A state observer based fault tolerant control method for gas turbine

Abstract

The application discloses a gas turbine fault-tolerant control method based on a state observer, relates to the technical field of gas turbines, and constructs a system state equation of the gas turbine by taking fuel flow as an input quantity, taking actual rotating speed of the gas turbine as a state quantity, and taking measured rotating speed of the gas turbine measured by a rotating speed sensor as an output quantity, designs a sliding mode approaching law in combination with the state observer, determines a fault-tolerant control law, takes the set rotating speed of the gas turbine as the value of the state quantity, and performs fault-tolerant control on the fuel flow according to the fault-tolerant control law when the rotating speed sensor of the gas turbine fails. The fault-tolerant control method has strong robustness to external interference and rotating speed sensor failure, has low dependence on model precision and is not prone to divergence, and can correctly acquire the state of the gas turbine and perform control after the rotating speed sensor fails.

Description

Technical Field

[0001] This application relates to the field of gas turbine technology, and in particular to a fault-tolerant control method for gas turbines based on a state observer. Background Technology

[0002] A gas turbine is an internal combustion engine that uses a continuously flowing gas as its working fluid to drive a high-speed rotating impeller, converting the energy of fuel into useful work. During operation, a high-speed permanent magnet generator, a high-pressure compressor, and a power turbine rotate coaxially. The high-pressure compressor compresses the intake air, and the compressed high-pressure air is exhausted through the power turbine and enters the combustion chamber to burn with the high-pressure gas, producing high-temperature, high-pressure gas. This high-temperature, high-pressure gas then enters the power turbine, doing work to drive the blades to rotate at high speed, which in turn drives the high-speed permanent magnet generator to generate electricity.

[0003] To ensure the stable operation of a gas turbine, the fuel supply needs to be adjusted in real time based on power demand and changes in turbine speed collected by a speed sensor. This closed-loop control of the turbine speed ensures stable power generation. However, speed sensors operate in harsh environments such as high temperatures, oil mist, and vibration for extended periods, making potential malfunctions unavoidable. If a speed sensor fails, the control system cannot accurately obtain the turbine's status, potentially leading to catastrophic consequences such as reduced operating conditions or emergency shutdowns. Summary of the Invention

[0004] In response to the aforementioned technical problem and requirement that existing gas turbine control systems cannot correctly obtain and control the gas turbine state after a speed sensor malfunction, the applicant proposes a gas turbine fault-tolerant control method based on a state observer. The technical solution of this application is as follows:

[0005] Using fuel flow rate as input u(t), actual gas turbine speed as state variable x(t), and measured gas turbine speed obtained by speed sensor as output variable y(t), the system state equation of the gas turbine is constructed based on the aero-thermodynamic model of the gas turbine. The system state equation includes fault-related terms n(t), and n(t) = 0 when the speed sensor is working normally, and n(t) ≠ 0 when the speed sensor fails.

[0006] A basic state observer is designed based on the system state equations of a gas turbine to obtain state quantity estimates.

[0007] State quantity estimation error based on the basic state observer Define the extended sliding surface s2 and design the sliding mode reaching law as follows: sgn() is the sign function, and k2>0 is the adjustment coefficient.

[0008] Based on the aero-thermodynamic model of the gas turbine, a fault-tolerant control law based on the sliding mode reaching law is determined;

[0009] The set speed of the gas turbine x1(t) is used as the value of the state variable x(t), and the fuel flow is controlled in a fault-tolerant manner according to the fault-tolerant control law when the speed sensor of the gas turbine fails.

[0010] A further technical solution is to estimate the state variables based on the state observer's error. Define an extended sliding surface s2, including:

[0011] Based on the state quantity estimation error e(t) and the system state equation of the gas turbine, a method for estimating extended state quantities is designed. Extended state observer; design of extended sliding surface s2=z L (t).

[0012] A further technical solution involves designing an extended state observer as follows:

[0013]

[0014] in, For z L The derivative of (t), A, B, C, and D are known invertible matrices of appropriate dimension obtained by fitting an aerodynamic thermodynamic model; n(t) is the fault-related term; and f(x) is a known nonlinear continuous function. Let f(x) be the estimated value, t be the time value, L be the observer gain, and Hörwitz stabilizing matrix be used.

