A design method for quadratic optimal controller of variable cycle engine

Abstract

The present invention discloses a method for designing a quadratic optimal controller for a variable cycle engine, which relates to the technical field of aero-engine control, and comprises the following steps: Step 1, given a linear model of a variable cycle engine and its state transfer matrix, a cost functional form and its weight matrix, an initial state and a target state vector, a Riccati algebraic equation, an optimal gain and an optimal performance are obtained; Step 2, using the idea of ​​input increment control, introducing input increments, and obtaining a generalized state vector and a weight matrix; Step 3, calculating and obtaining an augmented state transfer matrix; Step 4, solving the Riccati algebraic equation according to the augmented state transfer matrix and the weight matrix to obtain an augmented optimal control gain value. The present invention designs an augmented state space description, takes the augmented state as the optimization target, and determines the optimal controller gain by solving the algebraic Riccati equation. The method is concise and sophisticated, and is convenient for engineering practice.

Description

Technical Field

[0001] The invention belongs to the technical field of aviation engine control, and in particular relates to a design method for a quadratic optimal controller of a variable cycle engine. Background Art

[0002] The variable cycle engine, also known as the adaptive cycle engine, is a new type of aviation gas turbine engine designed to maintain efficient operation under complex flight conditions (subsonic, transonic, subsonic) and is an ideal propulsion device for the next generation of supersonic aircraft. The thermodynamic cycle is changed by changing the geometry, size or position of some adjustable parts of the engine. By using the variable cycle to change the engine cycle parameters such as the compression ratio, turbine inlet temperature, air flow and bypass ratio, the engine can have good performance under various flight and working conditions.

[0003] Although a large number of adjustable components have brought good performance to the variable cycle engine, they have also greatly increased the complexity of its controller design. At this stage, it can be concluded from the public literature that the research on variable cycle engines in China mainly focuses on modeling, algorithm optimization, principle analysis and other technical details, and the control method used is mainly PID control. The basic principle of PID control is to measure the error between the actual output of the control system and the expected output, and calculate the control quantity according to the three parameters of proportion, integration and differentiation, and adjust the input of the system to make the output as close to the expected value as possible. For example, in the article "Multivariable Fuzzy PID Control of Aircraft Engines", a fuzzy PID control scheme is used to control the aircraft engine, a multivariable system that is difficult to accurately model, and it is shown that this fuzzy PID control system not only has certain robustness, but also has good decoupling. However, due to its insufficient dynamic performance, it cannot fully meet the technical index requirements of variable cycle engines. In the article "Adaptive PID Control of Aircraft Engine Speed ​​Based on Similarity Theory", adaptive PID control is used to reduce the overshoot of PID control to a certain extent, improve adaptability and robustness, but the oscillation in its adjustment process is still too obvious, and it still cannot fully meet the technical index requirements of variable cycle engines.

[0004] In fact, although traditional control theories represented by PID control have been widely used in the field of aircraft engine control, their shortcomings such as insufficient robustness, poor adaptability to complex systems and inability to fully exert system performance have not been overcome.

[0005] Therefore, the industry still needs to solve the problem of being able to maximize the performance of the control system through appropriate control methods while ensuring the reliability of engine operation under various operating conditions. Summary of the invention

[0006] The technical problem to be solved by the present invention is to provide a method for designing a quadratic optimal controller for a variable cycle engine. The optimal control theory represented by the quadratic optimal control adopted by the present invention not only has higher robustness to ensure the reliability of the engine, but also can deeply explore the performance potential of the engine with its higher adaptability to complex systems and better optimization function.

[0007] In order to solve the above technical problems, the present invention comprises the following steps:

[0008] A method for designing a quadratic optimal controller for a variable cycle engine comprises the following steps:

[0009] Step 1: Given the variable cycle engine linear model and its state transfer matrix, cost functional form and its weight matrix, initial state and target state vector, obtain the Riccati algebraic equation, optimal gain and optimal performance;

[0010] Step 2: Using the idea of ​​input increment control, the input increment is introduced to obtain the generalized state vector and weight matrix;

[0011] Step 3, calculate the augmented state transfer matrix;

[0012] Step 4: Solve the Riccati algebraic equation based on the augmented state transfer matrix and the weight matrix to obtain the augmented optimal control gain value.

