Delta Modulator Based Quantised State-Feedback Control of Networked Linear Systems

In this study, the design of quantised state-feedback controller using a Delta-Modulator (Δ-M) for linear networked systems is proposed. This modulator can be considered as one type of sliding mode quantiser (SMQ) and has several advantages such as lower design complexity, lower cost, and less noisy. In this study, the stability conditions of both the continuous-time and the discrete-time Δ-M based quantised control systems are derived. The switching function bounds, which ensure that the steady state behaviour of the system is periodic are derived. The effectiveness of the Δ-M based quantised control system is investigated using a ZigBee protocol based experimental communication network with inherent imperfections associated with real-time network. It is shown that the designed quantised state-feedback controller gives the desired performance.


I. INTRODUCTION
During the past few years, sliding mode techniques have been popular in control applications such as controller design [1], [2], observer design [3] and sliding mode quantiser (SMQs) [4], [5] due to their robustness and simplicity in converting input signals to switching signals which are described by a sequence of binary values. Different types of SMQs can be used to generate these binary values such as delta modulator (∆-M), sigma-delta modulator (Σ∆-M), delta-sigma modulator (∆Σ-M) [6]- [8], pulse-width-modulator (PWM) based sliding mode [9], where these are used in many control applications [10]- [18].
However, the switching components in SMQs add more complexities to the system. If these quantisers are to be used effectively in control applications, they should satisfy both the stability and equivalence conditions [9]. In [5], [9] detailed investigations on the stability of SMQs in continuoustime (CT) systems have been carried out where it has been found out that the SMQs can be used in high-speed switching. However, the gain of the switching function affects the stability of the closed loop system. Under the assumption of ideal sliding mode and high switching frequency, the SMQs are equivalent to the inputs. This ensures the equivalence conditions [17], [19].
Most of the modern controllers are often implemented in the discrete-time domain where discretisation would introduce delays into switching components. In real-time, the effects of discretisation and the factors which affect the implementation under high switching frequency (for SMQs) have been reported in [20]- [22]. However, the results presented in [20]- [22] do not explicitly address the issues related to equivalence conditions. Therefore this has been the primary motivation of this study where the performance of SMQs designed using equivalence conditions is investigated. Amongst various quantisers, this study focuses on using Delta-modulators (∆-M) as a quantiser. This modulator is extensively being used in power converters [6], [17] and in applications where bandwidth utilisation of the communication channel is necessary. Some further advantages of , the ∆-M include its simplicity in implementation, requirements of fewer hardware resources, lower complexity, cost-effective operations, lower noise, and many others [23]- [26].
In this study, a ∆-M based quantised state-feedback controller is designed for multi-input-multi-output (MIMO) systems; both in CT and discrete-time (DT) domains. The main contributions of this paper are highlighted as follows: (i.) 1) Established the stability conditions for both CT and DT ∆-M based quantised state-feedback systems. 2) Derived sufficient conditions for the existence of periodic orbits (or zig-zag behaviour) of the switching function and states. 3) Determined the bound of the switching function which results in periodic orbits. 4) Validated the theoretical findings using a real ZigBee protocol based communication system. The rest of the paper is organised as follows. Section-II and section-III discuss, CT-∆-M and DT-∆-M respectively and the stability conditions are derived. The effectiveness of the proposed control strategy and the theoretical findings are validated using two simulation examples in section-IV followed by conclusions on section-V.
For the rest of the paper, ∥x∥ and ϱ(A) respectively denote the standard Euclidean norm of the vector x ∈ R n and the spectral radius of the matrix A ∈ R n×n .

II. CONTINUOUS-TIME SWITCHED CONTROL SYSTEM A. REVIEW OF CONTINUOUS-TIME DELTA MODULATOR
The block diagram of a MIMO CT ∆-M is shown in Figure. 1. It consists of two integrators, one on the feedback path and the other in the decoder side. Input of the quantiser is the difference between the output of the integrator on the feedback path and the input signal. This is a static quantiser and the output δ(t) is a single-bit signal which is transmitted through a communication channel before received by the decoder. Under ideal conditions, the output of the decoder x(t) is equivalent to the input signalx(t) [10], [19], [27].
The MIMO CT ∆-M is described as: where x ∈ R n ,x ∈ R n , s ∈ R n and Θ ∈ R n×n respectively denote the input signal, the quantised signal, the switching signal and the gain of the 2-level quantiser. Further, sgn(s(t)) ∈ {−1, 1} , · · · , {−1, 1} T ∈ R n , where, The communication channel between the encoder and the decoder carries a single-bit n-dimensional signal which is expressed as: where The quantiser gain Θ is often selected as a positive definite diagonal matrix for design simplicity. This is expressed as: For proper operation of the MIMO CT ∆-M, the quantisation gain Θ must be same in both the encoder and the decoder [10], [19]. In [28], it has been shown that the existence of the sliding mode ( i.e. s(t) Tṡ (t) ≤ 0 ) is ensured provided ∥ẋ(t)∥ ∞ ≤θ;θ = min (θ ii ).

