Efficient Control of a SmartWheel via Internet with Compensation of Variable Delays✩ In´es Tejadoa , S. Hassan HosseinNiaa , Blas M. Vinagrea , YangQuan Chenb a
b
Industrial Engineering School, University of Extremadura, 06006 Badajoz, Spain Center for Self-Organizing and Intelligent Systems, Utah State University, 84322 Logan, UT, USA
Abstract Control through networks requires effective strategies due to the presence of induced phenomena. This paper deals with the compensation of the effects of time-varying delays when controlling a delay-dominated system, called SmartWheel (SW), over the Internet in the framework of fractional order control (FOC). In particular, the gains of a nominal fractional order PIα controller, designed for a local purpose, are adapted with respect to the current delay condition to perform an efficient control over the network. The quadratic stability of the controlled system is proved by a frequency domain method on the basis of switching systems theory. Experimental results are given to show the effectiveness of the proposed control strategy, referred to the significantly better system performance in comparison with the application of the nominal PIα , especially for large Internet-induced delays. Keywords: Networked control system, Variable delay, Gain scheduling, Fractional order PI controller, Stability, SmartWheel NCS testbed. 1. Introduction Networked control systems (NCS) have been one of the main research interests in industrial applications for many decades, especially due to ad✩
This work was supported by the Spanish Ministry of Science and Innovation under the research project TRA2008-06602-C03-01. Email addresses:
[email protected] (In´es Tejado),
[email protected] (S. Hassan HosseinNia),
[email protected] (Blas M. Vinagre),
[email protected] (YangQuan Chen) Preprint submitted to Mechatronics
February 8, 2013
vances on data networking technologies, which introduced the concept of remote control of a system, and the networks potential for real-time high performance control. However, the insertion of a communication network in the feedback loop imposes additional challenges to the control system, mainly concerning time-varying transmission delays, packet losses or the consideration of band-limited channels (refer to e.g. [1, 2, 3, 4] for more details). Network-induced delays are the major drawback analysed in the technical literature since they may degrade the control performance, or even cause the system instability. Among the existing networked control strategies (see a survey in [5]), recently one effective trend is to compensate the delay effects in real-time with gain scheduling (GS) (e.g. [6, 7, 8, 9, 10, 11]). GS offers some advantages in comparison with other real-time strategies. Firstly, it allows to keep the original structure of a designed (nominal) controller, i.e., avoids the replacement of a widely used controller, which can be costly and time-consuming, extending its capability to efficient control over a network [6]. Moreover, GS can be simple and easy to implement in real applications. Nowadays, it is well-known that fractional order control (FOC) and its applications are becoming an important research topic since they translate into more tuning parameters, i.e., more adjustable time and frequency responses of the control system, allowing more robust performances to be attained (refer to e.g. [12] and references therein). In the last few years, fractional order strategies have been successfully applied to network-based control [13, 14, 15, 16, 17, 18]. In particular, the compensation of timevarying delays with fractional order strategies, by means of GS [19, 20] and gain and order scheduling [21, 22, 23], is one of our recent research interests. Inspired by the aforementioned results, the intention of this paper is to design and validate an effective strategy to control a delay-dominated system, called SmartWheel (SW), through the Internet with robustness to variable network delay. More precisely, a fractional gain scheduled controller (FGSC) will be designed to compensate the effect of time-varying delays. The contributions of this paper are: 1) the application and experimental validation of the GS approach proposed in [19] to the SW to compensate Internet-induced delays taking into account the previous characterization of the used Internet link –a preliminary version of this work appeared in [20], and 2) the stability analysis of the previous controlled system, on the basis of the theory of switching systems. The rest of the paper is organized as follows. In Section 2, the problem of the network-based scenario studied in this paper is stated. Section 2
