^{1}

^{*}

^{1}

^{1}

^{2}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

In many control applications, the sensor technology used for the measurement of the variable to be controlled is not able to maintain a restricted sampling period. In this context, the assumption of regular and uniform sampling pattern is questionable. Moreover, if the control action updating can be faster than the output measurement frequency in order to fulfill the proposed closed loop behavior, the solution is usually a multirate controller. There are some known aspects to be careful of when a multirate system (MR) is going to be designed. The proper multiplicity between input-output sampling periods, the proper controller structure, the existence of ripples and others issues need to be considered. A useful way to save time and achieve good results is to have an assisted computer design tool. An interactive simulation tool to deal with MR seems to be the right solution. In this paper this kind of simulation application is presented. It allows an easy understanding of the performance degrading or improvement when changing the multirate sampling pattern parameters. The tool was developed using Sysquake, a Matlab-like language with fast execution and powerful graphic facilities. It can be delivered as an executable. In the paper a detailed explanation of MR treatment is also included and the design of four different MR controllers with flexible structure to be adapted to different schemes will also be presented. The Smith's predictor in these MR schemes is also explained, justified and used when time delays appear. Finally some interesting observations achieved using this interactive tool are included.

A multirate sampling (MR) system is defined as a hybrid system composed of continuous time elements, usually the plant, and some discrete time components, usually the controllers or the filters, where two or more variables are sampled or updated at different frequencies. It can be also considered that the discrete actions are not equally spaced on time and/or delayed. Moreover, in a great number of computer control applications the approximation of a regular pattern of sampled signals is assumed.

A non-very restrictive assumption to simplify the treatment is to consider that the sampling pattern is periodic. That is, the process variables are sampled and/or updated at different and/or irregular intervals, but there is a global period _{0} with cyclic repetition. It may be also considered that there is a delay between the sampling and the updating of variables, but a global periodicity is still assumed. The case of asynchronous sampling/updating, with a random occurrence of the discrete actions, is much more complicated and it will not be considered in this paper.

In a basic digital control system, a perfect uniform sampling and updating pattern of the involved variables is assumed, but it should be pointed out that, in practical applications, the synchronicity of the set of discrete actions is not perfect or it can be modified in order to improve the performances. Thus, MR is an important issue not only for research purposes but also from a practical point of view. MR may be present in a wide range of applications and the users must understand its consequences in an easy way. Chemical analyses, or samples obtained by artificial vision with post-processing requirements need a time interval that for a real-time process control request could be long. Some other similar problems appear when the sensors are spatially separated from the controller algorithm device: distillation columns, UAVs, network based control schemes,

The control target is to achieve similar performances to those the faster single rate controller would provide. However, in these cases, the theoretical analysis of the controlled system performances is much more computationally involved. The modeling, analysis and design steps can consume a great amount of engineering time.

In order to analyze and study the different characteristics of the dynamic behavior of a MR, it is common to use time and frequency techniques and tools. These will provide a complete and global picture of the system behavior, showing up the interrelation among the different controlled plant performances.

The combined use of control system design tools and dynamic system simulation, leads to the computer aided control system design environments, simplifying the design task. Additionally the possibility of simultaneous visualization in various windows of the effects in different performances of some design parameter changes helps to observe with more flexibility the change gradient over the system [

In this sense some years ago, Åström and colleagues at the Lund Institute introduced some valuable concepts for control education task aid. In this context the significance of concepts like dynamic pictures and virtual interactivity must be highlighted. This original idea was implemented in packages as Ictools and CCSdemo, that Johansson

In the MR case this kind of application appears indispensable. Some key concepts in order to model, analyze and design MR systems are overcome by the use of this interactive application. The working principles of MR are easily understood using this procedure. A specific Sysquake application was implemented for MR. This tool takes advantages of the fast execution and excellent graphic features that the use of Sysquake provides. In summary, the main motivations in writing of this paper are the following: (1) to study, in a very direct way, how the modification of the different sampling periods that are involved in a MR system can lead to unexpected behaviours; (2) to present in a unified way the performance of different MR controllers in order to choose by means of this tool the most appropriate design in each case; (3) to provide a tool that is not only useful to do research on this kind of systems, but also to improve the learning of MR concepts for beginners.

The structure of the paper is as follows: the next section introduces some preliminary material and foundations, definitions and notation; Section 3 presents the problem statement and provides the detailed design procedure of the assumed controllers. Section 4 describes the developed application using interactive simulation techniques, and then some interesting examples derived with this tool are given in Section 5. A conclusion section closes the paper.

