Method for controlling a pulp and paper process plant using data-driven tracking

Abstract

A method and a process plant directed to implementing a production process of a pulp and paper product.

Description

[0001] METHOD FOR CONTROLLING A PULP AND PAPER PROCESS PLANT USING DATA-DRIVEN TRACKING

[0002] FIELD

[0003] The present disclosure relates to a method for controlling a pulp and paper process plant. The control uses data-driven tracking. The present disclosure further relates to a pulp and paper process plant and to a computer program product to control a pulp and paper process plant.

[0004] BACKGROUND

[0005] A pulp and paper process plant requires a complex control, in particular a control of a wet end process carried out in the pulp and paper process plant. For example, an active retention control is required to minimize fluctuations in the retention rate of fibers in a wire section of the pulp and paper process plant, i.e. the ratio of fibers that stays on the wire vs. those that tall off the wire needs to be adequately controlled. It is furthermore necessary to ensure the stability of the process and an optimal use of chemicals, while reducing ash variability.

[0006] An exact model of the pulp and paper process plant is usually not available and traditionally system identification methods needs to be carried out to obtain a mathematical model of the pulp and paper process plant based on measured data of the pulp and paper process plant. The model is then used for the control.

[0007] In alternative or in addition to traditional system identification methods, also machine learning and artificial intelligence solutions may be used to obtain a model of the pulp and paper process plant for the control of the pulp and paper process plant.

[0008] All these known solutions require a model learning phase that requires effort and time and that are in particular not suitable for a pulp and paper process plant that often changes operating point, for example for producing papers of different paper grade.

[0009] In feet, each operating point requires a dedicated model, for example as a linear dynamical system based on a linearization at the operating point of the pulp and paper process plant. Therefore, a change of operating point would require a new model learning phase that is often unfeasible due to excessive time and effort.

[0010] Data-driven control is known that avoids the need of a system identification. Nevertheless, known data-driven solutions also require a system that remains time-invariant, i,e. are not suitable for a system that often changes operating point and / or where the operating point may vary and / or drift over time.

[0011] There is therefore a need to improve the control of a pulp and paper process plant based on a data-driven control adapted to operating conditions that often change.

[0012] SUMMARY

[0013] The invention is defined by the independent claims. The dependent claims define further embodiments of the invention.

[0014] According to an aspect, the present disclosure provides a method for controlling a pulp and paper process plant, the method comprising:

[0015] - Identifying a set of inputs of 'the pulp and paper process plant, the inputs in the set of inputs being physical quantities defining process variables of the pulp and paper process plant that are controlled;

[0016] Identifying a set of outputs of the pulp and paper process plant, the outputs in the set of outputs being physical quantities measured by sensors of the pulp and paper process plant;

[0017] Obtaining data of a first trajectory of the pulp and paper process plant related to a first operating condition of the pulp and paper process plant, the first trajectory recording a sequence of inputs in the set of inputs and a corresponding sequence of outputs in the set of outputs obtained when operating the pulp and paper process plant according to the first operating condition;

[0018] Obtaining data of a second trajectoty of the pulp and paper process plant related to a second operating condition of the pulp and paper process plant, the second trajectory recording a sequence of inputs in the set of inputs and a corresponding sequence of outputs in the set of outputs obtained when operating the pulp and paper process plant according to the second operating condition;

[0019] Controlling the pulp and paper process plant with a first control scheme related to the first operating condition, the first control scheme repeatedly comprising: o Finding a first optimal finite sequence of inputs in the set of inputs that minimizes a penalty of tracking a first reference trajectory related to the first operating condition using data-driven tracking based on data of the first trajectory, the first reference trajectory, a first penalty of a deviation of a trajectory of the pulp and paper process plant from the first reference trajectory related to the first operating condition and data of a resulting first present trajectory of the pulp and paper process plant related to an present controlling of the pulp and paper process plant according to the first operating condition; o Controlling the pulp and paper process plant seting the first optimal finite sequence of inputs as set points for the inputs of the pulp and paper process plant; o Updating the data of the first trajectory and the data of the first present trajectory based on a measured trajectory of the pulp and paper process plant during the controlling with the first control scheme and updating the first reference trajectory; o Repeating the first control scheme based on the updated data of the trajectories to complete a first operation phase related to the first operating condition;

[0020] - Controlling the pulp and paper process plant with a second control scheme related the second operating condition, the second control scheme repeatedly comprising: o Finding a second optimal finite sequence of inputs in the set of inputs that minimizes a penalty of tracking a second reference trajectory related to the second operating condition using data-driven tracking based on data of the second trajectory, the second reference trajectory, a second penalty of a deviation of a trajectory of the pulp and paper process plant from the second reference trajectory related to the second operating condition and data of a resulting second present trajectory of the pulp and paper process plant related to an present controlling of the pulp and paper process plant according to the second operating condition; o Controlling the pulp and paper process plant seting the second optimal finite sequence of inputs as set points for the inputs of the pulp and paper process plant; o Updating the data of the second trajectory and the data of the second present trajectory based on a measured trajectory of the pulp and paper process plant during the controlling with the second control scheme and updating the second reference trajectory; o Repeating the second control scheme based on the updated data of the trajectories to complete a second operation phase related to the second operating condition;

[0021] Alternatingly carrying out the first control scheme, based on the updated data of the first trajectory, the updated data of the first reference trajectory, the first penalty and the updated data of first present trajectory; and the second control scheme, based on the updated data of the second trajectory, the updated second reference trajectory, the second penalty and the updated data of the second present trajectory; wherein the second operating condition is different from the first operating condition; and wherein the data of first trajectory and of the first present trajectory includes measured inputs and outputs when controlling the pulp and paper process plant with the first control scheme; and wherein the data of second trajectory and of the second present trajectory includes measured inputs and outputs when controlling the pulp and paper process pant with the second control scheme.

[0022] The present disclosure further provides a pulp and paper process plant having inputs and outputs, comprising:

[0023] -a measurement and data system configured to measure inputs of the pulp and paper process plant;

[0024] -sensor to measure outputs of the process plant;

[0025] -a control system configured to control the pulp and paper process plant automatically executing methods according to the present disclosure. Aspects and advantages of the present disclosure will be described in detail in the following detailed description and claims and drawings that exemplarily illustrate embodiments of the present disclosure.

[0026] BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Embodiments of the present disclosure will be exemplarily illustrated with reference to the following figures:

[0028] Figure 1 shows a schematic example of a pulp and paper process plant according embodiments of the present disclosure that includes a paper machine tor wet end processing.

[0029] Figure 2 illustrates how a Hankel matrix is obtained from a trajectory of the pulp and paper process plant according to embodiments of the present disclosure.

[0030] Figure 3 illustrates a trajectcny tracking of a pulp and paper process plant according to embodiments of the present disclosure.

[0031] Figure 4 illustrates a method for controlling a pulp and paper process plant according to embodiments of the present disclosure.

[0032] DETAILED DESCRIPTION OF EMBODIMENTS

[0033] A detailed description of embodiments of the present disclosure is presented in the following.

[0034] Different embodiments can be combined with each other, unless otherwise stated or unless they are mutually exclusive.

[0035] The expression “in one embodiment”, “in some embodiments”, “according to an embodiment”, “according to embodiments” refer to general teachings of preferred features which are combinable (at least if not otherwise stated or mutually exclusive), Le. features of embodiments of the present disclosure are combinable with other features and embodiments described herein, and said expressions are not intended (if not otherwise explicitly stated) as explicitly listing a specific delimited combination of features. Common features of different embodiments may not be explicitly repeated in order to obtain a compact and readable description.

[0036] The expression “embodiment” refers to embodiments of the present disclosure. Features of methods of the present disclosure can be combined with features of corresponding systems of the present disclosure.

[0037] Features of systems of the present disclosure can be combined with features of corresponding methods of the present disclosure.

[0038] The detailed description firstly discussed deterministic dynamical system and the representation of a pulp and paper process plant as a dynamical system. The state space representation is discussed together with known problems, in particular in view of the complexity of a pulp and paper process plant . Data driven fbrnralation to overcome the limitations of the identification of a state space representation are then introduced with the discussion of why known data driven formulations cannot be applied to pulp and paper process plants that repeatedly or altcmatingly operate in at least a first and a second operating condition.

[0039] Then the solutions according to the present disclosure are introduced in detail, showing how the limitations of known data driven formulations are overcome by embodiments according to the present disclosure.