3 addresses the design of the gain scheduled controller, including its corresponding stability analysis. Experimental results are given in Section 4 to demonstrate the effectiveness of the proposed control strategy. Section 5 draws the main conclusions of this paper. 2. Problem statement The experimental set-up to be considered is the Internet-based control of the SW, placed at the Center for Self-Organizing and Intelligent Systems (CSOIS), Utah State University, USA, from the University of Extremadura, Spain. The platform (see Fig. 1) consists of a self-contained wheel module, with separate drive and steering motors, and a linear actuator, which enables the wheel to be moved along the z-axis. The communication with this platform is performed through a serial server, which converts messages from RS232 format into TCP/IP format and vice versa. An Internet camera, located near the SW assembly, enables one to view the motion of the wheel axis in real-time during testing. In this paper, only the angular position of its steering axis will be actuated, whose dynamical model is given by the following first order plus delay transfer function [17]: G(s) =
K 0.1484 −0.592s . e−Ls = e Ts + 1 0.045s + 1
(1)
Figure 1: Photo of the SmartWheel NCS Testbed at CSOIS, Utah State University (USA) System dynamics can be conveniently characterized by the normalized L time delay τ , τ = L+T , which is essentially the classical controllability ratio 3
L/T [24], but belonging to the range (0, 1). This parameter can be used to characterize the difficulty of controlling a process. Roughly speaking, processes with small τ can be considered easy to control and the difficulty in controlling the system increases as τ increases, which is the case of the SW. In [16, 17], the performance of the SW through a network was examined by using both integer and fractional order strategies. It was demonstrated that its best performance can be obtained by using a PIα controller –henceforth, also referred to as FOPI– of the form Ki zc C(s) = Kp + α = Kp 1 + α , (2) s s where zc = Ki /Kp is the ratio between controller gains, designed based on a fractional Ms constrained integral gain optimization method (F-MIGO) to fractional order controllers [25]. Thus, taking into account the system transfer function (1), the FOPI controller is given by Kp = 2.1586, Ki = 5.9853 and α = 1.1. Figure 2 shows the system response when applying the nominal FOPI controller for a speed reference of 2 rad/s. The solid blue line corresponds to the experimental filtered data, whereas the dashed red and dotted black ones refer to the simulated response and the reference, respectively. As can be observed, the experimental response is stable but there exists a permanent oscillation in steady state due to the cogging effect and the asymmetry of the SW stand. Likewise, delays of multi-purpose networks are time-varying, random, non-uniform [26] and depending on the number of users connected to the network. For this application, time-varying delays were measured and modelled to characterize the Internet link between the University of Extremadura and CSOIS networks. In particular, six different sets of round trip time (RTT) measurements were obtained [18]. Figure 3 illustrates the RTT delay histograms of the data measured for two sets, in which PDF refers to their probability density distribution function. As can be observed, the histograms skew to the left, indicating the higher probability to have delays shorter than the median and mean delays. Under those circumstances, it was shown in [20] that the controlled system performance can be degraded if network-induced delay is increased when applying the nominal controller. This issue can be also demonstrated by means of the concept of jitter margin, i.e., how much
4
additional delay the control loop tolerates, which can be calculated as jm =
φm π , 180ωc
(3)
where φm is the phase margin at the crossover frequency ωc [27, 28]. Inherently, the SW is a delay-dominated system, feature which makes its control difficult itself. Indeed, the situation would be even worse when control it through Internet. Given this motivation, a strategy which enjoys both performance advantage and practical easiness is desirable. Thus, the aim of this work is to perform an efficient network-based control of the SW avoiding the replacement of the PIα controller (2) due to its simple structure and appropriate behaviour for the system. As a result, a gain scheduled approach to compensate the effects of time-varying delays s required in order to achieve an adequate control performance of the SW through the Internet.
