Numerical evaluation of sine-Gordon chain energy control via subdomains state feedback under quantizationand time sampling

The paper is devoted to the numerical performance evaluation of the speed-gradient algorithms, recently developed in (Orlov et al., 2018; Orlov et al., 2019) for controlling the energy of the sine-Gordon spatially distributed systems with several in-domain actuators. The influence of the level quantization of the state feedback control signal (possibly coupled to the time sampling) on the steady-state energy error and the closed loop system stability is investigated in the simulation study. The following types of quantization are taken into account: sampling-in-time control signal quantization, the level quantization for control, continuous in time; control signal quantization on level jointly with time sampling; control signal transmission over the binary communication channel with time-invariant first order coder; control signal transmission over the binary communication channel with first order coder and time-based zooming; control signal transmission over the binary communication channel with adaptive first order coder. A resulting impact on the closed-loop performance in question is concluded for each type of the quantization involved.

During the last years the energy control problem of spatially distributed systems has been widely studied and proper solutions have been proposed and rigorously justified. In the series of papers the speedgradient method of [Fradkov and Pogromsky, 1998;Fradkov, 2007;Fradkov et al., 1999] was used to design energy control algorithms for spatially distributed systems, using the boundary control [Dolgopolik et al., 2016;Dolgopolik et al., 2018] and distributed control [Orlov et al., 2017b;Orlov et al., 2017a]. The authors of [Orlov et al., 2017b] analyzed energy control problems for linear wave partial differential equation (PDE) and nonlinear sine-Gordon PDE where the distributed yet uniform over the space control was chosen. The speed-gradient method for energy control of Hamiltonian systems was developed and justified for the above partial differential equations (PDEs). An infinite dimensional version of the Krasovskii-LaSalle principle was validated for the resulting closed-loop systems. By applying this principle, the closed-loop trajectories were shown to either approach the desired energy level set or converge to a system equilibrium. In [Orlov et al., 2017a] a linear wave equation, governing 1-D (onedimensional) string oscillations, was considered. A distributed control input, independently enforcing the underlying string over its entire spatial location, was assumed to be available. The speed-gradient method was developed and justified for the considered PDE model. Capabilities of the proposed speed-gradient algorithm of reaching the energy goal were supported by numerical simulations. In [Orlov et al., 2018] an energy control of the sine-Gordon chain driven by several in-domain actuators was considered. The speedgradient method was generalized to the in-domain actuation such as in [Christofides, 2001;Fridman and Blighovsky, 2012;Fridman and Am, 2013;Pisano and Orlov, 2017] for the purpose of pump/dissipate the energy of the model to a desired level. This result has been extended to control via output feedback by means of developing the Luenberger-type spatial observer, which got measurement data from the sensors, placed within small spatial plant subdomains in [Orlov et al., 2019]. Some important issues have not yet been analytically studied due to nonlinearity and complexity of the systems under consideration. One of the challenging task is to evaluate the closed-loop system performance in the presence of external disturbances and measurement and data transmission errors. The main aim of the paper is obtaining an impression on the limitations of the energy control for a class of nonlinear distributedparameter physical systems (the sine-Gordon chains) by means of the spatially-discretized control with time and level quantization. In the present work, a step is made for clarifying some of the mentioned issues, namely the effect of level quantization separately, or jointly with a time sampling, is investigated for the control signal applied to the nonlinear chain with the state feedback control, developed in [Orlov et al., 2018].

Model of Controlled Plant
First, let us briefly recall the key points of [Orlov et al., 2018].
The system of interest is the 1-D chain, described by the following dissipation-free sine-Gordon PDE where t is the time instant, r ∈ [0, 1] is the scalar spatial variable, x = x(·, t) is the instant state of the system, the parameter κ is the elasticity of the system, F 0 stands for the torque gain, u(r, t) is for the manipulable input. The PDE (1) is accompanied with the Dirichlet boundary conditions is in play to control the sine-Gordon model (1) which is thus representable in the form As in [Orlov et al., 2019], the spatial domain [0, 1] is uniformly partitioned into m = 10 subdomains [r i , r i + h i ] of lengths h i = 0.1, i = 1, . . . , 10 so that r i = 0.1(i − 1). Within each subdomain the corresponding i.e., the first and the last actuators are located in the distance 0.02 from the left and right boundaries, respectively, whereas the neighboring actuators possessed a slot of the length 0.04 between them.

Control Objective
The control objective is to pump or dissipate the energy of the sine-Gordon system (1)-(3) to a prespecified level E * 0 for guaranteing the limiting relation on the solutions x(r, t) of the closed-loop sine-Gordon model (1), (2).

