Technical Field

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

In particular, the invention relates to a method for controlling an analog physical system with at least one M-level input signal(s) as well as to a device for carrying out this method.

Background Art

The problem of computing a binary, ternary, or M-level input signal to steer an analog physical system along some desired trajectory, or to make it produce some desired analog output signal, appears both in automatic control and in analog-to-digital conversion. This problem is not trivial (even for linear systems), and the computational complexity of known algorithms for computing “optimal” such input signals grows exponentially with the planning horizon [1].

In model predictive control (MPC), many pertinent methods have been developed [2, 3, 4, 5], with different trade-offs between “optimality” within a fixed (small) planning horizon vs. the ability to handle a long planning horizon. However, for applications requiring a long planning horizon, existing methods are often unsatisfactory,

One method for analog-to-digital conversion is to feed a digitally computed binary (or M-level) signal into an analog linear filter, which then produces (an approximation of) the desired analog signal. A standard method to compute the required binary (or M-level) signal uses a digital DS modulator, which, however, limits the planning horizon to a single step in time. Longer planning horizons can be achieved with the same methods (and with the same limitations) as in MPC.

In order to solve the limited-horizon problem, the disclosed method uses ideas originating from sparse Bayesian learning [6, 7, 8, 9] and variational representations of non-Gaussian distributions as in [10, 11, 12],

Disclosure of the Invention

The problem solved by the invention is to compute a binary, ternary, or M-level input signal to steer a linear (or linearized) analog physical system along some desired trajectory, or to make it produce some desired analog output signal, by a method that can handle an arbitrarily long planning horizon. Moreover, a preference for sparsity can easily be added so that the number of level switches in the discrete input signal(s) is kept small.

The key idea is to represent the M-level constraint by a combination of suitably parameterized Gaussian distributions (as detailed in Section 4), which enables the actual computation of the M-level input signal by means of iterations of Kalman-type recursions (as detailed in Section 5). In consequence, the computational complexity per iteration grows only linearly with the planning horizon. In formal terms, the invention is a method for controlling an analog physical system with input signal to follow a given target trajectory , wherein at least one component of said input signal is discrete-valued with M levels, M ∈ { 2, 3, . .} wherein

- is the input signal of the analog physical system at time k ∈

- (normally 1) is a planning horizon,

- L is the number of input signal components,

- ℓ (with 1 ≤ ℓ ≤ L) is the index of said discrete-valued component of u ,

- is the given target trajectory at time k, - H is the number of components of the target trajectory, wherein said method uses an extended system model with output signal for some , and with input signal for some L, such that for some matrix S and all k ∈ {1, . . . , K}, and with time-dependent parameters θ = {θ ₁ , . . . , θ _k ) and optional additional parameters ζ = (ζ ₁ , . . . , ζ _K ), wherein ( ) is determined by iteratively executing, in iterations i = 1, 2, 3, . . . , the steps of

(a) executing a maximization step in said extended system model with parameters θ fixed to θ ^(i-1) and ζ fixed to thereby obtaining a candidate input signal with optional precision information ), and an output signal with optional precision information , and

(b) determining new parameters and optionally , where is a function of and and is a function of and as computed in Step (a), and wherein said extended system model can be expressed as wherein

- is a probability density function in and with parameters θ and ζ,

- is the target trajectory of said extended system model comprising and optionally additional trajectories,

- “α ” denotes equality up to a scale factor,

- said analog physical system is modeled by a discrete-time linear state space model with recursions for k ∈ {1, . . . , K}, and with state vector , initial state x ₀ , noise , model output and matrices A _k , B _k , C _k , D _k , - said method uses an extended version of model (3) and (4) which is given by the state space recursions for k ∈ {1, . , . , K}, and with state vector , initial state , noise with , and wherein the matrices are derived from A _k , B _k , C _k and D _k , is given by where W _k is a real non-negative weight matrix and where is the extended system model output signal determined by (5) and (6),

- ) is an optional factor in (7), which means it is either 1, or if present, it may take the form such that, for fixed is a Gaussian probability density function in _k , up to a scale factor, - ) is a Gaussian probability density,

- factors as where each ) is a Gaussian probability density,

- ) factors as such that, o for fixed is a Gaussian probability density function in , up to a scale factor, and o for at least one index contains as a factor the product of J ≥ 2 Gaussian probability density functions with different means m ₁ , . . . , m _J and with variances determined by θ _k . The invention also relates to a device for carrying out this method. This device comprises:

- an output interface for generating said input signal u to the analog physical system and

- a digital control unit connected to said output interface and adapted and structured to carry out the above method. Finally, the invention also relates to a use of the method or device for at least one of the following applications:

- digital-to-analog conversion by feeding a discrete-valued signal into an analog low-pass filter,

- digital-to-analog conversion by feeding a discrete-valued signal into an analog band-pass filter,

- controlling an electric motor with at least one discrete-level input signal,

- controlling an electric power converter with at least one discrete-level input signal.