[0040] - DYNAMICAL SYSTEMS

[0041] A deterministic dynamical system, like for example a pulp and paper process plant, is a deterministic subsystem of the universe that can be described in terms of a set of states, identified by a vector x e Rnof n state variables. Analogously, a vector u e Rmof m inputs describes quantities of the surrounding universe that influence the dynamical system and that can typically be applied or imposed to the system and a vector y e Rpof p outputs describes quantities of the dynamical system that influence the surrounding universe and that are typical related to a desired function implemented by the dynamical system, like for example related to a production process of a pulp and paper process plant .

[0042] - STATE SPACE REPRESENTATION

[0043] Knowing the laws of physics and the structure of the dynamical system, it is in principle possible to obtain a state space representation that relates a change of the state variables to the input variables and the output variables and that relates the output variables to the state variables and the input variables, in the form of a set of differential equations:

[0044] (x = f(x, u) ly = h(x, u)W The symbol x denotes the derivative of the state variables with respect to time and f and h are functions that define a model of the system that completely describes the system, i.e. know ing an initial state of the system at an initial time instant and the inputs applied to the system ( for each time instant alter the initial time instant), the future states and outputs are determined.

[0045] For a convenient numerical handling or if the system is configured to operate in discrete steps, like tor example a digital system, it is typically convenient to consider discrete time instants instead of a time continuum.

[0046] A pulp and paper process plant is a complex deterministic dynamical system and usually a general description (1) is not possible or at least not feasible for a proper control of the pulp and paper process plant.

[0047] Figure 1 shows a schematic example of a pulp and paper process plant 100 according to embodiments of the present disclosure that includes a paper machine for wet end processing.

[0048] The pulp and paper process plant 100 includes at least one wet end paper processing machine configured to transform cleaned and bleached pulp into a wet paper sheet that is subsequently processed into e.g. paper of a desired paper grade by the paper process plant.

[0049] To cany out the wet end processing, the pulp and paper process plant 100 is configured for example to receive a mix 101 of pre-treated pulp or stock coming from a stock preparation area, chemicals and fillers used to control the retention rate.

[0050] I he retention rale is a ratio of fibers that forms the paper sheets over those that fell in a collection pit for recirculation.

[0051] The mix 101 may also be used to control the strength and quality of the final paper product, white w ater (the recycled mix of w ater and fiber that fell from the w ire) and broke ( i.e. semifinished paper product that is recycled, for example due to a sheet brake during production).

[0052] A headbox 102 is configured to disperse the mix 101 , in particular together w ith w ater, fibers and chemicals onto a rotating wire mesh 103, to be dried by a combination of gravity and suction.

[0053] Fibers that are retained on the rotating wire mesh 103 flow to the next machine section 104 of the pulp and paper process plant, becoming part of a continuous paper sheet, while the nonretained fibers and the water and chemicals mix (called w hite water) forms material 105 falling into a collection pit 106 to be recycled and sent back to the headbox 102 by a white water recycling system 107,

[0054] A control system manipulates for example the valves and pumps that bring water, pulp / stock, the different chemicals and fillers, white water and broke to the headbox to control the flowrate of each of these components, with the objective of for example stabilizing the process and reducing the variability in retention rate and product quality, while minimizing the use of chemicals to reduce costs. The retention rate may be calculated from three main measurements performed using sensors installed in the paper machine: headbox consistency (or ash content), white water consistency (or ash content) and paper ash content (for the retained fibers that form the paper sheet). Flowrate sensors may be used to measure the flow of pulp, chemicals, filler, broke and white water. The position of valves and the speed of motors dri ving pumps and the drums of the wire section may be measured. All these measurements may be accessible to the control system. The control targets are for example process stability, low product variability, low retention rate variability, low chemicals use, high machine efficiency and runnability, low amount of sheet breaks and product rejection during the later stages of the paper machine process.

[0055] For example, a pulp and paper processing plant 100 for wet end processing may be modeled as an input / output process where the inputs are the manipulated variables, Le. the variables whose values can directly be manipulated by the control systems, and the outputs are the target variables that the control system tries to stir to the desired beha\ ior by acting on the manipulated variables. Both inputs and outputs can be considered measurements, coming either from sensors and / or actuators equipped with measurement instruments installed in a pulp and paper process plant.

[0056] Inputs, e.g. manipulated variables, may be for example retention aid flow, flow of chemicals, filler flow, white water flow, pulp stock flow, broke flow, machine speed and / or any combination thereof Output variables, e.g. target variables may be white water consistency, headbox consistency, paper ash and / or any combination thereof Figure 1 fiirther illustrates typical input and outputs.

[0057] Due to the complexity of a pulp and paper process plant, e.g. the pulp and paper process plant 100, a general description of the pulp and paper process plant and / or of the wet end of the pulp and paper process plant in the form (1 ) is therefore generally not feasible. Often in practice, a deterministic dynamical system, like a pulp and paper process plant, operates locally near an operating point or equilibrium point, and it is possible to locally describe the dynamical system as a linear system that forms a good approximation if the operating conditions remain in a neighborhood of the equilibrium point. Therefore, instead of (1 ), it is often possible and convenient to use a state-space model with discrete time describing a linear system / linear approximation around the operating point:

[0058] J 6 RnKnis the system matrix, B the input matrix, C 6 RVxnt|ie 0U(pUtmatrix and D G Rpxmthe feedforward matrix. In (2) the state x G ffn, the input u G Rmand the output y G RP are typically defined as a relative displacement from an equilibrium point.

[0059] The state-space model (2) can be obtained from the model (1) with a linearization near an equilibrium point and is sometimes called small-signal model.

[0060] Before controlling a pulp and paper process plant that operates in an operating condition / operating point, known solutions demand a system identification to determine a linear state-space model (2) that holds at least locally near the operating point of the pulp and paper process plant.

[0061] The system identification according to known solutions require for example a model learning and a tuning phase, in particular reliably determine the system matrix A, the input matrix B, the output matrix C and the feedforward matrix D of the linear state space model (2).

[0062] When the operating point changes, it is typically necessary to redetermine a new linear approximation, identifying a new linear state-space model (2) for the new operating point.

[0063] In addition, if an operating point is variable over lime, for example drifting over time, then the state space model (2) does not adequately describe the system behavior.

[0064] In particular, if when alternating an operation between different operating conditions, for example when alternating between a first operating condition and a second operating condition, when resuming a first operating condition after having operated the pulp and paper process plant in a second operating condition following another previous or initial operation in the first operating condition, the resumed first operating condition may result in a state space model (2) that is slightly different than the state space model of the previous or initial operation in the first operating condition.

[0065] In other words, although a given operating condition is resumed after another operating condition carried out in the meanwhile, the state-space model, although related to the same operating condition, may still vary after the resumption due to some drift of the equilibrium point.

[0066] This implies that when operating a pulp and paper process plant in an initial first operating condition described by a first state space model, and then in a second operating condition described by a second state space model, and then resuming the first operating condition, the state space model related to the first operating condition after the resumption may be e.g. slightly different than the first state space model of the initial first operating condition due to some drift or variability of the equilibrium point such that a model learning phase would be indeed required after each alternation of an operating condition, even when a previous operating condition is resumed for which a state space model was previously already determined, i.e, a redetermination of the state space model is required to take into account a drift of the equilibrium point when resuming a previous operating condition.

[0067] Overall, the time and effort required to repeatedly obtain a linear state-space model (2) using known system identification methods is very high and often prohibitive in the case of a pulp and paper process plant that operates altematingly in a plurality of operating conditions that would require a repeated system identification to identify a state-space model (2) for each operating condition and after each alternation or switching to a different or subsequent operating condition, i.e. after each change of an operating point when transitioning to a different operating condition tor example related to a different paper grade that is produced by the pulp and paper process plant.

[0068] The system matrix A, the input matrix B, the output matrix C and the feedforward matrix D in the linear state-space model (2) are in general unknown due to the difficulties in determining the operating point and due to the fact that a general model (1) remains unknown or too complex.

[0069] - DATA DRIVEN FORMULATION

[0070] To avoid the use of system identification to identify a state-space model (2), it is alternatively possible to use a data-driven formulation, where the information is extracted from trajectories of the system without an identification of a state-space model (2) and / or of any equivalent model, like for example based on identification of transfer functions.

[0071] Data-driven formulations therefore avoid the need to obtain any model of the system, either as a state-space model or as any equivalent model like for example based on transfer functions.

[0072] We will now describe known data-driven solution, starting from the observation that the (unknown) system (2) is characterized by the behavior B of the system, defined as the set of all possible trajectories of the system

[0073] The state dimension n is called the order of representation. The smallest possible order of representation is called the order of the system and indicated with n(B).

[0074] The operator col returns a column vector and in the foltowing may be omitted if from the context it is clear that a column vector is demanded.

