Method and apparatus for M-level control and digital-to-analog conversion

Abstract

A method is disclosed for guiding a physical analog system (e.g., an electric motor) using discrete-level (e.g., binary) control signals. The discrete-level control signals are calculated by an iterative method that can handle long time horizons. A preference for infrequent level switching can be taken into account. The quality of the fit to the target trajectory can be expressed by other norms as well as quadratic error. The method can also be used for digital-to-analog conversion.

Description

[Technical field]

[0001] The present invention relates to finite control set model predictive control (FCS-MPC) and digital-to-analog conversion by using methods derived from sparse Bayesian learning.

[0002] In particular, the invention relates to a method for controlling an analogue physical system by means of at least one M-level input signal, and to a device for implementing this method. [Background technology]

[0003] In both automatic control and digital-to-analog conversion, the problem arises of computing binary, ternary, or M-level input signals to guide an analog physical system along some desired trajectory, or to cause it to produce some desired analog output signal. This problem is non-trivial (even for linear systems), and the computational complexity of known algorithms to compute "optimal" such input signals grows exponentially with the time horizon [1].

[0004] In model predictive control (MPC), many related methods have been developed with various trade-offs between "optimality" within a fixed (short) time horizon and the ability to handle long time horizons [2, 3, 4, 5]. However, for applications requiring long time horizons, existing methods are often inadequate.

[0005] One method for digital-to-analog conversion is to feed a digitally calculated binary (or M-level) signal to an analog linear filter, which then creates an approximation of the desired analog signal. The standard method for calculating the required binary (or M-level) signal uses a digital ΔΣ modulator, but this limits the time horizon to the time of a single step. Using the same method (and the same limitations) as MPC, a longer time horizon can be achieved.

[0006] To solve the time-limited problem, the method disclosed here uses ideas from sparse Bayesian learning [6, 7, 8, 9] and variational representations of non-Gaussian distributions as in [10, 11, 12]. [Prior art documents] [Non-patent literature]

[0007] [Non-Patent Document 1] [1] AHLand and AG Doig, "An automatic method for solving discrete programming problems," ECONOMETRICA, vol.28, no.3, pp. 497-520, 1960. [Non-Patent Document 2] [2] J. Richalet, A. Rault, J. Testud, and J. Papon, "Model predictive heuristic control", Automatica (Journal of IFAC), vol. 14, no. 5, pp. 413-428, 1978 [Non-Patent Document 3] [3] T. Geyer, Model Predictive Control of High Power Converters and Industrial Drives, pp. 3-23, John Wiley & Sons, 2016 [Non-Patent Document 4] [4] P. Karamanakos, T. Geyer, N. Oikonomou, F. Kieferndorf, and S. Manias, “Model predictive control in power electronics: Strategies to reduce the computational complexity,” IECON2013-39th Annual Conference of the IEEE Industrial Electronics Society, pp. 5818–5823, IEEE, 2013. [Non-Patent Document 5] [5] Y. Zhang, W. Xie, Z. Li, and Y. Zhang, “Model predictive direct power control of a PWM rectifier with duty cycle optimization,” IEEE Transactions on Power Electronics, vol. 28, no. 11, pp. 5343–5351, 2013. [Non-Patent Document 6] [6] METipping, "Sparse Bayesian learning and the relevance vector machine", Journal of Machine Learning Research, vol.1, no.Jun, pp. 211-244, 2001. [Non-Patent Document 7] [7] METipping, A. Faul, and JJTAvenue, "Fast marginal likelihood maximization for sparse Bayesian models," Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, pp. 3-6, 2003. [Non-Patent Document 8] [8] DPWipf and BDRao, "Sparse Bayesian learning for basis selection", IEEE Transactions on Signal Processing, vol.52, no.8, pp. 2153-2164, 2004. [Non-Patent Document 9] [9] DPWipf and SSNagarajan, “A new view of automatic relevance determination,” Advances in Neural Information Processing Systems, pp. 1625–1632, 2008. [Non-Patent Document 10]

[10] J. Palmer, K. Kreutz-Delgado, D. Wipf, and B. Rao, “Variational representations of non-Gaussian priors”, UCSD, San Diego, 2005. [Non-Patent Document 11]

[11] J. Palmer, K. Kreutz-Delgado, BDRao, and DPWipf, “Variational EM algorithms for non- Gaussian latent variable models,” Advances in Neural Information Processing Systems, pp. 1059–1066, 2006. [Non-Patent Document 12]