The scope and purpose of this work has been exposed in the previous section. In this section, the general problem, basic notations and operations among signals and processes are going to be introduced. After exposing the kind of problems that a practitioner finds when consider this topic, the elemental signal change of frequency operations and its properties are presented. Another subsection is devoted to the notations in process transfer functions in the MR topic, some elemental transformations between polynomials as well as the available relations between fast-skipped and slow or slow-expanded and fast signals. Finally the discrete lifting, traditionally introduced in an internal representation way, is adapted to our algebra. It is a section that is a survey to follow the design procedure in Section 3. First of all, it must be noted that the systems this contribution deals with are known as MultiMate Systems, that is, systems where there are sampled or discrete signals referred to two or more different frequencies. An initial scheme could help to understand different issues related to this kind of systems (see

One option in order to describe the different signals and systems in these environments is to use notation with superscripts. The signal (or system, when it is the case) ^{T}^{NT}

This single example enables one to understand that the sampling period transformation between discrete signals or the sampling operations involving blocks of different nature is quite common in MR systems. One way to achieve a proper handling of this kind of systems requires management of the great common divisor (gcd) and the least common multiple (lcm) of every sampling period occurring in the studied MR scheme. With these magnitudes, every sampling period is going to be repeated an integer multiple of times in one lcm, _{0}. There also will be a base period (gcd), _{B}_{0} = _{B}_{0}_{B}_{0}/

Once the general MR problem has been introduced, the multirate input control case (MRIC), that is when the output sampling period is strictly greater than the input sampling period, will be assumed in the rest of this work. In these conditions, it is adequate to assume two sampling periods, input

Some operators adapted for the transformation between periods with integer multiplicity must be introduced. In this sense, it is useful to consider basic operations such as “skip” and “expand”, introduced by [

If the

the

the skip (downsampling) operator creates a

The skip operation applied to the

Some known skip-expand properties usually considered in this work are:

As an example guide to demonstrate every other property, the pattern “skip operation does not commute”,

Once some basic operations have been introduced, it is also very interesting to expose the application over transfer functions. From here, the expressions will be managed without the subindex “1” for the variable _{N}^{N}

Assuming the continuous time process in

The fast sampling DT (FSDT) model is defined by:
^{T}^{T}^{−1}. Following the notation in _{i,T}_{i,T}_{0,}_{T}_{0,}_{T}^{−1}.

For the same process, a slow sampling DT (SSDT) model can be similarly defined by:
^{NT}_{N}^{NT}_{N}^{−}^{N}

When the operation [•]^{Ti}

The FSDT transfer function poles are denoted by _{i,T}

If the SSDT poles are _{i}_{,}_{NT}

Note that, dealing with the same continuous time system, _{i}_{,}_{NT}_{i}_{,}_{T}^{N}

Two technical assumptions are made:

If α is a pole of ^{T}^{T}

All poles are different, _{i,T}_{j,T}

Assumption (a) is required to avoid aliasing, [

The FSDT model may be also expressed by:
^{T}_{T}_{T}_{P}_{T}_{T}

In what follows, some important results are proved:

From ^{NT}^{T}^{NT}

In ^{T}U^{T}^{NT}

The opposite situation is viable: the transformation of a slow frequency DT sequence into a fast frequency DT sequence. The dual rate zero order hold (DRZOH) device is defined by:

In this case, it is possible to obtain a transfer function of the process plus the DRZOH device:

Thus, a dual rate discrete time (DRDT) operator is defined by:
^{T,NT}

Using a similar notation, the DRZOH operation, ^{T}^{,}^{NT}

When facing the modeling step of a MR system, most authors suggest the use of the so-called “Lifting” or “Discrete Lifting” method [^{T}

That is, every sequence _{l}_{,}_{i}^{N}

In this way it is possible to say that:

It must be noted that

Anyway, there are different options for Lifting application. All of them assume the technique of “Vector Switch Decomposition” [

As exposed in Section 2.3, it is viable to express a fast signal as a sum of

As it is proved, the terms can be separated. Actually, it is a laborious procedure but it assures the feasibility to get a fast sampling period modeling from a dual-rate closed loop, as it will be shown in Sections 3 and 4.

The problem this contribution deals with appears when a computer control application, where the control algorithm implementation should be a _{p}

Based in this MRIC structure, that is slow measurement and fast control updating, in [^{T}^{NT}^{NT}^{T}

As

Nevertheless, if the dual rate modelling is considered at slow rate:

With a similar procedure, the following expression is reached:

In order to avoid the denominator complexity;

These expressions are useful when the simulation tool is implemented. Once the dual-rate closed loop modeling has been established ^{NT}^{T}R^{T}^{NT}^{NT}R^{NT}^{NT}^{T}

a fast part given by:

a slow part given by:

a rate converter with the form:

It is easy to observe that the dual rate controller is tuned for one type of command signal

In this case, the design procedure is based on a continuous closed loop including a controller _{R}^{T}^{NT}

If the process is non-minimum phase, the cancellation of unstable pole-zero pairs must be avoided. Thus, the fast part of the controller could be alternatively computed by

From the continuous time closed loop, it is possible to follow another design method. If some desired

With the proposed structure

For more information, reader is referred to [

The RST controller [

In this case,

This slow frequency control signal is modified by a dual rate part obtained from

Thus, the multi-rate controller can be redefined as follows:

In addition,

Leading to:

In the MR control design field it is difficult to find contributions about processes with time-delay. The projected interactive simulation application tackles this problem, even in the general case when the delay is integer or non-integer with respect to the fast

Revisiting the control scheme already shown in

With respect to the

If the process includes a transport delay

That is, the branch to subtract in order to avoid the effect of the delay.

