A method for switching control of limiting protection for turbofan engines based on dwell time

Abstract

A dwell time-based limiting and protection switching control method for turbofan engines is proposed. The method treats the fan speed tracking control loop and the high-pressure turbine outlet total temperature limiting loop as two subsystems. A switching LTI system is established using a state-dependent switching law based on dwell time constraints as the switching rule. The limiting and protection control of the turbofan engine is described as output tracking of a constrained switching linear system. A controller based on the internal model principle is then designed for each subsystem. Based on the switching rule, the solution of the controller parameters is described as an optimization of a set of linear matrix inequalities. By solving this set of LMIs, the controller parameters that stabilize the switching system are obtained, achieving turbofan engine speed tracking and upward limiting of temperature parameters. This invention achieves limiting and protection control of temperature parameters in turbofan engines, has the advantage of low-frequency controller switching, reduces damage to turbofan engine actuators, and provides better transient performance for fan speed and high-pressure turbine outlet total temperature.

Description

Technical Field

[0001] This invention belongs to the field of turbofan engine control technology, and relates to the application of switching control in turbofan engine control systems, particularly the application of a dwell time-based switching control method in turbofan engine limitation and protection control systems. Background Technology

[0002] Turbofan engines are complex power machinery systems, and their control systems are designed to achieve high efficiency and reliable operation. To ensure the safe and stable operation of turbofan engines in harsher environments such as high temperature and high pressure, the control system needs to limit and protect the turbine temperature. By increasing temperature limits, the system ensures that the engine achieves the desired performance without exceeding the safe operating temperature range.

[0003] Currently, aero-engine limiting and protection control laws commonly employ a combination of PID controllers and low-select / high-select (Min / Max) switching. If a variable exceeds its limit, the corresponding variable regulator is activated through the low-select / high-select switching law, limiting the output value to a specified range. It's worth noting that for systems with control law switching, if the switching signal is not properly selected, significant control signal chattering may occur at the switching moment. Litt and Yu, in "The Case for Intelligent Propulsion Control for Fast Engine Response" and "Multiobjective Robust Regulating and Protecting Control for Aeroengines," analyzed the mechanism of the Min / Max switching rule through simulation, demonstrating its conservatism and its susceptibility to control signal chattering or excessively high switching frequencies. They also revealed the drawbacks of this rule: the saturation characteristics generated by the limiter slow down the system response, increase overshoot, increase switching frequency, and degrade the control system performance.

[0004] To simplify the complex calculations in switching laws and thus the design of corresponding turbofan engine controllers, introducing dwell time into the switching law is a common research approach. The dwell time method first appeared in "Switching in Systems and Control," and scholar Feuer demonstrated in "Potential benefits of hybrid control for linear time invariant plants" that the transient performance of switching systems can be improved through dwell time-based switching strategies. Introducing a dwell time-based switching law into the design of aero-engine limit protection control systems can achieve low-frequency switching between the aero-engine main control loop controller and the limiter, thereby reducing control signal chattering near the switching point. To simultaneously achieve fast response, anti-interference, and low overshoot, the subsystem controller design can select a controller gain that can be solved using switching system research theory. Based on existing literature, this paper explores an improved switching rule from both the switching law and switching controller perspectives to simultaneously satisfy the above control objectives, which is of great significance for fully realizing the performance potential of the engine. Summary of the Invention

[0005] To address the problems existing in the background technology, this invention provides a dwell time-based constraint protection switching control strategy for turbofan engines: taking the turbofan engine's fan speed tracking loop and high-pressure turbine outlet total temperature limiting loop as two subsystems, and using a state-dependent switching law based on dwell time constraints as the switching rule, a constraint protection switching LTI model for the turbofan engine is established; further, a controller based on the internal model principle is designed for each subsystem, and then the state variables of the built-in model are introduced into the state variables of the switching LTI model to obtain an augmented switching LTI model; the problem of solving the controller is transformed into an optimization problem of a set of LMIs through the multiple Lyapunov function method to obtain the controller gain, thereby obtaining the final constraint protection switching control strategy.