[0075] Alternatively, any convenient permutation or representation may be used.

[0076] An analogous notation will be used with respect to vectors and matrices in the following.

[0077] Analogously, we define the set of possible truncated trajectories of length T as

[0078] Alternatively, a convenient permutation my be considered, for example Brma\ equivalently be defined as:

[0079] We observe that the model (2) remains valid near an equilibrium point as long as the state and the output remain near the equilibrium point / operating point and as tong as the system remains time invariant, i.e. as long as a more adequate or complex description is not required to capture the fill! system behavior, for example in the presence of varying parameters.

[0080] A trajectory wTof length T of a system (2) is a sequence of length T of pairs of inputs and corresponding outputs in the form: wT= coi(w(l), w(2), .... w(T)) = col ([“®] , [“Q) ' . ' [y(T)]) ®

[0081] Analogously we write for the sequences of length T of inputs and corresponding outputs respectively for the system u(i)

[0082] (2) and we define for convenience w(i) = for i = 1,2, y(0

[0083] We will use analogous notations also for other trajectories and lengths in the present description.

[0084] The rows or wTmay alternat ively be permuted, for example to obtain a representation of the form wT= coi(u(l), ... ,u(T),y(D, ... ,y(T)) (3bis)

[0085] We will consider the representations (3) and (3bis) equivalent, using the same symbol. The trajectory wTis divided into L overlapping batches of T-L+l elements, to obtain a matrix of the form:

[0086] The matrix is known as Hankel matrix {wT') : Figure 2 illustrates how a Hankel matrix is obtained from a trajectory of the pulp and paper process plant according to embodiments of the present disclosure.

[0087] The Hankel matrix ?ft(ivT) is shown as matrix 202 with rows 202-1. 202-2, 202-3, 202-4 that are formed by slices of the trajectorywr shown as 210. For example, the first row 202-1 , formed by the first slice of the trajectory ivrmay include the values

[0088] For example, the second row 202-2, formed by the second slice of the trajectory wTmay include the values (w(2), ... , w(T — L + 2)).

[0089] For example, the third row 202-3, formed by the third slice of the trajectory wTmay include the values (w(3), ... , w(T - L + 3)).

[0090] For example, the last row 202-4, formed by t he l ast slice of the trajectory wrmay include the values (w(l), ... , w(T — L + Lj) — (w(l), ... , w(T)) .

[0091] Under assumptions discussed in the follow ing. the columns of the Hankel matrix contain enough information to describe any trajectory of the system of length L. i.e. the single trajectory w<T) captures all information about the possible behavior of the system and can be used to describe all possible other trajectories of length L.

[0092] With analogous notation we will consider also for input and output sequences forming the trajectory the corresponding Hankel matrices, defined as:

[0093] Therefore, the Hankel matrix may be expanded to

[0094] .Mternatixely. any permutation may be considered instead, for example

[0095] We will consider (4). (4bis), ( 4ter) as equivalent. If the Hankel matrix JfL(uT) is of foil row rank, then the input sequence u(l),u(2), ... , u(T) is called persistently exciting of order L. We will also say that the trajectory wTis persistently exciting of order L in this case.

[0096] Under the assumptions that U(1),M(2), ... ,u(T) is persistently exciting of order t + n(B), i,e. that for L = t + n(B), the Hankel matrixJftCur) if of full row rank, with n(fi) being the order of the system (2), and that the system (2) is controllable, then cotepan(Wt(wT)) = Bt(5) with Btbeing the set of all possible trajectories of length t of the system (2), considered as column vectors (for a given permutation of inputs and outputs in the trajectory).

[0097] In other words, under the described assumptions, any trajectory of length t can be writen as a linear combination of the columns of the Hankel matrix Mt(wT).

[0098] This result is known as fundamental lemma of Willems et al, as described in fill! detail in Markovsky and Paolo Rapisarda (2008), “Data-driven simulation and control”, International Journal of Control, 81(12), 1946-1959, DOI; 10, 1080 / 00207170801942170,

[0099] This means that for each possible trajectory wtof length t, there is a vector g E RT-t+1such that

[0100] Intuitively, if the assumptions of the fimdamental lemma of Willems are satisfied, then the Hankel matrix Xt(ivT) captures enough information about the system, i,e. contains enough representative slices of a trajectory, such that any other trajectory of length t can be described as a linear combination of the columns o f said Hankel matrix.

[0101] According to the present disclosure, other alternatives to the use of a Hankel matrix are possible.

[0102] For example, the trajectories may be alternatively spanned by trajectories / trajectory slices organized in a Page matrix instead.

[0103] I.e., alternatively, trajectoiy slices organized in a Page matrix may provide a basis for the trajectories of the system (2). The interested reader finds details on how a Page matrix is defined and used in J. Coulson, J. Lygeros and F. Dottier, "Distributionally Robust Chance Constrained Data-Enahled Predictive Control," in IEEE Transactions on Automatic Control, vol. 67, no. 7, pp. 3289-3304, July 2022, DOI: 10.1 109 / TAC.2021.3097706.

[0104] - DATA DRIVEN SIMULATION

[0105] The fundamental lemma of Willems, i.e. (6) allows to simulate the system (2) without explicitly knowing the state-space model, i.e. without knowing the system matrix A, the input matrix B, the output matrix C and the feedforward matrix D. A system identification is therefore not required.

[0106] The data-driven simulation problem is defined as follows: given a known trajectory wT= col (w(l), w(2), w(T)) of the system (2), a known trajectory tVp = coi(«r(l), «r(2), ®r'(P)) up to a present time instant P and a sequence of future of input signals — colftc(P + 1),«.(P + 2), ... , u(t)), then determine the future response of the system (2) up to time t when the fiiture sequence of input signals is applied, i.e. determine a trajectory such that:

[0107] I.e. the future output sequence (t))of outputs of the system (2) has to be determined that results when applying the sequence of fiiture inputs col(ti(P + 1), ti(P + 2), ... ,-«•({)) to the system (2) assuming the presence of a present trajectory = coi(-tr(l), w(2), «r(P)) and knowing a known trajectory wT= coi(w(l), w(2), ... , w(T)) such that the hypothesis of the fundamental lemma of Willems hold (note the differences oft and T with always t < T).

[0108] The trajectory up to the present time instant P is assured known, i.e. w(i) = is already known for i = 1,2, ... , P, with P denoting the present time instant, with P < t. In particular, the past inputs -u(l), ,«(P) and the past outputs are known >(!), ••• , >(P) and therefore «r(l), 'Ur(2), <£r(P) is known.

[0109] From the fundamental lemma of Willems, considering (6) it follows that it exists a vector g e

[0110] IRT~t+1such that

[0111] The vector g and the future output sequence coi(y(P + are unkno >wn in (7).

[0112] Alternatively. the row s of ( 7 ) may be permuted w ith any permutation, for example considering (4terl the follow ing equivalent form may be obtained (then w ith a corresponding permutation of^ e RT-t+1): coi

[0113] We w ill consider (7) and ( 7bis) to be equivalent, based on the equivalence of (4), (4bis) and (4ter).

[0114] (7) or equivalently ( 7bis) can be solved to obtain both g and 1), y.(P + 2), y(t)). i.e. to determine the future outputs of the system (2 ) from P+1 to t. without the need to learn or identify a model of the system (2 ). i.e, without the need to explicitly determine the matrices A,B,C,D in (2).

[0115] From ( 7bis) or equivalently from ( 7), we observe the presence of t-P equations ( the last t-P equations in ( 7bis )) that allow to determine y-(P + 1 ), y,(P + 2), ... , in function of g. The remaining equations ( the first lAP equations in ( 7bist) put restrictions on g and may or may not allow to determine g uniquely.

[0116] If g is uniquely determined, then the t-P equations related to >(P in (7) or (7bis) (see the last t-P equations in (7bisJ) allow a unique determination of

[0117] Thereby solving ( 7) or ( 7bis) allow s to determine the output sequence up to t. i.e. to determine

[0118] It can be shown that g is uniquely determined (i.e. results uniquely from the first t+P equations of (7bis)) under the assumptions that the system (2) is controllable, the input component uT= col(u(l), u(2), u(T)) of wris persistently exciting of order t + n(B), and P > 1(B), with 1(B) indicating the lag of the system (2), then (7) or equivalently (7bis) have a unique solution that satisfies the data-driven simulation problem, i.e. such that holds. Under the discussed assumptions, it is therefore possible to uniquely determine the full output response -pf= co / (-y.(l),>(2), ... also for the future tune instants P + 1, P + 2, ... , t that follow the present time instant P.

