-time (DIPDT) process model. The particular design steps are discussed in terms of intelligent use of all available information extracted from a database of control tracking and disturbance rejection step responses, assessed by means of speed and shape-related performance measures of the process input and output signals, and denoted as a performance portrait (PP). In the first step, the performance portrait method (PPM) is used as a verifier, for whether the pilot analytical design of the parallel 2DoF PID controller did not omit practically interesting settings and shows that the optimality analysis can easily be extended to the series 2DoF PID controller. This is important as an explicit observer of equivalent input disturbances based on steady-state input values of ultra-local DIPDT models, while the parallel PID controller, allowing faster transient responses, needs an additional low-pass filter when reconstructed equivalent disturbances are required. Next, the design efficiency and conciseness in analyzing the effects of different loop parameters on changing the optimal processes are illustrated by an iterative use of PPM, enabled by the visualization of the dependence between the closed-loop performance and the shapes of the control signals. The main contributions of the paper are the introduction of PPM as an intelligent method for controller tuning that mimics an expert with sufficient experience to select the most appropriate solution based on a database of known solutions. In doing so, the analysis in this paper reveals new, previously undiscovered dimensions of PID control design.
... is used, where w denotes a piecewise constant reference setpoint, y the process output and e the control error. The advantage over using an alternative option called "settling time" [40,41] is that the I AE does not have to define the width of the zone around the desired state after which the transients are considered to be terminated. Since I AE minimisation of optimal responses typically leads to overshoot of the output, the shape-related constraints are combined based on the concept of monotonicity. ...
... In the control of a double integrator [40,41], the shape of the ideal plant input (controller output) of a piecewise continuous control signal u(t), t ∈ [0 − , ∞), which is required to obtain ideal monotonic setpoint steps and 1P disturbance steps of the output, is given by the inverse plant dynamics. It can be shown that it specified by two extreme points u m1 , u m2 , which occur at times t 1 , t 2 ∈ (0, ∞), lie between the initial and final values u 0 and u ∞ and satisfy the condition ...
... In the following, we compare the results of the analytical tuning of the controllers based on sets of real dominant poles with the performance portrait method (PPM). PPM [40,41] assumes verification of loop properties at all possible relevant controller settings, which should be expressed by a grid of points whose spacing is chosen fine enough to capture all loop properties of interest. We simulated the loop (or control the process in real time) over this grid and store the obtained transient properties in a matrix of performance measures. ...
The paper discusses the proportional-integral-derivative (PID) controller from the viewpoint of (a) the analytical tuning of the PID controller for the double integrator plus dead time (DIPDT) model and (b) the numerical tuning using the performance portrait method (PPM). In the first case, the already published tuning with multiple real dominant pole, extended by integrated tuning procedures, which incorporate the inevitable low-pass filters by delay equivalences, is elaborated for modified sets of real poles. By considering several such modified sets of real poles, resulting in several new sets of controller parameters, the design can be better adapted to the requirements of the control tasks solved and to the limitations of the existing control loop hardware. In a noisy and uncertain environment, the balance between speed of setpoint and disturbance responses and acceptable excessive controller effort can thus be improved. The effectiveness of the analytical design can be evaluated using the numerical performance portrait method (PPM). For an already generated performance portrait (PP), it can offer a broad spectrum of controller settings that satisfy various design constraints. However, the results of the analytical design are still important as they facilitate the initial steps in the elaboration of the PPM and in explaining the nature of PID control. The developed controller tuning are compared using a new interpretation of PID controller as an extension of the stabilising PD controller by disturbance observer (DOB). The input disturbances reconstructed by DOB by evaluating the controller output of an integral process model in steady-state, can be estimated by a low-pass filter with a sufficiently long (integral) time constant. All analysed results are in full agreement with the proposed DOB interpretation, which furthermore contributes significantly to the explanation of the problems related to the optimal design of PID controllers.
... However, the ever-increasing demands placed on the performance of PID control lead to the fact that when evaluating its dynamics of transients, it will again be necessary to pay attention to control responses obtaining possibly a higher number of pulses. However, due to the requirement of smoothness of processes, these pulses may not be obvious at first glance and thus new performance measures may be required [28]. We will show this on the example of the system 1/(1 + s) 4 (33) (process E4 from [4]), from which we require monotonic closed loop setpoint step response with the time constant shortened from T 4 = 1 to T c4 = 0.18. ...
... As shown in [28,30], in case of the single integrator the controller output (plant input) corresponding to the setpoint and input disturbance steps have to ideally consist of two monotonic intervals forming an 1P shape (as y (1) (t) in Figure 2). Then, similarly as above, deviations of the plant controller output u(t) from an ideal 1P step response should be constrained in terms of TV 1 (u) measures. ...
... Thereby, controllers j R j (s) coincide with the solutions defined for j ∈ [1, 4] by the Equations (6), (7), (21), (28) and (32). Proof. ...
The article extends a model-based controller design to higher-order systems, focusing on the speed and shapes of the closed loop responses, including the noise attenuation. It shows that, to obtain simple but reliable results, it is necessary to pay attention to the initial process identification and modelling and also to modify the target closed-loop transfer functions, which must remain causal. To attenuate high initial control signal peaks, appropriate pre-filters are introduced. In order to work with as few parameters as possible, all higher-order transfer functions (process models, target closed loops, pre-filters and noise-attenuation filters) are selected in the form of binomial filters with multiple time constants. Consequently, the so-called “half-rule”, used to reduce too complex process transfer functions, has been modified accordingly. Because derived controllers can lead to different transient dynamics depending on the context of use, the article recalls the need to introduce dynamic classes of control to clarify the mission of individual types of controllers. Consequently, also the performance evaluation using the total variation (TV) criterion had to be refined. Indeed, in its original version, TV is not suitable to distinguish between reasonable and excessive control effort due to improper tuning and noise. The modified TVs allow evaluating higher order systems with multiple changes in direction of their control signal increase without contributing to the excessive control increments. The advantages of the proposed modifications, compared to the traditional approaches, are made clear through simulation examples.
... One of the main drawbacks of numerous optimal control methods is that they narrow the problem to evaluating the speed of transients. Among the many different alternatives [26,27], this can be quantified as the settling time, integral of squared error (ISE), or more frequently [28], using the integral of the absolute error (IAE). ...
... However, despite this clear historical evidence, the use of shape-based performance measures can be considered neglected. The works [26,27] in this direction represent one of the first attempts to unify all previous usages. ...
... The drawback of working with a performance measure TV is that it presents unstructured information: the "useful" control interventions required to achieve the control objective are mixed with excessive interventions resulting from model or measurement inaccuracies. Therefore, in the work [26], it was proposed to use modified criteria based on deviations of a given transient from monotonicity instead of the performance measure TV. Monotonicity is not only one of the fundamental mathematical concepts, but also closely related to several important physical properties and the shape of control transients. ...
The paper investigates and explains a new simple analytical tuning of proportional-integrative-derivative (PID) controllers. In combination with nth order series binomial low-pass filters, they are to be applied to the double-integrator-plus-dead-time (DIPDT) plant models. With respect to the use of derivatives, it should be understood that the design of appropriate filters is not only an implementation problem. Rather, it is also critical for the resulting performance, robustness and noise attenuation. To simplify controller commissioning, integrated tuning procedures (ITPs) based on three different concepts of filter delay equivalences are presented. For simultaneous determination of controller + filter parameters, the design uses the multiple real dominant poles method. The excellent control loop performance in a noisy environment and the specific advantages and disadvantages of the resulting equivalences are discussed. The results show that none of them is globally optimal. Each of them is advantageous only for certain noise levels and the desired degree of their filtering.
... The paper shows tuning of the Filtered Smith Predictor for higher performance requirements expressed in terms of the monotonicity at the plant and the controller output by the Performance Portrait method (Huba, 2010;2011a). ...
... The approach to designing FSPs for the FOPDT plants introduced in Normey- Camacho (2009), or Normey-Rico et al. (2009) considers compensation of a disturbance acting at the plant output by correction of the loop reference value, whereby the disturbance is reconstructed by using parallel plant model. (Huba, 2010;2011a) is briefly described to be used in: ...
... For a strictly MO response 1 = α . In this paper for evaluating deviations from strict monotonicity we are going to use a similar and very simple integral measure achieved by modification of the TV criterion that can easily be modified also for evaluating more complex shapes consisting of several monotonic intervals (Huba, 2010;. Since it is frequently used for evaluating the controller output ( ) t u , its definition is shown for this variable. ...
The paper shows tuning of the Filtered Smith Predictor for higher performance requirements expressed in terms of the monotonicity at the plant and the controller output by the Performance Portrait method (Huba, 2010; 2011a).
... Although the transients corresponding to limit values of the tuning parameters show strong similarities in behaviour of the PI-FSP and P-FSP, still there arise question, if this really holds for all possible working points. The best way to answer such question is to derive Performance Portrait (Huba, 2010;Fig. 6 Comparing P-F2SP and PI-F2SP controllers for the 20% mismatch in the plant gain K. 2011a) showing chosen performance measure over much higher number of operating points. ...
... together with the TV 0 and TV 1 performance indices (Huba, 2010(Huba, , 2011a defined by modification of the Total Variance (Skogestad, 2003) according to ...