[0006] To achieve the above objectives, the specific technical solution adopted by the present invention is as follows:

[0007] A method for switching the limitation protection of a turbofan engine based on dwell time includes the following steps:

[0008] Step 1: Establish a turbofan engine limiting protection switching LTI system;

[0009] Step 1.1: Establish the state dependency switching law σ(t) based on the residence time constraint:

[0010]

[0011] Where e(t) represents the "temperature exceedance" event, T represents the residence time, x(t) represents the system state, and P j,0 and P i,N Defined by the Lyapunov functions of the j-th and i-th subsystems, such as V j =x T (t)P j,0 x(t). Assume the switching system at the switching time... Place Then the next switching time Must meet:

[0012]

[0013] Where i, j belong to the set of integers {1, 2, ..., M}. This indicates the moment when the i-th subsystem is triggered by the event. This represents the moment when the j-th subsystem is triggered by an event. This is a switching law that depends on both time constraints and event triggering.

[0014] Step 1.2: Using fan speed N f Tracking circuit and high-pressure turbine outlet temperature T 48 The limiting loop is treated as two subsystems, and a state-dependent switching law σ(t) based on dwell time is used as the switching rule to establish a turbofan engine limiting protection switching LTI system:

[0015]

[0016] Where x(t) represents the system state, y(t) represents the system output (fan speed or high-pressure turbine outlet temperature), and u(t) = ΔW f The fuel flow increment is the controller output, and w(t) is the disturbance input, which can also represent the impact of engine performance degradation; A σ(t) B σ(t) C σ(t) D σ(t) and E σ(t) It is a constant matrix with appropriate dimensions.

[0017] Step 2: Design a controller based on the internal model principle for each subsystem of the switching LTI system, and convert the state variables x of the built-in model. r Introduce state variables of the turbofan engine limiting protection switching LTI system and establish an augmented switching LTI system for turbofan engine limiting protection.

[0018] Step 2.1: Design a controller based on the internal model principle, whose built-in model is a common unstable model of the reference input r(t) and the disturbance signal w(t). This built-in model can be expressed as:

[0019]

[0020] Where, x r (t) represents the state variable of the built-in model, B r Defined as [0…1] T y(t) is the output of the built-in model, and e(t) represents the difference between the output of the turbofan engine limiting protection switching LTI system and the reference input, e(t) = y(t) - r(t). r The characteristic polynomial is defined by the least common multiple of the characteristic polynomials of the unstable parts of the reference input r(t) and the disturbance signal w(t). For example, if the reference input r(t) is a step signal, then its characteristic polynomial is Φ. r If (s) = s, and the disturbance signal w(t) is a step signal, then its characteristic polynomial is Φ. w (s)=s, thus the least common multiple of the characteristic polynomials of the unstable parts of the reference input r(t) and the disturbance signal w(t) is Φ(s=s), then A r It is obtained from the standard form matrix coefficients of the state-space equation representing the characteristic polynomial.

[0021] x r (t) Introducing the turbofan engine limiting protection switching LTI system (3), we obtain the turbofan engine limiting protection augmentation switching LTI open-loop system:

[0022]

[0023] The full-state feedback switching controller for the augmented switching LTI system is designed as follows:

[0024]

[0025] Where σ(t) represents the switching law, K 1σ(t) K is the gain of the state feedback controller to achieve asymptotic stability of the system; 2σ(t) To track the compensation controller gain in order to achieve zero steady-state error tracking.

[0026] Step 2.2: From the augmented switching LTI open-loop system (5) and the full-state feedback switching controller (6), the turbofan engine limiting protection augmented switching LTI closed-loop system is obtained:

[0027]

[0028] Where χ(t) is the state variable [x(t) x] of the switched LTI closed-loop system. r (t)] T coefficient matrix coefficient matrix

[0029] Step 3: Based on the switching rules of the turbofan engine limiting protection augmented switching LTI system, solve a set of LMIs using the multiple Lyapunov function method to obtain the controller gain and complete the controller design;

[0030] Step 3.1: For the turbofan engine limiting protection augmented LTI switching system (7), when external input... When the value is 0, a set of LMIs is obtained through the multiple Lyapunov function method:

[0031]

[0032]

[0033]

[0034] in, This indicates the case where σ(t) = i. The value of μ i,l Let L ∈ {1, 2, ..., L} represent the set of positive integers, T represent the dwell time, and N and P represent positive integers. i,n P i,n+1 P i,N When using the multiple Lyapunov function approach, the Lyapunov function definition for the i-th subsystem is as follows:

