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This paper faces two of the main drawbacks in networked control systems: bandwidth constraints and timevarying delays. The bandwidth limitations are solved by using multirate control techniques. The resultant multirate controller must ensure closed-loop stability in the presence of time-varying delays. Some stability conditions and a state feedback controller design are formulated in terms of linear matrix inequalities. The theoretical proposal is validated in two different experimental environments: a crane-based test-bed over Ethernet, and a maglev based platform over Profibus.

Networked control systems (NCSs) [

In this work, neither packet dropouts nor packet disorder will be considered (as later detailed). In addition, devices will be assumed to be synchronized (by means of a suitable initial synchronization procedure or by implementing time-stamping techniques). Thus, only bandwidth limitations and time-varying delays will be faced. To solve these issues, some previous authors’ developments such as those in [

Regarding bandwidth constraints, this could be the case if the network configuration imposes a limitation in the control frequency (say, because of an excessive number of devices sharing the communication link). In this context, the consideration of a dual-rate controller [

As time-varying delays are assumed in an NCS, the control problem becomes a linear time-varying (LTV) one. Then, stability and design for arbitrary time-varying network delays must be carried out. Depending on what kind of information about the delay is provided, different stability analysis can be performed. So, if the probability density function of the network delay is unknown, a robust stability analysis must be proved. However, if the probability function is provided, stochastic stability can be analyzed. In this work, in order to prove any of these situations in a dual-rate NCS, linear matrix inequalities (LMIs) [

As a summary, the main novelties introduced by this work can be lumped as the followoing.

NCS analysis improvements: in [

NCS design improvements: firstly, in [

The paper is divided into five parts: in Section

Depending on the network configuration, three main options arise when integrating a dual-rate controller in an NCS.

The dual-rate controller is located at a remote side (with no direct link to the plant), and its fast-rate control actions can be sent from this side to the local actuator (directly connected to the plant) following a packet-based approach [

The dual-rate controller is located at the local side, being directly connected to the actuator [

The dual-rate controller is split into two subcontrollers [

From the last option, Figure

The sensor works in a time-triggered operation mode, sampling the process output

After a certain processing and propagation time has elapsed

Then, at the remote controller an event is triggered. As a consequence, after a computation delay

After a certain processing and propagation time has elapsed

Then, at the local side an event is triggered. Its main consequence is the application of N faster-rate control actions to the process. Such actions are scheduled to be applied taking into account the total delay:

Chronogram of the proposed NCS.

Regarding the

Consider a continuous linear time-invariant plant

When the input change and output sampling do not follow such a conventional sampling pattern, but they follow an arbitrary but periodic one with period

Consider the above system (

In a networked control framework, since

As described, an MRIC strategy is considered in this work, and hence only the first sampled output

In this paper, two different structures for the controller will be taken into account: a one-degree-of-freedom linear regulator

For the first case, the regulator

an output feedback regulator, whose lifted discrete realization will be

where set-points are considered to be zero, so

a state feedback regulator, with a gain

From the previous plant representation

For the state feedback regulator, the closed-loop realization will yield the following [

Regarding the second regulator, the hierarchical one

fast-rate local subcontrollers are designed by means of robust

a coordinating, slow-rate remote subcontroller is designed using a state feedback approach (to be detailed in Section

Then, the representation of each local subcontroller will be similar to (

Let us denote the lifted expression

As commented, time-varying delays can appear in an NCS framework. Thus, a variation in the instants where the outputs are measured (

Three different scenarios will be studied as follows.

Consideration of arbitrary delay changes with unknown probability: a robust stability analysis will be needed.

A probability density function of the network delay for each network situation is assumed known: a stochastic stability analysis can be independently carried out for each situation.

Several network states are considered, which are defined by different probability density functions and different performance objectives: a multiobjective analysis will be developed.

In order to prove robust stability of the discrete LTV system:

If

Now, a probabilistic model of the network delay

As the network delay is supposed to vary in a random way, stability of the closed-loop system will be analyzed in the mean square sense [

Replacing the closed-loop equations in (

For a generic probability distribution, working with the above integral may be cumbersome. For bounded

Note that the above results are more relaxed than those in the robust case. Indeed, in a probabilistic case there is only one LMI constraint (average decay) instead of one for each possible sampling period. In this way, temporal random increases of the Lyapunov function are tolerated as long as the average over time is decreasing. Hence, better results in stability analysis can be obtained; however, the gridding approach leaves intermediate points out of the analysis so, in rigor, the results are not valid unless the grid is very fine.

The above idea can be extended to considering several possible network states, say

Then, the Lyapunov decrescence conditions can be written as the following probabilistic LMI (expressed, for computation, in its discrete approximation):

It is well known that the optimal result of multiobjective analysis will be a Pareto front with the optimal performance

Note that the use of the shared Lyapunov function in (

Apart from stability and decay rate, well-known LMI conditions can be set up for pole region placement,

As previously shown, from a lifted model

Due to the existence of time-varying delays, the lifted state

In this section, a test-bed Ethernet environment is used to implement a dual-rate NCS, where the controller will be split into two parts. The proposed NCS includes the following devices (see Figure

an industrial crane platform (to be controlled) equipped with three cc motors (to actuate each axis:

Test-bed Ethernet environment.

Details on the crane characteristics can be obtained at

A local computer which is connected to the platform by means of a DAQ board, and where the local subcontroller is implemented.

Two PLCs and one computer working as interference nodes in order to introduce different load scenarios.

A switch shared by the previous devices to connect them to Ethernet.

A remote laptop computer where the remote subcontroller is implemented.

