In a networked control system (NCS) that is packet switched, when a periodically sampled variable is transmitted over the communication network it can be subject to the congestion related issues such as delay, delay variation, and packet drops as reported in Zampieri (2008), Donkers et al. (2011), and Premaratne (2014). In Wu et al. (2013), it is demonstrated that congestion can lead to increased packet drops. If the bandwidth intensive feedback loop from the plant output sensor to the correlator (Figure 1) is transmitted over a communication network, it would be subjected to the issues mentioned above.

A realistic example of such a control system is an extremum seeking control scheme such as that of Scotson and Wellstead (1990) being used for optimizing the ignition timing of a ship. Since, the sensor that measures the torque of the propeller shaft is often located at a significant distance from the engine timing controller as mentioned in Hao and Ji (2012), the use of a communication network to complete the feedback path is a justifiable solution. A furnace with a gas emission sensor located at a significant distance due to the operating temperature range of the sensor is another realistic example.

The rational for using sporadic (i.e., aperiodic) sampling is to mitigate congestion due to periodic sampling by decreasing the effective sampling rate. Studies by Arzen (1999), Astrom and Bernhardsson (2002), and Henningsson et al. (2008) have demonstrated the ability of sporadic sampling to reduce the effective sampling rate provided the exogenous disturbances of the control system have a bounded rate of change. Table V of Premaratne et al. (2017) provides empirical evidence that sporadic sampling can reduce the likelihood of packet drops due to decreased transmission. This is an additional advantage.

Possible sporadic sampling techniques include Memory-Based Event Triggering (MBET) proposed by Arzen (1999) and further developed in Lehmann and Lunze (2010) and Lunze and Lehmann (2010). In Event Triggered Adaptive Delta Modulation (ETADM) proposed by Premaratne et al. (2013), event triggering is combined with Adaptive Delta Modulation (ADM). The main motivation for investigation is the effect of the reconstruction error of the above sporadic sampling techniques on the vital correlation stage of the ESC scheme.

The contribution of this paper is the investigation of the convergence requirements and bandwidth performance of MBET and ETADM when used in the feedback path of a sinusoidally perturbed ESC scheme.

The background of the encoding schemes used is discussed under the Preliminaries (Section 2). In Section 3, the main results are discussed. These include the requirements for convergence of MBET (Section 3.1) and ETADM (Section 3.2). In Section 3.3, the bandwidth reduction of both encoding methods is compared.

The set of real numbers and positive integers (excluding zero) is represented by R and Z+, respectively. The absolute value of a variable z∈R is given by |z|. The letter k in square brackets is used to denote a periodic (clocked) discrete time sample. A class KL function is a two-argument function R2>0→R>0. The first argument is a function (known as a class K function) which is continuous, strictly increasing, R>0→R>0 and zero when the argument is zero. The second argument of a KL function is strictly decreasing. The definitions of K and KL functions are taken from page 144 of Khalil (2002).

The control system considered consists of an encoded feedback loop (Figure 1). The system description is given by, (1)ẋ=F(x,u)  with  x(0)=x0y=G(x) where x∈Rnx is the plant state, u∈R is the control input, and y∈R is the output (nx∈N+). Both F and G are globally Lipschitz with respect to their arguments. Typically G is a concave (or convex) function with a single isolated optimum point. The high pass and low pass filters are excluded due to possible interference with the bandwidth limiting encoding scheme.

Assumption 1. There exists a globally Lipchitz x*: R→Rnx such that for all u∈R, F(x*(u), u) = 0.

Assumption 2. F(x*(u), u) = 0 is uniformly globally asymptotically stable in u.

Let ϕ(u): = G(x*(u)) be the steady state plant output with an isolated maximum v. Assumptions 1 and 2 ensure the stability of the ESC scheme such that for any bounded input u, the state x will remain bounded and converge to the unique equilibrium point determined by the diffeomorphism x*(u).

Definition 1. From Tan et al. (2006) and Nesic (2009): The system ẋ = F(t,x,ε), where x ∈ Rⁿ, t ∈ R ≥ 0, and ε ∈ R^l > 0 are, respectively, the state of the system, the time variable, and the parameter vector, is said to be semi-globally practically asymptotically (SPA) stable, uniformly in (ε_i, …, ε_j), j ∈ 1, …, l, if there exists β ∈ KL such that the following holds. For each pair of strictly positive real numbers (Δ,ν), there exist real numbers εk∗=εk∗(Δ,ν)>0, k = 1, 2, …, j, and for each fixed εk ∈ (0,εk∗), k = 1, 2, …, j, there exist ε_i = ε_i(ε₁, ε₂, …, ε_i−1, Δ, ν), with i = j + 1, j + 2, …, l, such that the solutions of ẋ=F(t,x,ε)with the so constructed parameters ε = (ε₁, …, ε_l) satisfy: (2)|x(t)|≤β(|x0|,(ε1⋅ε2⋅⋅⋅⋅εl)(t−t0))+ν, for all t ≥ t₀ ≥ 0, x(t₀) = x(t₀) with |x₀| ≤ Δ. If we have that j = l, then we say that the system is SPA stable, uniformly in ε.