[0035] V i (x,t)=x T (t)P i (t)x(t) (11)

[0036]

[0037] Among them, V i (x,t) represents the Lyapunov function of the i-th subsystem σ(t), where n and N are derived from the time interval t∈[0,1,…,n,…N] when the subsystem is activated. This represents the nth moment within the time period during which the i-th subsystem is triggered by the event. express The (n+1)th time after that, triggered by a time constraint or an event. This represents the Nth (last) moment triggered by the time constraint in the i-th subsystem, σ(t) = i. This represents the 0th (starting) moment when the σ(t) = j-th subsystem is triggered by the event.

[0038] Step 3.2: Multiply both sides of the three solution formulas obtained in Step 3.1. and Thus decoupling the controller gain K i and P i And given The following set of LMIs is obtained. By solving this set of LMIs, the controller gain that satisfies the control objective can be obtained:

[0039]

[0040]

[0041]

[0042] in, K i =[-K 1i K 2i A i B i C i D i K 1i and K 2i A respectively σ(t) B σ(t) C σ(t) D σ(t) K 1σ(t) and K 2σ(t) The value of P when σ(t) = i. i,n P i,n+1 P i,N L i,n L i,n+1 L i,N and μ i,l All of these are parameters to be solved.

[0043] The beneficial effects of this invention are as follows: The dwell time-based turbofan engine limit protection switching control method proposed in this invention features low-frequency controller switching, thereby reducing damage to the turbofan engine actuators. Simultaneously, this switching control strategy can achieve excellent anti-interference effects when facing various forms of interference signals. The dwell time in this method offers greater flexibility and can be adjusted in real time according to actual engineering needs. This method can be extended to achieve limit protection control in various control system design problems that require switching between the main controller and the limit controller. Attached Figure Description

[0044] Figure 1 This is a flowchart of a turbofan engine limitation and protection control method based on dwell time;

[0045] Figure 2 It is a state dependency switching rule based on residence time;

[0046] Figure 3This is a diagram of the turbofan engine limit protection switching control mechanism based on dwell time;

[0047] Figure 4 This is a flowchart of the switching law's operation;

[0048] Figure 5 The control structure diagram of the subsystem based on the internal model principle is shown in (a).

[0049] Figure 6 The subsystem control structure diagram (b) is based on the internal model principle;

[0050] Figure 7 (a) is the fan speed output curve of a turbofan engine without any limiting control;

[0051] Figure 7 (b) is the output curve of the high-pressure turbine outlet temperature of a turbofan engine without any limiting control;

[0052] Figure 8 (a) is a comparison of fan speed output at FC01 using a CAMPSS 90K class engine with a dwell time-based limiting protection switching strategy and a switching control strategy combining a Min-Max switching law and a PID sub-controller; (DS and SS in the following results graphs represent the corresponding outputs under the dwell time-based switching control strategy, and Min-Max SS represents the corresponding outputs under the Min-Max switching control strategy.) Figure 8 (b) is a comparison of the total high-pressure turbine outlet temperature output at FC01 using a CAMPSS 90K class engine with a residence time-based limiting protection switching strategy and a switching control strategy combining Min-Max switching law and PID sub-controller.

[0053] Figure 8 (c) is the switching signal output curve of the CAMPSS 90K class engine at FC01, using a dwell time-based limiting protection switching strategy;

[0054] Figure 8 (d) is the switching signal output curve of the CAMPSS 90K class engine at FC01, which uses a switching control strategy that combines Min-Max switching law and PID sub-controller.

[0055] Figure 9 for Figure 8 (b) is a magnified view of a portion of the image;

[0056] Figure 10(a) is a comparison of fan speed output at FC05 using a CAMPSS 90K class engine with a time-based limit protection switching control strategy and a switching control strategy combining a Min-Max switching law and an internal model-based sub-controller.

[0057] Figure 10 (b) is a comparison of the total high-pressure turbine outlet temperature output at FC05 using a CAMPSS 90K class engine with a residence time-based limiting protection switching control strategy and a switching control strategy combining a Min-Max switching law and an internal model-based sub-controller.