In this example, the controller will be a dual-rate PID one. Its parameters will be retuned according to Ethernet network delays, leading to a gain-scheduling proposal. In this case, the scheduling follows a Taylor-series-based approach (see [

First of all, several experiments are carried out, where the number and complexity of the tasks developed by the interference nodes is modified in order to obtain different load scenarios ranging between the two extreme histograms on Figure

Experimental delay histograms.

Since the crane model (

Let us suppose that no information about probability distribution is known. Then, the worst-case behavior of the proposed PD regulator can be assessed by means of the LMIs in (

LMI decay rate for the dual-rate PD controller (robust stability).

Max. delay | Scheduled | Nominal |
---|---|---|

bound (in s) | PD | PD |

0.1 | 0.42 | 0.59 |

0.15 | 0.63 | 0.73 |

0.20 | 0.84 | 0.84 |

0.30 | 1 | 0.99 |

As a conclusion, the proposed gain scheduled regulator improves worst-case performance for small delays (up to 0.2 s). In large delays, the approach used for retuning the PD parameters (based on Taylor-series) loses precision and results are similar (marginally worse) than those of a nonscheduled regulator. In fact, there are delay distributions involving delays larger than 0.3 s which might render the system unstable.

Now, information about probability distribution provided by experimental tests is taken into account. So, stability of the setup in probabilistic time-varying delays can be assessed. The LMI gridding in (

Two cases are analyzed as follows:

firstly, considering each network situation separately (a different Lyapunov function for each load scenario), the LMI in (

secondly, a multiobjective analysis is performed by considering a unique Lyapunov function for both network load scenarios.

The second case is more conservative but allows stability guarantees for mixtures and random switching between both scenarios. The two proposed cases are somehow extreme situations from which would happen in a practical situation. If each of the network behaviors is very likely to remain active for a dwell time significantly longer than the loop’s settling time, then assumptions in case 1 will be closer to reality. If arbitrary, fast, network load changes were expected, then case 1 would be too optimistic and the analysis in case 2 would be recommended.

Regarding the first analysis, results are presented in Table

LMI decay rate for the dual-rate PD controller (with probability information).

Network context | Scheduled PD | Nominal PD |
---|---|---|

Only unloaded | 0.50 | 0.65 |

Only loaded | 0.68 | 0.83 |

Now, the second (multiobjective) study is carried out. Figure

LMI decay-rate for the multiobjective study.

In summary, from the analysis of both Tables

if the probability of large delays is low, the use of probabilistic information indicates (as intuitively expected) that the gain scheduling approach used in this example (based on Taylor-series) seems a sensible practical procedure, because of improving average performance.

if no likelihood of (transient) instability is required, then the network must be reconfigured so the maximum delay does not exceed 0.3 s, or the initial controller specifications must be changed (reducing gains to improve robustness).

Since the previous figures indicate only stability and decay rate, to complete the study the control system time response is obtained. So, other performance differences (such as overshoot) can be evaluated.

Figures

Experimental closed loop output (unloaded network).

Experimental closed loop output (loaded network).

In this example, the position of a triangular platform assembled by joining three maglevs is hierarchically controlled by means of a dual-rate controller over Profibus. The proposed NCS (see Figure

a levitated platform with an equilateral triangle shape where each maglev is located at each corner. The maglevs provide position information from an infrared sensor array in

a National Instruments CompactRio 9074 acting as local subcontrollers,

a desktop PC acting as a remote subcontroller,

a Profibus-DP network configured to work with a bus rate of 187.5 kBits/s, and with asynchronous operation mode. This enables sending a remote control action every 20 ms.

Hierarchical control structure.

In this example, a standalone, fast-rate local subcontroller is designed for each maglev by using robust

After carrying out several experimental tests, network-induced time delays are measured (see histogram in Table

Experimental network round-trip time delay histogram.

Delay | 5 ms | 10 ms | 15 ms |
---|---|---|---|

Occurrences | 123,154 | 1,084,502 | 292,357 |

Percentage | 8.21% | 72.3% | 19.49% |

Now, the linearized state space for a generic maglev

From this representation, and using the robust control toolbox in Matlab, a fast-rate, local

If the coupled global platform model is obtained (details omitted for brevity; more information in [

From the previous model, the resulting state feedback controller in (

As the design strategy contemplated in this example is the same than that used in [

Once the dual-rate controller is designed, the next experiment is carried out in the proposed network scenario (adding a nonstationary Kalman filter as a state observer).

The experiment starts with the platform in equilibrium point, as shown in Figure

Figure results for experiments 1 and 2.

Next, as position error is presented, a new remote subcontroller that includes accumulated error in system state is developed. So the controller is designed considering

Following this reasoning, the system state vector is expanded by adding the accumulated error for each one of the three maglevs. According to this new plant model, the feedback state controller is recalculated via LMI gridding, obtaining a new state feedback gain

In this paper, in order to face arbitrary time-varying delays in a dual-rate NCS framework, different stability conditions and a state feedback design approach are presented in terms of LMIs. Multirate control techniques are proposed to avoid bandwidth limitations.

Regarding the stability conditions, three scenarios are treated: the robust case, the probabilistic case, and its extension to the multiobjective case. With respect to the state feedback controller, it is designed to assure robust stability for any possible time delay measured for the considered network.

Experimental results from two different dual-rate NCS implementations (a crane system over Ethernet, and a maglev-based platform over Profibus) validates the applicability of these LMI-based dual-rate control techniques.

The authors A. Cuenca, R. Pizá, and J. Salt are grateful to the Spanish Ministry of Education research Grants DPI2011-28507-C02-01 and DPI2009-14744-C03-03, and Generalitat Valenciana Grant GV/2010/018. A. Sala is grateful to the financial support of Spanish Ministry of Economy research Grant DPI2011-27845-C02-01, and Generalitat Valenciana Grant PROMETEO/2008/088.