Brief Description of the Drawings

The invention will be better understood and objects other than those set forth above will become apparent when consideration is given to the following detailed description thereof. This description makes reference to the annexed drawings, wherein:

Fig. 1 shows the system model with an analog physical system and a control unit, comprising a digital control unit and an output interface.

Fig. 2 shows a third order low-pass filter in Sallen-Key topology.

Fig. 3 shows a numerical example of the DAC of Figs. 1 and 2.

Fig. 4 shows the cost function of eq. (22) for a two-level constraint.

Fig. 5 shows the cost function of eq. (25) for a three-level constraint.

Fig. 6 shows the cost function of eq. (28) for a four-level constraint.

Fig. 7 shows three numerical examples of electric motor control as in Section 7.

Fig. 8 shows a numerical example of MIMO electric motor control as in Section 9.1.

Fig. 9 shows a diagram illustrating the dynamical system of Section 9.2.

Fig. 10 shows a numerical example of trajectory planning as in Section 9.2.

Modes for Carrying Out the Invention

As mentioned, the key idea of the disclosed invention is to represent the M- level constraint by a combination of parameterized Gaussian distributions, as will be detailed in Section 4 below. However, this key idea requires a context of several other (mostly well-known) concepts, as will be explained in Sections 3, 5 and 6. Throughout Sections 3-5, a digital-to-analog converter as in Section 2.1 will be used for illustration. Further examples (i.e., further embodiments of the disclosed invention) will be given in Sections 7 and 9.

1 Notation

We denote the all-zero vector of length N by (Tv, the all zero matrix of dimension N x M by QN _X M, the all-ones vector of length N by 1 jy, the all-ones matrix of dimension A" x M by 1 N _X M and the identity matrix of dimension N x N by IN.

Further, the probability density function of a univariate Gaussian distribution in x e R with mean TO e R and variance s ² € R

The probability density function of a multivariate Gaussian distribution in x £ m e R^ and covariance matrix V € R ^NxN is denoted by

2 System Overview

As shown in Fig. 1, the present invention relates to techniques for controlling an analog physical system 1 by means of a device 2. For example, system 1 may be a low-pass or band-pass filter as part of a digital-to-analog converter, an electric motor, or the load in a electric power converter. System 1 has at least one input signal u that can take on at least M > 1 discrete levels and at least one output signal y.

Device 2 comprises an output interface 4 for generating the input signal u. For example, the output interface 4 may be a binary or ternary output port of device 2. Further, device 2 comprises a control unit 3, such as a microprocessor or FPGA, adapted to carry out the steps of the method described here. Device 2 is adapted to generate the input signal u in such a manner that the output signal y follows a desired trajectory (target trajectory).

2.1 Example

As a first example, consider a digital-to-analog converter wherein the analog physical system 1 of Fig. 1 is a simple (i.e., low-cost) analog low-pass filter. The analog filter is exemplified by the circuit in Fig. 2, which shows a Sallen-Key low-pass filter [13]. The task of the digital control unit 3 and the output interface 4 is to produce a two-level input signal u to the analog filter such that the resulting output waveform y closely approximates any given target waveform y, as illustrated in Fig. 3. This task is solved by the disclosed invention as will be detailed below. Moreover, in Sections 3 and 5, this example will be used to illustrate the concepts and methods of this disclosure.

3 Analog-System Model

The linear state space model (3) and (4) represents the given analog physical system. Such a state space model can be obtained by well-known methods (including discretization in time and linearization) from an underlying continuous-time state space model. However, the method utilizes an extended system model (5) and (6) which is an extended version of said analog physical system model and may contain additional inputs, states or outputs.

3.1 Example

In the example of Section 2.1, we have a single scalar input signal u — (m, . . ■ , UK) (i.e., L — 1) and a single scalar output y (i.e., H — 1). Between ticks and of the digital clock, the continuous- time input voltage is kept at the constant level u _¾ (cf. Fig. 3). For the specific circuit of Fig. 2, we obtain a state space representation of the form (3) and (4) with e R ³ as follows.