[0015] Its further technical solution is to determine the fault-tolerant control law based on the sliding mode reaching law, including:

[0016] Combining the expression for the extended sliding surface and the sliding mode reaching law, the fault-tolerant control law is obtained as follows:

[0017]

[0018] Among them, the σ() function is a symbolic function or a preset function.

[0019] The further technical solution involves determining the fault-tolerant control law based on the sliding mode reaching law, which also includes:

[0020] Based on the state observer and the estimated values ​​of the state quantities Obtain the output estimate by The fault-tolerant control law is obtained by estimating the fault-related term n(t):

[0021]

[0022] The further technical solution is that the expression for the preset function σ(s2) is:

[0023]

[0024] A further technical solution is that the fault-tolerant control method for gas turbine faults also includes:

[0025] Check if the speed sensor of the gas turbine is faulty.

[0026] When it is determined that the speed sensor of the gas turbine has failed, the procedure for fault-tolerant control of fuel flow is executed according to the fault-tolerant control law when the speed sensor of the gas turbine fails.

[0027] When it is determined that the speed sensor of the gas turbine is operating normally, the fuel flow is controlled according to the basic control law during the normal operation of the speed sensor of the gas turbine.

[0028] A further technical solution is that the fault-tolerant control method for gas turbine faults also includes:

[0029] The basic sliding surface is designed as follows: in, for The derivative of .

[0030] The basic control law is derived based on the basic sliding surface design.

[0031] The further technical solution is to design the basic control law as follows:

[0032] u1(t) = -k1|s1|·sgn(s1);

[0033] Where k1 is the adaptive adjustment parameter of the synovial membrane. Let β be the derivative of k1, and both β and δ be positive.

[0034] A further technical solution involves constructing the following system state equations for the gas turbine:

[0035]

[0036] in, Let f(x) be the derivative of x(t), f(x) be a known nonlinear continuous function, A, B, C, and D be known invertible matrices of appropriate dimension obtained by fitting an aerodynamic thermodynamic model, and t be the time quantity.

[0037] The basic sliding mode observer designed based on the system state equations of a gas turbine is as follows:

[0038]

[0039] in, for The derivative of Let f(x) be the estimated value, L be the observer gain, and Hörwitz's stable matrix be used.

[0040] The beneficial technical effects of this application are:

[0041] This application presents a fault-tolerant control method for gas turbines based on a state observer. The method constructs the system state equation of the gas turbine using fuel flow rate as the input, the actual speed of the gas turbine as the state variable, and the measured speed of the gas turbine obtained through a speed sensor as the output. A sliding mode reaching law is designed using a state observer to determine the fault-tolerant control law. The set speed of the gas turbine is used as the value of the state variable, and the fuel flow rate is controlled fault-tolerantly according to the fault-tolerant control law when the speed sensor of the gas turbine fails. This fault-tolerant control method exhibits strong robustness to external disturbances and speed sensor failures, has low dependence on model accuracy, and is less prone to control divergence. It can accurately acquire and control the gas turbine state after a speed sensor failure.

[0042] In designing the fault-tolerant control law, this application considers the characteristics of gas turbines, such as large inertia and strong time delay during actual operation. The controller output typically cannot directly land on the sliding surface but instead switches frequently on it, easily causing system chattering. Therefore, a preset function is designed to replace the sign function, thereby reducing system chattering caused by the sign function and effectively improving system stability.

[0043] This application also includes fault detection for the gas turbine's speed sensor. When a fault is detected in the speed sensor, fault-tolerant control is applied to the fuel flow rate according to a fault-tolerant control law. When the speed sensor is operating normally, the fuel flow rate is controlled according to a basic control law during normal operation of the gas turbine's speed sensor. The combined use of these two control laws allows for more coordinated control of the gas turbine, enabling rapid switching of control modes when the speed sensor transitions between normal operation and fault conditions, thus ensuring a high response speed.

[0044] Compared with traditional PID control, the fault-tolerant control method for gas turbines based on state observers designed in this application is more dependent on model accuracy and prone to divergence when the speed sensor fails. In contrast, traditional PID control is able to quickly track the actual speed output, and has the control effect of fast response speed and high control accuracy. It also has stronger robustness and smaller overshoot when the speed sensor fails. Attached Figure Description

[0045] Figure 1This is a schematic diagram of a gas turbine fault-tolerant control method in one embodiment of this application.

