Hybrid delta modulator: stability analysis using sliding mode theory

The present study proposes a new dynamic two-level quantizer, called as hybrid delta modulator (-M), which combines the features of both delta-modulator and delta-sigma modulator. In the transient state, the -M exhibits the dynamical behaviour of delta modulator (Δ-M) while in steady state, its behaviour is similar to delta-sigma modulator (-M). This study investigates about various dynamics of the proposed -M in both continuous and discrete-time domains. The stability conditions of -M are derived using the theory of sliding and quasi-sliding mode. The theoretical results are validated through extensive simulations. Abbreviations: Δ-M: delta modulator; -M: hybrid delta modulator; -M: delta sigma modulator; ADC: analogue to digital converter; CT: continuous-time; DAC: digital to analogue converter; DT: discrete-time; NCS: networked control system; QSM: quasi-sliding model; QSMD: quasi-sliding mode domain; SS: steady state; TP: transient process

CONTACT Dhafer Almakhles dalmakhles@psu.edu.sa It is worth to note that -M can be implemented both in continuous and discrete time domains. The choice between continuous-time (CT) -M and discrete-time (DT) -M depends on the specific application (i.e. ADC CT -M, DAC DT -M). For example, in applications such as NCSs, power converters etc., -M can either be implemented in CT or DT domain depending on the convenience of the designer .
The -M (also referred to as differential modulator) mainly consists of a transmitter (encoder or modulator) followed by a receiver (decoder or demodulator) where the dynamics of the input variables can be found directly. Note that the input signal is equivalent to the signals (single-bit) between encoder and decoder. However, it requires an integrator at the demodulation side to reconstruct the input signals (Haykin, 2000). The -M is also considered as dynamic quantizer and inherently contains relay components (i.e. two-level quantizers) which introduce nonlinearity and more complexity to these modulators/systems Azuma & Sugie, 2008;de Wit et al., 2009).
-M has the ability to directly reveal the dynamic of the input variables, where the single-bit signals between the encoder and decoder are ideally equivalent to the rate of change of the input signals.
-M contains all the useful information of the input signal in the low-frequency range and suppresses noise at the highfrequency range. This feature is known as noise shaping. The present study combines these features of the -M and -M. In the transient state, the H -M exhibits the dynamical behaviour of -M while in steady state, its behaviour is similar to -M. Thus the stability region is increased and the noise effect is significantly reduced.
The -M can roughly be classified into two categories; namely fixed step-size and adaptive step-size. When the step size of -M is adaptive, it is called as H -M (Haykin, 2000). It has been shown that the stability of -M is critically dependent on quantizer gain Xia & Chen, 2007;Xia & Zinober, 2006). The objective of this study is to investigate if the stability performance can be improved by making the step-size of the quantizer to be adaptive. Recently, in , the authors have investigated the performance of data-driven -M and -M. The present study investigates some inherent dynamical properties of datadriven hybrid H -M in CT and DT domains and establishes that the stability region of this modulator is larger compared to that of -M. The main contributions of this study are: (i) Derivation of stability conditions and periodicity for both data-driven fixed and hybrid H -M using QSM analysis; (ii) Accurate estimation of hitting-step is derived for both data-driven fixed and hybrid H -M; (iii) The quantizer gain for fixed -M which guarantees the stability, is derived using dynamics and bounds of the input signals.
The rest of the paper is organized as follows. Sections 2 and 3 discuss CT H -M and DT H -M, respectively. The effectiveness of the proposed H -M is demonstrated using simulations in Section 4 with conclusions in Section 5.

Continuous-time hybrid delta modulator ( H -M)
The schematic of a continuous-time H -M is shown in Figure 1. From this figure, the following relations can be established:ṡ ( 1 ) where where (1)-(4)) can be divided into three regions depending on the values of e(t) and φ. This is explained using binary sequence of the two-level quantizer as follows.
(1) Steady state region, ( ss : Remark 2.1: The behaviour of the H -M in the region ss is similar that of -M and in the regions I tp and II tp , it behaves like -M.