... In this context, a more detailed investigation of the role of the disturbance filters in influencing the achievable closed loop robust performance would be desirable that could e.g. be based on the newly developed method of the Performance Portrait Huba (2010Huba ( , 2011a. ...
This paper introduces modifications of the Filtered Smith Predictor for simple linear first order plants with dead time (FOPDT) and constrained input. Constraints may be simply considered due to replacing the PI controller by a two-degree-of-freedom (2DOF) P controller in the primary loop of the classical IMC structure with the dynamical feedforward and parallel plant model. Equivalence of the traditional and the new solutions in the proportional range of control is shown by the setpoint-to-output and disturbance-to-output transfer functions in the nominal case and by the computer simulations for systems with the plant-model mismatch.
... This paper verifies properties of previously proposed simple constrained gain-scheduled PD controllers for the Second Order Integrator Plus Dead Time (SOIPDT) plant (Huba, 2004a,b,c;2005) based on imposing dynamics characterized by complex closed loop pole and compensating control constraints effect by reference signal scheduling. Simultaneously, this paper broadens tuning possibilities of these solutions by the new method for poles specification based on the Performance Portrait (Huba, 2010; 2011a,b). Due to analyzing broad range of different operating points information of potential user about this solutions was reasonably enlarged ...
... In this paper the newly introduced Performance Portrait method (Huba, 2010;Huba, 2011a, b) enables to proceed in solving these problems in two points: a) Tuning guaranteeing faster output-monotonic and input-2P responses than the analytically derived tuning corresponding to the Tripple Real Dominant Pole is introduced; b) Performance of the gain-scheduling approach applied in deriving constrained pole assignment controller for the 2 nd order integrator + dead time is tested for a broad range of the loop parameters (control signal constraints related to initial conditions). ...
... A simple measure introduced by Åström and Hägglund (2005) for evaluating deviations from strict MO denoted as monotonicity index was defined as Besides of this integral measure for characterizing deviations from strict MO it is important to introduce also amplitude deviations measure based on defining an error band specified by the parameters y ε , or u ε around the reference MO signal (Huba, 2010(Huba, , 2011a. In general, − ε MO signal ( ) t f has to fulfil requirements ...
This paper verifies properties of previously proposed simple constrained gain-scheduled PD controllers for the Second Order Integrator Plus Dead Time (SOIPDT) plant (Huba, 2004a,b,c; 2005) based on imposing dynamics characterized by complex closed loop pole and compensating control constraints effect by reference signal scheduling. Simultaneously, this paper broadens tuning possibilities of these solutions by the new method for poles specification based on the Performance Portrait (Huba, 2010; 2011a,b). Due to analyzing broad range of different operating points information of potential user about this solutions was reasonably enlarged.
... The spectrum of possible situations includes loops with a disturbance observer based integral action, loops with a dynamical feedforward control and loops with a reference models applicable also to systems with a long dead time. In the tuning analysis, the traditional analytical method as the parameter space method [5], or the triple real dominant pole method [2] are combined with the numerical performance portrait method [1], [6]. A loop performance is evaluated by newly introduced measures for deviations from monotonic and two-pulse shapes of transients typical for control of plants with dominant 2nd order dynamics. ...
... Thereby, the closed loop performance will be characterized by the setpoint step responses, whereby the plant output y(t) will be required to change (nearly) monotonically (MO). For transients starting and finishing in steady states it means MO output with a shape of a "S-curve" [2], [6], [18]. By repeating arguments of the Theorem 1 in [2], it is possible to get following results: ...
... Deviations from strictly MO shape of a plant output y(t) with an initial value y 0 and a final value y ∞ [2], [6], [18] by ...
The paper deals with an optimal filter tuning for
a filtered PD control (FPD) with a specified tracking properties.
Within a simplified modular design [1]–[4], in a nominal case,
the filters required for a FPD controller implementation and
a noise attenuation may be generalized to several augmented
situations without necessity to repeat analysis of its optimal and
critical tuning. The spectrum of possible situations includes loops
with a disturbance observer based integral action, loops with a
dynamical feedforward control and loops with a reference models
applicable also to systems with a long dead time. In the tuning
analysis, the traditional analytical method as the parameter
space method [5], or the triple real dominant pole method [2]
are combined with the numerical performance portrait method
[1], [6]. A loop performance is evaluated by newly introduced
measures for deviations from monotonic and two-pulse shapes
of transients typical for control of plants with dominant 2nd
order dynamics. The analysis shows that for a broad class of
situations, a simplified design derived for a double integrator
gives acceptable results also for 2nd order integral systems with
one stable/unstable mode.
Keywords—Positional control, noise attenuation, filtered PD
control, disturbance observer, pole assignment control.
... Recently, a new performance portrait method (PPM) has been proposed for both optimal and robust controller tuning, that may be considered as a generalization of this method. Based on carrying out systematic simulation experiments on representative processes that were evaluated by chosen shaperelated performance measures (quarter amplitude damping), today, digital computers may be used for organizing and evaluating experiments, as well as for processing, visualizing, storing and recalling the achieved results for a large number of control loops typical in practice [3], [6]. Different qualitative and quantitative properties may be evaluated and stored in a computer database, to be chosen "on demand" and in different combinations by an engineer carrying out a design requiring particular specifications [4]. ...
... The performance portrait (PP) represents information about the closed loop based on performance specifications comprising of e.g. the output IAE values and tolerable deviations from the ideal shapes at the plant input and output. It may be expressed over a grid of normalized loop parameters corresponding to setpoint and disturbance step responses generated by simulation or real time experiments [3]. After carrying out a series of simulation experiments on the considered process the PP enables ondemand balancing of the trade-off between the required robustness, high speed of the setpoint tracking and good disturbance rejection. ...
... To evaluate the deviation of the plant output y having an initial value y 0 and a final value y ∞ from a MO shape we can use the T V 0 (y) criterion defined in [3] as ...
The performance portrait based optimal controller tuning may be considered as a generalization of the well known method by Ziegler and Nichols [1] that integrated an experimental plant identification with the controller tuning. Up to now, the performance portrait method (PPM) has been presented as a tool for robust and optimal controller tuning [4]-[6]. This paper presents its application as a new experimental identification and controller tuning tool built upon a precomputed closed loop performance portrait. Problems of choosing an appropriate plant model for evaluating experiments carried out under a chosen controller by using a performance portrait are discussed, together with problems of performance portrait quantization and rules for determining an optimal experiment scenario.
... In this paper, application of the 2-parameter plant model in robust design of stable SISO plants is supported by the new computer analysis method based on the Performance Portrait (Huba, 2010). The paper is structured as follows. ...
... For a strictly MO response 1 = α . Another such a measure (Huba, 2010) exhibiting deviations from MO denoted as TV 0 " was proposed as excess of the Total Variance (Skogestad, 2003) ...
... half-period) of the superimposed signal. Number of samples that need to be tested (Huba, 2010) may be decreased due to Theorem 1. ...
This paper shows new possibility of the robust design of simple controllers based on computer analysis of closed loop transient responses achieved over predefined grid of possibly normalized loop parameters by simulation, or by measurement. Evaluation of the collected transients according to chosen performance measures related to monotonicity and time yields the so called Performance Portrait (PP). Based on the PP, controller tuning satisfying desired robust performance may be derived what is illustrated by tuning simple integral (I) controllers for the simplest uncertain 2-parameter plant model. Achieved results that extend the simple tuning proposed by Skogestad (2003) to a broader spectrum of nearly monotonic shapes are compared with the analytical controller tuning based on the double real dominant closed loop pole and then, on an example of a higher order dead-time dominant plant, with the results achieved by the Filtered Smith Predictor. Possible extension to PI control is briefly mentioned.
... Thereby, the closed loop performance will be characterized by the setpoint step responses, whereby the plant output y(t) will be required to change (nearly) monotonically (MO). For transients starting and finishing in steady states it means MO output with a shape of a "S-curve" [8], [9]. By generalizing arguments of the Theorem 1 in [9], it is possible to get following results: ...
... Based on TV, new relative measures for deviations from MO, 1P and 2P shapes may be proposed [8], [9]. Deviations from strictly MO shape of the plant output y(t) with the initial value y 0 and the final value y ∞ may be characterized by ...
... Similarly, for determining optimal parameters of PD controller (9) with filter (14), requirement of a triple real dominant pole (TRDP) p o of characteristic polynomials (17) may be used. In [8] it has been shown that for PD controller tuning the new ; N = n(n + 1) ...
The paper represents first part of two contributions dealing with simplified modular design of filtered PD (FPD) and disturbance observer (DO) based filtered PID control (FPID) illustrated by example of positional servo control. This first part is devoted to analysis of the FPD controller tuned for different types of nonmodelled or filter dynamics. Openness of the approach means that different order filters required for implementation of these controllers and noise attenuation may be introduced without necessity to repeat in the nominal case analysis of the optimal and critical tuning. By simpler means, flexible approach enabling to fit filtering requirements of particular loop is proposed. Achieved performance is evaluated by newly introduced measures for deviations from monotonic and two-pulse shapes of transients typical for control of plants with dominant 2nd order dynamics.
... This work continues in confronting several traditional and some latest approaches to the PI controller design [10][11][12][13] with results offered by the new approach denoted as the performance portrait (PP) method [14][15][16] based on performance specifications comprising: ...