[12] H.-A. Loeliger, B. Ma, H. Malmberg, and F. Wadehn, "Factor graphs with NUV priors and iteratively reweighted descent for sparse least squares and more," 2018 IEEE 10th International Symposium on Turbo Codes & Iterative Information Processing (ISTC), pp. 1-5, IEEE, 2018 [Non-Patent Document 13]

[13] RPSallen and ELKey, “A practical method of designing RC active filters,” IRE Transactions on Circuit Theory, vol. 2, no. 1, pp. 74-85, 1955. [Non-Patent Document 14]

[14] H.-A. Loeliger, L. Bruderer, H. Malmberg, F. Wadehn, and N. Zalmai, “On sparsity by NUV-EM, Gaussian message passing, and Kalman smoothing,” 2016 Information Theory and Applications Workshop (ITA), pp. 1-10, La Jolla, CA, 2016. Summary of the Invention [Problem to be solved by the invention]

[0008] The problem solved by this invention is to compute binary, ternary, or M-level input signals to guide a linear (or linearized) analog physical system along some desired trajectory, or to cause a linear analog physical system to produce some desired analog output signal, in a manner that can handle arbitrarily long time horizons. Furthermore, a preference for sparsity can be easily added so that the number of level switches in the discrete input signal is kept low. [Means for solving the problem]

[0009] The main idea is to represent the M-level constraints by a combination of suitably parameterized Gaussian distributions (as detailed in Section 4), which allows the actual computation of the M-level input signals using iterative Kalman-type recursion (as detailed in Section 5). As a result, the computational complexity per iteration grows only linearly with the time horizon.

[0010] Formally speaking, the present invention provides an input signal u=(u1,...,u k ) to map an analog physical system to a given target trajectory.

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[0046] The system is modeled by a discrete-time linear state-space model having the method uses an extended version of models (3) and (4), the extended version being a state-space recursion for k ∈ {1, ..., K}

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[0011] The invention also relates to a device for carrying out this method, said device comprising: - an output interface for generating said input signal u to an analogue physical system; - a digital control unit connected to said output interface and adapted and structured to implement the method described above; Equipped with.

[0012] Finally, the present invention also relates to the following applications: - Digital to analog conversion by feeding the discrete valued signal into an analog low pass filter, - Digital to analog conversion by feeding the discrete valued signal into an analog band pass filter, - controlling an electric motor with at least one discrete level input signal; - controlling a power converter with at least one discrete level input signal; The present invention relates to the use of a method or device for at least one of the following:

[0013] The invention will be better understood and further objects will become apparent from a consideration of the following detailed description, which refers to the accompanying drawings, in which: [Brief description of the drawings]

[0014] [Figure 1] FIG. 1 shows a system model with an analogue physical system and a control unit with a digital control unit and an output interface. [Diagram 2] This figure shows a third-order low-pass filter in the Sallen-Key topology. [Diagram 3] FIG. 3 is a diagram showing an example of values ​​of the DACs in FIGS. 1 and 2. [Figure 4] FIG. 2 illustrates the cost function of equation (22) for a two-level constraint. [Diagram 5] FIG. 2 shows the cost function of equation (25) for a three-level constraint. [Figure 6] FIG. 2 shows the cost function of equation (28) for a four-level constraint. [Figure 7] FIG. 13 shows three numerical examples of electric motor control as in section 7. [Figure 8] FIG. 13 shows a numerical example of box-constrained trajectory planning as in Section 9. [Figure 9] FIG. 10.1 shows a numerical example of MIMO electric motor control as in Section 10.1. [Figure 10] FIG. 1 shows the dynamic system of Section 10.2. [Figure 11] FIG. 10.2 shows a numerical example of trajectory planning as in Section 10.2. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0015] As mentioned above, the main idea of ​​the disclosed invention is to represent the M-level constraint by a combination of parameterized Gaussian distributions, as detailed in Section 4 below. However, this main idea requires the context of several other (mostly well-known) concepts, as explained in Sections 3, 5, and 6. In Sections 3-5, a digital-to-analog converter similar to Section 2.1 is used for illustration purposes. Further examples (i.e., further embodiments of the disclosed invention) are given in Sections 7 and 10.

[0016] 1 Notation A vector of all zeros of length N is 0 N and a matrix of all zeros of dimension N × M is denoted by 0 N×M and a vector of all ones of length N is denoted by 1 N and a matrix of all ones with dimensions N × M is denoted by 1. N×M and the identity matrix of dimension N×N is I N As shown in Fig.