Stability analysis of continuous-time H -M
For the all three operating regions, the stability of CT H -M is proven in following proposition.
Proposition 2.1: For the system described in (1)-(4), following condition is valid for the sliding mode condition to be reachable within finite time: Proof: The proof is given considering three cases.
• Case-1: Stability analysis of continuous-time H -M in the region ss .
• Case-2: Stability analysis of continuous-time H -M in the region I tp .
If the increase of x(t) is such thatẋ(t) > 0, then this will result in x(t) >x(t) at the beginning. As a result of this, the error e(t) → φ. However, any further increase of x(t) (e(t) > φ) will force the operating regions to shift (from In the proceeding, the stability of the H -M in the region I tp is studied using equivalent control-based sliding mode. Let us replace the fast discontinuous component δ t in (7) by its equivalent slow components ϕ(t). The shifting of the operating regions from ss to I tp implies that Let us consider a Lyapunov function as Using (7) and δ t ⇔ ϕ(t) (i.e. equivalent control-based sliding mode) yieldsṡ Furthermore, by using (1)- (4) and (9), the following equation can be derived: Using (10), and (11), the derivative of V tp (t) becomeṡ Note that in the region I tp , e(t) > φ and therefore, e(t) − φ > 0. Furthermore, since (5) is true, thenV tp (t) < 0. Furthermore, when (5) is true, the rate of monotonous increment ofx(t) in (4) is higher compared to the rate of increment of x(t). As a result of this, the conditions required for the existence of sliding mode is fulfilled; asx(t) will force the operating regions to shift back from I tp to ss within finite time. This is defined as hitting time t f which will be estimated in the next proposition. The proof of Case-3 is similar to the proof of Case-2.

Computation of the hitting time (t f ) for continuous-time H -M
In this section, we estimate the hitting time t f , which equals to the time needed for the trajectory of the H -M to hit the sliding manifold.
Proposition 2.2: When (5) is true and |x(0)| 0, the upper bound of the hitting time t f of H -M, is given by Proof: In the following, the maximum value of the hitting time t f is estimated which is defined as the time required for the trajectory of the H -M to be forced back into the steady state region ss , from either of the operating regions I tp or II tp .
Furthermore (14) can be re-written as The upper bound of the hitting time t f for the case oḟ s(0) < −φ can be found out similarly. Finally, it can be concluded that the maximum value of the hitting time t f for the H -M to hit the sliding manifold (i.e.ṡ(t) = 0 ∀t ≥ t f ) when |ṡ(0)| φ can be expressed as This means that in the combined region tp = I tp ∪ II tp , the rate of change ofx(t) is higher compared to the rate of change of x(t). This will force H -M to change its operating region from tp to ss within finite time which is less than the maximum hitting time t f in (16).

Stability analysis of discrete-time H -M
The stability of DT H -M in all the operating regions is proven in the following proposition.

Proposition 3.1: For a system described by (17)-(21), if
then the system trajectory will converge, from any initial state e(0), to quasi-sliding mode domain (QSMD) which is defined by |e(k)| ≤ ε, where ε is bounded such that ε ≤ 2φ, ∀k > k * . Once the system trajectory enters into the quasisliding mode domain, it will remain in QSMD for all the subsequent time.
Proof: Stability of the H -M (17)-(21) will be studied in the two regions: ss and tp = I tp ∪ II tp .
• Case-1: Stability analysis of discrete-time H -M in the region ss .
• Case-2: Stability analysis of discrete-time H -M in the region I tp .
• Case-3: Stability analysis of discrete-time H -M in the region II tp .
Proof of Case-3 is similar to the proof of Case-2.