... Based on TV new measures appropriate to measure deviations from MO and 1P shapes of the setpoint and disturbance step responses may be derived [14,15]. For evaluating deviations from a strictly MO plant output y(t) having an initial value y 0 and a final value y ∞ , the TV 0 (y) measure is proposed as the total output variance decreased by the net output change |y ∞ − y 0 | as ...
... TV 0 (y) = 0 just for strictly MO response, else TV 0 (y) > 0. As shown by Theorem 1, in controlling unstable and integral plants the number of significant control pulses cannot decrease below the number of unstable poles [14,23]. To stress the contribution of the superimposed oscillation in transients with 1P dominant shape, it is appropriate to work with the TV 1 criterion defined for u(t) as ...
This paper considers PI controller tuning for the Integral Plus Dead Time (IPDT) plant subject to constraints on tolerable deviations from ideal shapes and guaranteeing minimal combined IAE (Integral of Absolute Error) measure composed of weighted IAE values of the setpoint and disturbance step responses. As the ideal responses at the plant output, monotonic (MO) transients are chosen. This implies one-pulse (1P) responses consisting of two monotonic intervals at the plant input. Optimal nominal tunings for the most important situations regarding the servo/regulation trade-off are determined by the newly developed Matlab/Simulink tool based on the performance portrait (PP) method. Generated results confirm Pareto-like limits of performance achievable under PI control and are used in classifying traditional tuning approaches into the setpoint and disturbance oriented ones. Principles of robust PP based tuning are briefly illustrated and conditions of invariance of the closed loop performance against the dead time uncertainty are formulated and checked by simulation.
... Therefore, it is also meaningful to start by considering shapes typical for possibly constrained control of single integrator (a = 0) and just then to specify possible deviations caused by internal plant feedback (a = 0). The closed loop performance is characterized by the setpoint step responses, whereby the plant output y(t) is required to be (nearly) monotonic (MO) [4]- [6]. For transients starting and finishing in steady states it means MO output with a shape of a "S-curve" (Fig. 1 above). ...
... In Matlab it may be simply computed by the command sum(abs(diff(u)). Based on TV, new measures for deviations from MO and 1P shapes may be proposed [4]- [6]. Deviations from strictly MO shape of the plant output y(t) with the initial value y 0 and the final value y ∞ may be characterized by yT V 0 criterion ...
... Later, many authors tried to improve this method by considering differently specified batch of processes, different performance measures and different design methods [3], [14]- [16]. This experimental approach was recently automatized by the performance portrait method [4], [5] enabling to look for optimal and robust tuning by specifying arbitrary performance restraint for given task. ...
The paper represents first part of two contributions dealing with simplified modular design of constrained P and disturbance observer (DO) based PI control with different filtering properties illustrated by example of speed servo control. This first part is devoted to analysis of the core structure of the P controller tuned for different types of nonmodelled or filter dynamics. Openness of the approach means that for approximating additional dynamics of the P-controller structure different filters may be used without necessity to repeat in the nominal case analysis of the optimal and critical tuning. By simpler means, flexible approach enabling to fit requirements of particular loop and simultaneously offering reasonably better performance than the traditional controller design based on Luenberger disturbance observer for reconstruction of the velocity signal is proposed. Achieved performance is evaluated by newly introduced measures for deviations from monotonic and one-pulse shapes of transients typical for control of plants with dominant 1st order dynamics.
... Development of the PPM has been based on formulating ideal requirements on shapes of the transients at the plant input and output expressed in terms of monotonicity [1] and on introduction of performance measures for quantifying deviations from such ideal shapes [2]. By formulating rules for generating and inspecting performance portraits of some chosen normed systems and controllers it may be extended to a broad class of systems which may be simplified to the chosen controller and plant models [3]- [6]. ...
... but also the prefilter parameter's number increased to 2. Furthermore, in numerous applications it is not acceptable to consider special sensors for measuring the output derivative (as in [2]) and, instead, use and design of inevitable suitable low-pass filters has to be preferred, considered e.g. in the form ...
The paper discusses approaches and results achieved by the performance portrait method in optimal and robust tuning of proportional-integrative-derivative (PID) controllers for the double integrator plus dead time (DIPDT) models. It extends results achieved in tuning proportional-integrative (PI) controller for the integral plus dead time (IPDT) plant models and proportional-derivative (PD) controllers applied to DIPDT plant models. The analysis and optimal tuning by the performance portrait method are using performance measures expressing deviations from piece-wise monotonic ideal output and input shapes. Detail analysis of different design aspects is necessary before starting time-consuming performance portrait generation.
... The performance portrait (PP) [2,3] contains and represents information about the performance of a closed loop consisting of a specified plant and controller. After a proper analysis the PP can be created for every control system. ...
... This criterion is suitable for evaluating shape deviations both from MO and 1P signals. To evaluate the deviation of the plant output having an initial value 0 and a final value ∞ from a MO shape we can use the 0 ( ) criterion defined in [2] as ...
The performance portrait method (PPM) may be considered as a representative of the second generation of optimal robust tuning methods. In contrast to the first generation robust tuning methods based on local loop properties, such as the maximal loop sensitivity or the complementary loop sensitivity, it guarantees chosen time or shape related properties over a whole uncertainty interval. It is based on and can be considered as a generalization of the well known experimental control design method by Ziegler and Nichols [1] enhanced by the power of available computers. Carrying out systematic measurements on representative processes we can store performance related measures in matrices given by different plant-controller parameter configurations. The performance measures may correspond to the needs of the considered application, usually involving shape and time related system signal measures. If the parameters defining the matrices of measured data are properly normalized, then the results can be applied to a large number of control loops. Different qualitative and quantitative measures may be evaluated and stored in a computer database, to be chosen on demand and in different combinations defined by a particular design or user.
The introduced method is illustrated by means of an online application that enables users to interact with several steps in the controller design. It is developed on the base of open software environments such as Octave and OpenModelica that are installed on the server.
... More information on the featured control structures can be found in e.g. [5], [6]. The exercises are based on real time experiments. ...
... The students get warning when out of limits performance measure occur. The Figs. [6][7] show an examples of the automated feedback to the students. For the correctly made task it shows only the points earned, the second example shows the feedback for the badly made experiment with a short explanation on what is wrong. ...
The paper describes the marking aids used in the constrained PID control course. The exercises are based on experimenting with the real plants which increases the difficulty of marking of the assignments. Several scripts are used to support the marking of the assignments. Originally the scripts were meant to be used by the teachers. The paper shows how the availability of these aids for the students influenced the quality of their work.
... may be analytically derived by the triple real dominant pole (TRDP) method (Huba, 2010). Instead of the optimal controller gains r 0 , r 1 , equivalent poles ...
... α em = −0.281/T d (module) (20) In (Huba, 2010) this tuning has been précised by the newly developed Performance Portrait method (PPM). It is control; output amplitude deviations from monotonicity (y 2 -MO), input amplitude and integral deviation from two-pulse function (uTV2 and 2P); tuning corresponding to TRDP (18) (black), optimal equivalent poles (21) (red) and real optimal poles (24) (blue); border or real equivalent poles (black) based on solving the optimization problem by sweeping all possible controller parameters yielding minimal IAE values of the step responses under shape related constraints put on the plant input, or output (Huba, 2013b), giving ...
The performance of a control system depends on its design and implementation. Embedded controllers are the controllers designed on compact embedded systems. Embedded controllers are gaining popularity in industrial applications as the embedded system provides benefits on power consumption, size, cost etc. The education on designing architecture for embedded controllers for (Field Programmable Gate Array) FPGA and Application Specific Integrated Circuits (ASIC) technologies helps in further customization. This paper presents the architecture designs for simple embedded controllers and gives a framework to design architectures for controllers. A novel approach to understand the designing architecture for embedded controller is proposed. This designing technique can be a part of a course on Embedded Control Systems.
... may be analytically derived by the triple real dominant pole (TRDP) method (Huba, 2010). Instead of the optimal controller gains r 0 , r 1 , equivalent poles ...
... α em = −0.281/T d (module) (20) In (Huba, 2010) this tuning has been précised by the newly developed Performance Portrait method (PPM). It is control; output amplitude deviations from monotonicity (y 2 -MO), input amplitude and integral deviation from two-pulse function (uTV2 and 2P); tuning corresponding to TRDP (18) (black), optimal equivalent poles (21) (red) and real optimal poles (24) (blue); border or real equivalent poles (black) based on solving the optimization problem by sweeping all possible controller parameters yielding minimal IAE values of the step responses under shape related constraints put on the plant input, or output (Huba, 2013b), giving ...
The use of simulation software allows control engineers to design better solutions, reduce development costs and avoid possible critical errors in the process to be controlled. In addition, simulation software can be used in education to provide students with realistic problems. Although there are a host of different solutions for continuous systems, there are few simulators of industrial processes focused on automation technologies. In this work, we present a new 3D simulation software of industrial systems for programmable logic controller (PLC) educational exercises. The application includes a physics simulation engine based on videogame technology, it can communicate with PLCs using the OPC protocol, and allows the student to test their designs in real time. Moreover, an extension of the application enables the automated assessment of student work, by operating in conjunction with the Goodle GMS e-learning platform. The simulation software has been used in two process automation courses during 2011 and 2012.