[0017] Furthermore, the average

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[0018] Mean Vector

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[0019] 2. System Overview As shown in Figure 1, the present invention relates to a technique for controlling an analog physical system 1 using a device 2. For example, system 1 can be a digital to analog converter, an electric motor, or a low pass or band pass filter as part of the load in a power converter.

[0020] The system 1 has at least one input signal u, which can take on at least M>1 discrete levels, and at least one output signal y.

[0021] The device 2 comprises an output interface 4 for generating an input signal u. The output interface 4 may for example be a binary or ternary output port of the device 2.

[0022] Furthermore, the device 2 comprises a control unit 3, such as a microprocessor or an FPGA, adapted to carry out the steps of the methods described herein. The device 2 is adapted to generate an input signal u in such a manner that an output signal y follows a desired trajectory (target trajectory).

[0023] 2.1 Examples As a first example, consider a digital-to-analog converter in which the analog physical system 1 in Figure 1 is a simple (i.e., low-cost) analog low-pass filter. The analog filter is exemplified by the circuit in Figure 2, which shows a Sallen-Key low-pass filter

[13] . The task of the digital control unit 3 and output interface 4 is to determine whether the resulting output waveform y corresponds to any given target waveform, as shown in Figure 3.

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[0024] 3 Analog System Model The linear state-space models (3) and (4) represent a given analog physical system. Such state-space models can be obtained by well-known methods (including time discretization and linearization) from an underlying continuous-time state-space model. However, the method makes use of extended system models (5) and (6), which are extended versions of the analog physical system models and may include additional inputs, states, or outputs.

[0025] 3.1 Examples In the example in Section 2.1, we consider a single scalar input signal u = (u1,...,u K ) (i.e., L=1) and a single scalar output y (i.e., H=1). Every tick of the digital clock, t k and tick t k+1 Between, the input voltage remains at a constant level u for a continuous period of time. k (See Figure 3.) For the particular circuit in Figure 2,

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[0026] The numerical example in FIG. 3 uses the following values ​​for the components in FIG. 2: R1=9.1kΩ, C1=16μF (14) R2=82kΩ, C2=4.3μF (15) R3=36kΩ, C3=2μF (16) The resulting frequency response is obtained using

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[0000] T The state-space matrices of the extended system model in this example are

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[0027] 3.2 Outside the scope of the examples In the above example, matrix A k , B k , C k , and D kdoes not depend on k, but in general it may. In fact, these matrices may even depend on the iteration step i (e.g. as a consequence of customary linear techniques). Furthermore, in general, analog physical systems may have multiple inputs (L ≥ 1) and / or multiple outputs (H ≥ 1) (see Section 10.1). It is not necessary to restrict all input signals to discrete levels. Furthermore, the system model may include stochastic noise, hence the term D in (3). k e k It can also be driven by

[0028] 4 M-level prior For simplicity of explanation, we assume a single scalar input signal (ie, L=1).

[0029] 4.1 Two-level priors Limiting the input signal u of an analog system to two levels {a, b} is S=1 (19) That is,

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[0030] function

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[0031] 4.2 Three-level prior distribution Restricting the input signal u of an analog system to three levels {b, 0, -b} is S =

[0011] (23) That is,

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[0032] In this case, the function (21) is as follows, as is clear from FIG.

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[0033] A three-level prior is a combination of two two-level priors, while S is

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[0034] 4.3 Four-level prior distribution Restricting the input signal u of an analog system to four levels {0,b,2b,3b} is S =

[0111] (26), that is,

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[0035] In this case, the function (21) is as follows, as is clear from FIG.

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[0036] 4.4 M-level The M-level prior distribution, where M>3, is

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[0037] 5. Extended System Model and Algorithms The analog physical system is modeled by a discrete-time linear state-space model with state recursion (3) and output relationship (4), and the matrix A k , B k , C k , and D k is obtained using well-known methods. The model consists of L input signal components, N states, and H output signal components, with at least one input signal component being discrete-valued. D k and e k For simplicity, in the given examples in Sections 5, 7, and 10, said system noise is omitted.

[0038] However, the disclosed method utilizes extended system models (5) and (6), which are extended versions of the analog physical system model and may include additional inputs, states, or outputs. In the example of Section 2.1, no additional input signal components, states, or output signal components are required, so the extended system model is trivially given by the analog physical model. Outside the scope of this example, additional input signal components may be used to model M>2 discrete-valued inputs, as shown in Section 4. Additional states and output signal components may be introduced, for example, to map input level differences to additional output signal components, as detailed in Section 7.