Assumption 3. The ESC scheme of (1) is SPA stable.

If a dynamical system such as equation (1) is SPA stable its output can start from an arbitrarily large initial bound Δ (||x₀|| ≤ Δ) and converging to an arbitrarily small bound μ such that ||y(t) − y_v|| ≤ μ where y_v = ϕ(v) is the desired steady state value of y(t) (i.e., the optimum). Assumption 3 extends the results of Assumptions 1 and 2 to the practical situation where the continuous periodic dither input results in an output perturbed by the bounded harmonics of the periodic input dither.

The purpose of an encoding scheme is to reduce the bandwidth of a signal by minimizing redundancy. A common redundancy in control systems is the high dependence of adjacent samples. This enables the low bandwidth difference between two adjacent samples to be used instead of the entire sample itself. However, according to Premaratne et al. (2013), the transmission of the difference is only beneficial in systems that use time division multiple access (TDMA). An alternative to this is to use sporadic sampling (aperiodic sampling) where the transmission takes place only when there is a significant change in the signal. This makes them more suitable for packet switched NCS. Both encoding schemes considered in this paper fall into this category.

When an encoding scheme is used it is necessary to sample the encoded variable (i.e., the output of the plant) at a sufficient rate T_S. The encoded variable will then have to be reconstructed and held for the sampling time T_S. This results in an error which can be modeled as a disturbance d(t) = ȳ[k]−y(t) (Figure 2) where k is the discretized time of the reconstruction due to holding. Due to this error further assumptions are necessary.

Assumption 4. For the sufficient sampling time T_S, there exists an error bound Δ_q where |y(t)−y[k]|≤Δq for t ∈ [(k − 1) T_S, kT_S). For convenience, Δ_q is taken as the upper bound of the quantization error.

Assumption 5. Ideal synchronized clocks are used for periodic sampling and reconstruction of y(t).

Assumption 6. The communication network uses packet switched multiple access with a negligible channel access time compared to T_S.

From Assumption 4, the ESC scheme of equation (1) need not be considered as a discrete system as in Choi et al. (2002). Assumptions 5 and 6 are necessary to neglect any effect due to delay and delay variation (jitter), which is inevitable in a packet switched communication network according to the results of Zampieri (2008), Donkers et al. (2011), and Premaratne (2014).

In Tan et al. (2010), it is mentioned that dither signal p(t) cannot be correlated to the disturbance d(t). This can be expressed as: (3)lim T→∞ ∫0Tp(t)d(t)dt=0 where T is the perturbation period and p(t) = b sin(ωt). If the two correlate, the direction of update u may be wrongly inferred. This is one of the assumptions used in Tan et al. (2010) to prove convergence of the system. In addition, it is also mentioned that in practice a small correlation can be tolerated provided that the integral of equation (3) is sufficiently small. Therefore, it is possible to relax the requirement of equation (3) such that (4)lim T→∞ ∫0Tp(t)d(t)dt=O(b2) will suffice for 1≫b. The requirement of equation (3) can be used to prove convergence when the reconstruction error can be obtained analytically as in the case of MBET (Lemma 1). When this becomes difficult as for ETADM; equation (4) or empirical correlation of the integrator input of (Figure 1) has to be used.

In this section, the two encoding schemes used (MBET and ETADM) are introduced in terms of their functionality and main parameters. Based on this in Section 3, the scenarios under which these encoding schemes fail for ESC are derived.

Lemma 1. The sinusoidally perturbed ESC system described by (1) that satisfies Assumptions 1 to 6 will converge under MBET encoding if there exists no u_x, uy∈R such that |ϕ(u_x) − ϕ(u_y)| < e_T and |(u_x − u_y)| > 2b where b > 0 is the amplitude of the sinusoidal perturbation.

Proof. At steady state for an initial value u₀ ∈ [u_x, u_y] such that u ∈ [u₀ − b, u₀ + b] ⊂ [u_x, u_y], no events will be triggered since |ϕ(u_x) − ϕ(u_y)| < e_T. Therefore, d(t) = y(t)−ȳ[k]=gx∗u0+bsin(ωt)−yS = ϕu0+bsin(ωt)−yS, where y_S is a constant since no events are triggered. Therefore from equation (3), lim T→∞ ∫0T p(t)d(t)dt=lim T→∞ ∫0T b sin(ωt)ϕu0+b sin(ωt)−ySdt=blim T→∞ ∫0T sin(ωt)ϕu0+b sin(ωt)dt≠0.