[0013] Furthermore, the given variable cycle engine linear model and its state transfer matrix, cost functional form and its weight matrix, initial state and target state vector, obtain the Riccati algebraic equation, optimal gain and optimal performance; specifically including:

[0014] Derive the Riccati algebraic equation from the given conditions:

[0015] The variable cycle engine linear model is a steady-state, uncoupled linear system, so the state space description of the given engine model is set to the following standard state space description matrix form:

[0016]

[0017] Among them, u is the system input vector, x is the system state vector, y is the system output vector, the first equation is the system state equation, and the second equation is the system output equation; here, the input quantity is u=[w f ,A8,A FVABI ] T , where w f is the fuel flow rate, A8 is the tail nozzle throat area, A FVABI is the flow area of ​​the front adjustable duct area ejector; the state quantity is x=[n L, n H , T4] T , where n L is the low pressure rotor speed, n H is the high pressure rotor speed, T4 is the temperature before the turbine; the output is y=[F N ,eprs,lepr] T , where F N is thrust, eprs represents engine pressure ratio, and lepr represents linearized engine pressure ratio;

[0018] On this basis, for the infinite time LQR problem, the cost functional of the problem design is:

[0019]

[0020] Among them, Q and R are known unit diagonal matrices, called weight matrices;

[0021] For a given engine target state x d , the system control input u is represented by the feedback gain K that needs to be solved, that is: u = -Kx d , bring it into the functional J and system equations, then we have:

[0022]

[0023]

[0024] Assume that the integral primitive function is in the form of:

[0025]

[0026] Among them, P is a symmetric matrix, that is, P = P T >0; Taking the derivative of both sides of this equation with respect to t and substituting it into the system equation, we can get the solution:

[0027] -x T (Q+K T RK)x=x T [(A-BK) T P+P(A-BK)]x

[0028] The equation about matrix P and gain K is obtained:

[0029] -(Q+K T RK)=(A-BK) T p+P(A-BK)

[0030] Solving this equation yields the expression for the optimal gain K that minimizes the cost J:

[0031] K=R -1 BT P

[0032] Substituting this into the equation about the matrix P and the gain K, we get the Riccati equation:

[0033] PA+A T P-PBR -1 B T P+Q=0

[0034] Combined with the actual application of engine control, we can further obtain the optimal control, optimal gain and optimal performance:

[0035] Without considering the Du term in the state space description of the engine model, it is assumed that the output is just a combination of state quantities y=Cx; at this time, the quadratic performance indicator functional becomes:

[0036]

[0037] Take its minimum value; Q is a semi-positive symmetric matrix, R is a positive definite symmetric matrix, and the system is completely controllable and observable, and the control quantity u is not constrained; the expression of the optimal gain K is obtained:

[0038] K=R -1 B T P

[0039] Substitute u=-Kx d , and get the optimal control u * :

[0040] u * =-R -1 B T Px

[0041] Substituting into the quadratic performance index functional, we can get the optimal performance:

[0042]

[0043] Where x(0) is the state vector of the initial state; the Riccati algebraic equation at this time becomes:

[0044] PA+A T P-PBR -1 B T P+C T QC=0.

[0045] Furthermore, the idea of ​​using input increment control is adopted to introduce input increment to obtain a generalized state vector and weight matrix. The specific steps are as follows:

[0046] Using the idea of ​​input increment control, define the input increment Δu at time k [k] for:

[0047] Au [k] =u [k] -u [k-1]

[0048] It represents the change of input during sampling time; using Δu [k] As a cost in the performance index J, the performance index can be set as:

[0049]

[0050]

[0051] Among them, e [k] is the reference error at time k, is the generalized state vector at time k, x [k] , x d[k] ,u [k-1] Represent the state at time k, the expected state, and the control input at the previous time; S a =C a T SC a , Q a =C a T QC a , R a =R is the augmented weight matrix, S, Q, R are the given weight matrices.