[0046] Figure 2 This is a block diagram of a gas turbine fault-tolerant control in one embodiment of this application.

[0047] Figure 3 This is the speed output curve of a power turbine speed sensor experiencing a constant gain signal fault in one embodiment of this application.

[0048] Figure 4 yes Figure 3 The control error comparison curves of sliding mode fault-tolerant control and PID control under the corresponding conditions.

[0049] Figure 5 This is the speed output curve of the power turbine speed sensor when a pulse fault occurs in one embodiment of this application.

[0050] Figure 6 yes Figure 5 The control error comparison curves of sliding mode fault-tolerant control and PID control under the corresponding conditions. Detailed Implementation

[0051] The specific embodiments of this application will be further described below with reference to the accompanying drawings.

[0052] like Figure 1 As shown, the fault-tolerant control method for gas turbines based on a state observer in this application includes:

[0053] S110, with fuel flow rate as input u(t), actual gas turbine speed as state variable x(t), and measured gas turbine speed obtained by speed sensor as output variable y(t), the system state equation of gas turbine is constructed based on the aerodynamic thermodynamic model of gas turbine. The system state equation includes fault-related term n(t), and when the speed sensor is working normally, n(t) = 0, and when the speed sensor fails, n(t) ≠ 0.

[0054] Optionally, the actual speed of the gas turbine includes the actual speed of the high-pressure compressor and the actual speed of the power turbine, and the speed sensors include the high-pressure compressor speed sensor and the power turbine speed sensor. The output quantity y(t) is in the same dimension as the state quantity x(t).

[0055] S120, Design a basic state observer based on the system state equation of a gas turbine to obtain state quantity estimates.

[0056] S130, State quantity estimation error based on the basic state observer Define the extended sliding surface s2 and design the sliding mode reaching law as follows: sgn() is the sign function, and k2>0 is the adjustment coefficient.

[0057] S140, combined with the aero-thermodynamic model of the gas turbine, a fault-tolerant control law based on the sliding mode approach law is determined; the set speed of the gas turbine x1(t) is used as the value of the state variable x(t), and the fuel flow is controlled in a fault-tolerant manner according to the fault-tolerant control law when the speed sensor of the gas turbine fails.

[0058] In this embodiment, the gas turbine fault-tolerant control method further includes:

[0059] The system checks for malfunctions in the gas turbine's speed sensor. If a malfunction is detected, it executes a fault-tolerant control procedure for fuel flow during this period, based on a fault-tolerant control law. If the gas turbine's speed sensor is functioning normally, it controls fuel flow during normal operation according to the basic control law.

[0060] Optionally, methods for detecting whether the speed sensor of a gas turbine is faulty include: establishing a nonlinear speed state observer using fuzzy theory, designing an adaptive fault threshold based on the speed fault law, and comparing the relative difference in speed estimation with the adaptive fault threshold to detect whether the speed sensor of the gas turbine is faulty. (For details, please refer to: Zhao Jun, Zhu Rongjia, Chen Peng. Research on fault detection of gas turbine speed sensor based on fuzzy state observer [J]. Journal of Sensor Technology, 2019, 32(08):1227-1231.)

[0061] The fault-tolerant control method of this application has strong robustness to external disturbances and speed sensor failures. After the speed sensor fails, it can correctly obtain the state of the gas turbine and control it, effectively avoiding the strong dependence of traditional control on model accuracy and the problem of control divergence when the speed sensor fails.

[0062] To more clearly illustrate the fault-tolerant control method for gas turbines based on state observers of this application, another embodiment of this application will be described in detail below with reference to the accompanying drawings.

[0063] S210, with fuel flow rate as input u(t), actual gas turbine speed as state variable x(t), and measured gas turbine speed obtained by speed sensor as output variable y(t), the system state equation of gas turbine is constructed based on the aerodynamic thermodynamic model of gas turbine. The system state equation includes fault-related term n(t), and when the speed sensor is working normally, n(t) = 0, and when the speed sensor fails, n(t) ≠ 0.

[0064] In this embodiment, the system state equation of the gas turbine is:

[0065]

[0066] in, Let f(x) be the derivative of x(t), f(x) be a known nonlinear continuous function, A, B, C, and D be known invertible matrices of appropriate dimension obtained by fitting an aerodynamic thermodynamic model, and t be the time quantity.