B. CONTINUOUS-TIME QUANTISED STATE-FEEDBACK CONTROL SYSTEM
The present study focuses on stabilisation problem of linear time invariant (LTI) systems via state feedback which is described by:ẋ wherex ∈ R n and u ∈ R m respectively denote the system states and the control signal. The pair (A, B) is assumed to be controllable and the closed loop system is expressed as: where A cl = A − BK. The feedback control gain K is designed such that the poles of the closed-loop system A cl are at desired locations. The solution of (6) gives: wherex(0) is the initial conditions ofx(t). It is assumed that The solution of (7) is therefore bounded and satisfies the condition: Using (1) and (5), the quantised state-feedback control system can be described as: where s(t) denotes the quantisation error. For the sake of convenience let us re-write (10) as: The objective is to lim t→∞ s(t) = 0 n in the equivalent sliding mode.
Proposition II.1. Assume x(0) is known and (8) holds. If there exists a quantiser gain Θ = diag{θ ii } such that then the quantised state feedback control systems (11), is stable and converges to the system (5).
Remark 2. Higher values of feedback control gain K will make the degree of stability of the system higher. However, it forces θ ii for all i = (1, 2, . . . , n) to be chosen high, which results in the chattering phenomenon. Remark 3. In Proposition II.1, an ideal sliding mode is assumed where infinite sampling switch is considered. As a result of the imperfection of switch elements, the switching manifold can be described as a boundary layer. The width of the boundary layer is defined as: where h denotes the sampling time. It is worth to note that, lim h→0 ϵ(h) = 0. By choosing the optimal quantiser gain Θ and reducing sampling frequency of the switching; the boundary layer (17) can be minimised, undesired oscillations (chattering phenomena) can be attenuated and the system can be stabilised.

III. DISCRETE-TIME SWITCHED CONTROL SYSTEMS A. EULER DISCRETISED SYSTEM DESCRIPTION
The discrete-time equivalent of the CT system (5), using Euler discretisation is expressed as: where h is the sampling period. Simplifying (18) gives, where The sampling time h ∈ R + is selected in the range [29]: This will ensure that the eigenvalues of Φ cl are at desired locations. Substituting (20) in (19) gives: where ∆x(k) =x(k + 1) −x(k). Re-write (22) as: If the feedback gain matrix K is selected such that, ∥Φ cl ∥ < 1, the it can be shown that This proves,

B. DISCRETE-TIME DELTA MODULATOR
The block diagram of a MIMO discrete-time ∆-M is shown in Figure. 2. From Figure. 2, the dynamics of the discretetime ∆-M can be expressed as: where, The communication channel between the encoder and the decoder carries a single-bit n-dimensional signal which is expressed as: For the proper operation of the MIMO DT ∆-M, the quantisation gain Θ must be same in both the encoder and the decoder [10], [19]. In [28], it has been shown that the existence of the sliding mode ( i.e. |s(k
Proof: Consider two cases: when s i (k) > 0 and s i (k) ≤ 0 From (32), when s i (k) > 0, Correspondingly, when s i (k) ≤ 0, From the above two cases |s i (k)| is bounded such that |s i (k)| < 2θ ii . Further, as k → ∞, s i (k) will be bounded as: Using Definition III.1, (31) can be expressed as: where α i and β i respectively denote the i th row of hA cl and Γ K. If both α i and β i can be expressed as α i = O(h) and β i = O(h), (33) can be satisfied with arbitrary small h. This also means that θ ii is a function of h (i.e. θ ii (h)) and θ ii → 0 as h → 0. Remark 4. For an arbitrarily small value of h, ∆x(k) tends to its counterpart of the nominal system ∆x(k) in (22). Considering (25), this also implies that ∥∆x(k)∥ ≤ h ϱ (A cl ) ∥x(0)∥ ≤ θ ii (h).