Figure 2: SW response when applying the nominal FOPI controller (experimental and simulated)
3. Control strategy This section includes the main issues involved in designing the FGSC to avoid a decreased control performance of the SW due to time-varying delays. The quadratic stability of the controlled system will be proved from the viewpoint of switching systems theory, specifically by using a frequency domain method. 5
Figure 3: Histograms of two sets of RTT delays measured between the University of Extremadura network to http://www.hust.edu.cn, which the SW is connected to: (a) Test 1 (b) Test 4
6
3.1. Design of the scheduled controller This strategy was developed inspired by [6] but introducing the Nyquist stability criterion instead of the root locus diagram during the optimization to do more precise the process of obtaining the optimal values of the scheduled variable with respect to the network condition. The way of estimating the network delay was also changed. Refer to [19] for the full description. Essentially, this approach deals with compensating the effects of variable network-induced delays by means of GS (see its structure in Fig. 4). It can be seen that the gain scheduler will modify the gains of the nominal controller –a PIα controller for the considered application–, via an external gain β, taking into account the current network condition. To do so, the optimal values of the external gain β (βop ) with respect to the current network delay τnetwork are required, and will be determined via an off-line closedloop simulation of the system by minimizing a defined cost function J. On the other hand, the function of the delay estimator (DS) is to estimate the current network condition, which is then utilized by the gain scheduler. To this respect, a feasible approach for real-time applications lies in obtaining the delay distribution, since the expectation value of the delay can be used at each sampling time to estimate the current network condition [29, 30].
Figure 4: Structure of GS for the networked SW For this application, J is assumed to be: ∞ e2 (t)dt, J = ISE =
(4)
−∞
where ISE is the integral of the squared error and e is the error signal. Its appearance, considering delay intervals of 100 ms, is of the form plotted in 7
Fig. 5 for different values of τnetwork . The FGSC is characterized by the values of βop represented in Fig. 6. Note that βop is not always lower than 1, as can be expected to reduce the closed-loop bandwidth, due to the different design criteria used for the nominal FOPI and FGSC controllers: the FOPI controller, which is taken as the non-scheduled controller, is based on the fractional Ms constrained integral gain optimization method (F-MIGO) method (see detailed information in [25]), whereas GS is calculated by using the ISE criterion. Refer to [20] for more details of the design. Concerning the distribution of Internet delays, and in accordance with [18], they can be better characterized by a random Gamma distribution as follows: 1 σ−1 −τnet P [τnetwork ] = , (5) τ e Γ(σ) net where σ is the standard deviation of the RTT measurements. This distribution will be used to estimate the current network delay in the experiments.
Figure 5: Cost function J
3.2. Stability analysis In the literature, NCS are commonly treated as hybrid and/or switching systems [4], so the stability problem of such systems can be analysed on the basis of switching systems. In our case of study, GS and networked control 8
Figure 6: Optimal values of β lead to the consideration of time-varying switching systems, where the system and the controllers can be represented as follows: 0.1484 −(0.592+τj )s e , (6) 0.045s + 1 5.9853 Cj (s) = βj 2.1586 + 1.1 , j = 1, 2, ..., 13, (7) s where τj refers to the network delay τnetwork and βj is the gain scheduler with the switching parameters given in Tab. 1. Hence, there are 13 subsystems to be considered. Gj (s) =
Table 1: System and controller parameters for each switching j τ β j τ β
1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 1.6 1.35 1.3 1.15 1 0.9 7 8 9 10 11 12 13 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.8 0.7 0.65 0.6 0.55 0.5 0.45
Consider a switching system given by x˙ = Ax, A ∈ co {A1 , ..., AL } , 9
(8)
where co denotes a convex combination and Ai , i = 1, ..., L, is the switching subsystem, and its characteristic polynomial of order n as: c(s) = sn + cn−1 s(n−1) + · · · + c1 s + c0 , with
⎡ −cn−1 −cn−2 ⎢ 1 0 ⎢ ⎢ 0 1 A=⎢ ⎢ .. .. ⎣ . . 0 0
⎤ · · · −c1 −c0 ··· 0 0 ⎥ ⎥ ··· 0 0 ⎥ ⎥. .. .. ⎥ .. . . . ⎦ ··· 1 0
(9)
(10)