Energy Control Synthesis Using State Feedack
Let us pick-up the goal functional as Following the speed-gradient design procedure [Fradkov, 2007;Fradkov et al., 1999], compute the time derivative of V (t) along the trajectories of system (1), (2), provisionally assuming that u(r, t) is constant on t. Differentiating (8) in time, integrating then the resulting equality by part, and employing a consequence x t (0, t) = x t (1, t) = 0 of the boundary condition (2) yield [Orlov et al., 2018] The control action in [Orlov et al., 2018] is specified in the form of a finite number of sampled-in-space actuators (3). ThenV reads aṡ As the second step of the speed-gradient procedure, one should derive the gradient ∇ uV ∈ R m of the resulting expression ofV with respect to the control components u i (t), i = 1, . . . , m, thus arriving at The third step of the speed-gradient procedure is to pick up a certain function η(x, x t ) which forms an acute angle with ∇ uV , i.e., satisfies an inequality η T ∇ uV 0. In [Orlov et al., 2018], it is chosen in the form According to the speed-gradient method, the control action u(x,ẋ) = −Γη(x,ẋ, t) is then constituted with the matrix design parameter Γ = diag{γ 1 , . . . , γ m }, composed of positive entries γ i , i = 1, . . . , m. Summarizing, the sampled-in-space actuation (3) is specified with where the energy error ∆E(t) = E * − E(t) is introduced, i = 1, . . . , m. The present research deals with an energy dissipation problem, where desired energy level E * is set to zero (in other words, the regulation problem is considered). In this particular case, ∆E(t) < 0, therefore sign ∆E = −1, and control law (11) takes the following form 4 Numerical Evaluation Setup 4.1 Computational algorithm Following the previous works [Orlov et al., 2017b;Orlov et al., 2017a;Orlov et al., 2018], in the numerical study, the PDE (1) is discretized in the spatial variable r ∈ R 1 by uniformly splitting the segment [0, 1] into N sub-intervals. The discretization step ν is introduced as ν = 1/N . The resulting system of N − 1 ordinary differential equations (ODEs) of the second order are then numerically solved over a time interval [0, T ] by applying the medium order variable step Runge-Kutta Method [Dormand and Prince, 1980], performed with the standard MATLAB routine ode45. At the discretization nodes r i = i·ν, i = 1, . . . , N −1, the second-order spatial derivatives of x(r, t) are approximately computed as The boundary values x(r 0 , t) = x(r 1 , t), x(r N , t) = x(r N −1 , t) are specified according to boundary conditions (2), thereby yielding the following values x(r 0 , t) = x(r N , t) = 0 at the boundaries. The remaining discrete values x(r 1 , t), . . . , x(r N −1 , t) are found by solving (N − 1) ODEs numerically.
To numerically find the values of definite integrals in (12), the MATLAB standard routine trapz of the trapezoidal numerical integration are used.

System Parameters and Initial Conditions
In the simulations of the closed-loop boundary-value problem (1)-(2), (12), the control gains were set to . . , m with γ 0 = 5 and the desired energy level was set to E * = 5. The initial states were pre-specified in the form with a certain "magnitude" parameter A. In the present study, A = 0.02 is taken. Parameters of (1) were specified to κ = 0.12, F 0 = 25. For avoiding consideration of the various combinations of initial and desired energies, for certainty, the desired energy level E * = 0 is taken, therefore actually, the stabilization problem is studied in the present work. The simulation results for various cases of the energy pumping/dissipation may be found in the above mentioned papers [Orlov et al., 2017b;Orlov et al., 2018;Orlov et al., 2019]. For given parameters κ = 0.12, F 0 = 25 and initial conditions (13) with A = 0.02, initial system energy E(0) is as E(0) = 12.47. A reasonably high number N = 2500 was selected for the PDEs (1) to discretize the spatial variable r and duration of the computation time T was confined to 15.

Numerical Results
Below, the several sources of the errors, caused by the discretization and time sampling are numerically studied for the system under consideration. In all of them, discrepancy of the "ideal" control signals given by (12) and the actual ones, denoted byũ i , i = 1, . . . , 10 is considered as an error source in the examined system.