[0052] Furthermore, the calculation to obtain the augmented state transfer matrix specifically includes:

[0053] For the augmented system state equation The general discrete form can be written as:

[0054] x a[k+1] =A a x a[k] +B a u [k]

[0055] Substituting the amount obtained in step 2, we get d[k+1] =x d[k] The system state equation under the premise:

[0056]

[0057] Among them, to match the current discrete form, A and B take the system matrix and input matrix of the original engine linear system state equation after discretization, and I is the unit diagonal matrix; the augmented state transfer matrix is ​​obtained from the system state equation:

[0058]

[0059] Furthermore, the Riccati algebraic equation is solved according to the augmented state transfer matrix and the weight matrix to obtain the augmented optimal control gain value, and the specific steps are as follows:

[0060] The augmented matrices A in step 2 and step 3 are a , B a , C a , R a , Q a Substitute the Riccati algebraic equation in step 1 into the equation:

[0061]

[0062] Solving this equation gives the augmented matrix P a The specific value of is brought into the analytical formula of the optimal gain K:

[0063]

[0064] The algebraic solution of the optimal gain K can be calculated; at this point, the design calculation of the LQR controller is completed.

[0065] The beneficial effects of the present invention are:

[0066] Aiming at the control problem of the design point of a variable cycle engine, the present invention designs a quadratic optimal controller based on the linear model of the engine; the design point state space description, reference error and control increment are set as augmented state quantities, and the augmented state space description is designed based on this; the augmented state is used as the optimization target, and the tracking problem of the reference input is converted into an optimization problem of solving the minimum value of the quadratic integral of the augmented state; the optimal controller gain is determined by solving the algebraic Riccati equation, and the method is concise and sophisticated and convenient for engineering practice; the present invention better takes into account key performance indicators such as steady-state error, adjustment time and overshoot, fully exerts the performance potential of the engine, and provides a new idea for the controller design of the variable cycle engine. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 It is the principle diagram of the LQR control structure proposed in the present invention;

[0068] Figure 2 It is the principle diagram of the design point LQR controller proposed in the present invention;

[0069] Figure 3 This is a diagram of the engine state simulation results for the design point proposed in the present invention. DETAILED DESCRIPTION

[0070] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the accompanying drawings in the embodiments of the present invention.

[0071] The main technology used in this invention is linear quadratic optimal control, and its basic principle is summarized as follows: Figure 1 As shown in the figure: The essence of optimal control is to make the system run at a minimum cost. When this cost is defined as a quadratic functional and the system is linear, then this problem is called a linear quadratic problem, and the designed controller can be called an LQR linear quadratic regulator. LQR uses full-state feedback to weight different states and obtain the optimal control quantity. It belongs to the category of optimal control and is a full-state feedback control structure. Its basic structure is as follows: Figure 1 The core idea is to construct a quadratic cost function of the state quantity change and actuator input quantity of the system in the future time period, and obtain the gain matrix K with the lowest total cost, so as to obtain the desired control sequence.

[0072] The present invention provides a method for designing a quadratic optimal controller for a variable cycle engine, and the specific implementation steps are as follows:

[0073] Step 1: Given the variable cycle engine linear model and its state transfer matrix, cost functional form and its weight matrix, initial state and target state vector, obtain the Riccati algebraic equation, optimal gain and optimal performance.

[0074] (1.1) Derive the Riccati algebraic equation from the given conditions:

[0075] The variable cycle engine linear model targeted by the present invention is a steady-state, uncoupled linear system, and the state space description of a given engine model can be set to the following standard state space description matrix form:

[0076]

[0077] Among them, u is the system input vector, x is the system state vector, y is the system output vector, the first equation is the system state equation, and the second equation is the system output equation. Here, the input quantity is u=[w f ,A8,A FVABI ] T , where w f is the fuel flow rate, A8 is the tail nozzle throat area, A FVABI is the flow area of ​​the front adjustable duct area ejector; the state quantity is x=[n L ,n H ,T4] T , where n L is the low pressure rotor speed, n H is the high pressure rotor speed, T4 is the temperature before the turbine; the output is y=[FN ,eprs,lepr] T , where F N is thrust, eprs is engine pressure ratio, and lepr is linearized engine pressure ratio.