A system described by (8) is quadratically stable if and only if there exists a matrix P = P T > 0, P ∈ Rn × n, such that ATi P + P Ai < 0, ∀i = 1, ..., L [31]. And, equivalently in the frequency domain, system (8) is quadratically stable if and only if |arg(c1 (jω)) − arg(c2 (jω))| <
π , ∀ω, 2
where c1 (s) and c2 (s) are two stable polynomials of order n corresponding to the subsystems x˙ = A1 x and x˙ = A2 x, respectively [32]. Let assume that switching between the controllers is not arbitrary and happens between each pair separately –step by step. Thus, the following 12 conditions should be satisfied to guarantee the quadratic stability of the controlled system given by (6) and (7): π , ∀ω, 2 π |arg(c2 (jω)) − arg(c3 (jω))| < , ∀ω, 2 .. . π |arg(c12 (jω)) − arg(c13 (jω))| < , ∀ω, 2 |arg(c1 (jω)) − arg(c2 (jω))| <
where cj , j = 1, 2, ..., 13, denotes the closed-loop polynomial of the system. It is important to remark that, due to the fact that the controller parameters and network delay are time varying, the stability analysis of the closed-loop system is difficult to establish for all possible cases of process switching. As 10
a matter of fact, it is a common practice to assume slow variations of the operating conditions and, consequently, suppose that the scheduled variable, the external gain β in this case, vary slowly [33] or step by step. Therefore, the stability method proposed here makes sense. Figure 7 shows the phase difference between each pair of characteristic polynomials of the closed-loop system for step by step switching. It can be observed that the system is quadratically stable, with the maximum phase difference equal to 30◦ , less than 90◦ . It should be remarked that the delay was approximated by using a third order Pad´e approximation in the frequency range of [0.01, 100] rad/s, so the results are valid in this range of frequencies.
Figure 7: Phase difference between each pair of characteristic polynomials of the closed-loop system, i.e., (c1 , c2 ), (c3 , c4 ), · · · , and (c12 , c13 ) for step by step switching
4. Experimental results This section concerns the validation of the previously designed strategy for the steering speed control of the SW through the Internet. For clarity purposes, the following tests have in common: • After initialization at the beginning of the process, the DS will periodically estimate the current network condition at every sampling time –Ts = 0.5 s– based on RTT measurements, in the same way as in [6], but using the Gamma distribution instead of the exponential. Then the gain scheduler will update the control signal to be sent to the remote system. 11
Figure 8: Summary of steps of the experimental set-up at every sampling time
• With respect to the FOPI controller (2), in order to preserve the integral effect, its integral part will be implemented as s−α = s2−α s−2 , where the term s2−α is approximated by the modified Oustaloup’s method (e.g. see [12]) of order equal to 7 (number of poles and zeros) in the [0.001, 1000] rad/s range. To perform the speed control of the SW over the Internet, a hardware in the loop based on Matlab/Simulink is needed, which allows the actual Internet delay to be reflected in the system automatically. A flow chart of the overall experimental set-up is illustrated in Fig. 8 (refer to [16] for more details). Next, the effectiveness of the FGSC, in comparison with the non-scheduled FOPI controller –controller with constant gains–, will be investigated considering different network conditions. More precisely, the condition of the net12
work will be changed at some stage of each experiment1 and, consequently, the delay will rise to τnetwork + τadd , where τnetwork is the real Internet delay –indeed, time-varying. Therefore, the experimental tests involve adding manually a constant delay τadd to the Internet-induced one so as to consider changes in its mean of values: (a) τadd,1 = 0.2 s (b) τadd,2 = 0.4 s and (c) τadd,3 = 0.6 s. It is important to remark that higher values of this additional delay are not added due to the risk for the experimental plant. Figure 9 shows the SW responses when applying the FOPI and FGSC controllers for the different network conditions τadd,1 , τadd,2 and τadd,3 , where the solid blue and dashed red lines correspond to the results obtained with the fractional gain scheduling strategy and the non-scheduled controller, respectively2 . It can be stated that the higher the values of τadd , the better system performance by using the FGSC in comparison with the non-scheduled controller. Due to this fact, FGSC can be an adequate strategy for efficient control in NCS, mainly for large network-induced delays.