The numerical example in Fig. 3 is obtained with the following values for the components in Fig. 2:

Ri = 9.1 kO, Ci = 16 pF, (14) 4.3 pF, (15) 2 pF. (16)

The resulting frequency response is output voltage 270.535

(17) input voltage (feu) ³ + 16.926(icj) ² + 108.160M + 270.535 with i = and frequency / = w/(2p). The corresponding discrete-time model of the analog physical system is given by

0.83941 -1.006 -2.48427 ^'

A _f c 0.00918 0.99485 -0.01278 (18) 0.00005 0.00998 0.99996 B _k = [0.00918 0.00005 0] ^T , C _k = [O 0 270.53523] , and D _k = [0 0 0] ^T . The state space matrices of the extended system model for this example are trivially given by .

3.2 Beyond the Example In the above example, the matrices A _k , B _k , C _k and D _k do not depend on k, but in general, they might. In fact, these matrices may even depend on the iteration step i (e.g., as a consequence of customary linearization techniques). Moreover, in general, the analog physical system may have multiple inputs (L ≥ 1) and/or multiple outputs ( H ≥ 1), cf. Section 9.1. Not all input signals need be restricted to discrete levels. In addition, the system model may also be driven by stochastic noise, hence the term D _k e _k in (3).

4 M -level Priors

For ease of exposition, we assume here a single scalar input signal (i.e., L = 1).

4.1 Two-Level Prior

Restricting the analog-system input signal u to two levels {a, b} can be achieved with S = 1 (19) in (1), i.e., and , and with and θ _k in (10) as and ). For fixed θ _k , the function (20) is a product of two Gaussian probability density functions, which is again a Gaussian probability density function, up to a scale factor. The function can be viewed as a prior on that strongly encourages u _k to lie in {a, b}, as is obvious from Fig. 4, which shows the function 4.2 Three-Level Prior

Restricting the analog-system input signal u to three levels {b, 0, —b} can be achieved with

S = [1 1] , (23) i.e., and , and with in (10) as with · Again, for fixed θ _k , the function (24) is a Gaussian probability density function, up to a scale factor. In this case, the function (21) can be viewed as a prior on u _k that strongly encourages u _k = Su _k to lie in { 6 , 0, —6}, as is obvious from Fig. 5, which shows the function cost(u _f c) = — log

The three-level prior is a combination of two two-level priors whereas S acts as a mapping between the two-level components of u and the three-level component of u.

4.3 Four-Level Prior

Restricting the analog-system input signal u to four levels {0, b, 2b, 36} can be achieved with

5 = [1 1 l] , (26) with Again, for fixed 9 _k , the function (27) is a Gaussian probability density function, up to a scale factor.

In this case, the function (21) can be viewed as a prior on u _k that strongly encourages u _k — Su _k to lie in {0, b, 2b, 36}, as is obvious from Fig. 6, which shows the function cost(u _f c) = — log

4.4 M-Levels

M- level priors with M > 3 can be achieved, in more than one way, by generalizations of (23) and (24) with with fixed means m _f - _· and with 9 _k comprising the variances Advantageously, the components u _{k j} are restricted to two levels and S is an all-ones vector.

5 Extended System Model and Algorithms

The analog physical system is modeled by a discrete-time linear state space model with state recur- sions (3) and output relation (4) where the matrices A _k , B _k , C _k and D _k are obtained using well-known methods. Said model consists of L input signal components, N states and H output signal components where at least one input signal component is discrete-valued. One may use D _k and e _k to model any present system noise. For the sake of simplicity, said system noise is omitted in the given examples of Sections 5, 7 and 9. The disclosed method, however, utilizes an extended system model (5) and (6) which is an extended version of said analog physical system model and may contain additional inputs, states or outputs. In the example of Section 2.1, the extended system model is trivially given by the analog physical model since no additional inputs signal components, states or output signal components are needed. Beyond this example, additional inputs signal components may be used to model discrete- valued inputs for M > 2, 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.

The extended system model comprises, besides (5) and (6), also the function (7), the prior r(ΰ, Q ), the initial state distribution p(x o), and the noise distribution p(e) and is summarized in (2).