[0078] On this basis, since the present invention is aimed at the LQR problem of infinite time, the cost functional of the problem design is:

[0079]

[0080] Among them, Q and R are known unit diagonal matrices, called weight matrices.

[0081] Since the system feedback is Figure 1 As shown, for a given engine target state x d , the system control input u can be represented by the feedback gain K that needs to be solved, that is: u = -Kx d , bring it into the functional J and system equations, then we have:

[0082]

[0083]

[0084] Assume that the integral primitive function is in the form of:

[0085]

[0086] Among them, P is a symmetric matrix, that is, P = P T > 0. Taking the derivative of both sides of this equation with respect to t and substituting it into the system equation, we can get the solution:

[0087] -x T (Q+K T RK)x=x T [(A-BK) T P+P(A-BK)]x

[0088] Furthermore, we can get the equation about matrix P and gain K:

[0089] -(Q+K T RK)=(A-BK) T P+P(A-BK)

[0090] Solving this equation, it is not difficult to find the expression of the optimal gain K that minimizes the cost J:

[0091] K=R -1 B T P

[0092] Substituting it into the above equation about matrix P and gain K, we get the Riccati equation:

[0093] PA+A T P-PBR -1 B T P+Q=0

[0094] (1.2) Combined with the actual application of engine control, the optimal control, optimal gain and optimal performance are further obtained:

[0095] Due to the limitation of engineering technology, in the actual application process, it is often impossible to fully feedback all state quantities, only part of the output feedback, so the Du term in the state space description of the engine model is not considered, and it is considered that the output is just a certain combination of state quantities y=Cx. At this time, the quadratic performance index functional becomes:

[0096]

[0097] Take its minimum value. Q is a semi-positive symmetric matrix, R is a positive symmetric matrix, and the system is completely controllable and observable, and the control quantity u is not constrained. Following the same method and idea in the previous step, the expression of the optimal gain K can be obtained:

[0098] K=R -1 B T P

[0099] Substitute u=-Kx d , and get the optimal control u * :

[0100] u * =-R -1 B T Px

[0101] Substituting into the quadratic performance index functional, we can get the optimal performance:

[0102]

[0103] Where x(0) is the state vector of the initial state. The Riccati algebraic equation at this time becomes:

[0104] PA+A T P-PBR -1 B T P+C T QC=0

[0105] Step 2: Use the idea of ​​input increment control to introduce input increments and obtain the generalized state vector and weight matrix. This step is as follows:

[0106] In order to achieve better tracking of the command signal at the design point, the conventional steady-state non-zero reference value control algorithm is mathematically difficult to eliminate the influence of the steady-state error. Therefore, the idea of ​​input increment control is adopted to define the input increment Δu at time k [k] for:

[0107] Δu [k] =u [k] -u [k-1]

[0108] It represents the change in input during the sampling time. [k] As a cost in the performance index J, this will affect the speed of the control input change, thereby achieving a smoother control input and avoiding the system changing too quickly. At this time, the performance index can be set to:

[0109]

[0110]

[0111] Among them, e [k] is the reference error at time k, is the generalized state vector at time k, x [k] , x d[k] ,u k-1] Represent the state at time k, the expected state, and the control input at the previous moment. a =C a T SC a , Q a =C a T QC a , R a =R is the augmented weight matrix, S, Q, R are the given weight matrices.

[0112] Step 3: Further calculate and obtain the augmented state transfer matrix.