The correlation is non-zero due to the fact that since ϕ is Lipschitz for all ϕ such that dduϕ(u)≠0, ϕ(u₀ + b sin(ωt)) consists of a fundamental at ω and harmonics.

Example 1. Consider the system of (1) with u = x and the plant input-output map: (6)y=G(x)=(x3+1)∕(x6+1)

From the plot of Figure 4, the optimum input, v ≈ 0.75, and the corresponding optimum output is ϕ(v) ≈ 1.2. The output map has a plateau at u = x ∈ [−0.21, 0.21] where y ≈1. When b = 0.2 and perturbation frequency f  = ω/(2π) = 10 Hz, the system fails to converge when e_T = 0.1 (Figure 5A) and succeeds only when e_T is reduced to 0.01 (Figure 5B).

Remark 1. In order to guarantee convergence when using MBET, prior knowledge of the non-existence of a plateau in the function ϕ(u) is required. This may not be possible in a time variant objective function.

Unlike in MBET, the error d(t) of ETADM is complex to model. This is due to the complex state transitions of the ETADM encoder that are highlighted in Theorem 2 of Premaratne et al. (2013). Therefore, it is necessary to ensure that the reconstruction ȳ[k] and p(t) are correlated unless y(t) = ϕ(v). Thus, if the perturbation frequency f can by tracked by a cycle of the encoder state S[k], this requirement can be satisfied. Taking the minimum number of samples to maintain a cycle as λ, the value of f would be given by (7)f≤1λTS

Finding a suitable value for λ requires insight into the cyclic behavior of S[k] of the ADM encoder based upon its implementation (Figures 6 and 7).

From the proof of Theorem 2 of Premaratne et al. (2013), the possible values of S[k] are discrete and range from S_max, −(S_max + ΔS), …, −S_min, and S_min, S_min + ΔS … S_max. Thus, these discrete values can be taken as individual states. By denoting the state (i.e., value of S[k]) at time k is denoted by the subscript S_k and the sign of e[k] as E_k = sgn(e[k]) = {−1, 0, 1} the state transitions can be represented as: (8)Sk+1=Sk+ΔSEk=1 and Sk≠SmaxSmaxEk=1 and Sk=Smax−(Sk−ΔS)Ek=−1 and Sk≠Smin−SminEk=−1 and Sk=Smin if S_k is positive and (9)Sk+1=−(Sk+ΔS)Ek=−1andSk≠−Smax−SmaxEk=−1andSk=−SmaxSk−ΔSEk=1andSk≠−SminSminEk=1andSk=−Smin if S_k is negative.

Since the cycle given by S[k] = − S[k − 1] = ± S_min would be blocked by equation (5), from equations (8) and (9), the cycle with the next largest step size (i.e., ±(S_min + ΔS)) is given by (10)Sk−3=Smin,Sk−2=Smin+ΔS,Sk−1=−Smin,Sk=−Smin−ΔS,Sk+1=Smin.

This cycle becomes {0, S_min + ΔS, 0, −S_min − ΔS} after blocking by equation (5).

From equation (10), the minimum value of λ is four. However, since the lossy integrator with an equation given by x[k + 1] = Kx[k] + KM(S[k]) (K < 1 with K being as close as possible to one) attenuates the signal when M(S[k]) = 0, additional transitions of S[k] = ± (S_min + ΔS) would be required to compensate for the attenuation x[k + 1] = Kx[k]. Thus, the perturbation frequency of equation (7) will have to satisfy the condition (11)f<1λTS=14TS

This result is verified for a bell function ϕ(u) = exp[−(u − 2)²/2] with a maximum at u = 2. Three sets of parameters for ETADM encoding (S1, S2, and S3) are selected such that S_min is comparable with the perturbation amplitude b (Table 1). For these results and the sampling time of 0.01 s, f  < 25 Hz.

Table 2 shows the results in terms of correlation, convergence, and bandwidth reduction. Out of the multiple frequencies used 50 Hz and 25 Hz are selected so that the perturbation cycle can only be reconstructed by two and four ETADM step values, respectively. For both of these frequencies, there is no significant correlation and convergence as predicted by equation (11). The frequency of 40 Hz is selected in between 50 Hz and 25 Hz for observation of the intermediate behavior. In this case, though there is an unexpectedly high correlation, the convergence time is very large compared to the remaining values. Consistent results are obtained for frequencies below 25 Hz where there is a sufficiently high correlation and rapid convergence.

Table 3 compares the performance of MBET and ETADM with e_T = S_min = 0.1 with a periodically sampled benchmark. The results show that for comparable parameters MBET outperforms ETADM in terms of bandwidth reduction. This confirms the previously published results of (Premaratne et al., 2013). However, though MBET is more bandwidth efficient than ETADM, it is increases the convergence time compared to the benchmark.