Sampled in Time Control Signal 5.1.1 Time Sampling Description
In the case of the control signal time sampling, controls u i (t) are calculated, accordingly to (12) at the uniformly distributed instants t k = kT 0 , where constant T 0 is for the given sampling interval, k = 0, 1, . . . denotes the step number (the discrete time). The resulting signals u i (t k ) are subjected to the first-order hold procedure which expands u i (t k ) fixing them over the sampling intervals [t k , t k+1 ). This leads to the following expression for actuator inputsũ i , where i = 1, . . . , 10:

Simulation Results for Time Sampling
The simulation results in the form of the energy error evolution ∆E(t), depending on the sampling interval The results show that the steady-state energy error is negligibly small at least for T 0 0.10, but if T 0 exceeds some threshold, loss of stability occurs.
are applied to the chain which are results of the uniform quantization of the signals u i , i = 1, . . . , 10 with a given quantization interval δ > 0. Hereinafter, · denotes the round function, which rounds each element of the input signal to the nearest integer.

Simulation Results for Level Quantization
The simulation results in the form of the energy error evolution ∆E(t), depending on the quanization interval δ ∈ {0.01, 0.20, 0.50, 1.0, 2.0, 5.0} are plotted on Fig. 2. The logarithmic scale is chosen along the y-axis (for the energy error values). The plots show that the energy tends in time to a certain steady-state value, depending on the corresponding δ. In Tab. 2, the steadystate relative energy error δE is given (in percents) for various quantization intervals δ, where t fin = 25 stands for the simulation final time.
One may notice that even in the worst case, the steadystate energy error does not exceed 1%. Therefore, simulation results show that the energy is stabilized with an affordable accuracy for the all chosen quantization intervals.    Table 3 shows that the steady-state error is less than 2 for the worst case with δ = 10.
Let us now consider the dependence of the system accuracy on sampling interval T 0 for fixed value of    Fig. 4 show that there exists a critical sampling time which should not be exceed to preserve the closed-loop system stability. In the presented example, the transients of the system have approximately the same shape and asymptotically tend to certain steadystate values with the exception of the case T 0 = 0.06, where the stability is lost and the process diverges. Table 4 exibits the dependence of the steady state error on sampling time T 0 for δ = 1. As seen, the steadystate error does not depend practically of T 0 and remains negligibly small before the stability is lost.

Control Signal Transmission Over Binary Communication Channel with Time-invariant
First Order Coder 5.4.1 Method Description In this Section, the control signal transmission is considered over the digital communication channel subject to a limiting data transmission rate. It should be noted that performance limitations under constraints, imposed by a finite capacity of communication channel, are well-recognized from the control literature, see [Nair et al., 2007;Andrievsky et al., 2010b;Matveev and Savkin, 2009;Matveev and Pogromsky, 2016] and the references therein.
To begin with, consider the following memoryless (static) binary quantizer for a certain signal y: where sign(·) is the signum function. Notice that given binary quantizer, each codeword symbol contains one bit of information. This quantizer is a part of the coder with a memory [Nair and Evans, 2003;Liberzon, 2003;Tatikonda and Mitter, 2004;Nair et al., 2007]. Let signal u(t) should be transmitted over the limited capacity communication channel to the actuator. For the first-order coder the predicted valueũ[k + 1] ≡ u (k + 1)T 0 is taken equal toũ[k] ≡ũ(kT 0 ), see [Tatikonda and Mitter, 2004;Fradkov et al., 2006]. In the coders with a memory, the sequence of centroids is transmitted at step k over the communication channel to the decoder. Then centroid c[k +1] is renewed by the following update algorithm: Equations (16)-(18) describe the coder algorithm. The same algorithm is realized by the decoder. Namely, the decoder calculates variablec[k] based on received codeword flow similarly to s[k] and on the value of M which is known on the receiver side.
As obtained in , in order to avoid the failure of the received signalũ(t) to track the transmitted signal u(t), the following inequality