... The plant output is frequently required to have a smooth monotonic (MO) transient preserving the direction of change. To evaluate the deviation of the plant output y having an initial value y 0 and a final value y ∞ from a MO shape we can use the T V 0 (y) criterion defined in Huba [2010] as ...
... This delay causes an output error before the (ideally MO) compensation starts. Thus the output will have an extreme point separating two monotonic intervals -an increasing and a decreasing one, denoted in Huba [2010] as a one-pulse (1P) transient. Therefore to evaluate the shape properties of the output y corresponding to the disturbance step we should use the T V 1 (y) criterion that describes deviation from a 1P transient. ...
The paper deals with a comparison of two possible approaches to the performance portrait method application for an optimal nominal tuning of an integral plus dead time (IPDT) plant. The first one relies on a more compact 2D approach used by Huba et al. [2009] inspired by the triple real dominant pole method [Víteček and Vítečková, 2008]. The second approach is a simpler but, at the same time, in some aspects a computationally more demanding 3D method. Pros and cons of both approaches are analyzed and discussed.
... For analytical controller tuning, requirement of the double real dominant pole s o of the characteristic polynomial A(s) of (13) may be used based on conditions A (s o ) = 0 and ˙ A (s o ) = 0. Despite it was shown to yield slightly overdamped transients (Huba (2010Huba ( , 2011aHuba ( , 2012b)) it is appropriate for any considered dynamics and yielding ...
... Since with long deadtimes the ability of the closed loop feedback to deal with plant/model mismatch is strongly decreasing, importance of a relevant robustness analysis increases. It is rigorously possible with the performance portrait method Huba (2010 Huba ( , 2011a Huba ( ,b, 2012c). Since in dealing with systems having long deadtime demands on precision of the plant ...
The paper discusses challenges related to design and education of the simplest integrating controllers for systems with the dominant first order plants and possible long dead time that obviously represent the most frequently treated control task. Based on experience achieved in research and education of the course ''Constrained PID Control'' (Huba and Simunek, 2007), three different approaches comprising the traditional PI control, the IMC approach and the newly developed disturbance observer (DO) based PI control are compared from different points of view. By designing and experimentally evaluating controllers for simple plants, students are encouraged to experience pros and cons of particular approaches. In this way, the modular design of constrained DO based PI control with different filtering properties based on deeper analysis of the core structure of the P controller with different types of nonmodelled dynamics approximations is shown as the most flexible and simultaneously sufficiently open and general alternative for stable, integral and unstable systems with possible long delays.
... As indicated by quotations of older references from one of the first textbooks of automatic control [17], the multiple real dominant pole (MRDP) method belongs to the oldest methods of optimal adjustment of controllers for time-delayed systems [18][19][20][21][22]. Although even with the help of a simple performance portrait method [23][24][25] it is possible to show that from the point of view of the resulting dynamics of the circuit, slightly better results can be achieved based on complex conjugate poles. More rigorously, similar problems have also been treated by the latest research in the field of the maximum decay rate of the closed-loop response for linear time-invariant time-delay systems [26,27], also giving a more detailed review of the latest research from this area. ...
The paper discusses the stability and robustness of the proportional-integral (PI), proportional-integral-derivative (PID), and proportional-integral-derivative-accelerative (PIDA) controller for the integral-plus-dead-time (IPDT) plants. To enable the implementation and measurement of noise attenuation, binomial low-pass filters are added to the traditional design of controllers with ideal transfer functions, and the impact of the low-pass filters on the robust stability of the circuit is studied in detail. The proposed controller tuning, which integrates the suboptimal controller and filter design, is based on explicit tuning formulas derived by using the multiple real dominant pole (MRDP) method. It is shown that by combining derivative actions with possibly higher-order low-pass filters, it is possible to either accelerate the transients or increase the closed loop robustness and that the problem of defining the robust stability area should be addressed at the stage of determining the process model. In addition, if wishing to maintain the closed loop robustness of unfiltered PI control, while increasing the degree of the derivative components, one needs to increase the filtering properties of the low-pass filter used accordingly. Simple analytical relations for setting filtered PI, PID, and PIDA controllers with equivalent robustness are derived.
... Design of a stabilizing PD controller for the DIPDT plant model based on the multiple real dominant pole [6], [48], [49], results in parameters ...
This paper illustrates the reference model-based dead-time compensator (RM-DTC) recently developed for first-order time-delayed systems using a real-time example, and extends this novel approach for a time-delayed double integrator system. RM-DTC enables fully decoupled setpoint tracking and disturbance reconstruction and compensation. The overall structure of RM-DTC includes feedforward control using either a transfer function based implementation or the primary loop which produces a filtered inverse of the model transfer function. The setpoint feedforward is complemented by a disturbance-observer-based input disturbance reconstruction that also uses the filtered inverse of the plant model. The reference models of setpoint and input disturbance feedforward control allow the introduction of a stabilising PD controller without compromising the nominally independent dynamics of setpoint tracking and disturbance rejection. The main advantage of retaining the full functionality of disturbance reconstruction in RM-DTC is that it enables monitoring and diagnostics of the controlled process, which is important in terms of full automation of running processes representing the fundamental goal of the Industry 4.0 campaign. This surpasses the approaches based on internal model control, which have been modified for unstable systems to achieve internal stability by eliminating the reconstructed disturbance signal from the control loop. The excellent properties of RM -DTCs are illustrated by simulation and real-time control of thermal process.
... Today, the problem of constraints is again in the focus of the research activities and it is hardly possible to give here just a brief overview of different approaches. For the sake of brevity, we will show here briefly one possible approach to controlling the coupled tanks combining constrained pole assignment control Huba and Bisták (1999); Huba (1999); ; Huba (2001Huba ( , 2005Huba ( , 2010Huba ( , 2011a with extended exact linearization method that is based on transformation of the existing nonlinear system by (5.9) into the double integrator one and in controlling such double integrator system by the PD 2 controller that fully considers the existing control saturation limits. Let us start with considering the double integrator system d 2ȳ (t) dt 2 = K sū (t) (5.11) ...
Textbook about Constrained PID control for the seminar at TU Brno
... For the sake of brevity, we will show here briefly one possible approach to controlling the coupled tanks combining constrained pole assignment control Huba and Bisták (1999); Huba (1999); ; Huba (2001Huba ( , 2005Huba ( , 2006Huba ( , 2010Huba ( , 2011a with extended exact linearization method that is based on transformation of the existing nonlinear system by (7.9) into the double integrator one and in controlling such double integrator system by the PD 2 controller that fully considers the existing control saturation limits. ...
This workbook was created within the NIL-I-007-d Project “Enhancing NOSK
Cooperation in Automatic Control” (ECAC) carried out in 2009-2011 by
university teams from the Slovak University of Technology in Bratislava and
from the Norwegian University of Science and Technology in Trondheim. As
it is already given by the project title, its primary aim was enhancing cooperation
in academic research in the automatic control area in the partner
institutions. This was achieved by supporting broad spectrum of activities
ranging from student mobilities at the MSc. and PhD. level, staff mobilities,
organization of multilateral workshop and conferences, joint development of
teaching materials and publishing scientific publications. With respect to the
original project proposal, the period for carrying out the foreseen activities
was reasonably shortened and that made management of the all work
much more demanding. Despite of this, the project has reached practically
all planned outputs – this workbook represents one of them – and we believe
that it really contributes to the enhancement of Slovak-Norwegian cooperation
and to improvement of the educational framework at both participating
universities. Thereby, I would like to thank all colleagues participating in
the project activities at both participating universities and especially professor
Sigurd Skogestad, coordinator of activities at the NTNU Trondheim,
associate professor Katar´ına ˇZ´akov´a for managing the project activities, and
professor Miroslav Fikar for the patient and voluminous work with collecting
all contribution and compilation of all main three project publications
(textbook, workbook, and workshop preprints).
... A broader collection of different approaches gives [27]. Quantitative criteria for evaluating deviations from ideal shapes of transients of process input and output signals have been developed based on deviations from monotonicity [14,28]. In the context of this work, we restrict the design of PI and PID controllers to obtaining a 1P control signal after setpoint and input disturbance steps. ...
This paper deals with the design of a DC motor speed control implemented by an embedded controller. The design is simple and brings some important changes to the traditional Ziegler–Nichols tuning. The design also includes a novel anti-windup implementation of the controller and an integrated noise-reduction filter design. The proposed tuning method considers all important aspects of the control, such as pre-processing of the measured signals and filtering (to attenuate the measurement noise), time delays of the process, modeling and identification of the process, and constraints on the control signal. Three important aspects of designing PI and PID controllers for processes with noisy output on Arduino-type embedded computers are considered. First, it deals with the integrated design of the input filter and the controller parameters, since both are interdependent. Secondly, the method of setting the controllers from step responses by Ziegler and Nichols is modified for the case of digital signal processing (without drawing the tangent), while it recommends the suitability of its modification in terms of the use of both integral and static models. Third, the most suitable anti-windup solution for the given controller structure is proposed. In summary, the paper shows that an appropriate design of the embedded controller can achieve excellent closed-loop performance even in a noisy process environment with limited control signals.
... In this context, a more detailed investigation of the role of the disturbance filters in influencing the achievable closed loop robust performance would be desirable that could e.g. be based on the newly developed method of the performance portrait [3]. The essential advantage of the new solutions is that they do not generate the windup effect due to the IMC structure [4] and due to replacing the PI controller typically generating windup by the 2DOF P controller. ...