[0067] Optionally, based on the characteristics of the gas turbine, it is determined that f(x) satisfies the first assumption, which is:

[0068] ||f(x1)-f(x2)||≤Γ1||x1-x2|| (2)

[0069] Where Γ1>0 is a constant, and ||·|| represents the norm of the function.

[0070] Optionally, the fault-related terms determined based on the characteristics of the gas turbine speed sensor satisfy the second assumption, which is:

[0071] ||n(t)||≤Γ2||x(t)|| (3)

[0072] Where Γ2∈R is a constant.

[0073] S220, a basic state observer is designed based on the system state equations of a gas turbine to obtain state quantity estimates.

[0074] In this embodiment, the basic sliding mode observer designed based on the system state equation of the gas turbine is:

[0075]

[0076] in, for The derivative of Let f(x) be the estimated value, L be the observer gain, and Hörwitz's stable matrix be used.

[0077] S230, State quantity estimation error based on the basic state observer Define the extended sliding surface s2 and design the sliding mode reaching law as follows: sgn() is the sign function, and k2>0 is the adjustment coefficient.

[0078] In this embodiment, the extended state variables are estimated based on the state variable estimation error e(t) and the system state equation of the gas turbine. Extended state observer; design of extended sliding surface s2=z L (t).

[0079] Determine the state quantity estimation error Then, the error dynamics are determined by combining formulas (1) and (4). for:

[0080]

[0081] The extended state observer is designed based on formulas (1) and (5):

[0082]

[0083] in, For z L The derivative of (t),

[0084] S240, combined with the aero-thermodynamic model of the gas turbine, determines the fault-tolerant control law based on the sliding mode approach law; the set speed of the gas turbine x1(t) is used as the value of the state variable x(t), and the fuel flow is subjected to fault-tolerant control in accordance with the fault-tolerant control law when the speed sensor of the gas turbine fails.

[0085] In this embodiment, the expression s2 = z of the extended sliding surface is used. L (t), Formula (6), and sliding mode convergence law The fault-tolerant control law is obtained as follows:

[0086]

[0087] Considering the large inertia and strong time delay characteristics of gas turbines during actual operation, the controller output is usually difficult to directly land on the sliding surface, but rather switches frequently on the sliding surface, which can easily cause system chattering. Excessive chattering can lead to reduced system control performance and even affect system stability. Since chattering is mainly caused by the sign function, in another embodiment, a preset function is used instead of the sign function, resulting in the following fault-tolerant control law:

[0088]

[0089] Optionally, the preset function σ() is a continuous nonlinear function, and its expression is:

[0090]

[0091] Since the fault-related term n(t) is usually difficult to obtain directly, its value needs to be determined through estimation. In one embodiment, the value of n(t) is estimated based on the state variables from the base state observer. Obtain the output estimate by The fault-tolerant control law is obtained by estimating the fault-related term n(t):

[0092]

[0093] In this embodiment, since the actual speed x(t) of the gas turbine cannot be accurately determined after the gas turbine speed sensor fails, in order to maintain high-precision and stable control of the gas turbine system, the set speed x1(t) of the gas turbine is used as the value of the state variable x(t), and the fault-tolerant control law is calculated by substituting it into formula (7), formula (8) or formula (10). The fuel flow is then controlled in a fault-tolerant manner according to the fault-tolerant control law when the speed sensor of the gas turbine fails.

[0094] like Figure 2 As shown, the estimated state variables are obtained by inputting the gas turbine output y(t) into the basic state observer. Based on state quantity estimates The state quantity estimation error is calculated from the set speed x1(t) of the gas turbine. The gas turbine set speed x1(t) and the state quantity estimation error e(t) are input into the extended state observer to obtain the extended state quantity. The fault-tolerant controller is based on the gas turbine set speed x1(t) and the extended state variable z. L The fault-tolerant control law u2(t) is obtained from the state quantity estimation error e(t). According to the fault-tolerant control law u2(t), the fuel flow is subjected to fault-tolerant control when the speed sensor of the gas turbine fails.

[0095] The fault-tolerant control method for gas turbines in this application further includes: detecting whether the speed sensor of the gas turbine is faulty; when it is determined that the speed sensor of the gas turbine is operating normally, controlling the fuel flow rate according to the basic control law during the normal operation of the speed sensor of the gas turbine.