[0119] It is therefore possible under the given assumptions, knowing a trajectory wT, to determine the unique future sequence of outputs col(^y,(P + 1),>(P + 2), ... ,>(t)) of the system (2), taiowing the applied future sequence of inputs «p+lt= the present trajectory up to the present time instant P. col( ic (l), to (2). ... ,«r(P)), that we indicate with the symbol , and without explicitly knowing the system matrix A, the input matrix B, the output matrix C and the feedforward matrix D of the linear state space model (2).

[0120] Solving (7) or (7bis) it is therefore possible to know the full trajectory wt, i.e. to know ®r(i) = r«(01 for all instants, i — 1,2, P, P + 1, P + 2, ... t and in particular for the future time instants from P + l to t.

[0121] For compactness, we introduce the foltowing notation: to indicate that is obtained from wTand with the consideration of P as present time instant by solving (7) or (7bis) as discussed with P < t < T, i.e. using data-driven simulation DDSIU.

[0122] Under the discussed assumptions, data-driven simulation DDSIMreturns a unique sequence of outputs, i.e. uniquely returns in particular >(P + 1),>(P + 2), future inputs t

[0123] We write with a notation analogous to (9) that

[0124] In particular, (9) and (10) may be applied repeatedly / iteratively, updating the present time instant P and the length of the trajectories t to cover a future time window of interest, and, if necessary to satisfy the requirements, also updating / redetermining the overall length T of the known trajectory and the known trajectory vvr, with P indicating the present time instant with P < t < T and such that the discussed assumptions hold.

[0125] We recall that the past inputs <(1), , u(P') and the past outputs y ( l ), •■• , y>(P) are assumed biown and therefore «r(l),«r(2), .. . , iv (P) is assumed known, also at each step of an iterative / repeated application of data drh en simulation.

[0126] With analogous notations as defined for the inputs, we write also y-P+-l t= col( y.[P + the future outputs and future trajectory slice, that are therefore know n by using data-driven simulation

[0127] - D.\ l \-DRI\ I \ Rl -.1- i -.Rl XCI- 1 R \.H ( I ORY I R ACK ING

[0128] Figure 3 shows a trajectory tracking of a pulp and papa process plant.

[0129] A present time instant P is show n and future time instants P^l, P+2, ..., t over a prediction horizon are illustrated.

[0130] Up to and including the present time instant P. a past trajectory formed by past inputs 310 and past measured outputs 306, i.e. w ( 1 ), iu(2), . . . , to f P) I formed by the combination of 310 and 3061 is known. Over the prediction horizon a reference trajectory 302 is gie en up to time instant t that we will indicate with rt= (r(l)fr(2), ... , r(t)).

[0131] The trajectory tracking problem is then to find future inputs 308 such that the resulting future outputs 304 (as predicted using data-driven methods) track as close as possible the reference trajectory 302 up to the future time instant t over the prediction horizon. A distance between the reference trajectory 302 and the future output 304 (that form a continuation of the past measured outputs 306 that is thereby included for determining said distance) is determined based on a cost matrix that quantifies a cost / penalty of a deviation of the future output 304 (together with the past output 306) from the reference trajectory 302.

[0132] While the past inputs and outputs are certain and can be measured, the future output can only be predicted using data-driven methods.

[0133] To control a pulp and paper process plant, it is often necessary or advantageous to implement a tracking of a reference trajectory 302, for example a liner quadratic tracking. Given a reference trajectory 302 rt= (r(l), r(2), ... , r(t)), the goal is to minimize a cost / penalty of tracking the reference trajectory, i.e. to find sequence of future inputs 308 -u.P+1>t= col(u(P + 1), 'tt(P + 2), , «-(()) to the system (2) that minimizes: life “ «fe Ik (11) with fo denoting a positiv e definite weight matrix that defines a cost / penalty of outputs (predicted future outputs 304 together 'with measured past outputs 306) deviating from the reference trajectory 302 fe.

[0134] In (11), the trajectory (i.e. predicted fiiture outputs 304 together with measured past outputs 306 together with future inputs 308 and past inputs 310) can be determined for given fiiture inputs 308 using data driven simulation (10) for the present time instant P.

[0135] Using data-driven simulation (10) it is therefore possible to find an optimal sequence of fiiture input values, optimum 308 that minimizes the cost / penalty (1 1 ) by using any known numeric optimization method, for example by using a downhill-simplex method or any suitable numerical optimization method to find a minimum of ( 1 1 ) based on a numerical evaluation of the cost / penalty (1 1) using repeatedly data-driven simulation (10).

[0136] The trajectory that results for the application of the optimal sequence of fiiture input values «*F+litwill be denoted as -w *tand minimizes the cost / penalty ( 1 1) thereby forming an optimal trajectory w*e.

[0137] The optimization method may take further into account constraints, lor example to restrict the inputs and / or the outputs to a feasible region and / or to handle equality and / or inequality constraints, for example modifying a downhill-simplex method to implement a proper constrained optimization.

[0138] For example, the cost / penalty function (11) may be modified adding additional barrier functions to constrain «P+Uand / or = col(>(P + 1),>(P + 2), to feasible regions.

[0139] Also, data-driven reference trajectory tracking can be applied iteratively / repeatedly, by updating P to the present time instant and adjusting t to cover a future time window and updating accordingly the reference trajectory rtand, if necessary, updating also the known trajectory wT. We recall that the past inputs «•(!), • • ■ , u(P) and the past outputs y ( 1), , >(P) are assumed known and therefore «r> = col(w(l), -w(2), ... , -MF (P)) is assumed known, also at each step of an iterative / repeated application of data drhen reference trajectory tracking.

[0140] As an alternative to a numerical minimization of (11), it is also possible to compute the optimal trajectory w*tusing data-driven linear quadratic tracking based on the computation of a basis for a zero initial condition sub-behavior of the system (2), then computing a free response of the system based on the fundamental lemma of Willems et al and finally determining the optimal trajectory based on the basis, the free response and tire positive definite weight matrix 0.

[0141] The interested reader is referred to Algorithm 8 of Ivan Markovsky and Paolo Rapisarda (2008), “Data-driven simulation and control”, International Journal of Control, 81(12), 1946-1959, DOI: 10. 1080 / 00207170801942170.

[0142] Further explanations about data-driven tracking are found in J. Coulson, J. Lygeros and F. Dorfter, "Data-Enabled Predictive Control: In the Shallows of the DeePC", 2019 18th European Control Conference (ECC), Naples, Italy, 2019, pp. 307-312, DOI: 10.23919 / ECC.2019.8795639.

[0143] To indicate that the optimal sequence of future inputs u *P+1 tis obtained by the described data driven tracking methods we will use the notation

[0144] Alternatively, or in addition, Page matrices instead of Hankel matrices can be used for data driven tracking, as described for example in L. Huang, J. Coulson, J. Lygeros and F, Dorfler, "Decentralized Data-Enabled Predictive Control for Power System Oscillation Damping," in IEEE Transactions on Control Systems Technology, vol. 30, no. 3, pp. 1065-1077, May 2022, doi: 10.1109 / TCST.2021.3088638. A Page matrix forms an alternative basis for describing the trajectories.

[0145] - DA I A l)Rl\ T\ IORMI LA I ION FOR A Pl l.P AM) I’APl R PROCESS Pl. \\ 1 ACCORDING TO EMBODIMENTS

[0146] The discussed known data driven fonnulation, i.e. the known data-driven simulation and / or known tracking is not suitable for a pulp and paper process plant that is switched between operating conditions, in particular that altematingly operates in at least a first operating condition and a second operating condition.

[0147] In fact, when using for example (7) and / or (7bis) during known data-driven simulation andfor known data-driven tracking, a trajectory ivrfrom time instant I up to time instant T, according to (3) and / or (3bis) is required, and similarly also the present trajectory w(l), w(2), . . . , w (P) up to the present time instant P has te) be known without interruptions.

[0148] When therefore the operating condition switch, i.e. when e,g, a new operating / equilibriuna point is reached, a new linear approximation (2) holds and the described known data-driven simulation and / or tracking methods ha\e to be reapplied anew and it is in particular necessary to obtain a new trajectory wrrelated to the new operating conditions, i.e. for the new linear approximation (2) describing the new operating point, in order to again apply data-driven simulation and / or tracking, and in particular to again make use of (7) andfor (7bis) when carrying out the data-drh cn simulation and / or tracking for the new operating condition.