Simulation Results for Signal Transmission with Time-invariant First Order Coder
The closed-loop system is simulated for various values sampling period T 0 . It is worth mentioning that for a binary quantizer (16) the communication channel bit-rate R (in bits per a time unit) is defined as R = 1/T 0 . This relation makes it possible to calculate the demanded communication channel capacity. For the present study L u = 30 has been experimentally found by the simulation of the "ideal" control system (1), (2), (6), (12) with the given initial conditions (13). Parameter α in (19) is set to α = 1.05. The simulation results are shown in Fig. 5, where time histories of the energy error ∆E(t) are depicted for the binary time-invariant coder with various sampling time T 0 . Since the transients do not tend to the steady-state values, the averaged limiting values are used to evaluate the system accuracy. The results are summarized in Tab. 5, where the dependence of the steady-state averaged relative energy error δE on the time sampling interval T 0 and data transmission rate R = 1/T 0 (in bits per time unit) is given. These results allow one to conclude that under a data rate less that 2.0 bits per time unit for every control channel i = 1, . . . , m lead to practically acceptable energy error. The data transmission scheme described in this section relies upon the Lipschitz constant L u of u(t).
Since the average rate of change for u(t) may vary on time (for example, it differs during the transients and in the steady-state mode, as demonstrated in [Orlov et al., 2017b;Orlov et al., 2017a;Orlov et al., 2018;Orlov et al., 2019]), calculation of M through this constant by means of (19) may be too conservative, reducing Table 5. Steady-state averaged relative energy error δE vs. time sampling interval T 0 and data transmission rate R. the data transmission accuracy. Therefore, when the transient is over, the coder saturation state and consequent failure of the signal transmission may be avoided for less values of M , than are given by (19). Potentially, reducing M on t makes it possible to ensure the asymptotical stabilization of the system. This approach is usually realized in the so-called zooming strategy [Brockett and Liberzon, 2000;Liberzon, 2003;Tatikonda and Mitter, 2004;Fradkov et al., 2006;Nair et al., 2007]. The values of M [k] may be precomputed (the time-based zooming), or, alternatively, current quantized measurements may be used at each step to update M[k] (the event-based zooming). Time-based zooming is described and evaluated in Sec. 5.5. The event-based zooming, in the form of the adaptive coding is considered in Sec. 5.6.

Control Signal Transmission Over the Binary Communication Channel with First Order
Coder and Time-based Zooming In time-varying quantizers [Brockett and Liberzon, 2000;Liberzon, 2003;Tatikonda and Mitter, 2004;Fradkov et al., 2006;Nair et al., 2007] the range M is updated with time and different values of M are used at each step, M = M [k]. Using such a "zooming" strategy it is possible to increase coder accuracy in the steady-state mode and at the same time, to prevent coder saturation at the beginning of the process. To this end, the following recursive algorithm for the quantizer range M [k] may be employed where 0 < ρ 1 is the decay parameter, M ∞ stands for the limit value of M  Fig. 6. One may notice that for the considered system an asymptotic stabilization has not been achieved due to non-zero value of M ∞ . The case of M ∞ = 0 and s = 0.15 is depicted in Fig. 7. The simulation results in the form of dependence of the steadystate averaged relative control error δE on time sampling interval T 0 and data transmission rate R = 1/T 0 (in bits per time unit) are summarized in Tabs. 6, 7.

Control Signal Transmission Over the Binary Communication Channel with Adaptive Firstorder Coder
The following adaptive coding procedure may be employed. It assumes adaptively changing quantizer range M [k] depending on the current measurements, cf. [Goodman and Gersho, 1974;Andrievsky et al., 2010a;Gomez-Estern et al., 2011;Goodwin et al., 2012;Andrievsky, 2013]. Range M [k] in (16)-(18) is updated adaptively by means of the following proce-  Table 7. Dependence of the steady-state averaged relative control error δE on time sampling interval T 0 and data transmission rate R for the coder with a time-based zooming; s = 0.6, M ∞ = 0.     In Tab. 8, the dependence of the steady-state averaged relative control error δE on the time sampling interval T 0 and data transmission rate R = 1/T 0 (in bits per time unit) is presented for the adaptive coder. The time histories of M [k] and control action u(t) are

Conclusions
In the paper, an impression on the limitations of the energy control for the sine-Gordon chains by means of the spatially-discretized control with time and level quantization is obtained. The speed-gradient-based state feedback energy control, developed in [Orlov et al., 2018;Orlov et al., 2019], is numerically evaluated for a sine-Gordon spatially distributed system. The closedloop system robustness is investigated under various types of disturbances, caused by: sampling-in-time control signal quantization, the level quantization for control, continuous in time; control signal quantization on level jointly with time sampling; control signal transmission over the binary communication channel with time-invariant first order coder; control signal transmission over the binary communication channel with first order coder and time-based zooming; control signal transmission over the binary communication channel with adaptive first-order coder. The results obtained demonstrate robustness of the examined regulation algorithm with respect to the considered disturbances. The steady-state error dependence on the quantization/sampling parameters is numerically evaluated by the simulations. Particularly, it is shown that the most critical role is played by the time sampling, which may lead to loss of the system stability once a certain threshold is overcome.
In the future it is planned to expand this study on the output-feedback energy control systems of [Orlov et al., 2019] where both the control and measurement signals may be affected by the quantization, and to evaluation of non-discontinuous type speed gradient control algorithms. Numerical study of robustness in the presence of quantization disturbances and extension to non-collocated actuation and sensing of other nonlinear distributed parameter models, e.g., Klein-Gordon PDE and chains [Dolgopolik et al., 2018;Kovaleva, 2016] are among challenges to be tackled within the adopted framevork.