Table 2: Values of ISE and overshoot
(a) (b) (c)
ISE error FOPI FGSC 3.96 3.96 4.84 4.64 6.01 5.42 8.00 7.05
Overshoot (%) I(%) FOPI FGSC I(%) − 2.00 2.00 − 4.6 12.65 12.6 0.4 9.8 22.4 13.5 39.5 11.8 46.9 22.2 52.7
For comparison purposes, Table 2 gives the values two indices measured to evaluate the performance of both strategies: the ISE error (or J) and 1
These changes could be originated, for example, from a network topology change, where a node has fallen off or moved in an ad-hoc network and the path (and delay) has become longer (see [30]). It can be also the way to model a communication band-limited link [2]. 2 For clarity purposes, experimental data in Fig. 2 and 9 were filtered by a first order low-pass filter with cut-off frequency ωcf = 2.5 rad/s.
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Figure 9: SW responses when applying FOPI and FGSC controllers for different network conditions: (a) τadd,1 = 0.2 s (b) τadd,2 = 0.4 s (c) τadd,3 = 0.6 s the overshoot. The improvement I of the FGSC with respect to the FOPI controller was defined as: indexF OP I − indexF GSC 100, (11) I(%) = indexF OP I where indexF OP I and indexF GSC are the ISE error and the overshoot when applying the non-scheduled FOPI and FGSC controllers, respectively. The examination of the previous step responses and results in Table 2 leads to the conclusion that the system performance with the FGSC controller is significantly better than with the non-scheduled one. Hence, FGSC can adapt its gains, and consequently the control signal, suitably to the current network condition. In addition, the higher the value of the network delay, the higher improvement in the system performance when applying this strategy, especially in terms of overshoot. 5. Conclusions In this paper, a fractional order strategy has been proposed for efficient control performance of a delay-dominated system via the Internet based on 14
gain scheduling. Thus, the gains of a local-purpose (nominal) PIα controller have been adapted with respect to the current network condition thanks to the minimization of a defined cost function. The quadratic stability of the controlled system has been proved in the frequency domain on the basis of switching systems theory. Experimental results have demonstrated that GS can adapt the controller gains suitably in real-time, maintaining the system performance in a satisfactory level and reducing deterioration of the system performance, especially in presence of large Internet-induced delays. To be more precise, the larger the value of the network delay, the higher the improvement obtained by the proposed fractional gain scheduled controller in comparison with the nominal PIα controller. Our future efforts will focus on towards analysing and testing different strategies to cancel the non-linearities presented in the SW. References [1] F.-Y. Wang, D. Liu, Networked Control Systems: Theory and Applications, Springer Publishing Company, Incorporated, 2008. [2] P. Bauer, New challenges in dynamical systems: The networked case, International Journal on Applied Mathematics and Computer Science 18 (3) (2008) 271–277. [3] J. Hespanha, P. Naghshtabrizi, J. Xu, A survey of recent results in networked control systems, Proceedings of the IEEE 95 (1) (2007) 138– 162. [4] D. Hristu-Varsakelis, W. Levine, Handbook of Networked and Embedded Control Systems, Birkh¨auser, 2005. [5] Y. Tipsuwan, M. Chow, Control methodologies in networked control systems, Control Engineering Practice 11 (10) (2003) 1099–1111. [6] Y. Tipsuwan, M. Chow, Gain scheduler middleware: A methodology to enable existing controllers for networked control and teleoperation Part I: Networked control, IEEE Transactions on Industrial Electronics 51 (6) (2004) 1218–1227. [7] Y. Tipsuwan, M. Chow, Gain scheduler middleware: A methodology to enable existing controllers for networked control and teleoperation 15
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