[0149] But obtainin a new trajectory wTfor the new operating conditions and in particular a trajectory with input component uT— co / (u(l), u(2), . . . , u(T)) that is persistently exciting of order as previously discussed is a time and resource consuming task that is not practicable if the system needs to be controlled at any time and in particularly also shortly after a switching to new operating conditions. In fact, during the time in which the trajectory wris determined for the new linear system ( 2 ) describing the new operating conditions, the system cannot be controlled based on a data-driven formulation and in particular data-driven simulation and data-driven tracking are not possible until a new trajectory wTis obtained / observed that satisfies the discussed requirements for the new system and in particular the requirement of having input sequence that is persistently exciting of order t + n(®).

[0150] For these reasons, the previously discussed known data-driven formulations, i.e. known data- driven simulation and knowm data-driven trajectory tracking are not applicable to a system, like a pulp and paper process plant that often alternates between operating conditions, e.g. between a first operating condition related te» the production of a first paper grade and a second operating condition related to the production of a second paper grade.

[0151] The present disclosure solves the problem of providing an improved control of a pulp and paper process plant based on data-driven tracking in the presence of an operation of the pulp and paper process plant that alternated betw eon different operating conditions, in particular between a first operating condition and a second operating condition.

[0152] The first operating condition may be an operating condition for which a first linear approximation (2) holds, i.e. for which the pulp and paper process plant may be described by a first (unknown) state-space model valid for time instants interval 21 :

[0153] The second operating condition may be an operating condition for which a second linear approximation (2) holds, i.e. for which the pulp and paper process ptant may be described by a second (unknown) state-space model valid for time instants t1-2’, tu)+ 1 e B in a second time interval Bt

[0154] Alternating between the first and second operating condition has to be intended in the sense that during a first time interval, the (unknow) state-space model (12) holds and subsequently during another second time interval the (also unknown) state-space model (13) hold as an approximation for the more general (unknown) description (1) and / or vice versa.

[0155] The first time interval A and the second time interval B do not intersect, and may be also separated by even other time intervals in v liich c\ en other (unknown) state-space models hold as an approximation for the system (1).

[0156] To avoid an excessively cumbersome notation, we define that the time t(1Jis measured by a clock that only advances when the first ( unknown) state-space model ( 12) holds and or when the pulp and paper process plant is in the first operating condition, while the time is measured by a clock that only advances w hen the second (unknown) state-space model (13) holds and 'or when the pulp and paper process plant is in the second operating condition, while being stopped when the respective models and / or operating conditions do not hold.

[0157] So for example, if the second time interval B follows the first time interval A, then for example the foltowing overall sequence may be observed overall: x(«(l),x<«(2), ... ,x®(l), x® (2), ...

[0158] When the first and second operating condition are repeatedly alternated, the (unknown) statespace models (12) and (13) may hold alternatingly, and for example a sequence of states in the from may be observed overall: xWClJ. xffifZ), . , xW(p), x®(l),xC2)(2), ... , x(^(c[)tx^(p + 1), ...

[0159] In the following, accordingly, quantities and trajectories with the superscript (1) are intended as related to the unknown state-space model ( 12) related to the first operating condition and with respect to a time as measured by a clock that ticks only when ( 12 ) holds, i.e. only when the first operating condition is present, while the clock does not advance when the pulp and paper process plant is in other operating conditions.

[0160] In particular, trajectories with the superscript (1) do not include any input or output or other quantities of the pulp and paper process plant operated in the second operating condition or in operating conditions different from the first operating condition.

[0161] In particular, trajectories with the superscript (2) do not include any input or output or other quantities of the pulp and paper process plant operated in the first operating condition or in operating conditions different from the second operating condition.

[0162] In the following, accordingly, quantities with the superscript (2) are intended as related to the unknown state-space model (13) related to the second operating condition and with respect te» a time as measured by a clock that ticks only when (13) holds, i.e. only when the second operating condition is present, while the clock does not advance when the pulp and paper process plant is in other operating conditions.

[0163] Figure 4 illustrates a method 400 for controlling a pulp and paper process plant according to embodiments of the present disclosure.

[0164] The method 400 for controlling a pulp and paper process plant comprises:

[0165] Identifying 402 a set of inputs w® of the pulp and paper process plant, the inputs in the set of inputs being physical quantities defining process variables of the pulp and paper process plant that are controlled; Identifying 404 a set ofoutputsy{1),y(2jofthe pulp and paper process plant, the outputs in the set of outputs being physical quantities measured by sensors of the pulp and paper process plant;

[0166] - Obtaining 406 data of a first trajectory = col of the pulp and paper process plant related to a first operating condition (12) of the pulp and paper process plant, the first trajectory recording a sequence of inputs in the set of inputs and a corresponding sequence of outputs in the set of outputs obtained when operating the pulp and paper process plant according to the first operating condition, w® (i)

[0167] Obtaining 408 data of a second trajectory = col (w®(l), w®(2), of the pulp and paper process plant related t) a second operating condition (13) of the pulp and paper process plant, the second trajectory recording a sequence of inputs in the set of inputs and a corresponding sequence of outputs in the set of outputs obtained when operating the pulp and paper

[0168] W(i)l process plant according to the second operating condition, w® (i) = for i =

[0169] (2>(0.

[0170] Controlling 410 the pulp and paper process plant with a first control scheme related to the first operating condition, the first control scheme repeatedly comprising: o Operating the pulp and process plant in the first operating condition, o Finding 412 a first optimal finite sequence of inputs in the set of inputs that minimizes a penalty of tracking a first reference trajectory related to the first operating condition using data-driven tracking based on data of the first trajectory tv , the first reference trajectory r1 11fi>s, a first penalty a deviation of a trajectory of the pulp and paper process plant from the first reference trajectory related to the first operating condition and data of a resulting first present trajectory P(D of the pulp and paper process plant related to a present P®controlling of the pulp and paper process plant according to the first operating condition; Controlling 414 the pulp and paper process plant setting the first optimal finite sequence of inputs as set [mints for the inputs of the pulp and paper process plant; Updating 416 the data of the first trajectory and the data of the first present trajectory wp<i) based on a measured trajectory of the pulp and paper process plant during the controlling with the first control scheme and updating the first reference trajectory Repeating 418 the first control scheme based on the updated data of the trajectories to complete a first operation phase related to the first operating condition; ling 420 the pulp and paper process plant with a second control scheme related cond operating condition, the second control scheme repeatedly comprising: Operating the pulp and paper process plant in the second operating condition; Finding 422 a second optimal finite sequence of inputs 'U^*pGa+tin the set of inputs that niinirnizes a penalty of tracking a second reference trajectory related to the second operating condition using data-driven tracking based on data of the second trajectory the second reference trajectory ll^^of a deviation of a trajectory of the pulp and paper process plant from the second reference trajectory related to the second operating condition and data of a resulting second present trajectory «rU ipi 2iof the pulp and paper process plant related to a present controlling of the pulp and paper process plant according to the second operating condition; o Controlling 424 the pulp and paps process plant setting the second optimal finite sequence of inputs i as set points for the inputs ofthe pulp and paper process plant; o Updating 426 the data of the second trajectory the data of the second present trajectory w^p(2) based on a measured trajectory of the pulp and paper process plant during the controlling with the second control scheme and updating the second reference trajectory r('2^ ; o Repeating 428 the second control scheme based on the updated data of the trajectories to complete a second operation phase related to the second operating condition;

[0171] - Alternatingly 430 carrying out the first control scheme, based on the updated data ofthe first trajectory, the updated data ofthe first reference trajectory, the first penalty and the updated data of first present trajectory; and the second control scheme, based on the updated data of the second trajectory, the updated second reference trajectory, the second penalty and the updated data ofthe second present trajectory; wherein the second operating condition is different from the first operating condition; and wherein the data of first trajectory and of the first present trajectory includes measured inputs ami outputs when controlling the pulp and paper process pant with the first control scheme; and wherein the data of second trajectory and ofthe second present trajectory includes measured inputs and outputs when controlling the pulp and paper process pant with the second control scheme.

[0172] In particular, the updating the data of the first trajectory and of the first present trajectory, includes updating a present time instant as measured by a first clock ticking only during the controlling ofthe pulp and paper process plant with the first control scheme.

[0173] In particular, the updating the data ofthe second trajectory and ofthe second present trajectory, includes updating a present time instant P^ as measured by a second clock ticking only during the controlling ofthe pulp and paper process plant with the second control scheme. The present disclosure therefore altcrnatingly carries out data driven tracking DDTRACKbased on first data that is updated only when carrying out the first control scheme related to the first operating condition, and carries out data driven tracking DDTRACKbased on second data that is updated only when carrying out the second control scheme related to the second operating condition.