[0096] In one embodiment, the basic sliding surface is designed as follows: The basic control law is derived based on the basic sliding surface design.

[0097] In one embodiment, the basic control law is designed as follows:

[0098] u1(t)=-k1|s1|·sgn(s1) (11)

[0099] Where k1 is the adaptive adjustment parameter of the synovial membrane. Let be the derivative of k1, β and δ are both positive, and |s1|>δ when t=0.

[0100] The following uses Lyapunov functions to prove that the basic control law and fault-tolerant control law designed in this application stabilize the gas turbine system.

[0101] (1) When it is determined that the speed sensor of the gas turbine is operating normally, the proof process that the basic control law makes the gas turbine system stable includes:

[0102] Determine by combining the basic sliding surface expression and formula (4)

[0103]

[0104] make Combining formula (12), we get:

[0105]

[0106] Optionally, the basic characteristics of the gas turbine and the first assumption are combined to determine and All are continuous and bounded functions, and when At that time, there exists a positive real number Γ M Z m Z M , making and

[0107] Determine the first Lyapunov function Where T represents transpose and γ>0.

[0108] Based on the first Lyapunov function and formula (11), determine

[0109]

[0110] in,

[0111] When |s1|>δ, the system is in the first state, given... Then l>0, and combining with formula (14) we get According to Lyapunov's stability law, for any initial state satisfying |s1(0)|>δ, the sliding diaphragm observer is asymptotically stable over a wide range, and s1 converges to the region where |s1(0)|≤δ in a finite time, proving that the gas turbine system is stable.

[0112] When |s1|≤δ, the system is in the second state, l<0. When, s1 converges to the region |s1(0)|≤δ in a finite time; when When |s1| gradually increases until it satisfies |s1(0)|>δ, the system enters the first state. s1 converges to the region where |s1(0)|≤δ in a finite time, proving that the gas turbine system is stable.

[0113] (2) When the speed sensor of the gas turbine is determined to be faulty, the proof process that the fault-tolerant control law makes the gas turbine system stable includes:

[0114] Determine the second Lyapunov function When the fault-tolerant control law includes a fault-related term n(t), combined with the extended sliding surface s2=z L (t), formula (6) and formula (10) determine

[0115]

[0116] When When estimating the fault-dependent term n(t) in the fault-tolerant control law, consider the extended sliding surface s2 = z. L (t), formula (6) and formula (8) determine

[0117]

[0118] Based on formula (15) or formula (16), it can be seen that when the state quantity estimation error e(t) approaches 0 and k2 is greater than or equal to the set threshold, This proves that the gas turbine system is stable.

[0119] To verify the effectiveness of the fault-tolerant gas turbine method designed in this application, the following uses the system state equation of the gas turbine as the gas turbine model, with the high-pressure compressor speed and the power turbine speed as state variables. Ignoring disturbance effects, the model is normalized, and the resulting parameter expressions include:

[0120] C is the identity matrix.

[0121] Simulation verification is performed using a typical fault that frequently occurs in the speed sensor of a power turbine as an example. Figure 3 This is a schematic diagram of the speed output when the speed sensor experiences a constant gain signal fault at the 30th second during the gas turbine startup process. Figure 4 yes Figure 3 The control error comparison curves of sliding mode fault-tolerant control and PID control under the corresponding conditions. Figure 5 This diagram illustrates the speed output caused by a pulse fault in the speed sensor at the 20th second during gas turbine operation, resulting in a drop of approximately 500 rpm in the speed sensor's sampled value due to the interference pulse. Figure 6 yes Figure 5 The comparison curves of control errors between sliding mode fault-tolerant control and PID control under the corresponding conditions are shown. This sliding mode fault-tolerant control is the gas turbine fault-tolerant control based on a state observer proposed in this application.

[0122] pass Figure 4It can be seen that after the speed sensor of the power turbine experienced a constant gain signal failure, although the PID control regulated the speed, it ultimately failed to constrain the target value due to a large deviation in the input speed, resulting in control divergence. In contrast, the sliding mode fault-tolerant control of this application, after a 10-second adjustment period, basically achieved tracking of the target speed, and the system tended to stabilize.