[0039] The extended system model includes, in addition to (5) and (6), the function (7), the prior distribution

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[0040] 5.1 Use of Linear Gaussian Estimation For fixed parameters θ (and ζ, if applicable), the augmented system model (2) is the usual linear Gaussian model. In particular, the posterior distribution

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[14] . Alternatively,

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[0041] Therefore, the input signal

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[0042] 5.2 Determining Variance by Alternating Maximizations One possibility for step (b) above is:

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[0043] Using (32) is effectively using the prior distribution (21).

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[0044] 5.3 Determining the variance by expectation maximization Another possibility for step (b) is the expectation maximization

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[0045] Real-life examples In the example in Section 2.1, function (7) is

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[0046] The numerical example in FIG. 3 is obtained by (41) and (42),

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[0047] 6 Online Operation and Feedback Control 6.1 Online Operation In many applications, the method of the present disclosure is used in an online mode as follows. In the first planning period (by the planning target period K), the method of the present disclosure is used to determine the input signals (u1,..., u K ). However, in practice, only an initial segment (u1,..., u r ) where 0 < r < K is supplied to the analog physical system. In the second planning period, the method of the present disclosure is used to determine the input signals (u r+1 ,..., u r+K ), but in practice, (u r+1 ,..., u2r ) is fed to the analog physical system. Similarly, in the nth planning period, using the method of the present disclosure, only the input signal (u (n-1)r+1 ,...,u (n-1)r+K ) is determined, and in fact, (u (n-1)r+1 ,...,u nr ) is fed to the analog physical system.

[0048] 6.2 Feedback Control In the on-line mode of section 6.1, measurements or additional output signals of the analogue physical system, if available, are used to determine the state of the system for each planning period (as described in step (a) of section 5.1) by the usual state tracking methods.

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[0049] 7 Sparse switching between discrete levels In many applications, switching an input signal between allowed discrete levels is costly (eg, due to heat dissipation) and should be done as infrequently as possible.

[0050] The method of the present disclosure can handle such a situation by an extended system model as follows. For simplicity of explanation, we assume a single scalar input signal (i.e., L=1) and a single scalar output (i.e., H=1). First, we consider the (A k , B k , C k , and D k Then, the state space model representation is derived as follows: k , B k , C k , and D kThe two states that result in the state-space matrices of the extended system model are

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[0051] The quadratic penalty for the level difference is

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[12] .

[0052] Example: Electric motor control As a concrete example, consider the following simple model of an electric motor. The model has a scalar input u (the control voltage applied to the motor coils) and a scalar output y (the motor coil current). The input is assumed to be a discrete-valued signal. The desired motor coil current

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[0011] , which strongly encourages the input signal to be {-1,0,1}. The target trajectory of the extended system model

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[0053] The numerical results in Fig. 7 are

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[0054] The first plot shows the desired target motor coil current.

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[0055] 8. Secondary fitting costs are out of scope For ease of explanation, this section will use scalar input and scalar output (i.e.,

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[0056] Any coefficient is

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[0057] As mentioned above, β k is a constant scale factor that is precomputed before the first iteration and is not updated during the iteration.

[0058] Arbitrary Coefficients

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[0059] Example: Absolute value penalty The absolute penalty (i.e., p=1) is

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[0060] 9 Soft Box Constraints For ease of explanation, this section will use scalar input and scalar output (i.e.,

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[0061] Any coefficient is

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[0062] The arbitrary parameters are

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[0063] Arbitrary Coefficients

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[0064] Example: Guiding an object through a tunnel As a concrete example, consider a control problem where the analog physical system is a third-order integrator chain. The task is to find the resulting output trajectory

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[0065] A discrete-time model of an analog physical system is

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[0066] prior distribution

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[0067] The numerical results in Fig. 8 are

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[0068] In the first plot, the desired boundary

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[0069] 10 Further Examples The following sections describe further embodiments of the present invention.

[0070] 10.1 Multiple Discrete Inputs and Multiple Outputs In the previous example, the analog physical system had a single input (i.e., L=1) and a single output (H=1). However, the methods of this disclosure can handle multiple inputs and outputs.

[0071] Example: MINO Motor Control This example generalizes the example from Section 7 to multiple inputs and outputs. Specifically, the motor in this example consists of three independently controlled motor coils. As a result, L=3 and H=3. The input signal u of the analog physical system has three components (u 1,1 ,...,u K,1 ), (u 1,2 ,...,u K,2 ), and (u 1,3 ,...,u K,3 ) is composed of the three signal components that represent the three motor coil currents. Similarly, the model output y contains three signal components that represent the three motor coil currents.