This paper represents continuation of the Part I. of the paper [2] devoted to the new structures of the Filtered Smith Predictor with the simplified primary loop using the 2 degreeof-freedom (2DOF) P controller. This Part II. brings computer simulation results for the 2nd order disturbance filter and concluding remarks.
... the deviations from monotonicity (MO) at the output (Huba, 2010) by modification of the Total Variance (Skogestad, 2003) ...
This paper deals with an experimental verification of the recently published theoretical analysis (Huba, 2013b,a) of the traditional two degree of freedom (2DOF) PI control and of a noise attenuation oriented predictive disturbance observer based filtered PI control (PDO-FPI). It is shown, how both controllers may be tuned to get the fastest possible monotonic outptu responses with the best IAE values, or to get an excellent noise attenuation.
... Primarily it is appropriate for system, where analytical solutions either do not exist, or they are too complex. The method has been proposed in Huba and Gerke, 2009) and from that time it has been used in numerous case studies, which include time delayed systems (Huba, 2010a(Huba, ,b, 2011cSóos and Huba, 2014;Huba, 2014a;Huba et al., 2016b), robustness issues (Huba, 2011a(Huba, , 2012(Huba, , 2013bSóos and Huba, 2014;Huba, 2015c), constrained control (Huba, 2011b,d), or disturbance-observer structures (Huba, 2013c(Huba, , 2014b(Huba, , 2015a. ...
The Performance Portrait Method enables to visualize influence of chosen control parameters on the loop performance of the selected dynamical system with given interval parameter uncertainties. The knowledge of this behavior can help to analyze and tune control algorithm and to fulfill expected requirements as far as robustness and control quality. The presented paper introduces a new web tool that in an interactive way enables to design filtered 2DOF PI controller for the integral plus dead time system models on the base of the performance portrait method. For a performance portrait calculated once in the 3-5D parameter space it enables to find optimal robust controller tuning. For given plant parameter uncertainties it yields the fastest possible transient responses satisfying with chosen tolerances the shape related constraints at the plant input and output. A web application developed to illustrate the problem offers much richer visual information than it is possible to offer by standard publications, interactivity and dynamical effects. The backend of the application is driven by Matlab software. This environment is used both for calculation of the performance portrait and for simulation of transients of the considered feedback control structure.
... The input disturbance is obtained according to the following formula (see e.g. [3]) ...
The paper presents laboratory model of unmanned aerial vehicle. The plant design, construction and building was a part of a project in the course on autonomous mechatronic systems. The goal was to make an exercise in mechatronics, resulting in plants which could be used in other project or classes, using the same hardware as common UAVs.
... výstupnej poruchy s kompenzáciou dopravného oneskorenia bola popri problémoch s analógovou realizáciou oneskorenia modelu absencia vhodnej metódy na jej robustné nastavovanie [157]. To sa zmenilo až vyvinutím metódy portrétu správania [59,60,64]. Touto metódou bola v prácach [61,69,170] analyzovaná ekvivalentná štruktúra navrhnutá pre vstupné dopravné oneskorenie. ...
Predkladaná publikácia prináša výsledky viac ako troch desatrocí výskumu
a experimentovania v oblasti najjednoduchších štruktúr riadenia. Vznikla sledovaním
troch hlavných cielov:
• experimentálne overovanie základných metód automatického riadenia jednoduchých
dynamických systémov s využitím laboratórnej tepelno-optickomechanickej
sústavy TOM1A,
• zovšeobecnenie riadenia známeho ako riadenie s vnútorným modelom (internal
model control, IMC) uvažujúceho výstupné poruchy aj na štruktúry
so vstupnou poruchou a s explicitným stabilizacným regulátorom a
• rozvoj tretieho stupna volnosti nastavovania regulátorov na modifikáciu
filtracných vlastností obvodu.
Kniha uvádza najznámejšie metódy automatického riadenia jednoduchých dynamických
systémov a skúma hranice ich možností. Komplexné spracovanie
problematiky ukazuje súvislosti medzi viacerými doteraz navzájom izolovanými
prístupmi a v urcitých momentoch aj potrebu ich modifikácií. To vedie
na súdržnejší a dôslednejší prístup k problematike, z coho nakoniec vyplýva
aj dosahovanie zvýšenej kvality riadenia. Metódy sú diskutované z hladísk:
• co najvyššej zhody ocakávaných a experimentálne overených procesov automatického
riadenia a
• co najrýchlejších prechodových dejov pri co najnižšej úrovni šumov merania,
nepresností modelu a s co najjednoduchšími postupmi.
Navrhnuté algoritmy identifikácie a riadenia sú zväcša hned aj experimentálne
overované riadením vhodných dynamických procesov v reálnom case. V tomto
sa jej obsah líši od prevažnej väcšiny matematicky orientovaných publikácií,
ktorých hlavným cielom spravidla býva sformulovat rigoróznu, kompaktnú a
pritom co najvšeobecnejšiu matematickú teóriu procesov automatického riadenia.
Už menej sa zaoberajú vyššie postavenými cielmi, ktoré sú dominantné
z hladiska spolahlivosti ich praktických aplikácií.
VI Predhovor
Zhoda medzi teoretickými predpokladmi a experimentálnymi výsledkami
bude teda základným kritériom posudzovania úspešnosti návrhu. Takýto postup
prináša viaceré špecifiká. Na jednej strane ukazuje, že miesto jednoliatej
teórie, ktorá by dokázala zvládnut všetky možné situácie jediným univerzálnym
vzorcom, je vhodnejšie využívat modulárny prístup. Ten umožnuje pružnejšie
prispôsobenie špecifikám riešeného problému. Na druhej strane bude
zásada experimentálneho overovania využívaných postupov vyžadovat širšie
spektrum reálnych dynamických procesov, ktoré by dostatocne reprezentovali
problémy vystupujúce v praxi. To v našom prípade pre vyclenené úlohy z dynamických
tried 0 a 1 aspon ciastocne zabezpecí tepelno-opticko-mechanická
sústava TOM1A. A bude treba aj matematický aparát zameraný na posudzovanie
zhody medzi predpokladanými a dosiahnutými regulacnými procesmi.
Po laboratórnej tepelno-optickej sústave TOS1A Model 2013 predstavuje
najnovšia tepelno-opticko-mechanická sústava TOM1A další clánok dlhodobého
vývoja, ktorého zaciatky možno datovat do roku 1980. Vtedy sme v diplomovej
práci [11] ako jedni z prvých na Slovensku obšírne experimentovali
s priamym císlicovým riadením jednosmerných motorov pocítacom RPP-16.
Krátko na to sme od roku 1982 s využitím pocítaca HP-85 podstatne rozšírili
okruh realizovaných experimentov na rôzne rýchlostné a polohové systémy
s jednosmernými motormi. Experimenty zahrnovali aj výrazné skokové zmeny
parametrov systému (momentu zotrvacnosti, ci zátaže). Od roku 1984 sme
s využitím HP-85 experimentovali aj s riadením tepelných sústav [9, 128].
Odvtedy možnosti využitia pocítacov na riadenie sústav neuveritelne
vzrástli. Otvorili sa nové možnosti obvodov typu FPAA (Field-programmable
analog arrays), FPGA (Field-programmable gate array), ci vnorených pocítacov
typu Arduino a Raspberry Pi umožnujúcich prácu s malými periódami
vzorkovania. V akademickom prostredí sa však pocítace ešte stále využívajú
skôr na simulácie ako na riadenie v reálnom case. S experimentálnymi prácami
sa už stretávame menej. Dôvodov k tomu môže byt hned niekolko. Zvycajne
ide o súbeh viacerých okolností: nedostatok vhodnej výpoctovej techniky a
pracovníkov na jej zvládnutie, ci nedostatok návrhárov algoritmov identifikácie
a riadenia s dostatocnými experimentálnymi skúsenostami. Problémom
môže byt aj nedostatok vhodných dynamických systémov na realizáciu širokej
škály experimentov. Simulácie samozrejme hrajú dôležitú rolu. V publikácii sa
ich snažíme systematicky konfrontovat s riadením reálnych procesov. V zmysle
vyššie uvedených cielov má táto viacero castí. Uvádza krátky prehlad:
• možných lineárnych a nelineárnych modelov uvažovaných sústav,
• opis základných typov regulátorov a metód na ich nastavovanie,
• softvéru na realizáciu a spracovanie experimentov,
• kritérií kvality na vyhodnocovanie prechodových dejov,
• najjednoduchších metód experimentálnej identifikácie a
• experimentov na experimentálne overenie jednotlivých postupov.
V prílohách možno nájst technické parametre, súhrn doplnujúcich informácií
a krokov potrebných na inštaláciu a používanie uvažovanej sústavy.
Predhovor VII
Z pôvodnej rozmernej, hlucnej a energeticky nárocnej tepelnej sústavy na
báze sušicov vlasov z roku 1984 vznikli v rokoch 2005-09 v niekolkých prototypoch
tepelno-optické sústavy známe pod typovým oznacením uDAQ28/LT.