[0113] For the augmented system state equation The general discrete form can be written as:

[0114] x a[k+1] =A a x a[k] +B a u [k]

[0115] Substituting the amount obtained in step 2, it is not difficult to obtain d[k+1] =x d[k] The system state equation under the premise:

[0116]

[0117] Among them, to match the current discrete form, A and B should be the system matrix and input matrix of the original engine linear system state equation after discretization, and I is the unit diagonal matrix. It is not difficult to obtain the augmented state transfer matrix from the system state equation:

[0118]

[0119] Step 4: Solve the Riccati algebraic equation based on the augmented state transfer matrix and the weight matrix to obtain the augmented optimal control gain value. The specific steps are as follows:

[0120] The augmented matrices A in the previous steps 2 and 3 are a , B a , C a , R a , Q a Substituting the Riccati algebraic equation in step 1, we get:

[0121]

[0122] Solving this equation gives the augmented matrix P a The specific value of is brought into the analytical formula of the optimal gain K:

[0123]

[0124] The algebraic solution of the optimal gain K can be calculated. At this point, the design calculation of the LQR controller is complete.

[0125] Aiming at the control problem of the design point of a variable cycle engine, the present invention designs a quadratic optimal controller based on the linear model of the engine; the design point state space description, reference error and control increment are set as augmented state quantities, and the augmented state space description is designed based on this; the augmented state is used as the optimization target, and the tracking problem of the reference input is converted into an optimization problem of solving the minimum value of the quadratic integral of the augmented state; the optimal controller gain is determined by solving the algebraic Riccati equation, and the method is concise and sophisticated and convenient for engineering practice; the present invention better takes into account key performance indicators such as steady-state error, adjustment time and overshoot, fully exerts the performance potential of the engine, and provides a new idea for the controller design of the variable cycle engine.

[0126] Experimental Example: The advantages of the present invention can be further illustrated by the following simulation experiment:

[0127] In this case, the initial state of a variable cycle aircraft engine is defined as x0 = [0 0 0] T , set the high pressure rotor and low pressure rotor 1% step signal input for tracking, then there is x d=[0.01 0.01 0] T ; The parameters of the variable cycle engine linear model in step 1 are as follows:

[0128]

[0129]

[0130] The weight matrix given in this case is as follows:

[0131]

[0132] Design and write the simulation program according to the method shown in steps 1 to 4 above. The program structure schematic diagram is as follows: Figure 2 shown.

[0133] The simulation results of this case are as follows Figure 3 :Simulation. The results are analyzed as follows:

[0134] Conclusion 1: From Figure 3 It can be seen that the design point system responds well under step input, the steady-state errors of the high-pressure rotor and the low-pressure rotor are controlled within 0.01%, the overshoot of the high-pressure rotor is less than 1%, and the overshoot of the low-pressure rotor is less than 0.6%. It has the advantages of smooth adjustment process and small oscillation amplitude, which meets the performance requirements of the engine controller.

[0135] Conclusion 2: From Figure 3 It can be seen that the high-pressure rotor adjustment time is 0.95s and the low-pressure rotor adjustment time is 0.8s, which meets the controller performance requirements for the variable cycle aircraft engine used in this case.

[0136] Conclusion 3: The simulation results show that the quadratic optimal controller designed by the present invention meets the expected goals, and the feasibility and effectiveness of the theories and methods proposed in this patent have been verified.

[0137] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for designing a quadratic optimal controller for a variable cycle engine, characterized in that: The steps include: Step 1: Given the variable cycle engine linear model and its state transfer matrix, cost functional form and its weight matrix, initial state and target state vector, obtain the Riccati algebraic equation, optimal gain and optimal performance; Step 2: Using the idea of ​​input increment control, the input increment is introduced to obtain the generalized state vector and weight matrix; Step 3, calculate the augmented state transfer matrix; Step 4: Solve the Riccati algebraic equation according to the augmented state transfer matrix and the weight matrix to obtain the augmented optimal control gain value; The idea of ​​input increment control is adopted, input increment is introduced, and a generalized state vector and weight matrix are obtained. The specific steps are as follows: Using the idea of ​​input increment control, define the input increment Δu at time k [k] for: Δu [k] =u [k] -u [k-1] It represents the change in input during the sampling time; using Δu [k] As a cost in performance index J, the performance index is set as: Among them, e [k] is the reference error at time k, is the generalized state vector at time k, x [k] , x d[k] ,u [k-1] Represent the state at time k, the expected state, and the control input at the previous time; S a =C a T SC a , Q a =C a T QC a , R a =R is the augmented weight matrix, S, Q, R are the given weight matrices.