[0174] When altematingly carrying out the first control scheme and the second control scheme, still other further control schemes may be carried out between the first control scheme and the second control scheme and / or vice versa and or still further operating conditions may be present between the presence of the first operating condition and the second operating condition and / or vice versa.

[0175] Thereby the present disclosure avoids the need to redetermine a new trajectory wTfor carrying out data driven tracking DDTRACKeach time the operating conditions arc changed and / or the second operating condition and the first operating condition alternate.

[0176] In fact, keeping, updating and reusing the first trajectory the second trajectory w®r(2) allows a fest switching between data driven tracking DDTRACKcarried out for the first control scheme and the second control scheme respectively, without the need to a lengthy redetermination of full trajectories anew after each alternating of the control schemes and thereby avoiding lengthy time intervals in which a control of the pulp and paper process plant with data driven tracking DDTMCKwould otherwise not be possible due to the need of a complete redetermination of the trajectory wTfor each operating condition, Le. of the foil first and second w^T(2) trajectory respectively, which would require a high amount of time in which the data driven tracking cannot be carried out.

[0177] Given that the time reference / clock for the first data indicated with superscript (1) only ticks during the first control scheme related to the first operating condition and that time reference / clock for the second data indicated with superscript (2) only ticks during the second control scheme related to the second operating condition, it is possible to alternate from controlling the pulp and paper process plant based on the first control scheme to the second control scheme, the alternating including: storing the data of the first trajectory w®T(i) and / or of the first present trajectory te> a storage and retrieving the data of the second trajectory of the second present trajectory uf from the storage, so that the data of the second present trajectory to and / or the second trajectory updated, and the data of the first present trajectory and / or of the first trajectory is not updated tat remains stored in an unaltered manner for a future controlling based on the first control scheme; and storing the data of the second trajectory of the second present trajectory w®pO) to a storage and retrieving the data of the first trajectory and / or of the first present trajectory from the storage, so that the data of the first present trajectory and / or the first trajectory jsupdated, and the data of the second present trajectory and / or of the second trajectory w®T(2) is not updated but remains stored in an unaltered manner for a future controlling based on the first control scheme.

[0178] The storing allows to reuse the stored trajectories without the need of redetermining said trajectories anew with an unacceptable time length in which the pulp and paper process plant would not be controllable due to the need of said redetermination.

[0179] In particular, if when switching between operating conditions the pulp and paper plant reaches and starts from an equilibrium state and zero inputs are applied (i.e. inputs of value zero) during the transition between operating conditions, for example reaching an idle state at the end and at the beginning of each operation in different operating condition, then when altemating / switching between different operating conditions the piecewise obtained trajectories and w®p(2) would be indistinguishable / identical to respective trajectories obtained in the absence of any switching of operating conditions.

[0180] It follows e.g. that the optimal finite sequence of inputs obtained according to embodiments of the present disclosure remain indeed optimal also in the presence of the alternating between operating conditions, i.e. the first optimal finite sequence of inputs remains indeed optimal when the first control scheme is altematingly resumed after the second control scheme (that is then not executed while the first control scheme is carried out).

[0181] Analogously, e.g. the second optimal finite sequence of inputs remains indeed optimal when the second control scheme is altematingly resumed after the first control scheme (that is then not executed while the second control scheme is carried out). Moreover, it does not matter if when alternatingly carrying out the list control scheme and the second control scheme, and then again the first control scheme, a plurality of other control schemes is carried out in-between, given that for example a switching of operating conditions in an idle state (in equilibrium state and with no applied inputs) always produces an ordered transitions terminating and resuming the first control scheme such that any other controls scheme has no effect on the trajectories related to the first control scheme that are therefore indistinguishable from trajectories for which only the first controls scheme is executed with the effect that the first optimal finite sequence of inputs remains indeed optimal also when alternating and resuming the control schemes.

[0182] Analogous result holds also for the second optimal finite sequence of inputs.

[0183] Methods and systems according to the present disclosure have also the advantage, that they can handle a drill of an equilibrium point related to an operating condition of the pulp and paper process plant.

[0184] In fact, when the data of the trajectories is updated, the data driven tracking will be based on the updated trajectories that therefore take implicitly into account a slight shift of an operating point (at least as long the Hankel matrix remains of full row rank and / or the input component of the trajectory ivrremains persistently exciting as discussed), whereas known model identification methods would need to be reapplied anew to learn a new model (2) of the system related to the shifted operating point, thereby being not applicable to slight ly shifting operating point, or would result in a systematic error and therefore provide a had tracking of the reference trajectory.

[0185] In some embodiments, when the pulp and process pant is controlled with the first control scheme, after being controlled with the second control scheme, the stored data of the first trajectory and the stored data of the first present trajectory is retrieved from memory and updated and used for the controlling, while the data of the second trajectory and the data of the second present trajectory not modified and remains stored and is not used for controlling; and, when the pulp and process pant is controlled w ith the second control scheme, after being controlled with the first controls scheme, the stored data of the second trajectory w and the stored data of the second present trajectory -urWp(2) is retrieved from memory and is updated and used for the controlling, while the data of the first trajectory ® "rni and the data of the first present trajectory c®1®!) is not modified and remains stored and is not used for controlling.

[0186] According to the present disclosure, updating the first present trajectory wp(i) means adding a inissing sequence of values w® (i) = up to the present time instant P*-15when operating the pulp and paper process plant in the first operating condition, that include only values related to the operation in the first operating condition, while values related to input and output values obtained during the operation in the second operating condition are discarded.

[0187] According to the present disclosure, updating the second present trajectory -ur^p(2) means adding a missing sequence of values up to the present time instant when operating the pulp and paper process plant in the second operating condition, that include only values related to the operation in the second operating condition, while values related to input and output values obtained during the operation in the first operating condition are discarded.

[0188] According to the present disclosure, updating the first trajectory' t > may mean obtaining a trajectory of the pulp and paper process plant when operated in the first operating condition that fulfills the requirements of the fundamental lemma of Willems et al. Said updated trajectory may be obtained updating the values of the previously stored first trajectory based on the newly observed sequence of inputs and outputs or in any other suitable way.

[0189] Alternatively, the updated trajectory may be identical to the previous first trjyectory if the requirements of the fundamental lemma of Willems et al. arc still satisfied.

[0190] Alternatively, or in addition, the an initial prefix of the first trajectory may be removed when updating the first trajectory.

[0191] According to the present disclosure, updating the second trqectory w®T(a) may mean obtaining a trajectory of the pulp and paper process plant when operated in the second operating condition that fulfills the requirements of the fundamental lemma of Willems et al. Said updated trajectory may be obtained updating the values of the previously stored second trajectory based on the newly observed sequence of inputs and outputs or in any other suitable way.

[0192] Alternatively, the updated trajectory may be identical to the previous second trajectory if the requirements of the fundamental lemma of Willems et al. are still satisfied.

[0193] Alternatively, or in addition, an initial prefix of the second trajectory may be removed when updating the second tnyectory.

[0194] The update ensures that data-driven tracking remains valid also in the presence of a drifting operating point, e.g. after a resumption of an operation according to an operating condition that may be in an operating point slightly different than during a previous operation before the resumption in the same operating condition of the pulp and paper process plant.

[0195] In some embodiments, the data-driven tracking comprises a data-driven linear quadratic tracking; and the first penalty is a first cost function based on a first positive definite weight matrix, the first cost being a scalar function related to consumption of physical resources during the operation of the pulp and paper process plant; and the second penalty is a second cost function based on a second positive definite weight matrix, the second cost being a scalar function related to consumption of physical resources during the operation of the pulp and paper process plant.

[0196] For example, the first cost / penalty function may be defined as: i,e. the penalty function is defined in terms of a quadratic form defined by cpW with respect to a future deviation from the reference trajectory, i.e. with respect to future time instants +

[0197] For example, the second cost / penalty function may be defined as: i.e. the penalty function is defined in terms of a quadratic form defined by with respect to a future deviation from the reference trajectory, i.e. with respect to future time instants + In some embodiments, in the first control scheme the sequence of inputs of the first trajectory is persistently exciting of order at least given by the sum of the length of the first present trajectory and of the length of the first optimal finite sequence of inputs and of a system order of the pulp and paper process plant; and in the second control scheme the sequence of inputs of the second trajectory is persistently exciting of order at least given by the sum of the length ofthe second present trajectory and of the length ofthe second optimal finite sequence of inputs and of a system order ofthe pulp and paper process plant.