5.1 Harnessing Linear Gaussian Estimation

For fixed parameters Q (and z, if applicable), the extended system model (2) is a customary linear- Gaussian model. In particular, the posterior distributions p(u\y, q , z) and p(y\y, q, z) are both jointly Gaussian, and the MAP (maximum a posteriori) estimates of u and y coincide with their MMSE (minimum mean-squared error) estimates. Moreover, the estimate of ΰ — (ΰi, . . . , ΰc) with {¾ = (ΰ _¾, i, . . . , u _k r), the estimate of y = (yi, . . . , yx) with y _¾ = (y _¾ .i, . . . , y _{k #} ) and the corresponding sequence of posterior variances . . , V _y _ ) can be computed by well-known recursive algorithms (variations of Kalman filter algorithms) with complexity linear in K. A preferred such algorithm is the MBF algorithm augmented with input signal estimation as described in Section V of [14]. Alternatively, the estimates of u and y (but not their variances) can be computed by steepest-descent or other customary optimization algorithms. Accordingly, the input signal u can be computed by alternating, in iterations i = 1, 2, 3, . . . , the steps of

(a) computing the estimates _■ . · , y _# ) and optionally the = (V^ , . . . , V^) for fixed parameters (b) determining new parameters based on (and perhaps V~^) and new parameters based on yW (and perhaps V~^), as computed in Step (a).

The following two Sections describe two ways to determine new parameters Q. Specific ways to determine the optional parameters z are given in Sections 7 and 8.

5.2 Determining the Variances by Alternating Maximization

One possibility for Step (b) above is to determine the new parameters by q which splits into argmax p _k (u^\ 9 _k ) (32)

^@ _k for k) E {1, . . . , K}, which can be expressed in closed form: for _/ ¾({¾, i, . . . , u _k 0 _¾ ) as in (29), we obtain

Using (32) amounts to effectively using the prior (21). The posterior variances are not required.

5.3 Determining the Variances by Expectation Maximization

Another possibility for Step (b) is to determine the new parameters by expectation maximization as follows: argmaxE logp(f/, y|0, C ^{(i 1}} ) (34) q where the expectation is with respect to p(u \ y, But (34) simplifies to

Q k argmax Sk for fc e {1, . . . , K}, where the expectation is with respect to p(u _k \ y, 0( ^l L· ^) as in (29), we obtain

(37)

Example

In the example of Section 2.1, the function (7) can be written as and

Determining and

The numerical example in Fig. 3 is obtained with (41) and (42), p(x o) = Af{xo] 0, 0.01), e _¾ = 0, and with w = 25. Larger values of w in (38) will yield better approximations of the target y. However, if w is too large, the computed input signal u may fail to comply with the two-level constraint. 6 Online Operation and Feedback Control 6.1 Online Operation

In many applications, the disclosed method will be used in an online mode as follows. In a first planning period (with planning horizon K), the disclosed method is used to determine an input signal (w,i, . . . However, only an initial segment (ui, . . . , u _r ) with 0 < r < K is actually fed into said analog physical system. In a second planning period, the disclosed method is used to determine an input signal (u _{r+ 1 ;} . . . , U _T+ K), but only (u _r+ 1, . . . , U2 _T ) is actually fed into said analog physical system. An so on: in the n-th planning period, the disclosed method is used to determine an input signal (tq _n _ i _)r +i, . . . , n· _(h-ΐ)t +k) _> °f which only (u _(n-1 ) _r+1 , . . . , u _nr ) is actually fed into said analog physical system.

6.2 Feedback Control

In the online mode of Section 6.1, measurements or additional output signals of said analog physical system, if available, can be used to improve p(x o) (i.e., the estimate of the initial state XQ), for each planning period (as described in Section 5.1, Step (a)) by customary state-tracking methods. 7 Sparse Switching Between Discrete Levels

In many applications, switching the input signal between the allowed discrete levels is costly (e.g., because of thermal losses) and should be done as infrequently as possible.

The disclosed method can handle such situations by an extended system model as follows. For ease of exposition, we will assume a single scalar input signal (i.e., L — 1) and a single scalar output (i.e. , H = 1). First, a state space model representation (with A _k , B _k , C _k and D _k ) of the analog physical system is derived. Then, the state space is extended by two states yielding the state space matrices of the extended system model derived from A _k , B _k , C _k and D _k , with u _{k —} u _k , e _k — e _k , and y _k = (¾ -¾-i, 2/ _f c) for k € {1, . . . , K}. By choosing C _k as above, the level differences u _k — u _k - 1 appear as an additional signal component y _{kt i} in the extended system model output signal y.