D. BOUNDARY LAYER OF SLIDING MODE AND PERIODICITY
In the following, we will estimate the boundary width of sliding mode which is associated to the quantization error.
[30] Let x(k), f s respectively denote the discretised samples of x(t) and the over-sampling frequency of ∆-M which satisfies where X , B X respectively denote the upper bound of x and the bandwidth of the closed-loop system and α is a positive number. Then, The bound of s(k) in the equivalent-based sliding mode, is presented in the following [19], [31].
Lemma III.5. If the diagonal elements of Θ is selected such thatθ and the initial conditions satisfy s(0) = 0, then the system (26) exhibits quasi-sliding motion and the switching function s(k) remain inside the boundary layer Ω indefinitely.
Proof: Consider the dynamics of s(k) in (32). Let us choose a positive-definite, DT candidate Lyapunov function, Using (43), the first difference ∆V(k) of the Lyapunov function V(k) can be written as: whereθ satisfies the dominance condition with respect to ∆x(k). Solving ∆V(k) with respect to s(k), suggests that s(k) is bounded such that ∥s(k)∥ ≤ Ω, where, Remark 5. When the switching function s(k) is bounded (i.e., ∥s(k)∥ ≤ Ω), the trajectory of the switching function will exhibit periodic behaviour [31]. The period is dependent on the choice of sampling time h and quantisation gain Θ.

A. EXAMPLE 1
Consider a continuous-time linear system with three number of inputs described by: where A and B are given by, The performance of the controller in stabilising the states are shown in Figure 4 and the behaviour of the switching function is shown in Figure 5. After designing the controller for continuous time systems, in the next phase, the ∆-M based quantised controller is designed for the discrete-time system. The discrete equivalent of the continuous-time system described in (49), at a sampling rate of h = 0.01 seconds, is given by:  Figure. 6, it is observed that all the states converge to zero within finite time. Further, the switching function starts inside the region Ω, as can be seen in Figure. 7, and stays there indefinitely with a period of 2 (see Figure. 8). It is worth to emphasis that although the period of the switching function depends on the choice of the quantisation gain and the sampling time, this remains always within the region Ω.

B. EXAMPLE 2
The effectiveness of the proposed controller is further shown considering an example of a mechanical system shown in Figure. 9. The continuous-time dynamics of this system is described as [32]:ẋ The values of various parameters of the system, used in the simulation, are:k 1 = 1; k 2 = 1; m 1 = 1; m 2 = 2; c 1 = 1; c 2 = 1. The design of controller and the quantisation gain follows same procedure as used in example 1.
The closed loop poles are placed at −2+1i, −2−1i, −2+ 0.5i, −2 − 0.5i and the initial condition x 0 are taken as [0.05; −0.5; −0.05; 0.5]. The controller and the gain of the ∆-M based quantiser are designed following the method described in section-II for the continuous-time system. The performance of the controller in stabilising the states are shown in Figure 10 and the behaviour of the switching function is shown in Figure 11.
After designing the controller for continuous time systems, in the next phase, the ∆-M based quantised controller is designed for the discrete-time system. The discrete equivalent of the continuous-time system described in (51) at a sampling rate of h = 0.01 seconds is given by:  Figure. 12, it is obvious that all the states converge to zero is within finite time. Similar to example 1, the switching function starts inside the region Ω, and remain there indefinitely with a period 2 (see Figure. 13 and Figure. 14).
From both examples, it can be seen that the performance of the quantised feedback controller is satisfactory. Although this study mainly focuses on quantisation related issues, the behaviour of the designed ∆-M based quantised control system is investigated under many networked constraints such as random packet delay, transmission delay and packet losses in the practical Zigbee protocol based networked control system. From the results, it is observed that the proposed one-bit quantised controller could maintain the stability of the overall system.

V. CONCLUSIONS
In this study, a delta modulator based single-bit quantised state-feedback controller is designed for linear networked control systems. The stability conditions ∆-M is derived for both in CT and DT domains. It is confirmed that the stability of delta Modulator is heavily dependent on the quantiser gain and on some properties of the input signals. The bound of the switching function is derived such that periodic behaviour in the steady-state is ensured. The effectiveness of the designed delta modulator based single-bit quantised control approach is illustrated using a practical ZigBee protocol based networked control system which inherent many network imperfections like packet losses, transmission delays, and bitrate constraints. The results of the experimental simulations were carried out using two examples and the results of the simulations confirm the theoretical findings. His current research interests include nonlinear system identification and control, machine learning and big data.