[0198] This ensures that the fundamental Iemma of Willems et al. holds when carrying out methods and systems according to the present disclosure. For example, when alternating between the first and second control scheme, it may be necessary or advantageous to update the first and / or the second trajectory to obtain most recent trajectories or most recent trajectory suffixes such that the persistency of excitation is still present, in particular also with respect to a slightly different operating point, i.e. in the presence of a slightly time variability ofthe system

[0199] In some embodiments, the first optimal finite sequence of inputs is obtained by computing a first basis for a zero initial condition sub-behavior related to the first operating condition, and computing a first free response related to the first operating condition and the first optimal finite sequence of inputs is based on the first basis and the first free response and the first positive definite weight matrix; and wherein the second optimal finite sequence of inputs is obtained by computing a second basis for a zero initial condition sub-behavior related to the second operating condition, and computing a second free response related to the second operating condition and the second optimal finite sequence of inputs is based on the second basis and the second free response and the second positive definite weight matrix. For example, the first and / or second optimal finite sequence of inputs may be obtained in accordance with Algorithm 8 of Ivan Markovsky and Paolo Rapisarda, Data-driven simulation and control, DOI: 10.1080 / 00207170801942170.

[0200] In some embodiments, altcmat h cly or in addition, the first optimal finite sequence of inputs is determined numerically by solving a first minimization problem that minimizes the penalty of tracking the first reference trajectory by data-driven simulation; and the second optimal finite sequence of inputs is determined numerically by solving a second minimization problem that minimizes the penalty of tracking the second reference trajectory by data-driven simulation.

[0201] In some: embodiments, altematingly carrying out the first controls scheme and the second control scheme, repeatedly comprises:

[0202] - carrying out the first control scheme; saving data of the first trajectory and data of the first present trajectory; carrying out the second control scheme;

[0203] - saving data of the second trajectory and data of the second present trajectory; retrieving the saved data of the first trajectory and the saved data of the first present trajectory;

[0204] - repeating the first control scheme based on the retrieved data of the first trajectory and the retrieved data of the first present trajectory; saving data of the first trajectory and data of the first present trajectory after repeating the first control scheme;

[0205] - retrieving the saved data of the second trajectory and the saced data of the second present trajectory; repeating the second control scheme based on the retrieved data of the second trajectory and the retrieved data of the second present trajectory;

[0206] - saving data of the second trajectory and data of the second present trajectory after repeating the second control scheme; wherein any saved data of a trajectory is not modified between saving and retrieving of the data.

[0207] In particular, when carrying out the first control scheme, and using data-driven tracking to compute the first optima finite sequence of inputs none of the second trajectory the second present trajectory and the second present time instant Pi 2 }are modified. In particular only the clock related to the first superscript (1), i.e. to the first control scheme advances, whereas the clock related to the second superscript (2), i.e. to the second control scheme does not advance. I.e. P^ advances while P^ does not advance, when carrying the first control scheme repeatedly.

[0208] The first control scheme may be carried out repeated ly. for example with a period between 1 second and 1000 seconds, for example between 60 s and 600 s, for example of 60 s. and after a certain number of periods, e.g. when a paper of a first paper grade is manufactured, then the first control scheme may be stopped / suspended and the pulp and paper process plant may be controlled according to the second control scheme.

[0209] In particular, when carrying out the second control scheme, and using data-driven tracking to compute the second optimal finite sequence of inputs none of the first trajectory w(13rd)5the first present trajectory and the first present time instant are modified. In particular only the clock related to the second superscript (2), i.e. to the second control scheme advances, whereas the clock related to the first superscript (1), i.e, to the first control scheme does not advance. I.e. P^'1advances while does not advance when carrying out the second control scheme repeatedly

[0210] The second control scheme may be carried out repeatedly, for example with a period between 1 second and 1000 seconds, for example between 60 s and 600 s, for example of 60 s, and after a certain number of periods, e.g. when a paper of a second paper grade is manufactured, then the second control scheme nay be stopped / suspended and the pulp and paper process plant may be controlled according to the first control scheme.

[0211] Between the first controls scheme and the second control scheme and vice versa additional control scheme, e.g. related to additional paper grades may be carried out.

[0212] In some embodiments, alternatingly carrying out the first control scheme and the second control scheme repeatedly comprises: carry out the first control scheme starting from a first idle state;

[0213] - repeatedly carry out the first control scheme to complete a first operation phase related to the first operating condition; bring the pulp and paper process plant in the first idle state when the first operation phase related to the first operating condition is completed;

[0214] - carry out the second control scheme starting from a second idle state;

[0215] - repeatedly cany out the second control scheme to complete a second operation phase related to the second operating condition; bring the pulp and paper process plant in the second idle state when the second operation phase related to the second operating condition is completed.

[0216] The idle state may be for example an equilibrium state of the pulp and paper process plant described according to (2) and / or (1).

[0217] The inputs may be zero in the idle state.

[0218] Starting and terminating the control in an idle state ensures that for example a sequence of states in the form is observed with x®(p + 1) = x®(p) = equilibrant state p e N and The inputs at time p may be zero.

[0219] Alternatively, a sequence in the form x(2)(l), x«C2), ... , x®(p), xC1)(D,x(«(2), . xW(q),xC2)(p + 1), ... may be observed with x®(p + 1) = x^(p) = equilibrum state, for some p E N. The inputs at time p may be zero.

[0220] This ensures e.g. that the trajectories related to the first and second control scheme obtained when alternating the control schemes are identical to trajectories obtained / obtainable only carrying out the first control scheme and the second control scheme, respectively.

[0221] In some embodiments, the first operating condition is the production of paper of a first paper grade and the second operating condition is the production of paper of a second paper grade. Other examples of different operating conditions maybe related to a change between electrical drying and steam-based drying depending on energy cost or availability, CO2 target footprint, etc.

[0222] In some embodiments, finding the first optimal finite sequence of inputs, controlling the pulp and paper process plant setting the first optimal finite sequence of inputs as set points, and updating data of the first trajectory and the data of the first present trajectory and updating the first reference trajectory is repeated in a first periodical execution, in particular with a first period in the range from 0, Is to 3600s, more in particular with a period in the range from Is to 60s; and finding the second optimal finite sequence of inputs, controlling the pu lp and paper process plant seting the second optimal finit e sequence of inputs as set points, and updating data of the second trajectory and data of the second present trajectory and updating the second reference trajectory is repeated in a second periodical execution, in particular with a second period in the range from 0.1s to 3600s, more in particular u ilh a period in the range from Is to 60s.

[0223] In some embodiments, the first period is based on responsiveness of the pulp and paper process plant in the first operating condition; and wherein the second period is based on responsiveness of the pulp and paper process plant in the second operating condition.

[0224] In some embodiments, the set of inputs comprises one or more of a retention aid flow, flows of chemicals, flows of fillers, a flow of white water, a flow of pulp stock, a machine speed; and the set of outputs comprises one or more of a. white-water consistency, a headbox consistency, a quantity of paper ash.

[0225] The present disclosure further provides, a pulp and paper process plant having inputs and outputs, comprising:

[0226] -a measurement and data system configured to measure inputs of the pulp and paper process plant;

[0227] -sensor to measure outputs of the process plant;

[0228] -a control system configured to control the pulp and paper process plant automatically executing the methods of the present disclosure. The present disclosure further provides a computer program comprising instructions which, when the program is executed by a computer configured to control a pulp and process pant, cause the comput er to carry out the methods of the present disclosure.