It must be noted that (_{T}G_{p}^{−}^{ms}^{T}

To better understand the multi-rate control aspects previously presented, an interactive simulation application has been developed. An executable version can be downloaded online from [

Being aware interactivity is difficult to be shown in a writing text, next the different sections of the interactive application will be described. The application user interface presents a main window which is split into two parts (see in

Regarding the process parameters, two radio buttons enable selection of the process model complexity (first or second order). Data entries to define the consequent process parameters can be carried out via sliders or text boxes. Below these entries, a checkbox named _{p}

With respect to the controller parameters, the input data depend on the chosen controller. Thus, firstly, for PID controllers (illustrated in _{r}

Regarding the

With respect to

In this section an example for each controller will be presented. Output responses for the single-rate and multi-rate cases will be compared and analyzed, focusing on multi-rate control benefits.

Let us consider the example shown in

An acceptable continuous-time PID controller is given by _{p}_{D}_{I}

From

Let us consider the example shown in

In order to decrease around 10 times the settling time, the next closed-loop transfer function is chosen:

If the sampling time is

Let us consider the example shown in

An acceptable continuous-time PID controller is given by _{p}_{D}_{I}_{p1} = 1 and _{PD} = _{p}, and

In this example, the dual-rate response reaches time indexes whose values are between those obtained by the single-rate controllers. If some process time delay were considered, for example

Let us consider the next example, where the process is defined by:

In order to eliminate the overshoot and slightly decrease the settling time, the next closed-loop transfer function is chosen:

If the sampling time is

One solution for the problem of control schemes with slow sensors is to assume a MR system, considering a restricted slow measurement sampling but also a faster control updating. the use of an interactive simulation tool in order to study multirate systems is a feasible option to make proper decisions about the correct control design. Different non-conventional controller structures have been provided. From application some unexpected results are achieved and explained. Even for an expert in this field the tool appears absolutely essential and time saving. For a beginner student or researcher it is entirely necessary if the study of this kind of systems is needed/desired.

This work was supported in part by the Spanish Ministry of Economy and Competitiveness under Project DPI2012-31303. The authors J. Salt, A. Cuenca, are grateful to Grant TEC2012-31506, from the Spanish Ministry of Education. The work of Á. Cuenca was supported in part by the Spanish Ministerio de Economía under Grant DPI2011-28507-C02-01.

The authors declare no conflict of interest.

An initial MR System.

Expand operation. Case

Skip operation. Case

Dual rate controller structure.

Block diagram for the control system: multi-rate controller including an RST stage.

Dual rate scheme with Smith's Predictor.

RST Controller in a loop with dead time.

Application user interface for the PID controller.

Application user interface for the Model-based (PID) controller.

Application user interface for the Cancellation controller.

Slow-rate and dual-rate root locus graphic.

Application user interface for the Model-based (PID) controller with no oscillations.

Application user interface for the PID controller with process time delay and no predictor.

Application user interface for the PID controller with process time delay and predictors.

Application user interface for the RST controller.

Parameters available in the interactive simulation application.

kp, Ti, Td | PID proportional gain, integral time, and derivative time, respectively. |

P, PI, PD, PID | Four radio buttons to select the PID-type controller complexity. |

K, T.C. | Static gain and time constant, respectively, for a first order model/process. |

K, D.F., Wn | Static gain, damping factor, and natural frequency, respectively, for a second order model/process. |

1st Order, 2nd Order | Two radio buttons to select the model/process complexity. |

Integrator | A checkbox to add an integrator to the process (when activated). |

Oscillations | A checkbox to eliminate process output oscillations in Model-based controllers (when deactivated). |

N | Multiplicity. |

T | Sampling time. |

D | Process time delay. |

Switch to Root Locus | A link to show the discrete-time root locus plots. |

Switch to Pole-Zero Map | A link to come back to the continuous-time poles and zeroes map. |

Manual, AutoAdjust | Two radio buttons to select operation mode for figure scales. |

Predictor slow | A checkbox to include a Smith's Predictor in the slow-rate control algorithm (when activated). |

Predictor fast | A checkbox to include a Smith's Predictor in the fast-rate control algorithm (when activated). |

Predictor mrate | A checkbox to include a Smith's Predictor in the multi-rate control algorithm (when activated). |