Computation of the hitting step k * for discrete-time H -M
In this section, we estimate the hitting step k * . The hitting step k * is defined as the maximum number of steps required by the trajectory of the H -M to hit the sliding manifold.
Proposition 3.2: For the system described in (17)- (21), if (22) is true, then the number of steps required for the trajectory of discrete-time H -M to cross the switching manifold s(k) = 0 (from any initial value s(0)), equals to k * where k * = m , and The floor operation is denoted by m .

Proof:
In the succeeding section the hitting step k * , where the system trajectory of the discrete-time H -M crosses the hyper-plane e(k) = 0, is estimated. Note that the results are derived under the assumption of e(0) > ε.
The mth iteration of (23) gives From Proposition 3.1, it is known that there exists a step where δ k changes from +1 to −1. Consider the case of maximum m where x(k) = X k . When e( m ) ≤ 0,  then m ≤ e(0)/(φ − X k ). Similar results can be derived for the case when e(0) < −ε. Hence from the both scenarios, (|e(0)| > ε), it can be concluded that the system trajectory of the discrete-time H -M requires k * number of steps to cross the surface e(k) = 0 where and k * = m . This completes the proof of Proposition 3.2.

Simulation results
In this section, the dynamical properties of H -M in both CT and DT domains are investigated. In the simulation, the input, output and trajectory of the modulators are computed and their behaviour in various operating regions are studied considering the signal where r(t) is the ramp signal.

Simulation results of continuous-time H
Consider the CT H -M, described in (1)-(4) with x(t) = y(t) (defined in (26)) as the input signal and s(0) = 0. The trajectories of this modulator are shown in Figures 3-5.
According to the stability condition (5), the CT H -M, with quantizer gain φ = 2, is stable for all |ẋ(t)| ≤ X t , ∀ X t = 2. It can be seen that when t ∈ [0, 5),ẋ(t) > 2. Therefore, the modulator becomes unstable (see Figure 3). Note that if this condition persists for a longer period, it will eventually lead to instability. For t ∈ [5, 10), the derivative of the input signal becomes equal to the quantizer gain, i.e.ẋ(t) = φ. Therefore, the system is marginally stable during this time (see Figures 4  and 5). In this region, both the input x(t) and the outputx(t) increase with the same rate which is evident from Figure 3. For t ∈ [10, 5), the operating region is switched to the stability region. Note that stability region consists of sliding mode and equivalent mode. The hitting time t f is calculated from (13) and has been found to be 12.5 s. The trajectory in the phase plane, shown in Figure 4, further confirms the results presented in Figures 3 and 5.

Simulation results of discrete-time H
Consider the DT equivalent of the signal in (26) (denoted as y(k)), ∀t ∈ [hk, (k + 1)h], where k and h denote the sampling step and period, respectively. Let the input of the H (17) be x(k) = y(k). The trajectories of H system  are shown in Figure 6 for φ = 0.4, h = 0.2 and s(0) = 0. According to the stability condition (22), it can be seen that H , with φ = 0.4, is stable for all |x(k)| ≤ 0.4. Thus, the system outputx(k) diverges from the system input x(k) when x(k) = 0.6, ∀0 ≤ k < 25 and it converges when x(k) ≤ 0.4, ∀50 ≤ k < 65; since it satisfies stability conditions. According to (24), the number of steps which the trajectory takes to make transition from marginal mode (which starts at k = 51) to equivalent mode, is 15. This gives k f = 66.

Conclusion
The stability conditions and accurate estimation of hitting-step for both data-driven H -M are derived analytically using QSM analysis. It is found that the stability and the upper bound of hitting-time, for H -M, are critically dependent on some properties of the input signals, quantizer gain, as well as the adaptive parameters in H -M. The simulation results confirm theoretical findings. Future research includes using H -M in applications such as in Power Electronics, in event-triggered NCS, load frequency control of multi-area interconnected power systems (Lu, Zhou, Zeng, & Zheng, 2019) and in the internet of things (IoT). Also, this adaptive algorithm can further be improved to tackle many challenges that we face in the aforementioned applications.