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[0072] The three motor coils of the analog physical system are modeled by three first-order integrators and have the following state-space representation with inductance α for k∈{1,...,K}:

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[0073] The state of the system is completely separate. k By filling in the off-diagonal entries of , we can easily incorporate coupling between states. The state-space matrices of the augmented system model are

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[0074] prior distribution

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[0075] The numerical results in Fig. 9 are

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[0076] 10.2 Trajectory Planning with Sparse Checkpoints In the previous example, the weight matrix W in (7) k did not depend on the time index k. However, time-dependent weights can be useful. For example, the target trajectory

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[0077] Example: Flappy Bird In this example, the analogue physical system consists of a point of mass m moving in the xy plane as depicted in Figure 10. The horizontal velocity v x is constant and is not of particular interest (it is not part of the system model). The vertical velocity v y is affected by the constant gravitational acceleration g which pulls the mass downwards. Any input fed into the system modifies its total impulse in the y direction and hence its vertical velocity.

[0078] The analog physical system is

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[0079] Function (7) is

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[0080] The numerical results in Fig. 11 are

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[0081] The four checkpoints at time instances k ∈ {60, 120, 180, 240} are illustrated by black crosses in the first plot.1,1 ,...,u K,1 ) is shown by a stem in the second plot. The resulting output trajectory y is depicted as a solid line in the first plot. As can be seen, the output trajectory y is an acceptable trajectory and does not collide with any obstacles.

[0082] Note While presently preferred embodiments of the invention have been illustrated and described, it is to be clearly understood that the invention is not limited thereto and can be variously embodied and carried out within the scope of the appended claims.

Claims

1. Input signal u = (u １ ,... ,u ｋ ) to map an analog physical system to a given target trajectory. [0010] at least one component (u) of the input signal, １，ｌ ,... ,u ｋ，ｌ ) is discretized with M levels, M∈{2, 3, . . .}, and the device: an output interface (4) for generating an input signal for said analog physical system; a digital control unit (3) connected to said output interface (4) and implementing a control method; Equipped with The control method includes: － [0025] is the input signal of the analog physical system at time k ∈ {1, . . . , K}, － [0030] is the planning period, L is the number of input signal components, l, with 1≦l≦L, is the index of the discrete-valued components of u, － [0045] is the given target trajectory at time k, - H is the number of components of the target trajectory; and There is an extended system model [0050] Against [006] The output signal is [0070] and for some matrix S and all k∈{1 ,...,K} [0080] Something like, something like [0090] Against [0010] An input signal that is ##EQU00011## and the time-dependent parameter θ = (θ １ , . . . , θ Ｋ ) and any additional parameter ζ=(ζ １ , . . . , ζ Ｋ ) and Assuming that, The digital control unit (3) (a) θ （ｉ－１） The parameters θ and ζ are fixed at （ｉ－１） , and performing a maximization step on the augmented system model with the parameter ζ fixed to , thereby obtaining any precision information ##EQU00012## Candidate input signals having ##EQU00013## and any precision information ##EQU14## An output signal having ##EQU00015## and obtaining (b) New parameters ##EQU00016## and optionally ##EQU00017## determining the formula: [0018] is calculated in step (a). [0019] and [0020] is a function of ##EQU00021## is calculated in step (a). [0022] and [0023] is a function of step and by repeatedly executing [0024] and the extended system model is adapted to determine [0025] It is assumed that the expression is － [0026] using parameters θ and ζ [0027] 、 [0028] 、 [0029] , and [0030] is the probability density function at － [0031] 、 [0032] but, [Equation 33] and optionally an additional trajectory, - "∝" denotes an equation up to a scale factor, the analogue physical system is a recursion for k∈{1, . . . , K}, [0034] And the state vector [Equation 35] , initial state x ０ ,noise [0036] , model output [Equation 37] and matrix A ｋ , B ｋ , C ｋ , D ｋ [0046] Assume that the inputs are modeled by a discrete-time linear state-space model with Use an expanded version of the model of Equation 83 and Equation 84, where the expanded version is a state-space recursion for k ∈ {1, . . . , K} [Equation 38] , and [0039] A state vector [0040] , initial state [0041] 、 [0042] The noise [0043] and the matrix [0044] 、 [0045] 、 [0046] , and [0047] But, A ｋ , B ｋ , C ｋ , and D ｋ is derived from － [0048] but, [0049] where W ｋ is a real, non-negative weight matrix, where [Number 50] is the output signal of the augmented system model, determined by Equation 85 and Equation 86; － [0051] is an optional coefficient in Equation 87, i.e., 1 or, if present, a fixed ζ ｋ Against [0052] up to the scale factor, [0,53] of the form [0,54] In some cases, － [0.55] is the Gaussian probability density, － [0,56] but [0.57] It is factored as [0,58] is the Gaussian probability density, － [0,59] but, [0.60] which results in ◇Fixed θ ｋ About [0061] but with a maximum scale factor of [0062] is the Gaussian probability density function in At least one index [0063] About [0,64] but, Different average values ​​m １ ,... ,m ｊ , and θ ｋ The variance value determined by [0.65] A product of Gaussian probability density functions with J≧2, [0.66] as a factor,device.