[0123] pass Figure 6 It can be seen that after a pulse fault occurs in the speed sensor of the power turbine, the PID control adjusts the speed and the system tends to stabilize, but its initial control error is large and the stabilization time is long. In contrast, the sliding mode fault-tolerant control of this application exhibits smaller system error oscillations after a pulse fault occurs and quickly tends to stabilize.

[0124] In summary, compared with traditional PID control, the fault-tolerant control method for gas turbines based on a state observer designed in this application can quickly track the actual speed output, and has the control effect of fast response speed and high control accuracy. It also has stronger robustness and smaller overshoot when the speed sensor fails.

[0125] The above descriptions are merely preferred embodiments of this application, and this application is not limited to the above embodiments. It is understood that other improvements and variations that can be directly derived or conceived by those skilled in the art without departing from the spirit and concept of this application should be considered to be included within the protection scope of this application.

Claims

1. A fault-tolerant control method for gas turbines based on a state observer, characterized in that, The gas turbine fault-tolerant control method includes: Using fuel flow rate as input The actual speed of the gas turbine is a state variable. The measured speed of the gas turbine, obtained through a speed sensor, is the output speed. The system state equations of the gas turbine are constructed based on the aerodynamic thermodynamic model of the gas turbine, and the system state equations include fault-related terms. And when the speed sensor is working normally When the speed sensor fails ; A basic state observer is designed based on the system state equations of the gas turbine to obtain state quantity estimates. ; State quantity estimation error based on the basic state observer The system state equations of the gas turbine are designed to estimate the extended state variables. Extended State Observer ,in, for The derivative of , , , , , A , B , C , D The known appropriately dimensiond invertible matrix obtained by fitting the aerodynamic thermodynamic model is... Fault-related items, Given a nonlinear continuous function, for The estimated value, where t is the time quantity. For the observer gain, a Herwitz stabilizing matrix is ​​chosen; an extended sliding mode surface is designed. And design the sliding mode reaching law as , For symbolic functions, This is the adjustment coefficient; Based on the aerodynamic-thermodynamic model of the gas turbine, a fault-tolerant control law based on the sliding mode reaching law is determined. This includes combining the expression of the extended sliding surface and the sliding mode reaching law to obtain the fault-tolerant control law as follows: ,in, The function is either a symbolic function or a preset function; Set speed of gas turbine As a state quantity The value of is determined, and the fuel flow is controlled in a fault-tolerant manner according to the fault-tolerant control law when the speed sensor of the gas turbine fails.

2. The gas turbine fault-tolerant control method according to claim 1, characterized in that, The determination of the fault-tolerant control law based on the sliding mode reaching law also includes: Based on the state quantity estimates from the aforementioned basic state observer Obtain the output estimate ,by Estimate the fault-related items The fault-tolerant control law is obtained as follows: 。 3. The gas turbine fault-tolerant control method according to claim 1 or 2, characterized in that, Preset functions The expression is: 。 4. The gas turbine fault-tolerant control method according to claim 1, characterized in that, The gas turbine fault-tolerant control method also includes: Check whether the speed sensor of the gas turbine is malfunctioning; When it is determined that the speed sensor of the gas turbine has failed, the step of performing fault-tolerant control on the fuel flow when the speed sensor of the gas turbine fails according to the fault-tolerant control law is executed. When it is determined that the speed sensor of the gas turbine is operating normally, the fuel flow is controlled according to the basic control law during the normal operation of the speed sensor of the gas turbine.

5. The gas turbine fault-tolerant control method according to claim 4, characterized in that, The gas turbine fault-tolerant control method also includes: The basic sliding surface is designed as follows: ,in, for The derivative; The basic control law is obtained based on the basic sliding surface design.

6. The gas turbine fault-tolerant control method according to claim 5, characterized in that, The basic control law obtained from the design is as follows: ； in, For synovial adaptive adjustment parameters, for The derivative of , All are positive numbers.

7. The gas turbine fault-tolerant control method according to claim 1, characterized in that, The system state equations of the gas turbine obtained are as follows: ； in, for The derivative of Given a nonlinear continuous function, A , B , C , D The known appropriate dimension invertible matrix is ​​obtained by fitting the aerodynamic thermodynamic model, where t is a time quantity; The basic sliding mode observer designed based on the system state equations of the gas turbine is as follows: ； in, for The derivative of for The estimated value, The observer gain is determined by the Herwitz-stabilized matrix.

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