A quadratic penalty on the level differences can be achieved by setting nonzero weights in W _k of the function (7) for the particular output signal component y _{k} i} - Penalties of other forms can be achieved using the optional factors ip _k iy _k^ ( _¾ ) in (8). Said factors i/J _k (V _k , C _k ) are proportional to Gaussian probability densities for fixed C _k . However, their variances may be unknown and determined in every iteration step. By choosing suitable factors ip _k (y _k , z _k ), the penalty function cost( can advantageously represent the absolute value function \u _k — ¾_i|, the Huber function, or convex- concave functions as described in [12].

Example: Electric Motor Control

For a concrete example, consider a simple model of an electric motor as follows. The model has a scalar input u (the control voltage applied to the motor coil) and a scalar output y (the motor coil current) where the input is assumed to be a discrete-valued signal. Given a desired motor coil current y (target trajectory), the goal is to determine a discrete- valued (three level) control voltage u such that the resulting motor coil current approximates the desired motor coil current. The motor coil is assumed to be an ideal inductor modeled by a first order integrator. The discrete-time model of the analog physical system is given by

A _k = 1, B _k — l/a, C _k = 1, D _k = 0 (46) for k G {1, . . . , K} and where a is the inductance of the motor coil. The extended state space ma- trices (43) and (44) are for k e {1, . . . , K}. The priors p _k (u _k , _k ) are chosen according to the three-level prior described in Section 4.2 with 6 = 1 and S = [l l] , strongly encouraging the input signal to lie in { — 1, 0, 1}. The first component of the extended system model target trajectory y is given by (yiy, ■ ■ ■ , {/k, _l ) = (0, . . . , 0), and the second component by ($1 _, 2, · · ■ , VKP) = y- The function (7) becomes with diagonal weight matrix W _k = dia.g where w _\ > 0 and W > 0 are real weights. This results in a quadratic penalty on the level differences with weight w _\ and a quadratic penalty on the approximation error with weight w _% .

The numerical results of Fig. 7 are obtained with p(xo) = O ₃ , 0.01 ^■ I ), e _k = 0, a = 10 and

K = 150, and with (41) and (42). The experiment was executed three times with different weights w _\ and W -

The desired target motor coil current y is shown in the first plot as dashed line together with the resulting motor coil current y as solid line. The plots below show the determined input signal u as solid line. Three different trade-offs between switching effort and approximation error are shown from top to bottom.

Top: A large w _\ leads to few level switches and a small ΐί¾ allows large approximation error.

Middle: A Medium and u¾ leads to more level switches but a fairly good approximation of y. Bottom: A small w _\ and a large w 2 leads to many level switches and a precise approximation of y. 8 Beyond Quadratic Fitting Cost

For ease of exposition, an extended system model with scalar inputs and outputs (i.e., L = L = H = H — 1) is assumed for this section.

If the optional factor y(ΰ, z) = 1, the function (7) results in a standard 2-norm penalty on the deviation between the extended system model target trajectory y and extended system model output signal y. However, the disclosed method can handle other p-norms with p > 1 ,p $ 2 (and essentially also quasi-norms with 0 < p < 1) by utilizing the optional factor ip(y , z) and updating z in every iteration step. The function (7) can be written as with scalar weight w _k . Choosing yields where the weights w _k are determined by ( _k and /¾ is a constant scale factor depending on the initial weight w _k and is defined by

The optional parameters are determined in every iteration i by f or k e { 1 , . . . , K } .

As denoted above, /¾ is a constant scale factor, precalculated before the first iteration and not updated during the iterations.

The utilization of the optional factor ip(y, z) results in the following cost term ) ) Example: Absolute-Value Penalty

An absolute value penalty (i.e., p = 1) is achieved by choosing with 0 _c = w _k an /¾ = w _k /2. 9 Further Examples

In the following sections, we describe further embodiments of the invention.

9.1 Multiple Discrete Inputs and Multiple Outputs

In the previous examples, the analog physical system had a single input (i.e., L = 1) and a single output (H = 1). However, the disclosed method can handle multiple inputs and outputs.