Claims

1. CLAIMS1 , A method for controlling a pulp and paper process plant, the method comprising:- Identify ing a set of inputs of the pulp and paper process plant, the inputs in the set of inputs being physical quantities defining process variables of the pulp and paper process plant that are controlled;Identifying a set of outputs of the pulp and paper process plant, the outputs in the set of outputs being physical quantities measured by sensors of the pulp and paper process plant;- Obtaining data of a first trajectory of the pulp and paper process plant related to a first operating condition of the pulp and paper process plant, the first trajectory recording a sequence of inputs in the set of inputs and a corresponding sequence of outputs in the set of outputs obtained when operating the pulp and paper process plant according tet the first operating condition;- Obtaining data of a second trajectory of the pulp and paper process plant related to a second operating condition of the pulp and paper process plant, the second trajectory recording a sequence of inputs in the set of inputs and a corresponding sequence of outputs in the set of outputs obtained when operating the pulp and paper process plant according to the second operating condition;- Controlling the pulp and paper process plant with a first control scheme related to the first operating condition, the first control scheme repeatedly comprising: o Finding a first optimal finite sequence of inputs in the set of inputs that minimizes a penalty of tracking a first reference trajectory related to the first operating condition using data-driven tracking based on data of the first trajectory, the first reference trajectory, a first penalty of a deviation of a trajectory of the pulp and paper process plant from the first reference trajectory related to the first operating condition and data of a resulting first present trajectory of the pulp and paper process plant refated to an present controlling of the pulp and paper process plant according to the first operating condition;o Controlling the pulp and paper process plant setting the first optimal finite sequence of inputs as set points for the inputs of the pulp and paper process plant; o Updating the data of the first trajectory and the data of the first present trajectory based on a measured trajectory of the pulp and paper process plant during the controlling with the first control scheme and updating the first reference trajectory; o Repeating the first control scheme based on the updated data of the trajectories to complete a first operation phase related to the first operating condition;- Controlling the pulp and paper process plant with a second control scheme related t) the second operating condition, the second control scheme repeatedly comprising: o Finding a second optimal finite sequence of inputs in the set of inputs that minimizes a penalty of tracking a second reference trajectory related to the second operating condition using data-driven tracking based on data of the second trajectory, the second reference trajectory, a second penalty of a deviation of a trajectory of the pulp and paper process plant from the second reference trajectory related to the second operating condition and data of a resulting second present trajectory of the pulp and paper process plant relatedan present controlling of the pulp and paper process plant according to the second operating condition; o Controlling the pulp and paper process plant setting the second optimal finite sequence of inputs as set points fcr the inputs of the pulp and paper process plant; o Updating the data of the second trajectory and the data of the second present trajectory based on a measured trajectory of the pulp and paper process plant during the controlling with the second control scheme and updating the second reference trajectory; o Repeating the second control scheme based on the updated data of the trajectories to complete a second operation phase related to the second operating condition;Altematingly carrying out the first control scheme, based on the updated data of the first trajectory, the updated data of the first reference trsyectory, the first penalty and the updated data of first present trajectory; and the second control scheme, based on the updated data of the second trajectory, the updated second reference trajectory, the second penalty and the updated data of the second present trajectory; wherein the second operating condition is different from the first operating condition; and wherein the data of first trajectory and of the first present trajectory includes measured inputs and outputs when controlling the pulp and paper process plant with the first control scheme; and wherein the data of second trajectory and of the second present trajectory includes measured inputs and outputs when controlling the pulp and paper process pant with the second control scheme.

2. The method of claim 1 , wherein alternating from controlling the pulp and paper process plant based on the first control scheme to the second control scheme includes: storing the data of the first trajectory a storage and retrieving the data of the second trajectory from the storage, so that the data of the second present trajectory is updated, and the data of the first present trajectory is not updated but is stored in an unaltered manner for a future controlling based on the first control scheme, storing the data of the second trajectory to a storage and retrieving the data of the first trajectory from the storage, so that the data of the first present trajectory is updated, and the data of the second present trajectory is not updated but is stored in an unaltered manner for a future controlling based on the second control scheme.

3. The method of claim 1 or 2, wherein, when the pulp and process pant is controlled with the first control scheme, after being controlled with the second controls scheme, the stored data of the first trajectory, and the stored data of the first present trajectory is retrieved from memory and updated and used for the controlling, while the data of the second trajectory and the data of the second present trajectory is not modified and remains stored and is not used for controlling;and •wherein, when the pulp and process pant is controlled with the second control scheme, after being controlled with the first controls scheme, the stored data of the second trajectory, and the stored data of the second present trajectory is retrieved from memory and is updated and used for the controlling, while the data of the first trajectory and the data of the first present trajectory is not modified and remains stored and is not used for controlling.4, The method of any of claims from 1 to 3, wherein the data-driven tracking comprises a data-driven linear quadratic tracking; and wherein the first penalty is a first cost ftmction based on a first positive definite weight matrix, the first cost being a scalar Junction related to consumption of physical resources during the operation of the pulp and paper process plant; and the second penalty is a second cost ftmction based on a second positive definite weight matrix, the second cost being a scalar function related to consumption of physical resources during the operation of the pulp and paper process plant.

5. The method of claim 4, wherein in the first control scheme the sequence of inpu ts of the first trajectory is persistently exciting of order at least given by the sum of the length of the first present trajectory and of the lengt h of the first optimal finite sequence of inputs and of a system order of the pulp and paper process plant; and wherein in the second control scheme the sequence of inputs of the second trajectory is persistently exciting of order at least given by the sum of the length of the second present trajectory and of the length of the second optimal finite sequence of inputs and of a system order of the pulp and paper process plant;6. The method of any of claims from 4 to 5, wherein the first optimal finite sequence of inputs is obtained by computing a first basis for a zero initial condition sub-behavior related to the first operating condition, and computing a first free response related to the first operating conditionand the first optimal finite sequence of inputs is based on the first basis and the first free response and the first positive definite weight matrix; and wherein the second optimal finite sequence of inputs is obtained by computing a second basis for a zero initial condition sub-behavior related to the second operating condition, and computing a second fee response related to the second operating condition and the second optimal finite sequence of inputs is based on the second basis and the second free response and the second positive definite weight niatrix;7, The method of claims from 4 to 5, wherein the first optimal finite sequence of inputs is determined numerically by solving a first minimization problem that minimizes the penalty of tracking the first reference trajectory by data-driven simulation; and wherein the second optimal finite sequence of inputs is determined numerically by solving a second minimization problem that minimizes the penalty of tracking the second reference trajectory by data-driven simulation8. The method of any of claims from 1 to 7, wherein alternatmgly carrying out the first controls scheme and the second control scheme, repeatedly comprises;- carrying out the first OMitrol scheme;- saving data of the first trajectory and data of the first present trajectory;- carrying out the second control scheme; saving data of the second trajectory and data of the second present trajectory; retrieving the saved data of the first trajectory and the saved data of the first present trajectory; repeating the first control scheme based on the retrieved data of the first trajectory and the retrieved data of the first present trajectory; saving data of the first trajectory and data of the first present trajectory after repeating the first control scheme; retrieving the saved data of the second trajectory and the saved data of the second present trajectory; repeat ing the second control scheme based on the retrieved data of the second trajectory and the retrieved data of the second present trajectory ;saving data of the second trajectory and data, of the second present trajectory after repeating the second control scheme; w herein any saved data of a trajectory is not modified bctw, een sin ing and retries ing of the data.

9. The method of any of claims from 1 to 8, wherein alternatingly carrying out the first control scheme and the second control scheme repeatedly comprises: carry out the first control scheme starting from a first idle state: repeatedly cany out the first control scheme to complete a first operation phase related to the first operating condition: bring the pulp and paper process plant in the first idle state when the first operation phase related to the first operating condition is completed;- can y out the second control scheme starting from a second idle state: repeated!) carry out the second control scheme to complete a second operation phase related to the second operating condition;- bring the pulp and paper process plant in the second idle state when the second operation phase related to the second operating condition is completed.

10. The method of any of claims from 1 to 9, wherein the first operating condition is the production of paper of a first paper grade and the second operating condition is the production of paper of a second paper grade.1 1 . The method of any of claims from 1 to 10, wherein finding the first optimal finite sequence of inputs, controlling the pulp and paper process plant setting the first optimal finite sequence of inputs as set points, and updating data of the first trajectory and the data of the first present trajectory and updating the first reference trajectory is repeated in a first periodical execution, m particular with a first period in the range from 0.1 s to 3600s. more in particular with a period in the range from 1 s to 60s; and wherein finding the second optimal finite sequence of inputs, controlling the pulp and paper process plant setting the second optimal finite sequence of inputs as set points, and updatingdata of the second trajectory and data of the second present trajectory and updating the second reference trajectory is repeated in a second periodical execution, in particular with a second period in the range from 0. Is hr 3600s, more in particular with a period in the range from Is to 60s.

12. The method of claim 1 1, wherein the first period is based on responsiveness of the pulp and paper process plant in the first operating condition; and wherein the second period is based on responsiveness of the pulp and paper process plant in the second operating condition.

13. The method ofany of claims from 1 to 12, wherein the set of inputs comprises one or more of a retention aid flow, flows of chemicals, flows of fillers, a flow of white water, a flow of pulp stock, a machine speed; and wherein the set of outputs comprises one or more of a white- water consistency, a headbox consistency, a quantity of paper ash.

14. A pulp and paper process plant having inputs and outputs, comprising:-a measurement and data system configured to measure inputs of the pulp and paper process plant;-sensor to measure outputs of the process plant;-a control system configured to control the pulp and paper process plant automatically executing the method of any of claims from 1 to 13.

15. A computer program comprising instructions which, when the program is executed by a computer configured to control a pulp and process pant, cause the computer to carry out the method of any of claims from 1 to 13.

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