2. In step (a), the candidate input signal [0.67] But the posterior distribution [0068] The device of claim 1 , wherein the average value of

3. In step (a), the output signal [0,69] But the posterior distribution [Number 70] The device according to claim 1 or 2, wherein the average value of

4. In step (a), the accuracy information [Equation 71] but, [Equation 72] posterior variance of [Equation 73] but, [74] A device according to claim 1 , which exhibits discrete value components of

5. In step (a), the accuracy information [Number 75] , k∈{1 , . . . , K} for all [76] for [77] The device according to claim 1 , further comprising a posterior variance of

6. In step (b), the new parameters [78] For k∈{1, . . . , K}, [79] The device according to claim 1 , wherein the

7. In step (b), the new parameters [Number 80] For k∈{1, . . . , K}, [Number 81] where the expectation value E[·] is [0082] A device according to any one of claims 1 to 5, for 【Request 8】 【Number 83】 where: [Number 84] , S=1, [Number 85] There are two levels [Number 86] and [Number 87] 8. A device according to claim 1 , wherein 【Request 9】 【Number 88】 and [Number 89] where: [Number 90] , S=[1 1], [Number 91] , L=1, and [Number 92] 9. The device according to claim 1 , wherein 10. The digital control unit according to claim 1, wherein the digital control unit is a discrete-valued input signal component (u １，ｌ ,... ,u Ｋ，ｌ ) to the sparsity of the number of level switches in at least one of the [Number 93] 、 [Number 94] 、 [Number 95] , and [Number 96] The level switching is performed by using an additional output signal component [Number 97] and at least one [Number 98] in the case of, [Number 99] 10. The device according to claim 1, wherein

11. For some k∈{k １ , k ２ ,... . .} ⊂ {1,... . . ,K}, then [Number 100] otherwise, [Number 101] 11. The device according to claim 1, wherein

12. Formula 87 is: [00102] and － [Number 103] and － [Equation 104] is a constant scale factor, and the initial weights [00105] Depends on [Number 106] is defined by p is the order of the p-norm used, p>0, p≠2, － [Equation 107] and Arbitrary parameter ζ ｋ For k∈{1, . . . , K}, [Number 108] The device according to claim 1 , wherein for each iteration i, it is determined by:

13. Specifically, the arbitrary coefficient in Equation 88 is the coefficient [Number 109] For k∈{1, . . . , K}, [Equation 110] By setting [Equation 111] The device of claim 12 , further comprising:

14. Formula 87 is: [Equation 112] where: － [Equation 113] and － [Equation 114] is the auxiliary weight, － [Number 115] is the width of the boundary at time k, － [Equation 116] is a constant scale factor, － [Equation 117] and Arbitrary parameter ζ ｋ For k∈{1, . . . , K}, [Number 118] The device according to claim 1 , wherein for each iteration i, it is determined by:

15. Specifically, the arbitrary coefficient in Equation 88 is the coefficient [Equation 119] For k∈{1, . . . , K}, [Number 120] By setting [Number 121] 15. The device of claim 14, further comprising:

16. S. [Number 122] 16. The device of claim 1 , wherein the mapping is a linear mapping between at least one bi-level component of x and at least one discrete-valued component of u.

17. - digital to analogue conversion by feeding the discrete valued signal to an analogue low pass filter; - digital to analogue conversion by feeding the discrete valued signal to an analogue band pass filter; - controlling an electric motor with at least one discrete level input signal; - controlling a power converter with at least one discrete level input signal; 17. Use of a device according to any one of claims 1 to 16 for at least one of the following:

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