[0058] Figure 10 (c) is the switching signal output curve of the CAMPSS 90K class engine at FC05, using a dwell time-based limiting protection switching control strategy;

[0059] Figure 10 (d) is the switching signal output curve of the switching control strategy at FC05 using the CAMPSS 90K class engine, which combines the Min-Max switching law with the internal model-based sub-controller.

[0060] Figure 11 for Figure 10 (b) is a magnified view of a portion of the image. Detailed Implementation

[0061] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, other embodiments obtained by those skilled in the art without creative effort are all within the scope of protection of the present invention.

[0062] This invention provides a dwell time-based control method for limiting protection switching of a turbofan engine, the specific process of which is as follows: Figure 1 As shown.

[0063] Step 1: Establish a turbofan engine limiting protection switching LTI system;

[0064] Step 1.1: Design the state-dependent switching law σ(t) based on the residence time constraint:

[0065]

[0066] Here, e(t) represents the difference between the actual temperature output value and the temperature limit value, and the occurrence of e(t) is an event trigger condition.

[0067] Step 1.2: Using the fan speed control loop and the high-pressure turbine outlet temperature limiting loop as two subsystems, and a state-dependent switching rule with residence time constraints as the switching law, establish a turbofan engine limiting protection switching LTI system:

[0068]

[0069] For the matrix coefficients in (2), system information from two different flight conditions of the CMAPSS-1 90k turbofan engine is used:

[0070] FC01: H(ft) = 0.00, Ma = 0.00, reference value N f (r / min) = 2388.00, baseline value T 48 (°R) = 2072.99;

[0071]

[0072]

[0073]

[0074]

[0075] FC05: H(ft) = 10000.00, Ma = 0.25, reference value N f (r / min)=2319.00、T 48 (°R) = 1947.00;

[0076]

[0077]

[0078]

[0079]

[0080] in, u=ΔW f For fuel flow, w(t) is the disturbance input, which can also represent the degradation of engine health parameters.

[0081] Step 2: Design a controller based on the internal model principle for each subsystem of the switching system, and convert the state variables x of the built-in model. r Introduce state variables of the turbofan engine limiting protection switching LTI system and establish an augmented switching LTI system for turbofan engine limiting protection.

[0082] Step 2.1: Taking the speed tracking loop subsystem of FC01 as an example, design a subsystem controller based on the internal model principle. Its built-in model is the common unstable model of the reference input r(t) and the disturbance signal w(t):

[0083]

[0084] Given that r(t) and w(t) are both step functions, and the controlled system is controllable. The structural characteristic model Φ of the input signal and the disturbance signal is used. r (s)=s、Φ w (s)=s, thus obtaining the least common multiple Φ(s=s, thereby obtaining A r =0, B r =1. Place A r and B r Substituting into equation (7), we obtain the common instability model as follows:

[0085]

[0086] The augmented LTI subsystem expression is obtained as follows:

[0087]

[0088] The full-state feedback controller for the augmented LTI subsystem is designed as follows:

[0089]

[0090] Where K1 is the gain of the state feedback controller to achieve asymptotic stability of the system; K2 is the gain of the tracking compensation controller to achieve zero steady-state error tracking.

[0091] Step 2.2: Based on the augmented LTI subsystem and the full-state feedback controller, the turbofan engine limiting protection augmented switching LTI system is obtained:

[0092]

[0093] Where χ(t) is the state variable [x(t) x] of the switched LTI closed-loop system. r (t)] T coefficient matrix coefficient matrix

[0094] Step 3: Based on the switching rules of the switching system, solve a set of LMIs using the multiple Lyapunov function method to obtain the controller gain and complete the controller design;

[0095] Step 3.1: For the turbofan engine limiting protection augmented LTI switching system (10), when external input... When the value is 0, a set of LMIs is obtained through the multiple Lyapunov function method:

[0096]

[0097]

[0098]

[0099] in, This indicates the case where σ(t) = i. The value of μ i,l Let L ∈ {1, 2, ..., L} represent the set of positive integers, T represent the dwell time, and P represent positive integers. j,0 Let N and P represent the starting time when the j-th subsystem is activated. i,n P i,n+1 P i,N When using the multiple Lyapunov function approach, the Lyapunov function definition for the i-th subsystem is as follows:

[0100] V i (x,t)=x T (t)P i (t)x(t) (14)

[0101]

[0102] Among them, V i (x,t) represents the Lyapunov function of the i-th subsystem σ(t), where n and N are derived from the time interval t∈[0,1,…,n,…N] when the subsystem is activated. This represents the nth moment within the time period during which the i-th subsystem is triggered by the event. express The (n+1)th time after that, triggered by a time constraint or an event. This represents the Nth (last) moment triggered by the time constraint in the i-th subsystem, σ(t) = i. This represents the 0th (starting) moment when the σ(t) = j-th subsystem is triggered by the event.