Súcasné laboratórne modely TOM1A vychádzajú zo sústav TOS1A. Od
uDAQ28/LT sa líšia vonkajším vzhladom aj použitou elektronikou [96]. Integráciou
tepelného kanálu do prístrojovej skrinky sme dosiahli zmenšenie
vonkajších rozmerov a celkovú kompaktnost, z ktorej vyplýva jednoduchšia
manipulovatelnost a prenositelnost. Zmenšil sa aj príkon použitého svetelného
a tepelného zdroja z 12V DC/20W na 5W. V súlade so súcasným trendom
vychádza koncepcia použitej elektroniky z vnorených pocítacov typu Arduino.
Zachováva sa priama komunikácia z programu Matlab/Simulink cez USB
port, pricom sa minimálna použitelná perióda vzorkovania skrátila zo zhruba
50 ms na cca 15 ms. Výrazne sa zlepšila regulovatelnost otácok použitého ventilátora,
co poskytuje novú triedu zaujímavých úloh. Zároven sme otvorili aj
dalšiu zaujímavú škálu úloh na riadenie tepelného kanálu prostredníctvom
ventilátora. V širšom rozsahu možno používat na riadenie optického kanálu
aj svetelnú diódu LED, co dalej rozširuje škálu riešitelných úloh.
Rastúca komunita používatelov tepelno optických, resp. tepelno-optickomechanických
sústav (ktorých pocet sa blíži císlu 100) a pribúdajúce skúsenosti
s ich riadením sa odrážajú aj na stále širšom spektre rôznych prístupov
a používatelských programov (pozri napr. [42, 79, 82–84, 95, 97, 101, 105–
107, 110, 111, 117, 123, 144, 186, 187, 190–192]). Jedným z cielov sledovaných
pri tvorbe tejto publikácie je vytvorit experimenty pre pedagogiku
aj výskum na ilustráciu podstatných krokov co najširšieho spektra základných
regulátorov. Radi by sme sa dostali na vyššiu úroven aj s ich opisom,
s ich optimálnym návrhom a kritickým zhodnotením. Pokial to bude možné,
budeme pri tom vychádzat z verejne dostupných publikácií v casopisoch a
zborníkoch významných konferencií a z návrhov nových patentových riešení
[100]. Novšie verzie programov a návodov sme sa snažili vytvorit tak, aby
boli využitelné aj na starších modeloch sústav. Samotné programy sú volne
dostupné prostredníctvom elektronického prostredia LMS Moodle na adrese
http://elearn.elf.stuba.sk/moodle/ v kategórii Projekty. Tu možno nájst aj zoznam
použitých skratiek a symbolov, elektronickú verziu publikácie a aktualizovaný
zoznam experimentov a k nim prislúchajúcich programov. Plánujeme
aj aktualizácie webovej stránky našich produktov na adrese http://www.eas.sk.
Záverom by sme chceli podakovat recenzentom prof. Ing. Miroslavovi Fikarovi,
DrSc. a doc. Ing. Danici Rosinovej, PhD za starostlivé precítanie rukopisu
a návrhy k jeho vylepšeniu. Rovnako dakujeme aj bývalým študentom T. Ballánovi,
A. Michalíkovi, M. Kamenskému, P. Kurcíkovi a F. Jelenciakovi, ktorí
spolupracovali pri vývoji predošlých verzií sústavy a prof. Hoyerovi a prof.
Gerkemu z FernUniversität Hagen za podporu výskumných aktivít vedúcich
k príprave publikácie v projekte hostujúceho profesora Mercator.
Bratislava, december 2015 autori
... The input disturbance is obtained according to the following formula (see e.g. [5]) • di is the input disturbance, • u(t) is system input, • K 1 is process gain from 3, • y(t) is the altitude system output. From the presented results there is obvious the tuning with the highest gains yields oscillatory behavior. ...
The paper presents laboratory model of aerial vehicle. The plant design, construction and building was a part of a project in the course on autonomous mechatronic systems. The goal was to make an exercise in mechatronics, resulting in plants which could be used in other project or classes.
... Deviations from the strictly MO shape of a plant output y(t) with the initial value y 0 and the final value y ∞ may be characterized by the relative yT V 0 measure [26], [28] ...
... For the first order plants, the closed loop performance is frequently specified by nearly monotonic (MO) output y s (t) corresponding to setpoint step changes [5]- [8]. For "S"-shaped MO output responses starting and finishing in steady states, inversion of the dynamics yields an "one-pulse (1P)" input. ...
... Based on TV, new measures for deviations from MO and 1P shapes will be preferred Huba (2010Huba ( 2011) that may be easily tested numerically by evaluating simulated or experimentally measured setpoint and disturbance step responses and are also appropriate for constrained control. For evaluating deviations from strictly MO plant output y(t) with the initial value y 0 and the final value y ∞ the T V 0 (y) measure is proposed as ...
This paper considers PI controller tuning for Integral Plus Dead Time plant (IPDT) by new Matlab/Simulink tool based on the performance portrait method. It enables to guarantee transient responses with specified deviations from ideal shapes at the plant output and input and to fulfill additional optimality specification, defined in terms of the minimal IAE (Integral of Absolute Error) values weighted for the setpoint and disturbance steps. As the ideal step responses at the plant output monotonic transients are considered, whereas at the plant input one-pulse responses consisting of two monotonic intervals are required. As an introduction to new generation of robust tuning approaches, optimal nominal tuning is firstly treated.
see https://www.researchgate.net/publication/265480509_Setpoint_Versus_Disturbance_Responses_of_the_IPDT_Plant
... Thus, it may require a designer decision and an interactive tool supporting his, or her activity. This paper presents one such a tool based on generating and using the closed loop PP ( [4]- [6]). The key question considered is, how to robustly design simple control loops with stable dead-time dominated plants to keep their setpoint and disturbance step responses nearly monotonic (MO) both at the plant input and the output. ...
The paper discusses online course materials used in the training for the simplest robust controller. It has been proposed for the plant that can be represented by simple gain and dead time. The controller design is realized by choosing appropriate model parameters and and by specifying the disturbance observer filter characterized by its order and by the time constant. All parameters have to be estimated in such a way that the loop will offer the fastest possible dynamics of the set-point and disturbance responses in a considered range of the plant parameter values. In order to help practitioners, engineers or other interested users to set the controller parameters optimally, a web application has been created. The interactive design procedure is required to see impact of an estimate of the uncertainty intervals for the dominant uncertain plant parameter on the controller parameter values. The user can e.g. choose the order of considered filters and then, by using slider, the analyzed plant parameter (gain or dead time) and to see the consequence of this choice (measured e.g. by integral criteria). This approach enables to better understand the controller influence on the loop performance.
... one-pulse (1P) and two-pulse (2P) shapes may be proposed (Huba, 2010(Huba, , 2013c. Deviations from strictly MO shape of the plant output y(t) with the initial value y 0 and the final value y ∞ may be characterized by ...
The paper shows Matlab based interactive tool for noise attenuation motivated filtered PD and disturbance observer based filtered PID controller performance analyses and design. The user can set several control loop parameters (plant model parameters and their uncertainty, measurement noise, controller, filter parameters) and evaluate various performance measures. The comparison with traditional linear PID controller design methods can be made. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
... For measuring deviations from the ideal shapes, new performance measures have been proposed [24] that may be interpreted as a relative Total Variance [25]. The first one defined as ...
This paper deals with robustness evaluation of the predictive disturbance observer based filtered PI (PDO FPI) control.
Robust performance of loops with integral plants, different order filters and with an uncertain dead time is examined by the performance portrait method. The analysis does not confirm the up to now accepted view that the best robustness corresponds to the 1st order filters. Features of robust performance measures in
the time domain are discussed.
... For the first order plants, the closed loop performance is frequently specified by nearly monotonic (MO) output y(t) corresponding to setpoint step changes (Huba, 2010(Huba, , 2012b. For "S"-shaped MO output responses starting and finishing by steady states inversion of dynamics yields an "one-pulse (1P)" input. ...
The paper shows organization and overview of assignments on nonlinear P and PI control of the liquid level in a single tank associated with exact linearization, its modification considering nonmodelled dynamics and nonlinear disturbance-observer (DO) based integral action. The assignments are based on the learning by doing and the learning by experimenting paradigms carried out in a quasi-authentic framework with a quasi-real plant. The laboratory model of coupled tanks system is introduced which can be used also for more complex tasks.
... Despite decades of development, the central tool of the course allowing optimal robust control design, was discovered only in recent years. Based on the so called performance portrait method [50]- [53], it shows, how it is possible to overcome the existing inflation of optimal controller tunings [11], [54]- [56]. Its essence is yet simple and easy to understand even for students without high prerequisites of their mathematical knowledge and skills typical for traditional robust control. ...
The paper brings a short overview of accomplished steps that promote development and enforcement of a new approach to PID control. Its concern is to unify numerous approaches and initiatives which emerged during the last hundred years in the field of control of the simplest dynamical systems and to organize them into a new modular building kit that would be more compact, consistent and effective than the existing one.
An essential role of different ICT tools and electronic media, corresponding learning environment, and a learning paradigm representing indispensable parts of the overall research framework are highlighted.
... This new computer analysis of control loops based on generating the closed loop PP [2], [3], [11]) is briefly described to be used in analyzing, designing and comparing DTCs and in visualizing their properties. The key question is, how to robustly design control loops with stable dead-time dominated plants to keep their setpoint and disturbance step responses nearly monotonic at the plant input and output also in the case of a possible plant-model mismatch. ...