2. A method for designing a quadratic optimal controller for a variable cycle engine according to claim 1, characterized in that: The given variable cycle engine linear model and its state transfer matrix, cost functional form and its weight matrix, initial state and target state vector, obtain the Riccati algebraic equation, optimal gain and optimal performance; specifically including: Derive the Riccati algebraic equation from the given conditions: The variable cycle engine linear model is a steady-state, uncoupled linear system, so the state space description of the given engine model is set to the following standard state space description matrix form: Among them, u is the system input vector, x is the system state vector, y is the system output vector, the first equation is the system state equation, and the second equation is the system output equation; here, the input quantity is u=[w f ,A8,A FVABI ] T , where w f is the fuel flow rate, A8 is the tail nozzle throat area, A FVABI is the flow area of ​​the front adjustable duct area ejector; the state quantity is x=[n L , n H , T4] T , where n L is the low pressure rotor speed, n H is the high pressure rotor speed, T4 is the temperature before the turbine; the output is y=[F N ,eprs,lepr] T , where F N is thrust, eprs represents engine pressure ratio, and lepr represents linearized engine pressure ratio; On this basis, for the infinite time LQR problem, the cost functional of the problem design is: Among them, Q and R are known unit diagonal matrices, called weight matrices; For a given engine target state x d , the system control input u is represented by the feedback gain K that needs to be solved, that is: u = -Kx d , bring it into the functional J and system equations, then we have: Assume that the integral primitive function is in the form of: Among them, P is a symmetric matrix, that is, P = P T >0; take the derivative of both sides of this equation with respect to t and substitute it into the system equation, then the solution is: -x T (Q+K T RK)x=x T [(A-BK) T P+P(A-BK)]x The equation about matrix P and gain K is obtained: -(Q+K T RK)=(A-BK) T P+P(A-BK) Solving this equation yields the expression for the optimal gain K that minimizes the cost J: K=R -1 B T P Substituting this into the equation about the matrix P and the gain K, we get the Riccati equation: PA+A T P-PBR -1 B T P+Q=0; Combined with the actual application of engine control, we can further obtain the optimal control, optimal gain and optimal performance: Without considering the Du term in the state space description of the engine model, it is assumed that the output is just a combination of state quantities y=Cx; at this time, the quadratic performance indicator functional becomes: Take its minimum value; Q is a semi-positive symmetric matrix, R is a positive definite symmetric matrix, and the system is completely controllable and observable, and the control quantity u is not constrained; the expression of the optimal gain K is obtained: K=R -1 B T P Substitute u=-Kx d , and get the optimal control u * : u * =-R -1 B T Px Substituting into the quadratic performance index functional, we get the optimal performance: Where x(0) is the state vector of the initial state; the Riccati algebraic equation at this time becomes: PA+A T P-PBR -1 B T P+C T QC=0。 3. The method for designing a quadratic optimal controller for a variable cycle engine according to claim 1, characterized in that: The augmented state transfer matrix obtained by the calculation specifically includes: For the augmented system state equation Its discrete form is written as: x a[k+1] =A a x a[k] +B a u [k] Substituting the amount obtained in step 2, we get d[k+1] =x d[k] The system state equation under the premise: Among them, to match the current discrete form, A and B take the system matrix and input matrix of the original engine linear system state equation after discretization, and I is the unit diagonal matrix; the augmented state transfer matrix is ​​obtained from the system state equation:

4. The method for designing a quadratic optimal controller for a variable cycle engine according to claim 1, characterized in that: The method of solving the Riccati algebraic equation according to the augmented state transfer matrix and the weight matrix to obtain the augmented optimal control gain value is as follows: The augmented matrices A in step 2 and step 3 are a , B a , C a , R a , Q a Substitute the Riccati algebraic equation in step 1 into the equation: Solving this equation gives the augmented matrix P a The specific value of is brought into the analytical formula of the optimal gain K: The algebraic solution of the optimal gain K is calculated; at this point, the design calculation of the LQR controller is completed.

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