[0103] Step 3.2: Multiply both sides of the three solution formulas obtained in Step 3.1. and Thus decoupling the controller gain K i and P i And given The following set of LMIs is obtained. By solving this set of LMIs, the controller gain that satisfies the control objective can be obtained:

[0104]

[0105]

[0106]

[0107] in, K i =[-K 1i K 2i A i B i C i D i K 1i and K 2i A respectively σ(t) B σ(t) C σ(t) D σ(t) K 1σ(t) and K 2σ(t) The value of P when σ(t) = i. i,n P i,n+1 P i,N L i,n L i,n+1 L i,N and μ i,l All of these are parameters to be solved.

[0108] Choosing a dwell time T = 0.08 and N = 1, solve (16), (17), and (18). The controller gains of the four subsystems at the two operating points are obtained using this method:

[0109] FC01:K 11 = [0.0046 0.0021], K 21 = -0.0220, K 12 = [0.0038 -0.0014], K 22 = -0.0143;

[0110] FC05:K 11 = [0.0253 0.0034], K 21 = -0.0703, K 12 = [0.0084 -0.0025], K 22 = -0.0934;

[0111] Other reference information is provided:

[0112] FC01: Fan speed reference value 2588r / min, high pressure turbine outlet temperature reference value 2226°R, residence time 0.08s;

[0113] FC05: Fan speed reference value 2519 r / min, high pressure turbine outlet temperature reference value 2101°R, residence time 0.09s;

[0114] Generally, the control cycle of an onboard control system for a turbofan engine is 20ms, so a dwell time of 0.08-0.12s is usually sufficient to meet tracking requirements. Existing literature indicates that there is an optimal dwell time, which minimizes the speed adjustment time and maximizes the dynamic response, and that the effect of dwell time on system response speed tends to be consistent.

[40] The switching frequency can be adjusted multiple times based on experimental results. This can be understood as follows: if the dwell time is selected too large, the dwell time in the limiting protection working circuit will exceed its necessary time, thus limiting the dynamic response of the tracking circuit; if the dwell time is selected too small (such as the Min-Max switching structure, which can be understood as a case of dwell time of 0), the switching between the speed controller and the limiting protection controller will be too frequent, which can easily lead to a decrease in the service life of the engine actuator.

[0115] To maximize tracking performance, in this section, the state-dependent switching rule setting based on dwell time constraints only selects to maintain a dwell time T in the temperature loop. That is, only time and state are restricted in system 2, while there are no restrictions on the speed tracking loop.

[0116] Figure 8 , Figure 9 , Figure 10 and Figure 11 The results show that the residence time-based switching controller reduces the temperature overshoot by about 2.8% and the number of switching events by more than 65% compared to the Min-Max switching controller. The residence time-based switching control strategy demonstrates good control performance in terms of both switching law selection and controller design.

[0117] Matters not covered in this invention are common knowledge.

[0118] The above embodiments are only for illustrating the technical concept and features of the present invention, and are intended to enable those skilled in the art to understand the content of the present invention and implement it accordingly. They should not be construed as limiting the scope of protection of the present invention. All equivalent changes or modifications made in accordance with the spirit and essence of the present invention should be covered within the scope of protection of the present invention.