This paper rediscovers and modifies the oldest and
the simplest dead-time compensator (DTC) introduced originally
by [1] and shows new possibility for its robust tuning by the
performance portrait method. This method comprising computer
based analysis, compressed characterization and visualization
of the closed loop properties achieved by evaluating set of
performance measures over chosen grid of loop parameters with
an optimization based on sweeping all possible stored data may
be successfully used for an optimal nominal or robust tuning
[2]–[4] based either on the known plant model with parameters
taking values upon uncertainty intervals, or directly on measured
experimental data [5]. In order to enable an easy comparing with
other possible solutions new robustness characteristics will be
derived showing how a model uncertainty influences decrease of
the loop performance
... Based on TV, new measures for deviations from MO and 1P shapes will be preferred Huba (2010 Huba ( 2011) that may be easily tested numerically by evaluating simulated or experimentally measured setpoint and disturbance step responses and are also appropriate for constrained control. For evaluating deviations from strictly MO plant output y(t) with the initial value y 0 and the final value y ∞ the T V 0 (y) measure is proposed as systems with 1P dominant control it is then appropriate to work with the T V 1 criterion defined for u m = max{u(t)} as ...
This paper considers PI controller tuning for Integral Plus Dead Time plant (IPDT) by new Matlab/Simulink tool based on the performance portrait method. It enables to guarantee transient responses with specified deviations from ideal shapes at the plant output and input and to fulfill additional optimality specification, defined in terms of the minimal IAE (Integral of Absolute Error) values weighted for the setpoint and disturbance steps. As the ideal step responses at the plant output monotonic transients are considered, whereas at the plant input one-pulse responses consisting of two monotonic intervals are required. As an introduction to new generation of robust tuning approaches, optimal nominal tuning is firstly treated.
... This tuning handicap may be considered (at least partially) to be solved by the recently published performance portrait method (PPM) [7], [9]- [11] that, for a control loop with a defined controller, allows to find its optimal tuning satisfying the given performance specifications, if such a tuning exists. This finding enabled full concentration on the analysis of existing control structures (required, for example, by [12]), which finally showed that by modifying the disturbance observer (DO) based control structures, huge performance improvements may be achieved [7], [11]. ...
The paper starts with a short overview of recent development in the MOOCs area. It discusses the role of feedforward and feedback control structures in automatic control. It points out that despite the relatively long history of the simplest and most frequently used PID control structures there is still huge space for their significant improvements accomplished by simple means. Then, making a parallel to the much more complex elearning, by examining a broad spectrum of different types and approaches of elearning up to the latest MOOCs courses, a similar role of appropriate feedforward and feedback control structures, as well as of their expected impact in automatizing learning processes is discussed. With respect to this, different forms and types of elearning ranging from the simple text based courses up to the newestMOOCs are analyzed and compared with equivalent engineering automatic control schemes. This comparison shows on high importance of automatic control in elearning courses.
... A second possible way to calculate the limit values K D , IAE sD and IAE dD arises from analysis of the optimal P controller gain by the PP method in [4], which yields for the tolerable shape related deviations constrained by = 0.001 the value K = K P K s T D ≈ 0.3974. From (37) one gets practically the same limit working point as (63) allowing to tune the controller by considering ...
This paper analyses the optimal nominal tuning of a new modification of predictive disturbance observer (PDO) based filtered PI control (PDO FPI) applied to a first order plus dead time (FOPDT) plant with exactly known parameters.
The impacts of applied filters on optimal controller tuning and on achievable closed loop performance are evaluated first of all. The limits of achievable performance are compared with those of traditional two degree of freedom (2DOF) PI control, with both controllers tuned by the multiple real dominant pole method. This comparison shows the potential of PDO FPI control to improve tracking and regulatory dynamics significantly, permitting the Pareto-like servo/regulator trade-off of 2DOF PI control to be removed.
Two PDO FPI tuning approaches are proposed, allowing optimal filter degree and time constants to be evaluated.
The first tuning scenario considers optimization of closed loop performance expressed in terms of the Integral of the Absolute Error (IAE) weighted alternatively by the relative total variance TV1 of the control signal. This is carried out by changing the filter order n under the constraint that a constant position of the dominant closed loop pole is maintained. This keeps the dynamics of the setpoint step responses almost unchanged. In the second tuning scenario the optimization is carried out under a constraint on constant speed of disturbance step responses.
All the main results are then numerically checked for the integral first order plant with dead time by the performance portrait (PP) method.
The analysis presented here shows that the new PDO FPI structure substantially enriches the spectrum of controllers applicable to simple control tasks.
Predkladaná publikácia sumarizuje výsledky viac ako štyroch desaťročí výskumu a experimentovania v oblasti najjednoduchších štruktúr riadenia s obmedzeniami akčnej veličiny. Vznikla sledovaním viacerých cieľov.
-Zameriava sa na výklad a experimentálne overovanie metód automatického riadenia jednoduchých dynamických systémov s obmedzeniami.
-Dôsledne využíva pojem dynamických tried regulácií a z neho odvodených kritérií kvality. Pritom sa zameriava hlavne na úlohy a štruktúry z dynamickej triedy 2.
- Identifikácia, modelovanie, linearizácia aj návrh riadenia reálnych fyzikálnych objektov sú realizované s uvažovaním dvoch typov lineárnych modelov sústav definovaných v okolí zovšeobecnených pracovných bodov.
-Do nastavovania regulátorov vnáša tretí stupeň voľnosti na modifikáciu filtračných vlastností obvodu.
-Zovšeobecňuje návrh známy ako \emph{riadenie s vnútorným modelom} (internal model control, IMC) charakteristický s uvažovaním výstupnej poruchy aj na štruktúry so vstupnou poruchou.
-Popri štruktúrach s dynamickým ovládaním od jednotlivých vstupov sa uvažujú aj štruktúry s explicitným stabilizačným regulátorom a s referenčnými modelmi.
The present publication summarizes the results of more than four decades of research and experimentation on the simplest control structures with the limitations of the manipulated variable. It was created by folowing multiple goals.
-It focuses on the interpretation and experimental verification of methods of automatic control of simple dynamic systems with constraints.
- Consequently uses the concept of dynamic classes of control and derived performance measures. In doing so, it focuses mainly on the tasks and structures of dynamic class 2.
- The identification, modeling, linearization and design of real physical object control are performed with consideration of two types of linear systems of systems defined around the generalized work points.
-To tune the controllers, it introduces a third degree of freedom to modify the filter properties of the loop.
-It generalizes a proposal known as internal model control (IMC), which is characteristic with regard to the output disturbances to structures with both input and output disturbances.
- In addition to structures with dynamic feedforward control from individual inputs, structures with an explicit stabilizing controller and reference models are also considered.
This paper deals with an optimal disturbance observer (DO) based filtered PID (DO-FPID) controller design based on the performance portrait method solved for a PD controller + double integrator + dead time loop configuration. By virtue of an equivalence of a dead time and n-th order filters used in DO-FPID control, the derived results may be used in a broad range of tasks originating in a control of plants with a dominant 2nd order dynamics, which will be illustrated by a noise attenuation motivated design for a positional DC motor control.
This tutorial deals with introduction to a new approach for an optimal and robust control design. Based on notion of dynamical classes of control, it combines new performance portrait based method for analysis, identification and design of robust control with modifications of two degree of freedom (2DOF) control structures with a reference model. These are enhanced by observers for input and output disturbances and by new corrective blocks communicating information about compensation of acting disturbances to the stabilizing controller. Furthermore, a new degree of freedom in their design may be introduced appropriate for achieving an improved noise attenuation and an increased loop robustness. Simulation and real time control examples of simple first order time delayed systems illustrate potential of dramatic performance improvements achievable within this new modular framework.
TENTATIVE SCHEDULE
15.00-15.30 M. Huba: Introduction to Robust Control Design by the Performance Portrait Method
15.30-16.00 P. Tapak: Interactive application to robust design of the simplest I controller according to [20] and its application to simple plant control [53].
16.15-16.45 D. Soos: Extension of the robustness analyses of 2DOF PI control [22] to FOTD systems in 4D parameter space.
17.00-17.30 I. Belai: Applications of 3DOF control structures from DC1 to the speed servo control [27], [30].
PREREQUISITES
The lectures are self-contained and suppose just basic background in automatic control. Introductory explanation will be illustrated by interactive examples carried out in Matlab/Simulink. The tutorial participants are supposed to come with own notebook and to be able to work with own license of Matlab/Simulink. They are recommended to upload Matlab/Simulink programs made available at a web repository (prepared in Matlab 2010b, or Matlab 2014b) and to adopt them to their own Matlab/Simulink version (if necessary).
Cited papers and presentation of the conference contributions are mostly available at ResearchGate. Other material, as presentation slides, computer programmes and performance portraits will be available in Moodle at the address
http://elearn.elf.stuba.sk/moodle/course/view.php?id=637
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The paper brings analytical design of a constrained
PD controller for the second order integral plants focused on
achieving the possibly best tracking dynamics considering the
control signal saturation. The linear pole assignment control and
relay minimum time control are included as its limit solutions.
The degree of freedom existing in choice of a distance definition
for evaluating deviation of the operating point from a required
reference braking trajectory is proposed to be used to achieve a
required performance in control of time-delayed systems
This paper discusses the issues connected with the real experiment utilization in the course Constrained PID Control. In this course the students work individually using real experiments, the “learning by doing” and “learning by experimenting” approach is applied. The anti-plagiary, automated marking system was developed to deal with the new problems the real experiments have brought to the course.