Example: MIMO Motor Control

This example generalizes the example of Section 7 to multiple inputs and outputs. Specifically, the motor in this example is composed of three motor coils which are controlled independently. As a consequence, L = 3 and H = 3. The input signal u of the analog physical system consists of three components (tii, _! , . . ·· , ti _f tr.i), (tii, _{2 >} · · · , UK _{, 2} ) and (« _1,3 , . . . , HK,Ά) describing the discrete-valued control voltages for the three motor coils. Analogously, the model output y contains three signal components describing the three motor coil currents as well as the target trajectory y contains three signal components, describing the three desired motor coil currents.

The three motor coils of the analog physical system are modeled by three first-order integrators, yielding the following state space representation with inductance a for k £ {1, . . . , K}. The states of the system are fully decoupled. Coupling between the states could easily be incorporated by populating the non-diagonal entries of A _¾ . The state space matrices of the extended system model are trivially given by A _k = A _k , B _k = C _k — C _k and D _k = D _k - The prior P _k {u _k , 9 _k ) is a product of multiple three-level priors, i.e., with 9 _k = (9 _k , ₁ , 9 _k , 2·, 9 _k .p), and where each factor in (60) is chosen according to the three-level prior described in Section 4.2 with 6 = 1. The parameter <¾. comprises all variances of (24). The matrix S is chosen to be

Ί 1 0 0 0 O ^'

S — 0 0 1 1 0 0 (61) 0 0 0 0 1 1 with L = 6 yielding The function (7) amounts to with weight matrix W _k = wl _$ and y — y.

The numerical results of Fig. 8 are obtained withp(zo) = lί(c o; O3, 0.01 - I3), e _k = 0, w = 5, a = 5 and K 200, and with (41) and (42). The desired motor coil currents y are shown in the first plot together with the resulting motor coil currents y as solid line. The plots below show the determined input signals components of u as solid lines.

9.2 Trajectory Planning with Sparse Checkpoints

In the previous examples, the weight matrix W _k in (7) did not depend on the time index k. However, time-dependent weights can be useful. For example, consider a situation where the target trajectory y consists of sparse checkpoints at times k _\ , k2, · · · C {1 , , K}, with no constraints between these checkpoints. This situation can be handled by choosing W _k = (½ _c ^ for k £ {/¾, /¾, · · ·}·

Example: Flappy Bird

In this example, the analog physical system consists of a point of mass m moving on a xy- plane, as depicted in Fig. 9. The horizontal velocity v _x is constant and not of particular interest (and not part of the system model). The vertical velocity v _y is influenced by a constant gravitational acceleration g, pulling the point mass downwards. Any input fed to said system modifies its total impulse in y-direction and thus its vertical velocity.

The analog physical system is modeled by a linear state space model with two-dimensional input u (i.e., L = 2) and scalar output y (i.e., H — 1) according to for k <£ , K} where T is the time constant of the discretization. The extended system model is given by A _k = A _k , B _k = B _k , C _k = C _k , D _k - D _k , and

S = hx2· (68)

The extended input u (and also u) comprises of two input components, i.e., L — 2. The component (ΰ^i, . . . , ΰk,i) is a two-level control input and its corresponding prior p _k {v _*k ,i, 0 _k ), k € {1, . . . , K}, is chosen according to Section 4.1 with a = 0 and 6 = 1. The component («1,2, · · · , fix, 2) is a constant input accounting for the gravitational pull and is defined as u _k ^ = -Tg for all fc e (1, . . . , K}. The combined prior amounts to

Pk(ti _k , @k) = Pk(u _k ,i, 0k)S(u _k,2 + Tg) (69) for k 6 {1, . . . , K} where d(·) is the Dirac delta. The function (7) is given by with weights with k ∈ {60, 120, 180, 240} representing the time instances of the prescribed checkpoints. The goal is to determine a binary input signal (u ₁ , ₁ , . . . , u _K,1 ) for said model such that the moving point mass passes through the checkpoints as closely as possible.

The numerical results of Fig. 10 are obtained with 1, m = 1, g = 0.25 and K = 250, and with (41) and (42). The four checkpoints at time instances k ∈ {60, 120, 180, 240} are illustrated by black crosses in the first plot. The determined input signal (u _{1, 1} , . . , u _K,1 ) is indicated by stems in the second plot. The resulting output trajectory y is depicted as solid line in the first plot. As can be seen, the output trajectory y is an admissible trajectory and does not collide with any of the obstacles.

Notes While there are shown and described presently preferred embodiment of the invention, it is to be distinctly understood that the invention is not limited thereto but may be otherwise variously embodied and practiced within the scope of the following claims.

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