Claims

1. A method for switching control of limiting protection for a turbofan engine based on dwell time, characterized in that, Includes the following steps: Step 1: Establish a turbofan engine limiting protection switching LTI system; Step 1.1: Establish the state dependency switching law σ(t) based on the residence time constraint: Where e(t) represents the "temperature exceedance" event, T represents the residence time, x(t) represents the system state, and P j,0 and P i,N Defined by the Lyapunov functions of the j-th and i-th subsystems, V j =x T (t)P j,0 x(t); Assume the switching system is at the switching time Place Then the next switching time Must meet: Event triggered, Where i, j belong to the set of integers {1, 2, ..., M}. This indicates the moment when the i-th subsystem is triggered by the event. This indicates the moment when the j-th subsystem is triggered by the event; This is a switching law that simultaneously relies on time constraints and event triggering; Step 1.2: Using fan speed N f Tracking circuit and high-pressure turbine outlet temperature T 48 The limiting loop is treated as two subsystems, and a state-dependent switching law σ(t) based on dwell time is used as the switching rule to establish a turbofan engine limiting protection switching LTI system: Where x(t) represents the system state; y(t) represents the system output, which is either the fan speed or the high-pressure turbine outlet temperature; u(t) = ΔW f The fuel flow increment is the controller output; w(t) is the disturbance input, or represents the impact of engine performance degradation; A σ(t) B σ(t) C σ(t) D σ(t) and E σ(t) It is a constant matrix with appropriate dimensions; Step 2: Design a controller based on the internal model principle for each subsystem of the switching LTI system, and convert the state variables x of the built-in model. r Introduce state variables of the turbofan engine limiting protection switching LTI system and establish an augmented switching LTI system for turbofan engine limiting protection. Step 2.1: Design a controller based on the internal model principle, whose built-in model is a common instability model of the reference input r(t) and the disturbance signal w(t), which is expressed as: Where, x r (t) represents the state variable of the built-in model, B r Defined as [0 … 1] T y(t) is the output of the built-in model, and e(t) represents the difference between the output of the turbofan engine limiting protection switching LTI system and the reference input, e(t) = y(t) - r(t). r It is obtained by defining the least common multiple of the characteristic polynomials of the unstable parts of the reference input r(t) and the disturbance signal w(t); x r (t) Introducing the turbofan engine limiting protection switching LTI system from formula (3), we obtain the turbofan engine limiting protection augmented switching LTI open-loop system: The full-state feedback switching controller for the augmented switching LTI system is designed as follows: Where σ(t) represents the switching law, K 1σ(t) K is the gain of the state feedback controller to achieve asymptotic stability of the system; 2σ(t) To track the compensation controller gain in order to achieve zero steady-state error tracking; Step 2.2: From the augmented switching LTI open-loop system of formula (5) and the full-state feedback switching controller of formula (6), the turbofan engine limiting protection augmented switching LTI closed-loop system is obtained: Where χ(t) is the state variable [x(t) x] of the switched LTI closed-loop system. r (t)] T coefficient matrix coefficient matrix Step 3: Based on the switching rules of the turbofan engine limiting protection augmented switching LTI system, solve a set of LMIs using the multiple Lyapunov function method to obtain the controller gain and complete the controller design; Step 3.1: For the turbofan engine limiting protection augmented LTI switching system of formula (7), when the external input w(t) is 0, a set of LMIs is obtained through the multiple Lyapunov function method: in, This indicates the case where σ(t) = i. The value of μ i,l Let L ∈ {1, 2, ..., L} represent the set of positive integers, T represent the dwell time, and N and P represent positive integers. i,n P i,n+1 P i,N When using the multiple Lyapunov function approach, the Lyapunov function definition for the i-th subsystem is as follows: V i (x,t)=x T (t)P i (t)x(t) (11) Among them, V i (x,t) represents the Lyapunov function of the i-th subsystem σ(t), where n and N are derived from the time interval t∈[0,1,…,n,…N] when the subsystem is activated. This represents the nth moment within the time period during which the i-th subsystem is triggered by the event. express The (n+1)th time after that, triggered by a time constraint or an event. This represents the last moment when the σ(t) = i-th subsystem is triggered by the time constraint. This represents the 0th (starting) moment when the j-th subsystem is triggered by the event; Step 3.2: Multiply both sides of the three solution formulas obtained in Step 3.

1. and Thus decoupling the controller gain K i and P i And given The following set of LMIs is obtained. By solving this set of LMIs, the controller gain that satisfies the control objective can be obtained: in, K i =[-K 1i K 2i ];A i B i C i D i K 1i and K 2i A respectively σ(t) B σ(t) C σ(t) D σ(t) K 1σ(t) and K 2σ(t) The value of P when σ(t) = i; i,n P i,n+1 P i,N L i,n L i,n+1 L i,N and μ i,l All of these are parameters to be solved.

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