By considering robust tuning of the PI controller for uncertain Integral Plus Dead Time plant (IPDT) this paper demonstrates possibilities of the new Matlab/Simulink tool based on the performance portrait method. For plants with parameters defined over uncertainty intervals it enables to guarantee transient responses with specified deviations from ideal shapes at the plant output and input and to full additional optimality specification, defined e.g. in terms of the minimal IAE values for the setpoint and disturbance steps, in terms of the maximal integral gain, etc. In difference to the robust tuning methods of the 1st generation considering typically controller parameters calculated from plant parameters specified by a single entry, in this new method uncertain plant parameters are specified by two entries characterizing their extreme values. As the ideal step responses at the plant output monotonic transients are considered, whereas at the plant input one-pulse step responses consisting of two monotonic intervals are required.
We demonstrate the feasibility of optical interconnection waveguides, including a Bragg grating and EO waveguides, formed by a self-trapping effect into a novel photopolymer. These technologies are useful for the integration of chip-scale photonic devices.
This paper deals with robust tuning of the Integral Plus Dead Time plant (IPDT), whereby it compares results achieved by the analytical tuning of the PI controller guaranteeing triple real dominant pole with those achieved with P controller extended by disturbance observer (DOB) under robust tuning based on experimentally achieved regions of parameters corresponding to the one-pulse control.
This paper presents design of P- and PD- controllers
for the single and double integrator plant models with dead time and constrained input.
The well known design procedure guaranteeing a double real closed loop pole used
for tuning of the P-controller for a single integrator system with dead time is extended
for the case of a double integrator with dead time controlled by a PD-controller. Its
parameters are determined to guarantee a triple real closed loop pole. A new notion
of equivalent poles is introduced to approximate the above design by the standard pole
assignment controller, whereby recommendation for the choice of the closed loop
poles in dependence on the dead time are proposed. The equivalent poles are
substituted into the near to minimum time pole assignment controlled derived for the
double integrator with constrained control signal without dead time. This procedure
gives good results also in controlling constrained systems with dead time.
This paper brings a brief overview of the blended-learning course "Constrained System Design" and "zooms in" some circumstances of its development.
This paper deals with numerical issues arising in robust design of PI controller for the Integral Plus Dead Time plant (IPDT) under robust tuning based on regions of parameters corresponding to nonovershooting, monot onic, or monotonic & onepulse control achieved by computer simulation.
The aim of this paper is to present analytic rules for PID controller tuning that are simple and still result in good closed-loop behavior. The starting point has been the IMC-PID tuning rules that have achieved widespread industrial acceptance. The rule for the integral term has been modified to improve disturbance rejection for integrating processes. Furthermore, rather than deriving separate rules for each transfer function model, there is a just a single tuning rule for a first-order or second-order time delay model. Simple analytic rules for model reduction are presented to obtain a model in this form, including the “half rule” for obtaining the effective time delay.
This paper presents new modular approach to the design of SISO
systems with constrained input. It combines the bang-bang minimum time
control and the linear pole assignment control into a new minimum time
pole assignment (MTPA) controller which is extended by an observer based
windupless I-action
This paper describes a blended learning approach in teaching Constrained Time-Delayed Proportional-Integral-Derivative Control. It is based on the learning by doing paradigm supported by several e-learning tools: by the interactive electronic course materials in Moodle; by the laboratory of low-cost and easy-to-manipulate plants; and by the virtual laboratory WebLAB. These open the access to study and experiment on real systems via Internet for 24 h/day, seven days a week. Synergy of the newly structured theory developed to deal with real-world constraints with the interactive content delivery and with active learning based on simulations, real-time experiments in laboratories, and tele-experiments via Internet offers stimulating and quasi-authentic learning environment. These give students multiple formative feedback, ranging from computer quizzes to plant reactions in a rich variety of authentic situations.
Almost every paper describing a new control algorithm attempts to prove superiority to established controllers such as the PID. However, there exist no accepted standards against which performance may be judged. Some claims turn out to be false, some misleading, in that they are presented under especially favourable conditions, and some not proven at all. In this paper, the author proposes a series of tests which may be used to compare one controller against another, and even against an absolute performance goal which is believed to be a limiting characteristic of feedback controllers in general. Dead-time dominant, self-regulating, non-self-regulating, and unstable processes are examined, each placing its own particular demands on the controller.
The paper describes two-degree of freedom analog PI and PID controllers tuning by a multiple dominant pole method for integral plus time delay plants. The effectiveness of the proposed techniques is shown in the example.
1. Overview of Control Systems.- 2. Some Current Techniques for PID Controller Design.- 3. The Hermite-Biehler Theorem and Its Generalization.- 4. Stabilization of Linear Time-invariant Plants Using PID Controllers.- 5. Optimal Design Using PID Controllers.- 6. Robust and Non-fragile PID Controller Design.- 7. Stabilization of First-order Systems with Time Delay.- 8. Constant Gain Stabilization with Desired Damping.- 9. Constant Gain Stabilization of Discrete-time Plants.- References.
The objective of most published tuning rules is “tight” or fast control, subject to avhieving acceptable robustness. This gives a maximum limit on the controller gain. In industrial practice, however, the objective is often “smooth” or slow control, subject to achieving acceptable performance in terms of disturbance rejection. This paper provides the corresponding minimum limit on the controller gain. In terms of appropriately scaled variables, the minimum controller gain is approximately 1, which justifies default settings commonly used in industry.
Model uncertainty is one of the major problems facing control system designers in practice. In this paper a general approach is outlined for the assessment of the effects of model uncertainty on control system performance. Also discussed are methods for the design of controllers which meet given performance specifications despite model uncertainty. The power of the new techniques as well as the failure of the traditional ones is illustrated on two examples widely studied in the process control literature: The design of Smith predictor controllers and the design of 2-point composition controllers for high purity distillation columns.
This chapter gives an overview of simple controllers for SISO systems based on the generalization of the linear pole assignment
method for constrained systems with dynamics ranging from relay minimum time systems to linear pole assignment systems. The
design is based on splitting the nth-order system dynamics into n first-order ones, which can be constrained without any problems with stability and overshooting. It requires a successive
decrease of the distance of the representative point from the next invariant set with lower dimension. Since the distance
of the representative point to such invariant set can be defined in many ways, the construction of the constrained controllers
is not unique. The controllers derived from the second-order integrator are simple, appropriate also for extremely fast application,
and easy to tune by a procedure that generalizes the well-known methods by Ziegler and Nichols, or ?Aström and Hägglund, respectively.
The Lambert W function has been used in an extremely wide variety of applications, including the stability analysis of fractional-order as well as integer-order time-delay systems. In this paper, we re-examine an application of using the Lambert W function through actually constructing the root distributions of the derived TCEs of some chosen orders. It is found that the rightmost root of the original TCE is not necessarily a principal branch Lambert W function solution, and that a derived TCE obtained by taking the nth power of the original TCE introduces superfluous roots to the system. With these observations, some deficiencies displayed in the literature are pointed out. Moreover, we clarify the correct use of Lambert W function to stability analysis of a class of time-delay systems. This actually enlarges the application scope of the Lambert W function, which is becoming a standard library function for various commercial symbolic software packages, to time-delay systems.
This paper presents an efficient numerical method for designing PI controllers. The design is based on optimization of load disturbance rejection with constraints on sensitivity and weighting of set point response. Thus, the formulation of the design problem captures three essential aspects of industrial control problems, leading to a non-convex optimization problem. Efficient ways to solve the problem are presented.
Advances in Three Term Control
- L H Keel
- Y C Kim
- S P Bhattacharyya
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World Congress Seoul, Korea, 2008.
Constrained pole assignment control, In: Current Trends in Nonlinear Systems and Control
- M Huba
- L Menini
- L Zaccarian
Huba, M. (2006). Constrained pole assignment control, In:
Current Trends in Nonlinear Systems and Control, L.
Menini, L. Zaccarian, Ch. T. Abdallah, Edts., Boston:
Birkhäuser, 163-183.
Analiza i synteza ukladów regulacji z opóznieniem. Wydawnictwo Naukowo Techniczne Warszawa
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Górecki, H. (1971). Analiza i synteza ukladów regulacji
z opóznieniem. Wydawnictwo Naukowo Techniczne
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metoda násobného dominantního pólu pro regulátory se dvema stupni volnosti a proporcionální soustavy s dopravním zpoždením Optimum settings for automatic controllers
- M Vítecková
- A Vítecek
- Vyšná
- Boca
- J G Slovak Republic Ziegler
- N B Nichols
Vítecková, M., Vítecek, A. (2010) metoda násobného
dominantního pólu pro regulátory se dvema stupni
volnosti a proporcionální soustavy s dopravním
zpoždením. Int. Conference Cybernetics and Informatics
February 10 -13, VYŠNÁ BOCA, Slovak Republic
Ziegler,J.G. and Nichols,N.B. (1942). Optimum settings for
automatic controllers. Trans. ASME, 759-768.
2010) metoda násobného dominantního pólu pro regulátory se dvema stupni volnosti a proporcionální soustavy s dopravním zpoždením
- M Vítecková
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